Transcript Learning mathematics with technology: Theoretical and empirical

Versatile Mathematical
Thinking in the Secondary
Classroom
Mike Thomas
The University of Auckland
Overview

A current problem

Versatile thinking in mathematics

Some examples from algebra and calculus

Possible roles for technology
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 problems

Consider
2x  3
1
6  4x
2x  3  6  4 x
2x  3
x  1 12
But, the LHS of the original is clearly one half!!
Concept not understood
A
B
C
2
2
(x  3)(x  2) x  5x  6 x  x  6
Which two are equivalent?
Can you find another equivalent expression?
A student wrote…
(x  2)(x  3)
…but he factorised C!
Procedural focus

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?
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
Examples 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

 2

lim x 4  or  1n2
x  2 x2 
n 1
tending to limit
value of limit
(x1, x2, Ι, xn)
vector shift
point in n-space
Lack of process-object versatility
(Thomas, 1988; 2008)
Procept example
f(x) = 2x+2 and
g(x) = 2(x+1)
are one and the same as processesΡ even though the arithmetic
procedures to compute them have a different sequence of operations.
They become objects when we can act on them as a single entity, eg
h(x) x2 1




 





 

h  2x  2  h 2  x 1  (2x  2)2 1 2  x 1
2
1
Procept example
f(x) = 2x+2 and
g(x) = 2(x+1)
are one and the same as processesΡ even though the arithmetic
procedures to compute them have a different sequence of operations.
They become objects when we can act on them as a single entity, eg
h(x) x2 1




 





 

h  2x  2  h 2  x 1  (2x  2)2 1 2  x 1
2
1
Effect of context on meaning for
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
dy
dx
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
f (x)
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
Surface (iconic) v deep
(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
interpret
external sign
‘appropriate’
schema
Interact
with/act on
Internal
world
Schema use
Perceived
Reality
interpret
Picture of
reality
interpret
Diagram
or
Drawing
interpret
Theoretical
Mathematical
Figure
Booth & Thomas, 2000
We found
e
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
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
Surface: only needs an = sign
All except b) are equations since:
Equation schema: only needs an
operation
Perform an operation and get a result:
The blocks problem
FRONT ELEVA TION
SIDE ELEVATION
Solution
1
2
1
2
Reasoning
The mi nim um number of blocks is 6, with 1, 2, 1, 2 along the diagonal as shown
above. After this it is possible to add a single box in any of the other squares above, or
a combination of boxes in those squares, without c hanging the elevations. The shaded
squares ca n each have up to 2 boxes without changing the elevation. Hence the
maxim um number of boxes is 16+4 = 20. Therefore there can be N boxes where
6² N²20.
Solve
x
50
e =x
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Solve

x
50
e =x
Check with two graphs, LHS and RHS
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Find the intersection
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How could we reason on this
solution?
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Antiderivative?
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What does the antiderivative look like?
Task: What does the graph of the
derivative look like?
Method easy
But what does the antiderivative
look like?
How would you
approach this?
Versatile thinking
is required.
Maybe some technology would
help

Geogebra
Geogebra
Representational 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...
"…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
Representations can lead to other
conflicts…
1 unit s quare
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 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 equation
ax2 + bx + c = 0 have? Explain.
1
6
Treatment and conversion
(Duval, 2006, p. 3)
Treatment or conversion?
25
Integration by substitution

2
1 x dx
x  sin


1
 cos cos d  2  cos2 1 d
Integration by substitution
Consider u 
 f (x)dx
Diff erentiating wrt t gives
du du dx
dx


 f (x) 
dt dx dt
dt
and integrating wrt t gives
u

dx
f (x)  dt
dt
Integration by substitution
Consider u 
 f (x)dx
Diff erentiating wrt t gives
du du dx
dx


 f (x) 
dt dx dt
dt
and integrating wrt t gives
u

dx
f (x)  dt
dt
Integration by substitution
Consider u 
 f (x)dx
Diff erentiating wrt t gives
du du dx
dx


 f (x) 
dt dx dt
dt
and integrating wrt t gives
u

dx
f (x)  dt
dt
Linking of representation systems

(x, 2x), where x is a real number
Ordered pairs to graph to algebra
Using 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 gesture
 Metaphoric – an abstract meaning is
presented as form or space

The task
Thinking with gestures
Creates a virtual space
Quic kT i me™ and a
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perpendicular
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converging
The 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?
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
etc
 assist with modelling
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 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)
Task

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–Generalising

Can we find the quadratics that meet at any
point (p, q), with any gradient k?
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Task–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 curves
 c  kp  q  2  2c  kp  2q 
y
x 
xc
2






p
p
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Another 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?
y  ax  bx  c
2

Consider

And its derived function
y  2ax  b
Solution

So these touch at one point
4a  b
y  ax  bx 
4a
2
2
y  2ax  b
2
For example
25
y  2x  3x 
and y  4x  3
8
2
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Extension: can you find any other functions with this
property?….
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
Why it may fail
Newton-Raphson

When is x1 a
suitable first
approximation for
the root a of
f(x) = 0?
Symmetry of cubics
3
2
y  x  3x  x  5

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Generalising 180˚ symmetry

In general (a, b) is mapped to (2p–a, 2q–b)
 Hence g(x)  2q  f (2 p  x)
(a,b)
e.g. q – y = b – q
(p,q)
(x, y) = (2p-a,2q–b)
Solving linear equations

Many students find ax+b=cx+d
equations hard to solve

We may only teach productive
transformations ax,  cx,  b,  d

But there are, of course, many more
legitimatetransformations kx,  k, k  R
10–3x = 4x+3

10–3x = 4x+3

10= 7x+3

7=7x

1=x
productive
10–x= 2x+3
8.37–x= 2x+1.37
legitimate
Legitimate: 10–3x = 4x+3
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Teacher 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 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 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
Email: [email protected]