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

This blog post was written by OU student Jim Darby.

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

 

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

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

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

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

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

It utterly changed my views on learning and doing mathematics.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

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Fostering Thinking in the Mathematics Classroom, by Nazanin Nikanjam

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

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

The plan

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

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

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

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

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

 

The experience

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

Image 1 – task instructions for 4.2.2 

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

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

Image 2 – screenshot of the GeoGebra worksheet

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

Image 3 – learner 3 using symbols to record the conjecture

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

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

Image 4 – task instructions for An Unusual Shape

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

Image 5 – the learners working on their own diagrams

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

Image 6 – the learner explaining her conjecture on the board

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

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

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

Reflection

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

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

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

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

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

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

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

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

 

References

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

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

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

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

 

 

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Why do I need to study maths? I’m not doing a maths degree

Written by Gerry Golding

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

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

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

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

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

 

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

 

How will we go about this?

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

 

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

 

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

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

 

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

 

What will we do with the data?

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

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

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

 

Thanks for reading!

Gerry & Andrew

 

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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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