versatile mathematical thinking in the secondary classroom mike thomas the university of auckland

Versatile Mathematical Versatile Mathematical Thinking in the Secondary Thinking in the Secondary ClassroomClassroom

Mike Thomas

The University of Auckland

OverviewOverview

A current problem

Versatile thinking in mathematics

Some examples from algebra and calculus

Possible roles for technology

What can happen?What can happen?

Why do we need to think about what we are teaching?– Assessment encourages:

Emphasis on procedures, algorithms, skills

Creates a lack of versatility in approach

Possible problemsPossible problems

Consider

But, the LHS of the original is clearly one half!!

Concept not understoodConcept not understood

€

A B C

(x + 3)(x − 2) x 2 + 5x − 6 x 2 + x − 6

Which two are equivalent?

Can you find another equivalent expression?

A student wrote…

€

(x − 2)(x + 3)

…but he factorised C!

Procedural focusProcedural focus

€

f (x ) =x 2

x 2 −1

Let

For what values of x is f(x) increasing?

Some could answer this using algebra and but…

Procedure versus conceptProcedure versus concept

′f (x) > 0

Procedure versus conceptProcedure versus concept

0.50 1.00 1.50 2.00 2.50 3.00-0.50-1.00-1.50-2.00-2.50

-1.00

-2.00

-3.00

1.00

2.00

3.00

4.00

For what values of x is this function increasing?

Versatile thinkingVersatile thinking in mathematics in mathematics

First…

process/object versatility—the ability to switch at will in any given representational system between a perception of symbols as a process or an object

Examples of proceptsExamples of procepts

symbol process object

3+2 addition sum

3+2x evaluation expression

y=f(x) assignment function

dy/dx differentiation derivative

f ( x ) dx∫ integration integral

lim

x → 2

x2

− 4

x − 2

⎛

⎝

⎜

⎞

⎠

⎟ or 1

n 2

n = 1

∞

∑ te nding to limit valu eof limit

(x1, x2, …, xn) vecto r shift point inn-space

Lack of process-object versatilityLack of process-object versatility

(Thomas, 1988; 2008)

Procept exampleProcept example

Effect of context on meaning for Effect of context on meaning for

Expression Rate ofchange

Gradientof

tangent

Derivative Term inan

equation

d y

d x

= 5 x 16 6 11 2

2 x +

d y

d x

= 1 3 0 5 8

d y

d x

= 4 y 7 4 7 1

z =

d (

d y

d x

)

d x

1 1 0 3

dy

dx

Process/object versatility for Process/object versatility for

Seeing solely as a process causes a

problem interpreting

and relating it to

d(dydx)dx

d2ydx2

dy

dx

dy

dx

Student: that does imply the second derivative…it is the derived function of the second derived function

))(( xff ′′

′′f (x)

Visuo/analytic versatilityVisuo/analytic versatility

Visuo/analytic versatility—the ability to exploit the power of visual schemas by linking them to relevant logico/analytic schemas

A Model of Cognitive IntegrationA Model of Cognitive Integration

Higher level schemas

Lower level schemas

C–links andA–links

Directed

conscious

unconscious

Surface (iconic) v deep Surface (iconic) v deep (symbolic) observation(symbolic) observation “

”

Moving from seeing a drawing (icon) to seeing a figure (symbol) requires interpretation; use of an overlay of an appropriate mathematical schema to ascertain properties

External world

Internal world

external sign

‘appropriate’ schema

interpretInteract

with/act on

PerceivedReality interpret

Picture ofreality interpret

Diagramor

Drawing

interpretTheoretical

MathematicalFigure

Booth & Thomas, 2000

We found e

Schema use

ExampleExample

This may be an icon, This may be an icon, a ‘hill’, saya ‘hill’, say

We may look We may look ‘deeper’ and see a ‘deeper’ and see a parabola using a parabola using a quadratic function quadratic function schemaschema

This schema may allow us to convert to algebra

Algebraic symbols: Equals schemaAlgebraic symbols: Equals schema

• Pick out those statements that are equations from the following list and write down why you think the statement is an equation:

• a) k = 5• b) 7w – w• c) 5t – t = 4t• d) 5r – 1 = –11• e) 3w = 7w – 4w

Surface: only needs an = signSurface: only needs an = sign

All except b) are equations since:All except b) are equations since:

Equation schema: only needs an Equation schema: only needs an operationoperation

Perform an operation and get a result:Perform an operation and get a result:

The blocks problemThe blocks problem

SolutionSolution

1

2

1

2

ReasoningReasoning

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Solve eSolve exx==xx5050

Solve eSolve exx==xx5050

Check with two graphs, LHS and RHS



Find the intersectionFind the intersection



How could we reason on this How could we reason on this solution?solution?



Antiderivative?Antiderivative?



What does the antiderivative look like?

Task: What does the graph of the Task: What does the graph of the derivative look like?derivative look like?

Method easyMethod easy

But what does the antiderivative But what does the antiderivative look like?look like?

How would you approach this?

Versatile thinking is required.

Maybe some technology would Maybe some technology would helphelp

Geogebra

Geogebra

Representational VersatilityRepresentational Versatility

Thirdly…

representational versatility—the ability to work seamlessly within and between representations, and to engage in procedural and conceptual interactions with representations

Representation dependant ideas...Representation dependant ideas...

"…much of the actual work of mathematics is to determine exactly what structure is preserved in that representation.”

J. KaputIs 12 even or odd? Numbers ending in a multiple of 2 are even. True or False?123?

123, 345, 569 are all odd numbers113, 346, 537, 469 are all even numbers

Representations can lead to other Representations can lead to other conflicts…conflicts…

1 unit square The length is 2, since we travel across 1 and up 1

What if we let the number of steps n increase? What if n tends to ?

Is the length √2 or 2?

Representational versatilityRepresentational versatility

Ruhama Even gives a nice example:

If you substitute 1 for x in ax2 + bx + c, where a, b, and c are real numbers, you get a positive number. Substituting 6 gives a negative number. How many real solutions does the equationax2 + bx + c = 0 have? Explain.

Treatment and conversionTreatment and conversion

(Duval, 2006, p. 3)

Treatment or conversion?Treatment or conversion?

25

Integration by substitutionIntegration by substitution

Linking of representation Linking of representation systems systems (x, 2x), where x is a real number

Ordered pairs to graph to algebra

Using gesturesUsing gestures

Iconic – “gestures in which the form of the gesture and/or its manner of execution embodies picturable aspects of semantic content” McNeill (1992, p. 39)

Deictic – a pointing gestureMetaphoric – an abstract meaning is

presented as form or space

The taskThe task

Thinking with gesturesThinking with gestures

Creates a virtual spaceCreates a virtual space

QuickTime™ and aYUV420 codec decompressorare needed to see this picture.

perpendicular

converging

QuickTime™ and aYUV420 codec decompressorare needed to see this picture.

The semiotic gameThe semiotic game

“The teacher mimics one of the signs produced in that moment by the students (the basic sign) but simultaneously he uses different words: precisely, while the students use an imprecise verbal explanation of the mathematical situation, he introduces precise words to describe it or to confirm the words.”

Why use technology?Why use technology?

It may be used to: promote visualization encourage inter-representational thinking enable dynamic representations enable new types of interactions with representations challenge understanding make conceptual investigation more amenable give access to new techniques aid generalisation stimulate enquiry assist with modelling

etc

It depends on how it is used…It depends on how it is used…

• Performing a direct, straightforward procedure• Checking of (procedural) by-hand work • Performing a direct procedure because it is too difficult

by hand• Performing a procedure within a more complex

process, possibly to reduce cognitive load• Investigating a conceptual idea

Thomas & Hong, 2004

Task Design – A keyTask Design – A key

Features of a good technology task: students write about how they interpret their work; includes multi-representational aspects (e.g. graphs and

algebra); considers the role of language; includes integration of technological and by-hand techniques; aims for generalisation; gets students to think about proof; enables students to develop mathematical theory.

Some based on Kieran & Drijvers (2006)

TaskTask

Can we find two quadratic functions that touch only at at the point (1, 1)?

Can you find a third?

How many are there?

Task–GeneralisingTask–Generalising

Can we find the quadratics that meet at any point (p, q), with any gradient k?



Task–GeneralisingTask–Generalising

Can we find two quadratics that meet at any point (p, q), with any gradient k?



Extending a task by A. Harradine

One family of curvesOne family of curves

y =c+ kp−q

p2

⎛⎝⎜

⎞⎠⎟

x2 −2c+ kp−2q

p⎛⎝⎜

⎞⎠⎟

x+ c



Another taskAnother task

Can we find a function such that its derived (or gradient) function touches it only at one single point?

For a quadratic function this means that its derived function is a tangent

How would we generalise this?How would we generalise this?

Consider

And its derived function

y =ax2 +bx+ c

y =2ax+b

SolutionSolution

So these touch at one point

y =ax2 +bx+4a2 +b2

4a

y =2ax+b

For exampleFor example



Extension: can you find any other functions with this

property?….

Newton-Raphson versatilityNewton-Raphson versatility

x2 =x1 −f(x1)′f (x1)

Many students can use the formula below to calculate a better approximation of the root, but are unable to explain why it works

Newton-RaphsonNewton-Raphson

x3 x2 x1

f(x) f(x1)

′f (x1) =f (x1)

x1 − x2

Why it may failWhy it may fail

Newton-RaphsonNewton-Raphson

When is x1 a suitable first approximation for the root a of f(x) = 0?

Symmetry of cubicsSymmetry of cubics

€

y = x3 − 3x2 + x − 5













Generalising 180˚ symmetryGeneralising 180˚ symmetry

In general (a, b) is mapped to (2p–a, 2q–b)Hence g(x) =2q− f (2p−x)

(p,q)

(a,b)

(x, y) = (2p-a,2q–b)

e.g. q – y = b – q

Solving linear equationsSolving linear equations

Many students find ax+b=cx+d equations hard to solve

We may only teach productive transformations

But there are, of course, many more legitimate transformations €

−ax, − cx, − b, − d

€

±kx, ± k, k ∈ R

10–310–3xx = 4 = 4xx+3+3

10–3x = 4x+3

10= 7x+3 10–x= 2x+3

7=7x 8.37–x= 2x+1.37

1=x

productiveproductive legitimatelegitimate

Legitimate: 10–3Legitimate: 10–3xx = 4 = 4xx+3+3



Teacher commentsTeacher comments

“I feel technology in lessons is over-rated. I don’t feel learning is significantly enhanced…I feel claims of computer benefits in education are often over-stated.”

“Reliance on technology rather than understanding content.”

“Sometimes some students rely too heavily on [technology] without really understanding basic concepts and unable to calculate by hand.”

GC’s “encourage kids to take short cuts, especially in algebra. Real algebra skills are lacking as a result”

PTK required of teachersPTK required of teachers

Pedagogical Technology Knowledge (PTK)– teacher attitudes to technology and their

instrumentalisation of it– teacher instrumentation of the technology– epistemic mediation of the technology – integration of the technology in teaching– ways of employing technological tools in teaching

mathematics that focus on the mathematics Combines knowledge of self, technology, teaching and

mathematics(Thomas & Hong, 2005a; Hong & Thomas, 2006)

Teaching implicationsTeaching implications

Avoid teaching procedures, algorithms, even with CAS—using CAS solely as a ‘calculator’ reinforces a procedural approach

Give examples to build an object view of mathematical constructs

Encourage and use visualisation Provide, and link, a suitable number of concurrent

representations in each learning situation Encourage a variety of qualitatively different

interactions with representations

Contact Contact

Email: [email protected]

versatile mathematical thinking in the secondary classroom mike thomas the university of auckland

Documents

values of x

versatile thinking

equation schema

arithmetic procedures

unknown versatility

single entity

propertiesbooth thomas

drawing icon