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What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical Thinking

What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

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Page 1: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

What we have met before:why individual students,

mathematicians & math educatorssee things differently

David TallEmeritus Professor in Mathematical Thinking

Page 2: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Euclidean geometry

ConceptualEmbodiment

Space&

Shape

Geometry

PlatonicMathematics

The Physical World

PracticalMathematics

Page 3: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

SymbolicMathematics

Euclidean geometry

ConceptualEmbodiment

Symboliccalculation &manipulation

Space&

ShapeCounting& Number

Arithmetic

Algebra

Calculus

Trigonometry

GraphsGeometry

Limits ...

Matrix Algebra

p4 =1- 1

3 + 15 - 1

7 +...

BlendingEmbodiment

&Symbolism

PlatonicMathematics

The Physical World

sinq=

opp

hyp

Functions

PracticalMathematics

Page 4: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

SymbolicMathematics

Euclidean geometry

Axiomatic Formalism

Axiomatic Geometry

ConceptualEmbodiment

Symboliccalculation &manipulation

Space&

ShapeCounting& Number

Arithmetic

Algebra

Calculus

Trigonometry

GraphsGeometry

Limits ...

Matrix Algebra

p4 =1- 1

3 + 15 - 1

7 +...

BlendingEmbodiment

&Symbolism

Axiomatic Algebra, Analysis, etc

Set-theoretic Definitions & Formal Proof

PlatonicMathematics

FormalMathematics

The Physical World

sinq=

opp

hyp

Functions

PracticalMathematics

Set Theory & Logic

Page 5: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

SymbolicMathematics

Axiomatic Formalism

Symboliccalculation &manipulation

PlatonicMathematics

FormalMathematics

The Physical World

PracticalMathematics

Embodiment

EmbodiedMeaningincreasing insophisticationfrom physical

to mentalconcepts

SymbolicMeaning

increasing powerin calculationusing symbols

as processes to do& concepts

to manipulate[procepts]

Formal Meaningas set-theoretic definition and deduction

Blendingembodiment& symbolism

to giveembodied meaning

to symbols&

symboliccomputational

powerto embodiment

Page 6: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Mathematicians can live in the world of formal mathematics.

Children grow through the worlds of embodiment and symbolism.

Mathematics Educators try to understand how this happens.

Fundamentally we build on what we know based on:

Our inherited brain structure (set-before our birth and maturing in early years)

Knowledge structures built from experiences met-before in our lives.

Page 7: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

The terms ‘set-before’ and ‘met-before’ which work better in English than in some other languages started out as a joke.

The term ‘metaphor’ is often used to represent how we interpret one knowledge structure in terms of another.

I wanted a simple word to use when talking to children.

When they use their earlier knowledge to interpret new ideas I could ask them how their thinking related to what was met before.

It was a joke: the word play metAphor, metBefore.

The joke worked well with teachers and children: What have you met before that causes you to think like this?

Page 8: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Inherited brain structure (set-before our birth and maturing in early years)

Examples:

Recognising the same object from different angles.

A Sense of Vertical and Horizontal.

Classifying categories such as ‘cat’ and ‘dog’.… triangle, square, rectangle, circle.

Practising a sequence of actions (see-grasp-suck).… used in counting … … column arithmetic … … adding fractions … … learning algorithms …

Set-Befores

May be performedautomatically

without meaning

Page 9: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Current ideas based on experiences met before.

Examples:

Two and two makes four.

Addition makes bigger.

Multiplication makes bigger.

Take away makes smaller.

Every arithmetic expression 2+2, 3x4, 27÷9 has an answer.

Squares and Rectangles are different.

Met-Befores

Different symbols eg and represent different things.

Page 10: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Two and two makes four

Addition makes bigger

Multiplication makes bigger

Take away makes smaller

Met-Befores

… works in later situations.

… fails for negative numbers.

… fails for fractions.

… fails for negative numbers.

An algebraic expression 2x+1 does not have an ‘answer’.

Later, by definition, a square is a rectangle.

Different symbols can represent the same thing.

Some are helpful in later learning, some are not.

Page 11: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

In practice, when we remember met-befores, we may blend together ideas from different contexts.

For instance, numbers arise both in counting and measuring.

Counting numbers 1, 2, 3, … have much in common with numbers used for measuring lengths, areas, volumes, weights, etc.

But they have some aspects which are very different ...

Met-Beforesblended together

Page 12: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Natural numbers build from counting

1 2 3 4 5...

The number track ...

The number line builds from measuring

The number line ...

1 2 3 4 5

Discrete, each number has a next with nothing between,starts counting at 1, then 2, 3, ....

Continuous, each interval can be subdivided,starts from 0 and measures unit shifts to the right.

Blending different conceptions of number

0 ...

Page 13: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Blending counting and measuring

Does the difference between number track and number line matter?

The English National Curriculum starts with a number track and then uses various number-line representations to expand the number line in both directions and to mark positive and negative fractions and decimals.

Doritou (2006, Warwick PhD) found many children had an overwhelming preference to label calibrated lines with whole numbers, with limited ideas that an interval could be sub-divided.

Their previous experience of whole number dominated their thinking and the expansion from number track to number line was difficult for many to attain.

Page 14: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Increasing sophistication of Number SystemsLanguage grows more sophisticated as it

blends together developing knowledge structures.Blending occurs between and within different aspects of embodiment, symbolism and formalism.Mathematicians usually view the number systems as an expanding system:

Cognitively the development is more usefully expressed in terms of blends.

NF

ZQ R C

Page 15: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Different knowledge structures for numbersThe properties change as the number

system expands.How many numbers between 2 and 3?

None

Lots– a countable infinity

Lots more – an uncountable infinity

None(the complex numbers are not ordered)

NQ

R

C

A mathematician has all of these as met-befores

A learner has a succession of conflicting met-befores

Page 16: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

From Arithmetic to Algebra

The transition from arithmetic to algebra is difficult for many.The conceptual blend between a linear algebra equation and a physical balance works in simple cases for many children (Vlassis, 2002, Ed. Studies).The blend breaks down with negatives and subtraction (Lima & Tall 2007, Ed. Studies).Conjecture: there is no single embodiment that matches the flexibility of algebraic notation.Students conceiving algebra as generalised arithmetic may find algebra simple.Those who remain with inappropriate blends as met-befores may find it distressing and complicated.

Page 17: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Embodied Symbolic

Blending different conceptions of number

Real Numbersas a multi-blend

Formal

RA mathematiciancan have a formal

view

A studentbuilds on

Embodiment & Symbolism

Page 18: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

From Algebra to CalculusThe transition from algebra to calculus is seen by mathematicians as being based on the limit concept.

For mathematicians, the limit concept is a met-before.

For students it is not.

A student can see the changing steepness of the graph and embody it with physical action to sense the changing slope.

slope +

slope zero

slope –

Page 19: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

CalculusLocal straightness is embodied:

You can see why the derivative of cos is minus sine

Page 20: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

CalculusLocal straightness is embodied:

You can see why the derivative of cos is minus sine

The graph of sineupside down....

slope is – sin x

Page 21: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

CalculusLocal straightness is embodied:

E.g. makes sense of differential equations ...

Page 22: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical

Reflections

As we learn, our interpretations are based on blending met-befores, which may cause conflict.

Learners who focus on the powerful connections between blends may develop power and flexibility, those who sense unresolved conflict may develop anxiety.

Mathematicians have more sophisticated met-befores and may propose curriculum design that may not be appropriate for learners.

It is the job of Mathematics Educators (who could be Mathematicians) to understand what is going on and help learners make sense of more sophisticated ideas.

Page 23: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical
Page 24: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical
Page 25: What we have met before: why individual students, mathematicians & math educators see things differently David Tall Emeritus Professor in Mathematical