1. Promoting Mathematical Thinking
Task Construction: lessons learned from 25 years of distance support for teachers
John Mason
Nottingham
Feb 2012
The Open University
Maths Dept
University of Oxford
Dept of Education

2. Outline Some Tasks OU Frameworks MGA, DTR, Stuck, EIS, …
APC or ORA: Own experience, Reflection on parallels, Apply to classroom
Systematics Frameworks
What makes a task ‘rich’?

3. Number Line Translations…
Imagine a number line with the integers marked on it
Imagine a copy of the number line sitting on top of it
Translate the copy line to the right by 3
I am thinking of a number … tell me how to work out where it ends up
Where does 7 end up?
Where does –2 end up? 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7
Chosen this because I have always enjoyed it, it is richly mathematical, and because of recent discussions with an OU student
Particular to General
Denote translation to the right by a, by Ta
What is Ta followed by Tb?
What about Tb followed by Ta?

4. Number Line Scaling…
Imagine a number line with the integers marked on it
Imagine a copy of the number line sitting on top of it 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7
Denote scaling from 0 by a factor of s by Ss
I am thinking of a number … tell me how to work out where it ends up
General to particular
What is Sa followed by Sb?
Denote scaling from p by a factor of s by Sp,s
What is Sp,s in terms of T and Ss?

5. Number Line Scaling…
Imagine a number line with the integers marked on it
Imagine a copy of the number line sitting on top of it
Scale the number line by a factor of 3
(keeping 0 fixed) 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7
I am thinking of a number … tell me how to work out where it ends up
Where does 2 end up?
Where does –3 end up?
Particular to general
Denote scaling from 0 by a factor of s by Ss
What is Sa followed by Sb?
Denote scaling from p by a factor of s by Sp,s
What is Sp,s in terms of T and Ss?

6. Number Line Rotations…
Imagine a number line with the integers marked on it
Imagine a copy of the number line sitting on top of it
Rotate the copy through 180° about the point 3 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7
I am thinking of a number … tell me how to work out where it ends up
Where does 7 end up?
Where does -2 end up?
Denote rotating about the original point p by Rp
What is Rp followed by Rq?
Rotate twice about 0 …
… to see why R–1R–1 = T0 = S1 and so (-1) x (-1) = 1

7. Diamond Multiplication
Chosen this because it always provokes immediate ‘work’ by audiences; illustrates beautifully the movement of attention

8. Differing Sums of Products
Write down four numbers in a 2 by 2 grid 4 5 3 7
Add together the products along the rows
= 43
Add together the products down the columns
= 41
43 – 41 = 2
Calculate the difference
Tracking Arithmetic
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

9. Differing Sums & Products4 5 3 7
Tracking Arithmetic
4x7 + 5x3
4x5 + 7x3
4x(7–5) + (5–7)x3
= 4x(7–5) – (7–5)x3
= (4-3) x (7–5)
So in how many essentially different ways can 2 be the difference?
What about 7?
Revealing Structure by attending to relationships not calculations
Numerals as placeholders
So in how many essentially different ways can n be the difference?

10. Patterns with 2
Embedded Practice
(Gattegno & Hewitt)

11. Structured Variation Grids
Tunja
Factoring
Quadratic Double
Factors
Sundaram

12. Put your hand up when you can see …
Something that is 3/5 of something else
Something that is 2/5 of something else
Something that is 2/3 of something else
Something that is 5/3 of something else …
Something that is 1/4 – 1/5 of something else

13. Remainders What is the remainder on dividing 5 by 3?
What question am I going to ask next?
What is the remainder on dividing 5 by -3?
What is the remainder on dividing -5 by -3?

14. Task Purposes To introduce or extend contact with concepts
To highlight awareness of human powers used mathematically
To focus attention on mathematical themes
To sharpen awareness of
study strategies
problem solving strategies (heuristics)
learning how to learn mathematics
evaluating own progress
exam technique
Purpose for students
Potential Utility
(Ainley & Pratt)

15. Learning from Tasks
Tasks –> Activity –> Actions –> Experience
But one thing we don’t seem to learn from experience …
is that we don’t often learn from experience alone!
–> withdraw from action and reflect upon it
What was striking about the activity?
What was effective and what ineffective?
What like to have come-to-mind in the future?
Personal propensities & dispositions?
Habitual behaviour and desired behaviour?
Fresh or freshened awarenesses & realisations?

16. Task Design Pre-paration Pre-flection Post-paration Post-flection
Content
(Mathematics)
Reflection
Interactions
(as transformative actions)
Tasks
Resources
Activity
When does learning take place?
In sleep!!!

17. Slogans
A lesson without opportunity for learners … to generalise mathematically … is not a mathematics lesson! to make and modify conjectures; to construct a narrative about what they have been doing; to use and develop their own powers; to encounter pervasive mathematical themes is not an effective mathematics lesson Trying to do for learners only what they cannot yet do for themselves

18. Modes of interaction Expounding Explaining Exploring Examining
Exercising
Expressing

19. Student
Content
Teacher
Exercising
Teacher
Student
Content
Expounding
Student
Content
Teacher
Exploring
Teacher
Student
Explaining
Content
Student
Content
Teacher
Examining
Student
Content
Teacher
Expressing

20. Activity Goals, Aims, Desires, Intentions
Tasks (as imagined, enacted, experienced, …)
Resources: (physical, affective, cognitive, attentive)
Initial State
Affordances– Constraints–Requirements
(Gibson)

21. Potential What builds on it (where it is going) Most it could be
Math’l & Ped’c essence
Role of reflection in achieving affordances
Least it can be
What it builds on
(previous experiences)
Affordances– Constraints–Requirements
(Gibson)
Directed–Prompted–Spontaneous
Scaffolding & Fading (Brown et al)
ZPD (Vygotsky)

22. Thinking Mathematically
CME
Do-Talk-Record (See–Say–Record)
See-Experience-Master
Manipulating–Getting-a-sense-of–Artculating
Enactive–Iconic–Symbolic
Directed–Prompted–Spontaneous
Stuck!: Use of Mathematical Powers
Mathematical Themes (and heuristics)
Inner & Outer Tasks

23. Frameworks Enactive– Iconic– Symbolic Doing – Talking – Recording
See– Experience– Master

24. Example From NNP project, pattern sequences to be counted
Stuck with providing first, second, third and only later recognising the dependency created
Unlocking potential
Universality of the Frame Theorem (Gaussian Curvature and Betti Numbers)
Counting squares, counting sticks, …
Counting weights

25. Example: Extending Mathematical Sequences Mathematically
“What is the next term …?” only makes sense when ...
Mathmematical guarantee of uniqueness
Geometrical or other construction source
Some other constraint

26. Painted Wheel (Tom O’Brien)…
Someone has made a simple pattern of coloured squares, and then repeated it at least once more
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 …
Theorem: a sequence is uniquely specified if you know that the repeating pattern has appeared at least twice.
Make up your own:
a really simple one
a really hard one
Provide two or more sequences in parallel

27. 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?
What shapes will have the same Border Numbers?

28. Extending Mathemtical Sequences
Stress in Thinking Mathematically and later on ‘specifying the growth mechanism before trying to count things’
Uniquely Extendable Sequences Theorem
Instance of general topological theorem (Betti numbers)
Attempts in two Dimensions!

29. 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?

30. Gasket Sequences

31. Attention Teahing and Learning is fundamentally about attention:
What is available or likely to come–to–mind when needed
What is available to be learned?
variation
Use of powers
Use of themes
Use of resources (physical, mental, virtual)
Structure of attention
Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

32. Follow-Up j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3
Designing & Using Mathematical Tasks (Tarquin/QED)
Thinking Mathematically (Pearson)
Developing Thinking in Algebra, Geometry, Statistics (Sage)
Fundamental Constructs in Mathematics Education (RoutledgeFalmer)
Mathematics Teaching Practice: a guide for university and college lecturers (Horwood Publishing)
Mathematics as a Constructive Activity (Erlbaum)
Questions & Prompts for Mathematical Thinking (ATM)
Thinkers (ATM)
Learning & Doing Mathematics (Tarquin)
open.ac.uk
mcs.open.ac.uk/jhm3