Exercising Control: Didactical Influences

Kathleen PineauMaître d’enseignement en mathématiques

École de technologie supérieureFrance Caron

Professeure au département de didactiqueUniversité de Montréal

Context

École de technologie supérieure (ÉTS) : “Engineering for Industry”

Our undergraduate students come from college technical programs

CAS-calculator mandatoryin Calculus since 1999

Exploratory study

Linking teaching strategies in Calculus with students’ practices in problem solving where the CAS is allowed

Elements of the didactical contract andtheir influences on student practices in their use of the CAS capacity to solve problems communication of results

Ideas to increase the tool’s contribution to the

students’ mathematical practice while minimizing risks associated with

its use

CAS as a tool for teaching and learning mathematics Pragmatic considerations

Existence and portability of the tool Engineering profession characterised by an

increasing complexity of problems and a diversity of technological tools

Epistemic considerations Heterogeneity of students’ backgrounds Potentialities, limits and risks associated with

the integration of such tools in understanding the mathematics being taught

Context of Study

Introductory Calculus Focused on 4 teachers and 212 of their students

All teachers and students had the same calculator

(TI-92 Plus/Voyage 200)

the same textbook (Hughes-Hallett et al., 1999)

Analysed Data

Teachers modes of integration Tasks (graded homework and exams): complexity,

instructions, need or relevance of calculator, scale used in grading

Interviews: relationship to mathematics and to their teaching

Students competencies analysed through their writing on the 2nd part of the common final exam (where the calculator is allowed) Characterisation of errors Interesting phenomena

Task Analysis:Competencies

Communication competencies

Evaluation competencies

Intervention competencies

De Terssac (1996)

Task Analysis:Competencies and Complexity

Levels of complexity

Communication competencies

Evaluation competencies

Intervention competencies

Level 4:

Reformulation

Level 3:

Organisation

Level 2:

Comprehension

Level 1:

Association

Establishing and justifying a property

Identifying distinct cases

Interpreting data or results

Combining complementary models

Recognising an object Applying a methodAssociating a property

De Terssac (1996); Caron (2001)

Adapting a method of resolution

Need or relevance of the calculator was also noted.

Choosing a method of resolution

Results - Teachers

Little difference in the distribution of the complexity of tasks given by

teachers

0%

20%

40%

60%

80%

100%

Alain Bernard Charlotte Diane Exam

Association

Comprehension

Organisation

Reformulation

Results - Teachers

Little difference in the distribution of the complexity of tasks given by

teachers in the need or relevance of the CAS to accomplish tasks

0%

20%

40%

60%

80%

100%

Alain Bernard Charlotte Diane Exam

Essential

Pertinent

Unnecessary

Results - Teachers

Little difference in the distribution of the complexity of tasks given by

teachers in the need or relevance of the CAS to accomplish tasks in the epistemic role the teachers give to the tool

focus on meaning through multiple representations (symbolic, graphic and numeric)

Subtle differences in what they like in mathematics, specifics in tasks given

to students and targeted competencies

Results - Teachers

What they like in math

Specifics in tasks Targeted competencies

Alain Precision, deduction,

calculation, analysis

Calculation, interpretation

Numerical Analysis

Intervention

Communication

Bernard Reasoning, logic,

structure

Deduction of properties

Geometric Modeling

Evaluation

Intervention

Charlotte Purity of expression,

complementarities of

representations

Translating

Justification

Communication

Evaluation

Diane Abstraction, rigour,

beauty, applicability

Explorations and

estimations

Applications and modeling

Evaluation

Communication

Intervention

Results – Students, final exam

Little difference in the comprehension of problems or concepts

Students having made a comprehension error

0%

20%

40%

60%

80%

100%

Alain Bernard Charlotte Diane

Question 1

Question 2

Question 3

Question 4

Question 5

Results – Students, final exam

Differences are a little more apparent in organisation and communication of results

Students having made an organisation error

0%

10%

20%

30%

40%

50%

60%

70%

Alain Bernard Charlotte Diane

Question 1

Question 2

Question 3

Question 4

Question 5

Results - Overall

Little difference in the distribution of the complexity of tasks given by

teachers in the need or relevance of the CAS to accomplish tasks in the epistemic role the teachers give to the tool

focus on meaning through multiple representations (symbolic, graphic and numeric)

in theirs students’ performance in the final exam

Reflect the use of a common textbook the common final exam the common culture

A Revealing Question

Part 2 – Question 2

Find the positive value k such that the area of the region between the graphs y = k cos x and y = k x2 is 2. Clearly specify the definite integral you use.

Expected resolution

Find numerically the x values of the intersection of the two curves,i.e. the roots x1 and x2 of the equation :

0cos 2 kxxk 0cos 2 xx (k≠0)

Then find, analytically, k such that

2

1

2)cos( 2x

x

dxkxxk

In principle, the student is allowed to use the calculator without restriction

“I mark off points when there is an abusive use of the TI.”

Alain

Intervention Competencies:pragmatic vs epistemic - tensions Variable reserve in using the tool

An attempt to demonstrate their capacity to determine the appropriate use of the tool.

Effect of teacher’s grading.

Expert (advanced) use of the tool Valorization by the teacher of the pragmatic

function of the tool in the symbolic register

Reflects the didactical contract specific to the teacher

Evaluation Competencies:legitimacy of registers

Graphical exploration and empirical approach Teacher granting status (value) to the

graphical register

From empirical to deductive reasoning Efforts to go beyond the graphical and numerical

registers

Reflects the didactical contract specific to the teacher

Communication Competencies:variable practices Incomplete documentation

Grading scale focused on the pragmatic function of tool Difficulty inferring the solving process and the control exercised

Detailed documentation Explicit instructions, consistent with assessment scale Refusal of expressions specific to the tool Students’ efforts in organizing their solution

Documentation that hides technical work Refusal of expressions specific to the tool Difficulty inferring the instrumentation process

Reflects the didactical contract specific to the teacher

Conclusions

Teachers used the CAS-calculator essentially as an epistemic tool complex problems and applications were not as

frequent as what could have been expected. Differences in students practices attributable to

teachers seem more the result of requirements for recording solutions (reflecting the

importance given to communication skills), methods and registers recognized as admissible for

solving problems.

Ongoing debates

How do we integrate in our math courses specificities of the communication protocol with the tool?

What form should be given to the communication of actions and thought processes? Math. languages and calculator expressions Public vs. private writings

What should we specify as requirements? Acceptable methods and registers Expected communication : equations, etc.

Ongoing debates

What role and what status should be given to heuristic exploration? Exploit the possibilities offered by the different registers

in problem solving Support by appropriate questioning the emergence of

rigour How can we encourage validation?

Through contexts and meaning Exploiting mathematical properties …

Intervention Competencies:variable reserve in using the tool

0%

20%

40%

60%

80%

100%

Alain Bernard Charlotte Diane

Limits and Integral

Integral only

Limits only

Intervention Competencies:variable reserve in using the tool

Restraint in using the tool disappears when outside of course content

One of Alain’s students

Intervention Competencies:expert use of the tool

One of Alain’s students

Intervention Competencies:expert use of the tool

Technical, numerical and structural control (variables and relations) Communication, more technical than mathematical

Transparency of technical work Algebraic control?

One of Alain’s students

Evaluation Competencies:exploration and empirical approach The parameter k caused problems for many students Some students got by through exploration

Rigour? Algebraic control? Approach potentially transferable to more

complex problems…

One of Diane’s students

Evaluation Competencies:from empirical to deductive

Graphical explorationGraphical exploration

Numerical exploration

Surprise Algebraicvalidation

« I always ask for answers in their exact form. Otherwise some

students will use graphs. »

Alain

One of Alain’s students

Communication Competencies:“fill in the blanks…”

Worked in what register? Difficult to infer the solving process

and the control exercised

Graph ? Equation? ?

dx = ???

?Private writings?

One of Alain’s students

Communication Competencies:detailed documentation

Solving with TI

Integration with TI

Solving without TI

One of Diane’s students

“ I tell them -Present commented solutions. I want to see more than just your calculations. I want complete phrases that describe your solving process…

Sometimes, I write: use mathematical syntax, not TI’s.”

Diane

Communication Competencies:What went wrong?

Refusal of expressions specific to the tool. Conflict with what is accepted by the graphing calculator. Impact on intervention competencies.

One of Charlotte’s students “I often tell them that I am not a TI.” Charlotte