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What is What is the Discipline the Discipline

of Mathematics Education?of Mathematics Education?Essential MathsEssential Maths

&&Mathematical EssencesMathematical Essences

What is What is the Discipline the Discipline

of Mathematics Education?of Mathematics Education?Essential MathsEssential Maths

&&Mathematical EssencesMathematical Essences

John MasonJohn Mason

Hobart 2007Hobart 2007

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

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OutlineOutlineOutlineOutline

Justifying “a problem a day Justifying “a problem a day keeps the teacher in play”keeps the teacher in play”

What mathematics is What mathematics is essential?essential?

What is mathematical What is mathematical essence?essence?

Justifying “a problem a day Justifying “a problem a day keeps the teacher in play”keeps the teacher in play”

What mathematics is What mathematics is essential?essential?

What is mathematical What is mathematical essence?essence?

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Grid MovementGrid MovementGrid MovementGrid Movement

7

?

+3

-3

x2÷2

((7+3)x2)+3

is a path from 7 to ‘?’.

What expression represents the reverse of this path?

What values can ‘?’ have: if exactly one - and one ÷ are used?Max value? Min Value?

What about other cells?Does any cell have 0? -7?Does any other cell have 7?

Characterise ALL the possible values that can appear in a cell

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ReflectionsReflectionsReflectionsReflections

What variations are possible?What variations are possible? What have you gained by working What have you gained by working

on this task (with colleagues)?on this task (with colleagues)?

What variations are possible?What variations are possible? What have you gained by working What have you gained by working

on this task (with colleagues)?on this task (with colleagues)? +3

-3

7?x2

÷2

What criteria would you use in What criteria would you use in choosing whether to use this (or any) choosing whether to use this (or any) task?task?

What might be gained by working on What might be gained by working on (a variant of) this task with learners?(a variant of) this task with learners?Tasks –> Activity –> Experience –> ‘Reflection’

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More Disciplined EnquiryMore Disciplined EnquiryMore Disciplined EnquiryMore Disciplined Enquiry What is the point? (What is the point? (Helen ChickHelen Chick))

– Outer task & Inner taskOuter task & Inner task What is the line? (What is the line? (Steve ThorntonSteve Thornton))

– Narrative for HoD, Head, parents, selfNarrative for HoD, Head, parents, self What is (the) plain?What is (the) plain?

– What awarenesses? What What awarenesses? What ‘outcomes’?‘outcomes’?

What is the space?What is the space?– Domain of related tasksDomain of related tasks– Dimensions of possible variation; Dimensions of possible variation;

ranges of permissible changeranges of permissible change

What is the point? (What is the point? (Helen ChickHelen Chick))– Outer task & Inner taskOuter task & Inner task

What is the line? (What is the line? (Steve ThorntonSteve Thornton))– Narrative for HoD, Head, parents, selfNarrative for HoD, Head, parents, self

What is (the) plain?What is (the) plain?– What awarenesses? What What awarenesses? What

‘outcomes’?‘outcomes’? What is the space?What is the space?

– Domain of related tasksDomain of related tasks– Dimensions of possible variation; Dimensions of possible variation;

ranges of permissible changeranges of permissible change

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DifferencesDifferencesDifferencesDifferences

17=16−142

Rehearsing

Checking

Organising

18=17−156

=16−124

=14−18

13=12−16

14=13−112

=12

−14

15=14−120

16=15−130

=12

−13

=13

−16

= 14

− 112

12=11−12

AnticipatingGeneralising

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Sketchy GraphsSketchy GraphsSketchy GraphsSketchy Graphs

Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines whose lines whose y-y-intercepts differ by 2intercepts differ by 2 Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines whose lines whose xx-intercepts differ by 2-intercepts differ by 2 Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines whose slopes differ by 2lines whose slopes differ by 2 Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines meeting all three conditionslines meeting all three conditions

Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines whose lines whose y-y-intercepts differ by 2intercepts differ by 2 Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines whose lines whose xx-intercepts differ by 2-intercepts differ by 2 Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines whose slopes differ by 2lines whose slopes differ by 2 Sketch the graphs of a pair of straight Sketch the graphs of a pair of straight

lines meeting all three conditionslines meeting all three conditions

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Chordal MidpointsChordal MidpointsChordal MidpointsChordal Midpoints

Where can the midpoint of a Where can the midpoint of a chord of your cubic get to?chord of your cubic get to?(what is the boundary of the (what is the boundary of the region of mid-points?)region of mid-points?)

What about 1/3 points or 4/3 What about 1/3 points or 4/3 points?points?

Where can the midpoint of a Where can the midpoint of a chord of your cubic get to?chord of your cubic get to?(what is the boundary of the (what is the boundary of the region of mid-points?)region of mid-points?)

What about 1/3 points or 4/3 What about 1/3 points or 4/3 points?points?

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Justifying ‘doing’ maths for Justifying ‘doing’ maths for oneself and with othersoneself and with othersJustifying ‘doing’ maths for Justifying ‘doing’ maths for oneself and with othersoneself and with others

Sensitise myself to what Sensitise myself to what learners may be experiencinglearners may be experiencing

Refresh my awareness of the Refresh my awareness of the movements of my attentionmovements of my attention

Remind myself what it is like Remind myself what it is like to be a learnerto be a learner

Experience the Experience the typetype of task I of task I might use with learnersmight use with learners

Sensitise myself to what Sensitise myself to what learners may be experiencinglearners may be experiencing

Refresh my awareness of the Refresh my awareness of the movements of my attentionmovements of my attention

Remind myself what it is like Remind myself what it is like to be a learnerto be a learner

Experience the Experience the typetype of task I of task I might use with learnersmight use with learners

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AwarenessesAwarenessesAwarenessesAwarenessesGive a family a fish Give a family a fish

and you feed them for a dayand you feed them for a dayShow them how to fish, Show them how to fish,

and you feed them and you feed them until the stocks run outuntil the stocks run out

Give a family a fish Give a family a fish and you feed them for a dayand you feed them for a day

Show them how to fish, Show them how to fish, and you feed them and you feed them

until the stocks run outuntil the stocks run out

Obtaining tasks and lesson plans gets you through some lessons …

Becoming aware of affordances, constraints and

attunements, in terms of

mathematical themes, powers & heuristics

enables you to promote learning

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More Or Less Altitude & AreaMore Or Less Altitude & AreaMore Or Less Altitude & AreaMore Or Less Altitude & Area

Draw a scalene triangle

more

same

less

moresameless

areaaltitud

e

Same altmore area

more altsame area

more altmore area

less altmore area

less altless area

more altless area

same altless area

less altsame area

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More Or Less Rectangles & More Or Less Rectangles & AreaAreaMore Or Less Rectangles & More Or Less Rectangles & AreaArea

more

same

less

moresamefewer

area

No. of rectangles

same rectsmore area

more rectssame area

more rectsmore area

fewer rectsmore area

fewer rectsless area

more rectsless area

same rectsless area

fewer rectssame area

Draw a rectilinear figure which requires at least 4 rectangles in any decomposition

How many can have the same perimeter?

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More Or Less Percent & More Or Less Percent & ValueValueMore Or Less Percent & More Or Less Percent & ValueValue

50% of something is 20

more

same

less

moresameless

%Value

50% of 40 is 20

50% of 60 is 3040% of 60 is 24

60% of 60 is 36

40% of 30 is 12

60% of 30 is 20

40% of 50 is 20

40% of 40 is 16

50% of 30 is 15

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More Or Less Whole & PartMore Or Less Whole & PartMore Or Less Whole & PartMore Or Less Whole & Part

? of 35 is 21

more

same

less

moresameless

WholePart

3/5 of 35 is 21

3/4 of 28 is 21

6/7 of 35 is 30

3/5 of 40 is 24

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Magic Square ReasoningMagic Square ReasoningMagic Square ReasoningMagic Square Reasoning

51 9

2

4

6

8 3

7

– = 0Sum( ) Sum( )

Try to describethem in words

What other configurations

like thisgive one sum

equal to another?2

2

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More Magic Square More Magic Square ReasoningReasoningMore Magic Square More Magic Square ReasoningReasoning

– = 0Sum( ) Sum( )

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Graphical AwarenessGraphical AwarenessGraphical AwarenessGraphical Awareness

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Multiplication as ScalingMultiplication as ScalingMultiplication as ScalingMultiplication as Scaling

If you stick a pin in Hobart in a map of If you stick a pin in Hobart in a map of Australia, and scale the map by a factor Australia, and scale the map by a factor of 1/2 towards Hobartof 1/2 towards Hobart

And if a friend does the same in Darwin, And if a friend does the same in Darwin, scaling by 1/2 towards Darwinscaling by 1/2 towards Darwin

What will be the difference in the two What will be the difference in the two scaled maps?scaled maps?

If you stick a pin in Hobart in a map of If you stick a pin in Hobart in a map of Australia, and scale the map by a factor Australia, and scale the map by a factor of 1/2 towards Hobartof 1/2 towards Hobart

And if a friend does the same in Darwin, And if a friend does the same in Darwin, scaling by 1/2 towards Darwinscaling by 1/2 towards Darwin

What will be the difference in the two What will be the difference in the two scaled maps?scaled maps?

What if one of you scales by a factor of 2/3 towards Hobart and then by a further 1/2 towards Darwin, while the other scales by 1/2 towards Darwin and then by a further 2/3 towards Hobart?

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Raise Your Hand When You Raise Your Hand When You See …See …Raise Your Hand When You Raise Your Hand When You See …See …

Something which is 2/5 of something;

3/5 of something;

5/2 of something; 5/3 of something;

2/5 of 5/3 of something; 3/5 of 5/3 of something;

5/2 of 2/5 of something; 5/3 of 3/5 of something;

1 ÷ 2/5 of something;1 ÷ 3/5 of something

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Essential Conceptual Essential Conceptual AwarenessesAwarenessesEssential Conceptual Essential Conceptual AwarenessesAwarenesses

—Choosing the unitChoosing the unit—Additive actionsAdditive actions—Multiplicative actionsMultiplicative actions

—Scaling; multi-ply & many-fold, repetition, lots of; …

—Coordinated actions Coordinated actions —Angle actionsAngle actions

—Combining—Translating

—Measuring actionsMeasuring actions—Comparing lengths; areas; volumes; (unit)—Comparing angles

—Discrete-ContinuousDiscrete-Continuous—RandomnessRandomness

—Choosing the unitChoosing the unit—Additive actionsAdditive actions—Multiplicative actionsMultiplicative actions

—Scaling; multi-ply & many-fold, repetition, lots of; …

—Coordinated actions Coordinated actions —Angle actionsAngle actions

—Combining—Translating

—Measuring actionsMeasuring actions—Comparing lengths; areas; volumes; (unit)—Comparing angles

—Discrete-ContinuousDiscrete-Continuous—RandomnessRandomness

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Essential Mathematical-Essential Mathematical-nessesnessesEssential Mathematical-Essential Mathematical-nessesnesses

Mathematical Mathematical Awarenesses Awarenesses underlying topics underlying topics Movement of Movement of AttentionAttention Mathematical ThemesMathematical Themes Mathematical PowersMathematical Powers Mathematical Mathematical StrategiesStrategies Mathematical Mathematical DispositionsDispositions

Ways of working on these constitute a (the) discipline of

mathematics education

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Movement of AttentionMovement of AttentionMovement of AttentionMovement of Attention

Gazing (holding wholes)Gazing (holding wholes) Discerning DetailsDiscerning Details Recognising RelationshipsRecognising Relationships Perceiving PropertiesPerceiving Properties Reasoning on the Basis of PropertiesReasoning on the Basis of Properties

Compare SOLO & van Hiele

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Mathematical ThemesMathematical ThemesMathematical ThemesMathematical Themes

Doing & UndoingDoing & Undoing Invariance in the midst of ChangeInvariance in the midst of Change Freedom & ConstraintFreedom & Constraint Extending and Restricting MeaningExtending and Restricting Meaning … …

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Mathematical PowersMathematical PowersMathematical PowersMathematical Powers

Imagining & ExpressingImagining & ExpressingSpecialising & GeneralisingSpecialising & GeneralisingConjecturing & ConvincingConjecturing & ConvincingClassifying & CharacterisingClassifying & Characterising……

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Mathematical Mathematical Strategies/HeuristicsStrategies/HeuristicsMathematical Mathematical Strategies/HeuristicsStrategies/Heuristics

Acknowledging ignorance (Mary Boole)Acknowledging ignorance (Mary Boole) Changing view pointChanging view point Changing (re)presentationChanging (re)presentation Working BackwardsWorking Backwards … …

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Mathematical DispositionsMathematical DispositionsMathematical DispositionsMathematical Dispositions

Propensity to ‘see’ the world Propensity to ‘see’ the world math’lymath’ly Propensity to pose problemsPropensity to pose problems Propensity to seek structurePropensity to seek structure PerseverencePerseverence … …

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Essential Pedgaogic Essential Pedgaogic AwarenessesAwarenessesEssential Pedgaogic Essential Pedgaogic AwarenessesAwarenesses

Tasks Tasks – initiate activity; initiate activity; – activity provides immediate experience;activity provides immediate experience;– learning depends on connecting learning depends on connecting

experiences, often through labelling experiences, often through labelling when standing back from the actionwhen standing back from the action

Tasks Tasks – initiate activity; initiate activity; – activity provides immediate experience;activity provides immediate experience;– learning depends on connecting learning depends on connecting

experiences, often through labelling experiences, often through labelling when standing back from the actionwhen standing back from the action

Mathematics develops from engaging Mathematics develops from engaging in actions on objects; and those in actions on objects; and those actions becoming objects, …actions becoming objects, …

Actions need to become not just Actions need to become not just things done under instruction or things done under instruction or guidance, but choices made by the guidance, but choices made by the learnerlearner

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ChoicesChoicesChoicesChoices

What pedagogic choices are What pedagogic choices are available when available when constructing/selecting constructing/selecting mathematical tasks for learners?mathematical tasks for learners?

What pedagogic choices are What pedagogic choices are available when presenting available when presenting mathematical tasks to learners?mathematical tasks to learners?

What criteria are used for What criteria are used for making those choices?making those choices?

What pedagogic choices are What pedagogic choices are available when available when constructing/selecting constructing/selecting mathematical tasks for learners?mathematical tasks for learners?

What pedagogic choices are What pedagogic choices are available when presenting available when presenting mathematical tasks to learners?mathematical tasks to learners?

What criteria are used for What criteria are used for making those choices?making those choices?

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What mathematics is What mathematics is essential?essential?What mathematics is What mathematics is essential?essential?

Extensions of teaching-mathsExtensions of teaching-maths– Experience analogously something Experience analogously something

of what learners experience, but of what learners experience, but enrich own awareness of enrich own awareness of connections and utilityconnections and utility

Extensions of own mathsExtensions of own maths– Experience what it is like to Experience what it is like to

encounter an unfamiliar topicencounter an unfamiliar topic

Extensions of teaching-mathsExtensions of teaching-maths– Experience analogously something Experience analogously something

of what learners experience, but of what learners experience, but enrich own awareness of enrich own awareness of connections and utilityconnections and utility

Extensions of own mathsExtensions of own maths– Experience what it is like to Experience what it is like to

encounter an unfamiliar topicencounter an unfamiliar topic

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It is only after you come to know It is only after you come to know the surface of things the surface of things

that you venture to see what is that you venture to see what is underneath; underneath;

but the surface of things but the surface of things is inexhaustible is inexhaustible ((Italo Calvino 1983Italo Calvino 1983))

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Human PsycheHuman PsycheHuman PsycheHuman Psyche

Awareness (cognition)

Behaviour (enaction)

Emotion (affect)

Mental imagery

Only awareness is educable

Only behaviour is trainable

Only emotion is harnessable

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What Can a Teacher Do?What Can a Teacher Do?What Can a Teacher Do?What Can a Teacher Do? Directing learner attentionDirecting learner attention

by being aware of structure of own attentionby being aware of structure of own attention(amplifying & editing; stressing & ignoring)(amplifying & editing; stressing & ignoring)

Invoking learners’ powersInvoking learners’ powers Bringing learners in contact with Bringing learners in contact with

mathematical heuristics & powersmathematical heuristics & powers Constructing experiences which, Constructing experiences which,

when accumulated and reflected upon, when accumulated and reflected upon, provide opportunity for learners to provide opportunity for learners to educate their awareness and train educate their awareness and train their behaviour through harnessing their behaviour through harnessing their emotions.their emotions.

Directing learner attentionDirecting learner attentionby being aware of structure of own attentionby being aware of structure of own attention(amplifying & editing; stressing & ignoring)(amplifying & editing; stressing & ignoring)

Invoking learners’ powersInvoking learners’ powers Bringing learners in contact with Bringing learners in contact with

mathematical heuristics & powersmathematical heuristics & powers Constructing experiences which, Constructing experiences which,

when accumulated and reflected upon, when accumulated and reflected upon, provide opportunity for learners to provide opportunity for learners to educate their awareness and train educate their awareness and train their behaviour through harnessing their behaviour through harnessing their emotions.their emotions.

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http://mcs.open.ac.uk/jhm3

I am grateful to the organisers for affording me the opportunity and impetus to contact, develop and

articulate these ideas

For this presentation and others

and other resources see