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

2. Some Sums 1 + 2 = 3 4 + 5 + 6 = 7 + 8 9 + 10 + 11 + 12 = 13 + 14 + 1516 + =
Generalise
Say What You See
Justify
Watch What You Do

3. 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

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.

5. CopperPlate Calculations

6. Structured Variation Grids

7. 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

8. 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?

9. 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?

10. 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?

11. 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

12. 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?

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

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

15. 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

16. 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

17. MGA & DTR
Doing – Talking – Recording

18. 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

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

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

21. Consecutive Sums
Say What You See

22. 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