Computer Algebra as an Instrument: Examples of Algebraic Schemes

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Computer Algebra as an InstrumentExamples of
Algebraic Schemes
Paul Drijvers Freudenthal Institute Utrecht
University Utrecht, The Netherlands www.fi.uu.nl p
.drijvers_at_fi.uu.nl
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Sources of this talk

• Drijvers Van Herwaarden, 2000
• PhD dissertation (2003) www.fi.uu.nl/pauld/disse
  rtation
• 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.

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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

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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)

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Attitudes towards ICT use

• Fear for the integration of ICT The students
  dont have to do anything anymore
• Optimism concerning the integration of ICTNow
  we can leave the work for technology, and focus
  on higher order skills, modeling, realistic
  application
• Tendency to separate skills and
  understandingICT for the procedures, the
  student for the conceptual understanding
• Concern about the relation learning ICT
  paperpencil The students are not able to
  carry out anything by hand / by heart anymore

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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

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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
  userThe tool shapes the scheme, and the
  students knowledge shapes the tool
  (instrumentation and instrumentalization)
• Different kinds of utilization schemes
• Usage schemes
• Instrumented action schemes
• Schemes are invisible, but techniques are!

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In a picture
students mental schemes
artefact
Type of tasks
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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.

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An example
A sheaf of graphs of y x2 bx 1 Find the
equation of the curve through the minima.
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One students 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?

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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.

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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

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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.

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One students 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.

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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 variableexpression2, 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.

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Transfer to paper pencil work
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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

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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.

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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

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Students work nested method

• The nested method
• Substitute en Solve in one line
• DifficultySolving with respect to the wrong
  unknown
• Adding an extra pair of brackets helps

Solve ((x2 y2 252 y 31 x), x)
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Students work errors

• Circular substitution
• Non-isolated substitution

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Students work variations

• Isolate twice(cf. pp method)
• Use and(not foreseen)

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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.

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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?

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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

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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

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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.

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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?

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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?