# data[].attributes.attachment_link.title
Woman's Hour - Parenting: Maths anxiety - BBC Sounds

# data[].attributes.attachment_link.description
Catch up on your favourite BBC radio show from your favourite DJ right here, whenever you like. Listen without limits with BBC Sounds.

# data[].attributes.headline
Mental Health & Wellbeing today and tomorrow – six models to help and explore stress

# data[].attributes.subject
Mental Health & Wellbeing today and tomorrow – six models to help and explore stress

# data[].attributes.attachment_link.readable_text
Member only Health and wellbeing SEND

Wellbeing-Driven Productivity for Teachers Who Are Neurodivergent (or Think They Might Be)

In this webinar, Martine Ellis shares her personal experience of a late autism diagnosis and explore the intersection between neurodiversity, wellbeing, and productivity. If you are interested in any of these topics, then this webinar is for you.

# data[].attributes.attachment_link.title
Mental Health & Wellbeing today and tomorrow - six models to help and explore stress

# data[].attributes.attachment_link.description
Stuart Rimmer, CEO at East Coast College Group, offers practical tips and takeaways on wellbeing and mental health in teaching, with a focus on dealing with stress, change, and the pressures placed upon colleges during the current pandemic.

# data[].attributes.headline
inTuition taster: Support roleTeachers may find themselves facing learners suffering from stress, anxiety or depression. Knowing how to respond and where to direct them to is vital, says Kiechelle Degale.

# data[].attributes.subject
inTuition taster: Support roleTeachers may find themselves facing learners suffering from stress, anxiety or depression. Knowing how to respond and where to direct them to is vital, says Kiechelle Degale.

# data[].attributes.attachment_link.readable_text
Teachers may find themselves facing learners suffering from stress, anxiety or depression. Knowing how to respond and where to direct them to is vital, says Kiechelle Degale.

There is no doubt the Covid-19 pandemic has taken its toll in many ways on mental health, not just for teaching staff but on our learners too.

As teaching staff, it's important to sharpen the tools in our toolbox so we can deliver the best service to our learners. One of the most common and ever-growing areas to support would be our response to mental health in the classroom. However, it's important to note that we are not being asked to respond as medical professionals but rather in a support role, in the way we recognise signs and symptoms and understand how to guide and signpost.

The following tips will help you support learners with their mental health in the classroom.

1. Undertake Training

Consider having some mental health training or becoming a mental health first aider. You will be trained on how to recognise the warning signs and symptoms for mental health issues and also on how to recognise a crisis and what to do.

Some of these developing conditions can be picked up in language and by practising active listening. For instance, some examples that may indicate a red flag may be:

Feeling very sad or withdrawn for more than two weeks

Intense worries or fears that get in the way of daily activities

Severe mood swings or drastic changes in behaviour or personality

Talking about feeling helpless or hopeless and not wanting to be around any more

Using alcohol or drugs as a coping mechanism

Ongoing breaking of commitment, not showing up or continuous lateness. This can initially appear as poor time management but can also be a sign of developing anxiety

If a learner is showing signs of a crisis, for instance actively self-harming or talking about suicide, it's critical to get appropriate professional help. An ambulance may need to be called so they can have access to their local mental health team for an assessment.

2. Know where to go for help

Ensure you know who to turn to or where to find support: your line manager, HR, safeguarding lead, local mental health referral services or mental health team, other local specialist support or simply encouraging your learner to go to their GP to access appropriate professional help.

Sometimes supporting learners who are experiencing mental health issues can take its toll, so it's important you also make time for yourself. Your own self-care is paramount.

3. Ask learners what help they need

If learners have ongoing mental health issues or an existing diagnosis, they may already be aware of what has worked well in the past. If they already use some good and helpful coping strategies, it's better to let them lead on what works best.

If they feel overwhelmed, listening to them and helping them to think about a few things they can incorporate into their lives based on their own likes and dislikes can be a useful approach. Setting realistic and achievable goals can give them a boost and raise their confidence in the classroom.

4. Let them know you're there

Have a place learners can go to talk about mental health and a strong open-door policy. I am a great believer in the statement 'conversations can change/save lives'. Being present and actively listening to someone in emotional pain can provide much-needed support and help to take stress levels down.

Let them know lots of people have also experienced these feelings, and that with the right support, recovery is possible and likely. Also explain that it's okay if they don't feel okay right now and you are there to listen.

5. Create a positive environment

Create a safe, positive learning environment encouraging inclusive language that actively reduces the stigma of mental health. The everyday language we use, sometimes without much thought, can have a huge impact on someone experiencing mental health issues.

It can be a helpful distinction to separate the person from the illness. For instance, for someone showing signs of developing depression it may be more helpful to say 'you are experiencing depression right now' instead of saying 'you are depressed'. A simple tweak in language can make the difference in how someone communicates the illness to themselves.

6. Promote helpful resources

Gather resources, websites, helplines and apps (see Useful resources) and make sure they are easily accessible on various platforms such as posters in key areas, and the organisation's intranet and website. You can also display resources in safe areas, such as the back of toilet doors, so they can be accessed discreetly.
Kiechelle Degale is lecturer and mental health first aid instructor at Waltham Forest Adult Learning Service.
Useful resources

# data[].attributes.attachment_link.description
Teachers may find themselves facing learners suffering from stress, anxiety or depression. Knowing how to respond and where to direct them to is vital, says Kiechelle Degale.

# data[].attributes.headline
Intuition Taster: Combatting anxiety: Three tools for all maths teachers and learners

# data[].attributes.subject
Intuition Taster: Combatting anxiety: Three tools for all maths teachers and learners

# data[].attributes.attachment_link.readable_text
When learners tie themselves into knots about doing maths, it can be so debilitating. Teachers can feel powerless too. But helpful techniques are at hand, write Sue Johnston-Wilder and Janet Kilpatrick Baker.

Maths anxiety is disabling and prevalent, affecting roughly 30% of 15-year-olds. If not addressed, it interferes with learning maths. Learners often feel helpless, or defiant. Maths anxiety is a psychological injury from which learners can be emancipated if we know how. In the process, behaviour often improves.

For some learners, a maths classroom is a very unsafe place, to be avoided if possible. Maths anxiety is on a spectrum from mild tension to a strong fear.

High maths anxiety causes learners to feel 'pain' when thinking about the subject. Sufferers report historic experiences of isolation, humiliation, being ignored, shouting, and unkind, negative or discouraging things being said. Such experiences lead to opting out, poor attendance, last-minute studying and poor performance.

We want to share three tools: the growth zone model (GZM); the hand model of the brain; and the relaxation response. They help teachers become more aware of maths anxiety, reduce its impact and build resilience. They enable learners to manage emotions in maths challenges.

The growth zone model

We often hear people talking about "moving outside the comfort zone". But, for many maths learners, this means moving straight to the anxiety or panic zone. We can introduce an explicit in-between zone: the growth zone. The growth zone model enables learners to name and communicate their current feelings - a step towards managing them. They can be encouraged to use their own words to describe past feelings when faced with challenge and threat.

Working on familiar tasks in the comfort zone helps to build self-confidence and provides opportunities for developing automaticity. New learning happens in the growth zone (see link below for more).

In this zone it is important that learners need to feel safe when making mistakes or getting stuck and asking for support. They are encouraged to develop explicit strategies to deal with being stuck: underlining unfamiliar words, accessing resources online, talking with a peer, or preparing a question for the teacher.

By contrast, the anxiety zone is where the learner feels that what is being asked is not within their reach at the moment, even with support. Stress increases, cognition decreases, and little or no useful learning takes place. With an element of control (Bandura, 2007) to encourage self-safeguarding, learners in the growth zone can build confidence and develop their mathematical resilience.

Siegel's hand model of the brain

Maths anxiety can be invisible; students may feel suddenly or increasingly stupid, angry, fearful or blank. The hand model helps explain, in an accessible way based in neuroscience, why anxiety may lead to such feelings, as emotion takes over and the thinking part of the brain goes 'off-line' (see link below for more on the model).

Benson's Relaxation Response

The 'relaxation response' is a quick, effective way to switch off the brain's anxiety response by engaging the parasympathetic nervous system (one of two systems regulating the body's unconscious activities, such as digestion).

The learner can trigger the release of chemicals and brain signals that slow muscles and organs down, increasing blood flow to the brain. For example, even in exams, learners can use longer out-breaths, such as 5/7 breathing, to trigger the relaxation response. In some cases, to switch off the anxiety response, a learner may need to leave the room with a 'permission to have a break' card, walk to a safe place or seek one-to-one support.

Conclusion

Faced with learners' maths anxiety, and correlates such as learned helplessness or hostility, teachers tend to remove challenge, give the answer and avoid giving any tasks that lead to struggle. But, once they know and understand about maths anxiety and learners have learned how to communicate any perceived threat effectively, challenging tasks can be undertaken safely.

References

Bandura, A. (2007) 'Reflections on an agentic theory of human agency'. Tidsskrift for Norsk Psykologforening, 10, 995-1004

Dowker et al (2016) 'Mathematics Anxiety: What Have We Learned in 60 Years?'; available from goo.gl/KERzq5

Siegel, D. Dr Daniel Siegel presenting a Hand Model of the Brain goo.gl/TpQQif

Benson, H. (2001). The Relaxation Response. HarperCollins

Marshall, E., Mann, V., Wilson, D., & Staddon, R. (2017). Learning and teaching toolkit: Maths anxiety. goo.gl/ztV3A9

Lee C. and Johnston-Wilder, S. (2018) 'Getting into and Staying in the Growth Zone', NRICH; available from goo.gl/AznfVg

Sue Johnston-Wilder is associate professor of mathematics education at the Centre for Education Studies, University of Warwick. Janet Kilpatrick Baker is a post-graduate researcher at the University of Warwick.

# data[].attributes.attachment_link.title
inTuition taster - combatting anxiety in maths | Society for Education and Training

# data[].attributes.attachment_link.description
Maths anxiety is disabling and prevalent. If not addressed, it interferes with learning maths. This article discusses three tools to help prevent anxiety when teaching maths.

# data[].attributes.headline
Reducing anxiety about learning maths in a workplace setting

# data[].attributes.subject
Reducing anxiety about learning maths in a workplace setting

# data[].attributes.attachment_link.readable_text
In my new role working with learners on workplace apprenticeships, I observed how they often 'shy away' from doing their Functional Skills mathematics. Some will make every excuse they can possibly think of to avoid doing the tasks that need to be completed.

My research aim is to develop new maths teaching, and learning techniques and resources tailored for learners' particular workplaces. My learners vary from customer service personnel to practical workshop specialists across a wide variety of businesses in Derby, Nottingham and Birmingham. All my learners are training in management, team leading, customer service or business administration, and it is essential their maths skills are up to standard.

By developing maths resources for particular workplace specialisms, the aim is for apprentices to share these with colleagues. Specialisms include: Industry, security industry, manufacturing companies and service centres. My research methodology is a mixed method approach, including a learner survey and case studies of individual learners who have a fear of maths.

Reducing the scope for shying away from maths

While my research is still ongoing, I've made a number of observations about what works in teaching Functional Skills maths in the workplace. One of the most effective approaches, but which can be easily overlooked, has been to ensure that I allocate specific and adequate time to work with learners on their maths. This is so important as it makes the learner face up to, and confront, their weaknesses. It reduces the scope for shying away from maths.

By setting aside adequate time, I've been able to go back to absolute basics and gradually work through each topic. This approach can help trainers get to the bottom of deep-seated and long-standing anxieties over maths, many of which are related to gaps in basic knowledge. Feedback from my learners on this back-to-basics approach has been positive.

One said: "It wasn't so bad after all. I've always avoided maths, but now I have more confidence." I have achieved good results using images, diagrams and practical tasks to demonstrate mathematics problems. Many of my students relate very well to the practice side of maths, especially if it is related to their jobs and workplaces.

Creating calm teaching and learning environments

While many workplace learners say they do not use maths in their everyday work, I've found this is not really true and that they use it in many different ways, often without realising. Again, feedback from my learners has been encouraging. One told me: "I never realised how much maths was actually embedded in my work." Other findings to emerge from my research include the importance of creating calm teaching and learning environments in busy workplaces.

These oases of learning have helped learners feel more comfortable with their learning, encouraging them to work through areas where they are not so strong. Another key point is the importance of being flexible and understanding. I'm sure most workplace tutors understand the importance of this, but I've found there is a limit and that you also need to keep busy workplace learners on task.

As for my professional support, I've drawn on the resources and ideas offered by bodies such as the Association of Teachers of Mathematics (ATM) and the National Centre for Excellence in the Teaching of Mathematics (NCETM). There is more work to be done as I want to continue with the project and try out further resources and techniques to encourage a better response to Functional Skills maths in the future. But the interim findings have been encouraging, indicating that there is a range of different and effective strategies that can be employed specifically in workplace settings.

Steph Taylor is an assessor with Acorn Training (Stoke-on-Trent). She has previously taught GCSE English in secondary schools and spent 11 years in offender learning as a tutor and a manager. Her research was one of the three winners of the 2017 Researching Numeracy Teaching and Learning competition run by SET. Steph is a Member of SET.

This article first appeared in InTuition 23, spring 2016.

# data[].attributes.attachment_link.title
Reducing anxiety about learning | Society for Education and Training

# data[].attributes.attachment_link.description
Providing calm spaces to study, allowing enough time to work with learners, using practical examples and being flexible can all help people grapple with an often daunting subject.

# data[].attributes.headline
Mental health, Wellbeing and resilience in Further Education

# data[].attributes.subject
Mental health, Wellbeing and resilience in Further Education

# data[].attributes.attachment_link.readable_text
"Mental health, wellbeing and resilience are huge in the further education context," writes Jennifer Lindsell, Graphic Design BTEC L2/3 and Art teacher at Joseph Chamberlain College.

Studies have shown that as many as a third of teachers have experienced mental health problems, and many more are leaving education because of the workload. Both teachers and students alike deserve the right support, as well as time to explore and find techniques to help them control their own emotions.

As a Recently Qualified Teacher, I decided to study my own mental health as well as my interests in art education. This study was designed for me to look into my own emotions, and how they can affect my day-to-day life within the education sector.

We are exasperated with cuts, more demanding work and, in some cases, an extreme lack of support from our senior leadership. I have watched first-hand how easily trainee teachers can lose their excitement for joining the education sector because of the struggles with vigorous observations and unrealistic feedback.

To carry out this study, I produced one sketch every hour of every day for a week. These sketches were unplanned, produced on any material I could find in that moment, and where meant to show exactly how I felt during that time, in the rawest form. During this week, I was teaching, spending time at University, taking students on trips and observing trainee teachers. The aim to show whether there is a correlation between certain aspects of my job as an educator, and how I feel throughout the day, and answering the questions: will art help me? And if it does, will it help the education sector?

The impact of this has been huge. What I found was that art helps in more ways than can be explained. From the beginning, I struggled to come to terms with having to think about how I felt on an hourly basis. As a teacher, you always put your students first. I thought of them, and only them when in session time. Outside of that, I have buried myself into work and study. Completing online courses, never dealing with what was there in front of me: that quite often I burn myself out, and that my mental health will not go away.

Towards the end of the week, producing these hourly illustrations became a saviour for me. I was able to take a step away from avoidance, think and deal with what I needed to. This art production became my release and I looked forward to being able to spend 30 seconds, or five minutes, producing something that was for me.

As a teacher, this has now shown me that this style of release is so important for us to be able to cope. We need to be able to pull ourselves completely away from the world we live and work in, and we need to be able to fully release our feelings in the rawest form. I think this study has allowed me, in more ways than one, to realise the full potential of art education, and creativity, to help teachers and students alike.

Now that this study has been completed, and the incredible impact this had on myself has been disseminated, it is so important to continue looking at, and completing research on the topic of mental health within the education system. Teachers need support in this area, as well as new coping mechanisms to help them, not only to stop them leaving their careers, but to help them realize how easy it may be to improve their mental health. Students need support to help them to continue their studies, understanding the struggles of life a lot more, and have the means to move forward from struggles they have.

The next step from this is to complete an almost identical study with both students and teachers, to look at the effects of art on their resilience as adolescents and educators on a larger scale. Alongside this, it is important to look into the fundamental problems within the education system, specifically the FE sector, and look at what improvements can be made to help adolescent and teacher mental health.

The main question I wanted to ask myself in this study was: Does art help? Yes, it does. It helps more than I think this study shows. I think it helps more than any study will ever be able to explain.

"Indeed, that's what creativity, whether its poetry, pictures or music and whether you're witnessing or creating, does best: bypassing the thinking mind and going straight to the heart, with the power to move and touch a person in a way rational argument can't," (Tes, 2018).

Bashford, S., 2018. The arts can boost teachers' mental health. TES November 2018.

About Jennifer

Jennifer teaches Graphic Design BTEC L2/3 and Art at Joseph Chamberlain College. Jennifer is taking the first MA in Art and Education at Birmingham City University.

# data[].attributes.attachment_link.title
Mental health, Wellbeing and resilience in Further Education

# data[].attributes.attachment_link.description
"Mental health, wellbeing and resilience are huge in the further education context," writes Jennifer Lindsell, Graphic Design BTEC L2/3 and Art teacher at Joseph Chamberlain College.

# data[].attributes.attachment_link.title
Loughborough University Maths Education Network

# data[].attributes.attachment_link.description
Share your videos with friends, family, and the world

# data[].attributes.headline
How to Support Girls in Forming a Positive Math Identity

# data[].attributes.subject
How to Support Girls in Forming a Positive Math Identity

# data[].attributes.attachment_link.readable_text
A look at one way high school math teachers can guide female students to overcome stereotypes and develop a positive sense of their math abilities.

From left: IanDagnall Computing / Alamy, Science History Images / Alamy, INTERFOTO / Alamy

Notable mathematicians (left to right) Emmy Noether, Maria Agnesi, and Sofya Kovalevskaya

Identity formation, including mathematical identity, is a lifelong process. Supporting high school students as they continue to develop a positive mathematical identity is essential for their-long term enjoyment of, and success in, mathematics. This entails-among other things-fighting some strong cultural associations we have with the nature of math abilities, such as the stereotype that certain people are more likely to possess innate math skills.

As a math teacher, I've always felt an obligation to build welcoming and inclusive math communities. Teaching in an all-girls' school requires me to be particularly thoughtful and intentional about instilling and building a sense of belonging to the math community with this particular group of young mathematicians-women-that has been disadvantaged in the learning of and access to high-level mathematics.

All students need to feel part of the math community, because their identity as mathematics learners shapes their willingness and ability to engage in mathematics work. Without a strong positive math identity, they have little motivation to engage with math that's challenging and to persevere when they face challenges. This is never more true than in calculus classes, when the challenges become real even for the most dedicated learner.

One of the classroom practices and strategies that support students' development and strengthen their positive identity as learners and doers of mathematics is confronting the stereotypes of who can be a mathematician. To this end, a project that centers on the lives and achievements of three historical women mathematicians helped my calculus class discover and reflect not only on these mathematicians' multifaceted talent but also on the many biases and societal expectations that hampered these women's careers.

Spotlight on Women Mathematicians

The National Math Festival is a source of endless inspiration and resources, and when they sent me some information about the play The Witches of Agnesi, I knew I had found a jewel: This is a play, as the site explains, "that brings together three women mathematicians-Maria Agnesi, Sofya Kovalevskaya, and Emmy Noether-from three different centuries in a mysterious meeting place outside of ordinary time and space, where they share a pot of tea and tell the stories of their lives and careers."

The play, which viewers can watch as a stand-alone video or with a panel conversation that follows the play, is sparingly staged and performed-which for me movingly adds to the representation of the important work that women mathematicians do and questions why women have faced so many barriers. The title of the play comes from the name of the most important curve that Agnesi studied, and which she named " versiera," later somewhat playfully mistranslated as "witch."

Reflection and Response

What have you learned from this play? Did you know of any of these mathematicians?

Do you see common threads among the lives and works of these three women?

What was your reaction to this play? Did you like it? Why or why not?

Please share your thoughts about these women mathematicians and their stories. What do you think of each of these three people?

If you could meet one of these mathematicians for dinner, who would you like to meet and why? What questions would you have for her, and what would you two talk about?

After asking the students to watch the play, I provided them with a few reflection questions, such as these:

After watching the play, the students responded to some prompts and to each other's entries in the Women in Math blog that I created for this specific project. Their interesting, heartfelt, and uplifting blog posts were inspiring and continue to motivate me to develop similar tasks that encourage the students' sense of belonging to the larger mathematics community and strengthen their identity as members of that community.

For evaluating this project, I created a rubric-there are many ready-made rubrics on the internet, including these from Intel Teach Elements and the University of Wisconsin. There are also tools that allow educators to create and shape their own rubrics around the competencies they want to foster and assess.

As a class, we also had a final conversation that helped evaluate what the students got out of this project. Teachers can develop and deepen this type of conversation into an interdisciplinary exercise with educators of other subjects (such as history) or topics (some particular aspect of the mathematics that the three mathematicians in question studied).

Finally, we initiated peer reviews, in the form of a short guided or structured paragraph on a limited number of peer entries and a rubric on a larger number of peer posts. Besides being helpful to the teacher, these are a very effective tool for student engagement and further reflection.

In closing, math teachers can support their students' identity simply by pondering the question posed by the NCTM president in a December 2021 communication: "When you consider your mathematics community, whom does it include?"

# data[].attributes.attachment_link.title
How to Support Girls in Forming a Positive Math Identity

# data[].attributes.attachment_link.description
A look at one way high school math teachers can guide female students to overcome stereotypes and develop a positive sense of their math abilities.

# data[].attributes.attachment_link.description
Do you have a Casio fx-83 ES scientific calculator (or a compatible model) and want to learn how to use it? This free course, Using a scientific calculator, will help you to understand how to use...

# data[].attributes.headline
Ep 14: English as an Additional Language

# data[].attributes.subject
Ep 14: English as an Additional Language

# data[].attributes.attachment_link.readable_text
Evidence into Action is a podcast from the Education Endowment Foundation (EEF). This podcast welcomes experts in the field of education - fantastic researchers with important things to say, alongside a wealth of brilliant teachers and school leaders. Our mission is straightforward: to tackle the most important educational topics and offer you plenty of evidence-based ideas to reflect on and, hopefully, be able to put into action.

# data[].attributes.attachment_link.title
Ep 14: English as an Additional Language | Evidence into Action

# data[].attributes.attachment_link.description
In this instalment, hosts Alex Quigley and, EEF Senior Content Manager, Kirsten Mould, are joined by guests including experts in linguistics and education, along with education leaders, giving their perspective, as they discuss how best to support pupils with English as an Additional Language.

# data[].attributes.headline
EEF blog: Scaffolding High-quality Talk in Maths

# data[].attributes.subject
EEF blog: Scaffolding High-quality Talk in Maths

# data[].attributes.attachment_link.readable_text
Using open questions such as, ​' What's the same, what's different?' and then scaffolding pupils' responses using the sentence stems ​' all, some and none' can support pupils to uncover mathematical structure.

When I used this activity with my class, responses included: multiples of 3 and 4 are even numbers as the products have a ones digit which is even e.g. 0, 2, 4, 6 or 8". multiples of 3 are even and some are odd."

" None (No) multiples of 4 are odd because the products never end in 1, 3, 5, 7 or 9."

By scaffolding pupils' talk using this set of sentence stems, I could assess pupils' fluency, but crucially, it also provided an opportunity for pupils to demonstrate their understanding and reasoning, by elaborating and building on their own ideas as well as those of others.

Sentence stems like these have become one of the strategies I use to promote high-quality exploratory talk. They scaffold the ability to articulate reasoning, equipping pupils to think and talk like mathematicians.

# data[].attributes.attachment_link.keywords[]
but they are an indispensable part of my teaching practice.  according to gaunt and stott (2019)

# data[].attributes.attachment_link.keywords[]
sentence stems like these are the ‘simplest and most effective technique for scaffolding talk’. but what exactly are they

# data[].attributes.attachment_link.keywords[]
and why should we use them to help develop high quality talk in the maths classroom?  sentence stems scaffold pupils’ talk by supporting them to start a sentence. this enables pupils to concentrate on  what  to say rather than  how  to say it.   eef’s ​​‘improving mathematics in key stages 2 and 3’ guidance report  highlights how ‘discussion and dialogue can be useful tools for developing metacognition

# data[].attributes.attachment_link.keywords[]
but pupils may need to be taught how to engage in discussion’.   the told framework  provides four key principles for promoting high-quality talk in maths:  t ake part  o pportunities  l inks  d ebate  the use of sentence stems is one way that we can support pupils to ‘take part’

# data[].attributes.attachment_link.keywords[]
by providing scaffolding to prompt and extend their thinking.    sentence stems can encourage pupils to engage in ‘exploratory talk’

# data[].attributes.attachment_link.keywords[]
express contrasting views and opinions in a respectful way

# data[].attributes.attachment_link.keywords[]
whilst also explaining their reasoning.  let’s consider these examples from the dfe (2020) maths guidance.    framing how children can explain their reasoning

# data[].attributes.attachment_link.keywords[]
it could be ’ helps scaffold high-quality talk and debate.  for example

# data[].attributes.attachment_link.keywords[]
when asked to estimate the first missing number in the example above

# data[].attributes.attachment_link.keywords[]
pupils in my class responded with:  “ it must be a number between 200 and 300 but a value that is nearer to 300.”  “ it can’t be a number less than 250 because the point is after halfway between 200 and 300.”  “ it could be 290 because 290 is greater than 250 but less than 300.”  the sentence stems enable pupils to create ‘links’ and build upon each other's ideas in order to extend their thinking.  we also know the importance of creating ‘opportunities’. this links  to  recommendation 2  of the  ​‘improving mathematics in the early years and key stage 1  guidance report

# data[].attributes.attachment_link.keywords[]
and can also provide rich opportunities to support pupils’ understanding of mathematical structure (  recommendation 4  of the  improving mathematics in key stages 2 and 3  guidance report  ).    using open questions such as

# data[].attributes.attachment_link.keywords[]
what’s different? ’ and then scaffolding pupils’ responses using the sentence stems ‘ all

# data[].attributes.attachment_link.keywords[]
some and none ’ can support pupils to uncover mathematical structure.  when i used this activity with my class

# data[].attributes.attachment_link.keywords[]
responses included:  “all multiples of 3 and 4 are even numbers as the products have a ones digit which is even e.g. 0

# data[].attributes.attachment_link.keywords[]
6 or 8”.  “some multiples of 3 are even and some are odd.”  “ none (no) multiples of 4 are odd because the products never end in 1

# data[].attributes.attachment_link.keywords[]
7 or 9.”  by scaffolding pupils’ talk using this set of sentence stems

# data[].attributes.attachment_link.keywords[]
it also provided an opportunity for pupils to demonstrate their understanding and reasoning

# data[].attributes.attachment_link.keywords[]
by elaborating and building on their own ideas as well as those of others.  sentence stems like these have become one of the strategies i use to promote high-quality exploratory talk. they scaffold the ability to articulate reasoning

# data[].attributes.attachment_link.keywords[]
equipping pupils to think and talk like mathematicians.   references  dfe (2020) mathematics guidance : key stages 1 and 2 non-statutory guidance for the national curriculum in england.  gaunt

# data[].attributes.attachment_link.keywords[]
a. (2018) transform teaching and learning through talk: the oracy imperative . rowman & littlefield.  mercer

# data[].attributes.attachment_link.keywords[]
l. (1999) children's talk and the development of reasoning in the classroom.  british educational research journal

# data[].attributes.attachment_link.title
EEF blog: Scaffolding High-quality Talk in Maths

# data[].attributes.attachment_link.description
Techniques for scaffolding talk in mathematics lessons

# data[].attributes.attachment_link.description
Grace Coker explains how to develop mathematical talk within classrooms

# data[].attributes.headline
Blog: Using picture books to promote high-quality conversations in maths: the ShREC Approach

# data[].attributes.subject
Blog: Using picture books to promote high-quality conversations in maths: the ShREC Approach

# data[].attributes.attachment_link.title
Blog: Using picture books to promote high-quality conversations in...

# data[].attributes.attachment_link.description
Fliss James explains how picture books can be used to promote high-quality conversations in maths

# data[].attributes.headline
Review of the evidence on numeracy skills interventions for adults Jan23

# data[].attributes.subject
Review of the evidence on numeracy skills interventions for adults Jan23

# data[].attributes.attachment_link.readable_text
Skip to main content

Cookies on GOV.UK

We use some essential cookies to make this website work.

We'd like to set additional cookies to understand how you use GOV.UK, remember your settings and improve government services.

We also use cookies set by other sites to help us deliver content from their services.

Accept additional cookies Reject additional cookies View cookies

A review of the evidence base for interventions intended to improve numeracy skills among adults with below Level 2 qualifications in mathematics.

Applies to England

Documents

Ref: ISBN 978-1-83870-455-1, RR1319 PDF, 1.07 MB, 127 pages

Ref: ISBN 978-1-83870-451-3, RR1319 PDF, 732 KB, 33 pages

Details

This research aims to inform the development of policy in this area as part of the UK government's Multiply programme.

The review looks at:

the lessons that can be learnt from adult skills policy in this area over the past 20 years

what is known about adults (19+) in the UK who have limited numeracy skills (below Level 2)

the approaches that seem to be the most and least successful in supporting different groups of adults to improve their numeracy skills, up to and including Level 2

Thank you for your feedback

Report a problem with this page

# data[].attributes.attachment_link.title
Review of the evidence on numeracy skills interventions for adults

# data[].attributes.attachment_link.description
Skip to main content We use some essential cookies to make this website work. We'd like to set additional cookies to understand how you use GOV.UK, remember your settings and improve government services. We also use cookies set by other sites to help us deliver content from their services.

# data[].attributes.attachment_link.description
Gresham provides outstanding educational talks and videos for the public free of charge. There are over 3000 videos available on the Gresham website. Your support will help us to encourage people’s love of learning for many years to come.

# data[].attributes.attachment_link.description
This event will focus upon mathematics as expressed in different languages and cultures.

# data[].attributes.headline
Degrees and Radians - those Babylonians and 360

# data[].attributes.subject
Degrees and Radians - those Babylonians and 360

# data[].attributes.attachment_link.title
Degrees and Radians – Circles and Pi – Mathigon

# data[].attributes.attachment_link.description
Introduction, Degrees and Radians, Tangents, Chords and Arcs, The Circle Theorems, Cyclic Polygons, Spheres, Cones and Cylinders, Conic Sections

# data[].attributes.headline
Language in Use in Mathematics Classrooms: Developing Approaches to a Research Domain

# data[].attributes.subject
Language in Use in Mathematics Classrooms: Developing Approaches to a Research Domain

# data[].attributes.headline
Mathematics in England’s further education colleges: who is teaching what, and why it matters

# data[].attributes.subject
Mathematics in England’s further education colleges: who is teaching what, and why it matters

# data[].attributes.body
May22

# data[].attributes.attachment_link.description
This resource from SMILE consists of sets of question cards, which are differentiated by level, and include answers which can be photocopied onto the back of the card. They are best used for group work to promote discussion and collaborative learning. Different models of using the cards are suggeste…

# data[].attributes.attachment_link.readable_text
Charles Dietz examines the work of Guy Claxton, professor of education at Bristol University and fellow of the British Psychological Society.

Professor Guy Claxton's interest in 'learning to learn' goes back to his own schooldays. After obtaining mediocre O-level results he began to pay attention to his study habits, strengthening the most effective ones. Academic success in the form of a double first from Cambridge and a doctorate in psychology from Oxford ensued. Subsequently, over more than 25 years, he has brought together his interests in learning skills and psychology in numerous books, lectures, workshops and research projects.

Claxton thinks that the widespread view of intelligence as a fixed, innate property is pernicious. It puts a ceiling on what students can be expected to achieve and tells them that they are either bright or stupid. It leads to resignation among low achievers and a lack of resourcefulness on the part of the 'bright' producing school leavers unequipped for a rapidly changing world. What students require is not to be spoon-fed facts, many of which they will never make use of, but the ability to know what to do when they don't know what to do. This, for Claxton, is true intelligence - something that comes from somewhere deeper than the conscious, rational mind.

Identifying the problem

In Hare Brain Tortoise Mind (1997) Claxton brings together evidence that much of learning is unconscious, done by osmosis or imitation, and that too much thinking can get in its way. This doesn't mean that he denigrates reflection; indeed, he thinks that modern society overvalues speedy thinking, producing 'premature articulation'. To this Claxton juxtaposes the idea that 'slow is smart' - that the best ideas are the product of contemplation. Logical thinking has its place, but many of the problems we encounter in life call for imagination and intuition - indeed, he argues, the latter quality may be almost a pre-requisite for winning a Nobel prize.> Wise Up (1999) Claxton goes on to outline his view of what makes a good learner in the environment of the early 21st century. The particular relevance of his approach for G&T pupils can be seen from a piece of American research that he quotes in his book.

Professor Carol Dweck gave a class of high-school girls workbooks to complete. Some of them contained extra questions that were above the girls' level. The girls not given the inserts performed normally on the test, but among the others the normal pattern of high and low achievement was actually reversed. The bright girls, who usually coasted along, panicked when they came up against the unknown - their confidence, based on the idea that they were 'clever', crumbled when they didn't automatically know the answers. Claxton agrees with Dweck that what makes someone a resilient learner is their underlying belief that ability is not fixed but expandable, that as we learn things we can become smarter.

Theory into practice

Claxton's Building Learning Power (2002) deals with the practical application of his ideas. In it he introduces us to four new 'Rs' that he would like to see replace the three traditional ones:

Resilience: 'being ready, willing and able to lock on to learning'. Being able to stick with difficulty and cope with feelings such as fear and frustration.

Resourcefulness: 'being ready, willing and able to learn in different ways'. Having a variety of learning strategies and knowing when to use them.

Reflection: 'being ready, willing and able to become more strategic about learning'. Getting to know our own strengths and weaknesses.

Relationships: 'being ready, willing and able to learn alone and with others'.

Claxton wishes to progress beyond some of the flaws he sees in the pioneering approaches to 'learning to learn'. Study skills and thinking-skills methods have tended to be treated as 'clip-ons', last minute additions to learning that are unable to develop a major shift in the way students think. Learning-styles approaches such as visual audio kinesthetic (VAK) have failed to take account of the fact that people do not act the same way in all situations. A student labelled as a 'kinesthetic learner' may fail to take up certain challenges.

Learning Coaches

Claxton thinks students need to be 'ready and willing' as well as 'able'. To this end teachers need to become 'learning coaches'. Good coaches support athletes in pursuing goals; as good learning coaches teachers can give students feedback, encourage them to stretch themselves and provide a good personal model. Students cannot be forced to want to learn, but a good learning coach can raise their motivation. And this, says Claxton, is what is happening in the dozen LEAs where his type of approach is being used.

Taking responsibility for learning

Methods such as self-report questionnaires enable both student and teacher to track progress in learning skills. Students can see where their strengths and weaknesses lie, what they need to work on and where a strength in one subject might be transferred to another.

From here students can be helped to develop their own learning programs. They are encouraged to think about their feelings towards learning and weigh up its costs and benefits.

Meanwhile, teachers no longer need to be seen as infallible: as learning coaches they can say, 'I don't know the answer, let's work it out together'. Schools that adopt a culture of learning allow students and teachers alike to become more creative.

Claxton doesn't want to throw out the traditional 'content curriculum'. What he proposes is a 'learning curriculum' that would shadow it at every stage. His ultimate emphasis is on equipping students to lead more fulfilling lives, but the promise is that these methods will also lead to results in the classroom.

References

Hare Brain, Torotise Mind: Why Intelligence Increases When You Think Less, Fourth Estate: London (1997)

Wise Up: Learning to Live the Learning Life, Network Educational Press: Stafford (1999)

Building Learning Power: Helping Young People Become Better Learners, TLO: Bristol (2002), www.buildinglearningpower.co.uk

# data[].attributes.attachment_link.title
Guy Claxton and 'Learning to Learn' - Teaching Expertise

# data[].attributes.attachment_link.description
Charles Dietz examines the work of Guy Claxton, professor of education at Bristol University and fellow of the British Psychological Society. Professor Guy Claxton's interest in 'learning to learn' goes back to his own schooldays. After obtaining mediocre O-level results he began to pay attention to his study habits, strengthening the most effective ones.

# data[].attributes.attachment_link.description
Breaching experiment to determine whether mathematicians expect equations to be written in one order over another. * Mathematicians show relative uniformity in their judgments of order sometimes explicitly calling attention to the non-normative order. * Mathematicians' order preference was influenced by how they saw the communicative purpose of the presented equations.

# data[].attributes.headline
Ditch Those Math Worksheets. The Case for Teaching Real-World Problem Solving in K-5 May22

# data[].attributes.subject
Ditch Those Math Worksheets. The Case for Teaching Real-World Problem Solving in K-5 May22

# data[].attributes.attachment_link.readable_text
The designers for Monte Vista Elementary School's proposed outdoor classroom had some key questions to answer: What materials could they use while staying under the district's $10,000 budget? How much square footage would work? How would they produce the scalable models the client wanted?

Those critical questions-which could have been considered at a local architecture firm-were part of just another math assignment in Robbi Berry's 5th grade class at the Las Cruces, N.M, school. The prototypes the students created in class ultimately helped inform the work of the professional architects hired for the job.

And the project helped Berry's students master some important 5th grade math concepts-such as adding and subtracting multidigit numbers with decimals. And the students did it without having to plod through boring math worksheets.

This problem-solving approach to teaching math to elementary school kids makes the subject much more intriguing and relevant, Berry and other educators say. It allows students to experience math the way it is used in real life, rather than as a pile of equations with no meaningful context.

It's never too early to begin having kids learn math this way, experts say.

Today's elementary school students need to learn how to analyze, reason, and make complex decisions to improve their chances of success later in life, said Jo Boaler, a professor of education at Stanford University.

In the world of work, "it is not very useful to be somebody who has memorized methods to [answer] textbook questions, because the work that is in our world is very different from that," she said. "In whatever job you go into, you need to problem solve, think critically, make connections between different areas. And we're just not helping our students develop those kinds of capabilities" in most schools.

The benefits of a real-world problem-solving approach can stick with students for years, Boaler said. For a study published in 2017 in the Journal for Research in Mathematics Education , she followed a group of high school students who were learning math through real-world problem solving and another group-with similar socioeconomic characteristics-who were taught math in a more traditional way. (Boaler has not conducted similar research with elementary students.)

Eight years later, the students who had been exposed to the problem-solving methods were in higher-skill jobs and more likely to be looking at upward mobility in their careers than those who were taught more traditionally. The ones who had learned the traditional way said they saw math all around them in their professional lives but felt that their K-12 education had left them unprepared to use it for professional success.

What's more, the problem-solving approach gives teachers the opportunity to introduce students to all kinds of possible careers.

Berry's students, for instance, wear all sorts of professional hats to tackle math problems that she makes up based on real problems people face in the working world.

They've been event planners tasked with coordinating transportation to a wedding with 76 guests and four cars that can hold differing numbers of passengers. They've designed a house for a very picky client-played by Berry, doing her best posh lady voice. The house had to be built to certain specifications: a particular range of square footage, number of rooms, and sizes for those rooms.

The strategy gives students insight into how professional mathematicians think, Berry said. "Real mathematicians [see] a problem in the world, and they use math to solve it."

More than one right answer, more than one right method

While those tasks make sense for 5th graders, kindergartners-and even preschoolers-can learn math through problem solving, Boaler said. She pointed to a lesson she created, called "foot parade," in which children see pictures of the feet of different types of animals and are told to select as many creatures as they need to reach a particular number of feet. A cat and a bird, for instance, would be a total of six feet.

In the Howard County public schools in Maryland, elementary school math classes typically kick off with a problem-solving exercise, such as a discussion of a level-appropriate "target" number. For instance, students have a few minutes to brainstorm as many equations as possible to reach the number 147. Correct answers could be 100 plus 47, 823 minus 676, or other combinations of numbers. Students then share their answers in groups.

They must collaborate and recognize there are usually multiple ways to solve a problem, skills people in the working world exercise every day.

The goal is to expose "kids to different ways of thinking, without telling kids, 'here's the different ways you have to think,' " said John SanGiovanni, the district's elementary math coordinator.

These math problems often have multiple correct answers, and multiple pathways for getting to a solution. That's exciting for kids, Boaler said. "One of the things that has [turned] off so many kids [to math] is that [they] think it's a subject with one answer and one method," she said.

Whether the math problems are as intricate as Berry's outdoor classroom challenge-or simpler, like Howard County's "target number" exercise-the underlying goal is to "create interest and wonder," and help students unlock the concepts behind operations like addition, subtraction, multiplication, and division, SanGiovanni said.

When students veer off in the wrong direction, it's a chance to learn, Berry said.

"Mistakes grow our brain, we celebrate them," she said. Going over a misstep is "when we have the best conversations."

If a kid gets tripped up, Berry will ask the student privately if they are OK sharing their process with the class as a "favorite mistake." If they aren't, Berry will still share their work, but pretend the blunder was her own.

Should teachers throw out their worksheets?

Teachers using the real-world problem-solving approach can present the problem without initially giving instruction in the operations they might use to solve it, such as multiplication, Boaler said. That's the reverse of how it's done in a typical classroom where teachers say, " 'here are your methods, now do some questions that practice them.' "

Flipping that traditional process on its head means giving students a rich problem first. Then, once they've given it some thought, a teacher can introduce methods that might help students find a solution.

"At that point, kids are interested. They're like, 'Oh, yeah, I need that method.' Whereas, when you show them the methods first kids are like, 'Why? Why am I doing this?'" Boaler said.

SanGiovanni agrees with that perspective, to a point. But in his mind, the occasional worksheet still has its place. "Ten [math] problems on a paper from time to time, it's probably good for maintenance of a skill," he said.

Plus, he emphasized that most kids won't be able to learn all they need to know just by solving a big problem, he said. "There has to be some explicit instruction about some of the math," he said. "You can't just discover it all."

The problem-solving approach can be problematic for some

Teaching math this way can be a lot to ask of teachers, students, and even parents.

Many teachers learned the subject through more traditional methods back in their student days, such as practicing math problems on worksheets before applying that knowledge to real world situations. They're skeptical that young kids can learn this way, SanGiovanni said.

"Some of our teachers [believe] that you can't solve this huge problem or this really cool problem, because you don't know how to add three plus four yet, or better yet, you don't know how to do it quickly," he said. They say they "don't feel like I can take the time" for the real world problem-solving approaches.

It's also tough for some teachers to give up control, said Latrenda Knighten, a mathematics instructional coach in Baton Rouge, La.

"It is scary because you don't know what answers you're going to get," she said. Children may get frustrated with the open-ended process, or a class discussion may veer off topic if every kid is encouraged to talk about the problem in their own way.

Another big hurdle: "There isn't always access to good materials that frame lessons in [real world] problems," SanGiovanni said.

While Boaler and others have created excellent resources, he said, "you teach kids for an hour and a half of math every day," he said. "That means you need 180 lessons, or 180 [real world] problems. And I know that some of those problems could be two or three days. But just the sheer volume isn't there."

Teacher preparation programs often don't help prospective educators teach math through rich problem solving, particularly at the elementary level, he added.

District and school administrators may also push back against the approach. They worry students taught this way aren't going to perform well on standardized tests, though those assessments increasingly measure students' problem-solving abilities, Knighten said. "If we allow children to really internalize the concept, so that it sticks, you're going to see [good test scores] because they're going to make sense of it."

Teachers get grief from parents, too, SanGiovanni said. "A lot of parents say they want their kids to be able to think and reason. They just don't want it to look different from the way they learned," he said.

Berry is quick to point out the real-world problem solving approach was not the way the subject was taught when she was a student. That, she said, is largely why she grew up hating math and now makes it her mission for students to love it by the time they leave her classroom.

"Sitting in a seat, just doing problems out of a textbook, or me walking the kids through step by step on how to do a procedural. That, to me, is not authentic learning," she said. "I don't want them just to be test takers. I want them to be productive citizens when they leave me."

# data[].attributes.attachment_link.title
Ditch Those Math Worksheets. The Case for Teaching Real-World Problem Solving in K-5

# data[].attributes.attachment_link.description
The designers for Monte Vista Elementary School's proposed outdoor classroom had some key questions to answer: What materials could they use while staying under the district's $10,000 budget? How much square footage would work? How would they produce the scalable models the client wanted?

# data[].attributes.headline
Low attainment in mathematics: an analysis of 60 years of policy discourse in England, 2022

# data[].attributes.subject
Low attainment in mathematics: an analysis of 60 years of policy discourse in England, 2022

# data[].attributes.body
https://bera-journals.onlinelibrary.wiley.com/doi/epdf/10.1002/curj.128#.YlRqB1h5X4w.twitter

# data[].attributes.attachment_link.provider_name
Erikson Institute Early Math Collaborative

# data[].attributes.attachment_link.description
The common definition of a cardinality of a set states, it’s the understanding that the last number word said when counting tells how many.

# data[].attributes.attachment_link.readable_text
Maryam Mirzakhani (مریم میرزاخانی‎, 1977 - 2017) was an Iranian mathematician and professor at Stanford University. She was the first woman to receive the Fields Medal, the highest award in mathematics.

Mirzakhani worked at the intersection of dynamical systems and geometry. She studied objects like hyperbolic surfaces and complex manifolds, but also contributed to many other areas of mathematics.

When solving problems, Mirzakhani would draw doodles and diagrams on large sheets of paper, to see the underlying patterns and beauty. Her daughter even described Maryam's work as "painting". At the age of 40, Mirzakhani died of breast cancer.

Born in Adelaide, Australia, Terence Tao (born 17 July) is sometimes called the "Mozart of mathematics". When he was 13, he became the youngest ever winner of the International Mathematical Olympiad, and when he was 24, he became the youngest tenured professor at the University of California, Los Angeles.

Tao has received the MacArthur Fellowship, the Breakthrough Prize in mathematics, as well as the Fields Medal, the highest award in mathematics, for "his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory".

Together with Ben Green, Tao proved the Green-Tao theorem, which states that there are arbitrarily long arithmetic sequences of prime numbers.

In 2003, the Russian mathematician Grigori Perelman (Григо́рий Перельма́нborn, born 1966) proved the Poincaré Conjecture, which, until then, was one of the most famous unsolved problems in mathematics.

The complex proof was verified by 2006, but Perelman declined two big awards that came with it: the $1 million Clay Millennium Prize, and the Fields Medal which is the highest recognition in mathematics. In fact, he said: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."

Perelman also made contributions to Riemannian geometry and geometric topology, and the Poincaré Conjecture is still the only one of the seven Millennium Prize problems to have been solved.

Yitang Zhang (张益唐, born 1955) was born in China and is now a professor of mathematics at the University of California.

Zhang discovered that there is a number k less than 70 million, so that there are infinitely many pairs of prime numbers that are exactly k apart. This was a groundbreaking discovery in number theory, for which he received the MacArthur award in 2014.

This is similar to the Twin Prime conjecture, which states that there are infinitely many pairs exactly 2 apart (for example 11 and 13) - but no one knows if this is true.

The British mathematician Sir Andrew Wiles (born 1953) is best known for proving Fermat's Last Theorem, which, until then, was one of the most famous unsolved problems in mathematics.

In 1637, Pierre de Fermat, wrote in the margin of a textbook that he had a wonderful proof that the equation a n n>2. Unfortunately, no one was able to find a proof - until Wiles, some 400 years later.+b n=c n has no integer solutions for

Wiles had been fascinated by the problem since the age of 10, and spent seven years working on it in solitude. He announced his solution in 1993, although a small gap in his argument took two more years to fix.

He was too old to receive the Fields medal, the highest award in mathematics, which has an age limit of 40. Instead, Wiles was awarded a special silver plaque for his work.

The problem asks for an algorithm to decide whether a given Diophantine equation (a polynomial equations with integer coefficients) has any integer-valued solutions.

William Paul Thurston (1946 - 2012) was an American mathematician and a pioneer in the fields of topology, manifolds and geometric group theory.

Thurston's Geometrization Conjecture is about describing the structure and geometry of different three-dimensional spaces. In 1982, he was awarded the Fields Medal for his study of 3D manifolds.

Paul Joseph Cohen (1934 - 2007) was an American mathematician who proved the continuum hypothesis, and that the axiom of choice is independent from the other Zermelo-Fraenkel axioms of set theory. He received the Fields medal for his work.

Alan Turing (1912 - 1954) was an English mathematician and is often called the "father of computer science".

During the Second World War, Turing played a critical role in breaking the Enigma code used by the German military, as part of the "Government Code and Cypher School" at Bletchley Park. This helped the Allies win the war, and may have saved millions of lives.

He also invented the Turing machine, a mathematical model of a general purpose computer, and the Turing test, which can be used to judge the ability of artificial intelligence.

Turing was gay, which was still a crime during his life, and meant that his groundbreaking accomplishments were never fully recognised. He committed suicide at the age of 41.

André Weil (1906 - 1998) was one of the most influential French mathematicians in the 20th century.

He was one of the founders of the Bourbaki group, a group of mathematicians working under the collective pseudonym Nicolas Bourbaki. The goal of the Bourbaki group was to unify all of mathematics with a formal, axiomatic foundation.

Weil believed that many problems in algebra and number theory had analogous versions in algebraic geometry and topology. These are known as Weil conjectures, and became the basis for both disciplines. They also have applications in fields like cryptography and computer science.

During the second World War, Weil fled to the United States and later joined the Institute for Advanced Study at Princeton University.

Kurt Gödel (1906 - 1978) was an Austrian mathematician who later immigrated to America, and is considered one of the greatest logicians in history.

At the age of 25, just after finishing his doctorate in Vienna, he published his two incompleteness theorems. These state that any (consistent and sufficiently powerful) mathematical system contains certain statements that are true but cannot be proven. In other words, mathematics contains certain problems that are impossible to solve.

This result had a profound impact on the development and philosophy of mathematics. Gödel also found an example of these "impossible theorems": the continuum hypothesis.

Mary Lucy Cartwright (1900 - 1998) was a British mathematician and one of the pioneers of Chaos theory. Together with Littlewood, she discovered curious solutions to a problem: an example of what we now call the Butterfly effect.

Elbert Cox (1895 - 1969) was the first African-American mathematician to receive a PhD. Universities in England and Germany refused to accept his thesis at the time, but Japan's Tohoku Imperial University did.

Cox taught at Howard University in the United States, he studied polynomial solutions to differential equations, generalised the Boole summation formula, and compared different grading systems.

Albert Einstein (1879 - 1955) was a German physicist, and one of the most influential scientists in history. He received the Nobel Prize for physics and TIME magazine called him the person of the 20th century.

Einstein triggered the most significant transformation in our view of the universe since Newton. He realised that classical, Newtonian physics was no longer enough to explain certain physical phenomenons.

At the age of 26, during his "miracle year", he published four groundbreaking scientific papers that explained the photoelectric effect and Brownian motion, introduced special relativity, and derived the formula E=m c 2, which states that energy ( E) and mass ( m) are equivalent.

G.H. Hardy (1877 - 1947) was a leading English pure mathematician. Together with John Littlewood, he made important discoveries in analysis and number theory, including the distribution of prime numbers.

In 1913, Hardy received a letter from Srinivasa Ramanujan, an unknown, self-taught clerk from India. Hardy immediately recognised his genius, and arranged for Ramanujan to travel to Cambridge where he was working. Together, they made important discoveries and authord numerous paper.

Hardy always disliked applied mathematics and expressed this in his personal account of mathematical thinking, the 1940 book A Mathematician's Apology.

Peano also developed a new, international language, Latino sine flexione, which was a simplified version of Latin.

The French mathematician Henri Poincaré (1854 - 1912) is often described as the last universalist, meaning that he worked in every field of mathematics known during his lifetime.

Poincaré is one of the founders of the field of Topology, and he came up with the Poincaré conjecture. This was one of the famous unsolved problems in mathematics, until it was proven in 2003 by Grigori Perelman

He also found a partial solution for the "three body problem", and discovered that the motion of three stars or planets in space can be completely unpredictable. This laid the foundation for modern Chaos theory.

Poincaré was the first to propose gravitational waves, and his work on Lorentz transformations was the basis upon which Albert Einstein built his theory of special relativity.

Charles Lutwidge Dodgson (1832 - 1898) is best know under his pen name Lewis Carroll, as the author of Alice's Adventures in Wonderland and its sequel Through the Looking-Glass.

However, Carroll was also a brilliant mathematician. He always tried to incorporate puzzles and logic into his children's stories, making them more enjoyable and memorable.

Richard Dedekind (1831 - 1916) was a German mathematician and one of the students of Gauss. He developed many concepts in set theory, and invented Dedekind cuts as the formal definition of real numbers. He also gave the first definitions of number fields and rings, two important constructs in abstract algebra.

While still in his teens, Galois proved that there is no general solution for polynomial equations of degree five or higher - simultaneously with Niels Abel.

Unfortunately, other mathematicians who he shared these discoveries with repeatedly misplaced or simply returned his work, and he failed his school and university exams while concentrating on much more complex work.

At the age of 20, Galois was shot in a duel (some say a feud over a woman), and later died of his wounds. During the night before his death, he summarised his mathematical discoveries in a letter to a friend. It would take other mathematicians many years to fully understand these letters, and realise the impact of his work.

Augustus De Morgan (1806 - 1871) was a British mathematician and logician. He studied the geometric properies of complex numbers, formalised mathematical induction, suggested quaternions, and came up with new mathematical notation.

The De Morgan laws explain how to transform logical relationships in set theory, for example A∩B‾A∪B‾=A‾∩B‾. =A‾∪B‾ and

William Rowan Hamilton (1805 - 1865) was an Irish mathematician and child prodigy. He invented quaternions, the first example of a "non-commutative algebra", which has important applications in mathematics, physics and computer science.

He first came up with the idea while walking along the Royal Canal in Dublin, and carved the fundamental formula into a stone bridge he passed: i 2=j 2=k 2=ijk=−1.

Hamilton also made significant contributions to physics, including optics and Newtonian mechanics.

Niels Henrik Abel (1802 - 1829) was an important Norwegian mathematician. Even though he died at the age of 26, he made groundbreaking contributions to a wide range of topics.

At the age of 16, Abel proved the binomial theorem. Three years later, he proved that it is impossible to solve quintic equations - by independently inventing group theory. This had been an open problem for over 350 years! He also worked on elliptic functions and discovered Abelian functions.

Abel spent his life in poverty: he had six siblings, his father died when he was 18, he was unable to find a job at a university, and many mathematicians initially dismissed his work. Today, one of the highest awards in mathematics, the Abel Prize is named after him.

In theory, these machines could automatically perform certain calculations stored on cards or tape. However, due to the high production costs, they were never fully completed during Babbage's lifetime. In 1991, a functional replica was constructed at the Science Museum in London.

August Ferdinand Möbius (1790 - 1868) was a German mathematician and astronomer. He studied under Carl Friedrich Gauss in Göttingen and is best known for his discovery of the Möbius strip: a non-orientable two-dimensional surface with only one side. (However, it was independently discovered by Johann Benedict Listing just a few months earlier.)

Many other concepts in mathematics are named after him, including the Möbius plane, Möbius transformations, the Möbius function μ n in number theory, and the Möbius configuration of two mutually inscribed tetrahedra.

Joseph Fourier (1768 - 1830) was a French mathematician, and a friend and advisor of Napoleon. In addition to his mathematical research, he is also credited with the discovery of the greenhouse effect.

While travelling to Egypt, Fourier became particularly fascinated with heat. He studied heat transfer and vibrations, and discovered that any periodic function can be written as an infinite sum of trigonometric functions: a Fourier series.

Adrien-Marie Legendre (1752 - 1833) was an important French mathematician. He studied elliptic integrals and their usage in physics. He also found a simple proof that π is irrational, and the first proof that π 2 is irrational.

Gaspard Monge (1746 - 1818) was a French mathematician. He is considered the father of differential geometry, having introduced the concept of lines of curvature on surfaces in three-dimensional space (e.g. on a sphere). Monge also invented orthographic projection and descriptive geometry, which allows the representation of three-dimensional objects using two-dimensional drawings.

During the French Revolution, Monge served as Minister of the Marine. He helped reform the French education system and found the École Polytechnique.

Maria Gaetana Agnesi (1718 - 1799) was an Italian mathematician, philosopher, theologian, and humanitarian. Agnesi was the first western woman to write a mathematics textbook. She was also the first woman to be appointed professor at a university.

Her textbook, the Analytical Institutions for the use of Italian youth combined differential and integral calculus, and was an international success.

Agnesi also studied a bell-shaped curve described by the equation y=a 3 x 2+a 2. This function is now called the Witch of Agnesi. The strange name might come from a pun in the Italian language, were the word "versiera" for "witch" sounds similar to the ropes used when sailing.

Émilie du Châtelet (1706 - 1749) was a French scientist and mathematician. As a women, she was often excluded from the scientific community, but she built friendships with renown scholars, and had a long affair with the philosopher Voltaire.

She applied her mathematical ability while gambling, and used her winnings to buy books and laboratory equipment, and made important advanced regarding the concepts like energy and energy conservation.

Around the age of 42, Du Châtelet became pregnant again. At the time, without adequate healthcare, this was very dangerous for women of her age. She was also working on a French translation of Newton's book Principia, which containes the basic laws of physics.

Du Châtelet was determined to finish the translation, as well as a detailed commentary with additions and clarifications, and often worked 18 hours per day. She died just a few days after giving birth to a daughter, but her completed work was published posthumously, and is still used today.

Christian Goldbach (1690 - 1764) was a Prussian mathematician and contemporary of Euler, Leibniz and Bernoulli. He was tutor of Russian Tsar Peter II, and is remembered for his "Goldbach Conjecture".

Abraham de Moivre (1667 - 1754) was a French mathematician who worked in probability and analytic geometry. He is most remembered for de Moivre's formula, which links trigonometry and complex numbers.

De Moivre discovered the formula for the normal distribution in probability, and first conjectured the central limit theorem. He also found a non-recursive formula for Fibonacci numbers, linking them to the golden ratio φ.

Pierre de Fermat (1607 - 1665) was a French mathematician and lawyer. He was an early pioneer of calculus, as well as working in number theory, probability, geometry and optics.

In 1637, he wrote a short note in the margin of one of his textbooks, claiming that the equation a n n>2, and that he had a "marvellous proof, which this margin is too narrow to contain". This became known as +b n=c n has no integer solutions for Fermat's Last Theorem, and one of the most famous unsolved problems in mathematics - until it was finally proven in 1994.

René Descartes (1596 - 1650) was a French mathematician and philosopher, and one of the key figures in the Scientific Revolution. He refused to accept the authority of previous philosophers, and one of his best-known quotes is "I think, therefore I am".

Descartes is the father of analytical geometry, which allows us to describe geometric shapes using algebra. This was one of the prerequisites, which allowed Newton and Leibnitz to invent calculus a few decades later.

He is credited with the first use of superscripts for powers or exponents, and the cartesian coordinate system is named after him.

Marin Mersenne (1588 - 1648) was a French mathematician and priest. Because of the frequent exchanges with his contacts in the scientific world during the 17th century, he has been called the "the post-box of Europe".

Today we mostly remember him for the Mersenne primes, prime numbers that can be written as 2 n−1. Most of the largest known primes are of this type. He also studied acoustics and the harmonics of a vibrating string, and wrote about theology and philosophy.

Galileo Galilei (1564 - 1642) was an Italian astronomer, physicist and engineer. He used one of the first telescopes to make observations of the night sky, where he discovered the four largest moons of Jupiter, the phases of Venus, sunspots, and much more.

Galileo, sometimes called the "father of modern science", also studied the motion of objects in free fall, kinematics, material science, and invented the thermoscope (an early thermometer).

He was a vocal proponent of Heliocentrism, the idea that the Sun was at the centre of our solar system. This eventually led to him being tried by the Catholic Inquisition: Galileo was forced to recant and spent the rest of his life under house arrest.

François Viète (1540 - 1603) was a French mathematician, lawyer, and advisor to Kings Henry III and IV of France. He made significant advances in Algebra, and first introduced the use of letters to represent variables.

Viète discovered the connection between the roots and coefficients of a polynomial, called Viète's formula. He also wrote books about geometry and trigonometry, including calculating π to 10 decimal places using a polygon with 393216 sides.

Pedro Nunes (1502 - 1578) was a Portuguese mathematician and astronomer. As Royal Cosmographer of Portugal he taught navigational skills to many sailors and explorers.

Nunes first noticed that if a ship always follows the same compass bearing, it won't travel on a straight line or great circle. Instead, it will follow a path called a rhumb line or loxodrome, which spirals towards the North or South pole.

Nunes also tried to calculate which day in the year has the fewest hours of sunlight, he disproved previous attempts to solve classical geometry problems like trisecting an angle, and he invented a system for measuring fractional parts of angles.

The Italian Gerolamo Cardano (1501 - 1576) was one of the most influential mathematicians and scientists of the Renaissance. He investigated hypercycloids, published Tartaglia's and Ferrari's solution for cubic and quartic equations, was the first European to systematically use negative numbers, and even acknowledged the existence of imaginary numbers (based on −1).

Cardano also made some early progress in probability theory and introduced binomial coefficients and binomial theorem to Europe. He invented many mechanical devices, including combination locks, gyroscopes with three degrees of freedom, and drive shafts (or Cardan shafts) that are still used in vehicles today.

Nicolaus Copernicus (1473 - 1543) was a Polish mathematician, astronomer and lawyer. During his life, most people believed in the Geocentric model of the universe, with Earth at the centre and everything else rotating around it.

Copernicus created a new model, where the sun is at the centre, and Earth moves around it on a circle. He also predicted that Earth rotates around its axis once every day. Afraid that it would upset the Catholic church, he only published the model just before his death - triggering what is now called the Copernican Revolution.

Copernicus also worked as a diplomat and physician, and made important contributions to economics.

Zhu Shijie (朱世杰, 1249 - 1314) was one of the greatest Chinese mathematicians. In his book Jade Mirror of the Four Unknowns, he showed how to solve 288 different problem using systems of polynomial equations and four variables (called Heaven, Earth, Man and Matter).

Zhu made extensive use of Pascal's triangle. He also invented rules for solving systems of linear equations - predating our modern matrix methods by many centuries.

Qin first developed a method for numerically solving polynomial equations, which is now known as Horner's method. He found a formula for the area of a triangle based on the length of its three sides, calculated the sum of arithmetic series, and introduced a symbol for "zero" into Chinese mathematics.

Qin also invented Tianchi basins, which were used to measure rainfall and gather meteorological data important for farming.

Leonardo Pisano, commonly known as Fibonacci (1175 - 1250) was an Italian mathematician. He is best known for the number sequence named after him: 1, 1, 2, 3, 5, 8, 13, ...

Fibonacci is also responsible for popularising the Arabic numerals (0, 1, 2, 3, 4, ...) in Europe, which was still using Roman numerals (I, V, X, D, ...) in the 12th century CE. He explained the decimal system in a book called "Liber Abaci", a practical textbook for merchants.

Jia Xian (賈憲, c. 1010 - 1070) was a Chinese mathematician during the Song dynasty. He described Pascal's triangle, more than six centuries before Pascal, and used it to calculate square and cube roots.

Hasan Ibn al-Haytham (أبو علي، الحسن بن الحسن بن الهيثم, c. 965 - 1050) lived in Cairo during the Islamic Golden Age, and studied mathematics, physics, astronomy, philosophy, and medicine. He was a proponent of the scientific method: the belief that any scientific hypothesis must be verified using experiments or mathematical logic - centuries before European scientists during the Renaissance.

Al-Haytham was particularly interested in optics and visual perception. He also derived a formula for the sum of fourth powers (1 4+2 4+3 4+...+n 4), and he studied the link between algebra and geometry.

Zu Chongzhi (祖沖之, 429 - 500 CE) was a Chinese astronomer, mathematician, writer, politician and inventor.

He calculated Pi accurately to 7 decimal places - a record which was not surpassed until 800 years later. To do this, he approximated a circle with a 24,576-sided polygon.

Zu also discovered the formula 4 3 πr 3 for the volume of a sphere. His precise astronomical observations allowed him to create a new, more accurate calendar and to predict solar eclipses. He also calculated that Jupiter takes almost 12 years to orbit the sun.

While we know today this model is incorrect, Ptolemy's scientific impact is indisputable. He developed trigonometric tables with many practical applications, which remained the most accurate for many centuries. He also created detailed maps of the Earth, and wrote about music theory and optics.

Heron of Alexandria ( Ἥρων ὁ Ἀλεξανδρεύς, c. 10 - 70 CE) was a Greek mathematician and engineer. He lived in the city of Alexandria in Egypt, and is one of the greatest "experimenter" of antiquity.

His inventions include windmills, pantograph, as well as a radial steam turbine called aeolipile or Hero's engine. Hero's formula allows you to calculate the area of any triangle, using just the length of its three sides.

Hipparchus of Nicaea (Ἵππαρχος, c. 190 - 120 BCE) was a Greek astronomer and mathematicians, and one of the greatest astronomers of antiquity.

Hipparchus made detailed observations of the night sky and created the first comprehensive star catalog in the western world. He is considered the father of trigonometry: he constructed trigonometric tables and used these to reliably predict solar eclipses. He also invented the astrolabe and solved different problems in spherical trigonometry.

Archimedes (c. 287 - 212 BCE) was an ancient Greek scientist and engineer, and one of the greatest mathematicians of all time. He discovered many concepts of calculus and worked in geometry, analysis and mechanics.

While taking a bath, Archimedes discovered a way to determine the volume of irregular objects using the amount of water they displaced when submerged. He was so excited by this discovery that he ran out on the street, still undressed, yelling "Eureka!" (Greek for "I have found it!").

As an engineer, he built ingenious defence machines during the siege of his home city Syracuse in Sicily. After two years, the Romans finally managed to enter, and Archimedes was killed. His last words were "Do not disturb my circles" - which he was studying at the time.

Euclid taught mathematics in Alexandria, but not much else is known about his life.

Aristotle (Ἀριστοτέλης, c. 384 - 322 BCE) was a philosopher in Ancient Greece. Together with his teacher Plato, he is considered the "Father of Western Philosophy". He was also the private tutor of Alexander the Great.

Aristotle wrote about science, mathematics, philosophy, poetry, music, politics, rhetoric, linguistics, and many other subjects. His work was highly influential during the Middle Ages and into the Renaissance, and his views on ethics and other philosophical questions are still being discussed today.

Aristotle is also the first known person to formally study logic, including its applications in science and mathematics.

Zeno of Elea (c. 495 - 430 BCE) was a Greek philosopher who his known for his famous paradoxes, which have fascinated mathematicians for centuries.

One example is the paradox of motion: imagine that you want to run a 100 meter race. You first have to run half the distance (50 meters). But before doing that, you have to run a quarter of the distance (25 meters). Before running a quarter, you have to run 1 8th, 1 16th, and so on. This is an infinite number of tasks, which means that you'll never arrive!

Pythagoras of Samos (c. 570 - 495 BCE) was a Greek philosopher and mathematician. He is best known for proving Pythagoras' Theorem, but made many other mathematical and scientific discoveries.

Pythagoras tried to explain music in a mathematical way, and discovered that two tones sound "nice" together (consonant) if the ratio of their frequencies is a simple fraction.

He also founded a school in Italy where he and his students worshipped mathematics almost like a religion, while following a number of bizarre rules - but the school was eventually burned down by their adversaries.

The Ishango Bone is possibly the oldest mathematical artefact still in existence: it was discovered in 1950, in the Democratic Republic of Congo in central Africa, and is named after the region where it was found. It is dates back to the Upper Paleolithic period of human history, and is approximately 20,000 years old.

The bone is 10 cm long and contains a series of notches, which many scientists believe were used for counting. The grouping of the notches might even suggest some more advanced mathematical understanding, like decimal numbers or prime numbers.

In ancient Mesopotamia, almost 10,000 years ago, scribes and merchants started using small, three-dimensional clay objects as counters, to represent certain quantities, units or goods. Thousands of these were found on archaeological sites across the Middle East, like these from Tepe Gawra in Iraq (from around 4000 BCE):

The cone, sphere and flat disc were used to represent small, medium and large measures of grain. The tetrahedron probably measured the amount of work done in one day.

These two tablets from Susa in Iran were created around 3200 BCE and used a more advanced technique: the counters were pressed into the clay while it was still soft, to create a record:

Again, the triangular and circular impressions represent smaller and larger measures of grain. The patterns across the rest of the tablet were the official seals of the scribes.

These simple markings actually laid the foundations for cuneiform, one of the first writing system in history.

This is the oldest known clay tablet with mathematican computations - it was created around 2700 BCE in Sumer, one of the earliest civilisations that flourished in the Middle East.

It shows a multiplication table in cuneiform, which may have been used by student scribes to learn mathematics.

This tablet shows a multiplication table that was created around 2600 BCE in the Sumerian city of Shuruppak. It is one of the oldest mathematical tablets we have ever discovered.

The table has three columns. The dots in the first two columns represent distances ranging from around 6 meters to 3 kilometres. The third column contains the product of the first two, which is the area of a rectangle with the given dimensions.

Sumer was a region of ancient Mesopotamia in the Middle East. They invented Cuneiform as one of the earliest writing systems, by pressing small, wedge-shaped markers into clay tablets like this one. They also developed the base-60 number system.

This Babylonian clay tablet, called Plimpton 322, was created around 1750 BCE in Sumeria, during the reign of Hammurabi the Great.

The exact purpose of the tablet has been debated by archeologists. Some think that it was a "teachers aid", designed to help generate right-angled triangles. Others think it may be a very early trigonometry table.

This circular tablet from the Yale Babylonian Collection, called YBC 7289, was created around 1800 - 1600 BCE in ancient Babylon. It shows the geometric diagram of a square with its diagonals.

The cuneiform numerals indicate that one side of the square is 30 units long, and show how to find the length of the diagonal: units.

The tablet shows that Babylonian scribes knew Pythagoras' theorem, more than 1000 years before Pythagoras was even born. They were also able to calculate square roots and had an estimate for accurate to 6 decimal places. It is the highest computational accuracy ever seen in the ancient world!

While this simple tablet may have just been a practice exercise by a novice scribe, its mathematical and historical importance is enormous.

These two clay tablets from the Yale Babylonian Collection were created between 1800 and 1600 BCE, and contain exercises by student scribes, to calculate the area of different geometric shapes.

Tablet YBC 7290 shows how to calculate the area of a trapezium, by multiplying the average of the bases and the average of the sides.

Tablet YBC 11120 shows how to calculate the area of a circle, using the approximation

The Rhind Papyrus is one of the most famous mathematical documents from ancient Egypt. It was written around 1550 BCE by a scribe called Ahmose, who is maybe the earliest contributor to maths in history, whose name we still know today.

The papyrus is around 2 meters long and contains 84 problems about multiplication, division, fractions, and geometry. It was probably used as a kind of "textbook" by other scribes.

One of the most notable sections is a table. This shows how you can write rational numbers of the form where n is an odd number, as sums of unit fractions.

The papyrus is named after Scottish antiquarian Alexander Henry Rhind, who purchased it in Luxor, Egypt. Today, most of its remains are located at the British Museum in London.

Menna was a chief scribe in ancient Egypt, and in charge of measuring the size of fields for farming, inspected crop yields, reporting to the Pharaoh's central field administration, and calculating taxes.

The wall paintings in his tomb show the different measuring and calculating techniques used more than 3,000 years ago. For example, in the first row, you can see how long distances were measured using ropes with knots at regular intervals.

The tomb was built around 1420 BCE in the Valley of the Kings.

Here you can see a set of 21 Bamboo Strip that were created around 2300 years ago in China. When arrenged correctly, they form a multiplication table in base 10, written in ancient Chinese calligraphy.

While earlier civilisations like the Babylonians created multiplications tables in base 60, this is by far the oldest known decimal multiplication table - and it looks very similar to what we still use today.

Around 300 BCE, Euclid of Alexandria wrote The Elements, collection of 13 books that contained mathematical definitions, postulates, theorems and proofs, and covering topics like geometry and number theory.

It is one of the most famous books ever written, and one of the most influential works in the history of mathematics. Copies were used as textbooks for thousands of years and studied all around the world, with thousands of new editions published

No original copies of the Elements still exist today. This small papyrus fragment dates back to around 100 AD, and may be a part of the oldest existing copy of Euclid's work.

A palimpsest is a scroll or parchment from which the text has been washed or scraped off so that it can be reused. This method was common in the Middle Ages - even for documents by brilliant scientists and mathematicians.

Archimedes of Syracuse lived in the 3rd Century BCE and was one of the greatest mathematicians in history. A Greek copy of some of his work, created around 1000 CE in Byzantium, was later overwritten by Christian monks in Palestine. More recently, forgers added pictures to increase the value of the documents.

In 1998, scientists started studying the Archimedes Palimpsest, and used X-rays, ultraviolet and infrared light to uncover the hidden original text.

The Suàn shù shū (筭數書), which means Book on Numbers and Computation, is one of the oldest mathematical manuscripts from China. It was written around 200 BCE and consists of 200 strips of bamboo.

There are 69 problems, each with a solution, covering topics like arithmetic, fractions, integer factorisation, geometric sequences, inverse proportions, unit conversion, and error handling. Geometry problems show how to find the area of circles and rectangles, as well as the volume of three-dimensional solids, while assuming that

The inscription on this stone includes the oldest known use of the number zero: it dates back to the Khmer civilisation in Cambodia, around the year 683 CE.

Part of the text contains the number 605. Can you spot the dot that represents the zero? The inscription was only recently rediscovered by Amir Aczel.

Many ancient civilisations, like the Greeks and Romans, did not have a "zero" in their numeral system. From Cambodia, the concept was passed to India, where the Hindu-Arabic numeral system originated. From there, it spread to the Middle East and Europe, and we still use it today.

Some ancient American civilisations like the Maya also used zero in their calendars, but their numbers systems did not survive colonisation.

The title of the book Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa'l-muqābala (الْكِتَابْ الْمُخْتَصَرْ فِيْ حِسَابْ الْجَبْرْ وَالْمُقَابَلَة, short just __Al-Jabr__‎) translates to The Compendious Book of Calculations by Completion and Balancing.

It was written by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī around 820 CE, and established Algebra as a new area of mathematics. In fact, the name algebra derived from the word al-ğabr in the title of the book.

Al-Khwārizmī is often called the father of algebra. In the book, he shows how to solve linear and quadratic equations, how to calculate the area and volume of certain geometric shapes, and he introduces the concept of "balancing" when solving equations.

Maqalah fi al-jabra wa-al muqabalah, which means Demonstration of Problems in Algebra, is a manuscript written by the Persian mathematician Omar Khayyam, around 1100 CE.

Khayyam managed to classify and solve all cases of cubic equations, using the intersection of conic sections. For example, on this page he shows how to solve equations of the form using the intersection of a circle and a hyperbola.

He also explored a triangle of binomial coefficients. In Iran, this triangle is called the Khayyam triangle, while in Europe and America it is more commonly known as Pascal's traingle.

The Lilāvatī is the first volume of a series of books written by Bhāskara II, one of the greatest mathematicians and astronomers in medieval India. It was published around 1150, when he was 36 years old.

Bhāskara wrote the book for his daughter, and the title actually means "playful". He writes about problem-solving, number sequences, Pythagoras' theorem, combinatorics, and many other topics.

These two pages show a problem about a pet peacock standing on a column, which can be solved using Pythagoras' theorem.

In the following volumes, Bhāskara also writes about algebra and astronomy. The combined work is called Siddhānta-Śiromani, which is Sanskrit for Crown of Treatises.

Very few Mayan documents have survived until today: one of them is the Dresden codex. It was created in the 13th century and describes Mayan mathematics and astronomy.

The Mayan number system had base 20 - using both fingers and toes for counting. Every digit from 1 to 19 consists of circles (representing 1) and horizontal lines (representing 5). Can you work out what all the numbers on this page are?

The Dresden Codex was used as a divination almanac, to record the date of astronomical events important for certain rituals. This fragment may contain the dates of eclipses of the planet Venus.

The Liber Abaci, Latin for Book of Calculation, was published in 1202 by Leonardo Fibonacci, the son of an Italian merchant. Together with his father, he spent his youth travelling around the Mediterranean.

He studied mathematics from Islamic scholars and learned about new ideas like algebra and the Hindu-Arabic numerals, both of which greatly simplified business transactions. When he returned to Italy, Fibonacci wrote a book about everything he learned.

He first introduced our current number system to Europe, which was still using Roman numerals at the time, and explained how to convert between both systems. In later chapters, he explains how to calculate profit and interest, how to approximate irrational numbers, how to determine whether a number is prime, and many other topics in mathematics. Most famously, he shows how rabbit populations might grow using the numbers 1, 1, 2, 3, 5, 8, ... These numbers are now known as Fibonacci numbers.

The Siyuan Yujian (四元玉鉴), which means Jade Mirror of the Four Unknowns, is a masterpiece of Chinese mathematics, published in 1303 by Zhu Shijie. It consists of four individual books and 288 different problems.

Zhu shows how to solve problems using systems of polynomial equations with up to four unknown variables, 天 (Heaven), 地 (Earth), 人 (Man) and 物 (Matter). He explains how to eliminate variables and how to find the side length of two and three-dimensional shapes given their volume or area.

To solve some of these problems, Zhu even used the numbers in Pascal's triangle, more than 300 years before Pascal was born!

Zhu also published a number of other mathematics texts, like the Suanxue Qimeng (New Arithmetic Enlightenment) in 1299. This textbook is written in verse, like many similar books at the time, which makes it wasy to memorise the arithemtic calculations.

Quipu are a recording system that was used by the Incan civilisation in South America around 1400 - 1560. They consist of many strings with small knots, all of which are attached to one larger rope. The type and position of the knots, as well as the colour of the strings, was used to record numbers, dates and maybe even text.

The Incans used a decimal number system like we do today. The position of a knot indicates the place value (ones, tens, hundreds, ...). Different types of knots (e.g. figure-8 knots and long-knots) represents the digit from 0 to 9.

When the Italian mathematician Luca Pacioli needed illustrations for his book De divina proportione (published in 1509), he asked Leonardo Da Vinci, a renown artist and former student.

Da Vinci created 60 different images of polyhedra. He often made a solid version, as well as a transparent version that only shows the edges, which was a completely new way to represent these 3-dimensional solids.

The Codex Mendoza is a description of the Aztec civilisation, which was commissioned in 1541 by Antonio de Mendoza. Its three sections explain the history and daily life of the Aztec people and list the different rulers and towns that were conquered.

The codex also contains examples of the Aztec calendar system, which you can see along the blue bar. Each of the symbols represents a date, and consists of a small image combined with several small circles.

The Aztec calendar used 20 day signs represented by a small image (crocodile, wind, house, lizard, snake, rabbit, water, etc.), together with up to 13 circles. This gives a cycle of 20 × 13 = 260 days.

Can you see which dates are be represented by the symbols on this page?

# data[].attributes.attachment_link.description
Travel through time and explore the greatest mathematicians and biggest mathematical discoveries in history.

# data[].attributes.attachment_link.description
I'd been teaching maths for over ten years before I even heard the word 'variation' in relation to my job. Now, the word, and what lies behind it, is in my mind almost all the time. I consider it from the moment I conceive a lesson to the moment the students walk out of the door when the bell rings.

# data[].attributes.attachment_link.provider_name
Erikson Institute Early Math Collaborative

# data[].attributes.attachment_link.description
Quantity is a particular amount of something, expressed as a number. Quantity cards for young children can have pictures of small sets such as dots, finger patterns, or 5- or 10- frames.

# data[].attributes.headline
New Face to Face offer for GFE's, 16Dec21

# data[].attributes.subject
New Face to Face offer for GFE's, 16Dec21

# data[].attributes.headline
Using your MiDES report to improve English and Maths in your college (Code: C1387)

# data[].attributes.subject
Using your MiDES report to improve English and Maths in your college (Code: C1387)

# data[].attributes.body
16/03/2022 16:00 - 18:00

# data[].attributes.attachment_link.readable_text
This course looks at the data featured in MiDES reports and explores key aspects of what the report shows including learner starting points and the progress that learners make.

MiDES reports provide unique insight into learners' maths and English grades and the progress that learners make, from pre-enrolment to exams

MiDES is a useful tool to inform planning, delivery and evaluation

MiDES data can provide supporting evidence for Ofsted's Intent, Implementation and Impact.

Upcoming

Who is this for?

Training aim

Outcomes

# data[].attributes.attachment_link.description
This course looks at the data featured in MiDES reports and explores key aspects of what the report shows including learner starting points and the progress that learners make.

# data[].attributes.headline
Maths courses   ALL ARE NOW FULLY FUNDED please book on these courses up until end March 2022

# data[].attributes.subject
Maths courses   ALL ARE NOW FULLY FUNDED please book on these courses up until end March 2022

# data[].attributes.body
Maths workshop (Code: C1261)

Experiences a range of methods and fill your mathematical toolbox with alternative approaches - suitable for both GCSE and Functional Maths, Express your interest

- Tips and resources for maths teaching 30/03/2022 16:00 - 17:00 and 31/03/2022 10:00 - 11:00

Algebra in maths (Code: C1226)  Express your interest,  2hrs
Application of ratio and direct proportion (Code: C1215) | Express your interest, 2 hrs

Culture, Gender and Maths (Code: C1405) Express your interest, 2hrs

Functional number (Code: C1230) 30/03/2022 13:00 - 15:00, 2hrs
Getting things in proportion in GCSE maths (Code: C1253) Express your interest, 2hrs
Graphs in maths (Code: C1257) Express your interest, 2hrs
Impact of language in maths (Code: C1256)  Express your interest, 2hrs
Integrating maths into vocational courses (Code: C1254) 26/04/2022 15:00 - 17:00, 1hr
Key factors in individual maths lesson planning (Code: C1251)  31/03/2022 15:00 - 17:00, 2hrs
Manipulating and using numbers (Code: C1237)  Express your interest, 2hrs
Maths in a triangle (Code: C1258)  Express your interest, 2 hrs
Maths in more than one dimension (Code: C1239)  Express your interest, 2hrs
Maths is beautiful (Code: C1249)  Express your interest, Watch the maths is beautiful video
Maths problem solving (Code: C1255)   30/03/2022 10:00 - 14:00 and 06/04/2022 14:00 - 16:00, 2hrs
Maths puzzles (Code: C1250)  31/03/2022 12:00 - 14:00, 2hrs
Modelling and abstraction (Code: C1259)   Express your interest, 2hrs
Money maths (Code: C1240)  29/03/2022 09:30 - 11:30, 2hrs Watch the money maths video
Number at the core of maths (Code: C1227)  Express your interest,  1hr
Presenting statistical data using graphs and charts (Code: C1238)  Express your interest, 2hrs
Supporting maths learners from Entry level 3 to Level 2 (Code: C1229)  29/03/2022 13:00 - 15:00,  2hrs Watch the E3 to L2  video
The role of assessment in teaching maths (Code: C1228) register your interest and click to check dates, 1hr
Visualising maths (Code: C1260) Express your interest, 2hrs

Maths Functional Skills teaching (Level 5 module) (Code: C1305) Express your interest

Maths GCSE Teaching: Develop your Practice (Level 5 module) (Code: C1304) Express your interest

Teaching Functional Skills Maths Online course on Future Learn

Other support
Mathscoe Mini Webinar Series - open access, put together by CfEM.  I am going to work through these myself and we can discuss at our next network meeting

Foundation Online Learning modules

The  multi-modal and pdf guide ‘Improving the delivery of maths and English in apprenticeships’ is still available on the Excellence Gateway.

# data[].attributes.headline
THE DFE KS3 MATHS GUIDANCE – WHAT YOU NEED TO KNOW March22

# data[].attributes.subject
THE DFE KS3 MATHS GUIDANCE – WHAT YOU NEED TO KNOW March22

# data[].attributes.attachment_link.readable_text
Classroom-facing materials

The guidance is packed with examples for use in the classroom with your students. In addition, you will find details of difficulties and misconceptions that you are likely to come across when teaching each topic, so you can plan a curriculum which tackles these issues head on.

Our recent feature Using the DfE's KS3 maths guidance in secondary teaching focuses on percentage change and gives a step-by-step approach to using examples from the guidance in lesson planning.

Creating a curriculum

As well as providing student-facing content, the guidance also supports departments in creating a coherent and connected curriculum. It could be used in a departmental meeting to review your existing curriculum and ensure that it is well-resourced and uses appropriate representations consistently. There is even a sample curriculum framework that you could base yours on or check yours against. Each school term is broken down into topics and sub-topics using an easy-to-follow approach.

Additional materials

If you'd like to hear directly from a teacher starting to use the guidance, listen to our recent podcast episode. A maths lead from a multi-academy trust getting to grips with the guidance talks to Carol Knights, the NCETM Director for Secondary, and one of the authors of the document.

For further supplementary classroom-facing materials, we have created a series of key ideas PowerPoints. They contain activities and professional development stimuli based on the 52 exemplifications found in the guidance.

And don't forget, you can follow us on Twitter to be the first to hear about new resources we publish to complement the guidance. Our recent Week in Focus looked at what the KS3 guidance has to offer. You can find out more by searching #NCETMWeekInFocus on social media.

# data[].attributes.attachment_link.title
The DfE KS3 maths guidance - what you need to know

# data[].attributes.attachment_link.description
The guidance is packed with examples for use in the classroom with your students. In addition, you will find details of difficulties and misconceptions that you are likely to come across when teaching each topic, so you can plan a curriculum which tackles these issues head on.

# data[].attributes.attachment_link.description
A self-marking exercise in three levels on solving direct and inverse variation problems.

# data[].attributes.attachment_link.readable_text
Directly proportional: as one amount increases, another amount increases at the same rate.

Example: you are paid $20 an hour

How much you earn is directly proportional to how many hours you work

Work more hours, get more pay; in direct proportion.

This could be written:

Earnings ∝ Hours worked

If you work 2 hours you get paid $40

If you work 3 hours you get paid $60

etc ...

Constant of Proportionality

The "constant of proportionality" is the value that relates the two amounts

Example: you are paid $20 an hour (continued)

The constant of proportionality is 20 because:

Earnings = 20 × Hours worked

This can be written:

y = kx

Where k is the constant of proportionality

Example: y is directly proportional to x, and when x=3 then y=15.
What is the constant of proportionality?

They are directly proportional, so:

Put in what we know (y=15 and x=3):

Solve (by dividing both sides by 3):

The constant of proportionality is 5:

y = 5x

When we know the constant of proportionality we can then answer other questions

Example: (continued)

What is the value of y when x = 9?

What is the value of x when y = 2?

Inversely Proportional

Inversely Proportional: when one value decreases at the same rate that the other increases.

Example: speed and travel time

Speed and travel time are Inversely Proportional because the faster we go the shorter the time.

As speed goes up, travel time goes down

And as speed goes down, travel time goes up

This: y is inversely proportional to x

Example: 4 people can paint a fence in 3 hours.

How long will it take 6 people to paint it?

(Assume everyone works at the same rate)

It is an Inverse Proportion:

As the number of people goes up, the painting time goes down.

As the number of people goes down, the painting time goes up.

We can use:

t = k/n

Where:

t = number of hours

k = constant of proportionality

n = number of people

"4 people can paint a fence in 3 hours" means that t = 3 when n = 4

So now we know:

And when n = 6:

So 6 people will take 2 hours to paint the fence.

How many people are needed to complete the job in half an hour?

So it needs 24 people to complete the job in half an hour.
(Assuming they don't all get in each other's way!)

Proportional to ...

It is also possible to be proportional to a square, a cube, an exponential, or other function!

Example: Proportional to x2

A stone is dropped from the top of a high tower.

The distance it falls is proportional to the square of the time of fall.

The stone falls 19.6 m after 2 seconds, how far does it fall after 3 seconds?

We can use:

Where:

d is the distance fallen and

t is the time of fall

When d = 19.6 then t = 2

So now we know:

And when t = 3:

So it has fallen 44.1 m after 3 seconds.

Inverse Square

Inverse Square: when one value decreases as the square of the other value.

Example: light and distance

The further away we are from a light, the less bright it is.

In fact the brightness decreases as the square of the distance. Because the light is spreading out in all directions.

So a brightness of "1" at 1 meter is only "0.25" at 2 meters (double the distance leads to a quarter of the brightness), and so on.

# data[].attributes.attachment_link.title
Directly Proportional and Inversely Proportional

# data[].attributes.attachment_link.description
Directly proportional: as one amount increases, another amount increases at the same rate. How much you earn is directly proportional to how many hours you work Work more hours, get more pay; in direct proportion. This could be written: Earnings ∝ Hours worked If you work 2 hours you get paid $40 If you work 3 hours you get paid $60 etc ...

# data[].attributes.attachment_link.readable_text
Mathematical goals

This lesson unit is intended to help you assess whether students are able to:

Identify when two quantities vary in direct proportion to each other.

Distinguish between direct proportion and other functional relationships.

Solve proportionality problems using efficient methods.

Introduction

This lesson unit is structured in the following way:

Before the lesson, students work individually on a task designed to reveal their current levels of understanding. You review their solutions and write questions to help them improve their work.

At the beginning of the lesson, there is a whole-class discussion about key features of direct proportionality.

Students then work in small groups on a task related to the assessment task. They write and solve their own questions on direct proportion, exchange questions with another group, assess each other's work, and write suggestions for improvement.

In a whole-class discussion, students share their questions and solution methods, generalizing to identify criteria for identifying direct proportion.

In a follow-up lesson, students use their learning and your questions to review their work.

Materials required

Each individual student will need a calculator, a mini-whiteboard, a pen, an eraser, a copy of the assessment task, Getting Things in Proportion, and a copy of the review task, Getting Things in Proportion (revisited).

Each small group of students will need a copy of the lesson tasks, Direct Proportion or Not? and Swapping Questions.

Time needed

15 minutes before the lesson, a 60-minute lesson, and 15 minutes in a follow-up lesson. These timings are approximate: exact timings will depend on the needs of your class.

# data[].attributes.attachment_link.description
This lesson unit is intended to help you assess whether students are able to: Identify when two quantities vary in direct proportion to each other. Distinguish between direct proportion and other functional relationships. Solve proportionality problems using efficient methods.

# data[].attributes.attachment_link.title
How to draw direct and inverse proportion graphs

# data[].attributes.attachment_link.description
Explore how to draw and read graphs that show direct proportion and inverse proportion with BBC Bitesize Maths. For students between the ages of 11 and 14.

# data[].attributes.attachment_link.readable_text
it may be more appropriate to present ideas and tasks without the ppt offered (where they are)

ppts are provided with, hopefully, some interesting questions and so that teachers can adapt questions

sorry, I'm a unicase fan...

contents have been moved to this blog

apologies if this causes any difficulties e.g. with links

practice and GCSE sheets blog

ideas are aimed at those who teach mathematics to adolescents; who like tasks with some depth, novelty and a focus on generalising relationships and on transformationdon steward

MEDIAN

# data[].attributes.attachment_link.description
it may be more appropriate to present ideas and tasks without the ppt offered (where they are) ppts are provided with, hopefully, some interesting questions and so that teachers can adapt questions sorry, I'm a unicase fan... contents have been moved to this blog apologies if this causes any difficulties e.g.

# data[].attributes.attachment_link.description
Join the Pearson Maths team as we invite a host of special guests to join us to discuss the important issues for maths teachers in the UK.

# data[].attributes.headline
Does it matter if you don't get a C (or 4) grade in GCSE mathematics

# data[].attributes.subject
Does it matter if you don't get a C (or 4) grade in GCSE mathematics

# data[].attributes.attachment_link.title
Does it matter if you don’t get a C (or 4) grade in GCSE mathematics? - FFT Education Datalab

# data[].attributes.attachment_link.description
We look at the long term effects of missing out on a C/4 grade, including future educational outcomes, job prospects and overall life satisfaction.

# data[].attributes.headline
We're only able to mentally represent and exact number if we have a word for it

# data[].attributes.subject
We're only able to mentally represent and exact number if we have a word for it

# data[].attributes.attachment_link.readable_text
Babies, monkeys and even bees have a basic "sense of number". They can instantly perceive that there are one, two, three or four objects in a pile, without having to count them. They can also tell at a glance that a pile of 50 objects contains more than a pile of 20, say. But what explains the unique ability of older kids and adults to go far beyond this, and mentally represent quantities much bigger than four exactly? Some researchers argue that language must be key - that learning to count "one", "two", "three", and on and on, enables this cognitive feat. Others argue that language can't be fundamental to this "numerical" ability.

Now a striking new study in Psychological Science by Benjamin Pitt at the University of California, Berkeley, and colleagues comes down firmly on the side of language as being key. And this has a broader significance. It supports the hotly contested idea that language itself influences or even enables abilities that have been viewed as being completely independent - such as colour perception, or, in this case, understanding of number.

Earlier studies have suggested that a knowledge of number words is important for understanding exact numbers above four. But these studies have suffered from various drawbacks. Take work on the Pirahã, an indigenous group living in the Brazilian Amazon who have no words for any exact quantity, not even "one". Pirahã adults seem to be able to mentally represent only up to about four objects. But as Pitt and his colleagues note, they differ from WEIRD comparison groups in all sorts of ways, beyond differences in number language. Some researchers have suggested that their performance might reflect simply their society's "indifference" to exact large numbers, rather than an inability to form concepts of them, for example.

Pitt and his team turned to the Tsimane', a society of farmer-foragers who live in the remote Bolivian Amazon (but who, it has to be said, have featured in all manner of studies looking at cross-cultural differences in perception, including music perception; they may not be familiar with Western culture but they must be getting very familiar with Western researchers).

Ideally for the team's research purposes, the Tsimane' language features a full number system, but while some adults have an excellent knowledge of it and can count indefinitely, others have a very limited knowledge of number words. This allowed the team to compare the performance of "high" vs "low" counters within the same culture.

To do this, they gave adult participants pebble-matching tasks. The strongest test of exact number understanding was an "orthogonal-matching task". A line of white pebbles was placed vertically, stretching away from the participant. Their task was to use glass pebbles to create a matching horizontal line beneath. To succeed at this task, a participant had to understand the exact number of white pebbles, and place the same number of glass pebbles in their line.

The team first ascertained the participant's highest verbal word count. Then high counters (who could count above 40) started with a line of 10 white pebbles, while low counters started with two fewer pebbles than their highest verbal word count (which ranged from 6 to 20). From this point on, if they placed the correct number of glass pebbles to match the white ones, the team increased the number of pebbles in the next array by two. If they were wrong, they were given one fewer pebble in their next array. This continued until a participant did one of three things: they did not make a perfect match for the same number of pebbles three times; they correctly matched three arrays of 20 pebbles or more; they completed 20 trials.

The team were then able to compare each participant's verbal word count with the number of pebbles at which they started to make mistakes - the point at which they switched from exactly reproducing the target array to approximating it.

The results were clear: the switch point for high counters was far higher - averaging 28, compared with just 7 (and never exceeding 11) for the low counters. Importantly, with one exception, the participants' switch points were at or below their highest verbal word count - even for pebble sets smaller than ten. When the number of target pebbles approached or reached the number that they had a word for, they started approximating. And this led to the team to an important conclusion: within this group "the ability to represent exact numbers was limited to the part of the verbal count list they had mastered". This, they argue, "represents the strongest evidence to date that number words play a functional role in representing large exact numbers".

Given the debate over the extent to which language might underpin or enable very different abilities, this is a really significant finding. It doesn't mean that language is the only tool that can enable understanding of large exact numbers, however. Patterns of counters on an abacus, for example, work as wordless symbols of even extremely large numbers. But as the team writes, their new findings suggest that "whatever set of symbols people use, their ability to represent large exact numbers extends only as far as their mastery of those symbols".

- Exact Number Concepts Are Limited to the Verbal Count Range
Emma Young (@EmmaELYoung) is a staff writer at BPS Research Digest

# data[].attributes.attachment_link.title
We're only able to mentally represent an exact number if we have a word for it

# data[].attributes.attachment_link.description
Babies, monkeys and even bees have a basic "sense of number". They can instantly perceive that there are one, two, three or four objects in a pile, without having to count them. They can also tell at a glance that a pile of 50 objects contains more than a pile of 20, say.

# data[].attributes.body
No matter where you live, your world is full of geometric ideas. From the ways buildings are constructed, to streets and telephone poles, there are shapes, lines, and angles intersecting, overlapping, and partitioning space. We draw on this bounty in our Off You Go activities, in which we ask students to find concepts in their environment and bring them back to share and discuss with the class. This process helps sharpen students’ ideas about concepts and widen their lens for the many ways these concepts can show up in their worlds.

# data[].attributes.attachment_link.title
Where can you find find parallel lines or intersecting lines?

# data[].attributes.attachment_link.description
You can take a picture, make a sketch, or bring objects to show. Bring back at least one example of parallel lines or intersecting lines.

# data[].attributes.attachment_link.readable_text
Research
My research interests in mathematics education focus on the learning and teaching of mathematics in ways that support students' conceptual understanding. I am particularly interested in the design and use of rich tasks in the mathematics classroom, and in finding ways to enable students to develop the necessary fluency in mathematical processes to support them in solving mathematical problems. I am also interested in confidence assessment as a tool for formative assessment.

My main research focus is currently as a Co-Director of the Centre for Early Mathematics Learning, funded by the Economic and Social Research Council (ESRC), which is part of UK Research and Innovation. This ESRC Centre will transform understanding of children's mathematics learning during the early years and design effective educational activities to improve skills and knowledge. Related to this, in my role as Director of the Loughborough University Mathematics Education Network, I am leading a design-research project to construct the LUMEN Curriculum, a completely free, research-informed, full set of mathematics curriculum resources for schools. This builds on a recent Economic and Social Research Council (ESRC) project Exploring socially-distributed professional knowledge for coherent curriculum design, carried out in collaboration with Professor Geoff Wake, Dr Fay Baldry and Professor Keiichi Nishimura.

I am continuing to build on my research in the , which is a study into the development of mathematical fluency, and also the Mathematical Etudes Project Confidence Assessment Project, which is exploring how asking students to indicate how sure they are about the answers they give in low-stakes mathematics tests can contribute to formative assessment.

I have twice spoken about my research as a guest on the and Mr Barton Maths Podcast: click here to listen.

I supervise PhD students across a variety of topics within mathematics education.
Current research students
Abbas, Suyi (PhD): The effect of confidence assessment on confidence and attitude to mathematics.

Britton-Drewry, Emily (PhD): Optimising the learning of multiplication and division facts in schoolchildren.

Casha Sammut, Melanie (PhD): Gifted and talented students in primary mathematics.

Gleadow, Tanya (PhD): Evaluation and development of a data science curriculum to increase motivation and ambition for further study and careers in computing and STEM.

Mahak, Sheeza (PhD): The role of anxiety, confidence and metacognitive skills in mathematical problem solving.

Makri, Depy (PhD): Content-based reflection as a structural part of online workbooks for learning mathematics.
Completed research students
Strauss, Jacob (2023). Developing secondary school students' procedural fluency with Cover, Copy and Compare. Unpublished PhD dissertation. Loughborough University.

Bustang, Bustang (2022). Probability conceptions and metacognitive judgements of Indonesian secondary school students and in-service mathematics teachers. Unpublished PhD dissertation. Loughborough University.

Guimarães, Rita Santos (2019). Investigating mathematics teachers' changes in practice during a professional development initiative. Unpublished PhD dissertation. University of Nottingham.

Barichello, Leo (2018). An investigation into how low-achieving secondary students learn fractions through visual representations. Unpublished PhD dissertation. University of Nottingham.

Calleja, James (2018). Learning to teach mathematics through inquiry: A case study of continuing professional development in Malta. Unpublished PhD dissertation. University of Nottingham.

McCance, Lorna (2014). How do students' solutions to a problem-solving task change after attempting to understand an algebra-focused sample solution? Unpublished MA dissertation. University of Nottingham.

Peake, Eleanor (2014). When and how do Year 2 pupils use the Bar Model to solve mathematical word problems? Unpublished MA dissertation. University of Nottingham.

Roberts, Katherine (2013). Which aspects of mathematical problem-solving lessons do pupils value and why? Unpublished MA dissertation. University of Nottingham.
Conferences and seminars
I have also regularly participated in the Institute of Mathematics Pedagogy .

Mathematics Education Blog

I posted at blog.foster77.co.uk every fortnight during my year as President of the Mathematical Association (2022-23).

These 26 posts continue to be available below:

Foster, C. (2022, April 14). Intro-ducing and outro-ducing methods [Blog post]. https://blog.foster77.co.uk/2022/04/intro-ducing-and-outro-ducing-methods.html

Foster, C. (2022, April 28). Tangible contexts for mathematics [Blog post]. https://blog.foster77.co.uk/2022/04/tangible-contexts-for-mathematics.html

Foster, C. (2022, May 12). Learning times tables efficiently [Blog post]. https://blog.foster77.co.uk/2022/05/learning-times-tables-efficiently.html

Foster, C. (2022, May 26). Are two cars better than one? [Blog post]. https://blog.foster77.co.uk/2022/05/are-two-cars-better-than-one.html

Foster, C. (2022, June 9). Motivation for measurement [Blog post]. https://blog.foster77.co.uk/2022/06/motivation-for-measurement.html

Foster, C. (2022, June 23). Lines of not-very-good fit [Blog post]. https://blog.foster77.co.uk/2022/06/lines-of-not-very-good-fit.html

Foster, C. (2022, July 7). A football on the roof [Blog post]. https://blog.foster77.co.uk/2022/07/a-football-on-roof.html

Foster, C. (2022, July 21). Making rounding interesting [Blog post]. https://blog.foster77.co.uk/2022/07/making-rounding-interesting.html

Foster, C. (2022, August 4). Misremembering Goldbach's conjecture [Blog post]. https://blog.foster77.co.uk/2022/08/misremembering-goldbachs-conjecture.html

Foster, C. (2022, August 18). The Differentiator [Blog post]. https://blog.foster77.co.uk/2022/08/the-differentiator.html

Foster, C. (2022, September 1). Interactive introductions [Blog post]. https://blog.foster77.co.uk/2022/09/interactive-introductions.html

Foster, C. (2022, September 15). Always simplify your answer [Blog post]. https://blog.foster77.co.uk/2022/09/always-simplify-your-answer.html

Foster, C. (2022, September 29). Tasks that can't fail [Blog post]. https://blog.foster77.co.uk/2022/09/tasks-that-cant-fail.html

Foster, C. (2022, October 13). How open should a question be? [Blog post]. https://blog.foster77.co.uk/2022/10/how-open-should-question-be.html

Foster, C. (2022, October 27). Butterfly effects when adapting tasks [Blog post]. https://blog.foster77.co.uk/2022/10/butterfly-effects-when-adapting-tasks.html

Foster, C. (2022, November 10). Is area more difficult than volume? [Blog post]. https://blog.foster77.co.uk/2022/11/is-area-more-difficult-than-volume.html

Foster, C. (2022, November 24). Fractions as factors [Blog post]. https://blog.foster77.co.uk/2022/10/fractions-as-factors.html

Foster, C. (2022, December 8). Dividing into thirds [Blog post]. https://blog.foster77.co.uk/2022/12/dividing-into-thirds.html

Foster, C. (2022, December 22). Mixing the dimensions in models of number [Blog post]. https://blog.foster77.co.uk/2022/12/mixing-dimensions-in-models-of-number.html

Foster, C. (2023, January 5). Proportionality [Blog post]. https://blog.foster77.co.uk/2023/01/proportionality.html

Foster, C. (2023, January 19). Is zero really a number? [Blog post]. https://blog.foster77.co.uk/2023/01/is-zero-really-number.html

Foster, C. (2023, February 2). Non-expository video clips [Blog post]. https://blog.foster77.co.uk/2023/02/non-expository-video-clips.html

Foster, C. (2023, February 16). Don't forget the units? [Blog post]. https://blog.foster77.co.uk/2023/02/dont-forget-units.html

Foster, C. (2023, March 2). Are probabilities and inequalities approximate? [Blog post]. https://blog.foster77.co.uk/2023/03/are-probabilities-and-inequalities.html

Foster, C. (2023, March 16). Crocodiles and inequality signs [Blog post]. https://blog.foster77.co.uk/2023/03/crocodiles-and-inequality-signs.html

Foster, C. (2023, March 30). Teaching specific tactics for problem solving [Blog post]. https://blog.foster77.co.uk/2023/03/teaching-specific-strategies-for.html

© Colin Foster 2023

# data[].attributes.attachment_link.description
Research My research interests in mathematics education focus on the learning and teaching of mathematics in ways that support students' conceptual understanding. I am particularly interested in the design and use of rich tasks in the mathematics classroom, and in finding ways to enable students to develop the necessary fluency in mathematical processes to support them in solving mathematical problems. I am also interested in confidence assessment as a tool for formative assessment.

# data[].attributes.attachment_link.description
Dynamic graphing and geometry for the web

# data[].attributes.attachment_link.readable_text
Noticing smaller groupings within the set is the beginning of Conceptual Subitizing- when you quickly perceive smaller groups within the set and use these to determine the amount. Check out this at mathisvisual.com for a fun example based on the ideas of Jo Boaler.

Another idea I play with my children is "Fingers, Fingers," developed by Kristen Reed and Jessica Mercer Young. Check out their great article about games: https://www.naeyc.org/resources/pubs/tyc/oct2017/play-games-learn-math-explore-numbers ; Youtube video

*One additional thought. It is recommended that children primarily practice Subitizing with simple dots. Consider how you can incorporate dice and dot cards into your daily routines and math games.

MATH LANGUAGE

Quantity- the amount of something. The "how many or how much."

Perceived- to understand or realize something.

Subitizing- ability to quickly perceive a number without counting.

Conceptual Subitizing- when you quickly perceive smaller groups within the set and use these to determine the amount.

Set- items that are grouped together in a meaningful way.

Number sense- the ability to understand the quantity of a set and the name associated

with that quantity.

Resources
Erikson Institute's Early Math Collaborative (2014). Big Ideas of Early Mathematics. Pearson Education.
Clements, D.H., & Sarama, J. (2017)*

Reed, K., & Young, J.M. (2017)*

Nguyen, Laski, Thomson, Bronson, & Casey, 2017*

Mathisvisual.com

* full citations can be found on the Research Page

SHARE YOUR EXPERIENCES WITH OTHERS IN THE COMMENTS SECTION

# data[].attributes.attachment_link.description
Noticing smaller groupings within the set is the beginning of Conceptual Subitizing- when you quickly perceive smaller groups within the set and use these to determine the amount. Check out this at mathisvisual.com for a fun example based on the ideas of Jo Boaler.

# data[].attributes.attachment_link.readable_text
Article by Dr Sue Gifford University of Roehampton

Download pdf
Young children have a remarkable skill: they can recognise numbers of things without counting. This is called subitising, and it develops from a very early age. Very young babies can not only tell the difference between one and two but also between large numbers of dots when there are twice as many in one group, as with 16 and 8 (Sarama and Clements, 2009). Young children also have powerful visual memories and some may find it easier to remember images than words: three-year-olds can recognise three things, although they may not say the word. Subitising can help children to build images for numbers, to visualise and to learn number facts. For instance, most four-year-olds readily learn to recognise five dots on a dice, which helps them to understand the cardinal value or 'howmanyness' of five, which they can link to the word and symbol for 5. Structured images like this also help children to begin to see numbers inside numbers, for instance seeing four and one within five.

How do you teach subitising?
Children's ability to subitise can be developed by teaching. Games provide repeated experience of interpreting the dot images, either to decide how many spaces to move, as in our activity Playing Incey Wincey Spider, or how many things to collect, as in our activity Dice, where the dotty number determines how many animals to put in your pen. (To find both of these activities, scroll down the collection of number based EYFS tasks.) For children who are unfamiliar with dot images, dice can be adapted to just show numbers up to three. Large dotty dice are good for outdoor games, like jumping along the track. Spinners with dots on provide an alternative, as with the child in the picture, who has his own spinner on a lanyard: he spins it and then makes that many jumps around the circuit track, then does the same again and again and again..! Having your own dice or spinner is a good option for younger children and avoids waiting for turns.

Large floor dominoes invite children to match number patterns, indoors or out.

Fingers also provide subitisable images for numbers, with the added advantage that they are embodied in muscle memory. Showing all-at-once finger numbers is the key skill here: Marton & Neuman (1990) found that older children with maths difficulties tended to count fingers one at a time, rather than using all-at-once 'finger numbers'. You can ask young children to first 'grow', then 'show' and finally to 'throw' finger numbers.

Using a range of objects and apparatus in different arrangements helps to make children familiar with different images for numbers, while also emphasising different number properties. For instance, Numicon presents numbers as even or odd, whereas linked cubes show numbers as lengths: connecting all these images helps children build a rich network of understanding to link to number words and symbols.

Subitising games can involve less common arrangements of numbers of objects. Games which involve revealing hidden objects will prompt young children to say how many they see (see the NRICH activity Hidden Jewels). One game involves upturned dishes hiding three, four and five objects, rapidly swopping them around and challenging children to keep track of the dish with four. When you stop and briefly lift the dish, children intuitively subitise the number revealed. This activity is taken from Clements and Sarama's very useful website, which presents a learning trajectory for subitising, supported by videos of activities for each stage.(If you have not already done so, you will need to create an account before accessing the free resources.)

https://www.learningtrajectories.org/math-activities/fantastic-five-tricky-two

Activities include auditory subitising, or recognising numbers of sounds, such as claps or drum beats, linking with movement and music.

Conceptual subitising
Subitising can also help children learn addition and subtraction facts. In some of the images below the total number of five is recognised by adding subgroups, like three and two, or four and one (Back, 2014 - see the NRICH article Early Number Sense). This involves part-whole awareness, which enables children to simultaneously see the whole number and numbers which make it up. The ability to think about numbers as compositions of other numbers was described by Resnick (1983) as 'probably the major conceptual achievement of the early school years'.

Recognising how many there are by combining numbers of smaller subgroups is called conceptual subitising: this is more advanced than perceptual subitising, which involves instant recognition (Clements, 1999). Seeing numbers as combinations of other numbers helps children to develop part-whole awareness and build familiarity with number combinations. It is also much more effective than giving children number sentences to complete by counting cubes and hoping they will remember the numbers involved!

Number talks
Showing children 'quick images' and asking them how many they see is one way of developing subitising. An alternative way of developing conceptual subitising is to show children a number of dots and ask them to say how they see the number (see the NRICH activity Number Talks). Teacher Janine Davenall engages young children by showing an image and asking, 'What numbers can you see hidden inside this number?'

They may reply, 'I see three and four', or 'six and one' or 'two and two and two and one'. This is an inclusive activity with many 'right' answers and the opportunity to discover that others see the same thing differently. Children also learn about all the different combinations which can make the same number. An alternative is to use giant magnetic buttons on a tray, which you turn over to secretly rearrange and show different combinations of the same number.

( www.teachingchannel.org/videos/visualizing-number-combinations)

Not only do younger children respond more to real objects than to images on a screen, but they can also see that the number of buttons stays the same despite being rearranged.

More complex images can present greater challenges for some children. Mulligan and Mitchelmore (2009) showed young children a 6 dot triangle and asked to draw it from memory: they found their responses varied significantly, with some just drawing a lot of dots.

Even when asked to copy the image, many five-year-olds produce dotty triangles like the one on the right, apparently finding it hard to focus on the number and the arrangement of the dots (Gifford & Thouless, 2018). This implies that some children will need a lot more practice in looking and talking about number patterns, in order to see how they are organised. Daily 'number talks' can help children to analyse and visualise number images, building an understanding of cardinal values, like the 'sixness' of six.

Reception children enjoy creating their own arrangements for numbers like seven and nine, and seeing which their friends recognise most easily. Older children with mathematics difficulties found the 'lollipop seven' image most memorable, at the same time learning a key number fact, that four and three make seven (Emerson & Babtie, 2010).

In Hungary, children are taught to recognise such numbers as 'five and a bit', using the dice image for 5 and linking with finger numbers (see the NRICH article From Objects and Images to Mathematical Ideas).

In the Netherlands, researchers developed bead-strings and rekenreks (counting frames) using groups of five, as they found these helped children to 'see' numbers more quickly (Gravemeijer, 1991). Being able to subitise groups supports children's understanding of larger numbers and later multiplicative thinking: for instance, recognising larger numbers by counting groups of two, three or five.

(The above images are taken from the Number Talk Images website.)

Subitising with everyday objects is more challenging than with simpler shapes, like dots. However, it can be more engaging for children to discuss images which are of particular interest and relevance to them, or even to create their own images for number talks.

Acknowledgments:

Janine Davenall
Buckinghamshire, Berkshire & Oxfordshire Mathematical Pedagogy Work Group

References:

Back, J. (2014) Early Number Sense
Clements, D.H. (1999) Subitising: What is it? Why teach it? Teaching Children Mathematics 5(7) 400-405
Emerson, J. & Babtie, P. (2010) The dyscalculia assessment: Emerson House Mathematics London: Continuum.
Thouless, H. & Gifford, S. (2018) 'Dotty triangles: two different approaches to analysing young children's responses to a pattern replication activity' in Golding, J., Bretscher, N., Crisan, C., Geraniou, E., Hodgen J. and C. Morgan (Eds). (2018) Research Proceedings
of the 9th British Congress on Mathematics Education (3-6 April 2018, University of Warwick, UK) 175-182 Online at www.bsrlm.org.uk/bcme-9/
Gravemeijer, K.P.E. (1991) An instruction-theoretical reflection on the use of manipulatives in Streefland, L. Realistic mathematics education in primary school. Utrecht: Freudenthal Institute 57- 76
Marton, F. & Neuman, D. (1990) 'Constructivism, phenomenology and the origin of arithmetic skills' in Steffe, L.P. & Wood, T. (1990) Transforming children's mathematics education: international perspectives NJ: Lawrence Earlbaum Associates 62- 75
Resnick, L.B. (1983) 'A developmental theory of number understanding' in Ginsburg, H. (ed) The development of mathematical thinking. London Academic Press 109 - 181.
Sarama, J. S., & Clements, D. H. (2009). Early childhood mathematics education research. London: Routledge.

here.

# data[].attributes.attachment_link.description
In this article for EY practitioners, Dr Sue Gifford outlines what we mean by subitising and how we can teach it.

# data[].attributes.attachment_link.readable_text
ATM experts are sharing ideas through Maths Snacks videos, to support children and parents whilst children are away from school for whatever reason or for use in the classroom. The videos are a mix of tasks, puzzles, challenges and games; selected to support and enrich mathematics taught at school.

Many videos can be accessed by very young children, with a little bit of adult support, and all are relevant to pupils across the school age range. There's hardly ever a single right answer and there are many ways of working on tasks. The videos are designed to entice and to focus on working in the ways mathematicians do; such as exploring situations, looking for patterns and rules, asking questions and seeking answers, and trying out and testing ideas.

If you find these videos useful and want to support our work you can become a member, or make a donation via our Paypal link

Accessible for all

Accessible for those who have a little mathematical experience

Parents can work with their children. All family members can join in. This isn't cheating! Encourage children to have a go; ask them what they are noticing and what they have found out, ask them if there are more ways of completing the task. Why not make another similar task using different shapes or numbers? Make their maths snacking enjoyable!

Maths Snacks can supplement the work being sent home from school or, for those unable to access the work from school, these Maths Snacks provide a healthy alternative. Children will need a pencil and paper; and they might need a pair of scissors, dice, counters or buttons and a calculator.

By keeping mathematically fit, children on their return to school will be in good mental shape to reconnect and progress with their school mathematics courses. Who knows? They may continue maths snacking!

Thank you for choosing to use these materials in your organisation. These videos are Copyright 2020 by the Association of Teachers of Mathematics, and they are free to use and redistribute under the terms of the CC-BY-3.0 licence (see https://creativecommons.org/licenses/by/3.0/ for full details). In particular, you must ascribe the source of these materials; you can do this by quoting the italicised text below.

A small group of people, Geoff Faux, Pip Huyton, Peter Lacey, Mike Ollerton, Colin Taylor and Helen Williams, all of whom are members of the Association of Teachers of Mathematics, voluntarily created a collection of 2-minute videos, which the ATM have freely circulated; they were initially for use during COVID-19 Lockdown. However, because of the positive responses the ATM have received, we believe the videos might also be used as resources in mathematics classrooms. If you like these then why not join with us in the ATM community at www.atm.org.uk/Membership-Types Any feedback about how children work on these snacks would be greatly appreciated.

Thank you to the following for their contribution to Maths Snacks.

# data[].attributes.attachment_link.description
ATM experts are sharing ideas through Maths Snacks videos, to support children and parents whilst children are away from school for whatever reason or for use in the classroom. The videos are a mix of tasks, puzzles, challenges and games; selected to support and enrich mathematics taught at school.

# data[].attributes.attachment_link.readable_text
This section will contain evidence from various research carried out during the CfEM project in order to find out what works for maths in the Further Education (FE) and Training sector.

It will be of interest to researchers and policy makers, but also to maths teachers and managers who are seeking to find evidence to support the promotion of mastery teaching in FE.

For now, it contains the video overviews and detailed reports on action research projects conducted by practitioners in the 21 CfEM colleges and their networks during 2020-21, accompanied by a synthesis of the reports, entitled Changing the Experience of Maths Resits.

By the end of March 2023, this section will be developed to include:

A full report on the teaching for mastery randomised control trial, carried out by the University of Nottingham during 2021-22.

A full report on a larger-scale research project, being carried out by the ETF and Pye Tait during the autumn term on 2022. This is based on a scale-up of four promising interventions developed as part of CfEM colleges' action research.

Further action research reports from 2021-22 and a final synthesis of the projects.

# data[].attributes.attachment_link.title
CfEM evidence - The Education and Training Foundation

# data[].attributes.attachment_link.description
Evidence from various research carried out during the CfEM project in order to find out what works for maths in the Further Education (FE) and Training sector.

# data[].attributes.attachment_link.readable_text
Why do we use ​'Think Aloud'?

If we want pupils to tackle open ended problem solving in my school, we'll often begin with Think Alouds.

Recommendation 3 of the EEF ​' Improving Mathematics at Key Stages 2 and 3' guidance report highlights that, to help pupils develop problem-solving skills, we need to provide opportunities for them to use and compare different approaches, and to monitor and reflect upon their effectiveness. Think Aloud approaches meet this end.

When I approach Think Alouds, I explore what the problem is asking me to do, and what I still need to find out. I narrate my thought processes to show I knew that I needed to work systematically to find multiple solutions to a problem. I then model how to organise my working to record solutions using a table. I model monitoring and evaluating my approach, showing pupils how I knew that I had found all possible solutions.

Using a Think Aloud in this way gave my pupils insight into how I - as an ​'expert' learner - approach problem-solving, making my thought processes clear and transparent. Ultimately, this allowed pupils to become self-scaffolders, giving them a script to replay when it came to their own independent problem-solving.

When should we use ​'Think Aloud'?

At Amberley primary school, we use Think Alouds frequently to model both the problem-solving process as a whole, and specific elements of this that we need teach explicitly. For example, this can be used to demonstrate how to get started with a problem, how to select an appropriate approach, or how to monitor progress or evaluate the strategies used.

Think Aloud models aren't just used for whole class teaching. They're also used when supporting small groups or delivering interventions. Staff then encourage pupils to narrate their thought processes when working with an adult. This helps our staff team to understand the strategies children use and whether or not these are effective. It also supports formative assessment by allowing us to check for understanding, which then informs future teaching.

What do we say in a ​'Think Aloud'?

What I say during a Think Aloud depends on the stage of the problem-solving process I am modelling. I use the first person to ask myself questions whilst working. This shows learners the types of questions that they can ask themselves so that they can later apply this in their own independent problem-solving.

I use the metacognitive cycle - plan, monitor and evaluate - to structure my Think Alouds.

When planning my approaches, I ask myself questions such as, ​' What do I notice?', ​' What information is given and how could this help me?' and ​' Have I seen something similar to this before?'

At this stage, I model choosing a strategy such as bar modelling to represent the problem and help me identify the steps needed to solve it. For pupils, this highlights the importance of considering prior knowledge and how to identify a way into the problem in their independent work.

I model monitoring my problem-solving approaches by asking: ​' Is my approach working?', ​' I'm stuck; what could I do now?', or ​' Does my solution make sense?'

I've found that modelling this stage of the problem-solving process has been extremely valuable to show pupils how to check that their work is on track and makes sense in the context of the problem. This helps pupils to review their work more frequently and prevent pupils from blindly pursuing a strategy even when it becomes apparent that this just isn't working.

Finally, I model evaluating my approach to problem-solving by asking myself: ​' Did the approach work? Why?' or ​' What would I do differently next time?' I also ensure that I model asking, ​' What have I learned about myself and my learning?' When considering my own practice, this evaluation is the element of the problem-solving process which I may previously have skipped over or missed out. However, I've come to realise this is vitally important because it helps pupils make connections and develop their understanding of themselves as learners more broadly.

For example, pupils in my class have learned to identify specific gaps in their own understanding, or areas where they feel more confident than others. For many children, this has helped them to identify their own targets for learning, for example in relation to particular multiplication facts, or mathematical concepts which they find challenging.

For us, using Think Aloud models to develop problem-solving has been the first step in helping learners to understand how to overcome challenges in their own learning, by showing how we - the adults working with them - do this ourselves.

Providing a Think Aloud model helps pupils to develop a toolkit of strategies which inform their future work, equipping them with the knowledge and skills they need to become independent problem-solvers in the future.

# data[].attributes.attachment_link.keywords[]
why do we use ‘think aloud’?  if we want pupils to tackle open ended problem solving in my school

# data[].attributes.attachment_link.keywords[]
we’ll often begin with think alouds .  recommendation 3 of the eef ‘ improving mathematics at key stages 2 and 3 ’ guidance report highlights that

# data[].attributes.attachment_link.keywords[]
to help pupils develop problem-solving skills

# data[].attributes.attachment_link.keywords[]
we need to provide opportunities for them to use and compare different approaches

# data[].attributes.attachment_link.keywords[]
and to monitor and reflect upon their effectiveness. think aloud approaches meet this end.  when i approach think alouds

# data[].attributes.attachment_link.keywords[]
i explore what the problem is asking me to do

# data[].attributes.attachment_link.keywords[]
and what i still need to find out. i narrate my thought processes to show  how  i knew that i needed to work systematically to find multiple solutions to a problem. i then model how to organise my working to record solutions using a table. i model monitoring and evaluating my approach

# data[].attributes.attachment_link.keywords[]
showing pupils how i knew that i had found all possible solutions.    using a think aloud in this way gave my pupils insight into how i – as an ‘expert’ learner – approach problem-solving

# data[].attributes.attachment_link.keywords[]
making my thought processes clear and transparent.  ultimately

# data[].attributes.attachment_link.keywords[]
this allowed pupils to become self-scaffolders

# data[].attributes.attachment_link.keywords[]
giving them a script to replay when it came to their own independent problem-solving.    when should we use ‘think aloud’?    at amberley primary school

# data[].attributes.attachment_link.keywords[]
we use think alouds frequently to model both the problem-solving process as a whole

# data[].attributes.attachment_link.keywords[]
and specific elements of this that we need teach explicitly. for example

# data[].attributes.attachment_link.keywords[]
this can be used to demonstrate how to get started with a problem

# data[].attributes.attachment_link.keywords[]
or how to monitor progress or evaluate the strategies used.    think aloud models aren’t just used for whole class teaching. they’re also used when supporting small groups or delivering interventions. staff then encourage pupils to narrate their thought processes when working with an adult. this helps our staff team to understand the strategies children use and whether or not these are effective. it also supports formative assessment by allowing us to check for understanding

# data[].attributes.attachment_link.keywords[]
which then informs future teaching.   what do we say in a ‘think aloud’?   what i say during a think aloud depends on the stage of the problem-solving process i am modelling. i use the first person to ask myself questions whilst working. this shows learners the types of questions that they can ask themselves so that they can later apply this in their own independent problem-solving.    i use the metacognitive cycle – plan

# data[].attributes.attachment_link.keywords[]
monitor and evaluate – to structure my think alouds.  when planning my approaches

# data[].attributes.attachment_link.keywords[]
‘ what information is given and how could this help me? ’ and ‘ have i seen something similar to this before? ’  at this stage

# data[].attributes.attachment_link.keywords[]
i model choosing a strategy such as bar modelling to represent the problem and help me identify the steps needed to solve it. for pupils

# data[].attributes.attachment_link.keywords[]
this highlights the importance of considering prior knowledge and how to identify a way into the problem in their independent work.  i model monitoring my problem-solving approaches by asking: ‘ is my approach working? ’

# data[].attributes.attachment_link.keywords[]
or ‘ does my solution make sense? ’   i’ve found that modelling this stage of the problem-solving process has been extremely valuable to show pupils how to check that their work is on track and makes sense in the context of the problem. this helps pupils to review their work more frequently and prevent pupils from blindly pursuing a strategy even when it becomes apparent that this just isn’t working.  finally

# data[].attributes.attachment_link.keywords[]
i model evaluating my approach to problem-solving by asking myself: ‘ did the approach work? why? ’ or ‘ what would i do differently next time? ’ i also ensure that i model asking

# data[].attributes.attachment_link.keywords[]
‘ what have i learned about myself and my learning? ’ when considering my own practice

# data[].attributes.attachment_link.keywords[]
this evaluation is the element of the problem-solving process which i may previously have skipped over or missed out. however

# data[].attributes.attachment_link.keywords[]
i’ve come to realise this is vitally important because it helps pupils make connections and develop their understanding of themselves as learners more broadly.

# data[].attributes.attachment_link.title
EEF blog: The what, why, when and how of using a 'Think Aloud' in...

# data[].attributes.attachment_link.description
Why do we use ​'Think Aloud'? If we want pupils to tackle open ended problem solving in my school, we'll often begin with Think Alouds. Recommendation 3 of the EEF ​' Improving Mathematics at Key Stages 2 and 3' guidance report highlights that, to help pupils develop problem-solving skills, we need to provide opportunities for them to use and compare different approaches, and to monitor and reflect upon their effectiveness. Think Aloud approaches meet this end.

# data[].attributes.headline
A Level Maths: Binomial expansion in pure and applied maths

# data[].attributes.subject
A Level Maths: Binomial expansion in pure and applied maths

# data[].attributes.attachment_link.readable_text
A Level Maths: Binomial expansion in pure and applied maths

Steven Walker, Maths Subject Advisor

The binomial expansion of ( a + b) is essentially focused on n = 2 at GCSE and then developed further in subsequent Level 3 maths qualifications ( FSMQ: Additional Maths - 6993, Mathematics A - H230/H240 and Mathematics B (MEI) - H630/H640), initially for positive integer powers of n, and then for any rational value of n.

Here I look at some of the common mistakes and misconceptions seen by examiners in student responses. I'll also look at the applications of binomial expansion that could form the basis of Level 3 problem solving and modelling questions.

When n is a positive integer

The FSMQ and AS qualifications limit binomial expansion questions to positive integer powers of n.

Included on the formulae sheet for each is:

for use when n is a positive integer.

The most common errors with this type of expansion are when there are multiples or negatives involved.

Find the binomial expansion of (1 − 5 x) 4,
expressing the terms as simply as possible.
Question 6 Core 1 - 4751 June 2016 MEI Mathematics 3895/7895

Whilst this question was done well in most cases, it was not unusual to see
1 - 20 x + 30 x... (forgetting to include the multiplying coefficient with each term) or to see
1 - 20 x - 150 x 2 ... (forgetting to include the negative sign with the indices).

Careful presentation can help avoid these common mistakes. In the classroom I would suggest students set out their work in a column format:

The terms can then be evaluated before writing out the single result of 1 - 20 x + 150 x 2 - 500 x 3 +625 x 4. There is less risk of missing terms, mixing indices or not applying the index correctly to both the sign and coefficient.

This technique may also be useful in questions only requiring the coefficient of a specific term.

Find the constant term in the expansion of
Question 6 H640/03 June 2018 Mathematics B(MEI) - H640

Students were generally successful here, but those writing out the full expansion would waste valuable exam time

Problem solving in pure maths

Questions where an unknown is included within a binomial expression are an example of problem solving.

It is given that, in the expansion of (3 + kx) 3, the coefficient of x 2 is equal to the constant term.
Find the possible values of k, giving your answers in an exact form.
Question 3 Core 2 - 4722 June 2016 OCR Mathematics 3890/7890 (edited)

Again, setting the response out in a column format allows students to identify the appropriate terms before any calculations are needed.

A common mistake on this question was assuming k > 0. Many students reached 27 = 9 k 2 but, having found k = √3, didn't include k = - √3.

Modelling accurate calculations

One application of binomial expansion is to determine exact results for calculations that would not be possible using a calculator.

For example, the exact result of the calculation (0.0103) 5 can be found using a binomial expansion such as (1 × 10-2) 5 (1 + 3 x) 5 with x = 0.01.

Other substitutions are valid, but students should ensure that the working is clear and they have confidence that the individual terms can be evaluated exactly.

For example, (0.0103) 5 could be written as (1×10-2) 5 (1 + x) 5 where x = 0.03. However, the alternative form (1 × 10-4) 5 (2 + x), with x = 101, would be problematic, since a standard calculator could only display an approximate solution for 101 5 ≈ 1.05101005 × 10 10.

There is no restriction on the values of a or b in the expression ( a + b). However, it is not unusual to expect the values to be limited to 1 significant figure in order to allow confidence with the numerical terms when evaluating exact results. Making either a or b equal to 1 also helps simplify the arithmetic.

Binomial distribution model

Binomial expansion is important in binary outcome experiments. Here, the expression used is often written as ( p + q) where p is the probability of a successful outcome, n is the number of attempts and q = 1 - p.

In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, X, which are defective.
If X > 2, the batch is rejected. It is given that 5% of mugs are defective.
Find the probability that the batch is rejected after just the first sample is checked.
Question 7 Probability and Statistics 1 - 4732 Jun 2013 OCR Mathematics 3890/7890 (edited)

When this question was set in 2013, students were expected to use statistical tables rather than using the expansion formula, whereas now the expectation is that students use the binomial distribution function on the calculator. For some students however, the written expansion will help avoid the common confusion between P( X = 2), P( X ≤ 2) and P( X ≥ 3). Using p = 0.95 or p = 0.05 is equally valid, but a sketched distribution curve may help avoid introducing further confusion in solutions.

The synoptic nature of the current A Level Maths means that the link between binomial expansion and binomial distributions may be more explicit.

Write down and simplify the first four terms in the expansion of (x + y)7. Give your answer in ascending powers of x.

Given that the terms in x 2 y 5 and x 3 y 4 in this expansion are equal, find the value of .

A hospital consultant has seven appointments every day. The number of these appointments which start late on a randomly chosen day is denoted by L. The variable L is modelled by the distribution B (7, ). Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.

Question 10 H230/01 SAM AS Mathematics A - H230

General expansion

In A Level Maths, the topic progresses to include fractional and/or negative indices. The A Level Maths formulae sheet includes

Use the expansion (1 + 3 x) 0.5 to find an estimate for √103, correct to five decimal places.
Question 2 H240/02 June 2019 Mathematics A - H240 (edited)

A common mistake on this question was to attempt x = 34, ignoring the constraint that | x|<1. Successful students generally spotted that 103 = 100 × 1.03 and so used √103 = 10 (1 + 3 x) 0.5 where x = 0.01.

Applications in calculus

A particularly useful application of binomial expansion is in finding approximations for complex functions, which can then be easier to differentiate or integrate.

Find the binomial expansion of and hence find .

Question 6 Core 2 - 4722 June 2014 OCR Mathematics 3890/7890 (edited)

Students generally avoided careless mistakes as long as they used brackets around each part of the binomial product and remembered to include the nC r component. A few did however express terms such as ( x 3) 4 incorrectly as x 7, or mistakenly evaluated terms such as

as 32 x 3.

With the expression successfully expanded, students generally found the subsequent integration relatively straight forward (though arithmetic mistakes were seen with negative indices and the perennial omission of '+ c ').

Support

For further past paper feedback, see the examiner reports on the 'Assessment' area of our qualifications' webpages ( A Level Maths A - H240 and A Level Maths B (MEI) - H640), where you can also find the papers themselves. (Papers for autumn 2021 are currently held on Interchange). Further questions can be found in the legacy papers available on ExamBuilder, our free question-building platform.

You can find resources such as delivery guides and section check in tests in the 'Planning and teaching' area for each qualification.

A set of introductory videos and exercises for this topic are available on the binomial theorem section on the Khan Academy site.

For a nice challenge, try the numerical problem in the section on counting and binomials on Underground Maths.

Stay connected

There are many great online resources available, so join the conversation by sharing your ideas and links to all your favourites in the comment box below.

If you have any queries you can email us at maths@ocr.org.uk, call us on 01223 553998 or tweet us @OCR_Maths. You can also sign up for monthly email updates to receive information about resources and support.

About the author
Steven Walker, OCR Maths Subject Advisor
Steven joined OCR in 2014 during the major qualification reform period and now primarily focuses on supporting the Level 3 maths qualifications. He originally studied engineering and then took an extended period to work and travel around the world before completing a PGCE in secondary mathematics. Steven began his teaching career with VSO in Malawi and has taught maths in both the UK and overseas.

Related blogs

# data[].attributes.attachment_link.title
A Level Mathematics - what's in spec A for you?

# data[].attributes.attachment_link.description
A Level Maths: Binomial expansion in pure and applied maths Steven Walker, Maths Subject Advisor The binomial expansion of ( a + b) is essentially focused on n = 2 at GCSE and then developed further in subsequent Level 3 maths qualifications ( FSMQ: Additional Maths - 6993, Mathematics A - H230/H240 and Mathematics B (MEI) - H630/H640), initially for positive integer powers of n, and then for any rational value of n.

# data[].attributes.headline
Using 'Think Alouds' to support pupils' mathematics problem solving in practice

# data[].attributes.subject
Using 'Think Alouds' to support pupils' mathematics problem solving in practice

# data[].attributes.attachment_link.readable_text
Emma Barker -assistant headteacher, maths lead and Year 6 teacher from Amberley Primary School- explains how she uses the ​"Think Aloud" approach to model metacognition and support pupils' mathematical problem-solving.

Read the blog then share your ideas in the discussions in the Teaching 11-19 Science and Primary STEM groups.

Why do we use 'Think Alouds'?

If we want pupils to tackle open ended problem solving in my school, we'll often begin with Think Alouds.

Recommendation 3 of the EEF ​ 'Improving Mathematics at Key Stages 2 and 3' guidance report highlights that, to help pupils develop problem-solving skills, we need to provide opportunities for them to use and compare different approaches, and to monitor and reflect upon their effectiveness. Think Aloud approaches meet this end.

When I approach Think Alouds, I explore what the problem is asking me to do, and what I still need to find out. I narrate my thought processes to show how I knew that I needed to work systematically to find multiple solutions to a problem. I then model how to organise my working to record solutions using a table. I model monitoring and evaluating my approach, showing pupils how I knew that I had found all possible solutions.

Using a Think Aloud in this way gave my pupils insight into how I - as an ​'expert' learner - approach problem-solving, making my thought processes clear and transparent. Ultimately, this allowed pupils to become self-scaffolders, giving them a script to replay when it came to their own independent problem-solving.
When should we use ​'Think Alouds'?
At Amberley primary school, we use Think Alouds frequently to model both the problem-solving process as a whole, and specific elements of this that we need teach explicitly. For example, this can be used to demonstrate how to get started with a problem, how to select an appropriate approach, or how to monitor progress or evaluate the strategies used.

Think Aloud models aren't just used for whole class teaching. They're also used when supporting small groups or delivering interventions. Staff then encourage pupils to narrate their thought processes when working with an adult. This helps our staff team to understand the strategies children use and whether or not these are effective. It also supports formative assessment by allowing us to check for understanding, which then informs future teaching.
What do we say in a ​'Think Aloud'?
What I say during a Think Aloud depends on the stage of the problem-solving process I am modelling. I use the first person to ask myself questions whilst working. This shows learners the types of questions that they can ask themselves so that they can later apply this in their own independent problem-solving.

I use the metacognitive cycle - plan, monitor and evaluate - to structure my Think Alouds.

When planning my approaches, I ask myself questions such as, ​'What do I notice?', ​'What information is given and how could this help me?' and ​'Have I seen something similar to this before?'

At this stage, I model choosing a strategy such as bar modelling to represent the problem and help me identify the steps needed to solve it. For pupils, this highlights the importance of considering prior knowledge and how to identify a way into the problem in their independent work.

I model monitoring my problem-solving approaches by asking: ​'Is my approach working?', ​'I'm stuck; what could I do now?', or ​'Does my solution make sense?'

I've found that modelling this stage of the problem-solving process has been extremely valuable to show pupils how to check that their work is on track and makes sense in the context of the problem. This helps pupils to review their work more frequently and prevent pupils from blindly pursuing a strategy even when it becomes apparent that this just isn't working.

Finally, I model evaluating my approach to problem-solving by asking myself: ​'Did the approach work? Why?' or ​'What would I do differently next time?' I also ensure that I model asking, ​'What have I learned about myself and my learning?' When considering my own practice, this evaluation is the element of the problem-solving process which I may previously have skipped over or missed out. However, I've come to realise this is vitally important because it helps pupils make connections and develop their understanding of themselves as learners more broadly.

For example, pupils in my class have learned to identify specific gaps in their own understanding, or areas where they feel more confident than others. For many children, this has helped them to identify their own targets for learning, for example in relation to particular multiplication facts, or mathematical concepts which they find challenging.

For us, using Think Aloud models to develop problem-solving has been the first step in helping learners to understand how to overcome challenges in their own learning, by showing how we - the adults working with them - do this ourselves.

Providing a Think Aloud model helps pupils to develop a toolkit of strategies which inform their future work, equipping them with the knowledge and skills they need to become independent problem-solvers in the future.

Did you like this blog? Scroll back to the top and hit the 'Recommend' button!

# data[].attributes.attachment_link.title
Using ‘Think Alouds’ to support pupils’ mathematics problem solving in practice

# data[].attributes.attachment_link.description
Using ‘Think Alouds’ to support pupils’ mathematics problem solving in practice

# data[].attributes.headline
Exclusively for the 2022 Advanced Information topics!🔺 Higher and Foundation 🔺 Full Video Tutorial for Each Booklet🔺 Completely FREE!🔺 Available Now 

# data[].attributes.subject
Exclusively for the 2022 Advanced Information topics!🔺 Higher and Foundation 🔺 Full Video Tutorial for Each Booklet🔺 Completely FREE!🔺 Available Now 

# data[].attributes.body
There is a series of AMSP online STEP sessions, priced at £160 in total

There are a couple of online resources available...

http://meikleriggs.co.uk/

https://maths.org/step/

There are the following non-STEP but perhaps useful problem-solving-based resources...

https://madasmaths.com/

https://www.cemc.uwaterloo.ca/resources/potw.php

https://mei.org.uk/maths-item-of-the-month/

https://www.ukmt.org.uk/competitions/solo/senior-mathematical-challenge/archive

Advanced Problems in Maths and More Advanced Problems in Maths by Stephen Siklos

https://undergroundmathematics.org/

http://www.risps.co.uk/

https://colmanweb.co.uk/problemsolving/index.htm

https://integralmaths.org/ritangle

# data[].attributes.attachment_link.readable_text
Video

US. Dept. Spec. Education & NCLD:

Saying dyslexia, dyscalculia, & dysgraphia in schools, an Interview with Michael Yudin, Assistant Secretary of Special Education & Rehab Services (2015)
2017 National Assessment of Educational Progress (NAEP) Report
52% of US 4 th-graders were not proficient in Mathematics;

56% of US 8 th-Graders were not proficient in Mathematics;

72% of US 12 th-Graders were not proficient in Mathematics (2015, NEAP).

In 2015, the USA, ranked 38 out of 71 countries in Math Achievement on the Program for International Student Assessment. Trends in International Mathematics & Science Study (TIMSS).

In 2015, 29% of 15-year-olds in the United States of America, scored deficient (in the lowest 10%) on international tests of mathematical literacy- 3 in every 10 students were below proficiency Level 2- worse than half (35) of the 69 countries tested. Only 6%, scored above 90% proficiency (Level 5), worse than 36 countries.

Source: NCES, 2016, p.23.

DSM-V: Specific Learning Disorders (SLDs) are defined in The Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5), published by the American Psychiatric Association. "Specific Learning Disorder-- with impairment in Mathematics (315.1), Reading (315.0), or Written Expression (315.2). SLD is "a neurodevelopmental disorder of biological origin manifested in learning difficulties and problems in acquiring academic skills markedly below age level and manifested in the early school years, lasting for at least 6 months, not attributed to intellectual disabilities, developmental disorders, or neurological or motor disorders."

VIDEO: Dyscalculia Signs & Research

* References

* Images

* Dyscalculia Videos

You're copying, " 15.17" thinking or saying, " fifteen seventeen," but you write, " 15.70". It sounds almost identical. You're copying, " 264.69" thinking or saying, " two sixty-four sixty nine," but you write, " 36 9. 4 9".

"Sixty" suggested 3, you retained sixty nine, and 4, and the 9 ending, and wrote " 369.49".

You can't seem to remember numbers, even important ones (multiplication tables, birthdays, dates, amounts).

When counting, you easily lose track, especially if distracted. You're overwhelmed by all of the directional sequences in long division, multiplication, arithemetic, and algebra. You struggle with visualizing, time, directions, layouts, sequences, scheduling, budgeting, figuring, procedures, and logistics.
Smart Thick Kid ~ Living with Dyscalculia by Loz Mac
Young children struggle with left and right, directionality, counting reliably, number-amount associations, memory of numbers and quantitative information, memory of instructions, short-term memory (working memory), time awareness, telling time, time management, schedules, organization, sequencing, procedures for arithmetic, place value, memory of addition and multiplication facts, memory of math rules, mental arithmetic, visualization, name-face memory, visual memory, and visual-spatial discrimination, interpretation, processing, and memory. They make unconscious errors with numbers and symbols when reading, listening, thinking (reasoning), copying, writing, and speaking. When doing math, they think slowly and carefully, and operate without confidence. When tasked in their deficit areas, children may demonstrate agitation, distress, anxiety, anger, avoidance, and resistance.

Children grow into dyscalculic adults who exhibit the same problems, but become better at hiding and managing their difficulties.

# data[].attributes.attachment_link.description
Video US. Dept. Spec. Education & NCLD: Saying dyslexia, dyscalculia, & dysgraphia in schools, an Interview with Michael Yudin, Assistant Secretary of Special Education & Rehab Services (2015) 2017 National Assessment of Educational Progress (NAEP) Report 52% of US 4 th-graders were not proficient in Mathematics; 56% of US 8 th-Graders were not proficient in Mathematics; 72% of US 12 th-Graders were not proficient in Mathematics (2015, NEAP).

# data[].attributes.attachment_link.title
What is Dyscalculia? - The Dyslexia Association

# data[].attributes.attachment_link.description
Developmental Dyscalculia, like dyslexia, is one of a family of specific learning difficulties.

# data[].attributes.headline
Using double number lines and bar modelling to teach the GCSE maths curriculum based on the Mastery approach.

# data[].attributes.subject
Using double number lines and bar modelling to teach the GCSE maths curriculum based on the Mastery approach.

# data[].attributes.headline
Can Mastery methods be applied online to the teaching of ratio?

# data[].attributes.subject
Can Mastery methods be applied online to the teaching of ratio?

# data[].attributes.headline
Using double number lines and bar modelling to teach the GCSE maths curriculum based on the Mastery approach.

# data[].attributes.subject
Using double number lines and bar modelling to teach the GCSE maths curriculum based on the Mastery approach.

# data[].attributes.headline
CHANGING THE EXPERIENCE OF FE MATHS CFEM INTERIM REPORT SYNTHESIS OF ACTION RESEARCH REPORTS 2020/21

# data[].attributes.subject
CHANGING THE EXPERIENCE OF FE MATHS CFEM INTERIM REPORT SYNTHESIS OF ACTION RESEARCH REPORTS 2020/21

# data[].attributes.headline
Probability and Its Limits - Professor Raymond Flood

# data[].attributes.subject
Probability and Its Limits - Professor Raymond Flood

# data[].attributes.headline
The Simpsons and Their Mathematical Secrets | Simon Singh | Talks at Google

# data[].attributes.subject
The Simpsons and Their Mathematical Secrets | Simon Singh | Talks at Google

# data[].attributes.attachment_link.title
Simon Singh: "The Simpsons and Their Mathematical Secrets" | Talks at Google

# data[].attributes.attachment_link.description
Simon Singh, author of the bestsellers Fermat's Enigma, The Code Book, and The Big Bang, offers fascinating new insights into the celebrated television series The Simpsons: That the show drip-feeds morsels of number theory into the minds of its viewers-indeed, that there are so many mathematical references in the show, and in its sister program, Futurama, that they could form the basis of an entire university course.

# data[].attributes.headline
Did you solve it? The viral maths video that will have you in stitches

# data[].attributes.subject
Did you solve it? The viral maths video that will have you in stitches

# data[].attributes.attachment_link.readable_text
{{#ticker}}

{{/ticker}}

{{heading}}

{{#paragraphs}}

{{.}}

{{/paragraphs}}{{highlightedText}}
{{#choiceCards}}

{{/choiceCards}}

We will be in touch to remind you to contribute. Look out for a message in your inbox in . If you have any questions about contributing, please contact us.

# data[].attributes.attachment_link.title
Did you solve it? The viral maths video that will have you in stitches

# data[].attributes.attachment_link.description
ticker}} {{/ticker}} {{#paragraphs}} {{.}} {{/paragraphs}}{{highlightedText}} {{#choiceCards}} {{/choiceCards}} We will be in touch to remind you to contribute. Look out for a message in your inbox in . If you have any questions about contributing, please contact us.

# data[].attributes.attachment_link.readable_text
Skip to main content

Cookies on GOV.UK

We use some essential cookies to make this website work.

We'd like to set additional cookies to understand how you use GOV.UK, remember your settings and improve government services.

We also use cookies set by other sites to help us deliver content from their services.

You have accepted additional cookies. You can change your cookie settings at any time.

You have rejected additional cookies. You can change your cookie settings at any time.

Accept additional cookies Reject additional cookies View cookies

Guidance for teaching mathematics at key stages 1 and 2 to help pupils progress through the national curriculum.

Details

This guidance is for:

school leaders

teachers and other school staff

governing bodies

You can use this guidance to help plan teaching the statutory mathematics curriculum in primary schools in England. This guidance:

identifies the core concepts that are needed for pupils to progress in their study of mathematics

demonstrates how pupils can build their understanding of these concepts from year 1 to year 6

The guidance links to non-statutory guidance on teaching mathematics at key stage 3.

Short videos that introduce and summarise this guidance are also available.

Teachers may find this guidance helpful in planning and prioritising the curriculum following the coronavirus (COVID-19) pandemic. This guidance is intended to support teaching. It can be used alongside other high-quality plans and resources, including textbooks.

The National Centre for Excellence in the Teaching of Mathematics has produced a range of resources to support teachers using this guidance. This includes:

Further support for maths teachers is available through our maths hubs programme.

# data[].attributes.attachment_link.description
Guidance for teaching mathematics at key stages 1 and 2 to help pupils progress through the national curriculum.

# data[].attributes.attachment_link.readable_text
At the start of the 2021/22 school year, the DfE published new guidance for teaching maths at KS3, written by the Secondary Team at the NCETM. The guidance is wholly aligned with all the individual resources and recommended teaching approaches that have appeared on the NCETM website at different times during the previous few years.

The short video below offers a summary of what's in the guidance.

# data[].attributes.attachment_link.description
At the start of the 2021/22 school year, the DfE published new guidance for teaching maths at KS3, written by the Secondary Team at the NCETM. The guidance is wholly aligned with all the individual resources and recommended teaching approaches that have appeared on the NCETM website at different times during the previous few years.

# data[].attributes.attachment_link.title
- Dyscalculia, math anxiety, and other number struggles. Raising awareness of dyscalculia.

# data[].attributes.attachment_link.description
Dyscalculia, math anxiety, and other number struggles. Raising awareness of dyscalculia.

# data[].attributes.attachment_link.readable_text
Some learning disabilities have received more attention than others. Educators are generally aware of dyslexia, autism, and ADHD. However many teachers and parents have never heard of dyscalculia, sometimes referred to as 'dyslexia with mathematics'. This is despite the fact that there might be as many as 3-6% of dyscalculics in the general population.

The result of this general lack of awareness means that many people with dyscalculia remain undiagnosed. A student can struggle with numbers all through life and simply think s/he is 'bad at maths'. A dyscalculic student might find that because mathematics is part of other subjects such as physics and chemistry, their entire future career could be in jeopardy.

If the signs of dyscalculia in a child go unnoticed, a student may continue through secondary school without receiving the support s/he/she needs. In fact, they may receive only blame instead. Students report being labelled as 'lazy' by their teachers, as well as experiencing deep feelings of inadequacy due to their lack of ability in handling basic number functions.

Dyscalculics who finally receive a diagnosis, even if not until secondary school, often experience a feeling of relief. Once the cause of their struggles with mathematics is known, dyscalculics can receive intervention and support. It is therefore critical that all teachers become more aware of the causes and signs of dyscalculia.
Download a dyscalculia resource for teachers, which outlines some of the main indicators of dyscalculia.
For more information on dyscalculia, see Christian Vögeli blog and software:

www.dyscalculia-blog.com
www.dybuster.com

# data[].attributes.attachment_link.description
Some learning disabilities have received more attention than others. Educators are generally aware of dyslexia, autism, and ADHD. However many teachers and parents have never heard of dyscalculia, sometimes referred to as 'dyslexia with mathematics'. This is despite the fact that there might be as many as 3-6% of dyscalculics in the general population.

# data[].attributes.attachment_link.description
We are starting to see more and more questions at GCSE asking students to spot and correct the mistake made. Clumsy Clive and Erica's errors problems by @a...

# data[].attributes.headline
How modern mathematics emerged from a lost Islamic library

# data[].attributes.subject
How modern mathematics emerged from a lost Islamic library

# data[].attributes.attachment_link.title
How modern mathematics emerged from a lost Islamic library

# data[].attributes.attachment_link.description
Centuries ago, a prestigious Islamic library brought Arabic numerals to the world. Though the library long since disappeared, its mathematical revolution changed our world.

# data[].attributes.attachment_link.title
Why Wordle Works, According to Desmos Lesson Developers

# data[].attributes.attachment_link.readable_text
Filtered research on mathematics education

Every two months we bring you an Espresso - a small but intense draught of filtered research on mathematics education, expressly designed with teachers in mind. Each Espresso considers one particular issue in mathematics education, and how the latest good-quality research can provide helpful guidance or further reading. Perfect with a biscuit; sip gently and combine with a drop of professional judgement for a benevolent stimulant effect.

# data[].attributes.attachment_link.description
Each month we bring you an Espresso - a small but intense draught of filtered research on mathematics education, expressly designed with teachers in mind.

# data[].attributes.headline
What Is Math Modeling? (And Why Should You Care About It?)

# data[].attributes.subject
What Is Math Modeling? (And Why Should You Care About It?)

# data[].attributes.attachment_link.readable_text
Demonstrating the value of mathematics in our communities and lives can be difficult for educators; understanding math modeling can help.

Demonstrating the value of mathematics in our communities and lives can be difficult for educators; understanding math modeling can help.

Mathematical modeling refers to the process of creating a mathematical representation of a real-world scenario to make a prediction or provide insight. There is a distinction between applying a formula and the actual creation of a mathematical relationship. Some graphical illustrations of the modeling process can be seen on this one-page flyer.

Real-world, messy problems can be approached with mathematics, resulting in a range of possible solutions to help guide decision making. Students and teachers are sometimes uncomfortable with the notion of math modeling because it is so open-ended. So much unknown information seems prohibitive. And how do you decide which factors are most relevant? But it is this open-ended nature of real-world problems that leads to building and applying problem solving skills, creativity, and innovation in mathematics.

Math modeling obliterates the question "Why do I need math?" by demonstrating the value and importance of math in approaching big problems found in our communities, regions, and world. Identifying the important variables and quantifying them-even with assumptions and incomplete information-can lead students to insights and understanding that have reason and structure.

Identify the Problem-Be specific in defining what you want to find out.

Make Assumptions and Identify Variables-It is impossible to account for all the important factors in a given situation; you must make choices about what to incorporate in your representation of the real world. Making assumptions helps reveal the variables to be considered and reduces their number by deciding not to include everything. Relationships between variables will emerge based on observations, physical laws, or simplifications.

Do the Math-Eventually, a relationship between input and output will allow for a solution to be found.

Analyze and Assess the Solution-Consider the results and insights gained from the model. Does the answer make sense?

Iterate-Usually the model can be refined, and the process can be repeated to improve performance.

Implement the Model and Report Results-Make the model understandable to others.

It's a Process

Mathematical modeling can be thought of as an iterative process made up of the following components. (Note that the word "steps" is intentionally avoided: there is no prescribed ordering-some may occur simultaneously, and some may be repeated.)

Pitfalls to Avoid

The biggest pitfall in developing a reasonable model is often time management. When modeling is new to students, it is easy for them to become overwhelmed, and they may spend too much time "in the weeds." To define a succinct problem statement, students need to brainstorm, and they should be encouraged not to get rid of any ideas. However, there are times when students may get caught up in trying to include variables or relationships in their model that are not tractable or where data is unavailable. If this happens, students should assume and move on, making sure they come back to those assumptions after they have gone through the entire modeling process. Having said that, sometimes students include unneeded assumptions in their documentation that are never explicitly used in the modeling process. This can cause them to lose valuable time and could detract from the presentation of the solution. Students must try not to get off track while creating models, particularly by making choices or assumptions that undermine the quality of the solution.

When students are in a time crunch, apprehension may lead to mathematical relationships that are removed from reality being introduced, even to the point of being nonsense! For example, students may form an additive relationship between the key variables they identified but the units are meaningless (for example, adding dollars to time to get a model for resources). Coefficients are often used in models that also do not reflect units properly or there is no justification as to why they were chosen. Other times, students may have a sound idea for a mathematical relationship, but they overcomplicate it to make the mathematics look more sophisticated than it is (for example, introducing a triple integral when really addition is appropriate). This is another reason leaving time for reflection is critical, so that students can read over their entire solution and ask themselves "Does this make sense?".

Dealing with data can also be overwhelming. Students may have a brilliant idea for a model but cannot find the data they need to move it forward (again, at this point they should make an assumption and stop wasting time looking). Other times, datasets may be prohibitively large, and students are not equipped with the tools to interpret key trends. Linear regression or high-degree polynomials are often used to fit data without any sound reason and then used as predictors. The connection to the underlying physical problem that instigated the data fitting can get lost, or the quality of the fit is ignored completely due to relief at having done something that seemed to work.

All the above pitfalls (which are by no means an exhaustive list) can be amended if the team reflects on the quality of their work. If an assumption seemed way off base, students could report out the identified weaknesses of their approach and point the way toward improvements, even if they do not have the means or access to the information to do so. Even better, a sensitivity analysis can help a student assess the robustness of their model and make comments on its applicability. Much of this goes back to time management.

An Opportunity for Students to Engage in Math Modeling

MathWorks Math Modeling Challenge (M3 Challenge) is an annual free online contest in which high school juniors and seniors and sixth form students work in teams of 3-5 to solve an open-ended problem about a real-world issue in 14 continuous hours over Challenge weekend. The Challenge problem is completely unknown until teams download it during Challenge weekend. The use of online collaboration tools and resources is encouraged.

Scholarships totaling £75,000+ (will be awarded in 2022. Extra credit awards are available for teams who choose to write or employ outstanding code as part of their solution. The competition has given $1.65 million+ to date.

A program of Society for Industrial and Applied Mathematics ( SIAM) that is funded by leading technical computing software developer MathWorks, M3 Challenge motivates students to study and pursue careers in applied mathematics, computational science, data science, and technical computing. It has been a virtual competition since its inception in 2006. Past competition topics include issues such as the transition of trucking from diesel to electric, universal internet access, substance abuse, food insecurity, plastic recycling, and car sharing.

From Judges, Coaches, and Past Participants

Submissions are judged by an international panel of 150 primarily Ph.D.-level mathematicians, and almost every team receives specific feedback on their submission from the judges who read their papers during rigorous assessments. "The questions can be approached in a variety of ways depending on students' skills and experiences," says problem development committee member and judge Dr. Katie Kavanagh of Clarkson University in Potsdam, New York. "For example, participants may use simulation and programming, probability and statistics, or even algebra and basic math to model aspects of the open-ended problem. We are always eager to see how the students approach the issue," says Kavanagh. "They offer unique perspectives on what factors are important to consider, particularly at that critical, transitional time in their own lives and influenced by their unique circumstances."

Joy Qu, a member of a 2021 finalist team from Adlai Stevenson High School in Lincolnshire Illinois, was asked about the secret to the team's success. "We always give a line of reasoning behind our equations and how we derive them. So I think reasoning is really important because if you can support your model, it increases the validity of your model."

David Stein is a teacher at Montgomery Blair High School in Silver Spring, Maryland who has coached multiple finalist teams in M3 Challenge. He believes that teachers can influence students to study and pursue STEM careers and can demonstrate the interdisciplinary nature of the sciences by basing rigorous lessons on real-world issues and problems, focusing students' attention on communication and writing, and providing opportunities like M3 Challenge for them to showcase their work.

M3 Challenge has had a profound impact on many of its participants, including Dr. Chris Musco, a 2008 M3 Challenge finalist who is now head of the competition's technical computing judging and a member of the problem development committee. "Being involved in this program as a judge and using my experience and perspective to help shape the problem the students will work on is really exciting," says Musco, assistant professor in the Tandon School of Engineering at New York University. "M3 Challenge changed the trajectory of my college major and career, and it's great to play a part in doing that for the students who are participating now."

"For M3 Challenge, students work together and put forward ideas and models-some teams even include computer code - toward a solution or an approach to the problem presented. Every year we are blown away by the work produced by teams in just 14 hours," said Dr. Suzanne Weekes, executive director at SIAM. "Problem topics are relevant and meaningful - for example the 2021 topic of reliable access to the internet for all, which became so prominent and important during the pandemic. Students are solving real problems by asking questions, digging for information, making assumptions to get started, getting creative in their approaches, and then providing insight about the issue that can be used to make decisions going forward."
Register by February 18, 2022 Challenge weekend (Friday through Monday): February 25-28, 2022 *Much of the content for this article was written by Katie Kavanagh and Ben Galluzzo of Clarkson University, and is based on ideas presented in : Guidelines for Assessment and Instruction in Mathematical Modeling Education, Second Edition, Sol Garfunkel and Michelle Montgomery, editors, COMAP and SIAM, Philadelphia, 2019.
"Every year without fail, we hear from students who refer to their participation in M3 Challenge as a transformative experience that helped open their eyes to how important, useful, and valuable the application of mathematics can be," said Michelle Montgomery, M3 Challenge program director at SIAM. "I am confident that this year's Challenge will do the same."

For rules, resources, and to register visit M3Challenge.siam.org.

# data[].attributes.attachment_link.title
What Is Math Modeling? (And Why Should You Care About It?)

# data[].attributes.attachment_link.description
Demonstrating the value of mathematics in our communities and lives can be difficult for educators; understanding math modeling can help. Demonstrating the value of mathematics in our communities and lives can be difficult for educators; understanding math modeling can help.

# data[].attributes.attachment_link.description
Assess your confidence in teaching the content of the KS3 maths curriculum

# data[].attributes.body
Advanced & Core maths
Anxiety and motivation (boys and girls etc)
Applied maths and science
Area and volume
Art
Assessment
Awarding organisation material
British Value
Calculators
CfEM
Competencies
Connections
Data literacy
DESMOS
Digital
Displaying information
Dyscalculia
EEF
Embedding
Estimation
Events
Factorial
Family and community
Finance
Fractions
Functional maths
GCSE
Geometry
Gifs
History of maths
Images and maths
Influential maths peeps
Key Stage 3
L5 maths links
Language and maths
Learning
Level 3&5 materials
Mastery and basics
Maths English and ESOL
Mathematical modelling
Mathematical reasoning
Mathematical talk - see also vocab
Maths, English & ESOL
Maths books
Memory and recall
Misconceptions
Padlets
Percentages
Podcasts
Posters
Practice papers
Probability
Problem solving
Questioning
Research
Schemeta Padlets
SEND
Starters
Statistics and probability
STEM project summer 2021
STEM videos and Padlet
Swain and Swan
Tackling questions
Tools
Topics
T&L
Upskilling for maths teachers
Variation theory
Videos
Vocab and
Websites 1-4
Women (and men) in mathematics
YouCubed

# data[].attributes.headline
Our maths content specialist, Kirstin Mulholland, explores how practitioners can support pupils to problem-solve effectively using metacognition

# data[].attributes.subject
Our maths content specialist, Kirstin Mulholland, explores how practitioners can support pupils to problem-solve effectively using metacognition

# data[].attributes.attachment_link.readable_text
I feel as though every time I have a conversation about maths with a teacher or school leader, problem-solving is identified as something we want to work on.

Pupils find it difficult. They don't know where to start. They can't apply learned strategies to a new context and can often resort to guesswork when their way ahead seems unclear. Sometimes, they even just freeze, like rabbits in the headlights, unable to take a step or move their learning forward without significant support from a peer or adult.

To make matters worse, there appears to be some evidence that this issue may be more pressing now than ever before. Partial school closures, and the need for remote learning due to the Covid pandemic, may have resulted in fewer opportunities for pupils to engage in mathematical talk, metacognitive activities, or to receive formative feedback. Consequently, many teachers seem to be finding that children may be less independent, and are experiencing even greater challenges in applying their learning to problem-solving contexts. commissioned by the EEF indicated that mathematics learning was disproportionately affected by the impact of the pandemic - with a negative impact of -3months. Not only that, the impact appears to be more marked for pupils from disadvantaged backgrounds. More recently, some positive evidence, has shown that Year 1 children were only 1 month behind expectations in mathematics.

Is the problem specific to problem solving?

Part of the issue could well lie in the very nature of problem-solving. For example, ​'What Works Clearinghouse' defines problem-solving as ​'the movement from a given state to a goal state with no obvious way or method for getting from one to the other' (IES, 2018, p. 7).

I think this is useful because it emphasises the potential unpredictability of problem-solving. And because problems have no set routine or fixed and known procedure which can be used to solve them, this makes our job as teachers inherently difficult.

If problems have no fixed routine or problem-solving strategy and can consist of a potentially limitless combination of smaller steps, drawn from across multiple mathematical concepts, how can we possibly teach children how to solve them?

Recommendation 3 of the EEF ​' Improving Mathematics and Key Stages 2 and 3' guidance report suggests some key actions that can help pupils to improve their skills in problem-solving. We need to:

- teach pupils to use and compare different approaches to problem-solving;
- show them how to interrogate and use their existing knowledge to solve problems;
- use worked examples to enable them to analyse the use of different strategies;
- encourage pupils to monitor, reflect on, and communicate their problem solving.

To me, this means that we need to spend more time talking about we are doing and .

For instance, rather than spending an hour's lesson in completing a sequence of multiple different problems, with pupils working independently to correctly (or incorrectly) work through a series of tasks, we need to look at fewer problems, but in significantly greater detail. This would allow us to invest time in really drilling down into which strategies pupils use to solve specific elements of problems, and why, comparing these and reflecting upon which approaches are most effective.

By talking more - and perhaps ​'doing' less - we can provide the kinds of models that pupils need to think about the rationale behind some of the choices that they make. They can see how other learners approach problems, learn to critique them, and also gain insider knowledge into a broad range of ways in which peers, and adults, think about problems.

Working in this way could enable pupils to really move beyond learning how to solve a single, unique problem, and towards a more comprehensive - and ultimately useful - understanding of how problem-solving works, so that they can learn how to learn and to best utilise strategies for problem-solving. We can also begin to make those invisible processes of thinking and learning visible, demystifying these so that pupils can see first-hand what successful problem-solving looks like.

School teachers and leaders are agreed that we want to work on problem solving, so let's concentrate on the ​'what' and the ​'why'.

# data[].attributes.attachment_link.keywords[]
i feel as though every time i have a conversation about maths with a teacher or school leader

# data[].attributes.attachment_link.keywords[]
problem-solving is identified as something we want to work on.  pupils find it difficult. they don’t know where to start. they can’t apply learned strategies to a new context and can often resort to guesswork when their way ahead seems unclear. sometimes

# data[].attributes.attachment_link.keywords[]
unable to take a step or move their learning forward without significant support from a peer or adult.  to make matters worse

# data[].attributes.attachment_link.keywords[]
there appears to be some evidence that this issue may be more pressing now than ever before. partial school closures

# data[].attributes.attachment_link.keywords[]
and the need for remote learning due to the covid pandemic

# data[].attributes.attachment_link.keywords[]
may have resulted in fewer opportunities for pupils to engage in mathematical talk

# data[].attributes.attachment_link.keywords[]
or to receive formative feedback. consequently

# data[].attributes.attachment_link.keywords[]
many teachers seem to be finding that children may be less independent

# data[].attributes.attachment_link.keywords[]
and are experiencing even greater challenges in applying their learning to problem-solving contexts.  early research commissioned by the eef indicated that mathematics learning was disproportionately affected by the impact of the pandemic – with a negative impact of –3months. not only that

# data[].attributes.attachment_link.keywords[]
the impact appears to be more marked for pupils from disadvantaged backgrounds. more recently

# data[].attributes.attachment_link.keywords[]
has shown that year 1 children were only 1 month behind expectations in mathematics .  is the problem specific to problem solving?  part of the issue could well lie in the very nature of problem-solving. for example

# data[].attributes.attachment_link.keywords[]
‘what works clearinghouse’ defines problem-solving as ‘the movement from a given state to a goal state with no obvious way or method for getting from one to the other’ (ies

# data[].attributes.attachment_link.keywords[]
p. 7).  i think this is useful because it emphasises the potential unpredictability of problem-solving. and because problems have no set routine or fixed and known procedure which can be used to solve them

# data[].attributes.attachment_link.keywords[]
this makes our job as teachers inherently difficult.  if problems have no fixed routine or problem-solving strategy and can consist of a potentially limitless combination of smaller steps

# data[].attributes.attachment_link.keywords[]
drawn from across multiple mathematical concepts

# data[].attributes.attachment_link.keywords[]
how can we possibly teach children how to solve them?  recommendation 3 of the eef ‘ improving mathematics and key stages 2 and 3 ’ guidance report suggests some key actions that can help pupils to improve their skills in problem-solving. we need to:  - teach pupils to use and compare different approaches to problem-solving; - show them how to interrogate and use their existing knowledge to solve problems; - use worked examples to enable them to analyse the use of different strategies; - encourage pupils to monitor

# data[].attributes.attachment_link.keywords[]
and communicate their problem solving.  to me

# data[].attributes.attachment_link.keywords[]
this means that we need to spend more time talking about  what  we are doing and  why .  for instance

# data[].attributes.attachment_link.keywords[]
rather than spending an hour’s lesson in completing a sequence of multiple different problems

# data[].attributes.attachment_link.keywords[]
with pupils working independently to correctly (or incorrectly) work through a series of tasks

# data[].attributes.attachment_link.keywords[]
but in significantly greater detail. this would allow us to invest time in really drilling down into which strategies pupils use to solve specific elements of problems

# data[].attributes.attachment_link.keywords[]
comparing these and reflecting upon which approaches are most effective.  by talking more – and perhaps ‘doing’ less – we can provide the kinds of models that pupils need to think about the rationale behind some of the choices that they make. they can see how other learners approach problems

# data[].attributes.attachment_link.keywords[]
and also gain insider knowledge into a broad range of ways in which peers

# data[].attributes.attachment_link.keywords[]
think about problems.  working in this way could enable pupils to really move beyond learning how to solve a single

# data[].attributes.attachment_link.keywords[]
and towards a more comprehensive – and ultimately useful – understanding of how problem-solving works

# data[].attributes.attachment_link.keywords[]
so that they can learn how to learn and to best utilise strategies for problem-solving. we can also begin to make those invisible processes of thinking and learning visible

# data[].attributes.attachment_link.title
EEF blog: The problem with problem-solving in maths

# data[].attributes.attachment_link.description
I feel as though every time I have a conversation about maths with a teacher or school leader, problem-solving is identified as something we want to work on. Pupils find it difficult. They don't know where to start. They can't apply learned strategies to a new context and can often resort to guesswork when their way ahead seems unclear.

# data[].attributes.attachment_link.readable_text
People have always found symmetry aesthetically pleasing and examples of it are seen in the earliest art. This lecture will look at how we can understand symmetry using mathematics, and explore how the rules of symmetry can deepen our appreciation of beautiful works of art and design.
Introduction
Throughout history, across different times and cultures, it's clear from art, design, and architecture, that humans find beauty in regularity and symmetry. We see it, for example, in the exquisite mosaic ceilings at the Lotfollah mosque in Isfahan, Iran, and the Moorish tilings at the Alhambra Palace in Spain. We see symmetry in architecture: the Taj Mahal, often said to be the most beautiful building in the world, has a left-right, or "bilateral" symmetry with a vertical mirror line. By cleverly placing a shallow lake in the gardens in front of the building, the impression is given of an additional horizontal mirror line, as the building is reflected in the still waters below. Meanwhile if we look at the famous Renaissance Villa Rotonda at Vicenza in Italy, designed by the architect Palladio, we see that it has not only a pleasing bilateral symmetry on its individual façades, but that in fact the whole building is symmetrical - the four sides of the building form a square, with a central rotunda, so that the whole building (from the outside at least) has a four-fold rotational symmetry. Looking at the original plan, we can see that the corners of the building lie on a circle, which also just touches the centres of each of the colonnades.

Human beings clearly like symmetry. But why? There are two main views on this. The first is called the "Perceptual Bias" view. Symmetrical or regular designs are literally easy on the eye, in the sense that the brain has to make less effort to process them. A chequerboard floor has an instantly comprehensible pattern that the brain can analyse very quickly. Deviations from the symmetry require special notice and can have a jarring effect. Nobody wants a tiled floor with one tile out of place (an example of which I'll show you in the lecture - sorry). The second suggestion as to why we like symmetry is called the "Evolutionary Bias" view. Our primitive animal brains, goes the theory, subconsciously size up potential mates with a "survival of the fittest" bias. Symmetrical bodies may be perceived as healthier and thus a better bet to pass on your genes. Whatever the reason, the fact that we do perceive highly symmetrical objects as beautiful has led us to look for them, and then to associate them with goodness, with virtue, even with the divine.
Measuring Symmetry
Plato, in the Timaeus[1] (360 B.C.), wrote about how the universe was made, by "a method with which your scientific training will have made you familiar. Fire, Air, Earth, and Water are bodies and therefore solids [...] we must determine what are the four most beautiful figures." Which are the four most beautiful figures, then, that we must associate with the four elements? It turns out that they are the tetrahedron, the octahedron, the cube, and the icosahedron. How do we know which figure should be associated with which element? Well, when we touch fire it hurts, and the sharpest of these polyhedra is the tetrahedron - so that must be fire. By contrast, water slips through our fingers without any unpleasant sensation, so it is icosahedral. The cube is the only one of these solids that can be stacked up neatly without gaps, so clearly earth is the cube, which leaves the octahedron to be air. The illustration is the one from Book 5 of Kepler's 1619 Harmonices Mundi - Harmony of the World.

But, there is another solid in Plato's armoury - if you have heard of the Platonic solids you'll know this. He says that thereis "a fifth figure (which is made out of 12 pentagons), the dodecahedron - this God used as a model for the twelvefold division of the Zodiac". This is the "fifth element", the quintessence. So, how did Plato come to have five "most beautiful figures" to assign special roles to? To answer that, we need to think about how to measure symmetry.

We say that a symmetry of a shape is, informally, something you can do to it that leaves it looking the same. A bit more formally, we can define it to be a distance-preserving transformation of space (such as a reflection, rotation or translation) that maps a shape to itself. This is not how Plato or Euclid would have thought about it, but we'll say a little more about that later.

As an example, an isosceles (but not equilateral) triangle has two symmetries. It can be reflected in its one line of mirror symmetry, so that every point is mapped to the point the same distance from the line on the other side. The other symmetry is easily overlooked, but the identity map, where every point is simply mapped to itself, is a symmetry of every shape. Given that a symmetry has to preserve distances, it must map edges to edges, so the fact that one edge has a different length from the other two limits the possibilities. That odd edge can only be mapped to itself. Similarly, two edges that meet at a specific angle must still meet at that same angle after the symmetry operation has been completed. That leads us to suspect that we might get more symmetry in a triangle where all the sides are equal lengths and all the angles are equal. Sure enough, when we analyse an equilateral triangle, we find that there are now six symmetries. We get three reflections, two rotations (through 120° and 240° ) and of course the identity map (which we can think of as a rotation through 0 degrees). Thinking about polygons in general, the same reasoning tells us that the most symmetrical n -gon is a regular n -gon, that's one with equal sides and equal angles. A square has eight symmetries: four reflections and four (if you include the identity) rotations. A regular pentagon has ten. A regular n -gon has 2 n symmetries. If we take the limiting case, we end up at a circle - every rotation about the centre, and every reflection in a line through the centre, is a symmetry.

Moving up a dimension, let's think about polyhedra. These are three-dimensional shapes made with straight edges, whose faces are polygons. Again, for maximum symmetry we would want all edges to be the same length, and all angles to be equal. That is, we would like the faces to be regular polygons, and the same number of faces to meet at each vertex. What are the possibilities for these "regular polyhedra"? If the faces are equilateral triangles, how many could meet at each vertex? Two is not enough, because they collapse on each other and we cannot get a three-dimensional shape. Three, four, and five are candidates. But six equilateral triangles will give a total angle of 6×60°=360° , which would just be completely flat. Again, we wouldn't obtain a three-dimensional shape. Testing out the three candidates does indeed result in three regular polyhedra: the tetrahedron, the octahedron, and the icosahedron. With squares, we are more constrained. Two is again not enough. The internal angle of a square is 90° , so four squares round a point makes 360° , which would be flat and not give a three-dimensional shape. So our only hope is three squares round a point. This works: we get a cube. The internal angle of a regular pentagon is 108° . So three round a point is possible. It gives the dodecahedron. Four regular pentagons wouldn't even be flat: we'd actually have them overlapping. Anything higher than pentagons doesn't work because two is not enough, and three is too many to fit round a point - you end up with an angle of at least 360° . And that's why we have exactly five regular polyhedra - known as the five Platonic solids.

For me, this highlights one of the precious things about mathematics. Our scientific understanding of the world is completely different now from what it was two thousand years ago, and we might smile at the shoehorning of the five solids into a scientific theory of the universe. (It has to be said, though, that modern humans are hardly immune from the temptation to coerce data to fit preconceived ideas.) However, the fact that there are exactly five regular polyhedra is as true today as it was then. Mathematical proof is eternal. Incidentally, those limiting cases where we end up with exactly 360° from a whole number of regular polygons are not meaningless - they are precisely the possibilities for what's called regular tilings of the plane: tessellations where each tile is the same regular polygon. There are three of these: the triangular, square, and hexagonal.
The history of symmetry
Periodically the claim will resurface that hundreds of years before Plato, Neolithic tribes knew about the Platonic solids. The evidence for this is the hundreds of carved stone balls found at various sites in Scotland, dating back to around 2000BC. There is a compelling image of five of these stone balls with the edges emphasized by strips of white paper, looking very much like the five platonic solids. Unfortunately there does seem to be quite a bit of wishful thinking here. The stone balls all have bumps in them, and these bumps can be interpreted as either faces or vertices. So we can interpret something with six or eight bumps as a cube (six faces, eight vertices) or an octahedron (six vertices, eight faces). Tetrahedra have four faces and four vertices. For icosahedra and dodecahedra these numbers are twelve and twenty. Of the hundreds of stone balls found, the number of bumps ranges from 3 to over 100 (I've seen both 135 and 160 stated as the maximum number). The ones that happen to have four, six, eight, twelve, or twenty bumps don't seem to have any particular significance or prevalence. Around half the balls have six bumps. Just eight have twelve bumps, and only two have twenty bumps, with lots of other numbers represented that have no apparent significance. This is not very compelling evidence for Neolithic Platonic solids. These are very interesting archeological finds already, without imposing on them symbolism that almost certainly isn't there. If you want Platonic solids in archeology, you can have them, but you have to wait a little longer, probably until the metal dodecahedra found in Roman settlements. Their purpose, as with the Neolithic balls, is unclear. Theories include them being knitting aids, gauges for pipes (the holes all have different diameters), or gaming pieces. We may never know. But there's no doubt that they are dodecahedra.

I have mentioned that the modern definition of symmetry would not be familiar to Euclid and Plato. Ancient Greek had the words σύμμετροζ ( symmetros) and σύμμετρία ( symmetria). This is made up of the root words "syn" (together) + "metros" (measure). This word appears in Euclid's Elements (Book 10, Def 1), but not in a way that has anything to do with our idea of "symmetry". Two magnitudes are symmetros if they have a "common measure" (that is, they are both whole number multiples of a given amount). Platonic solids are constructed in Book 13 of Euclid, but they are not described as being symmetrical. Rather, they are "the solids made of equilateral figures equal to one another". When Greek texts were being translated into Latin, a decision had to be made about how to translate symmetros, because, according to Pliny in his Natural History (77AD), "non habet Latinum nomen symmetria": there is no Latin word for symmetria. Translations into Latin, and much later into English, of this rather technical geometric meaning of the word symmetria, use "commensuratio" and "commensurable" (having a common measure). However, this same word symmetria was also used in Greek in a broader aesthetic sense of magnitudes being connected or related: being in proportion. Aristotle (384-322BC) in his Metaphysica tells us that "the main species of beauty are orderly arrangement ( taxis), proportion ( symmetria), and limitation ( horismenon), which are revealed in particular by mathematics". This meaning transferred to Latin, along with the word. Vitruvius in De Architectura says that symmetria is "the appropriate harmony arising out of the details of the work itself; the correspondence of each given detail to the form of the design as a whole".

By the late 17th century we start to get clear evidence that the word "symmetry" had changed its meaning. The architect Claude Perrault (1613-1688), probably most famous now for designing the Louvre colonnade, translated the works of Vitruvius into French, along with explanatory notes. "Symmetry", he wrote, "does not signify in French what Vitruvius understands by Proportion [symmetria]. Symmetry, in French, signifies the relation, for example, that windows have one to another, when they are all of an equal height and equal breadth; and that their number and distances are equal to the right and the left; so that if the distances be unequal of one side, the like inequality is to be found in the other. [Whereas] the word Simmetria signifies in Latin and Greek relation only. As for example, as the relation that windows of eight foot high, have with other windows of six foot, when the one are four foot broad, and the other three. For our symmetry is properly the equality and the parity which is found between opposite parts, that is, if one makes, for example, an eye higher or larger than the other, or columns closer together to the right than to the left, or [their] number or size is not the same [to the right and to the left], one says that it is a defect of symmetry according to us: rather than, if a capital is larger or a cornice protrudes more than the rules of order for a column allow, this is a defect of symmetry according to the ancients." This Latin and Greek "symmetria" is nowadays translated as "proportion". That is the subject of my third Gresham lecture of this academic year, in February 2022. What we are discussing today is Perrault's "French" definition of symmetry, where different parts are alike or equal, such as the left-right bilateral symmetry that is common in classical architecture.
Groups
Symmetry, in its new "French" meaning, initially meant bilateral symmetry. Architectural manuals of the time would say, for example, that if you had a window or door on the left of a façade, you should have a corresponding one (fake if necessary) on the right. Thinking of symmetry like this makes it static: a property of an object rather than something you do. The definition of "a symmetry" as we have given it was not possible with this mindset. Mathematically, the crucial first step to our modern understanding was in a definition by the French mathematician Adrien-Marie Legendre, in his book Éléments de Géometrie, in 1794. "Two equal solid angles which are formed by the same plane angles but in the inverse order will be called 'angles equal by symmetry', or simply 'symmetrical angles'." What this does, for the first time, is to turn symmetry not into a property but a relation: "by symmetry". The symmetry is doing something rather than being something - it's turning into a verb! It's rather like what had happened a century or two earlier with the study of curves in geometry. Mathematicians like Descartes had started to think of curves not simply as static objects, but as representing an active relationship between variables, where we plot y as a function of x or maybe of time. This laid the path for the development of the calculus later on. So what can we do with the idea of symmetry as being active: something you do to an object that leaves it looking the same?

When you play around with symmetries, you notice that combining them results in more symmetries. With a square, for instance, the composition of any two of the reflection symmetries results in a rotational symmetry. If you combine a rotation with a reflection, you get another reflection. The same is true for any shape. If f and g are symmetries then f followed by g is also a symmetry. There are some other properties too. We already spotted that every shape has the "do-nothing" identity map as a symmetry. Every symmetry is also reversible - it has an "inverse" which is also a symmetry. For instance, rotation through 90 degrees anticlockwise can be undone by rotating through 90 degrees clockwise.

The set of symmetries of a shape is one example of the mathematical structure known as a group. In the 19th century, many different areas of mathematics started noticing and then using the same kind of structure, one in which elements of a set could be combined in some way and the outcome was still in the set. The first person to actually use the word "group" was Évariste Galois, in about 1830. He was studying the problem of whether there are formulae like the quadratic formula x= -b±b 2-4 ac 2 a for higher order polynomials like those involving x 3 (cubics), x 4 (quartics), or x 5 (quintics). What Galois did was to think about the roots of polynomials in terms of ways they could be shuffled around, or permuted, to still end up with roots. Just like shuffling a pack of cards, if you shuffle in one way and then shuffle in another, the net effect is another, perhaps more complicated, shuffle. So, there is this property of "closure", and that was how Galois defined a group. This kind of set-with-an-operation was appearing elsewhere in mathematics too. You find it in numbers, in matrices, in functions, everywhere. We first see groups appearing in geometry around the 1850s, with the work of people like Möbius and Felix Klein (of Möbius strip and Klein bottle fame). They and others realised that the key to understanding different kinds of geometry was first to work out what are the things that matter to you (so in Euclidean geometry that would be distances between points), and then look at the groups of transformations that preserve those things. Our symmetries are distance-preserving maps - they don't stretch space. But if you are studying topology or non-Euclidean geometry, you might want to focus on other characteristics. We tend to simply state the "four axioms of a group" as if recorded on stone tablets in antiquity. But actually, it took decades for the definition to be refined into its current form. Just for the record, here are the axioms, with the example of the set of integers (whole numbers), where the operation (the way of combining the elements) is addition, as an example.

A group is a set G with an operation * defined on it, so that the following four rules apply

All these axioms are true for the integers, with addition.

Group theory is incredibly powerful. Anything that we can prove starting with these four initial rules will give us information about any context where there are groups: numbers, geometry, symmetries, functions, and so on.
Friezes and tilings
Now we know what symmetry groups are, let's see how they can be used to understand and classify a particular kind of design seen in decorative art: the frieze. Mathematically speaking, we think of a frieze as a repeating pattern along a strip that could be continued indefinitely. So one symmetry that's always present is translation through a whole number of units (where a unit is the smallest distance after which the pattern repeats). What other symmetries there are depends on the design that's being repeated. We can actually classify the different possible patterns according to their symmetry groups. We begin by deciding the list of potential symmetries (apart from translations and the identity map). The central horizontal line must be mapped to itself by any symmetry. So the only rotations possible are rotations of 180° about points on that line. The only reflections possible are in vertical lines or in the central horizontal line. There's one other kind of symmetry that may be a bit less familiar: glide reflections. A glide reflection is a half-unit translation followed by reflection in the central horizontal line. Footprints have this kind of symmetry.

By the way, it can be shown - we probably won't have time for this in the lecture - that if there are rotations, then we can assume they are about points at the midpoints and endpoints of the pattern. The reason is that the product of two rotations of 180° about different points is a translation in the direction of the line joining them through twice the distance between them. So if there were two rotations closer than that, they would imply the existence of a translation symmetry through less than one unit. But we defined a unit to be the smallest repeating component. The product of two reflections in vertical lines is a horizontal translation through twice the distance between them. So the distance between any two vertical reflection lines can only be a half integer multiple of the unit distance.

The upshot of all this is that there are exactly four possible non-identity non-translation symmetries. They are:

(R) 180° rotations about the midpoints and endpoints of the design.

(V) Reflections in vertical lines through the midpoints and endpoints of the design.

(H) Reflection in the central horizontal line.

(G) Glide reflection

However, not all of these can happen independently, and this is because of the closure property of groups. I'll just give one example of this, but there are several other implications. If we have reflection in a vertical line and in the central horizontal line, then their composition must also be a symmetry - it is a 180° rotation. So V and H together implies R.

It takes a bit of work, but reasoning like this allows us to narrow down the possibilities to exactly seven frieze patterns. Here are real-life examples of the seven patterns, with the letters by them indicating the elements of the symmetry group. T represents translations, which every frieze group has, and included in this is translation through 0 units, which is the identity map.

T only

T + G

T + H

T + R + G

T + V

T + H + V + R

T + R

Friezes are everywhere, in decoration on the side of buildings, in ornamental borders of tiled floors, walls and ceilings: I encourage you to become a frieze pattern spotter in your everyday life.
The Utility of symmetry
At the same time as groups were appearing in mathematics, symmetry was becoming important scientifically. It was suddenly everywhere in the 19th century, and I'll just give a couple of examples. The French mineralogist René Haüy, often called the father of crystallography, propounded his Law of Symmetry in 1815. "The way in which nature produces crystals is always that of the greatest symmetry, in that opposite and corresponding parts are always equal in number, arrangement and shape." Crystals are three-dimensional, but they are built up of two-dimensional layers, and so an important step to understanding the possible symmetries of crystals is to understand the symmetries of repeating designs in the plane. It's a bit more effort, but it's possible to classify all possible symmetry groups of such designs, and it turns out there are exactly seventeen of them. The mathematical analysis of possible crystal shapes has a link to the decorative arts. We've talked about friezes, which cover one strip of space. But if you want to cover the whole plane, then you need tiling, or tessellation, or wallpaper. (These 17 groups are often called the 17 wallpaper groups.) When the artist M.C. Escher was experimenting with tilings following a visit to the Alhambra Palace, his older brother Berndt, a Professor of Geology, recognized that the patterns were like the ones crystallographers use to categorise different crystal structures. Berndt sent his brother papers about crystal structures, including ones by George Pólya, who had independently discovered the 17 patterns in 1924 (unaware that they had in fact first been classified by the Russian mathematician and crystallographer Evgraf Fedorov in 1891). It would take too long to list all these patterns, but one thing to note is that each has one of the three regular tilings underlying it, as we can make a lattice from the central points of each design. Thus, any wallpaper pattern or tiling will have an underlying triangular, square, or hexagonal structure. By the way, since crystal structures are actually three dimensional, to understand the full set of possibilities for those we need to know not just the symmetry possibilities for the two-dimensional layers, but for the whole three-dimensional structure. These are known as "space groups". This is an even bigger undertaking, but Fedorov managed to do it: there are 230 of them. Many artists since Escher have experimented with tessellation art. I'll show an example by digital artist Chris Watson, a lovely design with fish swimming around in a regular hexagonal tiling. Artists and designers continue to work with plane-filling designs, and we see them all around us. Think Orla Keily, Enid Marx (who designed many of the fabrics used on the London Underground), and of course William Morris with his wallpapers. Artists from the Islamic tradition, such as Zarah Hussain, have brought classical patterns up to date by incorporating projections of lights that change according to a carefully designed computer algorithm.

In the natural world, large animals usually have bilateral symmetry. Because of gravity our bodies are almost guaranteed to lack vertical symmetry. We move and see, so we have a front and back. This leaves less scope for symmetry, and bilateral is about the best we can hope for. But the fewer constraints there are, the more symmetry we tend to find. As microscopes improved, it became possible to study even the tiniest creatures. Ernst Haeckel (1834 - 1919) discovered and drew thousands of microscopic organisms with high degrees of symmetry, publishing books such as Morphological Symmetries in the Animal Kingdom. Haeckel was a German naturalist and philosopher who discovered thousands of new species and coined many terms in biology, including ecology, phylum, phylogeny and stem cell. His diagrams of radiolara (single-celled organisms around a tenth of a millimetre in diameter) show highly symmetrical, platonic solid-like forms. In biology then, the adding of constraints results in the breaking of symmetry. By the same token, in trying to find optimal solutions to any sort of physical problem, we can appeal to symmetry. Consider the legend of Dido[2], as recounted in Virgil's Aeneid. Dido's father was king of Tyre, and when he died her brother Pygmalion killed Dido's husband Sychaeus. The ghost of Sychaeus appeared to Dido in a dream, told her where his treasure was hidden, and that Pygmalion had killed him. Dido fled with the loot and ended up in Carthage (in what's now Tunisia, in North Africa). The local people agreed to her offer to exchange some of that fortune for whatever amount of land she could contain within the hide of an ox. Of course clever Dido cut the hide into strips and attached them at their ends to obtain effectively a long rope. The question then becomes: given a fixed perimeter with which to enclose an area, how do we maximise that area? How, in other words, can Dido get the most land? There might potentially be lots of solutions but suppose you have a candidate solution shape. Pick any point on the perimeter and draw a line L to the point halfway round the perimeter on the other side of the shape. Whichever half contains the most area, we can replicate on the other half, by just reflecting the design in L . So there is a solution at least as good that has L as a line of symmetry. But we could do this anywhere on the shape. Therefore, one optimal solution has the property that it is maximally symmetrical - it is a circle. A formal mathematical proof that this is the unique best solution takes a bit of work, but the underlying intuition is sound. Well, the upshot is that Dido became Queen of Carthage. Solving problems by exploiting symmetry is absolutely everywhere in mathematics, and it's one of the reasons why symmetry is useful and studied. And, of course, nature does this too - crystals, unaffected by the constraints of biology, have highly symmetric structures because the best solution in one direction is the best solution in all directions.
The symbolism of symmetry
I want to finish today with a glance at the symbolism of symmetry. Symmetry, to Plato, is about form, beauty and regularity - the icosahedron, say, is optimally regular, hence "good". Centuries later, Kepler's mysticism about number and geometry being at the heart of the creator's designs meant that he was absolutely certain there had to be a reason for every number in the solar system. He was so familiar with the Platonic solids that he recognised ratios of average diameters of planetary orbits as being, to a good approximation, the same as ratios of the diameters of inscribed and circumscribed spheres of Platonic solids, as long as you put them in the correct order. There's a beautiful illustration of this in his Mysterium Cosmographicum. This is a continuation of the Platonic idea that symmetry is good, the creator is good, therefore the creator will naturally use symmetrical things in his designs.

Symmetry is also associated with fairness and equality. Ebenezer Howard was an English town planner, and he founded the garden city movement which envisioned towns and cities designed for people to flourish and live in harmony with nature. The book "Garden Cities of Tomorrow" (1902) included a strikingly symmetrical design for a "group of slumless, smokeless cities" with homes in smaller towns around a central city, near the countryside, in a hexagonal design. Centuries earlier, town planners in Renaissance Italy were also thinking about the ideal, utopian town, where everyone would be equal and have fair access to the town's resources. Of course, such a town should be beautiful too, and thus a symmetrical solution must be the right approach. Such a settlement, called Palmanova, was designed in the late 16th century. Unlike Howard's smokeless city, this design was actually constructed, and it's still there today.

Once we know the rules of symmetry, we can choose, on occasion, to break them. This act can have its own symbolism. In the Islamic faith there is a traditional belief that only Allah can make something perfect, and to do so yourself is thus an insult to Allah. So, many handmade Persian rugs will contain a small deliberate mistake somewhere, a tiny break in the symmetry of the intricate design. Another, more earthly, example of symmetry-breaking is in the Baroncelli Polyptych, and we'll finish the lecture with this little puzzle. This is a series of five panels by Giotto, the only of his panel paintings that is still in the place it was designed for, the Baroncelli chapel in the church of Santa Croce in Florence. The central panel depicts the Virgin Mary being crowned Queen of Heaven, with an array of saints and angels symmetrically placed on either side. On each side there are 51 saints and ten angels. The use of symmetry gives a harmony to the work and draws the eye to the centre. All 61 figures on the right are looking towards the central panel. All 61 figures on the left are looking towards the central panel. Or are they? If we look closer, in fact one of the 51 saints on the left (but not on the right) is looking away! This is obviously a deliberate decision on the part of Giotto. Was it a self-portrait? Or a portrait of one of the Baroncelli family who commissioned the work? It's an intriguing mystery. There's an old Japanese proverb: if you want to break the rules of symmetry, you first have to understand them. I hope that today I've given you a bit more of that understanding.

© Sarah Hart, 2021

Follow me on Twitter @sarahlovesmaths
Further Reading

The classic text on symmetry is Hermann Weyl's Symmetry, published by Princeton University Press, ISBN 978-0691173252.

For a deeper look, try the beautifully illustrated 2008 book The Symmetries of Things by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. ISBN 978-1568812205

http://www.neverendingbooks.org/the-scottish-solids-hoax is an excellent article on Neolithic "platonic solids". See also the Encyclopaedia of Polyhedra https://www.georgehart.com/virtual-polyhedra/vp.html by George Hart (no relation!) which has many resources including a blog about the Neolithic stone balls.

Images shown in the lecture

Anything not mentioned below is either a diagram drawn by me or in the public domain with no usage restrictions.

The photograph of Palladio's Villa Rotonda a Vicenza is by Marco Bagarella, available on Wikipedia.

Leonardo Da Vinci's platonic solid drawings are from Luca Pacioli's 1509 book De divina proportione. The book is in the public domain, and the images are from the Pennsylvania State University Libraries facsimile edition of De divina proportione published by Silvana Editoriale in 1982. They are available online at https://www.maa.org/press/periodicals/convergence/leonardo-da-vincis-geometric-sketches-introduction.

Chris Watson's webpage https://www.tessellationart.com/ has lots of examples of his work, and he's releasing a free book, the Little Book of Tessellation, for you to try your own tessellation art. https://www.tessellationart.com/thelittlebookoftessellation/. His design "Fish Orange" is shown with permission.

Zarah Hussain's work has been featured in many exhibitions, including 2017's Numina at the Barbican Centre in London and the recent Breath exhibition at the Peabody Essex Museum in Massachusetts (until April 2022). If you are based in London, you can pop to Vernon Road in Walthamstow to see a mural of her work for the Wood St. Walls project. https://www.zarahhussain.co.uk/.

[1] The Timaeus is a dialogue in which several characters, Timaeus among them, discuss the creation of the universe.

[2] The picture shown in the lecture is Dido and Aeneas, by Pierre-Narcisse Guérin, c1815.
Available at the Web Gallery of Art (www.wga.hu)

# data[].attributes.attachment_link.description
People have always found symmetry aesthetically pleasing and examples of it are seen in the earliest art. The Platonic solids have been known to humanity for millennia, some possibly even to Neolithic man, as can be seen in the carved stone balls found by archaeologists.

# data[].attributes.headline
Anticipating Misconceptions – Understanding the Source?

# data[].attributes.subject
Anticipating Misconceptions – Understanding the Source?

# data[].attributes.attachment_link.readable_text
We talk about misconceptions all the time in subject knowledge - as if they are failures of knowledge, or epistemic understanding issues.

However, I'm not quite so sure it's as simple as that.

This is where I think misconceptions come from, which might potentially offer some insights in to where we might go to fix them...

Insecure Prior Knowledge

The easiest way to encounter misconceptions in subject knowledge is to make assumptions about what is already known. In a useful blog post, Tom Sherrington reminds us of the importance of a "check for understanding" to be all encompassing: often, teachers will make a check for understanding in to a compliance check "are you all ready?" "is everyone finished?" "has everyone got that?", rather than a genuine diagnosis of what vocabulary, conceptual information, or factual information has been covered.

I think this is important to focus on in observations, and explore in conversations. Be explicit as possible about the assumptions and pre-requisite knowledge needed to access the lesson; link that in to your do now and starter activity to check for understanding of those things. Plan effectively - consolidate effectively, and hopefully, we can avoid foundations of sand.

It's also possible that we have insecure knowledge of our own, as teachers. Geography is a big, broad discipline - and we might be teaching concepts at A Level that we have never studied before. Certainly, I can remember starting my teaching career with deserts, having never studied them at A Level or university (regrets about that, to this day - curses to Quaternary Science, I say!) - and only really getting it when I'd been teaching for a while, and then done some fieldwork in the Sahara. We might have big subject gaps and blindspots, which we only really expose in the cold light of an A Level lesson, when we run out of explanations to give...!

Ineffective Sequencing of Ideas:

Often, one of the challenges of misconceptions is that ideas have been put together in an order that doesn't make sense. Taylor (2017) explores ideas of curriculum planning in terms of step-by-step sequential knowledge development, but it may well be that we see problems with ordering material and explanations within a lesson, as well as within the curriculum. Teachers might introduce words without checking for understanding of meaning, or assume prior knowledge/contextual knowledge without considering how and why that knowledge needs to be built upon.

One of the most obvious ways this manifests is in presenting lesson content: perhaps the teacher shows a completed diagram, but the order in which they explain the stages to the students is not quite right, or leads to problems? They might try to teach evaluation, but without a coherent meta-structure to help students understand how to connect the ideas up. Structured thinking and lesson planning is key here: how do we model and reflect on what happens when you move this part of the lesson there and why that works, or doesn't. You may want to refer back to Allison & Tharby (2015)'s modelling of lesson sequence to help colleagues identify issues, if helpful.

Misconception of Place:

In the high tempo of a Geography curriculum under time pressure, it is difficult for nuance to be thoughtfully incorporated in to the classroom. It's easy for us to resort to quick articles, snapshots, and surface level understanding of case studies and illustrated points (Morris, 2021) - perhaps even chosen for pragmatic reasons about resources and available DVDs, rather than good Geography (Preece, 2021). At best, we end up with a slightly less optimal conception of place - perhaps a little date graphics, or a less high resolution video quality than we'd like to show.

At worst, we end up with out of date place knowledge; that risks the danger of a "single story" (Biddulph, 2011), and tells only a skewed version of the world (you may want to see this TED talk by Chimamanda Ngozi Adichie on the way that can be presented), and risks generating significant misconceptions in student learning about different places around the world.

The Geography has moved on:

There is an increasing increasing separation of university and school experience (Butt & Collins, 2018), and Enser (2021) reviews the changing academic leadership of the Geographical Association as an example of this in microcosm. During the 20 th century, the GA was led by multiple academic Geographers, but since 1973, no academic Geographer has held the role. The last post-holder - Professor Andrew Goudie - wrote of an increasing 'chasm between schools and universities' (Goudie, 1993), as teachers were increasingly responsible as curriculum makers (Brooks, 2013).

This may well mean that the subject content that you arrive with is contemporary, changed and modern - while the specifications, and textbooks that we are teaching with in the classroom reflect the pedagogical understanding of perhaps ten years ago. This can be significantly disconcerting for trainees, and new colleagues, and cause obvious discomfort and cognitive dissonance - you'll hear them plaintively ask why they are teaching something that's wrong?

There are often two different experiences in here:

Changing Concepts: some elements of our Geography content is unchanging and relatively confirmed through time. Our knowledge of rivers and the Bradshaw Model is not likely to be subject to a paradigmatic understanding of the way that water now works, for example. Other parts of the discipline are still being explored and emerging, and perhaps one of the best areas to illustrate this is in plate tectonics. Let's not forget, Einstein wrote his theory of General Relativity in 1917, while the theory of plate tectonics only emerged with the Vine-Matthews-Morley hypothesis in 1963. Since then, there has been significant change in the way we understand the mechanisms of plate movement, particularly as we apply seismic techniques to 'scan' inside the Earth. A trainee fresh out of a Geosciences degree will have a *very* different understanding of the way that the Earth moves, and the driving mechanics of convection currents in the mantle, of ridge push versus slab pull, and a number of more complex debates, than your typical A Level textbook. And here we face a challenge: do they teach what's "right" and "new", with the risk of it not being marked right in an exam? Or do they teach what's specified by the exam board, knowing that it's not quite up to date? There is no right answer to this - but it's easy to see how students might generate misconceptions and potentially inaccurate answers when being taught by someone whose knowledge now post-dates the specification and textbook!

Changing Utility of Concepts: in Teaching Geography there have been a recent series of articles about the changing use of the Burgess model in teaching urban morphology (Rawding, 2019; Puttick, 2020). While the model itself - originally developed based on 1920s Chicago - has not changed, the debate illustrates the way in which our pedagogical utility of the model has moved on. Some teachers argue that it should be taught, provided that you explain how and why thinking has developed; while others argue that it is out of date and leads to misconceptions when applied to the modern world. Unlike in our previous example, where the Geography and the science itself has changed, there is no new knowledge in the Burgess Model. Instead, Enser (2021) argues that this is an example of how concepts are being implemented in the classroom: can you still teach Burgess, but explain how the discipline has changed how we employ urban models?

A pragmatic and academic conversation is required with a colleague experiencing this dissonance. What is "right" versus "what is needed" is a difficult idealistic bridge to cross, and many Departments will want to explore this debate as part of mentor meetings and discussions over time. It can often be worth exploring Examiners' Reports with them, looking specifically for the evidence and analysis of case study and detail!

The world has moved on:

Many of us have an outdated and inaccurate view of the world in terms of data and situations. If you don't believe me, take a few moments to try this quiz. How did you do?

According to the Rosling Foundation, the successful team behind Gapminder, many of us have a statistical perception bias based on information that is no longer true. You can see them explain it in this TED talk, and read the mechanisms and causes in their superb book on Factfulness. But it leads to misconceptions in the classroom - students using outdated terminology (like "Third World"), and not sure how that's going to happen unless teachers reinforce it through their own teaching! Be alive to it in your lessons: consider what sources you're using for information, and challenge the use of outdated information, case studies and terminology with your colleagues if it occurs.

In this section, we also have outdated ideas about colonial curriculum, and the way that it's been structurally embedded in to our teaching. A number of excellent writers have explored specific issues on this theme - far better, and with more authentic understanding than I can hope to achieve - including this excellent article on why we should remove the word 'slum' from classrooms.

The Curse of the Expert

Experts, by their definition, have a different mental map (schema) of their discipline of knowledge than novices. Connections, terminology and sequences which are embedded in to their schema, and are therefore "obvious" to them can sometimes be impossible to 'unsee' and unpick, and they will presume much about others' ability in their audience. For teachers, this curse of knowledge (Dawn Cox) is difficult to move past; and can sometimes lead to "gaps" in the explanations and student misconceptions. They may see it reflected in marking, or in student explanations of their work - the part where the expertise explains something, or makes a connection will be absent from the lesson, as it was implicit not explicitly and directly said.

You can help colleagues address this by asking good coaching questions, and perhaps even co-planning with them. How do they script their explanations? What sequence do they assume, what prior knowledge do students have, how do they check for understanding of that - at heart, this is where the Rosenshine reminders are helpful for trainees and early career teachers!

Separation between threshold (core) and hinterland knowledge

Meyer & Land (2003) identified a conceptual variance between 'core' (threshold) and 'hinterland' concepts in education and pedagogy. The threshold knowledge is the core content of our subject - transformative, irreversible and the way we move our understanding forwards. However, the hinterland knowledge is the way in which we enable students to access that world. It may be through stories, videos, or examples that introduce salient points, critiques or places - but we need to be mindful of the potential for novice learners to confuse the core and the hinterland knowledge, and privilege the wrong bit! Many of my students will remember Mike Needleigh, the pig farmer, and his amazing jumpers. Fewer will be able to articulate the aspect of the Holderness case study example that his argument illustrated.

Telling too many stories, getting distracted in to anecdotes - or worse, getting students deliberately taking lessons off track by asking questions? Help colleagues to clarify the core knowledge, and the right sequence, and to carefully choose the hinterland they explore! Balancing 'relationships' in a classroom is critical!

The need for false balance

The intention of neutrality in topics, sometimes driven by the need to "evaluate" in specifications by presenting arguments for and against can occasionally lead to misconceptions driven by false narratives. For example, in the climate change debate, the science is now "unequivocal" about the causes - which equates to greater than 99% certainty. To present a "for and against" in this debate - as if there are equal weightings for e.g. human vs. physical causes of climate change; or that there are equal weightings of "yes it exists" vs. "no it doesn't" is to introduce a significant misconception to students. Be wary of false balances - they happen more often than perhaps we might like to think!

Can you think of others? Have I missed some?

Having considered how and where misconceptions may arise, I think it becomes increasingly important to reflect on how "planning" or "better subject knowledge" address these challenges. I don't think it's just as simple as "plan better" - it's a critical evaluation and engagement with the academic literature, the data and the changing world versus our perceptions of it, and this is a constantly evolving journey.

# data[].attributes.attachment_link.title
Anticipating Misconceptions - Understanding the Source?

# data[].attributes.attachment_link.description
We talk about misconceptions all the time in subject knowledge - as if they are failures of knowledge, or epistemic understanding issues. However, I'm not quite so sure it's as simple as that. This is where I think misconceptions come from, which might potentially offer some insights in to where we might go to fix...

# data[].attributes.attachment_link.readable_text
A new proposed idea regarding the art education debate is to integrate art with other core subjects to show art's importance and impact on other areas of study. Art advocates have, in particular, targeted math as a perfect candidate for integration with art. Historically, math has played a large role in art through mosaics, oriental rugs, stained glass windows, Gothic cathedrals, and a multitude of different art forms.

Cubists, abstract artists, and sculptors, throughout history, have relied heavily on geometry to turn their ideas into physical art. M.C. Escher was a particularly famous illusionist from the early 1900's, who was heavily influenced by math. He loved exploring the impossible in his works, which just radiates with math.

In spite of the obvious correlation between math and art, people often view art and math as two very different, distinct areas of study. Today, through computer-generated art, origami, tessellations, fractals, and illusions, math-inspired art continues yet often goes unnoticed. By integrating art and math on the classroom level, it will become plainly evident that participation in combined art and math courses can increase both one's math skills and one's art skills.

T hrough art one is able to visualize a math concept in a concrete and physical way. Also, in order to create more complex art forms, one needs a stronger math base, motivating students to gain a better understanding of math. On the other hand, math-based art can help students comprehend form and structure in their art. Math can also serve as a great basis for inspiration when designing a future piece. So many times have I gone to start a piece and realized that I would need to defy gravity to complete it. A basic understanding of gravity and weight distribution would tell me that I was heading in the wrong direction.

A few organizations have already formed to help connect the fields of art and math. One such organization, The Bridges Organization: Art and Mathematics holds an annual international conference to help "bridge" math and art. They hold workshops, paper presentations, math poetry readings, and a film festival. Their mission is to increase the abundance and integration of math, science, art, and architecture in our world. Another organization, The International Society of Arts, Mathematics, and Architecture, has an extensive website that acts as a database of math-related art and architecture. Simultaneous to the growth of these math related organizations, there are many mathematicians and artists, stepping into the fields of art and math, respectively.

One such mathematician artist, Helaman Ferguson is an algorist by profession, who makes crazy, cool, abstract sculptures. Rooted in his mathematical background, Ferguson expresses interesting equations and math wonders in artistic forms.Math artists, such as Ferguson, inspire students mathematically and artistically.

At the college level, students from art or math backgrounds could pursue a mathematical art minor. Math/art projects could be easily implemented into primary and secondary level classes, as well. Everything from learning shapes to deriving math proofs could be expressed through artistic means. As art education is threatened by budget cuts nationwide, it is crucial to emphasize the impact that math and art can have on each other. Obviously, math is crucial to education and will never be taken out of the curriculum. By promoting the integration of art and math, the public will be able to see art's importance for math and hopefully, one day, reaffirm art's individual importance.

Ultimately, I believe that art education is worth saving and that integration with math is a great way to achieve that. I love math and art and to see them combined in an endless number of flowing forms and shapes is so fascinating. In my high school art classes, I combined ceramics and outer space forms, which really taught me a lot about art and science that I did not know before. I often had to do a lot of research before I made a piece, but I did not find this research boring because I got to see how the research played out in my piece. Finding the right balance between scientific accuracy and artistic license was always a really fun task. Overall, cross-disciplinary study has been extremely rewarding for me personally, and I truly think that it could resurrect art education in the public school system.

Sources:
http://www.bridgesmathart.org http://www.isama.org https://www.sciencenews.org/article/when-art-and-math-collide http://mathcentral.uregina.ca/beyond/articles/Art/art1.html http://www.ams.org/mathimagery/
http://helasculpt.com/gallery/

# data[].attributes.attachment_link.description
A new proposed idea regarding the art education debate is to integrate art with other core subjects to show art's importance and impact on other areas of study. Art advocates have, in particular, targeted math as a perfect candidate for integration with art.

# data[].attributes.attachment_link.readable_text
The Bridges Organization oversees the annual Bridges conference on mathematical connections in art, music, architecture, and culture. Since 1998, Bridges has traveled to North America, Europe, and Asia, and has attracted participants from over thirty countries. The conference features invited speakers, regular and short paper presentations, hands-on workshops, a juried art exhibition, and arts performance events.

Bridges 2021

In light of the restrictions and challenges imposed upon us by the ongoing COVID-19 pandemic, Bridges 2021 will once again be held online. We plan to offer a 2021 virtual conference that includes more live presentations, shared events, and social interaction than we did in 2020. Please check our 2021 page regularly for updates on the conference, and watch the mailing list for announcements.

Recent conference reports

Katherine Wright's article about physics connections at Bridges 2021, from the journal Physics. (See also her article about Stepan Paul's Klein bottle research.)

Eve Torrence's article about Bridges in the European Mathematical Society Magazine.

Eve Torrence's report on Bridges 2018 in Nexus Network Journal.

Keep in touch

To contact the Bridges organization, please send email to bridges@bridgesmathart.org. You can also follow @BridgesMathArt on Twitter and look for the group "Bridges - Art and mathematics" on Facebook.

We use a mailing list for periodic announcements (such as forthcoming submission deadlines or updates on the conference). Use the form below to subscribe.

# data[].attributes.attachment_link.description
The Bridges Organization oversees the annual Bridges conference on mathematical connections in art, music, architecture, and culture. Since 1998, Bridges has traveled to North America, Europe, and Asia, and has attracted participants from over thirty countries. The conference features invited speakers, regular and short paper presentations, hands-on workshops, a juried art exhibition, and arts performance events.

# data[].attributes.attachment_link.title
2021 Bridges Conference Short Film Festival | Mathematical Art Galleries

# data[].attributes.attachment_link.description
Home › Exhibitions › 2021 Bridges Conference Short Film Festival

# data[].attributes.attachment_link.readable_text
When I joined the teaching profession I was surprised by two things:

no book existed about the common misconceptions in each topic.

no book existed about the various methods that can be taught in each topic.

I felt that both of these should be standard reference books for trainee maths teachers. But they didn't exist. I addressed the latter (see my book ' A Compendium of Mathematical Methods') but no one has fully addressed the former. We still don't have an easy-reference book or website featuring all the common misconceptions in one place. Instead, we repeat the mantra 'inexperienced teachers should learn about common misconceptions from their experienced colleagues'. That's a great idea, unless you find yourself working in dysfunctional maths department where no one talks maths. Then you just have to work it all out for yourself.

There have been a few occasions when I've tried to get conversations going on Twitter about misconceptions. Years ago, I set up a misconception sharing website but I didn't have time to maintain it. I also started a hashtag #misconceptionchat but I haven't used it often enough for it to gain traction. But I still believe that it's vital that this knowledge is shared widely. We all see misconceptions in the classroom every day. Some just need acknowledgment ('this is something students do, and our teaching should address this') whilst some are worthy of deep discussion.

There's a problem with sharing student misconceptions on social media though. Almost every time I've shared a misconception on Twitter, I've had replies from people criticising my teaching and suggesting that their approach is foolproof and would never lead to such misconceptions. Pfft. A more constructive dialogue (accompanied by a 'thanks for sharing this misconception, it's really interesting') would encourage teachers to share their observations more readily. Bear in mind that our students come to us with ingrained misconceptions that are not a reflection on our own teaching, but still need to be identified and addressed. We shouldn't hesitate to discuss these misconceptions with others - doing so enhances our pedagogical subject knowledge and makes us better teachers.

I think back to the time I tweeted a misconception that took me by surprise when I was marking a Year 7 fractions quiz.

I received a reply to my tweet from a teacher who told me I was making it up (unfortunately this teacher no longer has a Twitter account so his tweets have disappeared - I can't remember his exact words!). He said he'd give his students the same question the next day to see what happened, because he was adamant that this was not a mistake that any of his students would ever make. When he did so, he was absolutely shocked to find that they held the same misconception, and he later tweeted to apologise for not believing me. I guess it was a helpful misconception for me to share after all, as it alerted him to a gap in his students' understanding that he otherwise wouldn't have been aware of.

Just because we haven't personally seen a misconception, it doesn't mean it doesn't exist. Sometimes we just haven't asked the right questions.

Occasionally I see a student doing something in a lesson and realise it has resulted directly from my own flawed explanation. So I reflect on this, and I adjust my explanation the next time I teach it. I experienced this recently when I gave my Year 8 students a test on expanding double brackets. One student did well on every question, until he got to this one:

I was delighted that this student knew that 𝑥 2 ✕ 𝑥 -2 = 1, demonstrating his knowledge of index laws (other students wrote 𝑥 0 which was also acceptable).

However, a misconception relating to expanding arose that I wouldn't have spotted if I hadn't used this particular question.

Can you see what he's done? He seems to have assumed that the two diagonal cells can always be simplified. So there's a misconception in his underlying algebraic knowledge that needs to be addressed (i.e. that 4𝑥 12 + 5𝑥 can be simplified). There's also a step in the expansion procedure that he has fundamentally misunderstood. A couple of other students did something similar.

I take the blame for this one, even though I don't teach the popular 'sausage method' which could directly lead to this misconception. In case you don't know what the 'sausage method' is, check out the video here to here to see it in action (there are dozens more videos like this). Basically it involves drawing a ring around the two diagonal terms that can be simplified (the ring is sometimes referred to as a sausage, balloon or peanut).

Of course, the circled terms can only be simplified in certain expansions, so we should be careful to avoid implying that this ring can always be drawn.

So what did I do wrong in my teaching? Well in my initial modelling I always completed the grid then wrote out the four terms underneath the grid before simplifying them. I did that every time, until my students had built up some fluency in using the grids. Then I got a bit lazy and rushed my modelling, to the point where sometimes I skipped writing out the four terms underneath the grid, and took to drawing a ring around the like terms. I distinctly remember doing this when demonstrating a triple bracket expansion. So although I never said that 'drawing a sausage' is part of the procedure, and I certainly never said 'these two terms can always be simplified', I may have accidentally implied this in my modelling. Lesson learnt! Not every misconception is a direct result of a flaw in my explanation, but I think this one probably was.
Misconception baggage

I have a class of very strong mathematicians in Year 10, and one of the best students in the class asked me for help on this question:

'Write as a surd in simplest form: 10sin60 - 12tan30 + 8cos30'.

This was in a task on simplifying expressions involving exact trigonometric ratios, which is basically an exercise in manipulating surds.

Pre-empting misconceptions

"Miss, can I just check something with this question. I have to add 12tan30 and 8cos30 first, right? Because of BIDMAS?"

There's some debate about whether we should show our students common misconceptions up front. Some people think it might confuse them, but I'm all for it. In my teaching I directly show the mistakes people make and we discuss the reasons, and I make it clear what's gone wrong. That's not to say it always works though.

When teaching angles on a straight line, I'm careful about the wording (I always very specifically say ' adjacent angles on a straight line sum to 180') and I talk up front about how the angles need to be next to each other - they need to form a half-turn - in order to add up to 180. As a class we explore and acknowledge the common misconception. Yet still I have one or two students who do things like this:

Even though I don't have 100% success rate at eliminating all misconceptions, I still think addressing them head-on is worthwhile.

We can also address misconceptions through task design and curriculum sequencing. I recently had a Year 8 student who was factorising this expression:

He put up his hand to ask me whether the index in the first term only related to the c. A good question! It then occurred to me that I hadn't explicitly addressed this in my teaching, but I had previously designed a resource exactly for this purpose. I have an exercise on substitution that aims to draw out misconceptions around algebraic notation - see extract below.

"Frequently, a 'misconception' is not wrong thinking but is a concept in embryo or a local generalisation that the pupil has made. It may in fact be a natural stage of development."

# data[].attributes.attachment_link.description
When I joined the teaching profession I was surprised by two things: no book existed about the common misconceptions in each topic. no book existed about the various methods that can be taught in each topic. I felt that both of these should be standard reference books for trainee maths teachers.

# data[].attributes.attachment_link.description
Michael Anderson, Mathematics subject specialist at STEM Learning, shows you around some of our favourite Core Maths resources on the STEM Learning website.

# data[].attributes.attachment_link.description
Michael Anderson, Mathematics subject specialist at STEM Learning, shows you around some of our favourite A Level resources on the STEM Learning website.

# data[].attributes.attachment_link.description
Michael Anderson, Mathematics subject specialist at STEM Learning, shows you around some of our favourite GCSE resources on the STEM Learning website.