The tension between parametric registers and explicit patterns

The tension betweenparametric registersand 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 practicesPraxeology(Chevallard) • Tasks • Techniques • Technology (New) (Instrumented) (Discourse) 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).

Resource E-book for Teaching Analytic Geometry with CAS

1350 1200 600 450 A new task: viewing the parabola under an angle α or 180-α di rect r ix An unfamiliar relationship

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 outsourcingoperative 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.

ET RT OK Reflection, Operative Knowledge & Execution A ‘play’ with three actors(CAME, 2005) Reflective Thinking A mathematician Operative Knowledge A system engineer Execution Technician A Technician

ET ET RT OK locus of intersection points of perpend. tangents to the hyperbola We can plot the hyperbola in its implicit form

The equation of a tangent to the hyperbola through (p, q) is ET This system of equations can be solved for x and y RT RT OK OK

ET or RT RT OK OK Solves the system for x and y

The denominator ET RT RT RT OK OK OK asymptotes one solution The expression under the Square root sign should be zero tangent We can copy the expression and solve for m.

ET ET Copy and plot implicit tangents RT RT OK 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 ET ET RT OK OK Let’s go back to the original problem. In order for two lines to be perpendicular the product of their slope should be –1. looks like a circle Plot! Simplify!

ET RT OK Epistemological roles ofReflection, Operational Knowledge, andExecutionin developing new instrumented techniques inONE HEAD mediator

ET ET ET RT RT OK OK When do the two tangents touch the two branches of the hyperbola? We can actually view, in a dynamic way, pairs of tangents using a slider bar. 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

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.

Problem: What is the loci of points from which the two tangents to the hyperbola x•y = 1ouch the same branch / both branches are touched? 2a exterior interior

The Study Problem: What is the loci of points from which the two tangents to the hyperbola touch the same branch / both branches are touched? 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.

ET RT OK Parametric Register and OK Designed the animation Tension between RT and ET “The pair of tangents switches from touching one branch to touching both, and conversely.”

The Study • The teachers implemented slider bars to animate pairs of tangents to a hyperbolaand reported the results. • 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

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 • The teachers were asked to make explicitthe meaning of the symbolic expressions. • 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.

Conclusions • 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.

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. Thank you!

The tension between parametric registers and explicit patterns