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From Teaching Procedures To Thinking Mathematically:

Making Use of Students’ Natural Powers

The Open UniversityMaths Dept University of Oxford

Dept of Education

Promoting Mathematical Thinking

John MasonGothenbergNov 30 2012

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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!

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Tasks

Tasks promote Activity; Activity involves Aactions; 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

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Responsible teaching

Articulate about justifying choices of– tasks– ways of initiating mathematical thinking– ways of sustaining mathematical thinking– ways of concluding mathematical thinking

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Learning (Mathematics)

What is Avaibale to be learned (what is varying and in what ways)

What Actions are Initiated What Dispositions are Evoked What Powers are called upon

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My Way of Working Phenomenological-Experiential

– Try to generate an experience,– draw attention to it – label it in some way

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One More Than

What numbers can be presented as one more than the product of four consecutive numbers?

One natural response is to use algebra (if that is confidence-inspiring)– But that runs into obstacles

One natural response is to try some specific examples…– In order to locate a relationship that might be an

instance of a property!

Specialising Generalis

ing

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From Thomas Lingefjård

Given the numbers 1, 3, 4, and 6 - try to construct all numbesr from 20 to 30 by simple arithmetic (addition, subtraction, multiplication and division). No other way of combining or using numbers as power of is allowed. For instance: 1*6*3 + 4 = 22. In every calculation, all four digits must be present.

Try to find a number which consists of 769 digits, the sum of all the digits is 3693, every pair of consecutive digits is either a multiple of 17 or of 23 and all multiples of 17 or 23 in two digits is in the number.

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More or Less grids

More Same

Less

More

Same

LessPerimeter

Area

With as little change as possible from the original!

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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). Which object will you end on?

A B C D E

1 2 3 4 5

9 8 7 6

…

If that object is elimated, you start again from the ‘next’. Which object is the last one left?

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Substitution Pattern Generating

W –> WB

B –> W

How many squares will there be?

How many white squares will there be?

How many blue squares will there be?

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Substitution Relationships

W WBB WBB BW BW

WBB BW BW BW WBB BW WBB

WBB BW BW BW WBB BW WBB BW WBB WBB BW BW BW WBB WBB BW BW

⬆

⬆

⬆

⬆

⬆

⬆

⬆⬆ ⬆⬆

⬆ ⬆ ⬆ ⬆

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Gasket Sequences

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Sundaram’s Sieve

16 27 38 49 60 71 82

13 22 31 40 49 58 67

10 17 24 31 38 45 52

7 12 17 22 27 32 37

4 7 10 13 16 19 22

What number will appear in the Rth row and the Cth column?

Claim: N will appear in the table iff 2N + 1 is composite

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Circle Round a Square

Imagine a Square Now imagine a circle in the same plane as the

square, so that the two are touching at a single point

Now imagine the circle rolling around the outside of the square, always staying in touch

Pay attention to the centre of the circle as it rolls What is the path the centre takes, and how long is

it?

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Numberline Movements Imagine you are standing on a number line

somewhere facing the positive direction.(Make a note of where you are!)

Go forward three steps; Now go backwards 5 steps Now turn through 180° Go backwards 3 steps Go forwards 1 step You should be back where you started but facing to

the left.

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ThOANs

Think of a number between 0 and 10 Add six Multiply by the first number you thought of Add 4 Subtract twice the number you first thought of Take the square root (positive!) subtract the number you first thought of You (and everybody else) are left with 2!

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Ride & Tie Imagine that you and a friend have a single

horse (bicycle) and that you both want to get to a town some distance away.

In common with folks in the 17th century, one of you sets off on the horse while the other walks. At some point the first dismounts, ties the horse and walks on. When you get to the horse you mount and ride on past your friend. Then you too tie the horse and walk on…

Supposing you both ride faster than you walk but at different speeds, how do you decide when and where to tie the horse so that you both arrive at your destination at the same time?

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Ride & Tie

Imagine, then draw a diagram!

Does the diagram make sense (meet the constraints)?

Seeking Relationships

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Two Journeys Which journey over the same distance at two

different speeds takes longer:– One in which both halves of the distance are done at

the specified speeds– One in which both halves of the time taken are done

at the specified speeds

distance time

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Named Ratios

Now take a named ratio (eg density) and recast this task in that language

Which mass made up of two densities has the larger volume:– One in which both halves of the mass have the fixed

densities– One in which both halves of the volume have the

same densities?

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Counter Scaling

Someone has placed 5 counters side-by-side in a line

Someone else has made a similar line with 5 counters but with one counter-width space between counters.

By what factor has the length of the original line been scaled?

How many counters would be needed so that the scale factor was 15/8?

“Fence-post Reasoning”

Generalise!

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What’s The Difference?

What could be varied?

– =

First, add one to each

First, add one to the larger and subtract one from the smaller

What then would be

the difference?

What then would be

the difference?

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Ride & Tie

Imagine, then draw a diagram!

Does the diagram make sense (meet the constraints)?

Seeking Relationships

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Understanding Division

234234 is divisible by 13 and 7 and 11; What is the remainder on dividing 23423426 by 13? By 7? By 11?Make up your own!

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More or Less grids

More Same

Less

More

Same

LessPerimeter

Area

With as little change as possible from the original!

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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 What other fraction-actions can you see?

How did your attention shift?

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Put your hand up when you can see …

Something that is 1/4 – 1/5of something else

What did you have to do with your attention?

Can you generalise?

Did you look for something that is 1/4 of something else

and forsomething that is 1/5 of the same thing?

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Two Journeys Which journey over the same distance at two

different speeds takes longer:– One in which both halves of the distance are done at

the specified speeds– One in which both halves of the time taken are done

at the specified speeds

distance time

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Named Ratios

Now take a named ratio (eg density) and recast this task in that language

Which mass made up of two densities has the larger volume:– One in which both halves of the mass have the fixed

densities– One in which both halves of the volume have the

same densities?

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One Sum Diagrams

1

1

(1- )2

Anticipating,not waiting

1-2

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Reading a Diagram

x3 + x(1–x) + (1-x)3

x2 + (1-x)2

x2z + x(1-x) + (1-x)2(1-z)

xz + (1-x)(1-z)xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

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Outer & Inner Tasks

Outer Task– What author imagines– What teacher intends– What students construe– What students actually do

Inner Task– What powers might be used?– What themes might be encountered?– What connections might be made?– What reasoning might be called upon?– What personal dispositions might be challenged?

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Imagining

Basis of Geometric Thinking Basis of Anticipating Basis of ‘Realising’ Basis of Accessing & Enriching Example

Spaces Basis of Planning

Geometric ImagesATM

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Powers

Every child that gets to school has already displayed the power to– imagine & express– specialise & generalise– conjecture & convince– organise and categorise

The question is …– are they being prompted to use and develop those

powers?– or are those powers being usurped by text, worksheets

and ethos?

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Mathematical Themes

Doing & Undoing Invariance in the midst of change Freedom & Constraint Restricting & Expanding Meaning

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Reflection

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 the way the task is used Teachers can guide and direct learner

attention What are teachers attending to?

– Powers– Themes– Heuristics– The nature of their own attention

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Attention

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

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Motivation

Motivation is not a thing– Sense of gap or disturbance– Appropriate challenge + Trust in teacher

Phenomena to explain using mathematics Mathematical phenomena to explain &

appreciate

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The Problem about Problem Solving

It is not simply a Friday afternoon entertainment

It is not a ‘thing’ you (or the students) do It is an orientation to learning and doing

mathematics Change of Vocabulary:

– Teaching using exploration as one mode of interaction among many

– ‘teaching Investigatively’– Using Stdeunts’ Powers to teach Mathematics– …

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Pedagogic Strategies & Didactic Tactics

In how many different ways can you … Do as many exercises as you need to do in order

to be able to do any uestion of this type– Construct an easy, hard, peculiar, general question of

this type What is the same and what different about … If this is the answer, what questions of this type

would give the same answer? What sorts of answers can you get to questions of

this type? Presentation

– Particular General– General –> Particular –> Re-Generalise– Partly General –> Particular –> Re-Generalise

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Follow Up

mcs.open.ac.uk/jhm3j.h.mason @ open.ac.uk

Thinking Mathematically (new edition)

Designing and Using Mathematical Tasks (Tarquin)Questions and Prompts … (from ATM)Thinkers (from ATM)