Constructing Versatile
Mathematical Conceptions
Mike Thomas
The University of Auckland
Overview

Define versatile thinking in mathematics

Consider some examples and problems
Versatile thinking 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
Lack of process-object versatility
(Thomas, 1988; 2008)
Visuo/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 Integration
Higher level schemas
conscious
Directed
C–links and
A–links
unconscious
Lower level schemas
Representational Versatility
Thirdly…
representational versatility—the ability to
work seamlessly within and between
representations, and to engage in procedural
and conceptual interactions with
representations
Treatment and conversion
Duval, 2006, p. 3
Icon to symbol requires interpretation through
appropriate mathematical schema to ascertain
properties
External
world
interpret
external sign
translation or
conversion
‘appropriate’
schema
Interact
with/act on
Internal
world
Example
This may be an icon,
a ‘hill’, say
We may look
‘deeper’ and see a
parabola using a
quadratic function
schema
This schema may
allow us to
convert to algebra
A 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
Algebraic 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
Equation schema: only needs an
operation
Perform an operation and get a result:
Another 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
A formula
Linking of representation systems

(x, 2x), where x is a real number

Ordered pairs to algebra to graph
Abstraction 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
Abstraction of meaning for
dy
dx
Expression Rate of Gradient Derivative Term in
change
of
an
tangent
equation
dy  5x
dx
2x  dy 1
dx
dy  4y
dx
d( dy )
z  dx
dx
16
3
7
1
6
0
4
1
11
5
7
0
2
8
1
3
Process/object versatility for
dy
dx
dy
 Seeing
solely as a process causes a
dx
problem interpreting
2

and relating it to
dy
dx
d( )
dx
d y
2
dx
f ( f ( x))
Student: that does imply the second
derivative…it is the derived function of the
second derived function

b
a

g(x)dx  
b k
a k
g(x  k)dx
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)
Proceptual versatile thinking

If

3
1


,  8.6
f (t)dt
then write down the value of

4
2
f (t 1)dt
Versatile thinking–change of
representation system
nb The representation does not correspond; an
exemplar y=x2 is used
Newton-Raphson versatility

Many students can use the formula
below to calculate a better
approximation of the root, but are
unable to explain why it works
f (x1 )
x2  x1 
f (x1 )
Newton-Raphson
f (x1 )
f (x1 ) 
x1  x2
Newton-Raphson

When is x1 a
suitable first
approximation for
the root a of
f(x) = 0?
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.
Student 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
Reference
Thomas, M. O. J. (2008). Developing
versatility in mathematical thinking.
Mediterranean Journal for Research in
Mathematics Education, 7(2), 67-87.
From: moj.thomas@auckland.ac.nz
Proceptual versatility–
eigenvectors
Ax  x
Two different processes
Need to see resulting object
or ‘effect’ as the same
Work within the representation
system—algebra
Work within the representation
system—algebra
Same process
Conversion
u
v
v
u
Student 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
Conversions

Translation between registers
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Duval, 1999, p. 5
Representations
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Semiotic representations systems — Semiotic registers
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Processing — transformation within a register
Duval, 1999, p. 4

Epistemic actions are mental actions by
means of which knowledge is used or
constructed
Representations and
mathematics

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
Representation dependant ideas...
"…much of the actual work of mathematics is to
determine exactly what structure is preserved
in that representation.”
J. Kaput
Is 12 even or odd?
Numbers ending in a multiple of 2 are even. True
or False?
123?
123, 345,
569
113, 346, 537, 469
are all odd numbers
are all even numbers
Representational interactions
We can interact with a representation by:

Observation—surface or deep (property)

Performing an action—procedural or
conceptual
Thomas, 2001
Representational interactions
We can interact with a representation by:

Observation—surface or deep (property)

Performing an action—procedural or conceptual
Thomas, 2001

Procedure versus concept

Let
f (x ) 
x2
x 2 1

For what values of x is f(x) increasing?

Some could answer this using algebra
and f (x)  0 but…
Procedure versus concept
4 .0 0
3 .0 0
2 .0 0
1 .0 0
-2 .5 0
-2 .0 0
-1 .5 0
-1 .0 0
-0 .5 0
0 .5 0
1 .0 0
1 .5 0
2 .0 0
2 .5 0
3 .0 0
-1 .0 0
-2 .0 0
-3 .0 0
For what values of x is this function increasing?
Why it may fail

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
Icon, index, symbol
A
B
D
C
ABCD — symbol
Icon 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
Function