1. Reasoning Mathematically
Promoting Mathematical Thinking
Reasoning Mathematically
John Mason
Mathsfest Cork
Oct 2012
The Open University
Maths Dept
University of Oxford
Dept of Education

2. Specific Aims for Ordinary Level
an understanding of mathematical concepts and of their relationships
confidence and competence in basic skills
the ability to solve problems
an introduction to the idea of logical argument
appreciation both of the intrinsic interest of mathematics and of its usefulness and efficiency for formulating and solving problems

3. Conjectures
Everything said here today is a conjecture … to be tested in your experience
The best way to sensitise yourself to learners
is to experience parallel phenomena yourself
So, what you get from this session is what you notice happening inside you!

4. Tasks
Tasks promote activity; activity involves actions; actions generate experience;
but one thing we don’t learn from experience is that we don’t often learn from experience alone
Something more is required

5. Secret Places What is your best strategy to locate the secret place?
Homage to Tom O’Brien (1938 – 2010)
One of the places around the table is a secret place.
If you click near a place, the colour will tell you whether you are hot or cold:
Hot means that the secret place is within one place either way
Cold means that it is at least two places away
What is your best strategy to locate the secret place?

6. Counting Out
In a selection ‘game’ you start at the left and count forwards and backwards until you get to a specified number (say 37 or 177). Which object will you end on? A B C D E 1 2 3 4 5
How do you know?
Generalise! 9 8 7 6 10 …
If that object is eliminated, you start again from the ‘next’. Which object is the last one left?

7. Alternating Square Sums
Imagine a triangle
Imagine a point inside the triangle
Drop perpendiculars to the three sides of the triangle
Each side of the triangle comprises two segments
On each segment of each edge, construct a square
Alternately colour the squares yellow and cyan around the triangle
Conjecture: the sum of the areas of the yellow squares is the sum of the areas of the cyan squares.
For what hexagons is this the case?

8. Selective Sums
Add up any 4 entries, one taken from each row and each column.
The answer is (always) 6
Why? -2 2 -4 6 4 8 3 1 5 -1 -3
Example of (use of) permutations
Use fractions to urge practice with fractions
Get students to make up their own with their age or some other number as the answer
Example of seeking invariant relationships
Example of focusing on actions preserving an invariance
Opportunity to generalise

9. Selective Sums
Add up any 4 entries, one taken from each row and each column.
Is the answer always the same?
Why?

10. Chequered Selective Sums
Choose one cell in each row and column.
Add the entries in the dark shaded cells and subtract the entries in the light shaded cells.
What properties makes the answer invariant?
What property is sufficient to make the answer invariant? 2 -5 -3 -6 4 -1 9 3 -2 5
Why is it suffient that each 2 by 2 subsquare sums to zero for the property to hold?
How does this grid relate to a Covered up Sum grid?

11. Circles in Circles How are the red and yellow areas related? red
orange
yellow

12. Carpet Theorems
In a room there are two carpets whose combined area is the area of the room.
The area of overlap is the area of floor uncovered
In a room there are two carpets. They are moved so as to change the amount of overlap.
The change in the area of overlap is the change in area of uncovered floor

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

14. Perimeter Projections
The red point traverses the quadrilateral
The vertical movement of the red point is tracked.
What shape is the graph?
Given a graphical track of the vertical movement and the horizontal movement,
What is the shape of the polygon?

15. Could these all be squares?
Square Deduction
Could these all be squares?

16. Square Deduction: tracking arithmetic
(3x3+4)/3
3x4-3x3
3+3x4
3x3+4 3 4
3+2x4
2x3+4
3+4
Track the 3 and the 4:
Replace the 3 by a and the 4 by b

17. Square Deduction: acknowledging ignorance
(3a+b)/3
3a+b = 3(3b-3a)
12a = 8b
So 3a = 2b
3b-3a
a+3b
3a+b a b
For an overall square
4a + 4b = 2a + 5b
So 2a = b
a+2b
2a+b
a+b 2 3 5 8 7 9 11
For n squares upper left
n(3b - 3a) = 3a + b
So 3a(n + 1) = b(3n - 1)
But not also 2a = b

18. Reflection What aspects of reasoning… Stood out for you?
Involved some struggle
What actions …
Did you undertake?
Were ineffective (why?)
Were effective (why?)

19. Tasks
Tasks promote activity; activity involves actions; actions generate experience;
but one thing we don’t learn from experience is that we don’t often learn from experience alone
It is not the task that is rich …
but whether it is used richly
What matters more than the particular answer is …
how do you know?
what can you vary and still the same approach works?

20. Reminder Do–Talk–Record Generating need to communicate
Provoking Engagement
Surprise
Challenge (trust)
Promoting Mathematical thinking
How do you know? …
Why must …?
Active Students
Constructing
Extending

21. Institute of Mathematical Pedagogy
Follow Up
mcs.open.ac.uk/jhm3
open.ac.uk
Thinking Mathematically (new edition)
Designing and Using Mathematical Tasks
Questions and Prompts … (Primary version from ATM)
Thinkers
Institute of Mathematical Pedagogy
August
mcs.open.ac.uk/jhm3