Exercising Control: Didactical Influences Kathleen Pineau Maître d’enseignement en mathématiques École de technologie supérieure France Caron Professeure au département de didactique Université de Montréal

Context École de technologie supérieure (ÉTS) : “Engineering for Industry” Our undergraduate students come from college technical programs CAS-calculator mandatory in 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 and their 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 2 nd 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 objectApplying 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

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

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 tasksTargeted 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

Results – Students, final exam Differences are a little more apparent in organisation and communication of results

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 x 2 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 x 1 and x 2 of the equation : (k≠0) Then find, analytically, k such that 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

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 exploration Numerical exploration Surprise Algebraic validation « 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