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Didactical practices of computer algebra in mathematics education

Science Teaching Department Nurit Zehavi

Weizmann Instituteof Science, Israel

TIME 2008TIME 2008

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The MathComp Project (Mathematics on Computers) The MathComp is a R&D project that began in the

early 1990s, with the aim of integrating CAS into teaching to broaden learning opportunities and to promote greater mathematical understanding.

Noss & Hoyles, 1996Windows on Mathematical Meanings: Learning Cultures and Computers

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Issues discussed in CAME symposia

Chevallard’s anthropological approach Instrumentation Bridging the gap between techniques and

theory CAS & DGS Teachers in transit (professional development)

)2003(

http://www.lkl.ac.uk/research/came/

(1999)

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Chevallard’s anthropological approach to the didactics of mathematics

Didactics of mathematics as a domain of research is part of ‘anthropology of mathematical knowledge’ that is concerned with the emergence and growth of the mathematical knowledge in educational settings (“institutions”).

An educational setting can be characterized by its theory of practices, i. e.

a praxeology.

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The components of praxeology of a given topicThe components of praxeology of a given topic::

The constructed knowledge is strongly dependent on

the perspectives of the partners within a specific

educational setting on the above components.

(Types of) Tasks Techniques

“Technology” (Discourse of the techniques)

Theory

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Two Topics:

From a word problem to a family of word problems: In what ways can CAS help in solving word problems?

(Zehavi & Mann, IJCAME, 1999)

Justifications of geometric results using

CAS: Can an integration of features of DGS and CAS be useful for approaching proof?

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The School of Pythagoras

Pythagoras, who lived in the sixth century BC, ran a school.

He was once asked how many students are in his school.

After thinking awhile, he said:

1/2 of the students are now participating in a math class.

1/4 of the students are now in a science class.

1/7 of the students are now silently exercising their minds.

In addition to all the above, three students are in the garden.

How many students were in the school?

Study I: In what ways can CAS help in

solving word problems?

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Debugging the modelDebugging the model

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PD - Teachers’ contribution:

Tutorials for making explicit the implicit restrictions of the model, for example:

Change the problem so that the number of students in the School of Pythagoras will be 280 instead of 28.

x x xx 3

2 4 70

3280 30

28

3

28x n

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Diophantus, known as the 'father of algebra' – his epitaph“This tomb hold Diophantus Ah, what a marvel! And the tomb tells scientifically the measure of his life….” (Newman (ed.) The World of Mathematics, 1956)

How long did Diophantus live?

The equation is an object to explore; not just a tool for finding an answer

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1/9 of my life I solved math exercises by hand.

Then I was introduced to Derive and used it for 1/3 of my life .

Later, I knew everything! so I did math in my head for another 1/3 of my life.

In the last 26 years I did not learn any math, I just played music.

How long have I lived?

Math & Music

x

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Why do you want us to deal with ancient people?!

Let’s invent problems on someone more

interesting like… ET –

The Extra Terrestrial!!!

Students’ contribution

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Student and teacher contribution: A Basketball Game

Sport journalist:Eitan scored 40%Tom scored 25%Eyal scored 15%And Motti, as usual,made 5 three-pointShots.

Readers: The report must be wrong!

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Expanding the praxeology of learning word problems in a CASCAS environment

Didactical perspectives: Transferring the control over the process of

modeling to the student. By inventing problems the students share

responsibility, with the teachers, of the techniques and the validity of the established knowledge in class.

Enabling students to see any given problem as a member of a family of problems through its model.

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A new practice: resource e-book for teaching analytic geometry with CAS

A classical task: “Show that the locus of

points through which two perpendicular tangents are drawn to a given ellipse/hyperbola is ….."

The director circleThe director circle

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NewNew perspectives on conic perspectives on conic sectionssections

Classical textbook solution: Viète's formula

Users with basic CAS experience: Utilizing CAS for traditional techniques

Users with good mastering of CAS: New instrumented techniques further exploring the solution and the problem

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A new task: viewing the parabola under α or 180-α

600

1200

1350

450

di

rect

r

ix

An unfamiliar relationship

2y 4x

2 2(X 3) Y 8

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Illustration of tangents to hyperbola when sliding a point on the director circle

the pair of tangents switch from touching one branch to touching both

The relation between visual clues and formal proof

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Hanna, (ESM, 2000)

Experimental work with dynamic geometry could lead educators to question the need for analytical proofs.

The key role of proofs in the classroom is to promote mathematical understanding.

Exploration of a problem can lead one to grasp its structure and its ramification, but cannot yield an explicit understanding of every linkexplicit understanding of every link.

Proofs and dynamic geometry

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Laborde et al. ( PME Handbook, 2006)

Teaching and Learning Geometry with Technology:

When students were asked to justify conjectures the teachers did not mention the possibility of using Cabri to find a reason or elaborate a proof.

Proofs and dynamic geometry

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A significant difference between DGS and CAS

in DGS – The algebraic infrastructure that enables

construction and animation is hidden.in CAS – Users need to develop the algebraic

expressions for producing constructions and insert slider bars related to the parameters for animation.

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Proof and CAS

The algebraic expressions that create the animation can be used for justifying the visual results.

Mann, Dana-Picard and Zehavi (2007)

22

19 4

yx

Sliding…

and Proving.…

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a=3, b=2

2 2

2 21

x y

a b

1 2 0x x 2 24 9 0 !X Y 2 2 2 2

( ) ( )3 3 3 3

Y X Y X Y X Y X

Symbolic computation in CAS

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Symbol Sense in the messy expressionsSymbol Sense in the messy expressions

explicit understanding of every linkexplicit understanding of every link.

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Study II: Teachers’ justifications of geometric results using CAS (n = 43)

(a) teachers' views on the need for an algebraic proof of unfamiliar geometric results obtained by experimenting with slider bars in CAS, and

(b) the types of proofs they produced using the expressions that generated the animation.

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Goals To construct, with the teachers, elements of

a praxeology for integrating the new perspectives on conic sections into teaching.

To determine how a CAS, which enables both symbolic manipulation and animation, can promote mathematical understanding by bridging experimental mathematics with deduction.

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The tasksThe tasks

interior

exterior

Is it possible to draw two tangents to the hyperbola from every point in the plane, which is not on the asymptotes? Do two tangents to the hyperbola from specific points touch the same branch?Justify your answers

Part (a)

x•y = 1

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Implement two slider bars to animate a pair of tangents drawn from P(X, Y) to hyperbola x•y = 1 and identify the loci of points from which:

(The expressions were defined, but not displayed)

No tangent can be drawn;

A single tangent can be drawn;

Two tangents to the same branch can be drawn;

Two tangents can be drawn, one to each branch.

Rate the need for students to prove algebraically the results; explain your pedagogical arguments.

No need 1 2 3 4 5 6 need

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Part b: Here are the expressions obtained by the CAS while designing the animation of tangents through a general point P(X, Y).

)X, Y ≠ 0(

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Follow through the derivation of the coordinates of the tangency points,

and use these expressions to prove your findings.

Please rate (and explain) again (from 1 to 6), the need for students to provide algebraic proof of the partition of the plane into four loci.

)X, Y ≠ 0(

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Part (a)

Part (b)

1-2 3-4 5-6

1-23 2 1

3-42

6 7

5-64 6 12

Distribution of teachers’ rating (n = 43)

HH

22

20

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Teacher H: Answer (part a)

Rating: 2

Explanation

Intersection of same/different color tangents

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Teacher H: Answer (part b)

Reflection: Symbol Sense

(1 1 X Y) ( 1 X Y 1) X Y

I , III

X X0 or 0

Y Y x1 x2 0 or x1 x2 0

Xx1 x2

Y

1 1 1 11 ,

1 1 1 12 ,

[ ]

[ ]

X Y X YTP

Y X

X Y X YTP

Y X

explanation of every linkexplanation of every link

Rating: 6

Reflection

Explanation

respectively

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PD Practice: Distribution of proofs and arguments given by the teachers

Focal objects for proof Arguments for need of proof

No proof 15'Origin' of tangents P(X,Y) 13Coordinates of tangency points 10Location of tangency points 5

Conviction 32Explanation 21 Convention 15Reflection 9

)H(

Relevance to classroom practice

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The study focused on:

The coordination of algebraic, graphical and geometrical representations to deal with problems related to tangents to conic sections using a CAS.

The possibility of building a praxeology for learning about proofs based on these problems.

The particular place of actions with slider bars in this praxeology.

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Slider bars demonstrate, in a dynamic way, the effect of a parameter, in an algebraic expression, on the shape of the related graph.

The algebraic expressions encapsulate the relationships between the different parameters. For approaching proof these relationships need to be unfolded by means of symbol sense.

DiscourseDiscourse

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DiscourseDiscourse

Such symbol sense motivates not only qualitative exploration of the effect of changing the value of the parameter of the geometric representation;

but also quantitative explanation of the cause of the change.

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Conclusion

Teacher professional development within curricular R&D(of story problems, tangents to conics,

and other topics) can help in the didactical transposition of new practices that incorporate technology.

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k = 1.7

A new practice: Viewing hyperbolasViewing hyperbolas

No tangent

Obtuse angles

Acute angles

2

1

2

1