ATM EASTER CONFERENCE 2004 LOUGHBOROUGH OPENING SESSION subheads have underlining TAKE YOUR MARKS! GET SET! GO! [SLIDE 1] PETER LACEY [SLIDE 2] Isn't it strange how words mean different things to different people. To the athlete "take your marks; get set; go!" is clearly the start of a race. Years ago it meant something different to my daughter who, on the basis of her test scores in Y9, was put into the bottom set and spent a proportion of her next two years absconding from school. Take your marks; get set; go! I wanted to start this presentation with a witty joke about a mathematician, a teacher and a neuro-psychologist … but I couldn't think of one. Entries on a postcard please! - Better still - send it to MT! I chose these three occupations because they represent disciplines that have a bearing on the learning of mathematics but do not always recognise or communicate with each other. Sometimes they are antagonistic. Reading MT of November 1959 shows that the tensions were evident even then (MT11 p9 Puzzles in the classroom - Trevor Fletcher …P28 Number and Infants R.M.Fyfe) A number of university mathematicians continue to criticise what and how mathematics is taught in schools. They blame this on the dwindling supply and quality of what they call good mathematicians. The increasingly nationally determined curriculum, assessment and pedagogy frighteningly correlates with falling rates of teacher retention. The explosion in research and understanding into how the brain works and how different sorts of intelligence, including emotional intelligence, inter-relate constitutes an insignificant part of the initial and continuous training of teachers. However, If I take the wrapping off a mathematics lesson and examine its contents I inevitably see the teacher's interpretation of what they believe mathematics is, and is about, and is for. I see a lot of their interpretation of curriculum, pedagogy and assessment. I also see clues to their interpretation of theories of learning. Memorable lessons are those in which these interpretations are made explicit and somehow blended in a magical mix. I scanned a year's worth of "My Best Teacher" reports in the Times Educational Supplement and picked out the common features across all of them. Actually, if the truth were known, now in the twilight of my career I keep looking to see if anyone I taught has actually remembered me! The nearest hit so far is Steve Cramm writing about his best teacher in a school down the road from where I taught in the 1970s! Before I show the slide I want you to reflect on your best teacher. Think of his or her characteristics. [SLIDE 3] Characteristics of my best teacher I begin to see the mathematician, the teacher and the psychologist all rolled into one. Enough of the pre-amble: I want to share with you a series of nine classroom episodes (some of which are composites) where I have seen pupils stretch to new limits; achieve a new personal best. Attached to each episode are some reflections. After this, I will try to bring these reflections together in some coherent way. This is as much a celebration of the craft and talent of mathematics teachers as it is a personal learning journey. _________________________________________________ Episode 1 Jack I must start with Jack. He was five when I first met him in 1996, I asked him to tell me something he really knew in mathematics. He said “5 + 3 = 8”, I told him that we shared that knowledge. It’s interesting that knowledge isn’t like chocolate. If I share it with another person I am not left with only half of what I had! He was encouraged by my response and said he knew more. With a session on number bonds to 20 I tried to predict Jack’s next offering. He said “500 + 300 = 800”. He must have noticed my smile. “I know more!" he said, “I know that 5 000 + 3 000 = 8 000”. The game was on! “So what is 5 000 000 + 3 000 000?” I asked. “It’s 8 000 000” he replied. With a smile back I left him to his number bonds to 20. [SLIDE 4] Big ideas in mathematics Geoff Faux’s big ideas came to my mind that evening (MT 163 p12 What are the big ideas in mathematics - Geoff Faux) And could Jack see an image similar to the Gattegno Tens Chart? Had he seen the pattern? Had he seen numbers in formation. Had he seen number information? [SLIDE 5] Gattegno tens chart - whole numbers It was over a month later I realised the significance of what Jack hadn’t said - for example 30 + 50 = 80. I reflected on how our number name language, unlike Japanese, occludes the perfect structure of our number system. In this next slide note how the number names can be seen as 'co-ordinates' on the axes; or not. [SLIDE 6] Linguistic comparison of number names [SLIDE 7] Number names Jack’s story is now well known in my area and I have seen some interesting lessons that start off by saying if I know one particular fact what others can I deduce? For example, if I know that 5 + 3 = 8 then what else do I know? 8 - 5 = 3 8 - 3 = 5 80 - 50 = 30 [SLIDE 8] Membering - a mind map Note the process of membering - joining together. I am left wondering about fundamentals and consequentials. - and perhaps that subtraction comes before addition (if I start with five chocolates I always seem to have fewer left!) - and perhaps dividing things out is more realistic than multiplying them! Barbara’s report on observing her grandchild Bethany was about sharing things. (MT186 p22 ‘From the ATM’s Professional Officer) Of course the real fundamentals relate to the connectors rather than the nodes. They relate to the rules of combination, such as commutivity, associativity and distribution. Stimulate the reasoning: then introduce the rigour by hooking reasoning chains onto the axioms. Anyway, this single episode is beginning to take up too much time. So on to episode 2. Episode 2 The lesson on decimals, I once described in TES (in their series "Beating the Inspector”!), uses and extends a pattern or formation to develop new understandings and new information. I just want to take a snippet of the lesson. [SLIDE 9] Gattegno tens chart - decimals Demonstrate vertical transitions as multiplying & dividing by powers of ten This is the same image as before but it is now extended; back to Geoff's big ideas. Notice how pupils are able to glide into decimals and apply what they have already learnt about whole numbers. Episode 3 This lesson for Y9 students built on properties of triangles. Through practical exercises followed by discussion the students had deduced that the sum of the lengths of any two sides of a triangle had to be greater than the length of the third side. The exercise books showed a series of intersecting and non-intersecting arcs. [SLIDE 10] Triangle constructions Some constructions using Pythagorean triples have set the scene for a consideration of Pythagoras’ Theorem. The lesson demonstrated the thinking reputably undertaken by Pythagoras himself (using this slide). Students used squared paper to replicate the exercise with different groups selecting different dimensions of right-angled triangles. [SLIDE 11] Pythagoras demonstration Reinforcement of the conclusion a² + b² = c² was used as a basis for practice exercises. The final part of the lesson brought back the previous learning that a + b > c and then the teacher showed the following sequence of slides [SLIDE 12, 13, 14, 15] Hypotenuse in steps The reconnection with another human being’s thinking that took place over 2500 years ago was humbling. The discussion was fascinating. The dissonance was and is memorable. These students will take a critical view of limiting conditions when they encounter calculus. More modern proofs of Pythagoras may be convincing but I am left disconnected from original thought. Geoff Giles’ reflections in the last MT on Bob Burns’ session at last conference illustrates the power of connecting with original thought. (MT186 p3 Proof in Elementary Numberwork) I believe that creating cognitive dissonance can motivate and energise thinking as we struggle to make sense. Its as though we are born with a disposition to make sense of visual stimuli. I have seen visual images such as these in classrooms; they always generate discussion! Whilst these are impossible geometric objects they open the mind to the possibility of mathematical objects and non-mathematical objects. [SLIDE 16, 17, 18, 19 29, 21] Impossible objects I am left thinking about how traditional textbooks sanitise mathematics; avoiding the awkward bits. Taking the thinking out of mathematics does both mathematics and its learning a disservice. Even examination questions seem hooked on taking the thinking out. [SLIDE 22] Comparison of 16+ examination questions Spend five minutes working out the solution to these two problems. Discuss the difference between them. My view is that the second question is insulting. Episode 4 Putting the thinking into mathematics is demonstrated in this Y7 lesson, demonstrating a formula for finding the area of a triangle in square units is a classic. [SLIDE 23] Area of square and triangle in square units The altitude of the triangle is introduced as a significant clue. But what if the question is reversed to find the area of a square in triangular units? And this was set as an extension exercise! [SLIDE 24] Area of a triangle and square in triangular units Given the symmetry of the two questions, which linear measure is now the significant clue? Now the brain has to think for itself! This is an exercise in re-membering rather than memorising. The bits are reconnected in new and different ways. Spend five minutes working on the problem so that you feel what I have described as re-membering. Triangular dotty paper is provided . (you may need triangles lattice paper) Episode 5 Turning questions inside out is a feature of lessons that I recall where students have moved to deeper levels of understanding. In discussing lessons with teachers and learners, I talk about the balance between time spent at the cutting edge of learning - 'membering and ‘re-membering’ and time spent consolidating what is already learnt. This ‘lesson starter’ on mode, mean and median (ref to slide 25) required the brain to switch on and engage rather than to go down a well-trodden route. It means having to think 'backwards'. Re-membering strengthens the membering; and deepens the understanding. [SLIDE 25] Find the data sets Using a simple ICT drawing programme such as basic LOGO, angle properties of polygons are revealed and the relationship between the sum of the interior angles and the number of sides may be deduced A = 180n - 360 or ( 2n-4 ) right angles [SLIDE 26] Internal angle of an n-gon Turning this inside out offers the possibility of drawing polygons with 2.5 sides or -3 sides. I call these 'zany logical pursuits'. And this practical approach led to the following piece of pupil's reasoning. [SLIDE 27] Sum of internal angles of a triangle Is it a proof or just a line of reasoning? Such reasoning is the gateway to visiting geometric axioms: Identifying the fundamentals (as in number) and deriving consequentials. Stimulate the reasoning; then introduce the rigour by hooking reasoning chains onto the Euclidian axioms. [SLIDE 28] Squiggles Extending the learning territory allows new perspectives of the original territory. New formations; new information. Sometimes I need to go beyond the experienced territory to make sense of it. There is a note of caution here to textbook publishers who produce texts related to levels of the national curriculum - which I believe is an abomination. For example: ? Pythagoras' theorem is a special case of the cosine rule; ? An integral domain is a special case of a rational field; ? A bar chart may be a special case of a histogram; ? F(n) for n real is a special case of F(n) for n complex; ? F(x,y) is a special case of F(x,y,z) [SLIDE 29] Sense Episode 6 I was fascinated by an algebra lesson that started with a direct quotation from Euclid's Elements, notably that now ascribed as 11.4, written in 300 BC. [SLIDE 30] Euclid 11.4 If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. A geometric formation can be described using the language of algebra and algebraic relations can sometimes be seen in terms of shapes and configurations. The following relationships were all seen in lessons. The first in a Y6 lesson and remaining two in secondary schools. [SLIDE 32, 33, 34] Algebraic representations of patterns The first 'shows' the equivalence of 3n+1 with (n+1) + 2n. The second shows that for n = 1 to r, Sn = ½r(r+1). The third shows that n² - 1 = (n + 1)(n - 1). Having established the relationship it can be explored for non-integer values of n that cannot be expressed geometrically; going beyond the territory of visual representation. [SLIDE 35, 36] Extending the pattern Whilst geometric images can illustrate algebraic expressions and equivalencies, this is only a starting point. Coupled with arithmetic reasoning, possibilities for developing algebraic fluency are opened up. Develop the reasoning; then introduce the rigour. Different ways of expressing relationships can make them accessible to those with different ways of 'seeing' things. We may have personal ways of putting things in formation. Episode 7 Back in 1969 I ran a maths club. The students had been working on translations (SMP - green hard cover book 2!) They considered a line as the trace of a translated point, a parallelogram as the trace of a translated line and a parallelopiped as the trace of a translated parallelogram. [PP SEQUENCE 37] Building a hyperparallelopiped We spent hours building a 2-dimensional and then 3-dimensional model of a hyper-parallelopiped. (pipe cleaners and straws if you want to know). And, by the way, does the Euler relationship hold? Nodes + Regions - Arcs = 2 in 2D, Vertices + Faces - Edges = 2 in 3D. ……..! Mathematics shot through with infinity as Geoff Faux says (See Slide 4) New formations - new in formation From membering to re-membering Whilst geometric visualisations of hyperspace become increasingly tortuous with each added dimension (what does a hyper-sphere look like?), defining points in hyperspace, and distances between them, is straightforward. Any ordered set of n numbers can be represented as a point in n-space. Groups of n-sets describe a 'block' of n-space that can be explored for particular maximising or minimising characteristics. For example, the effect of different quantities in a cocktail of drugs for treating cancer, or AIDS. Episode 8 My next cluster of examples relates to lessons that have stretched thinking through problem solving. During the days of RAMP, Phil Dodd put together a collection of what he described as puzzles, and I have seen these and others used to good effect. (Mathematical Puzzles - Phil Dodd 1988) Although I have told this story before it serves as a reminder. I am not very good at DIY so, thoughtfully, my daughters buy me ever more sophisticated tools. I have in my garage all the tools a craftsperson could want but I still resort to the hammer to sort out minor problems - including reluctant screws. The point I am making is that we are only as good as our ability to use the tools we have - and this goes for mathematics. Practising using the tools at our disposal is part of mathematical development. Anyway, here are some problems that I have seen being worked on by groups of pupils in classrooms. Each was significant in that the discussion prompted reasoning - joining together existing pieces of mathematical understanding in a way that would reach a solution. It exemplified what I describe as re-membering. [SLIDE 38, 39, 40] Five problems Episode 9 I choose this last example to show that pupils can be problem posers as well as problem solvers. During the days of RAMP in the 1980s (Raising Achievement in Mathematics Project 1986 - 1989) a teacher researcher with her Y10 class was using a graph plotting function programme to explore transformations of the functions y = x². The students explained that y = x² + c, created a curve ‘parallel’ to the original. The consequent discussion concluded this was not the case as the lines got closer to each other the higher the value of y. [SLIDE 41] Transforming y = x² A discussion about railway lines and concentric circles led to someone asking how y = x² could be transformed to create a curve of constant normal distance. Well, actually they did not use that phrase but that is what they meant! One or two of the class worked at home to construct it on paper and demonstrated the lines might cross. Sixth form students picked up the problem and, with the help of their teacher, gained some experience of parametric equations. ______________________________________________ Pulling the reflections together Enough of classroom examples and reflections: I want to end by pulling some of these reflections together. My own awakening as a mathematician came after I left school. Although I was reasonably competent and passed some examinations I was not inspired. I had different coloured exercise books for arithmetic; algebra (brown); geometry (light blue and plain pages); graph work (green with alternating graph and lined paper); and trigonometry (red). It wasn’t until I started connecting these together that I began to see underlying form and shape. And then I began to see how I could connect and use this to describe observable natural and physical phenomena - from daisy heads and fir cones to heat and light transfer through different media. [SLIDE 42, 43, 44, 45, 46] Some descriptions of mathematics And then I began to use mathematical models to describe and predict outcomes in other disciplines: linking temperatures to the kinetic energy of atoms and linking maximising economic conditions to peaks and troughs on a mathematically defined surface. Thinking and behaving mathematically became the kernel of mathematics itself. It is this cross - discipline understanding that underpins being a mathematician. Kath Cross, in her address to this conference (Easter Conference 1989 - Sharing perspectives) talked about the links between the parts of mathematics. Establishing the arcs is as fundamental as knowing the nodes. [PP SEQUENCE 47, 48] membering & re-membering I have referred to arc building as ‘membering’. Developing more and different arcs is ‘re-membering’. [SLIDE 49, 50] Orion the hunter This process of membering and remembering is the constant quest for pattern and form. Putting things in formation allows the brain to make sense of experience. Hence the expression information processing. In this way mathematics is what mathematicians do. Jacotot said that to teach is to cause to learn (Essays on Educational Reformers - p417 - RH Quick 1902) It is our job not only to provide rich experiences to pupils but also to help them move from the superficial to the profound through a process of membering and re-membering in order to see and develop their own in-formation. [SLIDE 51, 52, 53] About learning In the way the slides are set out, learning is a process that moves us from the left-hand side to the right-hand side. The last slide shows my current view, that understanding and technical fluency have a symbiotic rather than sequential relationship. There is nothing new here for ATM but it may serve as a reminder of our aims and principles that bind us as an association in our quest to develop the mathematician in everyone. [SLIDE 54, 55] Aims of ATM If we focus on helping others to achieve and surpass their ‘personal bests’ then I am convinced world records can be broken. Whilst others may strive to take the thinking out of mathematics, misguidedly in order to pass examinations, we should be actively involved in putting it back - a hint here of our revolutionary and binding mission! This is the process of humanising mathematics - through recognising and developing the mathematician in each of us. Derek and Barbara Ball, building on the work in ‘Better Mathematics’, described last year features of rich mathematical activity. (HMSO 1987) (MT 183 Insert ‘Maths is more than you bargained for’) The current pre-occupation with determining what will be learnt by the end of a teacher-learner interaction dulls and limits the learning possibilities of an interaction stimulated by such rich mathematical activities. ATM’s series of publications ‘Points of Departure’ continue to challenge nationally determined current pedagogical thought that lessons should be designed around points of conclusion - or learning objectives. Recent publications continue to promote starting points and will have a consequently similar long shelf life. The lessons I have described have a common feature of stimulation; stimulating thinking; developing mathematical reasoning; bringing out the mathematician. And with this stimulation comes mental engagement and stretch. I thank the many talented and unacknowledged teachers for stimulating my thinking. I am pleased to be back here with colleagues and friends who put the aspirations and needs of learners first and at the same time who do not compromise on the challenge presented by promoting mathematical thinking. I started with: "Take your marks; get set; go!" We have arrived at the finishing tape. [OHT 56] Finish Have an exciting, stimulating and productive conference. Peter Lacey Deputy Director of Education in North East Lincolnshire.