the tension between parametric registers and explicit patterns nurit zehavi and giora mann
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The tension between parametric registers and explicit patterns
Nurit Zehavi and Giora Mann
Outline
• Theoretical and Practical Background
• Technological Discourse
• Deepening the Discourse: Parametric
Registers
• The Study, Results and Conclusions
Outline
• Theoretical and Practical Background
• Technological Discourse
• Deepening the Discourse: Parametric
Registers
• The Study, Results and Conclusions
Theory of didactical practices Praxeology (Chevallard)
• Tasks
• Techniques
• Technology
Technological Discourse is one of the components in Chevallard’s Praxeology, on which French researchers based their work on Instrumented Techniques in developing teaching with mathematical software (Artigue, and others 2005).
)New(
)Instrumented(
)Discourse(
Resource E-book for Teaching Analytic Geometry with CAS
A new task: viewing the parabola under an angle α or 180-α
600
1200
1350
450
2 2(X 3) Y 8 di
rect
r
ix
An unfamiliar relationship
2y 4x
New Perspectives on Conic Sections Using Instrumented Techniques
Geometric loci of points
from which a given
conic section is viewed
under given angles
Discourse: Analysis of Instrumented Techniques
• Peschek and Schneider introduced the notion of outsourcing operative knowledge (OK) as a didactical principle of CAS use.
• They regard operative knowledge as an object -means to generate new knowledge.
• In our studies we regard OK as a subject –that evolves while utilizing CAS for problem solving.
Outline
• Theoretical and Practical Background
• Technological Discourse
• Deepening the Discourse: Parametric
Registers
• The Study, Results and Conclusions
Reflection, Operative Knowledge & Execution A ‘play’ with three actors (CAME, 2005)
ET
RT
OK
Reflective Thinking A mathematician
Operative Knowledge A system engineer
Execution Technician A Technician
locus of intersection points of perpend. tangents to the hyperbola
x2
9 y2
41
ET
RT
We can plot the hyperbola in its implicit formOK
ET
RT
ET
OK
The equation of a tangent to the hyperbola through (p, q) is
y mx mp q
This system of equations can be solved for x and y
x2
9
y2
41 and y mx mp q
OK
RT
RT
OKOK
RT
x 3 2 m2 p2 9 2mpq q2 4 3m mp q
4 9m2
and
y 2 3m m2 p2 9 2mpq q2 4 2 mp q
4 9m2
or
x 3 2 m2 p2 9 2mpq q2 4 3m mp q
4 9m2
and
y 2 3m m2 p2 9 2mpq q2 4 2 mp q
4 9m2
Solves the system for x and yET
RT
ET
OKOK
RT
x 3 2 m2 p2 9 2mpq q2 4 3m mp q
4 9m2
and
y 2 3m m2 p2 9 2mpq q2 4 2 mp q
4 9m2
RTThe denominator
m
3
2asymptotes
tangent
one solution
The expression under the
Square root sign should be zero
OK We can copy the expression and solve for m.
RT
ET
OK
RT
3 3 p p
tangents 9(q 2 4) 4 p2 0
Copy and plot implicit
ET
The two values of m will be real if p^2-9≠0 and if the expression under the square root
sign is non-negative.
ET
OK
ET
OK
RT
ET
Let’s go back to the original problem. In order for two lines to be perpendicular the product of their slope
should be –1.
9 q2 4 4 p2 pq
9 p2
9 q2 4 4 p2 pq
p2 9 1
looks like a circle
Simplify!
p2 q2 5 and p 3 and p 3
Plot!
Epistemological roles of Reflection, Operational Knowledge, and Execution in developing new
instrumented techniques in ONE HEAD
RT
OK
ET
Outline
• Theoretical and Practical Background
• Technological Discourse
• Deepening the Discourse: Parametric
Registers
• The Study, Results and Conclusions
RT
ET
OKOK
RT
ETET
We can actually view, in a dynamic way, pairs of tangents using a slider bar.
When do the two tangents touch the two branches of the hyperbola?
New instrumented techniques
Representation registers (Duval, ESM 2006):Semiotic systems that permit
a transformation of representation
Two types of transformation
of semiotic representations
Treatment
within a register
e.g. solving an equation
Conversion
changing a register
e.g. plotting the graph of an equation
Deepening the Discourse
Semiotic repres.
registers
Discursive Operations
Non-Discursive
Multi-functional
No algorithms
e. g.
Explanation,
theorem, proof
e. g.
Sketch,
figure
Mono-functional
Algorithms
Computation
symbol proof
diagram
Graphs
Slider bars
A “Parametric” Register
A parametric register can be implemented in mathematical software in the form of
slider bars that enable to demonstrate, in a dynamic way, the effect of a parameter in an algebraic expression on the shape of the related graph.
Outline
• Theoretical and Practical Background
• Technological Discourse
• Deepening the Discourse: Parametric
Registers
• The Study, Results and Conclusions
ProblemProblem: What is the loci of points from which the two tangents to the hyperbola x•y = 1
ouch the same branch / both branches are touched?
2ainteriorexterior
The Study
The plane is partitioned into four loci: points through which no tangent passes, points through which a single tangent
passes, points through which two tangents to the
same branch of the hyperbola pass, and points through which two tangents pass,
one to each branch.
ProblemProblem: What is the loci of points from which the two tangents to the hyperbola
touch the same branch / both branches are touched?
Parametric Register and OK
“The pair of tangents switches from touching one branch to touching both, and conversely.”
ET
RT
OK
Designed the animation
The Study
1. The teachers implemented slider bars to animate pairs of tangents to a hyperbola and reported the results.
2. We asked the teachers to rate (from 1 to 6) the need to prove algebraically the results and explain their rating (part I).
Rate (from 1 to 6) the need to prove algebraically the visual results
No need123456need
3. Next, we exposed the expressions obtained by the CAS while we designed the animation of tangents through a general point P(X, Y).
The Study 4. The teachers were asked to make explicit
the meaning of the symbolic expressions.
5. Then we asked the teachers to rate (and explain) again, the need to provide explicit algebraic proof of the partition of the plane into four loci (part II).
Conclusions
Our findings from the pilot study elicit cognitive activities in the processing of slider bars, and also indicate that the tension created by the conversion between this parametric register and the symbolic (algebraic) register sharpen the way we think about parameters.
• OK evolves to his role as mediator between RT and ET in making it a habit to plot implicit equations and to implement slider bars. Slider bars operate on expressions.
• The expressions encapsulate the relationships between the different parameters, which need to be unfolded by means of advanced symbol sense.
ConclusionsConclusions
Conclusions
Such symbol sense motivates not only qualitative exploration of the effect of changing the value of the parameter on the geometric representation, but also quantitative explanation of the cause of the change.