Generalisation in Mathematics: who generalises what, when, how and why? John Mason Trondheim April 2009

Some Sums 1 + 2 = 3 4 + 5 + 6 = 7 + 8 9 + 10 + 11 + 12 = 13 + 14 + 15 16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24 Generalise Say What You See Justify Watch What You Do

Four Consecutives + 1 + 2 + 3 + 6 4 Write down four consecutive numbers and add them up and another Now be more extreme! What is the same, and what is different about your answers? + 1 + 2 + 3 + 6 4

One More What numbers are one more than the product of four consecutive integers? Let a and b be any two numbers, one of them even. Then ab/2 more than the product of any number, a more than it, b more than it and a+b more than it, is a perfect square, of the number squared plus a+b times the number plus ab/2 squared.

CopperPlate Calculations

Structured Variation Grids

Extended Sequences … Someone has made a simple pattern of coloured squares, and then repeated it a total of at least two times 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 … Make up your own: a really simple one a really hard one

Raise Your Hand When You Can See Something which is 1/4 of something 1/5 of something 1/4-1/5 of something 1/4 of 1/5 of something 1/5 of 1/4 of something 1/n – 1/(n+1) of something Direct perception, seeing through the particular to the general: 1/n – 1/(n+1) = 1/n(n+1); 1/n - 1/n(n+1) = 1/(n+1) 1/a – 1/b = (b – a)/ab What do you have to do with your attention?

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?

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?

Honsberger’s Grid 17 18 19 20 21 22 23 24 25 10 11 12 13 14 15 16 5 6 7 8 9 2 4 3 1 31 43 57 73 91 111 133 147 4

Painted Cube A cube of wood is dropped into a bucket of paint. When the paint dries it is cut into little cubes (cubelets). How many cubes are painted on how many faces?

Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of properties

The Place of Generality A lesson without the opportunity for learners to generalise mathematically, is not a mathematics lesson

Text Books Turn to a teaching page What generality (generalities) are present? How might I get the learners to experience and express them? For the given tasks, what inner tasks might learners encounter? New concepts New actions Mathematical themes Use of mathematical powers Rehearsal of developing skills and actions

Roots of & Routes to Algebra Expressing Generality A lesson without the possibility of learners generalising (mathematically) is not a mathematics lesson Multiple Expressions Purpose and evidence for the ‘rules’ of algebraic manipulation Freedom & Constraint Every mathematical problem is a construction task, exploring the freedom available despite constraints Generalised Arithmetic Uncovering and expressing the rules of arithmetic as the rules of algebra

MGA & DTR Doing – Talking – Recording

DofPV & RofPCh Dimensions of possible variation What can be varied and still something remains invariant Range of permissible change Over what range can the change take place and preserve the invariance

Some Mathematical Powers Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Ordering & Characterising

Some Mathematical Themes Doing and Undoing Invariance in the midst of Change Freedom & Constraint

Consecutive Sums Say What You See

For More Details Thinkers (ATM, Derby) Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby) Mathematics as a Constructive Activity (Erlbaum) Listening Counts (Trentham) Structured Variation Grids This and other presentations http: //mcs.open.ac.uk/jhm3 j.h.mason@open.ac.uk