Promoting Mathematical Thinking Task Construction: lessons learned from 25 years of distance support for teachers John Mason Nottingham Feb 2012 The Open University Maths Dept University of Oxford Dept of Education

Outline Some Tasks OU Frameworks MGA, DTR, Stuck, EIS, … APC or ORA: Own experience, Reflection on parallels, Apply to classroom Systematics Frameworks What makes a task ‘rich’?

Number Line Translations… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it Translate the copy line to the right by 3 I am thinking of a number … tell me how to work out where it ends up Where does 7 end up? Where does –2 end up? 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7 Chosen this because I have always enjoyed it, it is richly mathematical, and because of recent email discussions with an OU student Particular to General Denote translation to the right by a, by Ta What is Ta followed by Tb? What about Tb followed by Ta?

Number Line Scaling… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7 Denote scaling from 0 by a factor of s by Ss I am thinking of a number … tell me how to work out where it ends up General to particular What is Sa followed by Sb? Denote scaling from p by a factor of s by Sp,s What is Sp,s in terms of T and Ss?

Number Line Scaling… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it Scale the number line by a factor of 3 (keeping 0 fixed) 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7 I am thinking of a number … tell me how to work out where it ends up Where does 2 end up? Where does –3 end up? Particular to general Denote scaling from 0 by a factor of s by Ss What is Sa followed by Sb? Denote scaling from p by a factor of s by Sp,s What is Sp,s in terms of T and Ss?

Number Line Rotations… Imagine a number line with the integers marked on it Imagine a copy of the number line sitting on top of it Rotate the copy through 180° about the point 3 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7 I am thinking of a number … tell me how to work out where it ends up Where does 7 end up? Where does -2 end up? Denote rotating about the original point p by Rp What is Rp followed by Rq? Rotate twice about 0 … … to see why R–1R–1 = T0 = S1 and so (-1) x (-1) = 1

Diamond Multiplication Chosen this because it always provokes immediate ‘work’ by audiences; illustrates beautifully the movement of attention

Differing Sums of Products Write down four numbers in a 2 by 2 grid 4 5 3 7 Add together the products along the rows 28 + 15 = 43 Add together the products down the columns 20 + 21 = 41 43 – 41 = 2 Calculate the difference Tracking Arithmetic That is the ‘doing’ What is an undoing? What other grids will give the answer 2? Choose positive numbers so that the difference is 7

Differing Sums & Products 4 5 3 7 Tracking Arithmetic 4x7 + 5x3 4x5 + 7x3 4x(7–5) + (5–7)x3 = 4x(7–5) – (7–5)x3 = (4-3) x (7–5) So in how many essentially different ways can 2 be the difference? What about 7? Revealing Structure by attending to relationships not calculations Numerals as placeholders So in how many essentially different ways can n be the difference?

Patterns with 2 Embedded Practice (Gattegno & Hewitt)

Structured Variation Grids Tunja Factoring Quadratic Double Factors Sundaram

Put your hand up when you can see … Something that is 3/5 of something else Something that is 2/5 of something else Something that is 2/3 of something else Something that is 5/3 of something else … Something that is 1/4 – 1/5 of something else

Remainders What is the remainder on dividing 5 by 3? What question am I going to ask next? What is the remainder on dividing 5 by -3? What is the remainder on dividing -5 by -3?

Task Purposes To introduce or extend contact with concepts To highlight awareness of human powers used mathematically To focus attention on mathematical themes To sharpen awareness of study strategies problem solving strategies (heuristics) learning how to learn mathematics evaluating own progress exam technique Purpose for students Potential Utility (Ainley & Pratt)

Learning from Tasks Tasks –> Activity –> Actions –> Experience But one thing we don’t seem to learn from experience … is that we don’t often learn from experience alone! –> withdraw from action and reflect upon it What was striking about the activity? What was effective and what ineffective? What like to have come-to-mind in the future? Personal propensities & dispositions? Habitual behaviour and desired behaviour? Fresh or freshened awarenesses & realisations?

Task Design Pre-paration Pre-flection Post-paration Post-flection Content (Mathematics) Reflection Interactions (as transformative actions) Tasks Resources Activity When does learning take place? In sleep!!!

Slogans A lesson without opportunity for learners … to generalise mathematically … is not a mathematics lesson! to make and modify conjectures; to construct a narrative about what they have been doing; to use and develop their own powers; to encounter pervasive mathematical themes is not an effective mathematics lesson Trying to do for learners only what they cannot yet do for themselves

Modes of interaction Expounding Explaining Exploring Examining Exercising Expressing

Student Content Teacher Exercising Teacher Student Content Expounding Student Content Teacher Exploring Teacher Student Explaining Content Student Content Teacher Examining Student Content Teacher Expressing

Activity Goals, Aims, Desires, Intentions Tasks (as imagined, enacted, experienced, …) Resources: (physical, affective, cognitive, attentive) Initial State Affordances– Constraints–Requirements (Gibson)

Potential What builds on it (where it is going) Most it could be Math’l & Ped’c essence Role of reflection in achieving affordances Least it can be What it builds on (previous experiences) Affordances– Constraints–Requirements (Gibson) Directed–Prompted–Spontaneous Scaffolding & Fading (Brown et al) ZPD (Vygotsky)

Thinking Mathematically CME Do-Talk-Record (See–Say–Record) See-Experience-Master Manipulating–Getting-a-sense-of–Artculating Enactive–Iconic–Symbolic Directed–Prompted–Spontaneous Stuck!: Use of Mathematical Powers Mathematical Themes (and heuristics) Inner & Outer Tasks

Frameworks Enactive– Iconic– Symbolic Doing – Talking – Recording See– Experience– Master

Example From NNP project, pattern sequences to be counted Stuck with providing first, second, third and only later recognising the dependency created Unlocking potential Universality of the Frame Theorem (Gaussian Curvature and Betti Numbers) Counting squares, counting sticks, … Counting weights

Example: Extending Mathematical Sequences Mathematically “What is the next term …?” only makes sense when ... Mathmematical guarantee of uniqueness Geometrical or other construction source Some other constraint

Painted Wheel (Tom O’Brien) … Someone has made a simple pattern of coloured squares, and then repeated it at least once more State in words what you think the original pattern was Predict the colour of the 100th square and the position of the 100th white square … Theorem: a sequence is uniquely specified if you know that the repeating pattern has appeared at least twice. Make up your own: a really simple one a really hard one Provide two or more sequences in parallel

Gnomon Border How many tiles are needed to surround the 137th gnomon? The fifth is shown here In how many different ways can you count them? What shapes will have the same Border Numbers?

Extending Mathemtical Sequences Stress in Thinking Mathematically and later on ‘specifying the growth mechanism before trying to count things’ Uniquely Extendable Sequences Theorem Instance of general topological theorem (Betti numbers) Attempts in two Dimensions!

How many holes for a sheet of r rows and c columns Perforations If someone claimed there were 228 perforations in a sheet, how could you check? How many holes for a sheet of r rows and c columns of stamps?

Gasket Sequences

Attention Teahing and Learning is fundamentally about attention: What is available or likely to come–to–mind when needed What is available to be learned? variation Use of powers Use of themes Use of resources (physical, mental, virtual) Structure of attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

Follow-Up j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 Designing & Using Mathematical Tasks (Tarquin/QED) Thinking Mathematically (Pearson) Developing Thinking in Algebra, Geometry, Statistics (Sage) Fundamental Constructs in Mathematics Education (RoutledgeFalmer) Mathematics Teaching Practice: a guide for university and college lecturers (Horwood Publishing) Mathematics as a Constructive Activity (Erlbaum) Questions & Prompts for Mathematical Thinking (ATM) Thinkers (ATM) Learning & Doing Mathematics (Tarquin) j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3