Mathematics Education

Everyone loves a puzzle, don’t they?!

One of the pulls that drew me to maths is the satisfaction that comes from solving problems, that, and the fact it is fun!

In the May 2024 edition of the maths and stats (M&S) student newsletter, OpenInterval, there were two puzzles posed, sent in by one of our M&S associate lecturers, Bob Vertes:

Here’s my approach to Puzzle 1.

The use of different letters for the digits means 4 different digits, moreover single digits. This narrowed down my possibilities to the ten digits, 0 to 9.  Reflecting now, there are still a large number of options – over 5000 ways (permutations) of selecting 4 different digits from 10:

More information definitely required to reduce the possibilities!

I figured I would just start simple and try some values to see what happened.

Notice my use of the word ‘figured’. Figuring in this sense means to think or consider but can also mean to calculate; perhaps my choice of language is influenced by the fact I am solving a maths problem.  I also use the word ‘see’ – I am looking at the problem and noticing what happens when I try something; it is important to pay attention to what is happening to make some sort of structured progress, rather than just haphazardly working through a myriad combination of digits. This can be seen as an example of “Noticing Structure”, a module idea developed in our ‘Learning and Doing Algebra’ (ME322) module.

I began by summing the units column, just as taught when learning column addition.  This led me to think about extra digits to consider, those that get ‘carried over’. For the thousands digit in the answer to be a 2, I would need a 1 carried over to combine with my existing 1.  Alternatively, ‘A’ would need to be 2 and nothing carried over from the hundreds.  ‘A’ could not be larger than 2, possibilities reduced by 80% in one move!

Note: automatic assumption I seem to have made without realising, ‘A’ cannot be 0.

I moved my attention to the units column; what would happen if ‘A’ was 2?

Look at the hundreds column. If ‘A’ is 2 then A + B + ‘anything carried’ would need to end in a zero but this would force a carry over into the thousands which is not wanted.

‘A’ cannot be 2.

Confirmation, ‘A’ should be 1.

Possibilities halved.

For the thousands digit in the answer to be 2, another 1 carried from the hundreds column is therefore needed. At this point I merrily proceeded to find that ‘B’ is 9, only for my confidence to be shaken at a later stage.

With my certainty shaken, I continue carefully.

If I can fix ‘B’, my remaining possibilities will be reduced significantly.

A final box confirms I am certain with my solution but spot the little doubt that has crept in with my question.

Mathematics is about confidence, but it is also about questioning and analysing and unpicking; all mathematical thinking.

As a maths teacher for many years, I am only too aware of and, always intrigued with, the many different methods pupils use to approach and solve problems.  Whilst we can demonstrate particular ways to solve a problem, there is usually no single correct way.  By looking at the approaches of others, we develop our own understanding of maths and its interconnectedness making links between our existing knowledge and knowledge, methods and ideas that may be new to us.

The next time you see a puzzle that needs solving, rather than rushing to get to the answer, take time to notice the steps you take along the way.

Now for Puzzle 2…

Written By Dan Rust, Student Research Bursaries and EPSRC Internship Coordinator, and Ethan Shallcross, OU Research Bursary student.

A version of this article will also be available in the OU Open Interval newsletter.

Every summer,  the Open University (OU) school of Mathematics and Statistics hosts research placements for undergraduate students from across the world, giving them the opportunity to contribute to contemporary mathematics research. Projects are supervised by one or more experienced researchers in our school, covering topics in pure and applied mathematics, statistics and the history of mathematics. These include our own research bursary scheme and EPSRC DTP Vacation Internships.

This year, we had more students than we’ve ever had before, with a total of 10 undergraduates conducting research in a range of areas including: translating ancient Arabic mathematical texts; understanding the combinatorics of swarms of robots on a network; developing games that teach environmental dynamics; enumerating algebraic objects called ‘friezes’; identifying election fraud in the 2024 Venezuelan presidential election; and statistically analysing air quality data.

While most students met with their supervisors online, having weekly video chats and exchanging regular emails, some were able to visit the OU Milton Keynes campus at Walton Hall to work with their supervisors in person.

At the end of August, we celebrated a successful summer of research by hosting a hybrid event where many of the student researchers gave short presentations about their projects. It really was a joy to see the huge variety of activity that had taken part over just a short couple of months and the school couldn’t be prouder of each of the students that took part.

A great deal of work went into these placements. The students themselves showed incredible resilience in working on challenging problems, most of whom had no previous experience of research, and our school supervisors worked extremely hard to introduce their student researchers to the world of academic research in mathematics, statistics and history, helping them to achieve feats that include a variety of academic papers, talks and posters at international conferences, the awarding of external grants, blog posts for scholarly societies and more.

Below is an account from one of our 2024 students:

Ethan Shallcross 

Last December, I gratefully received one of the OU’s student research bursaries to work on a project which can be summarised as follows. Imagine stationing robots on a subset of the nodes in a network, which are permitted to move to adjacent nodes over a sequence of timesteps. You could think of the robots as delivery robots (like those found in Milton Keynes for delivering food) and the nodes as delivery locations. We were investigating the maximum number of robots able to visit all the nodes, whilst requiring that they can always communicate freely. We considered different variants of this problem such as requiring each node to be visited by each robot, allowing all the robots to move at each timestep, and altering the conditions needed for free communication.

Robots exploring a network when using one possible definition of ‘free communication’ 

As you might expect, I explored lots of fascinating new mathematical ideas. But I also improved my ability to communicate effectively. My supervisor provided helpful advice about writing proofs formally, clearly, and concisely. I enjoyed discussing my work with others during online video calls as well as through emails and texts. Whilst attending meetings with the combinatorics research group, I learned about other people’s work and had the chance to present my findings. I had the pleasure of working with mathematicians from both within the OU and other universities in Indonesia and Slovenia. I am thankful to have had the opportunity to meet and collaborate with so many people.

Over the course of the 8 weeks, I really got a feel for what doing research is like. This even included assisting with peer-reviewing a paper before its publication – an unexpected but very insightful opportunity. I liked learning about the new results and thinking critically about the proofs to provide feedback for the authors.

I experienced the lows of finding an issue with a construction at the final check, but also the highs of solving an open problem from the literature and proving new results. I gave a talk about these results at The International Conference on Graph Theory and Information Security VI – a great experience. The work will hopefully form part of a paper to be published, which I think is very exciting! But the hours spent toiling with different problems were interesting and fulfilling, regardless of the outcome.

A screenshot from a computer program that I wrote during the project.

I thoroughly enjoyed my research bursary experience and have learned a lot. It has confirmed to me that I would like to pursue a PhD after I finish my MSc next year. I would urge any student at the OU thinking of a career in mathematical research to apply for the next round. Finally, I would like to thank James Tuite for his advice and support, all the researchers and students that I have worked with, as well as the School of Mathematics and Statistics for the opportunity.

If you are an undergraduate student and are interested in taking part in a summer research project in 2025, keep an eye out for announcements and updates on the following pages:

Mathematics & Statistics Summer Research Bursaries – https://www5.open.ac.uk/stem/mathematics-and-statistics/research/student-research-bursaries (application deadline approx. December 2024 – OU students only)

EPSRC DTP Vacation Internships – https://stem.open.ac.uk/research/research-degrees/epsrc-dtp-doctoral-training-partnership/epsrc-dtp-vacation-internships (application deadline approx. March 2025 – open to external students)

The “Learning from Improvers” project was an in-depth study of 12 Open University undergraduate students who were identified as ‘improvers’ on mathematics education modules. These students were interviewed about the difficulties they experienced and supportive strategies they used while learning to write reflectively about mathematical activity as part of their module assessments. The findings from the project have already been incorporated into the production for two new modules, resulting in a new integrated approach to assessment, including new assessment designs, additional teaching about writing, and modelling ways of analytical thinking.

You can read the full report of this project on the eSTEeM website here.

*Note: the names in this blog (and full report) are pseudonyms and not the real names of the student participants.

Why did we carry out this study?

Mathematics Education (ME) modules involve reflective writing, a type of writing which applies academic analysis to personal practice. Writing for ME modules therefore requires a different approach to writing for either straight mathematics or humanities and arts subjects, as it involves combining specific inter-disciplinary elements. Students entering our level 3 modules, many with mathematical rather than education backgrounds, need to develop these ways of thinking and writing to succeed in their assignments. We know from speaking to students and tutors on our modules that students find reflective writing particularly challenging and we have long been interested in learning how to help them to develop and improve this skill.

We reasoned that our support for students in this new approach to writing needed to be strengthened and felt that the most relevant experiences to understand were not necessarily those of the most successful ME students, but those who had improved their marks during a ME module. So we set out to investigate what it was that helped those students to improve.

What did we find out?

Through interviewing our student improvers, we gained some insight into what it is like to be a student on our ME modules. Thematic analysis of the interview transcripts revealed some of the key difficulties faced by these students whilst working through ME modules, and the approaches they found most beneficial in their improvement in written assignments.

Difficulties

Through our analysis we identified three key themes in the difficulties reported by students in our sample:

• difficulty with mathematics
• difficulty with writing
• difficulty with the unfamiliarity of the module and assignments

Support 

Four key themes were identified as being supportive to student improvers:

• personalised support
• alternative explanations
• supportive ways of working
• support from module resources.

What does this mean for our students?

We have used the insights gained from interviewing the student improvers to help make changes to our new ME modules to support all students in improving their reflective writing. This has included:

• developing an integrated approach to assignments,
• narrowing the focus of early assignments,
• explicitly teaching about writing,
• modelling reflective ways of thinking.

The major change to module content was integrating assessment activities as part of study. Our two new ME modules (ME321 and ME322) each have a detailed activity planner and time for preparing assignments is built into this. Drawing on the improvers’ emphasis on ‘having two goes’ at assignments, preparation activities for each assignment start 4-6 weeks before deadlines and each question has scheduled time for starting and separate time for finalising. Reading feedback from assignments is also a scheduled activity before each new assignment, again considering the improvers view that reading and re-reading personalised feedback was a supportive way of working.

Improvers had expressed the value of having a summary of module ideas to refer to when planning and writing. The new modules therefore include an interactive ‘Module ideas map’ which allows students to find a short definition of each idea and where it is first mentioned.

Some changes have been made to module assignments to narrow the focus. For example: for the first assignment in ME322 students only write about their own mathematical work, and do not analyse learners’ work until the second assignment. The idea is to build up the level of challenge along the module, allowing time for development of understanding of module ideas and of a reflective voice.

In response to students reported unfamiliarity with writing analytically and reflectively about mathematics, new content has been written for each module, giving explicit advice about writing, including reading about the difference between writing types and structuring paragraphs.

The last findings that directly informed module design were the universal request for sample assignments, the critique of existing videos, and the improvement associated with seeing the value of the process. In addition to written resources about writing reflectively and analytically, other resources have been developed to support students’ writing through modelling ways of thinking. These include videos of school-aged children working on mathematics ‘live’ and interactive forum activities. These allow students to develop their analytic approach by providing examples close to their own experience of working with learners, but with the added value that these experiences can be shared with other students and discussed openly on the forum. These forum activities also allow for students to give peer feedback and to learn from other perspectives than their own, or their tutors. In Learning and Doing Geometry, students are asked to take part in such forum activity as part of their first assignment, increasing the participation in the overall cohort.

For us, the value of the Learning from Improvers project has been in using students’ knowledge to inform and improve future teaching.

We are lucky to be able to introduce two new tutors joining our Mathematics Education team as tutors this academic year, and this feels like a good opportunity to reintroduce you to the other tutors who are currently working on ME620, ME321 and ME322. No doubt you will get to know them yourself through tutorials and forum discussions.

Luke Bacon 

I’ve been teaching with the OU for just over two years on other Maths modules, and am really pleased to be joining the team on ME322 this year.

I studied Pure Maths and then completed my PGCE with the OU, before working in and around schools across London and managing a series of university outreach programmes for students from disadvantaged backgrounds. I spent some time at DfE working on Higher Education policy (ask me about the student finance system!) and have worked with lots of teachers on maths-specific CPD projects, as well as teaching on a specialist STEM PGCE course.

I also work in consulting, helping to make public services (education, healthcare, and others) a little bit better for everyone, and in my ‘spare’ time I write and examine A-level Maths and Further Maths.

Kellee Patterson

I am lucky enough to have had the privilege to teach mathematics to 11-18 year olds for a quarter of a century leaving the secondary classroom in 2022.   Teaching mathematics is both fascinating and frustrating but, most of all, fun.  Fun and enjoyment was the reason I became a maths teacher with my main goal being to make maths accessible to students by helping them see the beauty and relevance of mathematics.   

In addition to maths leadership in my teaching career, I worked with trainee and new teachers as the whole school lead for Initial Teacher Training (ITT) and Early Career Teachers (ECT) (was Newly Qualified Teachers (NQT)).  Working with those at the beginning of their career is a joy and it is a real pleasure to see so many enthusiastic and determined people achieve success as they start out in teaching. 

My undergraduate maths degree and PGCE are from the University of Exeter and then, whilst teaching, I gained a masters in education from the University of Winchester.  Working full-time whilst also studying is quite the juggling act particularly given the term time working intensity as a teacher.  I take my hat off to those of you studying whilst working. 

I now teach as an Associate Lecturer (AL) for the OU on MU123 Discovering mathematics, M140 Introducing statistics, M248 Analysing data and recently ME321 Learning and doing geometry.  I came to the OU through studying and started taking modules in 2018 before becoming an AL in 2020.  My original goal of making maths accessible to all has not changed and I hope, through my work with the OU, I can continue to encourage others to enjoy mathematics and statistics. 

When I am not doing maths, I enjoy travelling, cooking and walking (I am a Duke of Edinburgh Leader and still help with expeditions at my previous school).  I enjoy learning languages and am currently trying to teach myself Spanish.  Recently, I bought myself a keyboard in a bid to try and rekindle my piano playing skills from when I was a child – this one is definitely a work in progress that requires much more practice!  

Barbara Allen

My school teaching career was in middle schools in Worcestershire where I specialised in mathematics. I developed an interest in girls’ attitudes to mathematics and that became the focus of the dissertation for my MEd. My PhD focuses on Pupils’ Perceptions of Mathematics Classrooms and found the ways that pupils think they learn most effectively.

In 1994, I moved to the Open University as a Research Fellow and in 2000 I became the Director of the Centre for Mathematics Education. I continued as the Lead Academic for Mathematics Education until my retirement in May 2017.

I have written on a large number of OU modules from Access to Masters Level. For some reason, I always ended up writing the sections on fractions!

I am the co-author of the children’s book series The Spark Files and the writer of the children’s radio series The Mudds starring Bernard Cribbins and Mark Benton. Now available on iTunes!!

My main hobby has always been playing the clarinet. I play in Bewdley Concert Band and also play alto saxophone in the Wyre Forest Big Band. Now that I am retired, I am learning to play the xylophone and threatening to learn the drums. I also volunteer at Bewdley Museum and work with school groups that are learning about WWII.

Thabit Al-Murani

Hello, I am Thabit and I am an associate lecturer teaching on the ME322 course.

I have been involved in maths education for 25 years. Over this time, I have been a teacher, head of department, researcher, and more recently I started my own freelance business offering maths education consulting and specialised tutoring. My work has offered me the opportunity to live and work in several countries including the US, Sweden, Australia, Malaysia, and the UK.

I have a DPhil in Mathematics Education and my research interests are variation theory, the teaching and learning of algebra, and SEN mathematics education.

Ian Andrews

Hello, I’m Ian and I joined the OU as an Associate Lecturer in 2022.

I studied Maths and Statistics at university and once completed, I worked in data analysis in London for several years. Wanting a change of career, I started a PGCE in 2007 and have been a maths teacher ever since!

In 2015 I joined an 11-18 comprehensive secondary school in West Sussex as Head of Maths and member of the senior leadership team. I have also worked with the Sussex mathshub and lots of locality schools to improve maths in Sussex.

In 2019 I joined AQA (alongside my full time head of maths role) as Chair of Examiners for GCSE maths, GCSE statistics and L2 Further maths. This work involves running training and events for AQA and helping set grade boundaries for their exams.

I really enjoyed learning again when I took my PGCE and in 2013 I completed an MA in Education studies. Engaging with academic research whilst being a teacher has been hugely beneficial to my practice and my department.

When not teaching I enjoy music, playing football, running and acting as a taxi service for my 2 children.

Nick Constantine

Hello everyone, I am Nick Constantine associate lecturer for the Open University, I tutor on ME620, ME321 and ME322.  I also tutor on MU123 and MST124. I have been working with the OU for 9 years but I also used to tutor on the PGCE course from 2000-2002. I have had a very mixed career. My first degree was Astronomy and Astrophysics at Newcastle University, I then joined the Royal Navy as aircrew for a little while. I left way back in 1989 and did several ‘gap’ jobs before retraining as a Mathematics teacher and PE teacher.

My teaching career followed the standard path up to Deputy Head/acting Head but I always attended many mathematics training weekends with the ATM.  I also used to attend the MEI further mathematics conference in Nottingham for a few years. I was a Head of Mathematics in a 13-19 high school in Northumberland from 1998-2004 and enjoyed the process of organising and planning activities that reflected the fundamental philosophy of the OU ME(x) modules.

I also studied for a Master’s in Education from 2000-2002, one of my dissertations was ‘conjecture and proof in the most able’ (ME822), I really enjoyed designing my own research project and had a wonderful class to try some tasks with.  For me, if you can change the language of the mathematics in the classroom from a didactic controlling language to an atmosphere of questioning, conjecturing with learners and investigating relationships then you are really at the top of your game!

I now work part time as a teacher and OU lecturer.  I also work as a running and yoga coach and operate a small business where I organise retreats and workshops for private groups in Europe and in Scotland. Other hobbies are reading, radio 6 music, cooking and doing Maths problems!

Tom Cowan

I have been an Associate Lecturer with the OU since 2008 when I worked on the MEXR624 summer school each year in Bath.  When that ended I was lucky enough to be offered a chance to work on an earlier maths education module. I currently tutor on all the modules which we offer in the Mathematics Education suite of modules at Level 3 (ME620, ME321 and ME322) and also E209 – Developing subject knowledge for the primary years.

I completed my Master’s Degree with the OU in 2010 so remember what it was like to study at a distance and cramming in study whilst juggling other things in life.

My full-time role is as the Programme Lead of an initial teacher education programme at the University of Plymouth. I support the education and development of new Primary teachers on the BEd and PGCE – looking after those students with a specialism in mathematics.  Prior to this I was primarily involved with working with Secondary and Primary schools to support them with mathematics in challenging inner-city schools around Manchester and Salford.

I’ve never really left education and have found my next challenge in aiming to complete my Doctorate in Education in 2026! Hopefully I’m well on my way to becoming Dr. Cowan!

When I have some spare time, I enjoy going to the theatre, supporting Liverpool FC and Widnes RLFC and work as an officer with the Boy’s Brigade which keeps me in touch with further voluntary work (I did say spare time right?)

Jeffrey Goodwin

As well as being an Associate Lecturer for ME321, I also tutor on ME620. I first worked as a tutor for the OU in the 1980s on EM235 Developing Mathematical Thinking and returned to my current role of Association Lecturer in September 2014.

I was a classroom teacher for 10 years, working in Secondary and Middle schools. I moved into the advisory service in 1980 as Head of the Hertfordshire Mathematics Centre. I worked in Initial Teacher Training and running CPD courses for teachers; being head of mathematics education at Anglia Polytechnic University. For four years from 1986, I worked for the National Curriculum Council on a curriculum development project: Primary Initiatives in Mathematics Education (PrIME). I have always had an interest in assessment and testing and in 1998 was appointed to establish and lead the Mathematics Test Development Team at the QCA. We developed the end of key stage tests and other optional tests for all three key stages. I was in this role for eight years and then moved to become Head of Research at Edexcel and then Pearson Research and Assessment. In 2010, I became an independent consultant and worked with schools on making changes, particularly engaging with Japanese Lesson Study. For four years I was the Programme Director for the King’s College London MaST course.

I have seen it as important to make a professional contribution to education. This has involved being Secretary of the Mathematical Association, a member of the Royal Society Mathematics Education Committee and Chair of the coordinating committee for Primary Mathematics Year 1988. I have also been chair of governors of a primary school.

I have two main research interests: the role that Lesson Study plays in the profession development of teachers; and, a member of the research team at the UCL Institute of Education looking at The Nature, Prevalence and Effectiveness of Strategies used to Prepare Pupils for Key Stage 2 Mathematics Tests, a project funded by the Nuffield Foundation.

Rebecca Rosenberg

Hello. I have been working at the Open University since 2019, mainly on the development of the new Maths Education modules. 2022 is the first year I will be tutoring on one of these modules, and I’m really looking forward to putting all that hard work into practice!

Before joining the Open University, I worked as a maths education publisher, and before that I worked as a secondary maths teacher in Suffolk. I’m particularly interested in the way people talk about maths – both inside and outside the classroom; how do we form questions in maths lessons? How is maths discussed in popular culture and media?

In my spare time I knit, sew, cook, garden and binge-watch American high-school tv shows.

Jim Thorpe

I became a mathematics teacher through the accident of joining Bill Brookes’ PGCE course: suddenly I realised that much mathematical thinking could emerge from humble beginnings, numerical or geometrical, and realised that mathematics could make a major contribution to the intellectual and social development of adolescents if they were encouraged to function as young mathematicians within what John Mason calls a ‘conjecturing atmosphere.’

I have been committed to mathematics education for a long time, in the secondary classroom and then in a variety of ways supporting the work of mathematics teachers. My current occupation is mainly tutoring in mathematics and education, mathematics, and engineering for the Open University.

I am alarmed by much of what I see under the heading of teaching mathematics but remain unrepentant in proposing something richer than the all-too-frequent ‘training’ metaphor of communicating mathematics.

Why are multiple representations important when learning and doing mathematics?

One of my favourite starter problems which I picked up as a secondary mathematics teacher involves a spider, a fly, and a room in the shape of a perfect cube. The spider is sitting in one corner of the floor, whilst the fly is hanging off the ceiling in the opposite corner. If one side of the cube is three metres long, what is the shortest distance which the spider can travel to reach the fly?

Once I had clarified a few important details, (no year eight, spiders don’t wear jetpacks,) typical answers from the class would include travelling along three edges of the cube – a total of 9 metres – or crawling diagonally across the floor and then up one wall; using Pythagoras’ theorem this distance would be the square root of 18 plus 3 metres, a total of about 7.2 metres. However, the best possible answer is closer to 6.7 metres. The pictures below illustrate how this result comes about.

For many learners, switching from three dimensions to two, and representing the problem on a net of the cube is what gives rise to an ‘aha’ moment. It becomes transparent that the shortest distance is a straight line, and all that is left is the calculation.

The Value of Multiple Representations

The spider and fly puzzle is far from being the only mathematical problem where a change of perspective can empower and enable learners. This is why The Open University’s mathematics education module ME322 talks about learners meeting, choosing between, and connecting different representations. When learners can represent mathematical objects and concepts in more than one way, they develop more flexible understandings and have access to different avenues of investigation.

The value of representations can be seen throughout school mathematics, starting with some of our earliest work with number. Drawing multiplication facts as arrays can help young learners to see how multiplication is repeated addition (here 5 x 3 = 5 + 5 + 5) and to recognise the commutativity of multiplication (in this case 5 x 3 = 3 x 5).

There’s a subtle shift in how the representation is supporting mathematical thinking here. In the spider and fly puzzle the net representation helped to solve a specific problem, but now the array representation informs our mathematical understanding of multiplication and how it works. With appropriate prompts and experience, most learners will recognise that the visual switch in orientation keeps the product the same for any numbers, not just 3 and 5. Using representations to reveal structure is one of the central ideas in mastery approaches to teaching mathematics.

Taking Advantage of Multiple Representations

Alternative representations don’t have to involve pictures. Further along in their study of number, pupils meet decimals and fractions, two different symbolic representations of proportions or parts of a whole. Each has its own strengths in both application and calculation. For example, I find that school-aged learners would usually rather add 0.4 and 0.25 than 2/5 and 1/4. On the other hand, it’s quite demanding to square root 6.25 without a calculator, but rerouting through fractions makes it more straightforward:

Another topic where multiple representations are highly useful comes from A-level mathematics. The modulus (or absolute value) function f(x) = |x| returns the non-negative version of any number. For example, the modulus of 10 (written |10|) is 10 and the modulus of negative 10 (written |-10|) is also 10. This function can be defined as a piecewise function, represented by its graph, or considered as the distance from the origin. A visual metaphor for this is hitting a target in archery. To work out how many points you score, you only look at the coloured rings, which show the distance from the centre; it doesn’t make any difference whether you are off to the left or off to the right.

I find it interesting how (and when) learners switch between equation, graph and distance representations as they move through different types of modulus function questions. To solve the first equation in the box, a learner might typically call on a definition to reach the two answers of a = 8 and a = -8. However, the second question can be solved using different approaches. Some learners rewrite it as two separate linear equations: either b – 7 = 2 or b – 7 = -2. Others instead think about it in terms of distance: which two numbers satisfy the statement “the distance between b and 7 on the number line is 2”? The third question is perhaps a little too complicated to think of in terms of distance, so it would probably be most efficient to rewrite this as two quadratic equations, but multiple representations can still help understanding; sketching a graph can help learners to appreciate why this equation can have up to four solutions.

This progression might give the impression that symbolic (in this case algebraic) representations eventually become dominant, but other representations continue to be useful tools – and are sometimes shrewd choices. For example, when the modulus concept is extended to complex numbers, learners might be asked to sketch on an Argand diagram the set of all complex numbers  that satisfy the equation |z – (1+2i)| = 2. This question might seem intimidating, but it is perhaps less so when it is reframed in terms of distance: “sketch all of the complex numbers which are a distance 2 away from the fixed point 1+2i ”. This gives a big clue as to the shape of the final answer, recalling the archery target.

Probability, Phrasing and Perspective

Outside of number and algebra, multiple representations continue to be useful when thinking about probability. What representations come to mind when you think about probability in school? Venn diagrams offer a graphical way to distinguish between the union and the intersection. Possibility spaces and tree diagrams offer two complementary ways of representing combined events, each with its own advantages.

There are many more examples. One of our doctoral students is currently researching classroom language, including the teaching of gradient – a rich topic which can involve many representations. They are discovering how teachers’ choices about language can influence how secondary school learners connect ideas of gradient, slope, speed and velocity.

From spiders to speed then, it pays to consider your perspective. However, teaching multiple representations can take time, and teachers are often required to use their judgement to balance out cognitive costs with expected benefits. We might also ask how learners can learn (and teachers might teach) the skill of judging when to choose and use specific representations. For instance, whilst heuristic approaches such as drawing bar models or using a number line work well in many situations, they do not suit every problem; in some circumstances an adherence to a particular representation-based approach might slow down or frustrate certain learners.

What do you think? Have there been any representations which you remember helping you as a learner or a teacher? Let us know in the comments below.

FURTHER READING

More on Elizabeth Kimber’s work on language and gradient can be found at https://bsrlm.org.uk/wp-content/uploads/2022/08/BSRLM-CP-42-2-05.pdf.

The NCETM (National Centre for Excellence in the Teaching of Mathematics) offers guidance here about some useful representations for teaching mathematics at Key Stage 3 (ages 11-14).

Vinay Kathotia, one of our lecturers in mathematics education, writes about his work with an educational charity in Udaipur.

The Shiksha Sambal central maths team with Vinay,
from left: Jyoti, Neha (team lead), Vinay and Ruderaksh

Nestled within the Aravalli hills, one of the oldest mountain ranges in the world, there lies a vivid example of the extremes that exist side by side in India. Udaipur, the City of Lakes in India’s desert land of Rajasthan, has some of the most opulent palaces-turned-into-hotels in India. On the other hand, the neighbouring hills, strip-mined for centuries for zinc, lead, copper and other riches, are home to significant socioeconomic and educational disadvantage. Rural Rajasthan is one of the lowest-ranked regions for student enrolment and mathematics attainment (India’s Annual State of Education Report, 2022, p.67). In particular, there are substantial drops in retention for tribal communities and female students as they transition from primary to upper-secondary education (India’s National Education Policy, 2020, p.25). Rajasthan also ranks lowest on mothers’ schooling, which has an ongoing generational impact.

Working in the midst of this context is a multi-institution educational charity, Vidya Bhawan Society (VBS). VBS was founded in 1931 and has since grown to include an education resource centre, schools, a polytechnic, a rural (undergraduate) institute, teacher training colleges, agricultural and environmental centres. It also runs outreach programmes in the rural and tribal hinterland, aimed at fostering a more equitable society.

Vidya Bhawan’s Shiksha Sambal programme

Since April 2022, I have been collaborating – both online and in person – with the mathematics team of VBS’s Shiksha Sambal (Education Support) fellowship programme. This initiative, targeted at learners in Grades 6-8 (the first three years of secondary school) aims to support language and mathematics education (and some science and co-curricular activities) in a total of ten state schools across five rural and socioeconomically disadvantaged communities around Udaipur. Shiksha Sambal is funded by Hindustan Zinc Limited, a significant local employer and one of the world’s largest miners of zinc and lead, as part of its corporate social responsibility work.

The Shiksha Sambal fellowship programme is delivered by five pairs of ‘fellows’, who are all women and most are recent graduates of a four-year Bachelor of Elementary Education programme from Delhi University. Each pair of fellows (one for language and one for mathematics) is responsible for two schools. The fellows work together, three days per week in each of their two schools, complementing the work of the regular schoolteachers.

Shiksha Sambal fellows working with students in their schools. | Photo Credits: Vidya Bhawan Society

As part of the programme, some of the Grade 8 students also got to attend a residential summer camp at the VBS campus in Udaipur. For many of them it was the first time they had lived away from home. It was also the first time that many of the fellows had lived and worked away from home. Experiences like this can be particularly meaningful in a challenging, rural and patriarchal environment where girls are often kept home for domestic work, and there is prevalence of child marriage – even though it is illegal.

Now entering its second year, the Shiksha Sambal fellowship programme has been transformative for some of the students and for the fellows themselves. You can read more about the experiences of the fellows here.

The Open University’s contribution

My work with Shiksha Sambal has involved supporting the development of mathematics resources and working with the fellows on their professional development, both mathematical and pedagogical.

As part of this work, we have drawn on research in mathematics education, including approaches from Realistic Mathematics Education (RME). In particular:

• grounding mathematics in students’ experiences and contexts relevant and meaningful for them
• reinforcing the role of language and reasoning in mathematics
• using games, craft activities, and visual and hands-on materials.

These approaches were chosen to anchor learning and help the students see and use mathematics as a sense-making and expressive medium. As these approaches have to work for the particular learners and their environment, the collaboration with the fellows has been essential in trialling, adapting and improving the curricular materials and teaching strategies. One key resource for the learners is a workbook, which includes a mix of closed and open-ended mathematical tasks, puzzles, stories providing a narrative for the learning, and opportunities to create their own questions and stories.

Grade 7 students receiving their new workbooks. | Photo Credit: Vidya Bhawan Society

The first year of the fellowship programme has been full of rich experiences. As with most learning and research, it has led to more questions than answers! My colleagues in India and I are now developing research proposals to explore the exciting practice and emerging ideas, some of which are presented below.

Contrasts and overlaps between language and mathematical learning

The language and mathematics fellows naturally found themselves supporting each other’s activities in the classroom. This highlighted opportunities for joint work which are being integrated in this year’s activities. The language programme also involved songs and silly rhymes, performed as group embodied activities, rich in movement and gesture, which drew together students and fellows. The affective and learning gains of these song-led activities seem substantial. Can we develop similar activities for developing understanding and engagement in mathematics?

Dancing aloud ‘Ram Narayan baja bajata’ (Ram Narayan plays the music) | Photo Credits: Vidya Bhawan Society

Mathematical storytelling

The students’ workbook introduces the ‘Story of Ghanshyam’, a cowherd who develops symbols for counting to keep track of his cows. The story evolves to bring in manipulatives including bundles of sticks and a rudimentary ‘abacus’ (vertical rods which can accommodate nine beads each).

The Story of Ghanshyam: Pages from the Shiksha Sambal Grade 6 workbook | © Vidya Bhawan Society

We saw students using materials such as these and Dienes blocks to ground and enrich their understanding of place value. It was interesting to note that students not only used these materials as originally intended, but also co-opted them in games of their own. These variations need to be investigated further.

Counting sticks and Dienes blocks (used traditionally!) | Photo Credits: Vidya Bhawan Society

Learners extending their use of Dienes blocks | Photo Credits: Vidya Bhawan Society

Few of the students have books at home, so VBS runs a mobile library to support students. The students’ engagement with mathematical stories, games and manipulatives could point towards introducing mathematical materials and games into the mobile library, in a similar way to UK programmes such as the Letterbox Club.

Games that bring together dice, fractions, factors and multiples, and strategising (getting four counters in a line) | Photo Credit: Vidya Bhawan Society

Crafting mathematics

Alongside curricular work, students worked on a mix of puzzles and craft activities, including tangrams, modular origami and mobile origami, and this gave rise to engaging and natural contexts for geometry work. While some students were much engaged and enthused with these activities, others were frustrated when their constructions didn’t quite come together and didn’t have a positive or enabling experience. Some of the issues related to being able to fold accurately, the relevance of precise measurements, and being able to follow complex live or diagrammatic instructions. All of this may need careful unpicking. There may also be more fundamental issues at play here. Gandhi’s ‘Basic Education’ movement, which has craft as a basis for life-long learning, has never sat comfortably with traditional schooling.

Students constructing and sharing their rotating octagonal modular origami flowers | Photo Credits: Vidya Bhawan Society

Expressive and contextual mathematics

One of the problems that fellows and Grade 8 students worked on was Dragon-kali (originally, The sword of knowledge at www.mathfair.com/some-typical-projects.html)

“The dragon of ignorance has three heads and three tails. However, you can slay it with the sword of knowledge by cutting off all its heads and tails. With one swipe of the sword you can cut off one head, two heads, one tail, or two tails.

But . . .

When you cut off one head, a new one grows in its place.

When you cut off one tail, two new tails replace it.

When you cut off two tails, one new head grows.

When you chop off two heads, nothing grows.

 Help the world by slaying the dragon of ignorance.”

Fellows and students were also asked to devise their own similar problem and it was heartening to see their engagement. In their solutions, some used numbers, trial and improvement, symbols, or elaborate drawings, while others used pencils and erasers as part of physical enactments. Many learners were happy engaging with the challenge of devising their own problem, and some went on to develop their scenarios in some depth, naming their monsters, creating related imagery, and clarifying what mathematical principles they were using.

Three variants on the Dragon-kali problem

What supports or inhibits learners using mathematics and language expressively, and the variation across types of problems, contexts, students and groups is something that needs further study.

Navigating the classroom: practical barriers and stories of progress

The fellows had to grapple with a range of practical issues in their classrooms, schools and communities. Support from the existing school staff was variable. The staff were also under pressure to cover the necessary curriculum for the periodic assessments run by the state and for the final Grade 10 assessment which (as in England) serves as a gatekeeper for professions such as nursing or teaching. Fellows were sometimes asked, often with limited notice, to replace their work with assessment related ‘revision’. Some teachers and students saw the work with the fellows as recreation and not actual teaching and learning. Also, the requirement for using standard algorithms in the assessment limited work with sense-making approaches such as grid multiplication or ratio tables.

One of the most challenging aspects was the diversity of attainment and attitudes in their classes, including some students with severe learning needs but with no related diagnoses or support. Additionally, a few male students could be rude or dismissive towards the female fellows on occasion.

Nonetheless, the programme gave rise to a number of heartening successes. As fellows were living locally, they were able to visit the students’ villages and homes, better understand the local context and some of the underlying reasons for poor engagement and behaviour. Supported by this knowledge, a compassionate approach and, where feasible, collaboration with families, the fellows were able to develop and sustain strong and positive relationships with a number of vulnerable, unengaged or challenging students, and significantly improve their participation, behaviour and learning. Below are images of evolving work of one student who initially neither spoke, nor was able to read or write.

A learning journey| Photo Credits: Vidya Bhawan Society

The fellow who took this student under her wing learned that she had substantial learning needs, had been married early, and did most of the housework before and after school. Over a few months, through steady support, the student moved from scrawling over books, to making letter-like shapes, to being able to recognise and write particular letters and speak some words.

The fellows became role models, not only for all the students in their classrooms but for each other and for the VBS team and the wider community. An open question is now how these successes and the attendant community engagement might be strengthened, replicated, and sustained.

Looking forward

The work and development of the Shiksha Sambal fellows and their students has been uplifting and inspiring. I am looking forward to continuing to support their work and, with them, investigating some of these questions. Do get in touch if you have any thoughts, questions or suggestions – we would love to hear from you.

 Angie McConnell is a long-standing associate lecturer in Mathematics Education at The Open University. She has chaired several undergraduate and postgraduate modules, and is currently heading up ME620, Mathematical Thinking in Schools. Angie is retiring as a central academic this summer, so we pinned her down for her thoughts on mathematics, teaching, and what makes The Open University special.

Congratulations on your retirement!

Thank you very much.

How did you first get involved with The Open University?

I first joined as a student in 1979. I’d done a traditional degree at Liverpool University, but – due to far too many extra-curricular activities – I hadn’t done as well as I’d hoped, and I thought, “I’m better than that, I can do that.” So, I joined the OU and ultimately got a first which pleased me no end. As I student, I became involved with the Students’ Association through the 80s, through the Thatcher years, and a campaign to save the Open University. I was a student rep on the University Council for some time through the 80s and 90s and then I started tutoring on some postgraduate modules.

There’s a lot of history there. What are your favourite memories of working for The Open University?

I think my all-time favourite memories are from attending graduation ceremonies. I’ve attended quite a few over the years, and what makes them so wonderful is the cross-section of people. Everybody’s there: the young, the old, people with various disabilities… for me, The Open University was the university of the second chance, but for some people it’s also the university of the first chance, for people who wouldn’t have gone to a traditional university but are nevertheless capable of getting degrees.

What’s wonderful is when people walk across the stage, and as they shake hands with the people on stage, sometimes there will be a voice at the back going, “there’s my daddy!” or “there’s my mummy!” Someone wise once said to me that the only thing harder than being an OU student is living with an OU student, and you realise that behind each of these people who have just achieved a degree there’s a whole team of family and friends who have supported them, done the washing up, made cups of tea, deferred holidays when there’s been an exam and so on. And I love graduation ceremonies because you see this.

As well as your work with The Open University, you have a lot of teaching experience to your name in different sectors.

That’s right. I did ten years in a secondary school, an 11-18 comprehensive school, and then I did 30 years in further education.

What are some of the key differences in how you approach teaching mathematics in further education?

I enjoyed the move to further education because the learners were post-16, and by and large they were there by choice. In the latter years, I was teaching a lot of adult returners, and that was really rewarding. There were people who for all sorts of reasons hadn’t got GCSE Mathematics, and somewhere along the line they needed it for a job, or a promotion, or whatever. They would turn up in class and – not kidding you – the first session in September, they were white knuckled. One student told me they had come to battle their demons, and so many of them had bad experiences in the past. So it was my job to make them not scared of maths. I used a lot of humour to get through to them, and by and large it worked. They used to come out of those classes, and they weren’t scared of maths anymore.

I always think that further education colleges are sort of the Cinderella service of education – they do great work, but don’t get the same funding as higher education. Working with adult learners was joyous. What wasn’t joyous was when somebody higher up the food chain decided that everybody had to do GCSE Mathematics, whether they wanted to or not [in response to the Wolf review, 2011]. So you would get classes of resentful 17-year-olds who did not want to be there, and didn’t see the point. To my mind, this can make people hate maths. It doesn’t engender a love of maths.

You have seen a lot of changes in mathematics education. What do you think have been the most important changes, and what do you think is interesting about what is happening today?

I think mathematics teaching is evolving to become more collaborative. When I started in 1975, the teacher was expected to put an example on the board, ask the pupils if they understood, and then give them 20 more of the same. If they didn’t get it, they just got left behind. Now I think there is a much bigger move towards getting children in schools to understand.

There are a lot of good materials out there. There was a box of stuff which used to land in the further education college which had all sorts of interesting activities.

The Standards Unit?

Yes, that was it. That was great. I took that, and I used it a lot, and it was getting people to do the same mathematics, but in a more exciting, interesting, innovative, and collaborative way. In 1975 you wouldn’t get students to work in groups – the ideal was that they worked in silence!

I think getting learners to work in groups and spark off each other is where mathematics education is going, and that’s good. But I think, sadly, there aren’t enough mathematicians in schools and in further education colleges.

Do you mean mathematicians, or do you mean maths teachers?

Mathematicians who are maths teachers – there are a lot of maths teachers who aren’t confident mathematicians and are scared to innovate. We used to run a programme for primary teachers to get them more comfortable teaching mathematics, so that they would use novel ways of approaching the content. We said we could offer it anywhere in the country – and got the South West and the North East, which I think was someone’s idea of a joke! We would flit down to Plymouth and up to Newcastle to get primary teachers more comfortable teaching mathematics, so that they would like mathematics themselves, and communicate that to their pupils.

As you begin your retirement, what are you most looking forward to?

Lots more travelling! I don’t want to lose touch with mathematics, but I plan to do as much travel as I can. For years I’ve believed that you should have the holiday of a lifetime at least once a year!

One last question: what advice would you give to someone starting out with mathematics teaching? Continue reading →

What happens when we put Maths into reverse?

As a teacher, one of the things I appreciate most about the field of Mathematics Education is the number of simple and effective ‘big ideas’ that have been developed and shared by researchers. Many powerful ways of thinking about teaching and learning mathematics can be boiled down to an elevator pitch. Even though these ideas reward close inspection, it typically doesn’t take long to get started and see how they might connect to a wide range of topics and contexts.

A collection of these big ideas form the backbone of the Open University course, ME322 Learning and Doing Algebra, and one of my current favourites is the idea of doing and undoing. In a nutshell, this idea states that in mathematics there is often a way of working backwards as well as forwards, and that this ‘undoing’ can give rise to richer mathematics, additional insight, or creative opportunities (Mason, Graham and Johnston-Wilder, 2005, p.66).

A lot of number work can be thought of in this way. Undoing the addition of positive integers gives us subtraction, which brings negative integers into play. Undoing the multiplication of positive integers is in effect division, connecting the operations to fractions. Undoing squaring leads to square rooting, irrational numbers, and even the possibility of imaginary numbers.

Undoing a Puzzle

Doing and undoing can also be applied in all kinds of situations beyond arithmetic. For a more unusual example, consider a Sudoku puzzle. The goal of a Sudoku is to fill the gaps with digits from 1 to 9 so that every row, column and 3×3 square contains each of the digits exactly once.

The doing of a Sudoku is all about logical reasoning. For example, look at the grid above. The middle row with the gold square contains seven of the nine digits and is only missing a 1 and a 2. However, there is already a 2 in the central 3×3 square, so the gold square must contain the digit 1. If you want to try and complete the rest of the puzzle yourself, you’ve got one more paragraph to get it done!

Sudoku is certainly well-known – according to one recent survey, 98% of people had heard of Sudoku (YouGov, 2022). From a mathematics teaching perspective, the process of completing a Sudoku puzzle could be connected to A-level topics such as proof by contradiction and combinatorics. What is there to gain, though, by undoing a Sudoku? Let’s start with the answer to the puzzle above:

This time, the name of the game is removing numbers. If I remove the 1 in the gold square and pass on this puzzle to a friend, they would be able to complete it and end up at the right answer. In fact, I could get away with removing all the digits in the middle column, and still be sure that the remaining digits would lead to this exact solution. However, if I was to remove all 81 numbers, my friend could solve the Sudoku in multiple ways. This gives rise to an interesting undoing question: what is the largest number of digits that I can remove from the board without changing the solution(s)?

Have a go, or a guess, yourself. How might you make further progress? Thinking through this problem backwards is certainly more challenging and forces you to pay close attention to the Sudoku structure. I won’t give away the answer here, but it was reached by McGuire, Tugemann and Civario (2014), who found a way to the solution that required some clever use of computers. The act of turning a situation or problem around resulted in some rich and revealing mathematics.

Doing and Undoing in School

Doing and undoing can also be a great tool for teaching and learning algebra. If your experience of school mathematics was anything like mine, you were probably taught to solve linear equations by first writing out them out and then ‘doing the same to both sides’. However, in the past I’ve sometimes asked my pupils to start with an answer and work outwards instead.

This kind of ‘undoing’ activity can naturally give rise to some interesting questions. Could I redraw all the arrows to be two-way arrows? Is there a way of jumping between the two clouds on the left-hand side of the example above? What does this say about how equations like this might be solved?

Moving up the curriculum, consider these simultaneous equations:

What is the solution? Now, undoing the problem, can you find other sets of simultaneous equations which would have the same solution? Do you notice anything about their coefficients and constant terms? (This is a reversal of a fantastic task which connects to an unexpected topic – if you want to see another way of approaching this result, or want a hint, take a look at RISP 8 in Jonny Griffiths’s collection of rich starting points for A-level mathematics learners.)

The Benefits of Undoing

Undoing questions are frequently open-ended, allowing space for creativity and opportunities for differentiation. Consider turning the question “find the area of this triangle” into “if the area of this triangle is 24cm², what could the lengths be?” If the values are limited to positive integers, how many possible answers are there, and how do you know you have them all? What could the answer be if we insisted that at least one had to be a decimal, or a mixed number, or a surd?

The practice of undoing can also support learners’ understanding of concepts. I was recently working with a learner who had just learned about standard deviation. They had calculated that the standard deviation of 0, 1, 2 and 3 was √5/2 . Asking them to come up with other sets of four numbers with the same result reinforced their understanding of what standard deviation measured.

The extent to which doing and undoing permeates the curriculum at A-level and beyond continues to suggest the importance of this theme in mathematical thinking. Differentiation is taught next to integration; exponentials are quickly followed by logarithms. Beyond A-level the idea of an inverse is baked into the very definition of a group, and undoing seems relevant to many unsolved problems in mathematics such as Gilbreath’s conjecture.

Final Questions

I hope that this quick tour has illustrated some of the potential value of doing and undoing, but I would like to end with two questions.

First, what exactly do you consider is being undone in each of the examples above? Are we undoing the mathematics, the question, or both? Can one happen without the other, or does this balance depend on how each individual learner chooses to approach a problem?

Second, is doing always met before undoing? Take, for example, this differential equation:

This equation can be seen as an expression of an implicit doing: I have started with a function, then found that when I differentiate it and add the result to 2x times the original function, my answer is x. Solving this differential equation is therefore already an undoing, with the integrating factor being used to ‘undo’ the product rule. If the undoing is taught first, what value might arise from un-undoing?

Please share your thoughts and examples of doing and undoing in the comments!

REFERENCES AND LINKS

Mason, J., Graham, A. and Johnston-Wilder, S. (2005) Developing Thinking in Algebra. London: SAGE.

McGuire, G., Tugemann, B. and Civario, G. (2014) ‘There is no 16-clue sudoku: solving the sudoku minimum number of clues problem via hitting set enumeration.’ Experimental Mathematics, 23(2), pp. 190-217.

YouGov (2022) Sudoku Popularity and Fame. Available at: https://yougov.co.uk/topics/society/explore/activity/Sudoku. (Accessed: 1^st March 2023.)

Jonny Griffiths’s Rich Starting Points for A Level Mathematics (RISPS) can be found at www.risps.co.uk.

This post is written by Esther Wheatley, winner of the Stanley Collings prize in 2021.

The Stanley Collings prize is awarded annually by the School of Mathematics and Statistics. The prize is awarded to the student whose Mathematics Education assignment best combines innovation in devising materials suitable for learners and insightful analysis of their learning.

Congratulations Esther!

I always enjoyed maths and found it easy to focus on, even when Chronic Fatigue Syndrome reduced my school attendance. Having become a Teaching Assistant, I decided to study Maths further with a view to teaching. The Open University is great provider which allowed me to study around my circumstances and commitments: I highly recommend it. However, while I loved the maths modules, my decision to change to “Mathematics and its Learning” dropped me into reflective essay writing. Like others, I struggled with new ways of thinking, so I deferred a module while trying to figure it out. The deferment enabled development through each module, and my grades improved dramatically. However, I was extremely surprised to find myself cowinner of the Stanley Collings prize for tasks devised in the final assignment of the module Developing Thinking In Algebra. Here, I share modified reflections on the tasks I did with my adult learner, who I will call Amy. I have focused on one task, but allowed some comparison with the second to remain. The task I have chosen uses Algebra tiles, which are a great resource that I am pleased to introduce to a wider audience.

Amy and I worked together for my final two modules. Most lessons were on Zoom, which worked well because I could record and transcribe sessions, giving an accurate record to reflect on. Amy emailed me her memories after a couple of weeks. For health reasons, Amy missed a lot of school and so displays mathematical anxiety and low expectations. However, the Developing Thinking in Mathematics modules premise that every learner has the ability to specialise and generalise, imagine, express, organise, classify, conjecture and verify (referred to as mathematical powers.) I saw it as my role to encourage Amy in the awareness and use of these powers to learn and build her confidence.

The Plan

I naturally lean to giving direct instruction but, in adapting and transferring my understanding to teach others, a gap can be created between what they learn and the underlying concepts – this is known as didactic transposition. Therefore, attempts to impart my excitement and understanding often lead to behavioural, rather than conceptual, instruction or overwhelm my learners. To avoid this, I aimed for tasks that let Amy work independently and use her mathematical powers to discover generalities for herself, which is considered key to developing algebraic thinking. 

Algebra tiles

I devised the task around algebra tiles. Algebra tiles are manipulatives that attempt a concrete representation of variables and units.

Small squares represent units
Narrow rectangles represent a variable (e.g. x)
Large squares represent the variable squared (x²)
Each has a red side and a coloured side, with the red representing negative value. Hence the overall value in the image here is 0 as each red tile cancels out its counterpart.
My physical tiles have two variable lengths in different colours and therefore two sizes of larger square and also a rectangle representing the multiplication of the two variables (e.g., xy)

I used an electronic version to replace physical algebra tiles with manipulable images because they were more easily accessible with Zoom and less limited in number. The app can be accessed at https://didax.com/apps/algebra-tiles/

Enactive, Iconic, Symbolic and Manipulate – Get a sense of – Articulate

I first discovered algebra tiles when looking for manipulatives for intervention groups, having found that physical objects, pictures and practical examples made it easier for my students to understand and connect to topics. Like those learners, Amy shows apprehension with algebraic symbols and numbers, but is confident with pictures and physical objects, so I felt algebra tiles could provide an accessible link through familiar concrete materials to the symbolic representation that she found daunting.

The value of multiple representations and progression from concrete through iconic to symbolic representations is well documented. For instance, a concrete, pictorial, abstract approach, influenced by Piaget, is central in mastery curricula (Drury, 2018). Bruner (1986) also proposed three modes of representation which follow each other in cognitive development: Enactive (involving physical interaction and activity with concrete objects), Iconic (representation through pictures) and Symbolic (the use of language and symbols). The modes may relate to phases of development in terms of age, but also stages for any new learning. They underpin the manipulate-get a sense of- articulate (MGA) construct, where learners manipulate familiar objects or understanding, getting a sense of a new ideas that become clearer until a newly understood concept can be articulated. The process is cyclical, or spiral-like, as the new understanding can be further manipulated and built on to gain additional insights.

The Task

The aim of the activities was to simplify expressions by grouping like terms, but the tasks are also an example of diverting attention to automate, since they provided experience working with variables and alternative representations, which Amy needed practice in.

I first created examples with the algebra tiles, which are familiar-looking objects which could be virtually manipulated…

… The second section used numbers which are familiar to Amy, but she is less confident with numbers than objects.

Finally, I introduced algebraic symbols, which Amy is least confident with. These I gave first in zoom chat and then verbally.

I chose to make the initial questions in the first two sections completed examples, rather than questions to be answered (see above.) I then promoted the strategies of “say what you see” and “same and different” to encourage Amy to get a feel of and make sense, for herself, what was going on. I was concerned that Amy would make equivalences between the manipulatives and be confused by spacing, but I determined to let her create and revise her own conclusions rather than providing immediate guidance on this.

Learner Activity

Saying what she saw, Amy immediately recognised the equals sign as expressing a relationship, but sought confirmation.

“Well, you’ve got the… the top picture is like 2 different pictures linked together with the equals sign so they must be relationship between the left hand side and the right hand side umm… So are the ones on the left hand side supposed to be equal to the ones on the right hand side? Is there a mathematical equation between them?”

Other observations included size, shape, colour and position as well as possible connections, such as that a rectangle was worth four small squares.

“ … you’ve got three equal length rectangles 2 green 1 red and then you’ve got two little units like single units joined together uh, of red and yellow umm, which seem to be half the length of one of the rectangles so in a sense you’ve got three and a half of the rectangles there and then the other side you’ve got one rectangle.”

“I think in my mind I had thought it was the same value all the time but I couldn’t work out if I could put four single units together to equal the same amount as the rectangle”

Amy showed anxiety that she had to express the relationship and could not really see it but immediately conjectured that the red tiles were “minuses” apparently taking them to represent action on adjacent tiles, rather than value. Carefully giving my solution to an extra question enabled Amy to modify her conjecture, along with her understanding of the role of space and equivalence.

“I don’t think I could write it down. I can’t… I can’t see the relationship. I can see that y’know, there are like what you might call tens and units but I can’t see how unless you say umm, unless the reds are minuses if you like… ”

“the reds have eliminated the ones next to them and left the others”

After the new example:

“ok so I’m thinking that the red square deletes the blue square and the red rectangle deletes the green rectangle”

“the only other possible way you could do it I can see is if you could take a little square out of the big square, which I don’t know if you can do”

“those of the same value they eliminate”

That she had discovered this herself was the highlight of the lesson for Amy, along with making up her own examples, occurring after explaining the significance of the tiles.

“I felt jolly pleased with myself that I worked out it was minus”

Via email: reds were minuses which took out a rectangle or square of equal value… I was pleased to pick this up fairly quickly and be able to start to make up my own equations.

Showing her understanding of the general nature of the tiles, Amy gave examples of integers they could represent, extending her understanding when surprised by my example of a decimal.

“yeah so they’re all just different values but the size of them makes them the same value as others of the same size”

“well it could be anything really, 200, 100, 60, 80”

Me: 32.752

Amy: “pardon?! laughs, 5.5. But the little ones are always going to be one” (specialising from the general)

Moving to numerical questions, Amy’s attention was on the operations, and she performed the calculations, checking each side in given cases and either leaving a single answer or changing it to a similar form as the given cases on the rest.

“ok, so  2 x 6 is 6, plus 4 is 10, plus 5 is 15, minus 1 times 3, so one three is 3, so minus three is 12 = three plus nine is twelve, so that’s okay.

So four times five is 20, plus 3 = 23, minus that is 19, plus 3 times 5 that’s 15, hang on I’m just going to write some down (carries on working out totals). So now we’ve got the ones I need to fill in. That’s 20 plus 12 is 32, plus 2 is 34, minus that is 30, minus 8 is 22. So how do I write that down.. 22 equals 5 times 4 plus 2”

A key moment was Amy’s encounter with a more difficult question and the prompt to use algebra tiles to solve it.

“well this is the sort of thing you do in your puzzle book, which I’m hopeless at”

Me: try using your algebra tiles, see if it helps.

“I don’t know whether I can do it with the algebra tiles. Umm ok um, this is 10 (green tile) this is 27 (blue square) not going to fit them all in. Umm so now how do I do minus five of those? can I work it out on paper?”

Me: yeah, do whatever you like but do think about how you can do minus five of those

“well I could take away… oh I see, ok, ok umm” (erases three and puts in two reds)

From here, after initial worry, Amy engaged with algebraic questions, quickly becoming independent of algebra tiles, so that I wondered if it would have been better to have skipped the numerical examples. However, Amy felt that the combination of all aspects had been important and that without both she would have been unable to complete the algebraic questions, suggesting that the mind-set change as Amy moved from calculation to working out how to express with algebra tiles was important in building her understanding of the role of variables and symbols.

Amy: oh mate, hang on. You see this is where my brain suddenly goes whoosh. You see the others I could work out eventually with all the numbers in

Me: so try with your tiles

Amy: ok, right we’ll call this a, so we’ve got 2 a’s oh, yeah, two times a, plus 5 (putting out 5 yellows) plus four times a (puts out another 4 green) this isn’t going to work I don’t think minus three, ok (gets the eraser)

Me: I’d like you to show the minus three

Amy: well I’m trying to get rid… oh, ok silly me.

Me: And then show me the other side

***

Me: so write down on paper for me, 3b + 19 -2 +5b

Amy: I would say that equalled 8b + 17

Me: yeah do you think you would have got that at the beginning?

Amy: no

Me: do you think you’d have got it before you did the things with the numbers in?

Amy: no, probably not

After two weeks, Amy emailed me with her memories, having had no further exposure to the topic. She wrote:

…Reds were minuses which took out a rectangle or square of equal value. …gradually I could see how the equations could be put into algebraic form. So 2 x 25 + 25 = 75 could be written as 2P + P = 3P. Replacing numbers with letters allows you to use the same equation with different values…

Reflections

The mathematical powers of imagining and expressing, organising and classifying, conjecturing and convincing, and specialising and generalising, were evident throughout the task and, as hoped, activity is best described by the MGA construct. Imagining placement, action and modification of tiles, manipulating in her head, helped Amy make and imagine conjectures, gaining a sense of what was happening (maybe the reds are minuses) and convincing herself by comparing given special cases (individual examples) to develop a sense of generality (the reds always take away adjacent tiles). Verbal expression revealed thinking, while placing algebra tiles expressed/articulated conclusions which were compared again, sometimes modified as Amy made new conjectures (reds take away corresponding tiles), getting a sense of whether they fitted perceived relationships. Continuing cyclically as new special cases were introduced, the same processes enabled development to symbolic examples, Amy first expressing with tiles, manipulating to get a sense of how to simplify and relating back to symbolic forms, eventually manipulating symbols only. Easily overlooked was Amy’s power to classify according to shape, size and colour, and to organise and recognise order, which enabled her to determine which terms were equal and equate tiles to symbols. The powers intertwined in their support of each other. For instance, imagining supported developing conjectures and conjectures led to new mental images.

In comparison, during another task, Amy confirmed she had no mental image and could not “visualise.” This was a significant difference between the tasks. Rather than providing images, structure and completed examples, I had relied on Amy already having imagery for number (which I did not assume with algebraic symbols) and being able to apply her mathematical powers to this imagery and previous learning. Instead, Amy’s lack of imagery for numbers apparently prevented mental manipulation and sense of what was occurring, while absence of completed examples meant time to stop and make sense of relationship was not encouraged in the same way. As my own method for the other activity had built on mental imagery, and an MGA approach, I may be imposing that as an ideal, but the algebra tile task showed mental (and physical) imagery was important in driving progression and would likely have been useful in the other case.

The main similarity in Amy’s response to the tasks was that generalities retained followed connections made herself, through mathematical powers. This was particularly true where Amy tried to verify inaccurate conjectures and became stuck before further specialising prompted accurate generalities of which Amy could convince herself. The algebra tile task’s generalities were the general nature of the tiles and symbols and equivalency of like symbols, the first of which was embedded by Amy creating her own examples and applying to new questions, the second building on Amy’s classification of tiles, transferring to symbols.

Amy’s enjoyment and recall from using her own powers to discover generality and creating her own examples supports the principle I followed, that learners gain most actively exercising their mathematical powers. Boaler (2009, p30), states that students need to “engage, do… problem solve” – acting as mathematicians to experience mathematics as “living”. However, some problem solving in other tasks left Amy feeling frustrated, due to little strategy or previous experience of problem solving.  Christodoulou (2013, pp96-98) shows the pitfalls of expecting learners to behave like experts prior to conceptual understanding or, as Barton (2018, p300) points out, not knowing how to use it, while it is possible that inaccurate generalisations will occur and be internalised. The former was apparent in the less successful task where Amy showed seemingly limited conception of number and lack of strategies, until intervention provided tools for progress, which demonstrated my initial approach did not support her development. The latter was potential in both tasks, had I not provided or suggested new special cases. However, revising conjectures proved powerful for Amy, suggesting that creating and resolving conflict between initial ideas and new exposures deepens understanding, providing this does occur.

Although I felt the algebra tile task was successful, objections to my approach could be that imagery is provided instead of Amy building her own and that the structure is too closed, potentially causing behavioural rather than conceptual understanding. However, arguably providing some imagery is more important than leaving none. Willingam (2009, p88) avers, “we understand new things in the context of things we already know.” His recommendation of varied exposures suggests that images/manipulatives of varied forms, but same function, could improve grasp of abstraction. Even so, algebra tiles were powerful as a “mediating representational structure” (Bruner, 1966,p65) which could add to Amy’s imagery, something Bruner(p66) concluded was as important as a strong sense of underlying abstraction. This too, I believe answers concerns over behavioural instruction. Admittedly, red tiles do not logically delete corresponding ones in other contexts, but zero pairs are explicable, therefore if it is understood why the tiles are so used, they can mediate to activity and understanding, supporting concept as well as behaviour.

My experience of working with Amy throughout the module (and these tasks in particular) led me to believe that learners in early conceptual stages need empowering through structure, examples and imagery to make their own discoveries for effective development of algebraic thinking. However, opportunity must be created to reveal and challenge misconceptions.

References

Barton, C (2018) How I wish I’d taught mathematics. Woodbridge: John Catt Educational Ltd.

Boaler, J. (2009) The elephant in the classroom. London: Souvenir Press

Bruner, J (1966) Toward a theory of instruction. Cambridge, MA: Harvard University Press

Bruner, J (1986) Actual minds, possible worlds Cambridge, MA: Harvard University Press.

Christodoulou, D (2014) Seven Myths About Education Oxon: Routledge

Drury, H (2018) How to teach mathematics for mastery Oxford: Oxford University Press

Willingham, D (2009) Why don’t students like school? USA: Jossey Bass

Algebra tiles site: https://www.didax.com/apps/algebra-tiles/