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Videos, Practice Questions and Textbook Exercises on every Secondary Maths topic

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Welcome to Goal Free Problems, a place to find and share goal free problems for use with pupils in class and revision

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SSDD stands for Same Surface, Different Depth. They are a special set of problems that may look the same at first glance, but require different mathematical ideas to solve them.

SSDD Problems have become one of my key strategies to use with my students to help them become better problem solvers.

You can search for different topics using the tags on the right-hand side (or below if using a mobile device), and you can find out more about SSDD Problems on this page.

My books

I have written two books about my use of these types of activities.

My 2020 book, Reflect, Expect, Check, Explain: Sequences and behaviour to enable mathematical thinking in the classroom, goes into detail about making the most of these activities in the classroom. You can order the book from Amazon, John Catt, or wherever you get your books.

My 2023 book, Tips for Teachers: 400+ ideas to improve your teaching, details my most recent thoughts on using SSDD Problems effectively. You can order the book from Amazon, John Catt, or wherever you get your books.

My online courses

I have an online course store which offers low-cost, high-quality CPD on-demand. Each course contains short videos, resources, research, activities and questions to reflect upon. The courses most suited to the questions on the SSDD Problems website are:

Supporting this website

I do not make any money from this website, so if you did want to support its future development - and hopefully learn more about the pedagogy underpinning these ideas - then I would be so grateful if you would consider:

Thanks so much for taking the time to read this - now go and enjoy the site!
Craig

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Same Surface, Different Deep Structure maths problems from Craig Barton @mrbartonmaths

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This website is my attempt to assemble a collection of high-quality, sequences of questions and examples using key principles from Variation Theory. By holding key features between questions and examples constant, together with the mathematical behaviour that I call Reflect, Expect, Check, Explain and the supporting role of the teacher, I believe we can help students better understand key mathematical ideas.

You can find out more about the site and the different types of sequences available by visiting the Introduction.

You can access all the sequences of questions and examples from the Topics menu above.

My 2020 book, Reflect, Expect, Check, Explain: Sequences and behaviour to enable mathematical thinking in the classroom, goes into detail about making the most of these activities in the classroom. You can order the book from Amazon, John Catt, or wherever you get your books.

My 2023 book, Tips for Teachers: 400+ ideas to improve your teaching, details my most recent thoughts on using these sequences effectively. You can order the book from Amazon, John Catt, or wherever you get your books.

I have an online course store which offers low-cost, high-quality CPD on-demand. Each course contains short videos, resources, research, activities and questions to reflect upon. The courses most suited to the sequencers on the Variation Theory website are:

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Sequences and behaviour to enable mathematical thinking in the classroom - by Craig Barton @mrbartonmaths

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Since I started teaching three years ago, I've been well aware of the problems of traditional textbook exercises when introducing a class to a new concept, procedure. More often than not, textbook exercises became far too varied too quickly, with pupils becoming quickly unstuck, and it became difficult to identify the source of their confusion. Alternatively, it felt as though I was presenting students with 'just more practice of the same', which was deeply unsatisfactory (regardless of the prior attainment of the pupils).

Hence (and I am hugely indebted to Mr Barton and his book), creating sequences of intelligent, minimally varied practice questions seemed a reasonable step. When they are used to introduce pupils to a topic, they enable pupils to make connections and really focus on the underlying concept. The ones presented here are very much initial attempts - they are not a perfect model of how to write minimally varied exercises. All the same, I hope you find them of some use!

I am contactable at fortyninecubed (at) gmail . com, or on Twitter at @fortyninecubed.

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Since I started teaching three years ago, I've been well aware of the problems of traditional textbook exercises when introducing a class to a new concept, procedure. More often than not, textbook exercises became far too varied too quickly, with pupils becoming quickly unstuck, and it became difficult to identify the source of their confusion....

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Unfortunately, my father Don Steward passed away from Covid-19 on 3rd May 2020.

Throughout his working life he was passionate about improving the teaching of mathematics and has requested that all of his work remains freely available.

As per his wishes, I have uploaded all of his work to this Blog in the pages section which contains links to google drive where slides and presentations can be downloaded for free.

Don has also requested that those who use his work make a discretionary donation to charities supporting education in Africa.

We have set up a Justgiving fundraising page to support this wish below:
https://www.justgiving.com/fundraising/jessesteward

Many thanks for all of the kind messages we have received which are of great comfort to myself and the family.

Kind regards,

Jess

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ideas and resources for mathematics teachers of 10 to 16 year olds

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Maths Venns is my collection of Venn Diagram activities covering many mathematical ideas. Students must think hard to find examples that meet different combinations of criteria in order to complete the diagram. Are some regions impossible to fill, or have students simply not found an example yet?

You can search for different topics using the Topics menus above, and you can find out more about Maths Venns on this page.

My 2020 book, Reflect, Expect, Check, Explain: Sequences and behaviour to enable mathematical thinking in the classroom, goes into detail about making the most of these activities in the classroom. You can order the book from Amazon, John Catt, or wherever you get your books.

My 2023 book, Tips for Teachers: 400+ ideas to improve your teaching, details my most recent thoughts on using these sequences effectively. You can order the book from Amazon, John Catt, or wherever you get your books.

I have an online course store which offers low-cost, high-quality CPD on-demand. Each course contains short videos, resources, research, activities and questions to reflect upon. The courses most suited to the activities on the Maths Venns website are:

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Rich Venn Diagram maths activities from Craig Barton @mrbartonmaths

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SSDD stands for Same Surface, Different Depth. They are a special set of problems that may look the same at first glance, but require different mathematical ideas to solve them.

SSDD Problems have become one of my key strategies to use with my students to help them become better problem solvers.

You can search for different topics using the tags on the right-hand side (or below if using a mobile device), and you can find out more about SSDD Problems on this page.

My books

I have written two books about my use of these types of activities.

My 2020 book, Reflect, Expect, Check, Explain: Sequences and behaviour to enable mathematical thinking in the classroom, goes into detail about making the most of these activities in the classroom. You can order the book from Amazon, John Catt, or wherever you get your books.

My 2023 book, Tips for Teachers: 400+ ideas to improve your teaching, details my most recent thoughts on using SSDD Problems effectively. You can order the book from Amazon, John Catt, or wherever you get your books.

My online courses

I have an online course store which offers low-cost, high-quality CPD on-demand. Each course contains short videos, resources, research, activities and questions to reflect upon. The courses most suited to the questions on the SSDD Problems website are:

Supporting this website

I do not make any money from this website, so if you did want to support its future development - and hopefully learn more about the pedagogy underpinning these ideas - then I would be so grateful if you would consider:

Thanks so much for taking the time to read this - now go and enjoy the site!
Craig

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Same Surface, Different Deep Structure maths problems from Craig Barton @mrbartonmaths

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I am an experienced maths teacher at an inner-city secondary school. You are free to use anything shared in your teaching and as a part of your department. If you do, and enjoy it a lot, please let…

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CHALLENGING MATH PROBLEMS WORTH SOLVING DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL Get Our Favorite Problems Take The Online Workshop WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS? HERE’S OUR GROWING COLLECTION Get Google Slide Versions WANT TO SHARE OPEN MIDDLE WITH OTHERS? CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK Elementary Version

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Article by Colin Foster

The NRICH website is full oflovely rich tasks, attractively packaged, all ready to be used. And yet often when I go into mathematics classrooms, I see students doing dull, repetitive exercises. Why does this happen, when all these lovely NRICH problems are freely available, just a click away?

It seems that there is a battle going on in mathematics classrooms between the kind oflovely rich tasks that you see on the NRICH site (and in lots of other places too, of course) and what I refer to as tedious exercises.

Here's an example of what I mean by each:

You could easily imagine a page in a textbook containing tedious exercises like this:

1. What are the factors of 6?

2. What are the factors of 7?

3. What are the factors of 20?

and so on.

I call thesetedious exercises because there is no bigger picture here. You work your way through these, but you could stop at any point and it wouldn't matter. The textbook author perhaps decides how many to put in according to how much space they have available on the page. A student could do every other one; or just do the last one. You just slog through them; the answers don't matter - when you finish one you just move on to the next one until the time is up.

Contrast that with what I calllovely rich tasks, such as:

Find some numbers with exactly 4 factors.

I like this task very much.[1] It is wide open (there is certainly more than one number with exactly 4 factors), which means that different students can give different correct responses. It feels like a beginning rather than an end. As I accumulate numbers with 4 factors, I begin to ask, " Why do these numbers have 4 factors? What makes that happen?" If I find numbers with other numbers of factors, that's almost as useful, really, because the 4 is clearly arbitrary. The real question behind the question is "What determines how many factors a number has?" Exploring the number of factors that numbers have is a lovely investigation, which I have written about elsewhere (Foster, 2017). Unlike the tedious exercises, which might be too easy for some students, and consequently boring, this task can be mathematically interesting for almost anyone, and I have used it with students of all ages from 11 to 18, as well as with student teachers and experienced teachers. Everyone can get something out of it. Can you write down a number with exactly 30 factors? Can you find the smallest such number? You can go on and on posing and solving related problems.

Clearly, both of these tasks (thetedious exercises and the lovely rich task) have something to do with factors, and both involve working out factors of numbers quite a bit, but could they be in any sense alternatives to each other? Could I simply throw out tedious exercises, which I don't like, and replace them withlovely rich tasks? Will lovely rich tasks do the same job, or do we unfortunately need those tedious exercises? Do tedious exercises perhaps have a necessary place?

Mathematics teachers often think thatlovely rich tasks are all very well, and in an ideal world you would spend lots of time on them, but in the real world we have assessments! And we all know that the easiest thing to test on an examination is procedural fluency on short, closed questions. So, since they dominate high-stakes assessments, tedious exercises are bound to dominate in the classroom. Unless we prepare our students well for those high-stakes assessments, we do them a disservice. And presumably the only way to get good at doing tedious questions is to spend a lot of time practising tedious questions. So, for many teachers, they seem to be a necessary evil.

I recently carried out some research to explore whether this is true or not (Foster, under review). For some time, I have been working on devising what I call mathematical etudes (Foster, 2013, 2014). These are intended to be not just lovely rich tasks, but lovely rich tasks that force students' attention onto a particular, important mathematical procedure. Soetudes don't allow students a choice of method. Allowing students to choose what method they use is often a great idea, but if you want them to develop their skill at a specific technique, then you have to force them to use that technique, otherwise they can end up unwittingly avoiding their areas of weakness by (perhaps cleverly) solving the problem some other way than you intended. So,mathematical etudes are designed with a specific procedure in mind. But, rather than simply asking students to perform that procedure repeatedly (as with tedious exercises), in an etude there is some wider problem-solving goal - something a bit more interesting is going on at the same time!

You can see two examples ofmathematical etudes on the NRICH website, Almost One and The Simple Life, and there are no doubt many more NRICH tasks (and, of course, in other places) which follow the same kind of design, and which could, if you like, be called etudes.

Etudes are intended to be much more interesting to do than tedious exercises, but they entail at least as much repetition of the essential technique. So students should gain as much (if not more) practice by doing them. But, at the same time, students have to think about what they are doing, perhaps unpick the procedure to achieve a desired goal, perhaps think backwards, perhaps modify what they do this time in response to what happened previously. All of this means that theetude can operate at multiple levels of challenge. The answers obtained to the procedure matter for some wider purpose, and so students are more likely to carry out the procedure carefully and check what they have done. Wrong answers are more likely to be challenged by a peer if they matter for some wider purpose. And having something else to focus on beyond the procedure should help to promote the desired automaticity of carrying it out.

Several schools recently very kindly trialled some of theetudes that I have been developing, using etudes with one class and traditional exercises with a parallel class. The data showed that the etudes were no worse than the exercises for developing students' procedural fluency. Given that result, why would anyone usetedious exercises? One reason might be that they don't have ready access to lots of nice etudes! I like to see that as a challenge for the mathematics education community: if we were to work together, could we develop a plentiful supply of etudes - and say goodbye to tedious exercises forever?

[ 1] I feel free to say how nice this task is because, although I've used it a lot, I certainly didn't invent it. I've no idea where I first came across it. Maybe it's as old as numbers? I'm pretty sure that no one could claim that anyone in particular "invented it"!

References:

Foster, C. (2013). Mathematical études: Embedding opportunities for developing procedural fluency within rich mathematical contexts. International Journal of Mathematical Education in Science and Technology, 44(5), 765-774.
Available open access at: https://doi.org/10.1080/0020739X.2013.770089

Foster, C. (2014). Mathematical fluency without drill and practice. Mathematics Teaching, 240, 5-7.

Foster, C. (2017). Thirty factors. In Questions pupils ask (pp. 127-132). Mathematical Association. Available from: http://members.m-a.org.uk/Shop/product/1114

Foster, C. (2017). Developing mathematical fluency: Comparing exercises and rich tasks. Educational Studies in Mathematics, online first.
Available open access at: https://doi.org/10.1007/s10649-017-9788-x

Colin Foster is an Assistant Professor in mathematics education in the School of Education at the University of Nottingham. He has written many books and articles for mathematics teachers (see www.foster77.co.uk).

The Mathematical Etudes Project lives atwww.mathematicaletudes.com.

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This article by Colin Foster presents the idea of mathematical etudes as a way to develop fluency without tedium.