Mathematics Education

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/

Written by Senior lecturer in Mathematics Education, and co-organiser of the Aperiodic tilings exhibition, Charlotte Webb.

In this post, I will reflect on the Aperiodic tiling exhibition and workshop and offer the reader suggested activities and resources to use at home or in the classroom with learners.

An event inspired by, and in honour of Professor Uwe Grimm

In June 2022, the Open University hosted a collaborative art exhibition and interactive workshop inspired by the life and work of Professor Uwe Grimm, who sadly died in October 2021.

You can visit our online gallery here: Aperiodic tilings online exhibition | School of Mathematics and Statistics (open.ac.uk)

Uwe’s friend and colleague Ian Short wrote the following passage:

Uwe’s research career began in mathematical physics and later moved into mathematical biology, combinatorics and the theory of aperiodic order, which was the mathematical study of the artwork displayed in the exhibition. He was a world leader in aperiodic tilings. Uwe and co-author Michael Baake wrote and edited the beautifully illustrated series of mathematical books Aperiodic Order, which are invaluable resources for researchers in the field. Some of the images from the first text in this series were displayed at the exhibition and are available to view on the online gallery. These same images were used by Uwe to convey the beauty of mathematics to school students and the public in many outreach events.

Two of Uwe’s designs on display at the exhibition

Reflecting on the exhibition and workshop, there were three key features which made this event unique in my view: Collaboration, Creativity and Community.

Collaboration 

From the very start this event was collaborative in nature. The planning committee itself consisted of academics from different fields in mathematics and education, each bringing our own perspective and drawing on our experiences. We worked closely together to organise the three strands: a mathematical conference, an Aperiodic tiling inspired exhibition, and an interactive workshop for school children and members of the public.

The exhibition and linked conference were unique in bringing together mathematicians, engineers, artists, musicians, and educators from across the UK and beyond, all with a common interest in Aperiodic tilings and patterns. This resulted in a rich and diverse display of artworks and interactive workshop materials. For example, Liam Taylor-West, an artist-in-residence at the school of mathematics for the University of Bristol, brought musical installations based on aperiodic patterns which students were able to interact with and discover aperiodic wavelengths. Another artist, Natalie Priebe Frank, who describes herself as a mathematician studying self-similar tilings, brought handmade mobiles and ‘Self-similar Self-portraits’, demonstrating her mathematical research in a visual and accessible way.

A student visitor interacting with Liam Taylor-West’s light and sound installation: Wavelength.

Self-similar self-portrait by Natalie Priebe Frank

Feedback from another of the mathematicians who displayed artwork in the exhibition read: ‘I was greatly impressed by the range and depth of art displayed. For me, this was the largest collection of aperiodic art I have ever seen. For my own exhibit, a collection of items collected over my 25 years involvement in the field of aperiodic research, I received much appreciative feedback.  I was so pleased to return from the conference talks to see students playing with the aperiodic tiles I brought, making beautiful patterns.’ 

Finally, the workshop element allowed collaboration during the exhibition. Adult and school-student visitors worked together to solve geometric puzzles and tiling challenges, and to create collaborative art inspired by Islamic geometry with artist Maryam Smit for our live art wall (pictured below).

Students working together on an Aperiodic tiling puzzle

Maryam Smit and the start of our live art wall.

Uwe had inspired cross-curriculum collaboration throughout his career, having moved himself between disciplines. Most recently he had founded the NOVMAT (Novel superior materials based on aperiodic tilings) project, which brought together engineers and mathematicians to explore the potential applications of 3D aperiodic structures, including in acoustics, aerospace, automotive and medical engineering. Earlier in his career he had worked with artists and mathematicians to create outreach resources for the 2009 Royal Society exhibition.

One of the posters created by Uwe and his collaborators for the Royal Society exhibition.

A colleague of Uwe commented that this event was ‘a lovely and fitting memorial to Uwe, an amazing confluence of artists, mathematicians, outreach enthusiasts and their work, and a rare and incredible opportunity for all who could attend’.

 Community

In addition to bringing together the already established community of mathematicians, physicists and engineers involved in the research field of aperiodic order, the exhibition brought together a community of Islamic geometric artists, many of whom had never met in person, local school children, OU academics, and the Milton Keynes Islamic Artists Heritage and Culture group (mkiac.org). OU academics had run a workshop at the MKIAC Art in the Park festival earlier this year, inviting members of the public to create tiles for an Islamic geometry inspired tessellation, which was displayed at the exhibition and Art in the Park visitors were invited to view their work on display.

A community of Islamic geometric artists brought together through this event: Samira Mian, Clarissa Grandi, Maryam Smit, Richard Henry, Hasret Brown.

Artwork created at MKIAC’s Art in the Park on display at the exhibition

With around 700 visitors across the two days, this special event allowed two-way engagement between members of the public, artists and academics, where visitors could ask questions and interact with the artwork and puzzles on display. In particular, we worked closely with students at Kents Hill Park secondary school, providing them with learning materials to work on before the exhibition and inviting them to contribute artwork to the exhibition itself. Further details about their involvement can be read below.

One of the many positive outcomes of the event was the impact on the students of visiting a university site. Students commented that they were pleased to have the opportunity to visit the OU, which is situated only a few hundred metres from their school. Many students were surprised that staff and students were on their doorstep, highlighting the importance of this kind of community outreach for schools and colleges.

Creativity

The event was a hive of creativity, as can be seen in the online gallery, but this went beyond just viewing the beautiful works of art displayed in the exhibition. Participants were creative in the way they solved challenges – such as our giant soma cube and tetrahedron puzzle – having to try different approaches, and were able to create their own artwork, working closely with some of our exhibiting artists. Some student visitors commented on how they felt about taking part in the workshop activities: ‘I felt creative, and I thought hard about each activity’, ‘I felt excited, amazed, wowed, challenged when I tried the puzzles.’

Students working on Isometric patterns with artist Samira Mian

Prior to the exhibition, Kents Hill Park students were asked to complete a Pen-puzzle challenge, based on the famous Penrose tiling, and to create their own mathematical artwork inspired by this tiling using their own choice of colours and decorations. These collaborative pieces of artwork were displayed at the exhibition and are shown below.

KHP students’ Penrose inspired designs

The students reported feeling: proud, happy, excited, surprised, and confident when seeing their art displayed at the exhibition. They commented on the enjoyment of working together to create mathematical artwork and the communication and teamwork needed to solve the challenging puzzle, putting the tiles together correctly.

Many students also commented on the unexpected nature of the event and activities, suggesting they had not previously considered the links between mathematics and art. Their teacher commented: ‘Many of our pupils were surprised to learn that the exhibition was all based around maths and this really opened their eyes to maths as a much broader subject than perhaps the curriculum currently allows for. They certainly were given the opportunity to see maths in a very different way. The links between maths and art/music were obvious in the exhibition which allowed the pupils to explore the links between the subjects’.

Join in!

Create your own Penrose inspired artwork like the students of Kents Hill Park School. A PPT resource and printable templates for this activity can be found via the online gallery.

You can also create your own Islamic geometry inspired tessellation, similar to that created by the visitors at Art in the Park (link to resources below).

Share your creations with us on Twitter: @OUMathsStats – we would love to see what you come up with!

Further reading and exploration:

• You can visit our online gallery here: Aperiodic tilings online exhibition | School of Mathematics and Statistics (open.ac.uk)
• In this article for the ATM, I explore how Islamic geometric designs can be used in the classroom: pdf (atm.org.uk)
• You can find further detailed of classroom maths-art activities here: Art in the Mathematics classroom: Islamic Geometry (atm.org.uk)
• You can read another article written by artist and former mathematics teacher Samira Mian, exploring Islamic geometric patterns, here: pdf (atm.org.uk)
• You can read Professor Uwe Grimm’s publications here: Uwe Grimm’s Home Page (open.ac.uk)

It has been a while since we introduced our tutors formally on this blog and, in that time, some tutors are sadly no longer with us, and some new tutors have joined from other areas in the field of maths education.

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.

Suki Honey

I have been an associate lecturer with the OU since 2003. As well as being an AL for the OU, I also work for Plymouth University. I was a mathematics lecturer on the BEd and PGCE primary ITE programmes for 10 years, and am now a researcher in pedagogy.

Prior to working with Plymouth University and the OU, I taught mathematics at various secondary schools in Plymouth, Cardiff and Tring. As much as I love working with adults in HE, I really miss the younger students. They have such a wonderful outlook on life and mathematics that it keeps me on my toes. They also make excellent participants in my research, and I have worked with some amazing girls and young women on their experiences of being maths learners.

When I’m not working (that fleeting moment just after I finish marking ECAs), I can usually be found sitting cross-legged doing some mathematical origami or deeply engrossed with my nose in a book, cup of tea and some custard creams always close by.

Pete Kelly

I taught for fifteen years, first in London and then in South West England, before moving into higher education, and am now a researcher and tutor in comparative education and mathematics education. Long ago, I studied with the Open University, and so understand the benefits of online learning and some of the challenges our students face.

Angie McConnell

I’m Angie McConnell and I am currently a tutor on ME620 and ME322. I have been an OU tutor since 2005.

However, my connection with the OU stretches back a very long way – my first degree was from Liverpool University then in 1978  I started an OU degree and followed that with an MSc with the OU – all in Mathematics so I do understand what it is like to be an OU student. I taught for 10 years in a Secondary school then for 30 years in an FE college teaching a wide range of students but always Mathematics. The classes I loved were adult students, many of whom had a genuine fear of mathematics, and I am proud to say I converted many a ‘I hate maths’ student to an ‘it’s all right this maths stuff isn’t it?’ student.

In my spare time I love to travel and explore new places and countries and there’s usually an opportunity to do some maths. Sitting next to a young girl on a train in a remote part of Thailand I realised that she was struggling with her maths homework on quadratic equation. I offered to help and between us we sorted it. She spoke very little English and I speak even less Thai, but mathematics is a universal language.

I have two cats who are very helpful when it comes to marking TMAs and EMAs and love to walk across the keyboard as I am typing so if your feedback is returned with strange typos, blame the cats.

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.

This blog post was written by Jemimah O’Regan as part of a Summer Research Project under the supervision and guidance of Dr. Elsen Tjhung.

Jemimah O’Regan completed her BSc (Hons) degree in Mathematics and Statistics with first-class honours at the Open University in the summer of 2022. She felt that the Summer Research Project was a great learning experience, and found that it allowed her to apply and develop new skills.

The following blog post demonstrates an example of random processes in nature, which can be described mathematically using probabilities.

Consider, if you will, a particle placed inside of a box, which is then closed and filled with liquid. It is a tiny particle, so small as to be measured in micrometers, and it is subject to no forces other than the motion of the even smaller liquid molecules around it. If this box had a glass pane, and if one were able to see this tiny particle, one may note something curious. The particle will, of course, jostle and float as the water is poured into the box. Intuitively, though, one might expect that if one waits long enough, and keeps the box completely stationary, that the particle will eventually find a position to settle down in and stop moving. However, this is not what will happen. One could wait and wait, and yet this will not occur. This particle will continue to move.

In this simple case, it will not move due to gravity. There is nothing that the particle is attached to, no springs or strings acting on it and restraining or forcing its motion. There’s nothing obvious causing this motion, like the box being shaken or tilted, and it may leave one a little flummoxed that the particle refuses to settle.

Through examination, one may conclude that nothing larger than this particle is causing this motion, and one may even determine that it cannot be anything outside of the box making it move. Therefore, one may deduce, it must be something internal. Something very small, and an unobvious – uncommon – source of motion. The only thing inside of the box, other than the particle itself, is the liquid. The liquid which contains molecules even smaller than the particle – molecules which naturally move and collide in a manner that could be called random. When these molecules collide with the particle, they apply a force – ‘kicks’ – onto it, and those kicks push the particle in the liquid. These collisions happen all the time in other situations, but only really cause visible motion when the object being affected by the molecules is small enough to be displaced significantly by them.

This is the source of the particle’s continued motion, and the foundation of the concept of Brownian motion.

Brownian motion is, in a sense, an intersection between the fields of statistics and physics.

The particle’s motion can still be treated as a regular equation of motion, save in that it has a random component representing the random forces from the liquid molecules. As a result, its motion is not seen as deterministic, but rather, stochastic – that is, unlike many other cases in physics, one cannot guarantee exactly how the particle will move given the forces present on it.

Instead, due to the randomness introduced in the modelling, one may only draw conclusions on the motion the particle is likely to exhibit. This can be done using the concept of a probability distribution function from statistics, which provides the probability of (in this case) the particle being at a certain position at a certain time.

Say that the particle is at a certain position when one begins observation. At this point, which we take to be the beginning of the time scale, the particle is at its initial position with probability 1 – we know for certain where it is – and has the probability of taking any other position with value 0 – we know for certain where it isn’t.

After this first time point, however, the particle’s movement is not absolutely certain. It could start to move in any direction, it could collide with multiple particles and change directions and speed several times – as time increases, so too does the number of ways the particle could have moved since the beginning, and the number of positions it could end up in at the end of that time.

As time increases, the particle will explore the area of the box. Above is a representation of what the data over many such ‘runs’ of the physical situation and the probability distribution function at three times might look like for a particle constrained to move only in one dimension when the origin is zero – at small time values, the particle will most likely still be close to the origin, but as time increases, the particle’s probability of straying further away from its original position also increases. Similar reasoning holds for two dimensions.

There are many ways one can take this further – one can make the physical situation more complex (for example, by attaching the particle to a spring), or one could simulate the particle’s motion using computer software and analyze the data, one can find the trajectory of motion that the particle is most likely to take under certain conditions, and so on.

Motion is ubiquitous in life, and Brownian motion shows that it can occur even in situations one may assume would be stationary.

This blog post was written by our summer research student Xiang Fu, who was supervised by Andrew Potter and June Barrow-Green. This blog post focuses on the status of women in mathematics in Victorian times, and today.

Xiang Fu completed her Open University mathematics with statistics bachelor degree with first class class honours this July. She has commented that she absolutely enjoyed the study and appreciated the opportunity of the summer research project.

Flatland: A Romance of Many Dimensions by Edwin A. Abbott was first published under the pseudonym of A. Square, in 1884. It quickly gained popularity as a science fiction story for the introduction of higher dimensions; but also it is a satire criticising social inequality in Victorian Britain. The description by Square, the protagonist from Flatland, that women were regarded inferior to men was so vivid that the author was denounced by contemporary book reviewers as a misogynist. Abbott, writing in defence of A. Square in the preface to the second edition in 1885, had to explain that, as a historian, he had “identified himself (perhaps too closely) with the views generally adopted in Flatland and […] even by Spaceland Historians” (Abbott, 1884a, p.9), and in fact he believed that the straight lines (women) were superior to circles (the priests forming the top of the social caste) in many respects.  This essay will focus on how Abbott, using Flatland as a satire, exposed the injustice of women’s situations in Victorian times and Abbott’s effort to improve education, especially that for women.

Edwin A. Abbott (1838—1926) was a graduate of Cambridge, and worked as the headmaster of the City of London School (MacTutor, 2005).   Abbott’s written works covered various topics. Flatland was inspired by mathematician and science fiction writer Charles Howard Hinton (1853—1907), who was interested in the fourth dimension, and coined a word – tesseract – for the four-dimensional hypercube. Hinton, who was married to Mary Boole, the eldest of the five daughters of the renowned mathematician George Boole, taught Mary’s sister Alicia about tesseracts; and Alicia edited one of Hinton’s books about the fourth dimension (Chas, 2019). The fourth-dimension concept offered a great stage for drama and art in the late 19th century and first two decades of the 20th century (Ibanez, 2017).

Since women were viewed as disadvantaged in their intelligence level, they were excluded from Oxford or Cambridge until the 1860s to 1870s, and could not earn a degree from these two privileged universities until 1920 and 1948 respectively (BBC, 2019).  Female German mathematician Emmy Noether (1882—1935), who made a significant contribution to invariant theory, was discriminated against at different stages of her career due to her gender and Jewish origin. Like Alicia Boole, Russian mathematician Sofia Kovalevskaya (1850—1891) had to be tutored privately (Flood & Wilson, 2011, p.167).

In addition, the diminishing of women in the 19th century was supported by the widespread idea of the ‘missing five ounces’ of the female brain, though modern science has not found any significant difference in the functions of the brains of different genders (Eliot, 2019). In Flatland, one’s geometric properties decides one’s intelligence.  This can be understood as an analogy with the widespread idea of biological characteristics, such as the circumference of the head (measured with cephalometer) being supposedly linked with brainpower. French psychologist Gustave Le Bon (1841—1931) allegedly believed that women were the “most inferior forms of human evolution”, hence they “excel in fickleness inconstancy, absence of thought and logic and incapacity to reason” (Quotefancy, n.d.). Gustave Le Bon’s evolutionary theory about genders echoes Darwin’s conclusion that a man can attain “a higher eminence, in whatever he takes up, than women — whether requiring deep thought, reason, or imagination, or merely the use of the senses and hands” (Darwin, 1896, p.12).

That women held lower social status in Victorian Britain is mirrored in the narration about the social hierarchical system in Flatland by Square:  in Flatland each resident is a polygon; and one’s social status and intelligence are positively correlated with one’s number of sides and the equalness of one’s angles. The more sides one has, the higher one is positioned in the social pyramid. As a square, the protagonist is a middle-class mathematician. However, women are put in such a low rank that they are excluded from the two-dimensional hierarchy: women are nothing but “straight lines” (Abbott, 1884a, p.21); hence they are “needles” (p.25) from sideways and shrink to a point from the very front or back. Since women “have no pretensions to an angle”, they “are devoid of brain-power, and have neither reflection, judgement, nor forethought and hardly any memory” (p.27). Therefore, women have no “hope to elevate from the caste”; and “the very laws of Evolution seem suspended” (p.29) in their disfavour. Women are subject to emotions, not logic, and they identify others by feeling.

The “romance” narrative in Flatland develops with the key action in the second part of the book: Square is enlightened by a sphere from Spaceland, a higher dimensional world than the 2-dimensional Flatland. The sphere embarks upon a mission to Flatland to explain to Square about the existence of Spaceland. Square extrapolates his new knowledge of dimensions and insists that there must be a 4-dimensional terrain. However, this is rebuked by Sphere. The square offends both the sphere and the noble class in Flatland by challenging their superiority and privileges derived from the hierarchical system. The ruling class in Flatland sends Square to jail for his heretical proclamations about Spaceland.

The resistance from Sphere or from the governing class in Flatland to the acknowledgement of the existence of a higher dimension is assumed to come from their fear – the fear that the hierarchical system may be undermined, or even overthrown – leading to the loss of their social supremacy. It is the same kind of fear that drives the priests in Flatland to suppress the Colour Bill, an historical event detailed in the first part of the book. Starting from an unknown pentagon, a ‘chromatist’ – he painted himself and the new fashion spread quickly and widely in Flatland. When the lower-classed polygons begin to colour themselves too, the new colour culture in Flatland blurs the clear-cut social status based on being able to recognise by sight one’s number of sides and the equalness of one’s angles. The belief flies from mouth to mouth that “Distinction of sides is intended by nature to imply distinction of colours” (Abbott, 1884a, p.47). The lower-class polygons put forward the concept of social equality and asserted that there was not much difference between them and higher-class polygons. The Colour Bill was presented at the all-state assembly of Flatland, proposing that Priests would be painted in the same way as Women. The Colour Bill movement led to civil war and was ultimately suppressed.  The existence of the fourth dimension and the events of the Colour Bill revolt ridiculed Flatland’s (and therefore Victorian society’s) stiff social pyramid by exposing it as unjustified and illogical, which had been ingrained for generations as ‘natural’. The democratic movements in Flatland echo the fights for equality in the French Revolution, campaigns for political rights for working-class men in the 1840s, and fight for women’s rights  and the suffrage movement from the 1860s (Jann, 2008).

Flatland attracted attention from contemporary critics immediately after it was published. It was reviewed, in the journal Science in 1885, as “an amazing story” based on the “transcendental mathematical concept” of higher dimensions (Comment and Criticism, 1884). Robert Tucker in Nature recommended this humorous book for readers when they had a “leisure hour from their severer studies” (Tucker, 1884, p.77). Though puzzled and distressed with the “geometrical romance” of Flatland, (estimating only six or seven people in the US and Canada would enjoy reading it), the author of a review in the New York Times admitted that it made some apparent sense “in an appeal for a better education for women” (New York Times, 1885).

Though, as described in Flatland, women were confined by domestic duties and excluded from formal academic education, change involving women’s social status was happening in the 19th century. In 1846, the British government launched a teacher training program to take people older than thirteen years old into a teaching apprenticeship (Intriguing History, 2011). The paid teacher training scheme allowed women to enter the teaching profession, though female teachers received lower payment than their male colleagues. The suffragette movement, which fought for women’s voting rights, started in the 1860s. The Manchester Society for Women’s suffrage was established in January 1867. In 1866 and 1867, petitions promoting women’s suffrage were presented to the parliament by John Stuart Mill MP (1806—1873), who supported equality between the genders. However, all these petitions were turned down and women’s suffrage was not secured until August 1928, when Parliament passed the Representation of the People Act 1928 and British women won suffrage on the same terms as men.

In 1884, Abbott had his book Hints for Home Training and Teaching published to help parents who did home schooling with their children: boys and especially girls, since girls had much less chance to be admitted to the universities.  In the preface of the book, Abbott claimed that more educational opportunities then available for women “justifies the belief that in the next generation mothers will take a large part in the teaching and training of the young” (Abbott 1884b, p.12).  Abbott was one of the leaders of the local teacher training organisation. His effort in education reform was praised by the major female educators in Victorian times (Banchoff, 1991).  Thomas Banchoff (b. 1938) Professor Emeritus of Mathematics at Brown University argued that Abbottt was a strong believer in the equality of rights between men and women, especially in education (1991).

Flatland was written in the genre of science fiction (then called a ‘scientific romance’), introducing the mathematical concept of higher dimensions. By belittling women as one-dimensional line segments while the men were two-dimensional polygons, Abbott revealed the unfair adversity Victorian women were born into and had to live with. It is important to remember, however, that Flatland is a social satire. With his books and efforts in pushing female’s education reform, Abbott contributed to the fight against the social inequalities between men and women.  More than one hundred years on, the struggle for gender equality continues. For example, there still persists a gender wage gap (Advani et al., 2021); and in 2019, only 24% of the STEM professionals in the UK were women (Dossi, 2022). When reading Flatland today, enjoy the mysteries of the fourth dimension, but don’t forget the hidden message which strives for gender equality which is still as relevant today as it was in 1884.

 References:

• Abbott, E.A. (1884a) Flatland, a Romance of Many Dimensions. Oxford: Oxford University Press.
• Abbott, E. A. (1884b) Hints on home training and teaching. Available at : https://archive.org/details/hintsonhometrain00abbouoft/page/n9/mode/2up(Accessed: 15 August 2022).
• Advani, A., Smith, S,  Waltmann B.and  Xu, X (2021) Gender differences in subject choice leads to gender pay gap immediately after graduation. Available at:
  https://ifs.org.uk/publications/15657?gclid=CjwKCAjwo_KXBhAaEiwA2RZ8hLA1jyvTxlVOL5sbra60CbdtHxUA6FGah4s_l8QISyfhl7XqKUoAFBoCLYsQAvD_BwE  (Accessed: 17 August 2022)
• Banchoff, T.F (1991) Flatland: A New Introduction. Available at: https://www.math.brown.edu/tbanchof/abbott/Flatland/Publications/intros/banchoff.pdf  (Accessed: 17 August 2022).
• BBC (2019) Cambridge University marks 150 years of female students. Available at:
  https://www.bbc.co.uk/news/uk-england-cambridgeshire-49595057 (Accessed: 12 August 2022).
• Comment and Criticism (1885) Science, 5, pp.265—266.
• Darwin, C. (1896) The Descent of Man and the Selection in Relation to Sex. New York: D. Appleton and Company.
• Chas, M. (2019) The Extraordinary Case of the Boole Family, Notices of the American Mathematical Society, pp: 1853—1869.
• Dossi, S. (2022) How is the current STEM gender bias? Available at:
• https://www.imperial.ac.uk/news/233901/how-current-stem-gender-bias/#:~:text=Women%20are%20a%20minority%20in,much%20higher%20with%2046%25%20women (Accessed: 22 August 2022).
• Eliot, L. (2019) Neurosexism: the myth that men and women have different brains. Available at https://www.nature.com/articles/d41586-019-00677-x (Accessed: 12 August 2022).
• Flood, R. and Wilson, R. (2011) The Great Mathematicians: Unravelling the Mysteries of the Universe. Canterbury: Arcturus Publishing Limited.
• Ibanez, R. (2017) The Fourth Dimension: Is our Universe the Shadow of another? Barcelona: RBA Coleccioables, S. A.
• Intriguing History (2011) First teacher training scheme 1846. Available at:
  https://intriguing-history.com/first-teacher-training-scheme/ (Accessed: 17 August 2022).
• Jann, R. (2008). Introduction and notes: Flatland: A Romance of Many Dimensions Oxford: Oxford University Press.
• New York Times (1885). February 23, p. 3. Available at: https://www.math.brown.edu/tbanchof/abbott/Flatland/Reviews/1885nytimes.shtml  (Accessed: 28 August 2022).
• Quotefancy, (no date) Available at: https://quotefancy.com/gustave-le-bon-quotes  (Accessed: 11 August 2022).
• MacTutor (2005) Edwin Abbott Abbott [biography]. School of Mathematics and Statistics, University of St Andrews. Available at: https://mathshistory.st-andrews.ac.uk/Biographies/Abbott/ (Accessed: 27 August 2022).
• Tucker, R. (1884) Flatland, Nature, 27 Nov, pp.76-77.

This blog post was written by Roshani Senior, an OU mathematics graduate who undertook a 3 month virtual internship with the school of mathematics and statistics. Roshani’s placement was focused on developing external engagement materials based on mathematics.

In this blog post, Roshani writes about under represented voices in mathematics, which is a particular interest of hers. She reflects that as a school student of Indian origin, she learned about  European mathematicians but it was not until she left school that she learned about the many contributions of Indian mathematicians. She believes that mathematics should be taught in an inclusive way, sharing the experiences and achievements of mathematicians from across different cultures.

This blog focuses on the contribution of two Indian mathematicians to the concept of zero.

Aryabhata   

                                                                        Brahmagupta

The move from zero as merely a placeholder by the Mayans and Babylonians – a tool to distinguish larger numbers from smaller ones to a digit of its own was established in India by a man named Aryabhata in the 5^th Century. A mathematician and astronomer, Aryabhata contributed multiple mathematical concepts, crucial to maths as we know it today, including the value of pi being 3.14 and the formula for a right-angled triangle. The prior absence of zero created difficulty in carrying out simple calculations.

Gwalior, India

Following this in the 7^th century a man known as Brahmagupta, developed the earliest known methods for using zero within calculations, treating it as a number for the first time. The use of zero was inscribed on the walls of the Chaturbhuj temple in Gwalior, India. Carved into a wall the numbers 270 and 50 can be seen today and have been established as the second oldest recorded zeros in history. The city of Gwalior was designed so that the gardens around the temple were large enough so that each day the gardens would produce enough flowers to create 50 garlands for the employees of the temple. When the temple was built this was inscribed on the walls and it is this 50 that can be seen, annotated almost as we would write it today. 

What we now call zero in English, Brahmagupta named “shunya” or “sunya”, the Sanskrit word for emptiness or nothingness.

Aryabhata and Brahmagupta wrote their works in Sanskrit, an ancient and classical language of India. Their use of numbers would have looked quite different to what we use in English now. However, Sanskrit had a large

influence in how the English numeric system is written and so there are quite a lot of similarities.

Numbers 0 to 9 in Sanskrit 

Within Indian culture there is an idea of one having a “nothing” or a void inside of yourself. Long before the conception of zero as a digit, this philosophical concept was taught within Hinduism and Buddhism and practised through meditation.  The ancient Hindu symbol, the “Bindi” or “Bindu”, a circle with a dot in the centre symbolised this and was what probably led to the use of an oval as the symbol for the Sunya. It has been suggested that this cultural and philosophical influence on the concept of zero is what allowed India to develop what previous civilizations did not think of.

Brahmagupta was also the first to demonstrate that zero can be reached through calculation. He wrote these rules in his book the “Brahmasphutasiddhanta”. He was therefore able to make another important leap – in the creation of negative numbers, which he initially called “debts”. Brahmagupta placed small dots above numbers to indicate they were negative, unlike today where a minus symbol is used. The use of negative numbers was shown in “Brahmasphutasiddhanta”. Brahmagupta also demonstrated their use to produce the quadratic formula. and demonstrated rules for calculations involving both negative numbers and zero.

His rules were as follows:

Addition and Subtraction with zero and negative numbers:

• When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.
• A debt minus zero is a debt.
• A fortune minus zero is a fortune.
  Zero minus zero is a zero.
• A debt subtracted from zero is a fortune.
• A fortune subtracted from zero is a debt.

Division and multiplication with zero and negative numbers:

• Positive or negative numbers when divided by zero is a fraction with zero as denominator.
• Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
• Zero divided by zero is zero.
• The product of zero and a debt or fortune is zero.
• The product of zero and zero is zero.
  The product or quotient of two fortunes is one fortune.
• The product or quotient of two debts is one fortune.
• The product or quotient of a debt and a fortune is a debt.
• The product or quotient of a fortune and a debt is a debt.

When Brahmagupta attempted to divide 0 by 0, he came to the result of 0. However, most modern mathematicians would argue that 0 divided by 0 is undefined, or an “indeterminate form”. Despite this outlier, the rest of Brahmagupta’s grasp on the number zero is exactly how we conceptualize it today.

The concept of zero gradually moved East into China. Then West to reach the Middle East. And finally, over half a century from its conception, the zero made it to Europe, where its importance was finally recognised by the Western culture that previously frowned on the idea of nothing, referring to it as meaningless or even satanic. In 1200 AD, Italian mathematician Fibonacci, a man who has been considered the ‘most talented western mathematician of the middle ages’ wrote of Indian Mathematics and their use of zero:

Despite the number zero having quite literally no value, its concept has allowed mathematics to develop into what it is today. Its curation led to the three pillars of modern mathematics: algebra, algorithms, and calculus. The use of calculus (the mathematical study of continuous change), which the zero is crucial for, has allowed engineering and modern technology to be possible. The use of zero and one within the binary system is what made computing possible. So, without the invention of zero much of what we know today would not have been possible. The device you are reading this on would not have been able to be invented, if not for Aryabhata, Brahmagupta and India’s fascination with the idea of nothing.