Rebecca Would is  a year 12 work experience student who has visited the Maths and Stats department of the OU. She interviewed Karen Vines, a senior lecturer in statistics, to find out about her and her work

Maths and Stats students at the OU will have studied with Karen on modules such as M346 and M140.  She also wrote one of the M249 Practical Statistical Modelling books. 

This is Rebecca’s report:

Let’s start with my favourite opening question:  What is your favourite vegetable?

Karen: Avocado, or carrot.*

Karen in Orienteering mode

 Are there any particular bits of stats that are of interest to you?

I’m reasonably eclectic. There are many different areas and I’ve moved around a bit, I haven’t really got a main focus. Possibly the most fun has been what’s known as sonification, turning graphs into sound.

How do you do that?

Usually you start with a scatter plot or line graph, then equate pitches with the numbers, and each graph produces its own unique tune. [There is a 5-minute video about this project here].

I like to think as a statistician I have more fun than a lot of my colleagues, as I really can pick and choose, there’s data on everything. And it’s nice to know that if I were to leave I’d be able to find somewhere, everyone needs a statistician – I’m not going anywhere, but it’s good to know the option is there.

How do you get to be a statistician at the OU?

Mostly by accident. My career planning stopped at age 21: go to uni, get a degree. I did maths but no stats in my degree, unfortunately in first year no stats was offered, so when I started to become interested in second year I was told I couldn’t as I didn’t have the background. When I got to the end of my degree, I briefly flirted with the idea of becoming an actuary, and was completely put off when I went for an interview and someone said, “Now I’m more senior I get to play on the stock market!”. Then I thought about doing an MSc in statistics or operational research, and having done operational research in my degree, I chose to do statistics. After that I was fed up of being on the education treadmill, and so went to work as a practicing statistician at a medical research place (MRC Environmental Epidemiology Unit). There was a good mix of medics and statisticians and I got to learn a lot off my senior colleagues. They were doing a lot of fun things, some less fun things, but a lot of fun things. I stayed there for almost 2 years, by which point I decided that yes I did want to do a PhD.

What was the biggest difference between doing a masters in stats and an actual job, if there was much of one?

I think there was, yes. In a masters you could do this, you could do that, you could do a regression line, I could do a T-test… but it’s knowing what to do when someone comes along saying “here’s some data.” It’s a skill certainly, and one I hope I’ve picked up.

After that I knew I wanted to move, so applied to the OU and got in as a statistical advisor, with the intention of staying 1/2 years. Stuff happens, and here I am.

Having started with the aim of 1/2 years, how long have you been here?

23. And a half.

After this Karen and I talked a lot more about university and how it has changed over the years. It was really interesting talking to her, and she has a true enjoyment for everything. I’ll take away from this the drive to keep researching more, and her love of stats.

Thanks to Rebecca for writing this and Karen for taking part.

*Cathy comments here as a frivolous editor that we can say that Mathematics deals with certainty, while Statistics deals with uncertainty: Karen shows she is a true statistician in replying to a ‘what’ question with an ‘or’ answer.

Dr Cathy Smith, who leads the maths education team here at the OU, has recently had a paper published in the British Journal of Sociology of Education – congratulations Cathy! In a nutshell (if it’s possible!), the paper concerns the different forces at play when students make decisions about studying – and continuing to study – Mathematics and Further Mathematics at A Level. Here is a link to the full text of Cathy’s paper.

When I read the paper I started to think about how difficult it is for students (and teachers, and anyone else) to navigate the competing narratives that exist about mathematics, mathematicians and maths education. In so many areas of our society, we see messages that maths is the gateway to employability, financial success, boosting the economy, vital for functioning in society on any level. Yet we also see messages that maths is difficult, elite, beautiful, eccentric, invisible, unattainable – like some kind of fairy on a blackboard. Of course it is possible that maths is all of these things and more; “maths” can have many different interpretations and the distinction between “school mathematics” and “mathematicians’ mathematics” has been made numerous times in academic literature, but rarely in news stories.

The way we talk about and represent mathematics (in the news, online, in films, memes etc.) is important because it (re)produces stereotypes leading to a narrow, gendered or clichéd representation that could affect young people’s meaning-making, development and identification with mathematics and as mathematicians. In short – it affects what young people will choose to do next. As a society, we sometimes seem to determine a person’s value in the world too prominently by their profession, education and skill level. These are clearly important and the OU is established to support people in furthering their education, for whatever reasons they choose. But some politicians use the phrase ‘highly-skilled’ as shorthand for ‘the right kind of’ people (see Iain Duncan-Smith’s view here). In the context of young people making decisions about whether or not to study such a gateway subject as mathematics, the representations of maths become a matter of social justice.

For my masters thesis I studied the representation of mathematics and mathematicians in popular culture and the news media. This table shows some of the most common ways I found these two themes to be represented:

There are whole host of contradictions within these representations – maths is essential yet invisible, ubiquitous but inaccessible, rule-based but eccentric. The stereotypical mathematician is also a contradiction; a white, heterosexual, middle-aged, middle-class man is often the ‘default person’ – so far so good for that group of people – but when it comes to a mathematician, he has to have something ‘other’ about him as well, like messy hair, messy suit, silly heavy glasses or a reclusive personality. It goes without saying that most mathematicians are not represented by the stereotype.

Katherine Johnson, NASA

We are presented with a dilemma; on one hand, mathematics is a key to success and on the other, it is accessible and enjoyable for only a small section of society.  When adolescents are deciding whether to continue studying maths to A Level or at University, they need to consider not only what will help them ‘get ahead’ and what they will enjoy, but also whether they want to brand themselves as a ‘scruffy loner’ – in the case of white middle-class males – or as ‘the odd one out’ – in the case of everyone else. Dominic Cummings’ recent call out for ‘data scientists, project managers, policy experts, assorted weirdos…’ to apply for top jobs at No.10 strengthens the message that those who are destined for success have something unusual about them.

Another contradiction stems from the many different meanings of the word ‘maths’. What is being referred to in the message ‘you need maths to get a good job’ is most commonly school maths and, specifically, the actual qualification. Yet it is mathematician’s mathematics – maths play, exploration and investigation – we mean when we say ‘maths is beautiful and everywhere’. These two areas may overlap, but the curriculum and emphasis on exam results mean that maths qualifications and mathematician’s mathematics are often two very extreme ends of the spectrum.

Visitors using Zome at the Olafur Eliasson Exhibition, Tate Modern, London

Finally, there is the dilemma of ubiquitous maths. The theme for this year’s International Day of Mathematics is Maths is Everywhere. It is often said that maths can ‘explain the world around us’. As someone who has always enjoyed maths, I agree with this – I love finding out new patterns and puzzles in all aspects of life. But many people suffer from maths anxiety and, in saying ‘maths is everywhere’, are we also saying some people do not get to “understand” the world? If we are espousing the message that maths is everywhere, we should probably also make sure that it is there for everyone – practically and aesthetically – not just an elite few.

Bibliography

Barwell, R. and Abtahi, Y. (2015) Morality and news media representations of mathematics education, Proceedings of the eighth international mathematics education and society conference: 298-311

Civil, M. (2002) Everyday mathematics, mathematicians’ mathematics, and school mathematics: Can we bring them together?, In M. Brenner and J. Moschkovich (Eds.), Everyday and academic mathematics in the classroom. Journal of Research in Mathematics Education Monograph 11, 40-62

Criado Perez, C. (2019) Invisible Women: Exposing Data Bias in a World Designed for Men, London: Chatto and Windus

Damarin, S. (2000) The mathematically able as a marked category, Gender and education, 12(1), 69-85

Lewis, G. and Forsythe, S. (2018) Factors for and against choosing to study mathematics post-16, Mathematics Teaching, 262, 10-13

McLeod, D. B. (1992) Research on affect in mathematics education: A reconceptualization. In A. D. Grouws (Ed.) Handbook of research on mathematics teaching and learning, 575-596. New York / Toronto: Macmillan / Maxwell Macmillan Canada

Mendick, H. and Moreau, M.-P. (2012) New media, old images: Constructing online representations of women and men in science, engineering and technology, Gender and Education, 25(3), 325-339

Moreau, M.-P., Mendick, H. and Epstein, D. (2010) Constructions of mathematicians in popular culture and learners’ narratives: A study of mathematical and non‐mathematical subjectivities, Cambridge Journal of Education, 40(1), 25-38

National Numeracy, Manifesto for a numerate UK [Online] Available at: http://www.nationalnumeracy.org.uk/sites/default/files/media/manifesto_for_a_numerate_uk.pdf

Picker, S. and Berry, J. (2000) Investigating pupils’ images of mathematicians, Educational Studies in Mathematics, 43(1), 65-94

Smith, A. (2017) Report of Professor Sir Adrian Smith’s review of post-16 mathematics, London: Department for Education

Smith, C. (2019) Discourses of time and maturity structuring participation in mathematics and further mathematics, British Journal of Sociology of Education

Here in the OU Maths Education department we have been having some interesting discussions about the specific words we use when talking about teaching practice. In this post we discuss the variety of ways in which examples are used in mathematics teaching, either consciously or unconsciously.

As teachers, we use examples all the time – it is almost impossible to help someone learn a new concept without giving an example on which to hang it. Without examples, the mathematics, the definition or the concept is abstract – there’s nothing to see, touch or do, and we often can’t really tell whether it’s true or makes sense.

Here are some of the ways in which examples are used in classrooms. Thinking about them explicitly can help us to check we are making the best use of examples as a tool for understanding. (In a truly meta mode, we give examples of each type of example).

Modelling examples
These are the classic ‘Worked examples’ where a process or procedure is modelled to students so they can see it in action. For example, working through an application of the quadratic formula, using numbers instead of a, b and c.

Particular examples / specific examples
A particular example (sometimes referred to as specific example or specific case) is a single example where any generality has been removed. It can be used to illustrate a more general definition or concept. For example, a 3 × 3 square is a specific example of the more general class of squares. In the literature we sometimes see ‘special’ used in a similar way, since every object is special somehow. This means it is sometimes confused with ‘special cases’ or ‘special kinds’.

Special cases / special kinds
Special cases are those lovely examples that can sometimes catch us out and lead us to believe processes always ‘work’ (for example factorising only works for special kinds of quadratic equations). Special cases have interesting properties or additional constraints applied to them, and so we can view special cases as a subset or subclass of a larger set or class. For example, in Geometry, the squares are a special kind of rectangle. In Number theory, 2 is a special case of the prime numbers.

Peculiar examples
Peculiar examples are those  particular or specific examples where something interesting or unusual is happening. Let’s use the example of a dynamic triangle, where all the vertices can be manipulated.

You could drag one vertex so close to the opposite side that the triangle looks almost like a straight line but is, in fact, a very thin triangle. This type of peculiar example is sometimes called a degenerate example.

Another way of manipulating a triangle is to drag a vertex so that the triangle is isosceles. Of course what is ‘peculiar’ to one person might not be to another; it depends what examples each person considers as part of their concept image (Vinner and Tall, 1981).

Non-examples
These are brilliantly useful and often underused. Even outside of mathematics, it is sometimes clearer to define what something isn’t than what something is. Non-examples deliberately do not hold certain properties or do not meet required definitions, so looking at non-examples serves to clarify the boundaries of a concept or a definition (Bills et al., 2006) . Non-examples are most useful when presented alongside specific or particular examples, because we can see the contrast clearly.

Here are some examples and non-examples of polygons: A closed plane shape with straight sides.

Generic examples
A generic example is a specific example that is used to illustrate steps of reasoning that hold for the general case. The generic example is an object that is not there in its own right, but as a characteristic representative of the class. We often use these without thinking about it, e.g. we might draw a 3 by 4 rectangle and say it has 3 rows of 4 unit squares so its area is 12, and take that as a proof that this is the area formula for any rectangle.  We have not actually reasoned with a general rectangle, but with a specific one that takes the role of a generic example. To turn it in to a formal proof, we need to be sure that the same reasoning could be adapted to work with any rectangle. Reasoning with a generic example works when there are no peculiar cases to take into account. It is useful to do when it reduces abstraction for learners.

Counter-examples
Not to be confused with non-examples, counter-examples are mainly used when working with theorems, conjectures and proofs. They are specific cases for which a conjecture does not hold true and, as such, counter-examples may refer to special cases. For example, a counter-example for the often-held belief that all prime numbers are odd can be found in the prime number 2, which is even.

A counter-example for the conjecture that only quadrilaterals with line symmetry can be split into two congruent triangles is a parallelogram, which has rotational symmetry.

We hope you’ve enjoyed thinking explicitly about examples. If you have any questions or comments you’d like to share about your own experience of using examples, please use the comment space below.

References

Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A. & Zaslavsky, O. (2006).  Exemplification in Mathematics Education. In J. Novotna (Ed.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague, Czech Republic: PME.

Vinner, S., and Tall, D. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity, Educational studies in mathematics, 12(2), pp. 151-169

Rebecca Would is a sixth-form work experience student who visited the Maths and Stats department of the OU and supported the School Maths Conference. This is her report.

In July the OU ran a maths conference in conjunction with some local secondary maths teachers. The first day gave the 25 or so year 11s that turned up a brief introduction to some of what is covered in the year 12 A-level maths syllabus. This would have been a really beneficial experience for the year 11s, as it means the content will be less new in the first term and easier to understand.

We started off with surds and indices, the basis of all good algebra and geometry, and moved through quadratics, trigonometry and vectors giving the students the grounding in the core maths that’s needed throughout the A-level. The students got on really well, both with the work and together across different schools with new people.

The students were shown examples and then set tasks in a work book, while me and the other teachers went around offering advice and hints to the students when required and talking to them about their A-level options. I was able to share my experience at A-level with the students, which I hope was helpful, and also some of the things I’ve learnt from my A-level. Advice such as completing the exercises or keep your graphical calculator on you sounds like a teacher is being fussy but coming from another student presently doing their A-levels can actually be more compulsive. I was also interested to hear their thoughts on GCSEs, especially with the new content and grades that came in last year, although they haven’t received results back yet so this conversation went only so far.

I was also encouraged to offer more general A-level advice, and the students I spoke to seemed to appreciate these extra tips – I certainly would have done had I known them before entering year 12. These included:

• blocking out time at the start of the year for subjects so you know if you spend an hour chatting with friends, which is a perfectly reasonable thing to do, what subject you possibly should have been spending that time on this also helps to prevent a last minute rush for doing homework;
• not to start worrying if something you grasped at GCSE level throws a curveball at you and you can’t get your head around a particular topic;
• spend some of that time with friends, it is unreasonable to expect yourself to work at 100% every day,
• if you have a rough day do take some time to chat with friends or read a book to get your head back in the right mental space for doing work.

A guest speaker, Marc Pradas, joined us from the OU to speak about his research and chaos theory. I for one, found this fascinating as he went into some detail about the butterfly effect and why that works, as well as giving the background to ‘non-linear directive dynamic systems’ which is a lot more fun than it sounds!

For the second day  I was primarily based with the year 10s. This was for three schools and the session run by Kristen Coldwell from AMSP (advanced mathematics support programme). She started off with some logical thinking starter puzzles while everyone arrived. She then spoke for a bit about the benefits of maths as an A-level, and the different careers in which it can be used, as well as some of the courses it is needed for to get on at university.

After this Kristen ran a dragon race activity which required the students in teams to build a dragon by answering maths puzzles. This was a great team building exercise and the students did really well in answering the range of maths problems presented to them.

Post lunch the year 10s and 11s combined for a code breaking exercise run by Charlotte from the OU. This was good as we mixed the teams up between years 10 and 11 and the had a chance to discuss why they’d chosen maths or why they might choose maths, as well as had the opportunity to work together, which all students took really well and used the opportunity.

Overall the two days were really good and we hope that the students and teachers there too agree in the benefit for their students.

By Work experience student Rebecca Would

This blog post was written by OU student Jim Darby.

We have republished this (with permission) from Jim’s personal blog. Jim has encapsulated the learning outcomes of reflecting on his own thinking and appreciating the range of learners thinking. Jim has described a rich working relationship with the school he works with. Our students vary from those who have such opportunities to those who work with one or two learners perhaps from their own family.

After sending in my final EMA (End of Module Assessment) for ME625 I find myself reflecting on it all.

I ought to begin by introducing myself. My name is Jim Darby and I work full-time in the computing industry specialising in security of the computing infrastructure of large commercial enterprisies. I started with the Open University (OU) doing an Open Degree with Spanish as the first option because the local college had stopped all adult education. After completing Spanish I began to look for other things. Astronomy sprang to mind but I wanted to revise maths first. Maths (MST124, MST125 and MST224) followed then I moved onto Astronomy (S282) and Planetary Science (S283).

[Brief aside: For non-OU students you’ll see a lot of references to letters followed by numbers. These are the identifiers for specific courses, more accurately called modules. Also “student” refers to an Open University student and “learner” refers to someone learning mathematics that the OU student is working with.]

About this time I had begun volunteering at a local school, initially to work with mentoring learners (pupils). A new trust had recently taken over the school and were very keen to improve learner outcomes, with mathematics identified as a key area. The school had identified several learners in need of additional support but I felt woefully inadequate for the task, then I saw Developing Mathematical Thinking in Schools (ME620)…

Reading the course details it seemed pretty much ideal because it was the underlying thinking that I wanted to address and develop. I made the choice and took it.

It utterly changed my views on learning and doing mathematics.

I have to be very clear here. This is not a teaching course. I’ve found a fair amount of confusion about this: firstly myself and then from the teachers I’m lucky to work with.

What the course is about is the study and development of mathematical thinking both by the OU student taking the course and by the learners they’re working with. The modules (ME620 and the ME625) are focused on investigating and developing how we think about mathematics and how we learn it. The modules are based on highly reflective work where the student considers how they work to solve specific tasks and later on how the learners go about the same task. This is reflected in the assessments where questions are often in pairs to allow students to compare their processes with those of their (typically) younger learners when faced with a similar task.

There are major differences between these courses and teaching courses. These differences are very important. It would clearly be unacceptable to spend an hour long maths lesson focusing on a tiny proportion of a class and ignoring the rest. With the ME-series (Maths Education) modules we work with small groups or (most commonly) one-to-one to conduct an in-depth investigation of their learning. The emphasis is strongly on encouraging them to solve the problems their way with as little scaffolding (support) as is possible. In fact, revealing where their processes differ to that of us, the OU student, is an essential part of developing understanding of how everyone learns.

I am extremely lucky in having a great and highly-cooperative school to work with. Without their support I would not have been able to complete the courses. They lent me some amazing learners with whom it has been a pleasure to work. To be able to work well on the course you will need access to learners of mathematics (of any age) but they will need to be in a small group (often one-to-one) to allow the “deep dive” of what’s happening: a class of thirty just isn’t suitable.

Some of these learners had difficulty in accessing mathematics and presented with widely divergent levels of achievement, motivation and engagement. I was able to investigate their approaches to mathematical thinking and this helped me with the modules and (more importantly) the learners with their understanding of mathematics. Being able to “deep dive” their mathematical thinking using the ideas, concepts and models from both modules over the course of a year gave me a wide range of strategies to help them overcome some or their barriers.

It’s certainly possible to use just a single learner on the courses, but personally I found having varied learners in the school beneficial in contrasting mathematical thinking: an essential core of the modules. The point is to investigate how the learners’ approach solving mathematical problems and why they make the choices they do.

Having a basic understanding of the learner’s current achievements is essential to session preparation. They need to be challenged, but not too much.Getting the level right is often difficult, especially if you have a group you haven’t worked with before. Set it too easy and they’ll just march right through it revealing little about their problem solving processes. Set it too hard and you may find, as I did, that you’ll end up with a student sitting under the table glaring at you! If that happens you may need to reduce the task level…

However, once you’ve established a good working relationship with your learners then the ME courses are immensely rewarding. I found that working one-to-one with those learners having problems accessing classroom mathematics often helped them overcome the issues they had with learning mathematics and allowed them to make additional progress. I used many (if not all) of the modules’ concepts to analyse these barriers and assist the learners with breaking them down.

In analysing how effective various strategies were, I was able to gain substantial insight into how others access mathematics and the obstacles they face. The differences to my own learning processes were a great surprise and to me this was by far the most important end result of the modules.

Additionally, in a few cases the learner’s issues surfaced as behavioural issues, often borne of frustration. However once the learning issues were reduced their behaviour improved. Similarly for those becoming bored in classes and wanting a greater challenge I was able to provide tasks that deepened and broadened their understanding. Both of these are of great benefit to the learners, myself and the school.

I found one of the major parts of the modules is the one-to-one time to analyse the learner’s thinking in great depth. This would be substantially harder (if not impossible) in a class of 30-plus but it is an essential component of the ME courses. The initial analysis occurs during a session with a small (ideally one) number of learners. Later a more reflective account plays a central role in the course assessments. It is very much expected that this reflection will enhance the student’s understanding of how we think and learn mathematics in general (for ME620) and Algebra specifically (ME625). There are many “module ideas” and their use in developing this thinking, both in terms of practice and in terms of the reflection by the OU student. These ideas and techniques are useful both as approaches to support learning and to describe what actually happened.

I was able to have hour-long sessions with my learners and I would suggest that this is more-or-less the ideal length. Shorter and you don’t get enough time to go through a task, longer and your learner’s attention is going to fade. I believe that these small group sessions were well worth the time and effort because they enabled the less confident learners to better long term participate in mainstream education. With the more confident ones it allowed the exploration of topics in greater depth. Ultimately it is worth investing in for both the course student (you) and the learners.

Full-time teachers or TAs (Teaching Assistants) will find it difficult to make the time for these small-class sessions. You should be aware of this before beginning the course.

If you’re considering working (volunteering) with a school then it is essential to have a good working relationship with them. You should be familiar with how the school works in terms of lesson planning, timetabling and general ethos. It is a privilege to be able to work with learners so you’ll need to ensure that it all goes smoothly. You will almost certainly need to obtain records from the Disclosure and Barring Service (DBS) as well as being familiar with the school’s safeguarding process and principles. It is critical to be able to work well with the school’s Mathematics Department as well as its senior leadership team.

Returning to the theme of understanding how others learn and think about mathematics I would like to highlight an example of how my views were so radically changed. A few of the learners were finding fractions hard to work with. Before undertaking the modules I would have thought that this was “obvious” and have given a perfunctory (and ineffective) description. However, by employing skills learnt on the modules I was able to provide far more useful advice by first determining what they already knew and then working with them to expand that into a deeper and broader understanding. This was a very interactive approach often starting with physical models used to ensure that the core thinking was a sound foundation before building on that. At each step I would ensure that they were not repeating what I had just said but instead had grasped the underlying concepts. We would then use these new concepts and build upon them to the next stage. I really enjoyed the time that we had to explore how they thought about the concept of fractions and how they work.

I must end with a young learner’s comment made during my final EMA. After working with her for about three quarters of an hour she appeared most upset. However, she smiled as she said “You tricked me into learning ALGEBRA!” 

To me, that’s the ultimate aim of the ME modules.

This post was written by Nazanin Nikanjam, a mathematics education student who received the 2019 Open University Stanley Collings prize for her writing.

I have been teaching Mathematics and English as an Additional Language for the past 26 years, with the last 14 in schools offering the International Baccalaureate Programmes in London, UK and Bologna, Italy. When I was at university studying Electrical Engineering in the early 90s, I started a part-time job as a teacher. I fell in love with it and have remained in teaching ever since. I believe Mathematics is often taught in a way that discourages learners to think critically, and I have always been interested in creating resources and engaging my learners in developing a deeper understanding and appreciation for this subject. However, it wasn’t until I came across the two Mathematics Education modules ME620 and ME627 during my studies at the Open University (for the BSc in Mathematics and its Learning) that I was able to understand and analyse my own thought process and transfer this learning to my classroom. Also, prior to studying the ME627 module (Developing Thinking in Geometry), I had used GeoGebra only as a graphing tool, and was not familiar with its many other applications. I was quite excited to learn how to use other features of GeoGebra to enhance my learners’ experiences and facilitate their mathematical thinking. This is why I decided to use it in my End of Module Assignment task.

The plan

I worked with a group of 15 year-old learners for this task, the majority of whom had a fair understanding of shape and measure, but had mostly been practising algorithmically with a lot of repetition and substituting in formulae (e.g. to find the area of a composite shape). My interest in working with this group of students came from the fact that they had not previously been exposed to tasks that require higher order thinking. My aim was to engage them in seeing beyond the obvious.

I chose to focus on 3 learners who had very different approaches. I was curious as to how their different ways of thinking would influence one another. I planned to provide these leaners with one task that emphasized analysis, and the other synthesis. I chose task 4.4.2 (Sliding ladder) from the book Developing Thinking in Geometry (Johnston-Wilder and Mason, 2005), where learners are asked to make a conjecture about the path a ladder’s midpoint traces as it slips from the wall to the ground. I anticipated that my learners would be encouraged to use their powers to analyse (both the parts and the whole) throughout this task. For my second task I chose An Unusual Shape, an exercise I found on the nrich website at https://nrich.maths.org/2161. This task I believe calls for synthesis as it requires learners to bring together their understanding of several different concepts.

For the first task, I planned to extend it by asking my learners to explore what path any point on the ladder will trace out as it falls. I also created the stimulus on Geogebra for them. The aim was to facilitate the learners’ thinking and get them to move between at least two of Enactive-Iconic-symbolic modes (Johnston-Wilder and Mason, 2005).

I planned to start the second task by providing the diagram first without the instructions and asking the learners what they may be asked. I anticipated that this would encourage them to use their powers to ‘see’ connections, and ideas come to their attention organically and through sense-making. After this I would give them the instructions and the freedom to choose how they would like to approach it. I would scaffold their thinking by giving them prompts in the form of questions that would encourage reflection, such as ‘What else can you see?’ or ‘How do you know?’

Finally, I expected that Geogebra could help them visualize the different paths in the Sliding Ladders problem, so that they can make better sense of what was happening. I also expected that it would motivate my learners and allow them to focus on dimensions of possible variation by drawing their attention to what is changing and what is staying the same.

The experience

At the start of the Sliding Ladders problem, the learners were presented with a slide containing the task instructions that appeared one at a time. The first step was for them to discuss their thoughts. Two learners agreed that “the mid-point will trace out a straight line perpendicular to the ground”. The third learner suggested that “it will curve down … like a slide, in and out”. After a short discussion they each drew a diagram and then shared their thoughts again. They chose to start with a 5-meter long ladder (specialize) and each made a different generalization.

Image 1 – task instructions for 4.2.2 

It was clear to me that they were finding it easier to have the diagram to manipulate (iconic mode). One of them made a conjecture (and sounded very excited) that “it’s a circle! And it’s radius is half of the ladder!”

When they started using GeoGebra, there was a clear shift in how Learner 1 was engaging with the task, from being quite passive to seeming motivated, which emphasizes the importance of providing the opportunities for a variety of preferences. Using GeoGebra helped them explore the path for other points.

Image 2 – screenshot of the GeoGebra worksheet

They made a conjecture that “the path is an ellipse, except for the mid-point that follows a circle”. Learner 3 was the only learner who then moved to symbolic mode in recording her thoughts and tried to verify the conjecture for the mid-point “I’m going to find a rule”.

Image 3 – learner 3 using symbols to record the conjecture

We ended this task after a discussion in response to one learners’ question: “What happens if the ladder is not straight?”

In the second task the diagram was the only item on the board at first and then the rest of the instructions appeared one at a time. My first question with only the diagram on the board was “What do you think you may be asked about this diagram?” One learner responded: “how many trees can you plant in the light green?” and another one said: “find the area of the cut grass”. After displaying the instructions, the group started to discuss their approach and moved on with drawing their own diagram and dividing it into sections.

Image 4 – task instructions for An Unusual Shape

As learner 1 was finding it difficult to visualize the rope, I offered him a piece of string (enactive mode) which he started to manipulate and make sense of the now sectioned diagram.

Image 5 – the learners working on their own diagrams

Once they were convinced that they had found the answer to questions 1 and 2, learner 2 attempted to answer question 3 through trial and error, and learner 3 conjectured that “it will be a larger area if the rope was tied to a point in the middle of the shorter side”. She was excited about this conjecture and went to the board to explain her reasoning to everyone in the class: “Look! If you split the 10 ft side by two and tie the rope there, you’ll get the largest area!”

Image 6 – the learner explaining her conjecture on the board

Before question 4 was displayed, the group were already discussing it: “is there a point that gives an even bigger area?” They decided to think of the distance at which the rope is tied on the 15ft side of the shed and work out the area from there. Answering this question proved to be a challenge for them, so once they had an algebraic expression, they decided to use Google Sheets to conjecture: “It has to be tied to the corners!”

Image 7 – deducted formula used to calculate the area on Google Sheets

The final question sparked a very interesting discussion about a possible application of this task being for a gardener to find out where to install a sprinkler system, or the best position of a router in a building with an obstruction.

Reflection

My learners’ response to both tasks was positive with instances of surprise, engagement and a final sense of accomplishment. When leaving the classroom, a few learners exclaimed “I liked this!”, and this was my biggest reward. I believe there are a few factors that contributed to the success of this exercise. Firstly, the open-ended nature of these tasks and the opportunities to ‘discover’ enabled the learners to exercise their powers. They were encouraged to imagine and express their thoughts at the start of the first task, and the possibilities of what there is to be found in the second one. There were several opportunities to move from specializing, to generalizing, both to find a solution to the presented problem, and to satisfy their own curiosity and assumptions (e.g. “Is there a point that gives an even bigger area?”). They had opportunities to make conjectures (there was enough challenge to invoke their curiosity, but not too much to kill their interest) and to ‘talk’ to convince each other and even the whole group with confidence. The learners were also able to recognize and use their power to organize their thoughts, to help them come to a solution (using Googlesheets).

Another contributing factor was that in planning around these tasks, I had in mind the principles that make teaching more effective (NCETM, 2007). I believe the way the students engaged with both tasks is testimony to these principles. For example, I used cooperative small groups, both tasks involved higher order questions and encouraged reasoning, and they built on the knowledge the learners already had.

The tasks were similar in how they were both open-ended (the second task with more scope than the first) and facilitated the use of a strategy known as ‘Do, Talk, Record’, which in turn helped the learners progress through them by sharing ideas and building on each other’s powers and strategic thinking and not feel disempowered by their individual absences. Both tasks led the learners to use their imagination and to ask ‘what if …’ questions and look for ‘another and another’.

Both the learners and I found the second task more interesting. I was merely an observer during this task, with very little involvement. I felt all I had to do was to throw in another question or give them a nod to continue to explore. Perhaps what they found more interesting about this task was the presence of a more tangible context and the opportunity to think of their own authentic examples.

What surprised me about the learners was the many ways in which they stepped into the problem. For example, learner 1 seemed to find it difficult to imagine without having a physical stimulus (enactive), he also seemed to be more concerned with the whole, whilst learner 2 would rely on imagery almost all the time (iconic). She used visualizing to step into the problem, model it and plan ahead (Piggott and Woodham, 2009).

Learner 3 on the other hand was keen to express her thoughts in symbolic form, and it was evident that she was making intuitive jumps. She was also more reflective on her own thinking. Observing her reminded me how easy it is to take your own powers for granted and expect everyone to ‘see’ what you see. She also needed her space to work on her own in the recording stage, and join the group for the talking and doing.

My approach to geometry and developing thinking in geometry has changed as a result of completing this module and practising the ideas and framework provided. I used to lack the awareness of how we process concepts in geometry, and the terminology and structures that can help to identify these processes. I now find it easier to approach conceptual problems, from stepping into them to thinking of other ways to find solutions. As a result, I find myself to be a much more effective facilitator to my students and how they can develop their thinking. Reflecting on my development both as a leaner and an educator, and looking back at the introduction of this module, I was able to recognize that geometry is more than our understanding of space, and is linked to our brain power and innate ability to navigate, imagine and design. However, what I was unable to recognize then, was the way tasks and activities can be designed to help learners activate their powers and develop geometric thinking. I have learnt that simple pedagogic devices, such as asking learners to express what they see, can be a powerful tool in helping their thinking. Or leaving some ambiguity so that the leaners can make decisions and engage in chains of reasoning.

I would like to end by sharing that what I enjoyed most with these tasks is how the majority of the learners in class were able to ‘own’ the problem and to be involved a lot more in ‘doing’ what was needed rather than being told what to do and how. According to Dale (cited in Anderson), the most effective methods of learning involves direct, purposeful learning experiences. These tasks, although not completely hands-on, empowered the learners to lead their own learning. My aim as a teacher is to provide as many opportunities to myself and my learners to experience meaningful tasks and hence appreciate the beauty of Geometry. These Mathematics Education modules have helped me step closer to this aim and I have thoroughly enjoyed them.

References

Anderson, H. M. Dale’s Cone of Experience, accessed at http://www.queensu.ca/teachingandlearning/modules/active/documents/Dales_Cone_of_Experience_summary.pdf

Piggott.J., and L. Woodham, 2009. Thinking Through, and By, Visualising. [Online]
accessed at: https://nrich.maths.org/6447

Johnston-Wilder, S. and Mason, J. eds., 2005. Developing thinking in geometry. Sage.Jones, K., 2002. Implications for the classroom: Research on the use of dynamic software. Micromath, 18(3), pp18-20.

NCETM, 2007. Mathematics Matters: Deriving practices from what constitutes effective learning of Mathematics. pp. 13-14. accessed at https://www.ncetm.org.uk/public/files/30091176/Mathematics+Matters+-+The+Full+Report.pdf

Written by Gerry Golding

Hello, I’m Gerry Golding, deputy chair of Developing Statistical Thinking (ME626).

In this blog I would like to tell you about a new and exciting scholarship project that Andrew Potter and I are about to undertake. We are both members of the Discovering Mathematics (MU123) module team; a key level one introductory mathematics module carefully developed to meet the needs of students, who either wish to acquire a good foundation before studying further mathematics, or need to understand the mathematical aspects of their chosen subject areas if they are not following a specific mathematics qualification. It looks at a variety of mathematical topics such as numbers, statistics, graphs, algebra, trigonometry and associated techniques. It also introduces mathematical modelling and some problem-solving strategies. As well as ‘doing the maths’ students learn how to interpret their results in context and to explain their approach and conclusions. We often have parents studying MU123 as a standalone module just to be able to help their kids with their homework.

Andrew and I also have backgrounds in Mathematics Education. Andrew has a PGDE in Secondary Education and an MA in Online and Distance Education. He has taught in secondary schools, further education and higher education, and is interested in the transition between stages of mathematical study, and how mathematics is communicated. I completed my PhD in 2006 at the University of Limerick; I investigated “How adults learn advanced mathematics”. I was particularly interested in how mature students studying in a “service mathematics” environment coped with the demands of a system primarily designed for students who have just completed their schooling at age 18. Service mathematics can be described as the study of mathematics within another discipline, for example Science, Engineering, Computing, Business, etc.

Students’ perceptions of the usefulness of mathematics within their chosen degree pathway has the capacity to greatly influence their decision making and could potentially impact on their pass and progression rates. While looking at motivational factors during my PhD, I encountered varying perceptions of the usefulness of mathematics depending on the degree pathway the mature students were undertaking. Although not a focal part of my research at the time, I became aware that changing negative perceptions towards mathematics in general had a positive impact on their motivation to engage with the subject.

A secondary concern is that of students’ general perception of the relevance of first year/level one study. First year grades do not contribute directly to their degree award, leading to a risk of complacency and a lack of engagement that may come back to bite them later in their studies. We believe that a lack of understanding of the usefulness of the mathematics and the fact that first year study does not contribute to their degree award are intertwined, and any intervention must address both issues.

On MU123, we have approximately 2000 students per presentation, the majority of whom are studying this as a service mathematics module; approximately 60% are on some form of Computing and IT degree, 10% on Business and Economics related degrees and the remaining students vary across other subjects including the Mathematics Education pathway. This gives us a unique opportunity to harvest rich data about our students’ perceptions in relation to both issues. Through an analysis of tutor reflections on the tutor-student discourse at two key stages of their MU123 student journey, we will look to explore tutor’s understanding of their students’ perceptions of the usefulness of their mathematical studies on MU123 with the view to developing some good examples of typical MU123 student personas. Our project will form the first phase of a larger strategy within the School of Mathematics to enhance the provision of level 1 service mathematics.

How will we go about this?

The OU teaching model can be described as independent learning supported by a tutor. The tutors are called associate lecturers. We provide all the learning material both in hard copy and web based (in PDF format) and this material is designed to facilitate independent learning. MU123 has its own website which contains extra resources like screencasts (short videos) and interesting news items. There are interactive practice quizzes which the students can use to test their knowledge as they progress through the units. The tutors offer support in the form of tutorials, one to one email or phone support and give detailed feedback when marking assignments. MU123 students are required to complete four assignments and an end of module assessment. Each of the four assignments covers a number of units and the end of module assessment is based on the whole module.

Tutors on MU123 each have an allocated group of approximately twenty students. As well as providing correspondence tuition (by this we mean teaching in the form of feedback which arises from the marking of their students’ assignments and replying to emails etc.), tutors can monitor their engagement with the module website using an analytics tool which tells them how often their students log onto the MU123 website and attempt practice quizzes etc. Many tutors arrange a phone call with their students at the beginning of the module to get a general impression of each of their students’ educational backgrounds and their likely needs.

As shown in the diagram below, we perceive the tutors (or Associate Lecturers) as playing a key role in our project as they maintain the closest contact with the students. Over a number of cycles, we (as part of the module team) will read, analyse and implement tutor recommendations when required and we will feedback to the School and Faculty details of our interventions and their impact.

Initially, we plan to seek ten tutor volunteers. Each tutor will be asked to keep a professional journal in which they will be invited to reflect on their students’ engagement with MU123 at two key points in the module: after their first assignment, and after their last assignment before the end-of-module assessment. We plan to invite the tutors to a focus group discussion before the start of the module where we will seek their experienced opinions on the best indicators of how a student is performing on the module at these key assessment points. The first and last assignments also contain some reflective questions where students are asked to share with their tutors their perceptions of the usefulness of their studies to date. Tutors will be invited to reflect on how/whether increased awareness of their students’ perceptions facilitated a richer tutor-student dialogue and enabled more tailored student support.

Before they submit their journal, we will ask the tutors to reflect on their students’ overall journey based on what we decide at the focus group meeting before module start and to comment on whether in their opinion, the student has engaged sufficiently (even if they struggled) or displayed signs of complacency (just doing enough to pass). We plan to follow up with individual students who we feel may be able to add further value to our research.

What will we do with the data?

The success of the project will be determined by the quantity and quality of data which emerges from the tutor professional journals. The data will be subjected to a thematic analysis in order to identify and create student personas.

• We hope that our analysis will allow us, through the development of these typical student personas, to gain a better understanding of the diversity of students studying MU123 and their perceptions of the usefulness of their mathematical studies and level 1 study in general.
• We hope that the student personas will help the MU123 module team to inform the development of teaching and support interventions to better improve retention and progression of MU123 and other service mathematics modules.
• We would hope to explore to what extent assessment can be used, in itself, as a learning tool for facilitating richer dialogue between tutor and student.
• Finally, we hope that the impact of this project on students will be a greater awareness of how their studies at level 1 link with their chosen degree pathway and/or future career choices, leading to greater employability.

If you are aware of any other studies that we might draw upon, we would be delighted to hear from you. Please send comments to Gerry.Golding@open.ac.uk

Thanks for reading!

Gerry & Andrew

This blog is part of a conversation between Cathy Smith, Ruth Edwards and  Jayne Webster, discussing a ‘mastery approach’ lesson. We have taken some topics from the conversation: setting a context, careful choice of language, different representations, small steps  and reasoning.

Cathy Smith is from the OU, Ruth Edwards and Jayne Webster from Enigma Maths Hub based at Denbigh school in Milton Keynes. Ruth had invited Cathy to visit a lesson given by a visiting teacher from Shanghai in a local primary school. This is part of the NCETM UK-China Exchange national scheme where Shanghai and English teachers observe in each other’s classrooms. The lesson was about 3-digit column addition. It came at the end of a week of teaching in this year 3 class.

Setting a Context

Cathy: I noticed at the beginning of the lesson that the teacher set a context for asking the questions. There was a skipping competition between students and they wanted to add up to find the total jumps, first with 2-digit numbers, then with 3-digits. They set up a reason for making these additions.

Ruth: That was certainly a feature of every single lesson that I saw in Shanghai. It always started with a problem set in context.

Jayne: I saw that at secondary level too, for every lesson. Even algebra, for example in teaching like terms, the teacher started with ordering breakfast and they were working with the number of spring rolls and the number of pancakes.

Ruth: The context was developed through the sequence of lessons.  The first two columns here recapped the learning of the previous lesson, adding without and with exchange, and then the teacher’s jump column was a slight change to be the topic for this lesson. I notice that the Shanghai teachers use quite quick reviews  that reinforce points from previous learning – just two questions.

Careful choice of language

Jayne: What I liked here was the way the language that had been established in the 2-digit examples carried across to the 3-digit examples: Line up the digits. Start with the 1s.  It made it really clear that this was an extension of that previous work.

Ruth: Yes, they had rehearsed the language in the previous lessons through the use of a stem sentence.  Of course, if they were adding decimals they would start with the smallest digit and not the ones. So that isn’t quite general.

Cathy: It is one of the features that is emphasised in these lessons; choose your words and phrases carefully so that they communicate the mathematical thinking, and get the children speaking and repeating those phrases. But here, this is very procedural – line up the digits; start with the 1s. I can imagine my teachers from the 1970s and 80s saying exactly this. It is correct, and useful, but it is not helping the children put their mathematical reasoning into words.

Jayne: But look at the next one. Here you have a whole sentence, with an ‘If ‘, so that is mathematical language about conditions. And the children are using correct mathematical vocabulary ‘more than or equal to 10’.   They decided to use the word ‘carry’ rather than the mathematical word ‘exchange’ because that is what the children had met before.

Cathy: I agree: that’s a sentence with a mathematical structure.  It uses language that is very specific to the representation – it’s about the procedure and the columns.   Here the children are being told what to do, in mathematical language. But I’d say they are not reasoning yet.

Reasoning in different representations

Ruth: I think those representations are important for reasoning. The children had had a sequence of lessons that focused on their understanding of place value.  Pupils started by adding multiples of ten and they were saying 2 tens plus 3 tens is the same as 5 tens, and not using the language of  twenty , thirty, etc. The children then moved onto 2-digit addition, initially with no exchange and then with exchange.  So those mathematical small steps secured understanding of place value, then addition with no exchange, then with exchange, and then this lesson moved on to 3 digits.  The lesson before the teacher had noticed that the children were not putting the answer from the column addition back into a number sentence, so she was reminding them that writing the number sentences out was important.

Ruth: One of the common misconceptions for vertical addition we would expect is that children add 20 and 30 and write 50 in the column, so there is an extra 0.  Here in their previous learning they had talked about 2 ones and 7 ones is 9 ones, 2 tens and 3 tens is 5 tens so very few are making that error.

Cathy: But I am dubious as to whether we really want children to understand that as 4 tens rather than 40. There is an element of number sense in knowing that its 40.  I totally agree that we don’t want children to read that as just a 4, even as a 4 in the tens column.  We had such a lot of work in the 1980s about partitioning, about asking children to read 47 as 40 + 7 and 36 as 30 + 6. Then it can be added as 70 + 13 to make 83.  That keeps the idea of the size or the value of the number. I suppose that if they are now being asked to say 47 is four tens plus 7; it does still keep its value.

Jayne:  I think that’s what they do in Shanghai. In fractions, they stress the multiplication by the unit fraction. For example, it’s four of one-sixth and five of one-sixth is nine of one-sixth.

Ruth: And tied up with that, Cathy, in one of the earlier lessons they do both. They say four tens and three tens is seven tens when they do the column addition and say forty and thirty is seventy for the number sentence.  They practice that movement between representations.  The teacher emphasised that this worked for adding up any units – so it is generalised.  You have to remember that now we are nudged towards the compact method of column addition. When it was the expanded method, then children could write the units and tens under each other and then add up those rows.

Cathy : I suppose, before we used to add up 4 and 3 in the tens column but that gave us no idea of the size of the number. Then adding up 40 and 30 meant that the children might write an extra 0 into the answer and get  803 or 7013.  You are saying that adding up 4 tens and 3 tens keeps the size of the number and keeps the efficiency of the place value on the algorithm.

Ruth: In terms of conceptual variation, the children had been working with base ten materials. The exchange of a single ten to ten ones had been modelled physically in the concrete and then with pictures, and they had gone back and forward with those representations. That was part of the development.

Jayne: And comparing what’s the same and what different:  four tens and three tens, forty and thirty. That is varying representations.

Ruth: It’s about exposing the structure that is common across the additions. And choosing your stem sentences that reflect the structure.  So, you need good mathematical knowledge in order to choose those phrases.

Small steps  and reasoning

Cathy: I know you talk a lot with teachers about the idea of taking small steps. There seems to be two levels of steps – lesson by lesson, or within a lesson. What is the difference between those?

Ruth: I think teachers are more confident in planning out the steps between lessons. What is harder is planning the small enough steps within a lesson.   I think sometimes as teachers we know the task we want the children to get to by the end, but we haven’t planned the steps along the way. We have thought about task completion but not about what enables them to get there.  This is something which the Chinese teachers do really effectively. Within lessons the learning is ‘step by step’ with pupils building confidence and competence.

Cathy: So, is this what you would call the small steps here? First the move from 2-digit numbers to 3-digit numbers, then some examples where the children look for errors, then 3-digit additions with missing values.

Ruth: So, you see this is important about mastery. Traditionally some children would never have got the opportunity to work with missing digits. Through the small steps more children are enabled to access the questions that require mathematical thinking.

Cathy: Why is it that in England our teachers would not have used a missing digits problem in the middle of a sequence of problems?

Ruth: I think that in the past we have had a perception of lids on children. We would have differentiated. These children will get here, these won’t and these might. It’s historical perceptions of groups of children.

And sometimes I think we were expecting too much and supporting too much.  The small steps are also about the amount of time children are listening before they get to try for themselves. Sometimes a teacher would do some input maybe for fifteen minutes and be covering all the steps in one go, but then it’s a memory exercise for the children to remember all that when they go on to their independent work. For some of the children there is a nice adult who will sit and repeat it all with them, so they don’t need to listen, and they just stop expecting to be able to do it. With the small steps, most children are enabled to get it and they have the confidence ‘I can’ rather than ‘I can’t’.

Cathy: And I notice that here the missing number problem is put in when there is no exchange.

Jayne: Yes, to make it accessible. The reasoning question comes in earlier in the sequence of examples. Then they are more likely to go on and try something like this last problem that is more complex.

When I have done it with secondary, it has been the same. Everybody does the reasoning questions, a bit at a time and the whole class are at the same stage. The challenge will come, not just for the quick learners, but for everyone and when it is appropriate.  That’s the cumulation of the small steps.

Cathy: But what about if we teach everyone with small steps all the time, when are they going to work independently? When they have to make decisions about what to do?

Ruth: That’s us developing how smart we are about leading pupils to be able to think mathematically within lessons and how the next lesson builds on this one. Chinese pupils look forward to the challenge in their learning (sometimes called Dong Nao Jing) and they feel empowered to tackle the challenges.

Jayne: Or about when they bring the same idea back but in a different context.

Ruth: Or about quick intervention. We do have to say that in China, they do have time to have an intervention during lunch or break or during the school day to work individually with those children who have struggled with that idea.

Cathy:  Isn’t that consolidating the same skill? But what about if you wanted them to be able to do a reasoning question, like this one at the end, without that gentle warm up. At GCSE we want them to look at problems, something that is unfamiliar. Where is the transition between here, where everything is being made to seem familiar, to tackling unfamiliar problems?

Jayne: Yes, you do need to have that long-term view. You do need to introduce problem solving but I would say that here they are practising problem solving in a familiar context and that will make them more likely to be able to do it in an unfamiliar one.

Ruth: The big thing that I saw in all the lessons in Shanghai was that the steps are there for the children to take and the children take them. Here we are still at telling.

Jayne: Planning for the students to take the steps, not the teacher to take them.

Thanks to all involved for making the observation and the conversation possible.

Photo: Louise Cullen (Host teacher), Huihua Hu, Dr Debbie Morgan (NCETM), Yiyi Chen, Jayne Webster, Ruth Edwards.

Written by Brigitte Stenhouse, PhD student in History of Mathematics at the Open University.

The reactions I get when I tell people that my PhD is in History of Maths invariably involve some surprise: history and maths aren’t an obvious pairing, existing on separate sides of a perceived barrier between the humanities and the sciences. Beyond this, mathematics is commonly viewed as something static, unchanging, and the closest one can get to ‘truth’. Once you have given a mathematical proof, a mathematician’s job is done, right? So, what do we need history of mathematics for, when maths is the same now as it has always been?

However, the way we do mathematics today would be completely unrecognisable to Galileo, Newton, and their mathematical predecessors. Humans got along quite nicely for thousands of years before algebra was introduced (discovered? invented?), and although its utility was quickly recognised, it was plagued with philosophical objections for hundreds of years.

Mathematics is very much a human endeavour, and the progress of its development was and is strongly influenced by the idiosyncrasies of its practitioners. The transmission of knowledge between communities is affected by language barriers; political unrest; the circulation of journals, books and letters; transport and freedom of movement; and more. Thus, the history of mathematics can bring a different colour to the subject, and is a huge resource for alternative methods to solving problems, which students might not otherwise come across.

As such, when I was asked to give a Masterclass at Bletchley Park, I decided to run a workshop on the fluxional calculus, working through an extract from the 1736 English translation of Newton’s Method of Fluxions (below). BBC Bitesize was a great resource for finding out what the students would be expected to know, and what I would have to cover in the session before reading Newton. After revising equations of straight lines and giving examples of curves, we covered the relationship between tangents and gradients. We then looked briefly at Fermat’s method of drawing tangents to parabolas and discussed the benefits of having a general method which would work for all types of curves. This brought us neatly on to the calculus.

On first handing out the extract, I asked the students to underline all the words they didn’t recognise. After a few comments of “can I highlight the whole thing?”, there were soon conversations popping up about the strange typesetting of the ‘s’, and the difficulty of printing a fraction in the 18^th century. Together we read through the extract and translated the rules we needed to follow into understandable modern English:

1. Identify the variable unknowns in the equation (here only  and ).
2. Considering the variables one at a time.
   1. Put the terms in ascending order, depending on the power of the variable.
   2. Multiply by an arithmetic progression (here 1, 2, 3, …)
   3. Multiply each term by (or  when considering etc.).
3. Repeat for each variable.
4. Set the sum of the resulting terms equal to 0.

Newton’s method here gives an equation for what he calls the fluxions, and , in terms of and . However, in order to find the gradient of a line at a point we must go one step further; namely, we must rearrange the final equation into the form   .

With the assistance of a table to fill out for each step of the calculation, we applied these rules to an example together, (graph below).

Once we had calculated , we checked our answer by calculating the gradient and plotting the tangent at the origin .

Hence, at this point,  , and the equation of the tangent is . As we can see below, the line just touches our curve, as a tangent should.

Having never taught a maths lesson before, I had been a little worried about making sure the extract was accessible in 2.5 hours. It was thus quite exciting for me to ask a question to the room, and receive answers (often correct!) from multiple directions. After working through a second example together, the students completed a worksheet on their own, applying Newton’s method to a selection of curves. I found it very interesting discussing with some of the students what happens to the constant in an equation when you differentiate it; some of them reintroduced the constant at the end of the calculation because they were unhappy with it completely disappearing. But on considering how a curve is transformed when a constant is added, they soon understood why this happened.

Beyond the mathematics, we looked at the feud between Newton and Leibniz owing to their almost simultaneous development of the calculus, and how this negatively impacted the transmission of future work between mathematical communities in France and the UK. I was thus able to introduce my own doctoral research on the work of Mary Somerville (1780-1872), who played a key role in the in the dissemination of what was termed ‘French analysis’ in the 19^th century. Notably, she translated and adapted Pierre-Simon Laplace’s Traité de Mécanique Céleste in 1831 (retitling the work Mechanism of the Heavens), and advocated for the adoption of analysis in her 1834 book Connexion of the Physical Sciences. In both of these works she showcased the impressive results which can be gained by modelling natural phenomena using algebra and applying the calculus; for example predicting the motions of the planets and their moons, or even deducing the internal structure of the Earth.

I thoroughly enjoyed the chance to introduce these students to the history of mathematics, and look forward to re-running the session in the future!

There is a question in a recent Foundation level GCSE paper that asks:

How would you go about this question?

I can think of three ways. I could set up algebraic expressions for the length and width. I could draw a diagram and recognise that the rectangle must consist of two congruent squares. But what I really did was think “The length of a rectangle is twice as long as the width. Oh yes like 6 and 3. Area – yes 6 times 3. No, 8 times 4. ”

It’s that process I want to write about here – the process of using particular numbers to help you make sense of a general mathematical statement. In the tradition of OU mathematics education courses and books we call this ‘specialising’.

I have been in a thoughtful email conversation with one of the OU tutors over a couple of weeks, discussing exactly what we mean by specialising. We found definitions and descriptions of specialising in the literature written for researchers and teachers and used them to find opportunities for specialising and generalising in some GCSE questions. This blog shares what we found.

(At least) two purposes for specialising

The OU module ME620 introduces specialising as part of the analytic framework  ‘specialise – generalise – conjecture –verify’. This framework is a way of organising your thinking about mathematical activity.  The module text says “The value of specialisation lies in its role as a technique for getting insight into a problem and so for suggesting one or two sensible approaches.”

This agrees with my approach to the GCSE question. I used the numbers 6 and 3 as examples of numbers in which one is twice the others (and the examiners report suggests some 16-year-olds also chose 6 and 3 and stopped there). With these concrete numbers in mind, I knew how I would work out the area – by multiplying them.  Then 8 and 4 came to mind – aha! My method was close to ‘trial and improvement’ but I do not think it was, as I am not sure that I consciously decided that the answer for 6 by 3 was 18 and compared that with 32. But I certainly used specific numbers as a way of getting insight into a problem. That is one purpose for specialising.

The same module also suggests that specialising and generalising are complementary processes – they feed into each other.  We specialise in order to generalise. That is the second purpose.

What interested me about this question is that I felt I had specialised but I noticed that there is no general rectangle.  The question is about finding the one rectangle (OK you could swap length and width) that meets the given description.

What I actually did was use the numbers to make sense of the first statement in the task:

The length of a rectangle is twice as long as the width of the rectangle.

The statement needed interpretation because it named unknown quantities (the length and width) and gave a relationship between them.  If I examine it carefully I can see that it could be a general statement – a statement about a relationship that holds in general for some collection of cases, all of which are referred to as “a rectangle”. But I think that I – and most GCSE students – actually thought it was just about the rectangle in this question. In fact the second statement also gave a relationship between the unknowns but indirectly (because I also have to recall the relationship between area, length and width).  I think it’s even harder to read “The area of the rectangle is 32cm²” as a general statement about many rectangles since it is clearly about this one. So here, students are not generalising since they are not thinking about varying across a range of cases. Instead they have to make sense of these statements because someone else wrote them in mathematical language and decided to give the information via these clues.

So what we have are two slightly different ideas about the purpose of specialising. One emphasises getting insight into a problem – specialising for making sense; the other emphasises specialising as a starting point for seeing what is the same or different across several cases – specialising for generalising.

Is there an opportunity to generalise?

The OU modules are inspired by John Mason’s work since the 1980s. One of his sayings that is still yielding food for thought for many teachers is “A lesson without the opportunity to generalise mathematically is not a mathematics lesson”.   We might ask if there is an opportunity to generalise in the original question.

To get away from the single answer, I would have to explore one of the statements (and ignore the other). To explore the first statement I could draw my 6 by 3 rectangle and my 8 by 4 rectangle I could observe a common ‘look’, generalise that any such rectangle can be divided into two equal squares and then reason about what kind of numbers its area must be. Or I could notice that the area increases as the length increases and wonder if it goes up the same amount each time.   If instead I start with a rectangle with area 32 cm² , there is a lot of scope for the length and breadth, but what if I said that the length had to be a whole number multiple of the width?  There is no opportunity to generalise in the middle of your GCSE but by relaxing the constraints, this question could be the basis of an interesting mathematics lesson that takes in functional relationships between variables, spatial and geometric reasoning about area and number patterns.

Having decided that the rectangle question is mathematical, let’s look at how some others have approached these two ideas about specialising, and where these ideas fit in GCSE questions.

Specialising for making sense

Kaye Stacey collaborated with John Mason and Leone Burton in the 1980s.  In her 2007 paper ‘What Is Mathematical Thinking And Why Is It Important?’ she reviews their work on specialise – generalise – conjecture –verify. She offers two characterisations. First, she introduces specialising  as “trying special cases, looking at examples” (page 41). This characterisation seems to allow for specialising as making sense.  This type of specialising would come in useful for students attempting this GCSE question (you might like to think where your students would get stuck):

Obviously, put all thoughts out of your head about why Nadia needs more than one identical ruler, and whether the shop has limited supplies on the shelf. This is a maths question after all!

Did you specialise to  make sense? I don’t know about you but when I had worked out that Nadia had £3.80 left to buy rulers; I didn’t divide 380 by 30 (or 3.8 by 0.3) but thought she can buy ten for £3, then two more for 60p. At a high mathematical level that is the same operation, but I was using specific easy numbers (ten, two) and, instead of dividing, I was multiplying up to give me a sense of the situation.  Trying out numbers like this seems a very fruitful way to approach the problem; and the mental arithmetic involved is closer to what you would do in a shop than long division.

You could say that Nadia herself is specialising to make sense of a problem, since she has apparently arbitrarily decided to buy 15 pencils as a way of starting her shopping.

However in the Nadia question there is no emphasis on looking for a general method; just the particular answer.  Can it really be specialising if there is no generalising?   In her second characterisation Stacey emphasises this second purpose of specialising since she states:

“specialising – generalising: learning from examples by looking for the general in the particular.” (page 46).

Specialising for generalising

Polya writes about trying special cases as a strategy for problem solving in ‘How to solve it’. There is a famous quote where he explicitly connects specialising and generalising:

A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise.
Quoted in D MacHale, Comic Sections (Dublin 1993)

I am not sure that this throws any insight on what specialising and generalising actually are, but it suggests that Polya sees the two as necessarily connected.   If specialising means ‘looking at examples’, as Stacy first suggested, maybe the important word is ‘example’. These are not just any cases. They are chosen with a potential generalisation in mind. There has to be some notion – however vague – of what the case is an example of, or else we are not specialising.

I’ll end with two more GCSE questions and a comment. The first question seems a great example of when to specialise and it includes both purposes. You are trying to articulate a conjecture about  what happens in a general case and you need to check on the details, so why not choose some numbers for the distance and time, then vary them and see what happens:

The second question suggests that specialising is not always the appropriate focus.

In this question I could decide to put a as 10 then work out the answer 53, and a as 100 and get the answer 103. I might see a pattern that can be generalised. But this question wants the students to appreciate that arithmetic calculations can be carried out on unknowns as if they were numbers.   I want my students to recognise that we can add the 5 a’s and then take one and add 4, without even knowing what number a is.  The examiners report suggests that students did recognise that they were meant to write an equation using the given symbols but could not simplify or symbolise the result of adding 4 and subtracting 1.

My final comment is that I am struck by how rarely students were asked to generalise in these GCSE questions.  The reason specialising for making sense has such a high profile across these questions, and that specialising for generalising has a low one,  is that students are not often asked to make their own general mathematical statements. Instead they are being asked to make sense of ones that the examiner has written for them So it is the examiner who generalises and the student who specialises but only in response to the examiner’s words. The student does not complete the whole mathematical cycle.