Promoting Mathematical Thinking On the Structure of Attention

Promoting Mathematical Thinking On the Structure of Attention & its Role in Engagement & the Assessment of Progress The Open University Maths Dept 1 John Mason Oxford PGCE April 2012 University of Oxford Dept of Education

Attention / Macro – Locus, Focus, Scope / Micro – To be experienced / Meso – Student focus & disposition 2

Present or Absent? 3

Micro Attention / / / 4 Holding Wholes (Gazing) Discerning Details (making distinctions) Recognising Relationships (in the particular) Perceiving Properties (being instantiated) Reasoning on the basis of agreed properties

Find the error! 79645 64789 30 2420 361635 54242840 4230423245 28634836 497254 5681 63 5160119905 5 How did your attention shift?

Movements of Attention in Geometry A b x E G d y C F a D 6 c B

Rectangular Room with 2 Carpets How are the red and blue areas related? 7

Tracking Arithmetic Becomes Algebra 8

Differing Sums of Products / / Write down four numbers in a 2 by 2 grid Add together the products along the rows 28 + 15 = 43 / Add together the products down the columns 20 + 21 = 41 / Calculate the difference 43 – 41 = 2 / / / 9 4 7 5 3 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 / Tracking Arithmetic 4 x 7 + 5 x 3 4 7 5 3 4 x 5 + 7 x 3 4 x(7– 5) + (5– 7)x 3 = 4 x(7– 5) – (7– 5)x 3 = (4 -3) x (7– 5) / / / 10 So in how many essentially different ways can 2 be the difference? What about 7? So in how many essentially different ways can n be the difference?

Think Of A Number (THOAN) How is it done? How can we learn to do it? Tracking Arithmetic! 11

Club Memberships In a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both? 47 total poets 12 31 47– 29 47– 31 31–(47– 29) 29–(47– 31) 29 painters

Club Memberships (3) in a certain club there are 28 people. There are 14 poets, 11 painters and 15 musicians; there are 22 who are either poets or painters or both, 21 who are either painters or musicians or both and 23 who are either musicians or poets or both. How many people are all three: poets, painters and musicians? 13

In a certain club there are 28 people. There are 14 poets, 11 painters and 15 musicians; there are 22 who are either poets or painters or both, 21 who are either painters or musicians or both and 23 who are either musicians or poets or both. How many people are all three: poets, painters and musicians? 28 total 15 musicians 21 poets or musicians poets 14 23 musicians or painters 28– 22 28– 23 28– 21 11 painters 11+15– 22 poets or painters 23 14+11– 22 (14+15 -21) + (14+11 -22) + (11+15 -23) – (28– ((28 -23) + (28 -22) + (28 -21)) 14+15– 21 14 2

Tracking Arithmetic / / / 15 Engage in some ‘calculation’ but don’t allow one (or more) number(s) to be absorbed into the arithmetic Then replace those numbers by a symbol Use in any task that calls for a generalisation or a method or a use of algebra

Meso-Attention / / What do you enjoy about thinking mathematically? Could it be … – – 16 Getting an answer? Knowing your answer is correct? Using your natural powers? Encountering increasingly familiar themes?

Powers & Themes Powers u u Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Themes v v v Doing & Undoing Invariance in the midst of change Freedom & Constraint Are students being encouraged to use their own powers? or are their powers being usurped by textbook, worksheets and … 17

Teaching students to think mathematically … … involves developing a disposition to … think mathematically, to use powers mathematically, to be mathematical … to attend to situations mathematically v How often do you think mathematically with and in front of students? v v 18 What are they attending to? (and how? ) What are you attending to when interacting with students? (and how? )

Meso Level of Attention / Discrete & Continuous – Integers -> fractions -> decimals / / / / Additive & Multiplicative & Exponential Thinking Arithmetic as the study of actions on objects Finiteness & Infinity Rules & Tools Arbitrary (Convention) & Necessary It looks right => It must be so because … Procedures & Underlying Reasons Adolescent concerns – self in relation to the social; sex 19

Getting To Grips With Graphs / / 20 Imagine a square Imaging a point on the edge of the square, traversing the perimeter at a constant speed With your right hand, show the vertical movement of the point With your left hand, show the horizontal movement of the point

Perimeter Projections Imagine the vertical and horizontal movements of the red point as it traverses the perimeter Now imagine them being graphed against time 21

Ride & Tie 22

Elastic Multiplication / / / Imagine you have a piece of elastic. You stretch it equally with both hands … what do you notice? Hold one end fixed. Stretch the other so the elastic is four-thirds as long. Where is the midpoint? – Relative to the elastic – Relative to the starting position of the elastic 25

Straight Line Constructions / Sketch the graph of a pair of straight lines such that – – 26 Their slopes differ by two and their x-intercepts differ by two and their y-intercepts differ by two And the areas the triangles (origin, x-intercept, yintercept) differ by 2.

Tabled Variations 27

Structured Variation Grids Tunja 28 Factoring Quadratic Double Factors

Sundaram Grids All rows and columns are arithmetic progressions How many entries do you need to fill out the grid? 29

Spiral 30 4 3 4 2 41 4 4 2 1 20 4 5 2 2 7 4 6 2 3 8 4 7 2 4 99 4 0 39 19 6 1 2 18 5 4 3 38 17 16 15 14 37 36 35 34 33 48 2 25 5 1 0 1 1 1 2 1 3 3 2 49 4 9 2 6 2 7 2 8 2 9 3 0 3 1 5 0

Spiral 64 36 37 38 39 35 14 15 16 34 1 3 1 2 3 1 33 3 2 31 1 1 3 0 2 9 18 4 2 19 4 3 20 3 4 2 1 2 2 2 3 2 4 4 0 17 41 2 1 0 9 1 5 8 7 6 2 8 2 7 2 6 2 5 4 4 4 5 4 6 4 7 48 4 9 5 0 81

Structure of the Psyche Awareness (cognition) Imagery Will Emotions (affect) Body (enaction) Habits Practices 32

Structure of a Topic Language Patterns & prior Skills Imagery/Senseof/Awareness; Connections Root Questions predispositions Different Contexts in which likely to arise; dispositions Techniques & Incantations Emotion r ou vi ha Be Aw ar en es s Standard Confusions & Obstacles Only Emotion is Harnessable 33 Only Awareness is Educable Only Behaviour is Trainable

Attention / Macro – Locus, Focus, Scope / Micro – Holding wholes; discerning Details; Recognising Relationships; Perceiving Properties; reasoning on the basis of agreed properties / Meso – Student focus & disposition – Shifts in perception & conception 34

To Follow Up / http: //mcs. open. ac. uk/jhm 3 – – – / 35 Presentations Applets Structured Variation Grids j. h. mason@open. ac. uk