Transcript Pineau_Caron

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 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
Level 4:
Reformulation
Level 3:
Organisation
Level 2:
Comprehension
Level 1:
Association
Communication
competencies
Evaluation
competencies
Intervention
competencies
Combining complementary models
Adapting a method of resolution
Establishing and justifying a property
Choosing a method of resolution
Identifying distinct cases
Interpreting data or results
Recognising an object
Associating a property
Applying a method
De Terssac (1996); Caron (2001)
Need or relevance of the calculator was also noted.
Results - Teachers
Little difference
 in the distribution of the complexity of tasks given by
teachers
100%
80%
Association
60%
Comprehension
40%
Organisation
Reformulation
20%
0%
Alain
Bernard
Charlotte
Diane
Exam
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
100%
80%
Essential
60%
Pertinent
40%
Unnecessary
20%
0%
Alain
Bernard
Charlotte
Diane
Exam
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
Translating
Justification
Communication
Evaluation
Explorations and
estimations
Applications and modeling
Evaluation
Communication
Intervention
Charlotte Purity of expression,
complementarities of
representations
Diane
Abstraction, rigour,
beauty, applicability
Results – Students, final exam
Little difference
 in the comprehension of problems or concepts
Students having made a comprehension error
100%
80%
Question 1
Question 2
60%
Question 3
40%
Question 4
Question 5
20%
0%
Alain
Bernard
Charlotte
Diane
Results – Students, final exam
Differences are a little more apparent
 in organisation and communication of results
Students having made an organisation error
70%
60%
Question 1
50%
Question 2
40%
Question 3
30%
Question 4
20%
Question 5
10%
0%
Alain
Bernard
Charlotte
Diane
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 :
k cos x  kx2  0  cos x  x 2  0
(k≠0)
Then find, analytically, k such that
x2
2
(
k
cos
x

kx
)dx  2

x1
In principle, the student is
allowed to use the calculator
without restriction
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.
“I mark off points when there is
an abusive use of the TI.”
 Expert (advanced) use of the tool
Alain
 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
100%
80%
60%
Limits and Integral
Integral only
Limits only
40%
20%
0%
Alain
Bernard
Charlotte
Diane
Intervention Competencies:
variable reserve in using the tool
One of Alain’s students
Restraint in using the tool disappears when
outside of course content
Intervention Competencies:
expert use of the tool
One of Alain’s students
Intervention Competencies:
expert use of the tool
One of Alain’s students
 Technical, numerical and structural control (variables and relations)
 Communication, more technical than mathematical


Transparency of technical work
Algebraic control?
Evaluation Competencies:
exploration and empirical approach
 The parameter k caused problems for many students
 Some students got by through exploration
One of Diane’s students
 Rigour? Algebraic control?
 Approach potentially transferable to more
complex problems…
Evaluation Competencies:
from empirical to deductive
Graphical exploration
Numerical exploration
Algebraic
validation
One of Alain’s students
Surprise
« I always ask for
answers in their exact
form. Otherwise some
students will use
graphs. »
Alain
Communication Competencies:
“fill in the blanks…”
Graph ? Equation?
?
?
dx = ???
Private writings?
One of Alain’s students
 Worked in what register?
 Difficult to infer the solving process
and the control exercised
Communication Competencies:
detailed documentation
“ I tell them Present commented
solutions. I want to
see more than just
Solving with TI your calculations. I
want complete
phrases that
describe your
Integration with TI
solving process…
Solving without TI
Sometimes, I write:
use mathematical
syntax, not TI’s.”
Diane
One of Diane’s students
Communication Competencies:
What went wrong?
One of Charlotte’s students
“I often tell them that I am not a TI.”
Charlotte
 Refusal of expressions specific to the tool.
 Conflict with what is accepted by the graphing calculator.
 Impact on intervention competencies.