Computer Algebra as an Instrument: Examples of Algebraic Schemes

Computer Algebra as an Instrument:

Examples of Algebraic SchemesPaul Drijvers

Freudenthal Institute

Utrecht University

Utrecht, The Netherlands

www.fi.uu.nl

[email protected]

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:

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:

x y y x2 2 225 31 |

Type of tasks

student’s mental schemes

artefact

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 exampleA sheaf of graphs of y = x2 + b*x + 1Find the equation of the curve through the minima.

One student’s work

M: So you do =0 so to say, and then ‘comma b’, because you have to solve it with respect to b

O: Well, no.

M: 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= * 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?