Computer Algebra as an Instrument:
Examples of Algebraic Schemes
Paul Drijvers
Freudenthal Institute
Utrecht University
Utrecht, The Netherlands
www.fi.uu.nl
p.drijvers@fi.uu.nl
Sources of this talk
• Drijvers & Van Herwaarden, 2000
• PhD dissertation (2003):
www.fi.uu.nl/~pauld/dissertation
• Fey, J., Cuoco, A., Kieran, C., McMullin, L., & Zbiek,
R. M. (2003), Computer Algebra Systems in
Secondary School Mathematics Education. Reston,
VA: National Council of Teachers of Mathematics
• Guin, D., Ruthven, K. & Trouche, L. (in press). The
didactical challenge of symbolic calculators: turning
a computational device into a mathematical
instrument. Dordrecht, Netherlands: Kluwer
Academic Publishers.
Outline of the talk
1. Introduction to the instrumental approach
2. A scheme for solving equations
3. A scheme for substituting expressions
4. A composed scheme
5. Reflections on the instrumental approach
1. Introduction to the instrumental
approach
• Examples that make me think:
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Soft returns in a text editor
Cut-and-paste in a text editor
Viewing window in a graphing calculator
The left-hander and the pouring pan
(Trouche, 2000)
Attitudes towards ICT use
• Fear for the integration of ICT:
‘The students don’t have to do anything anymore’
• Optimism concerning the integration of ICT:
‘Now we can leave the work for technology, and focus
on higher order skills, modeling, realistic application’
• Tendency to separate skills and understanding:
‘ICT for the procedures, the student for the
conceptual understanding’
• Concern about the relation
learning – ICT – paper&pencil:
‘The students are not able to carry out anything by
hand / by heart anymore’
The instrumental approach
• … to learning mathematics in a technological
environment
• … distinguishes artefact / tool and instrument
• … stresses the process of instrumental genesis
• … which involves the development of mental
utilization schemes
• … sees the instrument as the combination of (part of
the) tool and scheme for a type of task
A bit more on schemes
• A scheme is an invariant organization of activity for a
given class of situations (Vergnaud 1987, 1996)
• In a utilization schemes, technical and conceptual
aspects interact
• A dialectic relationship between tool and user:
The tool shapes the scheme, and the student’s
knowledge shapes the tool
(instrumentation and instrumentalization)
• Different kinds of utilization schemes:
 Usage schemes
 Instrumented action schemes
• Schemes are invisible, but techniques are!
In a picture:
student’s
mental schemes
artefact
x 2  y 2  252 | y  31  x
Type of tasks
2. A scheme for solving equations
• As a scheme is individual, we cannot speak about
THE scheme for solving equations
• Or should we speak about ‘technique’ here?
• It seems so simple to use the solve command, but
observations show an interplay of technical and
conceptual knowledge
• Using the solve command for solving parameterized
equations requires an extended conception of what
solving means.
An example
A sheaf of graphs of y = x2 + b*x + 1
Find the equation of the curve through the minima.
One student’s work
M:
O:
M:
So you do =0 so to say, and then ‘comma b’,
because you have to solve it with respect to b
Well, no.
You had to express in b?
Elements in the scheme
1. Knowing that the Solve command can be used to express
one of the variables in a parameterized equation in other
variables;
2. Remembering the TI-89 syntax of the Solve command, that
is Solve(equation, unknown);
3. Knowing the difference between an expression and an
equation;
4. Realizing that an equation is solved with respect to an
unknown and being able to identify the unknown in the
parameterized problem situation;
5. Being able to type in the Solve command correctly on the
TI-89;
6. Being able to interpret the result, particularly when it is an
expression, and to relate it to graphical representations.
3. A scheme for substituting
expressions
• Substitution of numerical values for variables is easy
for students
• Substitution of expressions requires an object view
• The idea of ‘cutting an expression and pasting it into
a variable’ is powerful
v= a * h | a= p * r2
Example
If the height of the cylinder equals the diameter of the
base, so that h = 2r, the cylinder looks square from
the side.
Express the volume of this ’square cylinder’ in terms
of thee radius.
One student’s reaction
O: Now what exactly does that vertical bar mean?
T: It means that the left formula is separated from the
right, and that they can be put together.
O: And what do you mean by putting together?
T: That if you, that you can make one formula out of
the two.
O: How do you do that, then?
T: Ehm, then you enter these things [the two
formulas] with a bar and then it makes
automatically one formula out of it.
Elements in the scheme
1. Imagining the substitution as ‘pasting an expression into a
variable’;
2. Remembering the TI-89 syntax of the Substitute command
expression1 | variable=expression2, and the meaning of
the vertical bar symbol in it;
3. Realizing which expressions play the roles of expression1
and expression2, and considering expression2 in
particular as an object rather than a process;
4. Being able to type in the Substitute command correctly on
the TI-89;
5. Being able to interpret the result, and particularly to
accept the lack of closure when the result is an expression
or equation.
Transfer to paper & pencil work
4. Composed schemes
• A composed scheme consists of some elementary
usage schemes.
• The instrumental genesis of a composed scheme
requires high level mastering of the components
• Nesting commands is more difficult than a stepwise
method
• Example: Isolate – Substitute - Solve
Example
The two right-angled edges of a right-angled triangle together
have a length of 31 units. The hypotenuse is 25 units long.
a. How long is each of the right-angled edges?
b. Solve the same problem also in the case where the total length of
the two edges is 35 instead of 31.
c. Solve the problem in general, that is without the values 31 and 25
given.
Students’ work: stepwise method
The stepwise ‘ISS’ technique:
• Isolate one variable
• Substitute into other equation
• Solve the result with respect to
the variable
And finally calculate the value
of the other variable
Students’ work: nested method
The nested method:
• Substitute en Solve in one line
• Difficulty:
Solving with respect to the wrong
unknown:
• Adding an extra pair of
brackets helps:
Solve ((x2 + y2 = 252 | y = 31 – x), x)
Students’ work: errors
• ‘Circular’ substitution
• Non-isolated substitution
Students’ work: variations
• Isolate twice
(cf. p&p method)
• Use ‘and’
(not foreseen)
Elements in the scheme
1. Knowing that the ISS strategy is a way to solve the
problem, and being able to keep track of the global
problem-solving strategy in particular;
2. Being able to apply the technique for solving
parameterized equations for the isolation of one of the
variables in one of the equations;
3. Being able to apply the technique for substituting
expressions for substituting the result from the previous
step into the other equation;
4. Being able to apply the technique for solving equations
once more for calculating the solution;
5. Being able to interpret the result, and particularly to
accept the lack of closure when the solution is an
expression.
5. Reflections on the instrumental
approach
• Some conclusions
• What does it offer the teacher?
• What does it offer the researcher?
• How wide is its scope?
• Relations with other theoretical frameworks?
Some conclusions from my PhD
research
• Instrumentele genesis is a difficult process
• Indeed, a close relation is observed between machine
technique and mathematical conception
• Composed instrumented action schemes require high
level mastering of component schemes
• The instrumental approach is a fruitful perspective for
observing and understanding student behavior
What does the instrumental approach
offer the teacher?
• A framework to set up teaching which takes into
account the intertwinement of machine technique and
understanding, and to work out this relationship for a
particular ICT application.
• A perspective to keep in mind while
 Developing ICT-rich tasks;
 Teaching ICT-integrated courses;
 Helping students who encounter difficulties during
their work using ICT;
 Trying to capitalize on the opportunities that ICT
offers
What does the instrumental approach
offer the researcher?
• A framework to focus on the intertwinement of
machine technique and understanding, and to
investigate this relationship for a particular ICT
application.
• A framework to observe students working in an ICT
environment, to understand their difficulties and to
develop effective learning trajectories.
How wide is its scope?
• Can the instrumental approach be applied better to
‘pedagogy-free’ sophisticated mathematical tools,
than to pedagogical ICT tools?
• Can it be applied to other ICT environments than
computer algebra, such as DGS, applets?
• What would schemes be like for other ICT
environments?
Relations with other theoretical
frameworks?
• A theory ‘under construction’
• Difficult vocabulary, not all concepts are clearly
defined
• Is it an individual or a social perspective?
More attention for the relation between individual
schemes and collective instrumental genesis
• Elaborations concerning the didactical contract and
the orchestration by the teacher
• Articulation and coordination is needed with other
theoretical perspectives, such as socio-constructivism,
theories on symbolizing, CHAT
• How about didactical engineering and task design?