Assessing with technology in France

New problems and a new tool (Casyopée)

Jean-Baptiste LAGRANGE Equipe de Didactique des

Mathématiques, Université Paris VII France

Technology and (summative) evaluation

– Baccalaureate

– Teacher employment competition

Baccalaureate

• All calculators allowed• An experimental “two parts” exam paper (1998)

– Scientific stream– Questions about calculator use– Tasks without the calculator

• An exam involving spreadsheet knowledge (2000)– Non scientific stream– Traditional written exam

• A new assessment at the baccalaureate (2006)– Scientific stream– “Practical” assessment with computer

Teacher employment competition

• Secondary teachers– ‘Advanced calculators’ at the oral

examination (2004)

• Primary teachers– Possible questions about the use of tools

like a spreadsheet or dynamic geometry (2006)

An experimental “two parts” exam paper at the baccalaureate (1998)

On considère la fonction f définie sur IR: par f(x) = x Ln x - 2 Ln x - (Ln x)2, On note f ’ sa fonction dérivée et g la fonction définie sur R par g(x) = 2x. Voici ces trois courbes sur l’écran d’une calculatrice pour x compris entre 0 et 5. A- Étude de f.

Déterminer limx

x 0

f Montrer que f’(x) = 12

1

x

xln

3) En déduire le sens de variation de f. 4) Montrer que l’équation f(x) = 0 admet trois solutions.

Donner un encadrement de longueur 10-1 pour les deux solutions non entières. B- Intersection des représentations graphiques de f et de g On veut déterminer si la courbe représentative de f coupe la droite représentant g pour 0 < x < 7. Que peut-on, à l’aide de sa calculatrice, conjecturer ? Préciser les éléments qui permettent de faire cette conjecture. (noter le type de calculatrice utilisée).

A quite interesting attempt

• Distinguishing tasks with technology and tasks without

• Assessing specific calculator knowledge (instrumentation)

But…not continued…

An exam involving spreadsheet knowledge

• New curriculum for non scientific stream – Learning about linear and exponentional

sequences• from ‘real life’ examples

– Learning about the spreadsheet supporting Math learning

• Exploration of sequences’ properties• Spreadsheet symbolism prepares Math formalization

• Implementation by teachers– ??? (Lagrange & Erdogan, in preparation)

An exam involving spreadsheet knowledge

• Written exam (without computers)

In a class: 50% B2*0,045 (wrong) 50% B2*1,045 (correct)The actual difficulty is not the spreadsheet

Teacher employment competition

Correct answer???

=I2*8+B6*13or=I$2*8+$B6*13

A B C D E F G H I J

1

2 0 1 2 3 4 5 6 7

3 0 0 8 16 24 32 40 48 564 1 13 21 29 37 45 53 61 695 2 26 34 42 50 58 66 74 826 3 39 47 55 63 71 79 87 957 4 52 60 68 76 84 92 100 1088 5 65 73 81 89 97 105 113 1219 6 78 86 94 102 110 118 126 134

10 7 91 99 107 115 123 131 139 14711 8 104 112 120 128 136 144 152 16012 9 117 125 133 141 149 157 165 173

Books at 8€

Books at

13€

•School teachersWritten exam“ We bought books at 8€ each and others at 13€ each. We paid 150€.What did we buy ?…A solution with a spreadsheet. Question: What is the formula in cell I6?

Teacher employment competition

• Secondary teachers– Calculator more or less compulsory at the

oral exams– Function, sequences– Dynamic geometry– ‘Advanced calculators’

Teacher employment competition

Teacher employment competition

425 38 117 64 37 64 3

The new exam at the baccalaureate

Towards integration – (2004) Les technologies de l'information et

de la communication dans l'enseignement des mathématiques au collège et au lycée 

– (2006) in order to perform a real integration, the final evaluation should include the use of computers

Math exercises whose solution significantly involves technology

• Calculators, computers • Dynamic geometry, spreadsheet, CAS• Specific applications (preferably free)”

Rationales

• Technology use remains marginal– Because of evaluation

• Exam with calculator more and more problematic– Download whole math textbook– Cheating via wireless communication

• Important Math proficiencies not evaluated– Conjecturing– Self-inventiveness– Technology use

Practically• Each item in the bank

– A description • topic, • TICE proficiencies• Math proficiencies

– The student document– The teacher document

• Intentions• Possible use of

technology• Comments about

evaluation

– The evaluation document

• Exam in school• One hour• 1/5 of the mark• Teachers choose in a

“bank of exercises”• Teachers attend to 4

students during the exam

• They fill in an evaluation sheet

A description document• Optimizing pipes

We want to put pipes on the wall of a house to collect rain water.

This wall is rectangular. A vertical pipe has to reach the bottom at the middle of the wall. Two other pipes have to collect water from the sides of the roof.

We want to use the shortest total length of pipe. Find the position of point M that gives this minimum length.

• Proficiencies at stakeIn the use of technology

Building a figure using dynamic geometryUsing software to transpose a geometrical situation into a graphic.

In mathematicsEmitting a conjecture from various information:Elaborating a strategy to find the extremum of a function.

Student text: conjecturing

Student text: proving

A critical view

• Big differences between– Description

• Open problem• “Generic” problem

– Student text• Particular problem• No choice of variable• Separation between

– Conjecture– Proof

• Use of software– only for conjecture– numerical approach

-> some disappointment

• Why ?– Acceptance by teachers– Constraints of the

evaluation– No adapted tool

• Positive effects nevertheless– A process of preparing– A new interest for

technology(->teacher dev. Courses)– A new way of assessing(evaluation sheet)

Conclusion on assessment

• Assessing is always a compromise– New approaches and Constraints– Innovative views and teacher acceptance

• Bad and acceptable compromises– Written versus practical

• A prayer to teachers– Please, please, do not mimic evaluation

tasks when teaching with technology.

Casyopée

• Use of numerical tools (spreadsheet, dynamic geometry) is frustrating– Separation between conjecture and proof– No contribution of technology to important

activities in mathematics (proving, writing)

• Standard CAS integration is problematic• Our answer: building a new tool

– Geometric and algebraic– With help for

Exploring, modeling, proving, writing…

An example

• a functional dependency problem– being given:

• a, b : parameters• A(0,a) ; B(0,b) ; C free point on [oB]

– Study of the distance EC

Creating the objects

• symbolic objects

• geometric objects

Using Casyopée

3: geometric calculations

4: variable

5: co-variation

6: function 7: graph

8: algebraic proof

9: geometric proof

1: parameters

2: figure

Generalisation

1: numerical, graphical exploration

3: proof in a numerical case 2: geometrical construction

4: a general proof

Experimenting• First Experimentations

– 10th grade students – Variations and equalities of areas – Students took more initiative for modeling– Easy use of computer Algebra facilities– Students dealt with the notion of function

• interpreting co-variation• making sense of the domain• linking different representations

• Cross experimentation (France-Italy) going on.

Links

• About the teacher national competitionhttp://capes.math.jussieu.fr• About the “practical exam” at the

baccalaureatehttp://www.igmaths.net/• Debate about assessing mathematical

proficiencies especially in the context of technology.

http://educmath.inrp.fr/Educmath/en-debat/epreuve-pratique

/

Links

• About Casyopée– http://www.irem.univ-rennes1.fr/

recherches/groupes/groupe_aide_logiciel

• About ReMath– http://remath.cti.gr

More on my website soon to open-http://jb.lagrange.free.fr/site