"3 minute" reflections on a "Science of Education Working Group" day meeting (5th Nov 2011)

There are two main thoughts that I'm left with from the day.
Firstly, that beauty is important in an image. I chose to redescribe the part of the circles film that had given me an affective charge. That charge was very significant.
Secondly, that seeing changes happen provokes questions. My need to know was excited.  (But the rate of change is important. Some of the circle movements were too fast for me to get any sort of handle on. One needs an entry point into the challenge.)
Piers Messum

At the recent working group day, it was good to work with others, many of whom I had not met before, in a very supportive exploratory environment. I was enabled to gain new insights into some mathematical situations: for example, in the morning, a film made me see circles as I had not seen them before, particularly the sequence about generating circles from a pair of tangents.

But for me I think the most interesting part of the day was the time we spent considering the Leapfrogs poster Circle Packings. I hadn’t look at this poster for 20 (or maybe 30!) years, and it had never been one of my favourites but, sitting where I was and looking at the poster on the floor, I first saw lovely lines of stars I hadn’t really seen before. Then I listened, and contributed, to the discussion about what we saw in the poster. People spoke of six-sided holes and three-sided holes. After a while, someone spoke about five sided holes, to some gentle laughter – most of us ‘knew’ there were no five-sided holes! The speaker corrected himself, saying he meant six-sided holes, of course. But then later I, and I think many others, were taken by surprise to find there were indeed five-sided holes to be found. A wonderful insight into the poster and into how limited our vision can be without others to help to change our awareness of we see.

I look forward to many more such learning experiences in future meetings of the group.

Anne Haworth

Twenty four hours later and my thoughts are very much on the clash between my attempts to recount what it was that was in my imagination – the mental image I had evoked upon my distracted viewing of the circles film – and the image my colleagues thought they knew I was attempting to describe.  Instead of an attempt at construction of the image I was proposing it felt like there was an attempt at correction to make me describe the image which was already in their own imaginations, having experienced the film in a different way – as it transpired, correctly!

How does this relate to my experience in the classroom and that of my unwitting victims/students?  We each experience the world in our own ways and bring with us our own memories, awarenesses and sensations - whilst sometimes shared, always unique.  Are my classrooms a battle of wills where I attempt to meld an experience into the way in which I envision it or do I seek to find out how it was experienced by my students and develop from there?  How can than this be developed if there are thirty two different experiences?

David Lawrence

2(a+6)

How would you write this expression?

I have been working on reading and writing expressions a lot this half-term and using Grid Algebra in particular. The other day I was writing some expressions involving brackets by hand and not in front of a class and became suddenly aware that I was beginning by writing the a first. This was striking for me because I think I have always thought of this expression as something like ‘2 ‘lots of’ a+6’ or ‘2 times a+6’. Possibly because I have been thinking about eventually wanting learners to appreciate equivalence with an expanded form and although I have often worked on building expressions up, this is the first time I have noticed myself not writing from left to right as if some shift has occurred in me where I see the expression more as ‘a add six, times two’ at the moment. I am conscious that I can slide between the two images of 2(a+6) if I attend to it and this is something I would want my pupils to be able to eventually do but I was interested by the subconscious change in my writing of it.

As I was interested by it I asked some colleagues to write the expression and we compared similarities and differences. We then considered some other expressions such as ,100-a and 2(6-a). Dave then wrote an equation and we tried to read it starting from different places. This also raised new awarenesses for me. In particular, the language of ‘however many times seven goes into 6 – t...’

Tom Francome

On returning home today from the Science of Education meeting I have been reading the recent publication from Educational Solutions “The Gattegno Effect - 100 voices on one of history’s greatest educators” and the powerful idea of the ‘subordination of teaching to learning’. This is a phrase I have lived with for many years as a guiding idea in my own teaching and I have been considering the extent to which I embodied this today. I am particularly struck with a moment when I feel I did not subordinate my ‘teaching’ to the learning of others. In one of the last activities of the day, I laid on the floor a poster (created by the ‘Leapfrogs’ group) of circle packings. There felt like a rich discussion of what participants saw – and my invitation was for us to work on ‘seeing the same’, i.e., it was each individual’s responsibility, if something was said that they could not see, to ask a question. At one point there was an offer from a member of the group to see a square formed by the centres of four circles. I was aware of an urge to pursue this idea as, in my work on the poster, I knew that there was a richness to the poster from linking it to tessellations of regular polygons (which linking can be made by ‘seeing’ the shapes formed by joining centres of circles). I hesitated, someone else offered another thought and I chose not to pull the conversation back – laissez faire. As I reflect now, I see this as an abdication of my role of “teacher” in that session. Part of subordinating my teaching to the learning of others is a commitment to act when the moment arises.

Alf Coles

//4// Excerpts from the Recognitions 1 and 2 (1975) - selected by Laurinda Brown

RECOCT … To boil or cook a second time; also fig. to vamp or furbish up anew. So RECOCTION.

RECOGNITION … The act of recognising. 1. Payment on the conclusion of a bargain. The resumption of lands by a feudal superior.  3. Revision, recension. b. The form of inquest by jury in use in England under the early Norman kings. 4. The action of acknowledging as true, valid, or entitled to consideration; formal acknowledgement as conveying approval or sanction of something: hence, notice or attention accorded to thing or person …. (‘.. kind of publick reading, whereby the lives of such saints had .. solemn r. in the church of God.’ Hooker.) 5. The acknowledgement or admission of a kindness, service, obligation, or merit, or the expression of this in some way… not chiefly in phr.  in r. of. 6. The action or fact of perceiving that some thing, person, etc., is the same as one previously known; the mental process of identifying what has been known before; the fact of being known or identified… (‘I could not escape r.’) b. The action or fact of apprehending a thing as having a certain characteristic or belonging to a certain class. (The r. that certain things were not true’,)

…

RECOGNITOR …A member of a jury impanelled on an assize or inquest.

...

(Recognitions – page 1, no 1, April 1975)

COUNTING FROM ZERO                                                 Lucinda Bedford

Teacher: Draw me a line 3 cm long.

(The child, aged 9, drew a line 2cm long.)

Teacher: Let me draw one for you. Is that the same as yours?

Child:  No, yours is longer.

Teacher: You draw yours again.

Again the child (aged 9) drew a line 2cm long. I watched what she did, and found that she had started to draw from the line marked 1, and not from the 0 mark. She could not understand that you started measuring from a point marked 0; to her you began with 1. It was as if she were counting 1, 2, 3. She had no concept of 0 with regard to measurement.

I related the situation to the child; I asked her how old she was when she was born. She replied “Nothing”. So I said, “Yes, the same thing happens when you draw a line, you start at nothing and then go on to one. It is like having a birthday, you have to live one year before you are one.”

Child: If I had drawn a line 1 cm long in my way, it would have only been a dot.

(Recognitions – pages 8-9, no 1, April 1975)

Broadly speaking there are two strands in the writings submitted and being published so far. On the one hand there are anecdotes – accounts of some actual teaching encounter which someone at any rate feels to be sufficiently striking to be worth recording. These might be long or short. They may or may not have a point. Sometimes a reader can latch on to someone else’s observation; at other times he cannot and does not. But developing powers of observation is an important task and writing to share with others is a useful way of noticing what happens. Apart from anything else, a brief anecdote is something that everyone can submit – it requires no special literary skill or sustained analytic thought. An account of a classroom happening can be a start to reflective thinking – and writing; it is also a welcome reminder of where the action really is.

But observations are only the raw material of experimental science. They already reflect an implicit theory and we are surely concerned with using them to explore such frameworks in further depth. So on the other hand we have had articles that are reflective and exploratory. Research is a much abused and misunderstood word so that we hesitate to call these research articles. A sort of research that we are all depressingly familiar with is that which starts with a hypothetical construction and then seeks to confirm it, the construction. But in either case the scope of the exploration is predetermined. Real advance in any field seems to have an unpreconceived, unplanned and often zany, atmosphere about it. Probing the unknown is like walking towards the enemy lines in the dark – not at all the sort of thing that could be funded by a Schools Council. We need less ‘research’ and more recce. So we print articles that reconnoitre, make reconnaissance – well, allright, RECOGNITIONS.

(Recognitions – page 32, no 2, July 1975)

//3// Reflections on Teaching Matrices to Miss C’s year 8 (7/2/11) by Alistair Bissell

‘Matrices’ is a common task in my school’s scheme of work for year 8 (see Previous Post) where students investigate the effects of transforming shapes using different 2x2 matrices.  The starting point is to get students to plot points A to F (shown in blue below), and to transform them with the matrix : 

Thinking about how to get a class of mixed ability year 8 students to do this can be very daunting for teachers who have not taught this task before, but our department has found it to be very engaging for students once they get over the initial hurdle of being able to transform shapes. In discussing with another teacher (Miss C) how they might begin this task, I offered to teach the first lesson so that they could observe. I have previously had the opportunity to observe another teacher teach my class, and found it very influential – it is always helpful to observe other teachers, but to see what your own students do with another teacher can be fascinating and surprising! Having offered to do this, I now had the daunting task of planning a lesson for an established class of students that I’d not taught before.

In a first lesson with a new class I generally aim to get students working mathematically as quickly as possible so that I can comment on their mathematical behaviour, as this provides a way of giving students purpose and supporting them in working mathematically. I was aware that a first lesson on matrices would not necessarily be open enough for students to display many of the behaviours I consider to be mathematical; at least not for the majority of the lesson, while they need to learn (be able to reproduce) the process of transforming a shape with a 2x2 matrix. I was aware that these year 8 students were familiar with PLTS (Personal Learning and Thinking Skills) language of Creative Thinker, Self Manager, Team Worker, Independent Enquirer, Reflective Learner and Effective Participator, and would have experiences from year 7 which they would relate to these labels, so referring to these might provide strategies to support them in this lesson.

Most of my time in planning this lesson went into thinking about what the students would need in order to be successful in performing a matrix transformation, as this would provide things for me to focus my comments on in the lesson. I found it very difficult to pin down what was mathematical about transforming a shape using a 2x2 matrix. It definitely involves plotting coordinates, multiplying and adding, but there is more to working mathematically than this! I began to question what it was that I wanted these students to learn in this lesson, and while I definitely valued the learning that had taken place when I’d worked on this task in previous years, it began to feel like the main purpose of this first lesson was to set up a situation that students could go on to investigate in the following lessons. I have never really considered being able to perform a complex process to be particularly mathematical, but this seemed to be the most mathematical aspect of what the students would be doing for the majority of the lesson, and felt like something that I could comment on.

My school currently insists on having three (all, most, some) learning objectives on the board in every lesson, so I have been experimenting with these being aspects of working mathematically that I can focus my comments on. For this lesson I chose:

All:          Self Manager – Be able to carry out a complex mathematical process

Most:      Reflective Learner – Write a comment about what you notice

Some:     Effective Participator – Share your work with other students

I was surprised that the focus of ‘Self Manager – be able to carry out a complex mathematical process’ felt completely natural in the lesson, as I could link nearly everything the students did to this! Suddenly getting stuck became part of being mathematical, as did putting your hand up to ask a question, talking to other students and checking whether answers were correct.

I was able to comment when students were demonstrating behaviours that were helpful, for example when a student put up their hand to ask for help I was able to comment that ‘when mathematicians work on carrying out a complex procedure they might ask someone for help’. Usually when I am commenting on an aspect of working mathematically I am only able to do this when I see helpful behaviours, but with the focus of being able to carry out a complex mathematical procedure I was also able to comment when students were demonstrating behaviours that were unhelpful. For example, when a student called out ‘I don’t get it’ I was able to comment that ‘it’s okay to be stuck because this is part of what we’re working on today, what to do when we’re stuck. What strategies have we got for if we’re carrying out a complex procedure and we get stuck?’

Many of these behaviours came from students’ lack of confidence and not being sure they were doing things correctly, as the process is complicated. Labelling these behaviours as aspects of being able to carry out a complex mathematical process seemed to allow the students to feel like they were making progress even when they were stuck, and provided them with extra resilience and motivation.  I’m not sure about the purpose of the ‘Self Manager’ bit, but I suspect that the objective would not have been as powerful without it, as the students could recognise this as something they had worked on and thought about from ‘Learning Challenge’ lessons in year 7.  My sense was that because ‘being able to carry out a mathematical process’ was an aspect of both being a ‘Self Manager’ and being a ‘Mathematician’, it gave an extra sense of purpose to what the students were doing.  While the focus of being able to carry out a complex mathematical process was supportive for me in commenting on students’ behaviours, I feel that the focus of being a ‘self manager’ was more supportive for the students in finding strategies for carrying out a complex process.

I ended the lesson by displaying the image (shown above) of the shape before and after it had been transformed and asking students what had changed and stayed the same. It felt as though most students were ready and determined to have a go at performing their own transformation next lesson, and in this respect the lesson felt successful. While my sense is that most students had been challenged and had felt like they had become more at ease with working on things that they found hard, I’m still left pondering what mathematics these students learned.

//2// Matrices and Transformations (Notes from a Scheme of Work)

A first lesson:

(The teacher's voice is italicised.)

Draw a co-ordinate grid in your books from 0 to 12 on x and y ...

Draw one on the board as you say this, or have one already drawn.

... and draw this shape at exactly the same points.

What is it? A church, yes!

We are going to do a process involving these numbers that will change the shape.

   

We will do it point by point. But first, so we know what we are talking about we need to label each point. Call this point A, what are its co-ordinates? Okay, in this project we write co-ordinates vertically. We are going to work together until everyone can do this process. Copy this into your books as we do it.

1^st row, 1^st column, multiply in pairs and add                     

2^nd row, 1^st column, multiply in pairs and add          

                        A                                                   A’

Repeat these instructions for every point. Invite volunteers to complete.

Label and draw each of the new points:

What has happened to the shape?

You may want to introduce the language of stretches or shears and write up things like ‘What happens to the area?’ if students mention that kind of thing.

Try out your own numbers instead of    

... and see what happens to your own shape.

You may want to discuss sensible shapes to try.

Challenge: In 5 lessons time I will come in and write a set of four numbers (a matrix) on the board and your task will be to tell me what effect it will have just by looking at it.

Where this can go

You may need to share ideas about how to explore this challenge in an organised way. With less confident students the investigation can be limited to matrices of the form

Students may find several types of matrix that transform a 2D shape to a line. These are 'crushing' matrices, and it can be interesting to try and characterise them (e.g., conjecture that all matrices of the form ... will be crushes).

If students work on paper this can be pinned to a board. They must write what the matrix has done to their shape before they pin the paper to the board. Questions may arise from looking at the matrices others have tried. Someone could organise the class results and feedback how they did it. What do other people notice? So, what questions could we ask? Can you generalise or predict? What happens to the areas of your shapes?

At some point focus the class on the basic transformations: reflection, rotation, enlargement, stretch, identity, crush (ie when the shape becomes a line or point), shear. Ask if anyone has a matrix that does one of these things only. Create a space on the board for students to write up matrices that do these basic transformations, see if anyone can generalise, eg an enlargement scale factor n is: 

Someone could organise the class results under these headings.

Some students may even want to extend such generalisations, eg to 3D.

Get students to try and find the inverse matrix eg of an enlargement.

Guidelines for notes at the end of the topic

A reflection has a line of symmetry.

A rotation has a centre of rotation, a number of degrees and a direction (clockwise or anti-clockwise).

An enlargement has a centre of enlargement and a scale factor. If a shape is enlarged by a scale factor of n, each side becomes n times longer and the area becomes n² times bigger.

[You may want to draw illustrations of these points.]

Vocabulary

Reflection, rotation, enlargement, stretch, shear, crush, identity, inverse.

Further Reading

This activity & lesson start was taken from an original idea by Laurinda Brown, for a fuller write up see:

Brown, L. (1991) Stewing in your own juice. In  D. Pimm and E. Love (eds) Teaching and Learning School Mathematics. London: Hodder and Stoughton.

A similar activity is also written up by Don Cohen, see:

Cohen, D. (1995) Changing shapes with matrices. Champaign, Illinois: Don Cohen – The Mathman. (Also see: http://www.mathman.biz/)

//1// 'The 4 Square Perimeter Problem' by Nicola Waddilove

I got straight on with it today. I put the question on the board and quickly told them not to say anything once they had worked it out, but to be thinking of a ‘clue’ they could give other people to help them understand what was going on. I’ve been doing this a lot lately – seems to help stop them from calling out and give everyone a chance to think.

About 5 of them were sat there with their eyes bulging out of their heads and all ten fingers waggling at the ceiling, desperate to show me they knew.  Someone came to the board and traced around the edge of the shape, a good ‘clue’ I thought. Eventually we all understood what P was. I asked them what they noticed about the shapes, what was the same and what was different. I wanted them to tell me all the shapes had 4 squares, but I forced myself to give credit to all of the things they noticed, even if they were ‘irrelevant’ to where I was heading. I thought it was important to praise them for noticing things, and forced myself to keep taking asking answers even when I’d got what I wanted.  I think this was a good thing, although for some reason I found it hard.

I asked them what questions we could ask. I wasn’t hopeful - I had my own list of questions prepared. But this time I didn’t need them! Someone wanted to know what the perimeter of a 3D square was. Someone else wanted to know how many different shapes there were with 4 squares. One boy wanted to know how many right angles there were, whilst someone else asked what the smallest perimeter was that we could make! 

I was so impressed with their questions that I decided to let them all investigate any question they liked. I didn’t know whether this would work and questioned it a couple of times later on, but I wanted to encourage them to keep asking questions. I gave my list of questions to those that were stuck and spent my time going round the class asking people what they were working on. I felt like there were lots of people who were time wasting – not sure what to do or just drawing shapes with no purpose. But some of them were really into it. I stopped them and discussed with them how ‘being organised’ would help us spot things and notice patterns, we put some people’s results in a table. This helped focus some of them and I was soon informed that there were 4 right angles in each square so the number of right angles always had to go up in fours. I realised I would never have asked this question because it was too obvious to me, but clearly it was not obvious to everyone I taught. A few people found the maximum perimeter was four times the number of squares and they seemed very pleased with themselves.  I was too.

We ended the lesson by writing. I gave them some ideas to help structure their writing, but this seemed to confuse them more than anything. Perhaps it’s just best to allow them to write what they think the lesson was about.

I felt the lesson had 'worked', although slightly hectic since everyone was doing something different. I felt worried that some of them hadn’t really achieved anything. They hadn’t got the hang of what I meant by a question. One boy had just asked what ‘was the biggest perimeter you could find’, and kept drawing bigger and bigger shapes. I wasn’t sure how to help him understand what made a ‘good’ question.

I was more confident that we had been learning things today though. We had been using mathematical language – cubes, squares, perimeters, areas etc, and noticing patterns, making predictions and explaining ourselves. I feel that they got more out of today’s lesson than any of the traditional lessons we have done on ‘algebra’, especially in terms of enjoyment, but I still feel like I would be criticised for not having a clear learning objective on the board, and for not being able to demonstrate the ‘progress’ they had made in the lesson.