the tension between parametric registers and explicit patterns nurit zehavi and giora mann

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

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




Registers


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




Registers


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




Registers


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.

the tension between parametric registers and explicit patterns nurit zehavi and giora mann

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