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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” - PowerPoint PPT Presentation
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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