constructing versatile mathematical conceptions mike thomas the university of auckland
TRANSCRIPT
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Constructing Versatile Constructing Versatile Mathematical ConceptionsMathematical Conceptions
Mike Thomas
The University of Auckland
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OverviewOverview
Define versatile thinking in mathematics
Consider some examples and problems
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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
Not just procepts, which are arithmetic/algebraic
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Lack of process-object versatilityLack of process-object versatility
(Thomas, 1988; 2008)
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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
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A Model of Cognitive IntegrationA Model of Cognitive Integration
Higher level schemas
Lower level schemas
C–links andA–links
Directed
conscious
unconscious
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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
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Treatment and conversionTreatment and conversion
Duval, 2006, p. 3
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Icon to symbol requires interpretation through Icon to symbol requires interpretation through appropriate mathematical schema to ascertain appropriate mathematical schema to ascertain propertiesproperties
External world
Internal world
external sign
‘appropriate’ schema
interpretInteract
with/act on
translation or conversion
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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
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A possible problemA possible problem
The opportunity to acquire knowledge in a variety of forms, and to establish connections between different forms of knowledge are apt to contribute to the flexibility of students’ thinking (Dreyfus and Eisenberg, 1996). The same variety, however, also tends to blur students’ appreciation of the difference in status which different means of establishing mathematical knowledge bestow upon that knowledge.
Dreyfus, 1999
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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
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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:
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Another possible problemAnother possible problem
Compartmentalization “This phenomenon occurs when a person has
two different, potentially conflicting schemes in his or her cognitive structure. Certain situations stimulate one scheme, and other situations stimulate the other…Sometimes, a given situation does not stimulate the scheme that is the most relevant to the situation. Instead, a less relevant scheme is activated”
Vinner & Dreyfus, 1989, p. 357
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A formulaA formula
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Linking of representation Linking of representation systems systems (x, 2x), where x is a real number
Ordered pairs to algebra to graph
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Abstraction in contextAbstraction in context
We also pay careful attention to the multifaceted context in which processes of abstraction occur: A process of abstraction is influenced by the task(s) on which students work; it may capitalize on tools and other artifacts; it depends on the personal histories of students and teacher
Hershkowitz, Schwarz, Dreyfus, 2001
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Abstraction of meaning for Abstraction of 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
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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
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Student: that does imply the second derivative…it is the derived function of the second derived function
))(( xff ′′
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To see this relationship “one needs to deal with the function g as an object that is operated on in two ways”
Dreyfus (1991, p. 29)
g(x)dx =a+k
b+k
∫ g(x−k)dxa
b
∫
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Proceptual versatile thinkingProceptual versatile thinking
If ,
then write down the value of
€
f (t)dt = 8.61
3
∫
€
f (t −1)dt2
4
∫
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Versatile thinking–change of Versatile thinking–change of representation systemrepresentation system
nb The representation does not correspond; an exemplar y=x2 is used
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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
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Newton-RaphsonNewton-Raphson
x3 x2 x1
f(x) f(x1)
′f (x1) =f (x1)
x1 − x2
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Newton-RaphsonNewton-Raphson
When is x1 a suitable first approximation for the root a of f(x) = 0?
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Student V1: It is very important that the approximation is close enough the root and not on a turning point. Otherwise you might be finding the wrong root.
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Student knowledge constructionStudent knowledge construction
Learning may take place in a single representation system, so inter-representational links are not made
Avoid activity comprising surface interactions with a representation, not leading to the concept
The same representations may mean different things to students due to their contextual schema construction (abstraction)
Use multiple contexts for representations
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ReferenceReference
Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 67-87.
From: [email protected]
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Proceptual versatility–Proceptual versatility–eigenvectorseigenvectors
€
Ax = λxTwo different processes
Need to see resulting object or ‘effect’ as the same
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Work within the representation Work within the representation system—algebrasystem—algebra
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Work within the representation Work within the representation system—algebrasystem—algebra
Same process
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ConversionConversion
v
v
u
u
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Student knowledge constructionStudent knowledge construction
Learning may take place in a single representation system, so inter-representational links are not made
Avoid activity comprising surface interactions with a representation, not leading to the concept
The same representations may mean different things to students due to their contextual schema construction (abstraction)
Use multiple contexts for representations
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ConversionsConversions
Translation between registers
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Duval, 1999, p. 5
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
RepresentationsRepresentations
Duval, 1999, p. 4
Semiotic representations systems — Semiotic registers
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Processing — transformation within a register
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Epistemic actions are mental actions by means of which knowledge is used or constructed
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Representations and Representations and mathematicsmathematics Much of mathematics is about what we can
learn about concepts through their representations (or signs)
Examples include: natural language, algebras, graphs, diagrams, pictures, sets, ordered pairs, tables, presentations, matrices, etc. (nb icons, indices and symbols here)
Some of the things we learn are representation dependant; others representation independent
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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
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Representational interactionsRepresentational interactions
We can interact with a representation by:
Observation—surface or deep (property)
Performing an action—procedural or conceptual
Thomas, 2001
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Representational interactionsRepresentational interactions
We can interact with a representation by:
Observation—surface or deep (property)
Performing an action—procedural or conceptual
Thomas, 2001
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€
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
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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?
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Why it may failWhy it may fail
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We should not think that the three parts of versatile thinking are independent
Neither should we think that a given sign has a single interpretation — it is influenced by the context
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Icon, index, symbolIcon, index, symbol
A
D C
B
ABCD — symbol
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Icon to symbolIcon to symbol
“”
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
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Function