Discussing mastery: ‘small enough’ steps and what they add up to.

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.

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Fluxional calculus for fifteen-year-olds: A masterclass in the History of Mathematics

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 18th 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 19th 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!

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Getting to grips with specialising

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.

 

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Maths in a Zhen Xian Bao

Written by Hilary Holmes, Open University Staff Tutor in the School of Mathematics and Statistics.

One of the great things about working for the OU is that occasionally we are able to share lots of wonderful mathematical ideas with hundreds of people at national events, such as the Big Bang Fair in Birmingham. So when it was announced that the STEM faculty was having a stand at the New Scientist Live exhibition in London and that the theme was paper engineering, this was an opportunity not to be missed! The challenge was to come up with some paper maths that was unusual, fun to play with, but that was also fairly easy and quick to make. We already use some paper folding activities in our introductory maths course (MU123) to help with geometrical ideas and to illustrate how these ideas are used from space exploration to medicine, but something new was needed here!

A Zhen Xian Bao at the textiles fair

I had recently seen some amazing Chinese needle thread booklets (Zhen Xian Bao) at an international textiles fair. These beautifully decorated booklets consist of layers of interconnected collapsible paper boxes, often enclosed in an indigo cloth cover. Originally these booklets were made by Chinese Minority people, mostly in the southern Chinese provinces of Guizhou and Yunnan. They were used to store threads, tools and patterns for their richly embroidered clothes. Have a look at this short (40 s) clip to see both an original and a modern booklet in action.

https://www.youtube.com/watch?v=8roBJJbpWAk 

However, at the beginning of this century, the use of the booklets appeared to be dying out. Fortunately in 2012 after a lot of research, Ruth Smith and Gina Corrigan published their book `A Little Known Chinese Folk Art – Zhen Xian Bao’. This research has helped to revive the craft and it is now fairly easy to find online and other courses on how to construct a needle thread booklet.

A Zhen Xian Bao, full of mathematical ideas and puzzles instead of needles and threads, seemed perfect for the New Scientist exhibition!

The first step was to try and make a booklet. As you saw in the video, there are basically two types of collapsible boxes to construct:

  • the top layer boxes, which are either ‘twist boxes’’ or star boxes
  • rectangular boxes for the layers below. These have flaps which enable the layers of boxes to be connected together.

There are some instructions (about 30 mins) on how to create a flat thread booklet with star boxes here:   https://www.youtube.com/watch?v=9NzHjdjFXOM .

There are lots of mathematical ideas to explore in a thread booklet!

These range from simple counting and recognising patterns in the construction of the book, to describing these properties algebraically and describing the design of the top layer geometrically.

Open boxes on the top three layers

For example, if you try to make a booklet, the first step is to decide how many layers to have and then how many boxes will be needed. Working upwards from the base, each square box has two rectangular boxes in the layer above it then in the next layer each rectangular box has two square boxes.. In other words, starting from the base, the number of boxes double on each layer: 1, 2, 4, 8, 16 … . The booklet above has five layers and contains 31 boxes, of which 21 are square and 10 are rectangular.

Other sequences arise naturally too. For example, numbering the layers from the base, the layers with square boxes are 1, 3, 5, 7…. and the number of square boxes on each of these layers form the sequence 1, 4, 16, 64, 256 … .

Can you find any other sequences?  For example, where does the sequence 1, ½, ¼, … arise in the booklet?

If the top layer of a square booklet has 2^n rows, with each row having 2^n boxes, how many layers and how many boxes are there?

Next came the practicalities of constructing the boxes. What size of paper was needed for each box and how much paper was needed overall? What’s the relationship between the size of the base of each box and the size of paper needed? How can you cut the paper for the boxes most efficiently? The green and pink booklet above is made out of rather expensive but beautiful Himalayan Lokta paper and Japanese Chiyogami paper, so minimising the paper used was important. For the maths booklet below, I used scrapbook paper for the lower levels and wrapping paper for the star boxes on the top layer.

The maths booklet

Finally the design of the top layer gives many opportunities for discussing different patterns and symmetries, both of the overall design and of the individual boxes. Many of the original Zhen Xian Bao use ‘twist boxes’’ and are decorated with elaborate patterns, drawn or painted by hand or by using wooden stamps. However another option now is to design your own patterns for the boxes, using a computer drawing package.

Once the maths booklet was made, it just remained to fill the boxes with mathematical ideas and puzzles that might be suitable for a range of ages and mathematical abilities, ready for New Scientist Live. The ideas ranged from simple counting puzzles to unsolved problems such as the Collatz conjecture.

Hidden mathematics

The Zhen Xian Bao’s 31 boxes were filled with a range of puzzles and ideas, including:

  • traditional puzzles such as hexaflexagons and matchstick puzzles
  • tangrams, letter and area puzzles
  • exploring the mathematical rules for origami crease patterns
  • practical applications of origami.

Our mathematically filled Zhen Xian Bao was very popular with both adults and children alike at our OU STEM stand at NS Live. Many were fascinated by the hidden hexaflexagons and matchstick puzzles, others were struck by the patterns found within the different layers of boxes themselves.

Here is OU lecturer Charlotte Webb (@WebbMaths) with visitor and Zhen Xian Bao enthusiast, Dawn Denyer (@mrsdenyer) at our STEM stand at New Scientist Live.

 

Reference:

Smith, R. and Corrigan, G., (2012) A little Known Chinese Folk Art: Zhen Xian Bao, Occidor Ltd.              

 

 

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A sense of symmetry

In his famous book ‘The Descent of Man’ and as part of a discussion on the sense of beauty Charles Darwin commented:

“The eye prefers symmetry or figures with some regular occurrence”.

(Darwin, 1887, p. 93).

In mathematics the definition for symmetry is precise, eg a 2 dimensional object is symmetrical if it is invariant under a reflection about a line, named the axis or line of symmetry. Although the material world hosts a great number of examples of symmetry, the exact mathematical definition of symmetry is not adequate to describe most of them because the symmetry is not usually exact (Zabrodsky et al, 1992). Humans are sensitive to approximate symmetry an example of which is the human face or body (Palmer, 1985). It appears that humans perceive objects as having degrees of approximation to symmetry. However in mathematics an object either has true symmetry or not (Shepard, 1994).

Humans appear to show a preference for vertical symmetry and this probably arises from the experience of living in an environment where objects possessing vertical symmetry are more common or by an appreciation that it is more efficient to process information by considering vertical symmetry. The force of gravity is likely to play a part in this and certainly the higher species of animals, including humans, show close to a vertical symmetry in their bodily forms. Some-one once told me that near symmetry in an animal demonstrates a healthy organism.

Vertical symmetry also stems from the natural framework of reference that we hold for the world about us. Perceptions of the material world are affected by gravity and privilege upright shapes with their base on a level with the horizontal (Piaget and Inhelder, 1956). The paradox is that, due to the curvature of the planet earth’s surface, verticals are not all parallel and surfaces of liquids are actually curved. However humans behave as if we live within a vertical and horizontal framework.

Symmetry, and especially reflective symmetry with a vertical or horizontal axis thus appear to be important to us and we actively seek symmetrical patterns. Work that I did for my doctoral thesis (Forsythe, 2014) indicated that humans have a sense of symmetry and can use this to place objects either side of the line of symmetry by eye fairly accurately. Thirteen year old students in my study worked with a Dynamic Perpendicular Quadrilateral (DPQ) which had fixed length perpendicular diagonals which could be dragged inside the figure to generate certain shapes, two of which can be seen below.

The students tended to drag the diagonals inside the DPQ to generate quadrilaterals where one diagonal acted as the axis of symmetry (usually the vertical axis). They were able to drag the diagonals, by ‘eye’, keeping very close symmetry of the shape, to make isosceles triangles, kites, concave kites, and a rhombus.

I gave some students a version of the DPQ which was oriented at an angle to the vertical. The students were still able to drag maintaining near symmetry but asked if there wasn’t some way we could turn the figure ‘the right way up’!

I believe that a sense of symmetry is important to the way in which we view the world and that we should make greater use of symmetry as a powerful tool in geometrical thinking. For example the symmetry of a kite leads to the understanding of the kite as made of two congruent triangles, which infers the two pairs of adjacent equal sides and the two equal angles. We could define a kite as a quadrilateral with one diagonal as a line of symmetry. However, if we wish to be more inclusive and accept the rhombus as a special case of the kite then we must define the kite as a quadrilateral with at least one diagonal as a line of symmetry.

Looking at shape properties through the lens of symmetry helps us to think of shapes from ‘the inside out’ giving us fresh insight and a new perspective on shapes (Forsythe and Cook, 2012). This can free us to consider more flexible definitions of shapes which work better with the concept of inclusion of one set of shapes in another, like the example of the rhombus and kite above.

References

Darwin, C. (1887). The Descent of Man (2nd edition), John Murray, London

Forsythe, S. and Cook, D., 2012. Learning about Properties of 2-D Shapes from the Inside out. Mathematics Teaching, 226, pp.5-8.

Forsythe, S.K., 2014. The kite family and other animals: Does a dragging utilisation scheme generate only shapes or can it also generate mathematical meanings? Accessible at https://lra.le.ac.uk/bitstream/2381/28915/1/2014ForsytheSKPHD.pdf

Palmer, S.E. (1985) The role of symmetry in shape perception. Acta Psychologica vol 59 pp 67-90

Piaget, J. and B. Inhelder  (1956). The child’s conception of space. Routledge and Kegan Paul, London

Shepard, R. N. (1994). Perceptual-cognitive universals as reflections of the world. Psychonomic Bulletin & Review, 1(1), 2-28.

Zabrodsky, H., Peleg, S. And D. Avnir (1992) A measure of symmetry based on shape similarity. Computer Vision and Pattern Recognition, 1992. Proceedings CVPR ’92., IEEE

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Properties of a parallelogram, written by Stanley Collings Prize winner 2018 Ann Jehan.

This blog post was written by Ann Jehan, an ME627 student who received the 2018 Open University Stanley Collings prize.

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

The prize panel said “Ann’s learning design included a sequence of thought-provoking activities that combined open exploration and probing mathematical questions.”

Developing Geometric Thinking was the last of the four modules that I studied for the Graduate Diploma in Mathematics Education over a two-year period. I embarked on developing geometric thinking believing it would be one of the less challenging modules for me. I enjoyed geometry at school but hadn’t returned to it in any depth since then. I enjoy working visually and having had a previous career in illustrated publishing I really thought I was a visual person. Almost immediately I found this challenged; I had never been asked explicitly before to imagine a shape and manipulate it mentally. I couldn’t even see a square, let alone change it in some way. It was a complete revelation to me that other people could summon up a perfect image to order. I don’t have any internal imagery it would seem (though I do dream); rather I must think of shapes as a concept as I just don’t ‘see’ them. Frustrating, but interesting to know that I am not alone.

Unable to generate images to manipulate myself, I found using the software Geogebra particularly useful in exploring geometric ideas during this module. With this in mind, both my End of Module assignment tasks made use of its dynamic nature, to create images and allow my learner to manipulate them to test geometrical relationships. Following is an excerpt from my assignment describing one of these tasks. The learner was asked to investigate the properties of a parallelogram, by exploring the shape, (measuring sides and angles etc.) and making conjectures about which properties are invariant to all parallelograms. I chose this task because I like challenging learner’s misconceptions. As this learner was an adult, I was hoping for a particularly engrained idea of what a parallelogram is. It is always interesting when a long-held misconception gets highlighted. I know this all too well having made a fundamental error earlier in the module with the sum of internal angles of a pentagon. I know that it is 540°, but in my mind, 360° represents the ‘whole’ – an obviously deeply engrained and fixed idea left over from working predominately with quadrilaterals and circles in my school days.

 

Planning the task for my learner

My learner is a woman in her 40’s who works as an orthoptist. Whilst she uses angles as part of her diagnostic tools, she hasn’t had any exposure to formal geometry since school. I chose the task “Properties of a Parallelogram” to give her the opportunity to develop her mathematical powers of Conjecturing and Convincing (Johnston-Wilder and Mason, 2005). Everything she hypothesises will be just that, a conjecture, until she can convince herself that something is true. To support her in engaging these powers, I plan to use the pedagogic construct Structure of Attention (Johnston-Wilder and Mason, 2005). I want this learner to change her focus, shifting awareness from the outwardly simple nature of the tasks to discerning the underlying details. These details will be the start of her conjectures, and the process of convincing herself will help develop her geometrical thinking.

Figure 1: Properties of a parallelogram task

Figure 2: Module bookmark showing the key module ideas for Developing Geometric Thinking

To help her use her initiative, I will have a page of prompts borrowed from Watson & Mason (1998) (figure 3). I want to avoid the didactic tension described by Brousseau (1984), particularly with the parallelograms. It would be all too easy to guide her to the properties I want her to find. The more specific I am in signalling the results I want from her, the less likely it is she will have opportunity to exercise her powers of conjecture and convincing.

Figure 3: Prompts

With task “properties of a parallelogram” I plan to ask her to draw a four-sided shape, she has the freedom to draw anything she wants, but constrained by the number of sides. I will then constrain it further until eventually she can only draw a parallelogram. Along the way, we will have highlighted the properties she can see in her shapes. This mathematical theme of Freedom and Constraint (Johnston-Wilder and Mason, 2005) will allow me to access her sense of geometrical awareness before we move onto Geogebra. Using the properties found we will reconstruct one on the geometry software so she can use the tools to make reasoned conjectures about properties for all parallelograms. The dynamic nature of the software will allow her to drag a corner and see the shape adjust before her eyes and give her the opportunity to experience the power of specialising. She hasn’t used Geogebra before, so I am anticipating doing the constructing and then letting her have controls of the tools.

 

Observing my learner

The first shape my learner drew was a square, which is often the prototypical four -sided shape that springs to mind. I asked her then to draw a different shape but still with parallel lines, and she drew a kite, before realising that was incorrect.  She then drew a rhombus and then a parallelogram.

Figure 4: the learner’s four-sided shapes

When I asked her to name the parallelogram she correctly identified it, summarising the properties as two pairs of equal parallel lines. Asking her what she meant by ‘equal’, she clarified, ‘equal in length but also parallel’. Here her structure of attention is shifting from the whole, to starting to see it not only as a set of properties, but also how they relate to each other.

Moving to Geogebra, I constructed the parallelogram in front of her as planned, ensuring she felt we had included these properties. With a quick look around the tools of Geogebra she was keen to be very hands on. After making initial (accurate) conjectures about the internal and external angles, she was able to convince herself that they held true for all parallelograms by utilising the dynamic nature of the software and dragging different sides and vertices (figure 5). She rationalised it to herself that it was the invariant relationship of the parallel lines that ensured the angles summing to 180°.

Figure 5: using Geogebra

At my suggestion we looked at the angles of intersection formed by the diagonals. The learner then identified two congruent triangles, formed from two sets of scalene triangles (figure 6).

Figure 6: Learner conjectures that parallelograms must be constructed from scalene triangles

This finding was key to preventing her seeing other quadrilaterals as parallelograms later on in the session. Again, she was convinced as she was able to move the shapes around and the angles changed accordingly but retained the invariant relationships she had conjectured.

Figure 7: My learner’s conjectures of the properties of a parallelogram

Summarising the properties, I asked her to consider whether a square, rectangle or rhombus could be a parallelogram. She was adamant they couldn’t because the internal triangles weren’t scalene, and 90° angles didn’t fit with her mental figural image of a parallelogram. She saw parallelograms as slanting because that is they have always been presented to her, a figural concept, as coined by Fischbein (1993) that had taken on an unintended characteristic for the shape.

Figure 8: This learner’s reasoning as to why squares and rectangles are not parallelogram.

Needing to challenge her convictions, I encouraged her to create a parallelogram with isosceles triangles – if she could, then this property couldn’t hold. This wasn’t something I had prepared for, so was a challenge for both of us. Her first attempt was with equilateral triangles, which made a rhombus.

She dismissed that as a parallelogram since the sides were equal. The second parallelogram had a pair of isosceles triangles, thus finding a counterexample to disprove her conjecture (figure 9). (We ended up drawing the triangles first, reversing the construction. This pedagogic strategy Turning a doing into an undoing (Johnston-Wilder and Mason, 2005) was very useful to reinforce that two congruent triangles form a parallelogram).

Figure 9: The learner has found a counterexample to challenge her conjecture.

Whilst shifting the focus of her attention from the whole to discerning properties definitely supported her powers of conjecture and convincing, evident also was the mathematical theme of Extending and Restricting (Johnston-Wilder and Mason, 2005).  Reasoning purely on the basis of the properties she found, the learner found it very difficult to include the other quadrilaterals as special cases of parallelograms.  Accepting that squares, rectangles and rhombi do actually satisfy the requirements to be included as parallelograms, required her to relax her understanding of the properties.

She also found it difficult to see squares as rectangles. The specific qualities that the learner was attending to, and allowing her to classify shapes as parallelograms, were originally too restrictive. For example, ‘2 angles on the bottom line = 180°’ (she initially restricted that to all angles other than 90°, and ‘2 sets of lines which are equal lengths’ (figure 5) couldn’t include 4 lines of equal length). It took lots of prompts to shift her focus of attention and extend the meaning of these so that squares etc. could be included.

In doing this, she also extended and redefined the meaning of the word ‘parallelogram’ to her (figure 10). Not only is it the shape she holds in her mind’s eye but is also a list of properties to which squares etc. satisfy.

Figure 10: Redefining her understanding of quadrilateral shapes

This cycle of conjecturing, seeking a convincing solution, modifying or refining her conjectures until she was satisfied may well have been a new way of working for her but of course it could have been the use of Geogebra that made her hesitant. Asking her to reflect on the use of the ICT she commented that having the access to creating many diagrams, quickly, being able to correct as necessary, really helped her to work in an exploratory way, which of course was key to convincing herself of her conjectures. It also supported the theme of Freedom and Constraint. She had the freedom to create any different parallelogram by dragging vertices and edges but constrained by the way I had constructed it. Parallel equal sides remained invariant whatever she changed. On reflection though I wonder if I could have given this learner more responsibility to create her own from scratch, to fully ensure she appreciated the underlying construction and relationships in the parallelogram. In this sense the freedom she had was a fallacy.

 

I’ve really enjoyed the work I did with these four Mathematics Education modules and the opportunity they gave me to really reflect on my learners’ mathematical thinking and indeed on my own.   Recognizing when I or my learners are using a concept from the modules (see figure 2 for those in ME627 Developing Geometric Thinking) has been very satisfying and ensures that I work harder to provide opportunities for learners to enhance their mathematical thinking. I hope that this will lead to a deeper understanding of why and how the maths is, with learners becoming more confident and willing to take the initiative with their learning.

 

References 

Brousseau, G. (1984) The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In Steiner, H. (ed.), Theory of Mathematics Education, Paper 54. Institut fur Didaktik der Mathematik der Universitat Bielefeld, pp. 110-19.

Fischbein, E. (1993) “The Theory of Figural Concepts.” Educational Studies in Mathematics, vol. 24, no. 2, 1993, pp. 139–162. Available from: JSTOR, JSTOR, www.jstor.org/stable/3482943. (Accessed 13/2/18)

Johnston-Wilder, S. and Mason, J. eds., 2005. Developing thinking in geometry. Sage.

Watson, A. and Mason, J. (1998) Questions and Prompts for Mathematical Thinking. Derby: Association of Teachers of Mathematics.’

 

 

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A visit from Cambridge Mathematics

On Tuesday 8th May we were visited by Lucy Rycroft-Smith, a former OU student, who spoke to the academics in the School of Mathematics and Statistics about her work on the Cambridge Mathematics project, including researching and writing mathematics “Espressos”. In her seminar, she talked about how influential the Open University Mathematics Education modules have been in her career.

Lucy Rycroft-Smith (photo credit below).

The Open University

Lucy is no stranger to the Open University as she studied her undergraduate degree in “Mathematics and its Learning” with us and was also an associate lecturer within the School of Mathematics and Statistics.

Lucy said this, about her experience of studying our Mathematics Education modules:

“It is only since starting to work at Cambridge Mathematics and having the extraordinary opportunity to immerse myself in some of the significant and complex issues in maths education that I’ve fully realised how excellent a grounding I got from the OU in this area.  I still refer back to course materials that I studied ten years ago; I now have the privilege of conversing with some of the experts that wrote some of the materials.  The ideas remain powerful and well structured, and have influenced my thinking on mathematics education in the classroom and beyond.  I was delighted to be able to  visit the OU recently to discuss Espressos – our filtered research reviews for teachers – and start the process of completing the cycle, of beginning in my own small way to contribute something back to the field and to attempt to pay this debt in some way.  The OU Maths Education course planted seeds in my mind that are still growing, and some of which are only coming to fruition now, in the sunny open ground of working collaboratively with world-class experts and the time and space to reflect on my classroom experience. Connecting teachers not only with research, but with researchers and the powerful dialogue that needs to happen between them, is something I’m increasingly passionate about, and the OU has always done this well”. 

Lucy was recently a guest on Mr Barton’s podcast and during the interview she spoke* about about her experiences of studying at the Open University and particularly how the “Developing Mathematical Thinking” modules: ME625, ME626 and ME627 helped her grow as a learner and a teacher.

You can listen to the full interview here: http://www.mrbartonmaths.com/blog/lucy-rycroft-smith-cambridge-mathematics-setting-times-tables-anxiety/ 

(*Lucy spoke about the Open University about 13 minutes into the podcast).

 

Cambridge Mathematics

In her seminar, Lucy talked about the work she and her colleagues have done as part of the Cambridge Mathematics project. According to their manifesto, Cambridge Mathematics aims to secure “a world class mathematics education for all students from 5 –19 years old, applicable to both national and international contexts and based on evidence from research and practice“.

The project aims to support teachers in their planning and teaching by providing a well developed curriculum map, known as the “Framework”, which makes relevant links between mathematical topics from lower primary to upper secondary. The Framework will be a basis for planning curriculum pathways and will be linked to assessment and  professional development resources, including examples of effective pedagogies for particular mathematical topics.

Image showing part of the Cambridge Mathematics Framework

You can find out more about the Framework, and other elements of the project here: ttps://www.cambridgemaths.org/manifesto/

Research Espressos

“Educational research is like coffee…It can be invigorating, but only when it’s filtered, fresh and in the right quantity (Otherwise it can be overwhelming, overstimulating, or just leave you bloated and anxious)”

@CambridgeMaths

Cambridge Mathematics aims to support teachers in providing their students with a high quality, research and practice informed, mathematics education but, as Lucy discussed in her talk, there are many barriers to teachers using educational research, including lack of time and being overwhelmed by the sheer volume of existing educational research. In response to this issue, Lucy produces monthly “Espressos” for Cambridge Mathematics. These are “a small but intense draught of filtered research on mathematics education, expressly designed with teachers in mind” .

These “Espressos” are essentially short literature reviews of educational research topics which may be of interest to mathematics teachers in the classroom. Espressos are restricted to one (double sided) page of A4 and are written in accessible language, with busy teachers in mind. Each Espresso includes diagrams and references for further reading. Previous topics covered include: working memory, attainment grouping and effective feedback.

An example of an Espresso

Find out more

You can find out more about Lucy and her work on the Cambridge Mathematics project here: https://www.cambridgemaths.org.

This includes the filtered mathematical research “Espressos” and the Cambridge Mathematics blog “Mathematical Salad”, which Lucy edits.

Mathematical Salad: https://www.cambridgemaths.org/blogs/

 

Photo of Lucy taken at the Cambridge Mathematics London Conference. March 21 2018 (Matthew Power Photography www.matthewpowerphotography.co.uk
07969 088655 mpowerphoto@yahoo.co.uk)

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Teaching Mathematics – An OU tutor’s perspective

Written by OU tutor Nick Constantine.

I have been teaching for about 30 years, so it had to happen sooner or later. ‘You taught my Dad’, said the young man a couple of years ago. I looked at him and enquired about his Dad’s name. I could remember him ‘phew’ and then a few days later he came back to me and said, ‘you were his best teacher’ ‘phew’, I thought again. It made me think, was I a good teacher or was I a good mathematics teacher? You might think I am playing with these terms, but I think there is a point here; you can be a popular teacher, learners like you but you may not be an effective teacher of the subject.

After this long in education I can afford to be a little daring with my views but also share some reflections on the ideas of the modules I teach on, namely Mathematical Thinking in Schools (ME620), Developing Algebraic Thinking (ME625) and Developing Geometric Thinking (ME627).

In my experience mathematics is a wonderful subject to teach. You can pepper the whole lesson with spontaneous discoveries, share the learning and encounter surprise, but only if you allow this to happen with a deep awareness of the task choice, how it will be introduced, the tools used and choices that learners can share. Ahmed (1987), found in Open University module ME620, talks about rich mathematical tasks: the sense that learners have a lot of choices when engaged in a task and not simply ‘this is the way’, ‘my way or no way’. Choices allow a sense of empowerment and ownership and the best tasks have this built in.

My classroom is covered with investigations and ideas for learners to look at and question. These posters are from https://nrich.maths.org/. 

By allowing and encouraging learners to explore, question and conjecture you are building resilience. Good teachers explain and support, encourage and allow learners to accept the fact that a journey of discovery will result in wrong paths and dead ends. Why would you not want to foster resilience, an acceptance that to explore anything will result in obstacles and difficulties to overcome?

There are some other key ideas that have been introduced in the mathematics education modules that I find myself using a lot.

I want my learners to feel confident they can share what they sense and feel. In what ever form. They can draw it, explain it, share it or even wave their hands in the air! Conjecturing in the mathematics classroom needs to be encouraged and discussed, or in line with the mathematics education modules, getting learners to convince another person of their conjecture is an excellent way of encouraging and developing precise mathematical language. If teachers respond ‘nice idea John, but it is wrong’ then the principle of sharing any ideas will be diminished in the classroom. Sharing and discussing conjectures, observations and relationships is about developing a talking classroom. As learners are exposed to this their ability to be more precise with reasoning improves but only if guided by a skilled teacher.

Above my whiteboard are pointers to ensure mathematical thinking is always central to the lesson for example ‘convince me, convince another, convince your friends’ or ‘ a diagram, drawing or sketch can help you understand’.

The second experience I want my learners to meet is that how we imagine things is not fixed. Imagining and expressing is a key power in the mathematics education modules. How you imagine and what you imagine may be difficult to express initially. Teachers need to encourage and ask learners about what they imagine and try to get them to express this in a range of ways. However, perception is not the reality, it is A reality. How I perceive or sense something is how my mind internalizes the experience. This strongly relates to the cultural perspective and a full range of experiences (social and cultural) but it is worth stressing that if a teacher does not expose learners to using and developing their imagination, and how they express it, they lose a wonderful opportunity to gain some insight into the learner’s mind.

Linked to this key power, for me, is an often poorly used idea from many students studying our modules. Namely see, experience and master. In a Mathematics lesson, you show something, anything, and learners will only see certain features and rapidly interpret some form of understanding. The purpose of the teacher is to construct the learners’ experiences in a careful manner, perhaps using a range of techniques and tools. By doing this learners are encouraged to connect and develop relationships. By allowing learners a deeper experience of the richness of any topic, there is a greater chance they become more confident and indeed feel they have mastered the concepts. 

A range of tools can help learners see and experience the topic in diverse ways.

Finally, another story to share, in my early years as a new teacher I was teaching a class and I noticed a student looking out of the window. I asked him to ‘get on with his work’. He looked at me and just said ‘I am thinking’. Be careful that absence of evidence is not interpreted as evidence of absence. Teachers, and not entirely their fault, have been driven to distraction about written feedback for learners. I understand, it is useful but if a learner can explain me to what they have learnt, write down some form of summary and share with others their learning this is a vastly more powerful learning experience for the learner.

 

 

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Studying ME627 with Open University – Challenging, but worth it.

Written by OU student Christine Soerjowidjojo.

When I was at school, back in Indonesia, learning mathematics was about memorising formulas and the times tables. However, since beginning my journey at the OU, I have become aware that learning and understanding the theory behind the mathematical methods is more important than just memorisation. 


The tutor on ME627 stimulated our thinking by asking questions, to prompt us to widen our thinking, and as a result our knowledge became deeper. We have learned to think mathematically, by for example using frameworks and mathematical powers: specialising and generalising, conjecturing and convincing, to find invariant properties of geometrical shapes such as rectangles and triangles. Using GeoGebra as a learning aid was a wonderful experience.


Although the module content was not easy for me when reading straight from the text book, my struggles were made easier through conversing with the tutor via email or phone, and at times by being offered alternative accounts, or published papers or book references were suggested. As a mature student, with a non-teaching full time job, I sometimes found it difficult to manage my time, so the additional resources provided by tutor were very helpful.


There were good interactions between students and tutor via telephone chats/emails, which were lively and engaging, and so I thoroughly enjoyed taking part. The tutor was excellent, very knowledgeable, patient, caring and really helpful with a great sense of humour that makes me eager to learn.


The best parts of my learning experience was gaining confidence in myself leading to my own personal development and broadening my mathematical understanding of the module ideas and a new view of geometry through problem solving. 


In conclusion, studying with the OU is beneficial for my personal development and as a stepping stone to my planned move into teaching. OU study has had a largely positive impact on my life as a whole.

 

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Year 9 Mathematical Resilience Day at the Open University

On Friday 9th March, we hosted the very first Open University Mathematical Resilience Day for students in year 9 in the Hub Theatre on campus. The day was aimed primarily at girls with a focus on building their mathematical resilience. Students from schools in and around Milton Keynes were invited to attend the event, which was funded by a grant from the London Mathematical Society (you can find out more about the LMS here: https://www.lms.ac.uk/).

Throughout the day, our focus was to support students to develop as resilient learners, through working together and discussing strategies to help them when they found the mathematics challenging. Alongside working on mathematical problems, we discussed three key aspects of mathematical resilience:

  1. Growth zone model
  2. Growth learning theory
  3. Growing mathematical resilience

The students started the day by completed an attitude to learning questionnaire, based on the Fennema-Sherman Mathematics Attitudes Scales, encouraging them to think about the way they think about learning.

 

1. Growth Zone model

Clare introduced the Growth Zone model, a concept introduced in her work on mathematical anxiety and mathematical resilience

In the GREEN zone you are comfortable. You might be consolidating some ideas but you are not learning. 

In the YELLOW (or amber) zone you are learning, but learning feels risky and a bit uncomfortable, there are barriers to overcome, there is some struggle and you have to perserve to be able to “get it”. Stay here as long as you can, you will learn more that way, but don’t stay too long. 

In the RED zone the risks feel too great, you feel unsafe and anxious. The barriers feel too much, you just cannot do this. You need help. Don’t be afraid to ask for what you need.

Working on some mathematical problems using the Growth Zone model

The girls were encouraged to use their “growth zone” models throughout the day to chart how they felt about each activity and to help them to talk about their feelings in relation to the mathematics they were doing.

You might like to have a go at the task below yourself. If you do, why not chart your feelings using the Growth Zone model shown above?

Task: 9 coloured cube

If you have 27 cubes, 3 each of nine colours (e.g. 3 yellow, 3 blue etc).

Can you make a 3 by 3 by 3 cubes so that each face contains exactly one of each colour?

The image above shows a cube with two faces that meet this criteria, but unfortunately on the third face there are two green cubes and two black cubes, so this is not a solution.

You can find this task, and other similar problems here: Nrich.org.

 

2. Growth Learning theory 

Clare explained that, according to Growth Learning theory, there are two ways in which people think that they learn:

1.Fixed Theory of Learning: I can only learn just so much, no matter what help and support I get I really won’t make much progress in maths.

2.Growth Theory of Learning: I have the confidence that I can develop mathematical skills. I know that everyone can learn more mathematics with effort from themselves and support from others and that includes me.

Neuro-science tells us that the brain grows every time you use it – it makes more synapses and connections but if you don’t use it those connections are pruned. Grow your brain!

 

“Being Stuck is an honourable state”

John Mason, Emiritus professor at the Open University, wrote in 2014: “Everyone gets stuck sometimes, and it can be frustrating, even debilitating rather than stimulating. However, being stuck is an honourable and useful state because that is when it is possible to learn about mathematics, about mathematical thinking, and about oneself”.

We discussed the importance of “being stuck” in mathematics and talked about the fact that even very successful mathematicians get stuck sometimes. For instance, world-famous mathematician Andrew Wiles, who gained fame by solving a nearly 400-year-old, previously “unsolvable” problem: Fermat’s Last Theorem, recently spoke in an interview about the “state of being stuck”. In this interview, Andrew Wiles was asked what advice he would give to the general public about mathematics, his answer was “accepting the state of being stuck.

Having identified that “being stuck” would be inevitable at some point and reflecting on the way they worked together, the girls created “STUCK” posters, which were filled with advice for themselves and others, including “Don’t give up”, “Teamwork is key” and “keep practising”.

One group came up with an acrostic poem to help students when they are STUCK:

  • Stop
  • Think
  • Underline key words
  • Communicate
  • Keep trying

Having the confidence to question what you read: Numbers in the News

After lunch students were treated to a talk entitled “Numbers in the News”, given by Zoe Griffiths from @ThinkMaths who showed them misleading graphs and charts used in newspapers, leaflet campaigns and social media to give biased information. Her aim was to show the students that they can learn to use mathematics to critically review the information that they read and hear about.

Zoe also talked about the importance of knowing sample size and methods of data collection used before deciding whether a claim is credible.

For example, one newspaper claimed that 80% of people wanted to leave the EU. Looking into the detail of the poll, data was collected in only three constituencies and a leaflet campaign preceded the poll. In addition, out of 100,000 people who were asks, only 14,851 responded to the poll, of which 11, 706 voted to leave. This graph, published by the Independent online, demonstrates the newspaper’s claim and the actual quantities of responses.

To further exaggerate the point, Zoe showed a Tweet claiming that “100% think cats are the best”  (from a sample size of two: one person and their cat).

Zoe’s talk emphasised the importance of having confidence in our own mathematical abilities in order to interrogate data presented to us in the news, social media and advertising.

She finished the session with a demonstration on not being fooled by “special offers”! Girls were tempted with winning £20. I won’t spoil the game here, but you can get in touch with Zoe to find out more! @thinkmaths

You can find out more about the talks Zoe and her team offer for schools and events here: http://www.think-maths.co.uk/.

Student task: Analysing the validity of newspaper claims

Following Zoe’s talk on interpreting data in the news, the girls were given a task to identify whether there was enough evidence to support claims made. The students worked together, using strategies from their stuck posters to support each other, to organise and interpret the data given and to justify their decisions.

Have a look at some of the statements below

3. Growing mathematical resilience

Finally we invited the students to reflect on whether they have mathematical resilience. Clare explained that, if you have mathematical resilience you will:

  • seek to stay in your growth zone, get help when you need it and avoid the red zone where you know you cannot think;
  • feel part of an inclusive community of those who are learning mathematics and understand the value of doing so;
  • know that sometimes there are barriers to understanding mathematics which you have to struggle over;
  • persevere, knowing how to overcome difficulties, and how to get the help that you need;
  • work collaboratively with your peers giving and receiving help to push ideas forward;
  • work to use the language needed to express what you understanding, misunderstand and to ask questions;
  • have a growth theory of learning, that is you will know that the more you work at mathematics, with support, the more successful you will be.

To summarise, in order to be able to develop Mathematical Resilience you need:

  1. To believe that brain capacity can be grown (Dweck, 2000)
  2. To have an understanding of the personal value of mathematics
  3. To understand how to work at mathematics
  4. To have an awareness of the support available from the wider community, including: peers, teachers, school resources and the internet.

Reflecting on the day

The students reflected on what they had learned throughout the day, returning to an attitude to learning survey which invited the students to review whether their feelings towards leaning had changed as a result of the activities and discussions throughout the day.

Here are some of the questionnaire responses to the question: Has anything changed in your beliefs concerning learning mathematics? 

“That if I believe in myself I can do it.” 

“Maths is not that difficult if you ask for help and try it.”

“I shouldn’t give up! I should keep on trying!”

When asked to write down the three key ideas which the students felt helped them solve mathematical problems during the day, responses included:

  • Teamwork
  • Perseverance
  • Patience
  • Asking for help
  • Confidence
  • Your growth zone
  • Breaking up the question
  • Read the question carefully
  • Don’t give up straight away
  • Trial and error
  • Stop and think
  • Count to 10
  • Discussing worries/strategies
  • Writing down calculations
  • Having fun!

The students appeared to leave tired but happy, taking away with them the strategies used in each of the activities and a laminated growth zone model!

Further exploration

If you want to find out more about Clare’s work with Sue Johnston-Wilder on mathematical resilience, you can visit their website here: http://www.mathematicalresilience.org/.

 

 

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