Generalising in Mathematics

Generalising in Mathematics by Philip Higgins

 

 

One of the most revealing aspects of ME625 and ‘Developing Thinking in Algebra’ (Mason et al 2012) is that algebra is the search for the general rule and that this generality can be uncovered in most cases, and with some practise it becomes achievable. I say this because I never felt that way in my O’ Level school maths, back in the 1980’s. Most education critique that I currently read says that the curriculum should now be taught differently, with creativity being foremost. Searching out that general rule is a creative challenge. I have learnt all this aged fifty, having graduated in BSc Mathematics and its Learning. The following extract is from my ME625 End-of-Module assignment in which I tried to step up another level and actually teach generalising to a student. It was not easy.

Finding generality through Invariance and Change

My maths learner was my son, twenty years old, and in second year university on a Product Design course. He is educated to A’ Level in Mathematics. It was my difficulties in trying to teach him generalising during earlier work, which he found confusing due to my lack of structure, that prompted me to shake up ‘Question 13.1.5 Invariance’.

Here is the course textbook question:

Task 13.1.5 Invariance

What is changing and what is the same about the three following expressions as a whole?

3 + 5 = 2 x 4          5 + 7 = 2 x 6          3 + 7 = 2 x 5

 

I adapted it and split the whole task into six small parts so that steady progress could be made without feeling overwhelmed by suddenly having to generalise. The ‘Dimensions-of-possible-variation’ task shown at the end allowed him to create his own version.

Part One – finding changes

My learner explores change using spot-the-difference. The image is Paul Morphy, a famous Victorian chess player. My idea is to encourage visual interest.

He stumbled with finding the last three differences. It seems my tiniest of ‘Paul Morphy tweeks’ was challenging to him but this encouraged focus.

 

Part Two – visualising the nth term

In part two my idea was to familiarise my learner with the general number ‘ n ‘ but without the trauma of having to extract it from a ‘sequence of numbers’ or a ‘story question’. It is a simple approach.

Question

 

The ‘n&n’ task is undoubtably easy and sure enough it was for him but notice his use of ‘n-2’ and ‘n-4’. I prompted him for an alternative expression and he found ‘ n/2 ‘.

 

Part Three – moving from iconic to symbolic

For part three I asked my learner to explore change in a numerical context, so more symbolic than visual.

Question – Invariance & Change

What Stays the Same (invariance) and What Changes (change) for the three following expressions as a whole?

3 + 5 = 2 x 4, 
5 + 7 = 2 x 6,
3 + 7 = 2 x 5

On writing the headings ‘Invariance’ and ‘Change’ he was seeing the language patterns and using them. His articulation was valid. He noticed operators ( x and +), and values (3+5=8, and 4 is half of 8) but he did not see the 4 as being in the centre of 3+5. Learners see different things.

 

Part Four – using the nth term towards a general formula

Only now in part four did my learner need to generalise. The idea here is that he has settled into ideas of ‘change’ and ‘invariance’ without having had an algebraic equation launched at him.

Question – invariance & change

Think back to the “ n & n’s challenge “, earlier. We are going to use this idea now.

The aim is to find a general expression for our three equations above.

  1. Take your first equation.
  2. Let n=3.
  3. Now follow through the remaining numbers, term by term, for each of those terms that you found earlier that change, and in so doing, develop a general expression.

Those terms that did not change, the ‘invariant’ terms, will just stay the same.

  1. Repeat, to obtain a general expression for the remaining two equations.

Test it – exploring the range of possible change for “ n “

We suspect that ‘ n ‘ is an integer, since we made it so when we first set it at 3, but now ask if your general formula works for other types of number.

Question

  • Does it work for other whole numbers?
  • Does it work for fractions?
  • Does it work for square roots?

My learner’s generalising,

 

And his testing by specialising,

 

Part Five – can I create my own maths question?

School maths often ends there and the student thinks ‘so what?’ but now I encouraged my learner to push on and to create his own unique bit of maths using the theme of ‘Dimensions-of-possible-variation’.

Question – explore which bits of your general expression you can change

We want to create a whole new maths problem, which has the spirit of the existing, but which you alone have made.

  • Looking now at your general expression, which elements can you change?
  • Now change them.
  • Make sure both sides of the equation still equate to each other
  • Now insert some numbers to get three different expressions, much as the original question above did to start with.

This is your new maths problem. You could present these expressions to somebody and ask them “What Stays the Same and What Changes for these expressions as a whole?”

We reached a key learning moment; could he vary the generalisation? I did explain at this point what was being asked for (to adapt an element of your choosing in this general equation) but not what to vary. He hesitated, thought, and nervously said;
“The coefficient?”
I asked for more.
“n?” he said.
I told him to go ahead and our ‘n’ now morphed into 3n2

 

All that remained was to insert new numbers for ‘n’ and recreate the original question in his own vision. With care he substituted the numbers 2, and ⅓, and ¼ . Interestingly, despite squaring of fractions he did all of this without recourse to a calculator, which suggested competence.

Substituting for 2,

 

Substituting for ⅓, and ¼

 

This was his unique version of the original question. He had used ‘Imagination’ and he had ‘Got a Sense of’ the themes of Invariance and Change.

His rewritten version of Task 13.1.5

 

Part Six – The Conclusion – is generalising relevant to me?

This part of the task involved no mathematical work and no question to do. It was given as a simple analogy. The idea was to appeal my learner’s Product Design skills and to explain and inform him as to what we mean by ‘developing thinking’.

Mathematical thinking, as shown in the above process, can be summarised:

  • You are given a particular set of numbers in an equation.
  • You generalise that equation, so it works for any number ‘ n ‘.
  • You classify it as a simple linear relationship and explore its range.
  • You manipulate it by changing those dimensions which can vary.
  • You gain control over the structure of the problem.
  • This control enables you to create a new and extended problem. This new problem is your brand.

Here is a simple analogy:

“You dedicate time to studying some particular timber materials; Ash, Birch, Ebony, Iroko and others. You generalise them into ‘Hardwoods’. You classify the hardwoods in terms of texture. You manipulate that texture into your proposed design of a piece of furniture. This gives you control over the look of the finished product. This control and finish gain you kudos. This kudos informs your brand.”

 

My reflections on the above task

I tried to conduct this question in the spirit of Jo Boaler’s piece on Sarah Flannery, European Young Scientist of the Year (Boaler, 2013). She writes (using her italics),

“The first thing I realized about learning mathematics was that there is a hell of a difference between, on the one hand, listening to maths being talked about by somebody else and thinking that you are understanding, and, on the other, thinking about maths and understanding it yourself and talking about it to someone else.”

I conversed with my learner on Task 13.1.5, explaining why we were doing the steps and what it achieved. He proceeded smoothly, using Invariance and Change in both spot-the-difference and in his generalising, but it seems my approach backfired somewhat. He was simply listening to ‘maths being talked about by me’. My conversing had failed in the chance to let him talk and reveal his own understanding. I then questioned why he had done so well and he said,
“You explained quite a lot, which made it easy.”
I immediately sensed didactic tension and realised that I could not now undo all of that talk. Did he just display behaviours that I had asked of him? Was he generalising or did he just blindly convert numbers into n’s? I sense a lot of the latter, though he did work actively on Dimensions-of-possible-variation, creating his own set of three equations, and likewise the work on Invariance and Change was all his own. Fortunately, the ‘Heads & Tails’ task came next, so I then felt the urge to do just the opposite and maintain a steady silence whilst he experienced the mathematical struggle, intervening only where needed.

Finding generality through a recreational maths problem

Posamentier’s problem on Heads and Tails

You are seated at a table in a dark room. On the table, there are twelve pennies, five of which are heads up and seven of which are tails up. Now mix the coins and separate them into two piles of five and seven coins, respectively. Because you are in a dark room, you will not know if the coins you are touching were heads up or tails up. Then flip over the coins in the five-coin pile. When the lights are turned on there will be an equal number of heads in each of the two piles. How can this be possible?”

In ‘Heads & Tails’ I chose a question beyond the course material, and also as a way to challenge myself in applying the course techniques. I found the author’s original question (Posamentier, 2017) a little awkward (why sit in a dark room?) and he also revealed the solution, so in the spirit of the ME625 course textbook I developed it into a more eye-catching ‘Story Problem’, which related to my learner as a design student, and I kept the pattern hidden so I could register his surprise when he spotted the outcome of the puzzle, since if we could both take delight in his reaction then it acted as a motivator.

 

 

“You take a break from your studies, lean back in your chair, think about pizza, deadlines, assignments, software, formulae, design work, and you commit to acquiring a new set of watercolour markers for £11.99. You stare vacantly at the twelve pound-coins on your desk. The shops are shut. It’s late. Only the pizza place is trading.

You lean forward and playfully arrange five of the coins to form a group showing heads and the seven remaining coins to form a group showing tails. It looks concise, deliberate, and organised, but suddenly there’s a power cut. As the lights whine down you sit in the dark, waiting for something to reboot.

Nothing reboots!

The lights in the pizza parlour across the way have gone out too! Maybe it’s a temporary street-thing.

You think about reaching for your phone-torch but before you do you playfully shuffle the coins around in the pitch dark on your desk, and once again, by touch alone, separate them into a group of five, and a group of seven.

You flip over all the coins in the group of five.

The lights soon reboot, much to your delight, and you look down at the group of five and the group of seven coins, expecting something rather random.

Weirdly you notice something about the heads in each group!

How can that possibly be???”

Can you explain it;
i) in a sentence, that a friend or colleague could follow?
ii) in a diagram?
iii) using a table (ICT, not kitchen)?
iv) using a generalisation?

I had further adapted the task by asking the four questions shown above, three to help articulate, and a final challenging task of generalisation, with each step improving his understanding toward the next. The third question set up the problem on an Excel spreadsheet and my idea was to explore what awareness he had about the structure of the problem, which by this point should have been established. Would he see generalising or is it ‘just a spreadsheet’? At the final step he must generalise, but ‘finding x’ feels like ‘school algebra’. Would he still have absences in his grasp of it (you must first understand what ‘x’ is), despite progress on the earlier work?

The following photos show how the puzzle was played out after my learner had initially read the question and before he started on the written element.

 

This was the starting position, where heads were indicated by the white circles,

 

Now he shuffled the coins. We did not sit in the dark. I improvised a canopy to allow the coins to be hidden,

 

 

The coins are further shuffled prior to being flipped over, all still hidden under the canopy,

 

Joy was evident in this task. He whispered to himself upon reading the question, “So true …”. Idle time is universal. As the task unfolded for the very first time he said with sheer delight;
No Way!! No way does that happen every time!”
He did it again and again. Same result!

Question (iii) – using a table

In approaching the questions my learner did not explicitly use the theme of ‘Do / Talk’ to himself or draft sentences nor even replay the pieces. Instead he sat quietly and thought and eventually drew the table shown below. He had unexpectedly opted to answer question three first, despite my spreadsheet-in-waiting. His table lacked good labelling to identify the flip of the coins but you can see that ‘3 and 2’ becomes ‘2 and 3’. It favours the iconic with its visual cues like columns and place-holders for his numbers. He had the number ‘2‘ at the foot of each column, which was the correct answer for that shuffle.

 


Question (i) – using a sentence

My learner moved onto question one. Articulating through a sentence is tricky. I could not make full sense of his and there was a struggle to clarify the ratio found in the table. Again, there was no reference to the flip of the coins, so it lacked detail.

 


Question (ii) – using a diagram

Next he tackled question two but he was confused about a diagram. I had to first assist him on how to apply this diagram when moving from the enactive to the iconic. He followed my lead and then took a random selection of five circles, and then drew the flipped version, which allowed him to articulate the equal numbers of heads. You can see that his ‘Original Pile’ has two dotted circles and his ‘Taken Slot Inverted’ has the same.

 

Question (iii) – using my speadsheet

I now had to reintroduce question three with my pre-prepared spreadsheet. His task was to fill in the correct formulae in the cells. The Excel sheet is the key learning moment because inputting the cells is generalising and he recognised the need for the ‘Step-3 left cell’ to be a random number of heads (shown here as 32), and he succeeded with the formulae in Step-3 and Step-4 boxes with ease as they are simple subtractions or else unaltered. We did commence with 5 heads and 7 tails, but he soon experimented with 45 heads and tails.

 

Question (iv) – using a generalisation

Question four proved the trickiest for my learner despite him now having some grasp of the situation from the previous three.

His generalising version began with two variables, x and y, but see how he ended up with y = x, which got a bit confusing and he then scribbled out the y variable. When the coins are flipped he wrote down ‘ 7-x ‘ at the foot of the page and then scribbled out the ‘ 7 – ‘. He was almost there.

His first attempt,

 

In his following page the algebra still seemed difficult to grasp for him. Here he has 1 and 6, so no matching heads. There was no sense of the theme ‘Do / Talk’ to himself first in an effort to grasp the unknown before attempting to solve. In the bottom of his page I had by now prompted him to “think of what x is”. See how he succeeded but only by clinging on to Specialising at the same time, here treating x as 1, (despite having two heads). It shows more confidence is needed.

His further attempt,

 

My Reflection on the above task

The ‘Heads & Tails’ task embraced one of Posamentier’s ‘Effective Techniques’ (Posamentier, 2016),

Technique 4Entice the Class with a ‘Ghee-whiz’ Amazing Mathematical Result: One natural way to stimulate interest in mathematics among students is through the curiosity that nestles within all of us. Such curiosity can be awakened through new ideas, paradoxes, uncertainties, or complex results. Here the teacher’s talents come into play to find illustrations of easily understood situations that lead to unexpected results and leave the students intrigued (ghee-whiz), resulting in a motivation to pursue the topic further.”

Poems pare back word into some visceral cry and mathematical puzzles do likewise. This task offered that ghee-whiz result. My learner’s reaction, described above, resonated with this. It was sheer excitement at such a simple trick.

My learner Imagined-and-Expressed the process behind this trick in the form of a table. It was unexpected. I imagined he would sit for some time flipping and un-flipping coins (Do / Undo) to understand, but he thought it all through quietly in his mind. His table was accurate but see how his generalising fell apart in places. Where does the need for a second variable ‘y’ come from? It looked like a residue from linear equations (y=mx). No pause occurred to first grasp ‘x’ and follow its path from ‘5-x’ in one group, to ‘x’ in the other. It was hurried, evidenced by much scribbling. It was the last question-part so perhaps he was disengaging. Should I have intervened and talked more? It seems that conversing itself requires mastery. I asked him why, having found a general number in the spreadsheet, did he struggle with generalisation. Surely the spreadsheet simply ‘gave the game away’?
“It didn’t,” he said, “that was just filling in boxes, and besides those equations are all hidden anyway.”
So oddly, despite his setting up of those very generalised equations he could not easily connect that to generalising. Software, as powerful as it is, is not a panacea. It conceals. My view is that this is a problem of technique and method. He has not yet automated the concept of filling in Excel cells as a process of generality. More Do / Talk was required by him, both at the Excel and the generalising question. If I intervened at all it could have been to point this out. I was hoping to witness (maybe naively) the ‘Manipulate – Get a Sense of – Articulate’ spiral of increasing sophistication with each part. Instead it was curiously mixed.

Conclusion

Both tasks had the course textbook’s key ideas integral in their solutions and for me and my learner it showed that one can take ‘Developing Thinking in Algebra’ and apply it to the outside world, to problems randomly found in curiosity books. In other words, I could answer Root Questions using these general techniques.

Two things emerged from our solving:

  1. The level of verbal interaction in a lesson, and especially the learner talking of their maths understanding back to you, is extremely sensitive to how independently your learner will solve the problem.
  2. This conversation has to be tailored to your learner’s ability. This means that you have to know your learner (difficult in a large classroom).

It informed my approach to algebraic thinking, and serves my goal to seek control and power over the teaching process, and this is not dissimilar to the learner’s own quest for control and power over their maths problem.

Both draw inspiration from Kate Clanchy, on the poet Melissa Lee-Houghton, (Clanchy, 2019):

She says, frankly and simply, that she suffers from depression and poetry is not the cure for it, but that poetry can give her a way of understanding and formulating herself, both as she writes it, and as she reads herself back afterwards. It gives her some distance and control.
The kids are mesmerized by this, and so am I. Control. Not turnaround but control. This word has somehow never occurred to me before, in all my anxious considerations of poetry and therapy, but it seems the right one.”

She adds,

And if they dig deep, and find effective images, and make a good poem out of the truth of their lives, then that is not just control, but power. It’s different from being happy; it isn’t a cure for anything, but it is profoundly worth having. And actually, I don’t need anyone to tell me that; I know that from my own experience. I know it for myself.”

Care must be taken with such comparisons and the sense of control and power is by far an idiosyncratic thing but the spirit is the same; I want to understand, and dig-deep, and find effective methods, seek profundity, so as to be adept at producing the maths resource that converses, motivates and invokes the art of generalisation. Furthermore, what learner would refuse a chance at acquiring control and power; control in knowing how to start, where to start, preferred strategy, direction of travel and when to reverse, and of the power and delight when the problem yields because you fought for it, all of which alleviates (not cures) your maths fear.

Interestingly, none of it is for turnaround of the student, and the course textbook itself states, arguably, “I cannot change others”. The end desire is that the learner will change of their own volition, much as I have done in the course of my own algebraic thinking.

Seeking such personal power and control is rooted in human nature. It is enlightening to see ‘Developing Thinking in Algebra’ tie mathematical power to the humanist subjects of English and History, and by the prompting of learners to “develop their powers to imagine, and to express what they imagine to others”. We can rejoice that this notion is still vibrant thirty-three years after Davis & Hersh (Davis & Hersh, 1986) embraced it when the omnipresent I.C.T. was still cutting its coding teeth,

Metaphor and analogy exist in mathematics and physics as well as in poetry and in religion. Rhetoric exists in mathematics (despite claims to the contrary) as it exists in politics. Aesthetic judgement exists in mathematics as it exists in the graphical or performing arts.”

Those authors wholly reclaim mathematics as being a human institution, and if it ever ceases to be so then they argue that we must “let it decay”.

 

References

Boaler, Jo. 2013. The Elephant in the Classroom. Souvenir Press.

Clanchy, Kate. 2019. Some Kids I Taught and What They Taught Me. Picador.

Davis, Philip. Hersh, Reuben. 1986. Descartes Dream. Penguin Group.
The authors conclude on page 305, “If a synthesis cannot be achieved, if it comes to a showdown between man and mathematical science, then man would be best advised to stop the process. Let it fall into decay…”

Mason, John. Graham, Alan. Johnston-Wilder, Sue. 2012. Developing Thinking in Algebra. The Open University in association with Sage Publications Ltd.

Posamentier, Alfred S. Krulik, Stephen. 2016. Effective Techniques to Motivate Mathematics Instruction. Routledge.

Posamentier, Alfred S. 2017. The Joy of Mathematics. Prometheus Books.

 

 

 

 

 

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Developing Thinking in Learning and Doing Geometry

Developing Thinking in Learning and Doing Geometry.

 

You might have thought the title of this blog is unnecessarily complicated! It is a hybrid of the names for the current Open University geometry module (Developing Thinking in Geometry) and the new geometry module (Learning and Doing Geometry) which is currently being written by the Mathematics Education team at the OU. This blog post is about our developing thinking of how people learn and do geometry.

The current module

Developing Thinking in Geometry was one of a suite of modules originally developed as part of a post-graduate diploma for practising teachers and launched in 2005. It is now fifteen years old and is studied by level three undergraduate students, many but not all of whom intend to enter the teaching profession. The time has come to replace it with a new module written for contemporary students. When writing the module materials our picture of one typical student is of someone preparing to undertake PGCE in Secondary Mathematics or in Primary teaching. Other students may be working as teaching assistants or as unqualified teachers of mathematics who wish to upskill in the areas of mathematics subject knowledge and pedagogy. Other students may simply be interested in learning about learning and acquiring the specific skillset which allows them to do this.

The textbook which forms the basis of the study materials for the current module.

The production of the new module

The process of developing a new module for the Open University typically takes two years. The writing team have spent the last academic year writing the content of 30 weeks of study at 10 hours per week for this 30 credit module. Once this is completed, and has been reviewed by our critical readers, there is still a lot of work to be done in the second year as the module is gradually uploaded to the website; editing, production of high specification graphics, and videos, building the interactive content being amongst those. That’s not to mention the work done by people in the rights teams and the library to ensure we give students access to third party material and literature. All of this is being co-ordinated by our curriculum manager and the learning design team. Last but not least our External Assessor will oversee the materials to check that they are consistent with the standards of level 6 study (equivalent to OU third level undergraduate studies) in the Higher Education sector.

In writing Learning and Doing Geometry we aim to keep much of the theoretical underpinning of the older module but target it to the contemporary student demographic and move all resources online, with the exception of the task booklets. Important aspects of the module include the explicit connections made between presentation of pedagogic theories and their application to learner activities. Important ideas from selected theories in the field of mathematics education have been included in the module materials. Students studying Learning and Doing Geometry will learn about how geometric thinking develops.

Developing thinking about learning

Students studying the module will be asked to work on geometric tasks and to reflect on their own learning and their approaches to solving problems by making reference to a set of important module ideas. This aspect of the module is clearly important because the only mind you can look inside is your own. Being aware of how you learn and the approaches which work for you and linking them to important module ideas requires a degree of self-awareness and objectivity. The activity of thinking about your mental processes is known as meta-cognition and is considered to be the highest level of thinking. Students can use the insight gained from observing and reflecting on their own learning as a guide to ascertain how other learners learn geometry, whether this is in the context of the classroom or in smaller groups. It is important to be aware that not all people learn in the same way and that less experienced learners often need to work at a slower pace while they assimilate new knowledge and skills. Careful observation and listening to what they say are the best ways to explore how learners may be thinking about the geometry. Use of questions to prompt learners to talk about what they are doing and how they are thinking can shed more light onto their thought processes. It is the closest that you can get to looking into the mind of another person.

Perceptual and discursive reasoning

In order to reflect on geometric reasoning. we have identified important ideas and described how these can be used to support the analysis of learning and doing geometry. Two of the most important of these ideas in the context of geometry are perceptual reasoning and discursive reasoning. At the simplest level these two ideas describe how learners look at geometric figures and how they think and talk about them. Of course, it is more complex than that. When learners look at geometric figures they may notice all kinds of different aspects of the figure, emphasising what appear to be the important features and ignoring others. Learners may divide the figure into constituent parts and might imagine what happens if changes are made to it. When learners articulate aspects of the same geometric figure they may describe what they have noticed, attempt to justify this or ask questions about the figure.

Invariance and change

Another important idea in geometry is invariance and change. This idea can be used as a strategy by students to notice what is the same and what is different between two different figures, or two examples of the same figure. Students can notice what stays the same and what changes when they act on a figure, for example if the figure has been constructed in a dynamic geometry software program. The ability to identify invariance and change in geometry draws on learners’ natural tendency to spot differences and look for patterns and this gives it its power for learning. The example below demonstrates a typical task from the module where observing invariance and change can be used to as an approach.

Point Square

Place a point in a square as shown. Investigate areas A, B, C and D. What happens as the point moves?

 

This task, which is taken from one of the task booklets for Learning and Doing geometry, looks like a simple question. It becomes much more interesting as the learner begins to explore the areas A, B, C and D and to observe what happens to those areas as the point moves. Consider how perceptual and discursive reasoning come into play. How does asking ‘what is the same and what is different’ help the learner to notice invariance and change as the point moves?

Representations

Representations of geometric figures is another idea which is important in the module. In geometry we always work with representations, whether that be static diagrams drawn on paper, dynamic figures on the computer screen, or a worded definition of a figure. The type of representation and the context of the representation are important factors which influence the way that learners think and reason about the geometric objects being represented.

Content of the new module

These and other module ideas form a framework for students to reflect on their own geometric reasoning and to analyse the reasoning of learners which they work with. Other important theories that have arisen from research into mathematics education are presented in the module. We have aimed to provide an up-to-date picture of current educational know-how. Curriculum is mentioned briefly, but we are aware that the Mathematics Education modules are studied by students in England, Scotland, Wales, Northern Ireland and Ireland as well as by international students. There is a range of curricula in the various jurisdictions but most of the mathematics content is typical across them all. The mathematics content is presented at the level of secondary education for ages 11 to 18. What makes Learning and Doing Geometry and the other Mathematics Education modules into OU level three material is the depth and complexity of the Educational theory.

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What does it mean to understand probability: with diversions into parrots, children and sweets

One of the lockdown conversations that has stuck in my mind was with a colleague whose 8-year old does not really want to go back to school. One of his reasons is a tribute to her homeschooling arrangements: he has really enjoyed having more personalised learning activities, with adult attention. She was very conscious of the circumstances that make this possible.  Having time away from school has made all of us think more deeply about what school does and does not offer. Being in a room with  29 other children is probably not the best way to learn curriculum topics.  However, learning about other people, and learning how to get along without adult attention are also purposes of education.

The other reason for her son’s reluctance is that he is worried about catching the virus. They have talked about keeping safe, and what he can and cannot control in his environment. This is a time when it would be helpful for an 8-year old to have an understanding of low and high probabilities and low and high risk.      

What does it mean to understand probabilities? Coincidentally there has been a recent study suggesting that parrots can do probability  The researchers taught the parrots to take tokens from their closed hands: if the revealed token was black they gave them a treat, but an orange token gets them nothing. Then the parrots are shown two jars, each containing a mix of black and orange tokens – in different proportions.  The researchers pick a token out of each jar, hiding them in closed fists. The parrots consistently chose the fist corresponding to the jar that has higher proportion of black tokens. It is a fascinating study, and there is more to it. But are we sure that parrots understand probability? Maybe they understand ratio? Maybe understanding ratio supports them in making an aesthetic or emotional choice, based on preferring the look or the associations of the black-er jar. Are we sure their choice is based on predicting long term frequencies? How could we actually tell?

When we talk to children about probabilities of catching an illness, these differing aspects of probability are all involved.  One of the aspects they need to learn is ratio, and another is big/small numbers, because the chance of an 8-year old getting a severe case of COVID is very low. Another is using probability – a mathematical measure of chance –  to make predictions about long-term likelihood.

We often teach probability in schools through practical experiments. Experiments with dice and spinners are active and memorable and they help children establish a broad connection between theoretical probability and long-term expected outcomes. They work fairly well when we are interested in outcomes that are reasonably likely to happen. A popular activity is the ‘horse race’ modelled by adding the score on two two dice: horse #7 will usually win. The probability of throwing a 7 is  6/36, while the next most likely scores are 6 and 8 with probability 5/36. In my experience, after 15 throws, 7 does win. This aids children’s appreciation that it is more likely than any other score. ( Now I ought to go and calculate this theoretically – why 15 and not 10?).

But there are tensions in using practical experiments to appreciate low probabilities. A practical experiment ‘showing’ that a chance of 1 in 100 is unlikely to happen is necessarily boring to take part in.  It won’t maintain a child’s interest.  And there is an issue too that we are relying on the salience of experience, but you cannot control the outcome of a random event – it may be that the child does roll the dice and get something that feels like the very rare result more often than the probability suggests ( and sometimes children will make unexpected connections, for example ‘I didn’t get two 1s but I did get two 3s and that’s nearly the same’) .

Prof David Spiegelhalter estimates that the risk of an under 15 catching and dying of covid 19 is 1 in 5.3 million. That is very, very unlikely. It would be a very boring experiment to model.  But how to make sense of it. With older people you can use an analogy: their risk is the risk of driving for a year all bundled into a few weeks.

And risk – that is even more complicated than probability – as it takes into account the severity of the outcome as well as its probability.  For me the driving analogy works because I do know a few people who have been killed while driving – and I also know many, many people who haven’t. I can start to appreciate the scale and severity of risk. A teenager won’t necessarily  know of anyone, and may envisage low risk as no risk, or may be over-influenced by one close event.   In any case often when children act it is not because they are assessing risk, its because they are unconscious of it or ignoring it.

Our conversation about explaining risk ended up returning to sweets – the bags of mixed flavours that contain some you like and a few you don’t. There’s lots to talk about here:  If I offer you a sweet from the bag, would you take it? Even if you only know the nasty flavour when its in your mouth? Would you buy a whole bag for yourself? If you were making a bag of 100 sweets, how many nice ones would you put in and how many of the nasty ones?  Suppose someone buys a HUGE bag, enough for your whole school to have one,  but you know there is one nasty sweet in there – would you take one?   One in 5.3 million is a thousand children eating a sweet every day for one and a half years and finding just one bad one.

Of course the nature of the outcome matters as well – sweets aren’t frightening, although they can be imagined as very repulsive. I think that makes it  quite a nice context to play about with.  Going back to school has many things you do enjoy, and a small chance of something bad.

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…  even if you don’t get to a solution you’ll learn more about the environment around you

What makes some people love doing maths and others not? This is a question that inspires our research at the OU? We are always looking for stories of people who have come to maths in a surprising way.  In this blog we talk to Charlotte, who spent primary school struggling with times tables and calculations, and is now finishing her mathematics degree at Murray Edwards College in Cambridge. She reflects on how her dyslexia influences the way she approaches maths.

 

There are some points of Charlotte’s story that resonate with mathematics education research:

  • There is more than one way to do well in maths: you can be the person who sees connections, or a careful symbol-cruncher;
  • It’s very common to feel that you are struggling in maths, and not notice where you are succeeding (especially for quieter, slower workers);
  • Persistence – doing the same thing – is over-rated in mathematics; being willing to try another way is more successful;
  • Families and stories of people doing maths (even simple maths) are powerful motivators

Here is what Charlotte told us:

My earliest memories of doing maths involve me walking to primary school with my mum and trying to recite timestables. I was awful at timestables and at no point did I believe I could ever learn them, even with mum’s chants. I think she gave up quite quickly. I also remember being taught how to read analogue clocks in maths lessons. I could never do that either (although now I prefer them to digital).

Looking back the thing I found difficult was anything involving memory and/or calculations. That is pretty much everything in primary school. A particular memory that has stayed with me was during secondary school when I was 14. When we finished a lesson early we always did the same thing: we had specific numbers and operations and had to make every number up to 100 with them. Whenever we went back to it the boy next to me would laugh at how slow I was compared to everyone else, so that eventually I gave up and refused to participate. I dreaded maths lesson that entire year because of this.

My primary school split us into two maths sets. I was in the top one, but I couldn’t keep up. At secondary school I was put in the second set of five, but I found it very difficult and did not feel like I belonged. Everyone seemed to be faster than me.

That did change.  I remember the first time I finished a task first; the teacher came over to check my answers and started to show me the best strategy to use. Then we realised I had already used that strategy. It was a very good feeling.

Right at the end of year 9, just before the sets were locked for GCSEs, I was moved into the top set. Again I started near the bottom, but slowly moved up and was eventually the best girl in the set (not too hard-there were only 4 of us in a class of 30). This pattern repeated again in sixth form when I moved to a new (all girls) school; at the start I felt like everyone knew things I didn’t, but by the end I could correct the teacher.

I can see now that as I got older I got much better at maths compared to my peers; I think this reflects the change in maths from computation to more abstract concepts. I was planning on doing either a Physics or Computer Science degree until I did A level maths; this is when you start getting close to the fun stuff.

Also, I was diagnosed with dyslexia in year 12, when a teacher from my new school told me to get tested. My dyslexia still affects my maths a lot; My short term memory is very bad and I misread and/or mix up symbols in equations. This makes tasks like finding eigenvalues and simplifying equations literally impossible for me, when other people find it simple.  I can’t describe how annoying it is to solve a ‘trivial’ equation five times and get five different answers. I have had to avoid as many courses as I could which involve this sort of algebra, which has naturally pushed me to more abstract and pure maths.

I don’t spend all my time at university working.  I love playing rugby; it’s such a great balance between tactical thinking and physical strength. Also  my dad loves it so it’s in the family! 

I played for the Cambridge 1st team and was in the winning Twickenham squad last December, but sadly I am not playing rugby this term as I need to focus on my studies. I plan on going back to it once I’ve graduated.

I also spend a lot of time with CUBB, the university brass band and I’m the concert manager this year.  The friends I have made at CUBB have been some of the best people I’ve ever met, and I suspect they will be life long.

What I am hoping to do next is a Computational Biology masters, so that I can carry on studying maths, but start using it in a more practical way. We can learn so much about humans, plants and animals by studying their DNA and that is about finding mathematical patterns in data.  The computers do all the computation for me. Afterwards I’d maybe like to go onto a PhD and work in bioinformatics.

We asked Charlotte is she had any messages for others about learning maths, which she did, and she also has a Milton Keynes connection …

First message is that primary school/GCSE maths is ugly and boring; it gets so much better after that. It does also get harder though, which brings me to my next message. So much of maths is having the confidence to follow your intuition and try a solution, especially at the higher level (A-levels and beyond). Starting a difficult maths questions is often like exploring a dark pit; you have no idea what’s around you, so it’s easy to give up, but you just need to try things; even if you don’t get to a solution you’ll learn more about the environment around you so that your next attempt will be better. Maths is not about getting the solution (at least it isn’t when you get to the good stuff), it’s about exploring a weird abstract thing, and that is really fun. For example, I love to think about countability. It is fascinating that there are ‘more’ real numbers (decimals) squashed between 0 and 1 than there are whole numbers in total.  (Read more here).

A way of ordering the fractions between 0 and 1, but not the irrationals: the Calkin-Wilf tree

The final thing I’ll mention played a vital part in me deciding to apply for a maths degree in the first place. Bletchley Park is where codes were broken in the second world war by a bunch of mathematicians, and is where modern computation was invented. I didn’t really understand the machines back when I first visited but just the idea of them made me very excited, and I go often go back there with my family to this day.

 

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Meet an OU statistician

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.

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Mathematics: The unattainable key to success

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:

Mathematics Mathematician
Essential

Unappealing

Inaccessible

Invisible

Beautiful

Ubiquitous

Rule-based

Numbers

Male

White

Heterosexual

Middle-class

Old

Other / eccentric

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.

Image of Katherine Johnson, NASA physicist and mathematician.

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.

A group of adults and children create 3D obects using Zome.

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

 

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Using Examples in the Maths Classroom

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

 

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School Maths Conference at the OU – A-levels, Dragon Races and Codes

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

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Reflections on Mathematical thinking in schools (ME620) and Developing Algebraic Thinking (ME625)

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.

 

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Fostering Thinking in the Mathematics Classroom, by Nazanin Nikanjam

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

 

 

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