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Using Visual Maths and Errors as Learning Opportunities

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Using Visual Maths and Errors as Learning Opportunities

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Great Video!

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Teaching Students to Prove Their Mathematical Thinking through Questions, Charts, and Discourse

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Mona Iehl's third-graders at Polaris Charter Academy in Chicago develop their own understanding of how to solve math problems through cognitive guided instruction. They have time to grapple with problems and are guided to new understandings by Iehl's strategic use of questions. Following a period of grappling, students engage in discourse with each other and then work to prove their mathematical thinking to the group.

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Free Practice Website

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Thursday, 16 November 2017

At La Salle Education, we believe that pupils benefit enormously from having a deep understanding of multiplication and division facts, which can later be efficiently recalled for use in more complex problems.

A secure knowledge of times tables facts makes pupils able to engage in interesting mathematical problems without having to worry about working out basic facts first - these facts are part of the underlying mathematical grammar that pupils call upon to engage with mathematics throughout their learning and application of the subject.

But mathematics is not simply a list of facts to be remembered. At La Salle, we are interested in the interconnectedness of mathematical ideas. Most times tables practice is focused on simple rote learning and memorisation of the facts. This misses opportunities to build deeper understanding of multiplication and division and results in a superficial ability to simply regurgitate numbers. Our times tables app draws on variation theory to give multiple representations of multiplication facts, which builds more meaningful connections in pupils' minds and gives a greater chance of the facts becoming embedded in the long-term memory.

Through a variety of representations and metaphors, the Complete Mathematics Times Tables app gives pupils a better chance to 'meaning make' than traditional times tables apps.

Representations and metaphors

The Complete Mathematics Times Tables app deliberately intertwines a variety of ways of looking at and thinking about multiplication and division (and their connections to addition). The app includes standard recall prompts

but also makes connections to multiplication grids

and introduces pupils to arrays

The app also includes a pinboard manipulative, which not only connects the tables facts to multiplication grids, but also draws on the metaphor of multiplication and division as a view of area

Why no timer?

Becoming mathematically literate is not a competitive sport, it is a fundamental basic right for all. Although we want all pupils to be able to quickly recall times tables facts and be able to work efficiently with a wide range of problems that draw on these facts, we believe that - at the point of learning and embedding - it is far more important to carefully consider the problems and metaphors and to build a deeper understanding through meaningful practice.

Suggested uses

The Complete Mathematics Times Tables app is ideal for use in the mathematics classroom, at home, on the bus or... well... anywhere! Pupils can use the app on any device with a web browser.

With daily use, pupils will achieve a very secure knowledge of times tables facts. More than this though: unlike traditional times tables apps, which focus purely on the list of facts, using the Complete Mathematics Times Tables app daily, pupils will acquire a deep understanding of why the facts are true.

The times tables app could be used during tutor time, with pupils setting the quiz at 50 questions and recording each day how they are improving and which multiplication facts they need to continue to work on. Just 10 minutes per day for all pupils will help to drive up pupils' mathematical literacy across the school.

So, why not try the app today with your pupils and start a journey towards truly meaningful understanding of times tables rather than just fast regurgitation of meaningless numbers.

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Thursday, 16 November 2017 At La Salle Education, we believe that pupils benefit enormously from having a deep understanding of multiplication and division facts, which can later be efficiently recalled for use in more complex problems. A secure knowledge of times tables facts makes pupils able to engage in interesting mathematical problems without having to worry about working out basic facts first - these facts are part of the underlying mathematical grammar that pupils call upon to engage with mathematics throughout their learning and application of the subject.

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Reasources and research from Jo Boaler at Stanford University

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At youcubed, we inspire, educate and empower teachers of mathematics, transforming the latest research on mathematics learning into accessible and practical forms.

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Boost Engagement, Change Attitudes, Improve Results GET FREE MATH RESOURCES

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Great lessons and classroom challenges for Grades 6 and up

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress. The tools are relevant to any curriculum that seeks to deepen students' understanding of mathematical concepts and develop their ability to apply that knowledge to non-routine problems.

More about the Math Assessment Project

100 lessons for formative assessment, some focused on developing math concepts, others on solving non-routine problems. A Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

A set of 94 exemplar summative assessment tasks to illustrate the range of performance goals required by CCSSM. The tasks come with scoring rubrics and examples of scored student work.

Complete summative test forms and rubrics designed to help teachers and students monitor their progress using a range of task types similar to the 'Tasks' section.

5 Prototype modules that encourage groups of teachers to explore the practical and pedagogical concepts behind the materials, such as formative assessment, collaborative learning and the use of unstructured problems.

The Teaching for Robust Understanding of Mathematics (TRU Math) suite is a set of tools with applications in Professional Development and research based around a framework for characterizing powerful learning environments.

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress.

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Building number sense one day at a time

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Jo Boaler - Youcubed.org

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"How to Learn Math for Teachers" and "Mathematical Mindsets" are courses about the teaching of mathematics K-12. If you are new to the ideas shared on youcubed and in the book Mathematical Mindsets, "How to Learn Math for Teachers" is for you. If you are knowledgeable about the latest research and you already use number talks regularly, "Mathematical Mindsets" is the next course along. "Mathematical Mindsets" also shares the approach of our youcubed summer camps, with over 30 classroom videos. If you would like to learn about ways to integrate data science into your teaching, of any subject or grade, then our newest course, "21st Century Teaching and Learning," is the best choice. Click on any of the courses below to learn more!

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Online Courses for Teachers Archives - YouCubed

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"How to Learn Math for Teachers" and "Mathematical Mindsets" are courses about the teaching of mathematics K-12. If you are new to the ideas shared on youcubed and in the book Mathematical Mindsets, "How to Learn Math for Teachers" is for you.

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Time to find a better way to learn maths

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Learn and Teach Mathematics and Statistics

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The Numeracy Development Project has influenced the teaching of mathematics in New Zealand. It has changed the language people use to talk about mathematical understanding, introducing the terms "multiplicative thinking", "part-whole" and "proportional reasoning" to the teacher toolkit. It has empowered some teachers to think differently about the teaching of mathematics.

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All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime

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All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime

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Dr Keith Devlin is a mathematician at Stanford University

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If you are connected with the world of K-12 mathematics education, it's highly unlikely that a day will go by without you uttering, writing, hearing, or reading the term "number sense". In contrast everyone else on the planet would be hard pressed to describe what it is. Though entering the term into Google will return close to 38 million hits, it has yet to enter the world's collective consciousness. Stanford mathematician Keith Devlin explains what it is.
When I graduated with a bachelors degree in mathematics from one of the most prestigious university mathematics programs in the world (Kings College London) in 1968, I had acquired a set of skills that guaranteed full employment, wherever I chose to go, for the then-foreseeable future-a state of affairs that had been in existence ever since modern mathematics began some three thousand years earlier. By the turn of the new Millennium, however, just over thirty years later, those skills were essentially worthless, having been very effectively outsourced to machines that did it faster and more reliably, and were made widely available with the onset of first desktop- and then cloud-computing. In a single lifetime, I experienced first-hand a dramatic change in the nature of mathematics and how it played a role in society.

The shift began with the introduction of the electronic calculator in the 1960s, which rendered obsolete the need for humans to master the ancient art of mental arithmetical calculation. Over the succeeding decades, the scope of algorithms developed to perform mathematical procedures steadily expanded, culminating in the creation of desktop packages such as Mathematica and cloud-based systems such as Wolfram Alpha that can execute pretty well any mathematical procedure, solving-accurately and in a fraction of a second-any mathematical problem formulated with sufficient precision (a bar that allows in all the exam questions I and any other math student faced throughout our entire school and university careers).

So what, then, remains in mathematics that people need to master? The answer is, the set of skills required to make effective use of those powerful new (procedural) mathematical tools we can access from our smartphone. Whereas it used to be the case that humans had to master the computational skills required to carry out various mathematical procedures (adding and multiplying numbers, inverting matrices, solving polynomial equations, differentiating analytic functions, solving differential equations, etc.), what is required today is a sufficiently deep understanding of all those procedures, and the underlying concepts they are built on, in order to know when, and how, to use those digitally-implemented tools effectively, productively, and safely.

In many ways, that change in the society-required focus of mathematics education echoed a change in the nature of mathematics as an intellectual discipline that had taken placed in the late nineteenth century. Beginning in Germany (in particular, the small university town of Goettingen), the primary focus of mathematics shifted dramatically at that time from executing procedures to solve problems, to analyzing and understanding properties of, and relationships between, abstract mathematical concepts.

The most basic of today's new mathematical skills is number sense. (The other important one is mathematical thinking. But whereas the latter is important only for those going into STEM careers, number sense is a crucial 21st Century life-skill for everyone.) Descriptions of the term "number sense" generally run along the lines of "fluidity and flexibility with numbers, a sense of what numbers mean, and an ability to use mental mathematics to negotiate the world and make comparisons." The well-known mathematics educator Marilyn Burns, in her 2007 book, About Teaching Mathematics, describes students with a strong number sense like this: "[They] can think and reason flexibly with numbers, use numbers to solve problems, spot unreasonable answers, understand how numbers can be taken apart and put together in different ways, see connections among operations, figure mentally, and make reasonable estimates."

In 1989, the US-based National Council of Teachers identified the following five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for number, and referents for numbers and quantities.

Though to outsiders, mathematics teaching designed to develop number sense can seem "fuzzy" and "imprecise", it has been well demonstrated that children who do not acquire number sense early in their mathematics education struggle throughout their entire subsequent school and college years, and generally find themselves cut off from any career that requires some mathematical ability.

That outsiders' misperception is understandable. Compared to the rigid, rule-based, right-or-wrong precision of the math taught in my schooldays, number sense (and mathematical thinking) do seem fuzzy and imprecise. But the fuzziness and imprecision is precisely why that is such an important aspect of mathematics in an era where the rule-based, precise part is done by machines. The human brain compares miserably with the digital computer when it comes to performing rule-based procedures. But that human mind can bring something that computers cannot begin to do, and maybe never will: understanding. Desktop-computer and cloud-based mathematics systems provide useful tools to solve the mathematical aspects of real-world problems. But without a human in the driving seat, those tools are totally useless. And high among the "driving abilities" required to do that is number sense.

If you are a parent of a child in the K-12 system, there is today just one thing you should ensure your offspring has mastered in the math class by the time they graduate: number sense. Once they have that, any specific concept or procedure that you or they will find listed in the K-12 curriculum can be mastered (in the nature and to the degree required, given that procedural execution can be done by machine) quickly and easily as and when required. An analogous state of affairs arises at the college level, with the much broader notion of mathematical thinking in place of number sense.

Make no mistake about it, acquiring that modern-day mathematical skillset definitely requires spending time carrying out the various procedures. Your child or children will still spend time "doing math" in the way you remember. But whereas the focus used to be on mastering the skills with the goal of carrying out the procedures accurately - something that, thanks to the learning capacity of the human brain, could be achieved without deep, conceptual understanding - the focus today is on that conceptual understanding. That is a very different goal, and quite frankly a much more difficult one to reach.

This, by the way, is the new state of affairs that the mathematics Common Core was created to address. Outsiders, including politicians in search of populist issues to incite voters and others with an axe to grind, have derided caricatured portrayals of this important new educational goal, by describing it as "woolly" and "fuzzy". But I disposed of that uninformed red herring already. The fact is, number sense is (rightly, and importantly) the primary focus of 21st Century K-12 mathematics education that millions of children around the world are receiving today. Children who are not getting such an education are going to be severely handicapped in the world they are being educated to inhabit.

In a follow-up article, I look at some of the research behind the move away from computational mastery of procedures to the development of number sense in the K-12 education world, and say something about how it can best be taught.

This article is an expanded version of an essay published on Edge.org on January 1, 2017.

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All The Methods I Learned In My Mathematics Degree Became Obsolete In My Lifetime

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If you are connected with the world of K-12 mathematics education, it's highly unlikely that a day will go by without you uttering, writing, hearing, or reading the term "number sense". In contrast everyone else on the planet would be hard pressed to describe what it is.

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On fractional and proportional thinking

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Welcome to Splat! You are only moments away from a VERY POWERFUL, highly interactive number sense strategy that can be used at any grade level!

This post includes 50 (fifty!) free, downloadable PowerPoint math lessons!

Watch the video, download some (or all) of the lessons, and experience what happens when you unleash this resource in your classroom!

I've been excited to click the Publish button on this post for several weeks! It's time!
UPDATE: The Fraction Splat! Series (with 20 more downloadable lessons) is now available!
After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.

Splat Through 10
How many dots are under the Splat?

Splat Through 20
How many dots are under the Splat?

Multiple Splats
(Note: Splats that are the same color must cover the same number.)
Splat 3.1 - multiple splats Splat 3.2 - multiple splats Splat 3.3 - multiple splats Splat 3.4 - multiple splats Splat 3.5 - multiple splats
Instant Multiple Splat
(Note: An additional feature of this level is the opportunity to ask, "What could the total be?")

Splat 4.1 - Instant Multiple Splats Splat 4.2 - Instant Multiple Splats Splat 4.3 - Instant Multiple Splats Splat 4.4 - Instant Multiple Splats Splat 4.5 - Instant Multiple Splats
2-Color Splat
(Splats of different colors must have different values.)

Splat 5.1 2-color Splat Splat 5.2 2-color Splat Splat 5.3 2-color Splat Splat 5.4 2-color Splat Splat 5.5 2-color Splat Splat 6.2 Instant 2-color Splat Splat 6.3 Instant 2-color Splat Splat 6.4 Instant 2-color Splat Splat 6.5 Instant 2-color Splat
Instant 2-Variable Splat
(Splats of different colors must have different values. Splats of the same color must have the same value.)
Splat 7.1 Instant 2-variable Splat Splat 7.2 Instant 2-variable Splat Splat 7.3 Instant 2-variable Splat Splat 7.4 Instant 2-variable Splat Splat 7.5 Instant 2-variable Splat
Number Splat
(Note: Only the numbers circled on the screen may be used.)
Splat 8.1 - Number Splat Splat 8.2 - Number Splat Splat 8.3 - Number Splat Splat 8.4 - Number Splat Splat 8.5 - Number Splat Splat 9.1 - Instant Number Splat Splat 9.2 - Instant Number Splat Splat 9.3 - Instant Number Splat Splat 9.4 - Instant Number Splat Splat 9.5 - Instant Number Splat
Instant Number Splat with 2 Variables
(Note: Only the numbers circled on the screen may be used, and Splats of different colors must have different values.)
Splat 10.1 - Instant 2-Variable Number Splat Splat 10.2 - Instant 2-Variable Number Splat Splat 10.3 - Instant 2-Variable Number Splat Splat 10.4 - Instant 2-Variable Number Splat Splat 10.5 - Instant 2-Variable Number Splat
NOTE: I wrote each of these lessons using PowerPoint, so I recommend playing them in PowerPoint. If you do not have PowerPoint, you may want to download the PowerPoint viewer so that you can play the lessons.

UPDATE: The Fraction Splat! Series (with 20 more downloadable lessons) is now available!
Other Posts and Resources Which May Be of Interest Tiled Area Questions The Maze Hundreds Chart
Introducing Cube Connectors
Provide Massive Space to Notice
Primary Tile Questions
3 Powerful Tile Strategies (and 40 new downloadable pages)
Please take a moment to subscribe to the blog so you'll be notified each time new resources are posted!

I would treasure hearing about your experiences using Splat! to promote number sense in your classroom!

All my best,
Steve

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Splat! - Steve Wyborney's Blog: I'm on a Learning Mission.

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Welcome to Splat! You are only moments away from a VERY POWERFUL, highly interactive number sense strategy that can be used at any grade level!

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Classroom practice

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We don't use maths worksheets in our school. Ever. This might shock some people to hear, especially in a climate where teachers have to perform a juggling act: planning, marking, assessing and so on. But this decision was not taken lightly.

There are many fantastic resources out there in the form of worksheets, such as those produced by Nrich or NCETM. However, there are at least an equal number of not so useful worksheets.

We've all been there. Sunday evening, marking to finish and lessons to prepare. In these moments, a ready-made worksheet can seem to tick a lot of boxes. But now, more than ever, with the emergence of the mastery approach, we know how important the right questions are - not a variety of questions on a worksheet designed to target a range of learners at different levels, therefore maximising their audience. Don't get me wrong; I love that teachers share materials and appreciate having a resource ready. However, we need the appropriate variation that suits the needs of the specific class. This can't be pitched at the appropriate level by anyone other than the teacher of that class.

Just as we need a balanced diet for what we eat, children need a varied and balanced "diet" when they are taught mathematics. A constant diet of worksheets, where only the answer is required, is not healthy for the children or teacher. They are boring to do and make maths lessons boring for everyone in the room. Would you expect a diet of "fill the missing word in" for every English lesson? Of course not. Why, then, is this acceptable in a maths lesson?

Children 'more engaged in maths'

At my school, we have come up with a different approach. We develop a set of learning questions for a particular year group, making sure that they are open questions that allow all children to access the same learning, but at different levels. These low-threshold, high-ceiling activities help us to avoid the trap of recreating worksheets.

The practice and application part of the lesson normally involves open-ended activities designed to allow learners to take ownership of the learning and make connections across and within concepts. Our learning questions focus on the skill involved rather than the procedure. For example, we might have a learning question that reads: "How can we use our understanding of triangles to find missing angles?" This task will involve quadrilaterals and other polygons; however, the skill we are developing is the relationship between triangles and these shapes.

Our new approach to devising maths questions is time-consuming and it did take time to get everyone on board with it. But the benefits far outweigh the challenges.

Children are now more responsive to maths. The subject is no longer reduced to answering an unending list of questions. And if you finish your work, guess what? You get another task to work on. Pupils are no longer so keen to rush through the first activity and have become more resilient as a result. They realise that getting the right answer isn't always the most important thing and that what counts is whether they actually understand the maths involved.

I appreciate that people will have differing views on this, but I would suggest giving it a go - you might even enjoy designing your own lessons. Break the chains of worksheets and take control of your teaching.
Stevie Devlin is a Year 6 teacher and maths lead at Bursted Wood Primary School in Kent. He tweets @devlin_steven Want to keep up with the latest education news and opinion? Follow Tes on Twitter and Instagram, and like Tes on Facebook.

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We don't use maths worksheets in our school. Ever. This might shock some people to hear, especially in a climate where teachers have to perform a juggling act: planning, marking, assessing and so on. But this decision was not taken lightly.

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A Mathematician’s Lamentby Paul Lockhart

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A Mathematician’s Lamentby Paul Lockhart

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Why and what needs to change in maths learning

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Google Slides Manipulative- Make a Copy

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Great for discussions and NumberTalks

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Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It’s an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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Number sense through visual maths

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Bar modelling is often credited as a central pillar in the successful mathematics curriculum and pedagogy of Singapore and other East Asian nations. Furthermore, the Mastery approach evolving in the UK has led to an increased focus on bar modelling and the CPA method. The TES has several excellent articles on this. More recently, New Zealand schools have shown a growing interest in this method as a tool for developing learner agency and accessing problem solving. But, when and how should be introduced?

Is there a place for bar modelling in early maths?

As children move towards the abstraction of using numerals, using the bar model can help children combine the ordinal and cardinal aspects of number. In the example below, the brace/bracket used in bar modelling labels the number of objects. This could be a technique adults might use and model to children during early counting experiences.

Teachers familiar with Martin Hughes' seminal work might suggest the example above is too big a conceptual leap for children without one-to-one correspondence. His work suggested that children used pictographic and iconic marks before the symbolic use of numerals. The examples below show how children might label amounts of objects pictographically or iconically before moving to using numerals.

It could be argued that these images and experiences are not different to any early years classroom. Giving children rich experiences of counting, classification, sorting and identifying pattern are cornerstones of early mathematical experience. I'd argue that the bracket or brace encompassing the objects that are being counted is very powerful. I believe that this helps to cement the cardinal aspect of number, consolidating the 'threeness' of the objects.

Look at the image below, if we were to ask the question, 'where is the 3?' Would children point to the 3rd train on the right (assuming they have counted from left to right)? The brace helps us to show that once we have counted the third cube (whichever cube that may be) we know the number in the set.

The next step for young students is to begin bar modelling. Again, we encourage children to build on their counting experiences and manipulate concrete apparatus. Here the model is discrete, and still encourages and supports one-to-one matching. This also highlights the abstraction principle of counting. Children begin to see that they can count the pictorial representation of the block rather the block itself.

Developing counting strategies, calculation and understanding of pattern

The next step in the progression is to move from a model of 3 ones to a continuous model of one group of 3. Conceptually this is a significant change in understanding. I'd argue that it isn't sufficient to do this only by bar modelling. Educators need to consider whether they have grouped objects for students to explore. This is where Numicon and Cuisenaire are so powerful and, for me, a crucial tool for teaching.

Now that the 3 the in photograph above has been unitized we can work with it. So, we could place another 3 alongside it and begin thinking about multiplication as repeated addition. First though, we are developing counting strategies. In the calculation, '3+2' a developmental step would be to count all of the objects (or blocks in the bar) but now the 3 is continuous children have a scaffold to help them count on from 3, rather than begin by counting the objects within the 3.

The above image is similar to a number line. My issue with a marked number line is that children can still count from 0 to get to 5. Furthermore, with the progression shown above, children who are just moving past the one-to-one correspondence stage may be able to picture the 3 other cubes within the continuous bar. This type of mental imagery is vital in helping children develop number.

The next step would be to draw two continuous bars labelled 3 and 2 next to each other. At this point, I'd ask educators to consider how they move children away from the developmental stage of counting to calculate. The question could be rephrased as, 'Which manipulatives are you using to secure knowledge of number facts?' Numicon, Cuisenaire, tens frames and bead strings, all offer children a variety of experiences with number. Personally, I feel that Numicon is the most powerful - I'll save that blog for another day.

Where next with Bar modelling?

Once children have experience developing their additive reasoning with concrete apparatus and bar models, they can then be used to develop (or assess) their multiplicative understanding. Starting to use bar modelling and its principles early on, will mean students are familiar with it and its conventions for different aspects of number.

Bar modelling for young children - how is it beneficial?

Hopefully, this has answered some of your questions regarding using the bar. My previous post discusses bar modelling as a problem solving tool for increasing learner agency. Nick Hart's blog is also an excellent resource.

How do I get this into my classroom?

At first, including bar modelling in your classroom can be quite daunting. As with any new pedagogy, it is important to understand the principles behind the method. Sign up to a training course!

UK educators, go here. The White Rose training is exceptional. NZ teachers, you can access the same training by attending a CPA Maths workshop

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Bar modelling for young children - how is it beneficial?

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Bar modelling is often credited as a central pillar in the successful mathematics curriculum and pedagogy of Singapore and other East Asian nations. Furthermore, the Mastery approach evolving in the UK has led to an increased focus on bar modelling and the CPA method. The TES has several excellent articles on this.

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Gary S. Stager, Ph.D

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January 4, 2016
The Willows Community School - Culver City, California USA
By Gary S. Stager, Ph.D.

Math education has fascinated me for a very long time. I was always good at arithmetic and despite having a pretty bleak elementary school experience, I could do what they called, "math." Test scores in the 6 th grade indicted that I was mathematically gifted and earned me a place in something called Unified Math. "Unified" was an accelerated course intended to rocket me to mathematical superiority between grades 7 and 12. Rather than take discrete algebra, geometry, trigonometry, etc., Unified Math was promised as a high-speed roller-coaster ride through various branches of mathematics.

Then through the miracle of mathematics instruction I was back in a low Algebra track by 9 th grade and limped along through terrible math classes until my senior year in high school. In 12 th grade, I enrolled in a course called, " Math for Liberal Arts." Today this course might be called, " Math for Dummies Who Still Intend to Go to College." I remember my teacher welcoming us and saying, "Now, let's see if I can teach you all the stuff my colleagues were supposed to have taught you."

This led to two observations:

1. Mr. O'Connor knew there was something terribly wrong with math education in his school.

2. I looked around the room and realized that most of my classmates had been in Unified Math with me in 7th grade. These lifeless souls identified as mathematically gifted six years ago were now in the "Math for Dummies Who Still Intend to Go to College" class. If this occurred to me, I wondered why none of the smart adults in the school or district had observed this destructive pattern?

Two things I learned in school between 7 th and 12 th grade kept me sane. I learned to program computers and compose music. I was actually quite good at both and felt confident thinking symbolically. However, majoring in computer science was a path closed to me since I wasn't good at (school) math - or so I was told.

I began teaching children in 1982 and teachers in 1983. I was 18-19 years old at the time. While teaching others to program, I saw them engage with powerful mathematical ideas in ways they had never experienced before. Often, within a few minutes of working on a personally meaningful programming project, kids and teachers alike would experience mathematical epiphanies in which they learned "more math" than during their entire schooling.

In the words of Seymour Papert, "They were being mathematicians rather than being taught math."

Teaching kids to program in Logo exposed me to Papert's " Mathland," a place inside of computing where one could learn to be a mathematician as casually as one would learn French by living in France, as opposed to being taught French in a New Jersey high school class for forty-three minutes per day.

I met Seymour Papert in 1985 and had the great privilege of working with him for the next 20+ years.

Papert was a great mathematician with a couple of doctorates in the subject. He was the expert Jean Piaget called upon to help him understand how children construct mathematical knowledge. Papert then went on to be a pioneer in artificial intelligence and that work returned him to thinking about thinking. This time, Papert thought that if young children could teach a computer to think (via programming), they would become better thinkers themselves. With Cynthia Solomon and Wally Feurzig, Papert invented the first programming language for children, called Logo. That was in 1968.

What makes Papert so extraordinary is that despite being a gifted mathematician he possesses the awareness and empathy required to notice that not everyone feels the same way about mathematics or their mathematical ability as he does. His life's work was dedicated to a notion he first expressed in the 1960s. Instead of teaching children a math they hate, why not offer them a mathematics they can love?

As an active member of what was known as the Logo community, I met mathematicians who loved messing about with mathematics in a way completely foreign to my secondary math teachers. I also met gifted educators who made all sorts of mathematics accessible to children in new and exciting ways. I fell in love with branches of mathematics I would never have been taught in school and I understood them. Computer programming was an onramp to intellectual empowerment; math class was a life sentence.

It became clear to me that there is no discipline where there exists a wider gap than the crevasse between the subject and the teaching of that subject than between the beauty, power, wonder, and utility of mathematics and what kids get in school - math.

Papert has accused school math of "killing something I love."

Marvin Minsky said that what's taught in school doesn't even deserve to be called mathematics, perhaps it should just be called "Ma."

One of our speakers, Conrad Wolfram, says that every discipline is faced with the choice between teaching the mechanics of today and the essence of the subject. Wolfram estimates that schools spend 80% of their time and effort teaching hand calculations at the expense of mathematics. That may be a generous evaluation.

Time for Action
Over the years, I've gotten to know gifted mathematicians like Brian Silverman, David Thornburg, Seymour Papert, Marvin Minsky, and Alan Kay. I've even spent a few hours chatting with two of the world's most preeminent mathematicians, John Conway and Stephen Wolfram. In each instance, I found (real) mathematicians to embody the same soul, wit, passion, creativity, and kindness found in the jazz musicians I adore. More significantly, math teachers often made me feel stupid; mathematicians never did.

The 1999 National Council of Teachers of Mathematics Standards said, "50% of all mathematics has been invented since World War II." This is the result of two factors; the social science's increasing demand for number and computing.

These new branches of mathematics are beautiful, useful, playful, visual, wondrous, and experimental. Computing makes some of these domains accessible to even young children, and yet you are unlikely to find the likes number theory, chaos, cellular automata, fractal geometry, topology... in the K-12 math curriculum.

Hell, I dream of a day when a math textbook uses the symbol for multiplication used on computer keyboards for a half century. It makes my head explode when a high school student doesn't know how to ask a computer to multiply two numbers.

Since No Child Left Behind, parents, politicians, and educators have been engaged in a death match known as the Math Wars. The prescribed algorithmic tricks proscribed by The Common Core have thrown dynamite on the raging fire about how best to teach math. Ignorance, fear, and superstition are a volatile brew and impediment to learning.

Conrad Wolfram estimates that 20,000 student lifetimes are wasted each year by school children engaged in mechanical (pencil and worksheet) calculations. Expressed another way, we are spending twelve years educating kids to be a poor facsimile of a $2 calculator. Forty years after the advent of cheap portable calculators, we are still debating whether children should be allowed to use one.
Hard, Not Fun
We are allowing education policy and curriculum to be shaped by the mathematical superstitions of Trump voters. Educators need to take mathematics back and let Pearson keep "math."

When Barbie said, "Math is hard," the politically-correct class expressed their faux outrage, but Barbie was speaking a ubiquitous truth only tacitly acknowledged by the brave or those severely damaged my school math. If math is hard, fixing mathematics education is even harder.

Study after study tell us that kids hate math, computers are less likely to be used in a math class than anywhere else in school, teachers have little confidence in their own mathematical abilities and were poor math students, formidable gender gaps still exist - even the stupid test scores by which some measure "achievement" are static or worse.
Hope
Faced with an abundance of research, personal history, and good old-fashioned intuition while screaming from the rooftops that math education is a shambolic failure, we just double down on what does not work.

Against this backdrop of panic, misery, and despair there is room for optimism.

There is a renewed attention being paid to the importance of S.T.E.M. and S.T.E.A.M.

We live in a complex society awash in data. Citizenship depends on strong mathematical thinking and modeling skills.

Computational power has never been cheaper, easier to use, or portable. Today, you can ask your phone any question found in the K-12 math curriculum and receive an immediate answer. It can even "show all work." How will math education respond to this reality?

The maker movement has reenergized timeless craft traditions and supercharged such creative human expression with new tools and computational materials.
Why Progressive Educators Should Care About Reinventing Mathematics Education
Kids are miserable. Parents are fed-up. They are not only opting out of standardized testing, but rejecting that which is tested and the way it is taught.

I had a conversation with Dr. Papert in 2004 in which he was on-fire about the need to revolutionize math education with all the urgency our society can muster. When I asked if his focus on math education was because he was a mathematician, Papert rattled off more than a dozen reasons why this was a priority.

One argument in particular stayed with me while I have forgotten others.

Papert said that no pedagogical innovation of the past century has had any real impact on math education and if that were not disconcerting enough, it ultimately meant that in practice, no matter how progressive or learner-centered a school aspired to be, there was one point in the school day when "coercion was reintroduced into the system." Math class was when kids felt badly about themselves and were being taught irrelevant tricks they might need one day.

Papert argued that this scenario was corrosive to any other constructive efforts undertaken by a school, eventually undermining efforts like project-based learning, authentic assessment, student led inquiry, and other aspects of constructivist teaching. There is no way to make a noxious math curriculum more palatable.
Reinventing Mathematics Education
Papert would ask how math class could feel more like art class, where students would become lost in their work, think deeply, act creatively, and produce an artifact they were proud of?

Our January 4 th event, Reinventing Mathematics Education, assembles experts and practitioners together in Los Angeles for an action-packed day of learning by doing and conversations on how to proceed.
Reinventing Mathematics Education is different
Most discussions of math education define "reform" as devising a clever new teaching trick or test intended to fix the kid and make them understand what's in a textbook relatively unchanged since the advent of movable type.

Dr. Constance Kamii brings a lifetime of research on how children construct arithmetic understanding through play, thinking, arguing, and problem solving without being taught algorithms by the teacher. Her decades worth of peer-evaluated research and clever classroom strategies are deeply rooted in her fifteen years of work with Jean Piaget. Despite her diminutive physical stature, Kamii is a giant in the field of how teachers can create the context for student learning.

Dr, Ilana Horn is of a different generation and active in social media. Her work is centered on creating a productive context for learning and how older students can be supported to learn mathematics in a collaborative climate that fosters sustained habits of mind.

Conrad Wolfram is a mathematician, computer scientist, and educator who works with his brother Stephen, considered one of the world's greatest living mathematicians and scientists, at a company inventing many of the computational tools being used to propel science, mathematics, engineering, and the social sciences into an exciting and uncertain future. Conrad Wolfram is the force behind Computer-based Mathematics and works tirelessly to design materials that schools can use to bring real mathematical power to each child. In the words of Seymour Papert, Wolfram is "creating a new diet of mathematics for kids."

This event would not be possible without the leadership and generosity of my school, The Willows Community School. My colleagues understand the enormity of the task associated with teaching kids for their future, rather than our past. The January 4 th event is a professional development day for the entire Willows faculty and we are inviting you to join us for a one-of-a-kind world-class event.

Don't miss out. Register today!

Posted September 30, 2015 in: Event by gary

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The Personal Road to Reinventing Mathematics Education

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January 4, 2016 The Willows Community School - Culver City, California USA By Gary S. Stager, Ph.D. Math education has fascinated me for a very long time. I was always good at arithmetic and despite having a pretty bleak elementary school experience, I could do what they called, "math."

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Boost mathematical understanding by incorporating maths concepts into narratives

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​Whether you are an experienced teacher, a teacher trainee, a teacher educator, a parent, a researcher or a policy maker, we hope you will find this website interesting and useful. Specifically, we want to share with you the benefits of te﻿aching and learning mathematics through stories and creative writing, as well as to offer practical ideas for teachers and parents on how to embrace this approach.

This website has been designed and continuously developed based on the feedback we receive from the key stakeholders as noted above. There are plenty for you to do on this website, including, but not limited to:
Please take your time to explore the website using either the menu bar at the top of the page or from the menu below.

Finally, consider s ubscribing to our free quarterly newsletters, and please help spread the word about our project and its website :-)

Best wishes,

Dr. Natthapoj Vincent Trakulphadetkrai B.A. (Hons), M.Sc. (Oxford), M.Sc. (London, UCL), Ph.D. (Cambridge), QTS

@NatthapojVinceT Natthapoj.org

Founder & Principal Investigator
​MathsThroughStories.org

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​Whether you are an experienced teacher, a teacher trainee, a teacher educator, a parent, a researcher or a policy maker, we hope you will find this website interesting and useful. Specifically, we want to share with you the benefits of te﻿aching and learning mathematics through stories and creative writing, as well as to offer practical ideas for teachers and parents on how to embrace this approach.

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Includes links to teacher resources

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Paul Dirac had an eye for beauty. In one essay, from May 1963, the British Nobel laureate referred to beauty nine times. It makes four appearances in four consecutive sentences. In the article he painted a picture of how physicists saw nature. But the word beauty never defined a sunset, nor a flower, or nature in any traditional sense. Dirac was talking quantum theory and gravity. The beauty lay in the mathematics.

What does it mean for maths to be beautiful? It is not about the appearance of the symbols on the page. That, at best, is secondary. Maths becomes beautiful through the power and elegance of its arguments and formulae; through the bridges it builds between previously unconnected worlds. When it surprises. For those who learn the language, maths has the same capacity for beauty as art, music, a full blanket of stars on the darkest night.

"The slow movement of the Mozart clarinet concerto is a really beautiful piece of music, but I don't print off a page of the score and put that on my wall. It's not about that. It's about the music and the ideas and the emotional response," says Vicky Neale, a mathematician at Oxford University. "It's the same with a piece of mathematics. It's not how it looks, it's about the underlying thought processes."

Brain scans of mathematicians show that gazing at formulae considered beautiful by the beholder elicits activity in the same emotional region as great art and music. The more beautiful the formula, the greater the activity in the medial orbito-frontal cortex. "So far as the brain is concerned, maths has beauty just like art. There is common neurophysiological ground," says Sir Michael Atiyah, an honorary professor of mathematics at Edinburgh University.

Ask mathematicians about the most beautiful equation and one crops up time and again. Written in the 18th century by the Swiss mathematician, Leonhard Euler, the relation is short and simple: e iπ+1 = 0. It is neat and compact even to the naive eye. But the beauty comes from a deeper understanding: here the five most important mathematical constants are brought together. Euler's formula marries the world of circles, imaginary numbers and exponentials.

The beauty of other formulae may be more obvious. With E=mc ², Albert Einstein built a bridge between energy and mass, two concepts that had previously seemed worlds apart. Maggie Aderin-Pocock, the space scientist, declared it the most beautiful equation and she is in good company. "Why is it so beautiful? Because it comes to life. Now energy will have mass and mass can be put into energy. These four symbols capture a complete world. It's difficult to imagine a shorter formula with more power," says Robbert Dijkgraaf, director of the Institute for Advanced Study in Princeton, where Einstein was one of the first faculty members.

"One of the reasons there's almost an objective beauty in mathematics is that we use the word beautiful also to indicate the raw power in an idea. The equations or results in mathematics that are seen to be beautiful are almost like poems.The power per variable is something that is part of the experience. Just seeing a huge part of mathematics or nature being described with just a few symbols gives a great sense of elegance or beauty," Dijkgraaf adds. "A second element is you feel its beauty is reflecting reality. It's reflecting a sense of order that's out there as part of the laws of nature."

The power of an equation to connect what seem like completely unrelated realms of mathematics comes up often. Marcus du Sautoy, a maths professor at Oxford, has more than a soft spot for Riemann's formula. Published by Bernhard Riemann in 1859 (the same year Charles Darwin stunned the world with On the Origin of Species), the formula reveals how many primes exist below a given number, where primes are whole numbers divisible only by themselves and one, such as 2, 3, 5, 7 and 11. While one side of the equation describes the primes, the other is controlled by zeros.

"This formula turns these these indivisible prime numbers, into something completely different," says du Sautoy. "On the one side, you've got these indivisible prime numbers and then Riemann takes you on this journey to somewhere completely unexpected, to these things which we now call the Riemann zeros. Each of these zeros gives rise to a note - and it's the combination of these notes together which tell us how the primes on the other side are distributed across all numbers."

More than 2,000 years ago, the ancient Greek mathematician, Euclid, solved a numerical puzzle so beautifully that it still makes Neale smile every time it comes to mind. "When I think about beauty in mathematics, my first thoughts are not about equations. For me it's much more about an argument, a line of thinking, or a particular proof," she says.

Euclid proved there are infinitely many prime numbers. How did he do it? He began by imagining a universe where the number of primes was not infinite. Given a big enough blackboard, one could chalk them all up.

He then asked what happened if all these primes were multiplied together: 2 x 3 x 5 and so on, all the way to the end of the list, and the result added to the number 1. This huge new number provides the answer. Either it is a prime number itself, and so the original list was incomplete, or it is divisible by a smaller prime. But divide Euclid's number by any prime on the list and always there is a 1 left over. The number is not divisible by any prime on the list. "It turns out you reach an absurdity, a contradiction," says Neale. The original assumption that the number of primes is finite must be wrong.

"The proof for me is really beautiful. It takes some thinking to get your head around it, but it doesn't involve learning lots of difficult concepts. It's surprising that you can prove something so difficult in such an elegant way," Neale adds.

Behind beautiful processes lie beautiful mathematics. Well, some of the time. Hannah Fry, a lecturer in the mathematics of cities at UCL spent years staring at the Navier-Stokes equations. "They're a single mathematical sentence that is capable of describing the miraculously beautiful and diverse behaviour of almost all of Earth's fluids," she says. With a grasp of the formulae, we can understand blood flow in the body, make boats glide through the water, and build awesome chocolate enrobers.

In his 1963 essay, Dirac elevated beauty from an aesthetic response to something far more profound: a route to the truth. "It is more important to have beauty in one's equations than to have them fit experiment," he wrote, continuing: "It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress." Shocking at first pass, Dirac captured what is now a common sentiment: when a beautiful equation seems at odds with nature, the fault may lie not with the maths, but in applying it to the wrong aspect of nature.

"Truth and beauty are closely related but not the same," says Atiyah. "You are never sure that you have the truth. All you are doing is striving towards better and better truths and the light that guides you is beauty. Beauty is the torch you hold up and follow in the belief that it will lead you to truth in the end."

Something approaching faith in mathematical beauty has led physicists to draw up two of the most compelling descriptions of reality: supersymmetry and string theory. In a supersymmetric universe, every known type of particle has a heavier, invisible twin. In string theory, reality has 10 dimensions, but six are curled up so tight they are hidden from us. The mathematics behind both theories are often described as beautiful, but it is not at all clear if either is true.

There is a danger here for mathematicians. Beauty is a fallible guide. "You can literally be seduced by something that is not correct. This is a risk," says Dijkgraaf, whose institute motto, "Truth and Beauty" features one naked and one dressed woman. "Sometimes I feel that physicists, like Odysseus, must tie themselves to the mast of the ship so they are not seduced by the sirens of mathematics."

It may be that mathematicians and scientists are the only groups that still use the word "beautiful" without hesitation. It is rarely employed by critics of literature, art or music, who perhaps fear it sounds superficial or kitschy.

"I'm very proud that in mathematics and science the concept of beauty is still there. I think it's an incredibly important concept in our lives," says Dijkgraaf. "The sense of beauty we experience in maths and science is a multidimensional sense of beauty. We don't feel it's in any conflict with being deep, or interesting, or powerful, or meaningful. For the mathematician it's all captured by that one word."
Topics

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Magic numbers: can maths equations be beautiful?

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Paul Dirac had an eye for beauty. In one essay, from May 1963, the British Nobel laureate referred to beauty nine times. It makes four appearances in four consecutive sentences. In the article he painted a picture of how physicists saw nature.

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Mr Barton's Maths blog

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I have decided to move my "links to the best maths websites in the world" page to my blog to make it easier to update. To return to the Teachers section of my main website, just click here.

On this page I will list my go-to websites and blogs that have saved me hours and given me so many lovely lessons and ideas over the years. Some will be familiar to you, but hopefully there will be a few hidden gems in there. If you know of any I have missed out, please drop me an email.

Also, please check out my "Essential Freebies" page, for lots of lovely free maths goodies!
Index Single Purpose Maths Websites Rich Maths Problems Making Data and Statistics Interesting Best of the Maths Blogs Best of the Maths Resources Websites Best of the Rest
A pleasing trend in recent years has been the emergence of "single purpose" maths website - sites that do one specific thing, and do it very, very well. Here is a list of some of my favourites.

diagnostic questions
Okay, so I am (more than!) a little biased on this one as it is my own website, but it is free and I created it as there was nothing else like it on the web, so I hope you will forgive me! It is designed to help teachers identify key misconceptions in students' understanding and to help students all around the world learn from each other. See my "how to videos" or the "question of the week" feature for more information.

maths pret homeworks
Interesting, engaging, challenging homeworks, testing a full range of skills, all for free. See my video on this wonderful website here.

I absolutely love activities, investigations, puzzles and problems that challenge students of all ages and abilities to think deeper about their mathematics. Familiar topics presented in an interesting, original way tend to engage students a lot more. The following websites are my favourite places to find these.

3 act math
I've been obsessed with "3 act math" lessons for a few years now. Hook students in with an intriguing problem, let them request information and try to solve it, and then see how close they were. Click the links below for some of the best collection of 3 act math tasks around:
Dan Meyer demoing the pile of coins task
Dan Meyer post on running 3 act math tasks
Dan Meyer task collection
Andrew Stadel task collection
Mr Picc task collection
Kyle Pearce collection

I absolutely love statistics, and so it breaks my mathematical heart when I hear students (and teachers!) say they do not like it. Here are some of my favourite website to help get everyone loving a bit of lovely data!

nation master
With Nationmaster you can very quickly and clearly make comparisons between countries using thousands of interesting, up to date statistics. Its ability to then dig out both positive and negative correlations and direct you to related statistics make this website, in my opinion, a must visit before teaching any data handling topics. See my WebWhizz video here

The following Blogs are my favourites. I visit them at least once a week, and where possible I subscribe to them so that I get notified each time a new entry is posted. This is now my Number 1 way of keeping up to date with the latest ideas, resources and developments in teaching. They are fascinating reads by talented, generous people, that give me amazing ideas and save me bucket loads of time. I hope you find them useful as well.

resourceaholic
Jo Morgan shares her experiences in the classroom and documents her favourite resources. Her Maths Gems series in particular is well worth a read.

If you are looking for free maths resources on a variety of topics, then these are good websites to start.

brain-cells
Top quality maths resources for teachers and students. Some are available for purchase, but plenty of free stuff on there as well. TheGCSE Revision resources are definitely worth a visit.

And here are some other websites that didn't fit into the categorises above, but are definitely worth a visit!

wolfram alpha
There are few search engines that can rival Google, but for a mathematician, Wolfram Alpha certainly comes close. It is incredibly powerful and incredibly fun too. Are Craigs more popular than Kates and how often are people talking about mathematics? Wolfram Alpha has the answer. See my WebWhizz video here and check out my Wolfram Alpha Widgets

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Links to the best maths websites in the world! - Mr Barton Maths Blog

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I have decided to move my "links to the best maths websites in the world" page to my blog to make it easier to update. To return to the Teachers section of my main website, just click here .

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Great Resource Bank

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Great tasks you can do in 10-15 minutes every day...click the image and they're yours.

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Math Growth Mindset- Website with resources

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Lots of great ideas and resources

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Helping children to find maths everywhere and every day

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Our mission is to help you to see the maths that you use in your everyday life by opening your Maths Eyes. Unlock Your Maths Eyes.

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Rich maths tasks

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5.NBT.7 slant 5.OA.2 winter solstice HSS-IC.B.4 HSF-IF.C.8.b dollars spent to dollars earned cola per ounce HSN-RN Turkey total cost adding fractions with unlike denominators irrational numbers HSF.TF Starbucks HSF.IF.C.7.B Iditarod 5.MS Data conic surface area

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Complex Instruction - Raising Achievement Through Group Worthy Tasks

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Complex Instruction - Raising Achievement Through Group Worthy Tasks : nrich.maths.org

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How you can be good at math, and other surprising facts about learning | Jo Boaler | TEDxStanford

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How successful school approaches change students' lives- University of Toronto lecture

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Ontario Institute for Studies in Education of the University of Toronto When

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Jo Boaler Professor of Mathematics Education Stanford University

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Beautiful Maths How successful school approaches change students' lives Presenter: Jo Boaler Professor of Mathematics Education Stanford University Founder: youcubed May 8, 2014 Ontario Institute for Studies in Education of the University of Toronto

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UDL Guidelines in Practice: Grade 1 Mathematics

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National Center on Universal Design for Learning

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UDL Guidelines in Practice: Grade 1 Mathematics

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See and Learn the ABC’s of Deeper Instruction

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See and Learn the ABC’s of Deeper Instruction

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Great video of maths multilevel teaching

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By Libby Woodfin
When I first started teaching, back-to-school was always a complicated time. I was excited by the chance to turn the page on a new year and do things better than I had the year before. I was eager to get my supplies and organize my classroom with its freshly buffed floors.

And I found satisfaction in the deep and sustained focus I could give to my curriculum in the summer. I loved greeting my students knowing that I had a month of lessons in my back pocket.

But as soon as those students walked through the door, things always got complicated! Not only were they messy (not caring about my perfect, shiny floors), they often didn't respond in the way I expected to the lessons I had planned.

Their background and skills would sometimes surpass my expectations, but, often, they really struggled. And, always, always they asked me questions and had ideas that took me in unanticipated directions.

The downside of spontaneity

In some ways I loved these moments because their questions and ideas showed that they were engaged, and I enjoyed puzzling through things with them. But getting off track meant that I would often lose sight of the purpose of my lesson.

Unfortunately, rigorous thinking was usually the first thing to get lost in the shuffle. If I had planned for my students to engage in an evidence-based debate, for example, I would sometimes find myself running out of time and skip it in favor of a more straightforward Q and A format.

Spontaneity isn't necessarily a bad thing; in fact, it's an essential part of teaching. How could anyone expect things to go as planned when there are 25 students with 25 sets of needs sitting before us? In reality, bending our knees and changing plans is often one of the only things we can really plan for.

Given this reality, how can we ensure that we don't miss opportunities for deeper instruction that helps students think critically, collaborate, and engage in the kind of scholarly discourse we envisioned for them when we wrote our lessons? These ABCs of deeper instruction may help.

A. Ask Strategic Questions

In many ways, the questions you ask and the questions you answer define what and how students learn in your classroom. For this reason, questions deserve to be planned for strategically so that even if you need to change plans for your lesson as a whole, you can fall back on high-quality questions.

High-quality questions ensure that students move from simple recall to more sophisticated cognitive tasks, like analyzing and evaluating. It may be helpful to think about questions over an arc of lessons rather than just one lesson (e.g., if students are remembering information on Monday, that's okay, but try to get them analyzing information by Wednesday).

Thinking about and evaluating the cognitive rigor of your questions ahead of time will help you avoid asking too many low-level questions in the spur of the moment. Hess' Cognitive Rigor Matrix is a great tool for mapping out your questions.

It may also help you to develop a list of standard replies to student questions that put the cognitive work back on them and can be used in a wide range of circumstances. For example:

"Turn and talk about that question in trios. Let's see if a little conversation can warm us all up and help us find an answer."

"That's an insightful question and we all need to know the answer. I'm going to give all of you five minutes to go back to the text and find evidence that will help you answer it. After five minutes you'll share your ideas with an elbow partner and then we'll discuss it as a whole class."

"I bet there are multiple places where we could find that answer. As a table group discuss what you think is the best resource and be ready to defend your reasoning."

In this video, you can see a math teacher who has mastered the art of strategic questions and encouraging kids to grapple. If they look like 3rd graders - they are! But in my experience, the principles apply across the middle grades.
B. Balance Teacher Talk and Student Talk

Often, when the initial components of a lesson take longer than we think, we revert to doing more teacher talk and leaving less time for student talk. Often we do this to keep on track with pacing requirements, to maintain tight control of behavior, or to more efficiently correct misconceptions.

These are real and important considerations. However, it's possible to be efficient and give students plenty of time to think and do and to engage in quality discourse.

One great way to get students talking to each other, rather than only to you, is to identify a few high-leverage discussion protocols that you can teach them to use reliably. It's important to start with just a few protocols and practice them repeatedly before adding more to your repertoire.

Once students learn their roles and responsibilities during the protocol, and you've given them plenty of time to practice, you will likely find that protocols are an effective and efficient way to get students collaboratively making meaning of text, concepts, or ideas. The predictability of the "rules" of protocols, which demand participation from all students, engages students who may otherwise disengage with their learning.

Protocols can range from very simple and short, to longer lesson-length protocols. "Think-Pair-Share" is a classic short protocol in which students think about (or write about) a question or prompt for a minute or two and then share their thinking with a partner for one or two minutes each. After this warm-up, many more students will be ready to share with the whole class.

"Praise, Question, Suggestion" is an example of a longer protocol students can use to give each other feedback on a piece of work. See Praise, Question, Suggestion in action in this video from a middle school classroom.

C. Challenge Your Students to Grapple with Texts, Problems, and Ideas

Challenge is the heart of deeper instruction. A productive challenge stretches students to go beyond what they may think possible, and this stretch leads to new learning.

Absent a structured protocol or plan, the time for students to grapple with texts, problems, and ideas is hard to protect. It is tempting to think that when we "deliver" content to our students through a lecture, for example, and answer questions about that content, that students will learn more efficiently.

However, giving students time to grapple and make meaning first engages them more deeply and will likely make a lecture more productive and satisfying for teachers and students alike. Students will come to that lecture with questions and insights that are hard to "deliver," and this is what will help their learning last well beyond the lecture.
In this video, you can see the power of allowing students time to grapple - in this case, with a highly technical scientific figure. We've jumped to 11th grade biology class. Once again, grappling is the key to deeper learning, at every grade!
It's important to approach challenges like we see in the video together with students during class, rather than assigning them for homework, so that students can immediately discuss what they are learning and where they are stuck.

Creating a culture of grappling in your classroom and an ethos of playful exploration will help students approach challenges like these with a courageous spirit of curiosity.

____________________

Libby Woodfin, a former teacher and school counselor, is the director of publications for EL Education. She is an author of several books, most recently Learning That Lasts: Challenging, Engaging, and Empowering Students with Deeper Instruction, co-written with Ron Berger and Anne Vilen. She is also a co-author of the much-praised Leaders of Their Own Learning (2014).

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See & Learn the ABC's of Deeper Instruction

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By Libby Woodfin When I first started teaching, back-to-school was always a complicated time. I was excited by the chance to turn the page on a new year and do things better than I had the year before. I was eager to get my supplies and organize my classroom with its freshly buffed floors.

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Teaching Maths- evidence of best practice

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Teaching Maths- evidence of best practice

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Teachers can help parents support their child's maths learning at home. from www.shutterstock.com

In our series, Better Teachers, we'll explore how to improve teacher education in Australia. We'll look at what the evidence says on a range of themes including how to raise the status of the profession and measure and improve teacher quality.
" I'm just so bad at maths! " Too often we hear that claim uttered in fear and frustration.

It's not just students who say this, but also their parents and, in some cases, their teachers, particularly in primary schools.

Research on the topic of maths anxiety is inconclusive and scarce. But there are a few things we know.

A recent study found that "maths anxiety and maths performance can influence one another in a vicious cycle". Research is unclear as to whether poor maths performance triggers maths anxiety, or whether maths anxiety reduces maths performance. It seems likely to be a combination of both negatively reinforcing each other.

Psychologists have found there can be a very real physical response to maths in both adults and children. This includes the release of stress hormones like cortisol, which are characteristic of the fight or flight response.

One study even found that anticipating a maths test activates the brain's "pain matrix " - the regions that might light up if you had injured yourself.

It is an affliction that appears to affect females more than males. Cultural expectations may be to blame - girls are more likely to catch maths anxiety (particularly from female teachers), perhaps because of stereotypes that girls are "not very good" at maths.

What works in maths teaching?

When maths teachers see students struggling, they often give more of the same work, rather than going back and plugging the hole or gap in understanding.

No matter what age the student, before moving on, the teacher should always go back to the preceding skill, or take the concept back to the hands-on concrete phase, until the student has confidently mastered that skill or concept.
Learning the language of maths
It is important to teach children that maths is a language of its own. If students can't speak the language of maths fluently, they don't really understand the fundamental concepts.

Sentence frames, co-operative learning tasks and frequent problem-solving linked to real-world examples ensure students "talk" maths fluently and accurately to gain mastery over the language of maths.

Using a sentence frame such as "A ______ has four sides and four corners" or "one metre is equal to ___ centimetres" during warm-ups or plenary sessions allows students to develop fluency in maths vocabulary and a deeper understanding of the targeted mathematical concepts.
More emphasis on formative assessment
To reduce maths test anxiety, teachers should put more emphasis on formative assessment - on-the-spot monitoring and feedback - rather than relying mainly on summative assessment where a student is assessed at the end of a period of learning.

Giving specific, purposeful and timely feedback to maximise learning opportunities is more effective than missing key learning opportunities waiting for children to fail an end-of-unit assignment.
Mastering maths concepts
Being able to give students many opportunities to practise and master the concrete, pictorial and abstract phases of development for each maths concept is essential for them to learn successfully.
Understand which level each student is at
Some research suggests that girls rely more on manipulatives - using concrete materials to support their problem-solving - and that boys move on more quickly to mental cognitive strategies, such as mental strategies like doubles or bridge to ten.

Regular, informative assessment ensures the teacher has evidence of what each student knows and what they need to learn next when they are ready to move on.
Adapt teaching and resources
Effective maths teachers are able to differentiate their teaching practices, curriculum and resources to ensure all students are accessing the maths curriculum, feeling sufficiently challenged, but not overly anxious, and working to grow their potential in learning maths.

from www.shutterstock.com

Do we need specialist maths teachers in primary school?

In some states, such as New South Wales, there is a move to appoint specialist maths teachers in primary schools.

On the surface, this appears to be a tempting solution to raising student performance in maths, but what is the price to be paid?

The primary school system of the generalist classroom teacher intentionally differs from secondary schooling in order to meet the social and emotional needs of young children, as well as their academic needs.

Generalist teachers take responsibility for developing the whole child. A focus on coaching and feedback for primary teachers to improve their mathematics instruction would ensure the learning of maths remains the responsibility of everyone in a school. It would also combat the myth that some people simply can't do maths.

The federal government has committed A$54 million over four years to the Investing in Science and Maths for a Smarter Future initiative. In 2012, the Office for Learning and Teaching funded five research projects on the teaching of maths and science. These projects were funded in response to recommendations from Ian Chubb, then Australia's chief scientist, in his 2012 report Mathematics, Engineering and Science: in the national interest.

One of these projects focuses on the training of specialist maths and science teachers in primary and secondary schools. Findings from these projects are due to be delivered in 2017.

Confident maths teachers, confident maths students

Not all students will enjoy learning maths. But this is more likely to happen if maths is well taught from early childhood. And the key to confident maths students is confident teachers.

An instructional practices survey at a primary school in Western Australia found that 40% of teachers believed their confidence in teaching maths would benefit from explicit coaching and feedback.

In response to this, the school maths leadership team designed a series of professional learning sessions to build staff confidence; created mentors for other staff members to help reduce their levels of anxiety about teaching maths; and ran workshops for parents every few weeks to inform them of the latest research in maths, the common areas that children are having difficulty with, and provide practical ways that parents can support their children's maths learning at home.

By informing parents about key maths skills children are learning at school, teachers showed parents how everyday home activities, such as cooking and shopping, can help reinforce mathematical thinking.

Parents were made aware of the subliminal messages they might send to their children when they say things like, "It's ok you're no good at maths, because I wasn't either." Parents were then given alternative language to use with their children when they talked about maths.

As a result, teachers and parents at the school have gained confidence in their ability to teach maths and this is reflected in the students' learning and attitude to maths.

The school is analysing a range of data to see how improving students' confidence in learning maths can translate to improvements in maths performance.

Research from the Grattan Institute reinforces the point that teachers and schools need to collect and use evidence of students' learning achievements and progress over time to know what works to improve student learning and to change what doesn't.

Top tips for improving teaching in maths

To improve maths teaching, teachers should do the following:

If they suffer from maths anxiety, don't suffer alone. Identify the sources of anxiety and seek help from mentors and coaches to improve knowledge of maths concepts. Then use this expertise to build a broader repertoire of effective maths teaching strategies.

Demonstrate a growth mindset to students in their attitude towards teaching and learning maths.

Over time, regularly collect and analyse a range of evidence for each student's individual achievement and progress to understand what each student knows and what they are ready to learn next.

Join a professional maths teaching association and/or professional learning community to engage in regular discussion with other teachers about what teaching strategies are working for particular students and what are not.

* This piece was co-authored by Jacki McMahon, teacher at Makybe Rise Primary School and recent winner of a national ChooseMaths 16 teaching award, and Steph McDonald, principal of Makybe Rise Primary School in Western Australia. * Read more articles in the series

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Teaching maths - what does the evidence say actually works?

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In our series, Better Teachers, we'll explore how to improve teacher education in Australia. We'll look at what the evidence says on a range of themes including how to raise the status of the profession and measure and improve teacher quality. " I'm just so bad at maths!

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Using maths to investigate everyday experiences

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Using maths to investigate everyday experiences

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Fun maths mind benders

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This week's #edublogsclub prompt 12 asks us to embed something into a post.

No problems!

One of my favourite videos that I use in many of my workshops to do with differentiation and non-funded special needs is Ma & Pa Kettle's most convincing lesson in mathematics.

The problem: How many times does 5 go into 25?

I bet you said 5 ...

Can you think of any other numbers that can be manipulated to do the same?

No?

How about

Watch the video below. Costello (Albert and Costello fame) is pretty convincing...

Why are they so convincing?

I've used the Ma & Pa video in the past and asked students to convince me otherwise.

I would also imagine an open ended problem solving session where students explore the possibility that there may be other numbers for which we can present a similar argument.

How could you use these videos in your classes?

Feel free to share your ideas below.

Thanks for reading 🙂

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Ma & Pa Kettle and other mathematical dilemmas

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This week's #edublogsclub prompt 12 asks us to embed something into a post. No problems! One of my favourite videos that I use in many of my workshops to do with differentiation and non-funded special needs is Ma & Pa Kettle's most convincing lesson in mathematics.

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Full Class? Small Group? How Do You Decide?Posted on December 3, 2016

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Full Class? Small Group? How Do You Decide?Posted on December 3, 2016

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Blog reflection by Aviva

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Yesterday, we had a wonderful PA Day focusing on math. We started the day by watching this fantastic TEDx Talk by Jo Boaler all about math and the value of a growth mindset.

It's a talk that I think all educators, parents, and students need to hear to help change some perceptions around math. I thought of this talk a lot as we then moved into small group sessions that all focused on creating better math classrooms.

One of our sessions was on number talks. Please note here that I'm a big fan of number talks. I've used them in different ways in multiple grades and with much success. I think they help build students' understanding of numbers and belief that there's more than one way to solve problems. Compared to other times that I've participated in sessions on number talks, I was now doing so through the lens of a Kindergarten teacher with a new program document that emphasizes the importance of teaching literacy and math skills through play and reducing the time for full class instruction. Where do number talks fit into this?

During this session, we watched two Kindergarten number talks. The discussion around recognizing groups of up to 10 was fantastic, but both talks were done with the full class, on the carpet, for almost 10 minutes, and totally removed from the context of play. Here is what I observed as I watched the video clips.

Many students recognized the number amounts and could explain their thinking.

Some students determined the total in more than one way.

While many students were highly engaged in the lesson, some students were not involved at all.

It's this final point that really has me thinking. Instead of pulling the full class for these number talks, I wonder about doing them in small groups. While I'm sure that some students benefited from watching and listening to others participate, I wonder about those students for which this was way too difficult - past the zone of proximal development. What if they were linked to some of the math learning that's happening through play and occurred around the time that these math skills were demonstrated? What if they aligned with where each child is at and what each child needs to move forward? ? I also wonder about those students that have already mastered these targeted skills and may need to move onto something else. What was the value for them? What did they get out of this activity

I can't help but think again about Boaler's TEDx Talk. If we want to create entry points for all students, how are we doing so with this task? I also think about two key ideas in our new program document:

noticing and naming the learning.

constantly questioning, "why this learning, for this child, at this time?"

If we start by observing this math behaviour through play, naming it for children, and then extending it with discussions around subitizing skills, would this benefit children more? If we are really asking ourselves, "why this learning, for this child, at this time?," would we choose a full class lesson as the best way to meet various needs or would we look at small group options instead? During this Number Talks session, we had some discussions around the questions that I'm raising here, and as expected, people expressed many different viewpoints.

While I raise these points from a Kindergarten perspective, I wonder if they apply to just Kindergarten. This morning, I met an acquaintance of mine for brunch, and she started to talk to me about "small group instruction." She said that many educators she knows don't pull small groups because they "don't have time." This is when I asked, "what if we replaced full class lessons with small group instruction? Would this better benefit kids?" I can't help but think about one of my favourite blog posts by Kristi Keery-Bishop, where she discusses the value in letting things go: could some full class instruction be one of these things? As I continue to reflect on my teaching practices, I'm curious to hear what others do and what they think.

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Full Class? Small Group? How Do You Decide?

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Yesterday, we had a wonderful PA Day focusing on math. We started the day by watching this fantastic TEDx Talk by Jo Boaler all about math and the value of a growth mindset. It's a talk that I think all educators, parents, and students need to hear to help change some perceptions around math.