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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
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
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Solve eSolve exx==xx5050
Solve eSolve exx==xx5050
Check with two graphs, LHS and RHS
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Find the intersectionFind the intersection
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How could we reason on this How could we reason on this solution?solution?
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Antiderivative?Antiderivative?
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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.
1 6
Treatment and conversionTreatment and conversion
(Duval, 2006, p. 3)
Treatment or conversion?Treatment or conversion?
25
Integration by substitutionIntegration by substitution
Integration by substitutionIntegration by substitution
Integration by substitutionIntegration by substitution
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
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perpendicular
converging
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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?
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Task–GeneralisingTask–Generalising
Can we find two quadratics that meet at any point (p, q), with any gradient k?
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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
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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
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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
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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
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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]