Teaching as The Directing of Attention and Learning

Teaching as The Directing of Attention and Learning as the Transformation of Attention John Mason Stowarzyszenia Nauczycieli Matematyki Muszyna 2007 1

Outline: teaching using attention /Directing Attention Towards: – stressing, focusing – confronting /Directing Attention Away: – de-emphasising through exploring (integration through subordination) /Structure 2 of Attention

Directing Attention Towards 3

What’s The Difference? – = First, add one to each First, add one to the larger and subtract one from the smaller 4 Directing Attention Towards Whatdifference wouldthatmake to tothe thedifferences? What could be varied?

How many. . . are in. . . ? • How many 1 s are in one 2 ? ? third? one 2 1 ? ? fifth? blah oop 3 4. . . ? 2 3 1 s are in. . . 3 4 2 s are in. . . 3 What are you doing? 5 Directing Attention Towards

Find a … fraction which divides into both 14 10 and 15 21 an integral number of times. and another find the largest. 6 Directing Attention Towards

Find a. . . • fraction which both 14 10 and 15 21 divide into an integral number of times. and another find the smallest. 7 Directing Attention Towards

More doesn’t always go with More and Less doesn’t always go with Less /Draw a (non-square) rectangle – Now adjust it so that the area goes up while the perimeter [goes up [stays the same [goes down Area Perimeter Up Same Down Up Adjust = cut &/or paste Adjust = alter rectangle 8 Same Down Dina Tirosh 2007

Directing Attention Away (ignoring, de-emphasising; exploring) /In how many different ways can a unit fraction be written as the difference of two unit fractions? /In how many different ways can you decompose a given rectilinear region into non-overlapping rectangles? 9 Directing Attention Away

Learner Choice: Breadth & Depth Exercises 3 y = 12 – 2 x x=3 3 y = 13 – 2 x x=5 2 y = 14 + 5 x x = -2 2 x + 3 y = 12 23 x = 69 2 x + 3 y = 13 13 x = 65 -5 x + 2 y = 14 11 x = -22 2 x + 3 y = 12 21 x – 3 y = 57 2 x + 3 y = 13 11 x – 3 y = 52 -5 x + 2 y = 14 16 x – 2 y = -36 -3 x + 5 y = 14 20 x + 5 y = 60 2 x + 3 y = 12 7 x – y = 19 2 x + 3 y = 13 9 x – 6 y = 39 -5 x + 2 y = 14 8 x - y = -18 -3 x + 5 y = 14 4 x + y = 12 5 y = 14 + 3 x x=2 -3 x + 5 y = 14 23 x = 46 (Dave Herring 2007) 10

Using Attention 11

True or False What do might What you 8 yr do oldstododecide? to decide? 12 + 11 = 11 + 12 51 + 51 = 50 + 52 27 + 48 - 48 = 27 257 - 34 = 257 - 30 - 4 137 - 137 = 5 12 Directing Attention Towards Marta Mollina (Granada)

Raise Your Hand When You See. . . something which is 2/5 of something; 3/5 of something; 5/2 of something; 5/3 of something; 2/5 of 5/2 of something; 3/5 of 5/3 of something; 5/2 of 2/5 of something; 5/3 of 3/5 of something; 1 ÷ 2/5 of something; 1 ÷ 3/5 of something; 2/5 ÷ 3/5 of something? 3/4 ÷ 5/4 of something? 13 Directing Attention Towards Pat Thompson (2004)

Ride & Tie What is the same, and what different about. . . Which graph makes sense, and why? 14 Directing Attention Towards

Structures of Attention Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties 15

Structures of Attention Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties 16

Magic Square Reasoning 2 2 7 2 1 5 9 8 Sum( 17 6 3 4 )– Sum( ) Directing Attention Towards What other configurations like this give one sum equal to another? Try to describe them in words =0

More Magic Square Reasoning Sum( 18 Directing Attention Towards )– Sum( ) =0

Questions ★ Why is directing attention important? Need for confluence between learner and teacher attention Being aware of the structure and focus of your own attention is necessary if you are to direct the attention of learners appropriately Being aware of the importance of attention can alert you to reasons for what learners say and do 19

Summary ★ How is attention directed? • Towards (stressing, focusing) • Away (ignoring, de-emphasising) Strategies for Directing Attention Say What You See Watch What You Do In How Many Different Ways Can You. . . ? Show me that you see. . . What is the same, and what is different about … ? • Making Use of the Structure of Attention