Mathematics Education

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.              

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 (2^nd 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

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.’

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)

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.

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.

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/.

On January 16th the Mathematics and Statistics department at the Open University hosted a talk by Dr Clare Lee on mathematical resilience. The construct of mathematical resilience can help teachers to support learners of mathematics to develop the resilience to overcome barriers to mathematics learning such as anxiety about mathematics. It arises from what Bandura describes as self-efficiacy; the belief that one can overcome difficulties and successfully solve problems (such as learning hard things in mathematics).

Clare Lee, along with her colleague, Sue Johnston-Wilder, have developed a diagrammatic representation of three zones which can be used with learners to help them identify their current feelings towards mathematical learning. The comfort zone is an area where learners are probably not learning new mathematics, as this is an area of mathematics they have already mastered. The anxiety zone is an area where learners feel that the mathematics is too difficult and they tend to give up and not try. In between is the growth zone where learning occurs; there is enough challenge that the learner has to develop new skills, but the challenge is not too great as to cause the learner to panic. Using the diagram gives teachers and learners a way to promote staying in the growth zone so that learners can make progress in mathematics and develop the confidence to continue to make progress.

Find out more

Are you interested in finding out more about mathematical resilience?

The literature referred to in this blog post is here:

• Bandura, A. (1997). Self-efficacy: The exercise of control. New York: Freeman.
• Lee, C. & Johnston-Wilder, S. (2013) Learning Mathematics- letting the pupils have their say, Educational Studies in Mathematics, Vol 83 (2), 163-180

Clare Lee has also written about developing mathematical resilience in teachers which you find out about here: https://www.open.ac.uk/blogs/per/?author=63.

 Mathematical Resilience Event

On Friday the 9^th March, Clare Lee and Charlotte Webb will be hosting a Mathematical Resilience day on the Open University campus in Milton Keynes for year 9 students from the local area. The day is aimed primarily at girls and is targeted at those who feel under confidence with mathematics. During the day the students will use the growth model image shown above and will work on strategies to become resilient learners.

If you are a Teacher of mathematics or Head of department in the MK area and you are interested in bringing some students to this event, please get in touch via email: charlotte.webb@open.ac.uk.

Hi, I am Harry Gretton and, at present, a tutor on ME625 and other “M” courses. I have been an OU tutor since the Open University began! and have tutored continually on many varied “M,T and E” courses.

In parallel with this I worked for Sheffield Hallam University for even more years teaching mathematics, statistics and computing to a wide diversity of students and many different levels. Throughout this I have always being particularly interested in “what is doing mathematics now that technology is here”. https://www.gretton.net/Papers/ATCMA161.pdf  and the way mathematical skills are assessed.

The changes in the teaching and use of mathematics will always continue since “Some Mathematics becomes more important because technology requires it. Some Mathematics becomes less important because technology replaces it. Some Mathematics becomes possible because technology allows it” Bert Waits – 2000.

I used to play squash and rugby to a reasonable level but have retired due to some physical reconstruction. Since retiring from Sheffield Hallam University I fill my spare time? looking after a small holding and doing regular grand parenting duties. …

_______

Sue Forsythe

I have always liked solving maths puzzles and nowadays I am a fan of killer Sudoku. My Saturday morning routine is to solve the futoshiki and the killer Sudoku in the Guardian newspaper. I also like getting my hands on a good kakuru.

Mathematics in Art is another passion. I especially enjoy studying designs with symmetry in them. Symmetry appreciation is a human attribute which even babies partake in. Ask me for the reference sometime.

I also like playing the bass recorder in a recorder ensemble, knitting and occasionally creating a bit of craftwork. A sense of symmetry underlies both music and art and craft.

_______

Gerard Hayes

Hello everyone,

I have been a tutor on all ME modules since their inception. I have also tutored ME830 and ME831, and the residential school MEXR624.

Previously I have taught maths in secondary schools. I was appointed as an Advanced Skills Teacher prior  in to moving into advisory work then teacher education and training most recently on the secondary maths PGCE at Edge Hill university.

My  degrees were all obtained as a part time student while working full time, one was with the Open University.

My main hobbies are singing and operating as a volunteer at Lyme Park with the National Trust.

_______

Charlotte Webb

Before moving to the Open University, I was a secondary mathematics teacher in Bristol, Oxford and most recently in Madrid. I have also worked with primary and secondary aged children through delivering workshops and masterclasses with the Royal Institution and at Bletchley Park.

My Master’s degree in Mathematics Education looked at the way children use proof and reasoning in mathematics. I am interested in the way teachers can work with children to support them in becoming mathematicians and thinking mathematically, through using the ideas in our modules, such as conjecturing and convincing, within the classroom.

Alongside mathematics, I love painting, drawing and all things arts and craft. I am particularly happy when maths and art come together, for example in M. C. Escher’s tessellation pictures. I have just started an Islamic geometry art course, which I am excited to share with you in a blog post soon!

In my free time, when not getting crafty, I love to travel and enjoy seeing live music and comedy – the Edinburgh Fringe festival is one of my favourite places to be. I also have a miniature sausage dog called Ronnie!

_______

Jeffrey Goodwin

As well as being an Associate Lecturer for ME627, I also tutor on ME620 and ME626. I first worked as a tutor for the OU in the 1980s on EM235 Developing Mathematical Thinking and returned to my current role of Association Lecturer in September 2014.

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

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

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

_______

Angie McConnell 

I’m Angie McConnell and I am currently a tutor on ME620 and ME625. I have been an OU tutor since 2005.

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

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

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

_______

 Pete Kelly 

I have been a mathematics education tutor with the OU since 2013, and teach on all four of the Mathematics education modules. I have also studied with the OU myself, and so understand some of the challenges part-time distance-learning students face. As well as teaching with the OU, I am Reader in Comparative Education at Plymouth University. I am particularly interested in understanding mathematics classrooms, and have been lucky enough to travel to countries across Europe to compare how mathematics is taught. Before moving into higher education, I spent fifteen years working in primary and middle schools in London and South West England including five years as a deputy head and five as a head teacher.

_______

Wendy Troy

I tutor on both ME626 and ME627 modules, both since their inception. I have also tutored ME620 and ME830, and the residential school MEXR624, now sadly stopped.

Previously I have taught maths in secondary schools and a further education college. I then moved into advisory work then teacher education and training most recently on the secondary maths PGCE at IOE London and Goldsmiths.

My first degree was in Economics, my MA in Psychology of Education and later I gained an Honours Maths degree with the Open University.

The History of Maths is one of my enthusiasms together with maths from all over the world.

_______

Tom Cowan

I have been an Associate Lecturer with the OU since 2008 when I worked on the MEXR624 summer school each year in Bath.  When that ended I was lucky enough to be offered a chance to work on ME625 which led to tutoring on all the 4 modules which we offer in the Mathematics Education suite of modules at Level 3 and also working on the Masters module ME825.

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

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

I’ve never really left education and have found my next challenge in aiming to complete my Doctorate in Education in 2022! Hopefully I’m well on my way to becoming Dr. Cowan!

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

_______

Cathy Smith

I started as a secondary mathematics teacher in Hertfordshire and Suffolk, where my Head of Department noticed that my name could be anagrammed as itchy maths. I embraced this by moving roles within education, while always keeping a central interest in maths itself.

I have worked in the field of mathematics teacher education since 1993, in Cambridge, London and now at the Open University.  My professional and research interests lie in participation in advanced mathematics, and teaching for an inclusive mathematics.

My PhD was a poststructural analysis of the discourses of mathematics and further mathematics. I have been involved with British Society for Research in Learning Mathematics, the Further Mathematics Support Programme, and am a member of both MA and ATM.  Outside of work I like singing and making things (knitting, lace, Lego), sometimes simultaneously.

_______

Jim Thorpe

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

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

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

This is a time of year when I pay attention to ribbons and wrapping paper.

When I saw this  oak leaf ribbon, I had to buy some – just because of its beautiful symmetry.

But what kind of symmetry is it?  Thinking about this question reminded me of Bob Burn, a mathematics teacher educator, who introduced me to reasoning about algebra-geometry in a pleasurable, visual way . So that’s the thread of this blog, which is more about mathematics I enjoy than mathematics education.

Ribbons are like friezes, or decorative borders that appear on walls.  They become mathematical when they have a regular repeat in them. There is a section (the ‘motif’) that is ‘translated’ – let’s say moved horizontally – again and again to make the whole repeating pattern. It needs to be translated by the same distance each time: on this ribbon it’s about 3cm. Of course this is a sensible way of manufacturing them.

Every mathematical frieze has a basic translation symmetry defined by the pattern repeat; let’s call it T.  If I translate the frieze using T it looks the same, since all I have done is moved it along by exactly one repeat.  From now on I won’t worry about where the ends of the frieze are:  I am just going to think about its middle, or say it is infinitely long. Then I can say T ‘leaves the frieze invariant’.  In our OU mathematics education modules that’s called finding invariance in the midst of change: the translation is the change, but the overall appearance has not varied.

Here’s another ribbon with a repeating pattern of red and white checks. This time T is a translation of about 3.2cm. I can also see this ribbon has a horizontal line of reflective symmetry along its length. That means there is a horizontal reflection (call it H) that would just swap the top and bottom in the line of symmetry but again leave it looking the same overall.

It has other symmetries as well: I can see a vertical line of symmetry through the central one of each group of 3 white stripes, and through the middle of each red space.  So there are also vertical reflections V that leave this pattern invariant.

And there is rotational symmetry R as well.  Every frieze has rotational symmetry of 0ᵒ or 360ᵒ – that is not interesting – but it is possible to rotate this ribbon just by 180ᵒ and leave the pattern invariant.  What I have to do is choose the centre of rotation to be in the very centre of a red rectangle, or the centre of a middle white stripe. That is exactly where the vertical lines of symmetry meet the horizontal one (coincidence?).

This ribbon has four symmetries that leave it invariant: T and also H, V and R.  It’s got so much symmetry that it looks neat and crisp, but may be a bit boring.

There’s one more useful thing to notice about it though: in principle you could choose the motif as being any section of ribbon that is 3.2 cm long. I’ve put four example sections in the picture; each of these works since translating that section gives the whole ribbon. But only two of those motifs have got the same other symmetries as the whole ribbon, that is they have V,H and R. (Its the middle two)

It was Bob Burn who first told me about frieze symmetries. He often wrote about them in the ATM journals. For example, he gave Logo instructions to generate them in Micromaths (Spring 1995).  Bob was a dedicated scholar of mathematics education who worked at Homerton College. This was before it became a college of Cambridge University, and when it specialised in teacher training for undergraduates and postgraduates.  He later moved to Exeter University and was influential in thinking about the pedagogy of undergraduate mathematics education. At Homerton I taught several courses for teachers that Bob had helped design, and they always had two features.

First, they were advanced mathematics courses – but the mathematics included was the kind that underpinned the school curriculum. Teachers learnt about number systems, geometry and symmetry, how to categorise change and identify structures that stayed invariant, how to justify their reasoning and what was considered mathematically beautiful in those fields. It was not the same mathematics as in the school curriculum but it was intended to be the mathematics that would help teachers later, when they or their pupils raised questions about maths. (Now I realise we should have set up a research project to investigate this claim!) This is not a new idea: Felix Klein’s work in mathematics education before 1910 was titled ‘Elementary Mathematics from a Higher Standpoint’ and is definitely undergraduate-level mathematics.

Secondly, these courses for teachers were based on a pedagogic principle of suggesting just enough mathematics for people to start doing mathematics and asking their own questions.  I link that to the OU phrase ‘manipulate; get a sense of; articulate’.

When Bob drew the friezes he used a rather dull basic shape, like a flag, to make his motifs.  I decided to be more seasonal and see what motifs I could come up with. So that year a group of prospective mathematics teachers worked on identifying motifs and symmetries in these seven friezes:

Activity:

Which friezes have Horizontal symmetry? Vertical symmetry? Rotational symmetry? Which have more than one of these? None of these?

Can you find a motif for each frieze? Where there is a choice of motif can you find one that has exactly the same symmetries as the whole frieze?

Can you draw a motif that has V and R symmetries but not H?

One of the reasons I like this activity is that it creates – in me at least – a need for some new mathematics.   While I can ‘see’ horizontal and vertical symmetry (unlike some students) it always takes me a bit of time to appreciate rotational symmetry.  It helps to try and draw my own motifs.

I can also see that the reindeer pattern has some sort of symmetry, but it is not one of the ones that I know about – not H, V or R.  It’s the same kind of symmetry that makes my oak leaf ribbon beautiful, a horizontal reflection that is slightly out of step. Because I had noticed something and needed a name for it, I was delighted when Bob told me about the new bit of mathematics I needed: a glide reflection. First you reflect then you translate (glide) parallel to the reflection line.

The glide reflection G must fit perfectly with the basic repeat translation. The basic motif includes two reindeer, one facing up and one facing down, and I need to translate it through its full width for a pattern repeat. But if I do a glide reflection with the ‘glide’ part being exactly half the length of the motif, then it lands perfectly on itself, leaving the pattern invariant.  So this frieze has a glide reflection G and it has the basic symmetry T, but it does not have H or V or R.

The symmetry in the oak leaf ribbon is also a glide reflection: each leaf is reflected to the other side of the ribbon and moved down by 1.5cm, which is half the pattern repeat. There is also one in the frieze of angels, but that is not the only symmetry there. Once alerted to the existence of glide reflections, you will find you see them quite often. It’s a very attractive design feature, with a lovely sense of direction: a bit like footprints. I challenge you to find it in other decorations.

Last thought and activity: There’s a lot about friezes on the internet, usually pointing out that there are seven frieze groups. So: why seven? And particularly, why only seven?  It’s been easy to find friezes with no symmetry apart from T, and ones with just H, just V, just R, just G – that’s five types without even trying. But what about patterns that have H and V, G and R, … or three of these … or even all four? That is the kind of question about mathematical reasoning that I (and I guess Bob Burn) would want teachers to ask and encourage.

Activity

Can you prove that there are only seven distinct types of frieze? 

One approach is to consider all the systematic variations; try to draw an appropriate motif for each one, and see what happens. 

Bob’s articles in the ATM journals show some ways of constructing these motifs with flags although I prefer the creativity of finding images and playing with them in computer  ‘art’ or geometry packages.   Of course he gives no answers, because the point is to enjoy working it out. (If you can access a university library, there is an article by Belcastro and Hull that does). 

References

Belcastro, S.-M., & Hull, T. C. (2002). Classifying Frieze Patterns without Using Groups. The College Mathematics Journal, 33(2), 93–98.

Burn, R. (1995) Friezes with Logo. Micromaths 11:7-8.

Burn, R. (2008) Friezes. Mathematics Teaching 211:21.

Klein, F. (1908, 2016). Elementary Mathematics from a Higher Standpoint. Berlin, Heidelberg: Springer Berlin Heidelberg.