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PROBLEM SOLVING AND OPEN PROBLEM IN TEACHERS’
TRAINING IN THE FRENCH AND MEXICAN MODES
Alain Kuzniak*, Bernard Parzysz*, Manuel Santos-Trigo**, Laurent Vivier*
* Laboratoire de Didactique André Revuz, Paris-Diderot University, France
** Center for Research and Advanced Studies, Cinvestav-Mexico
Mathematics education may differ in traditions and theoretical approaches
throughout countries, but it is generally acknowledged that problems and tasks play
an important role in learning. Two research teams, one in Mexico and the other in
France, decided to work together in giving mathematics teachers in both countries a
same task to solve during a training session. This paper summarizes the results of this
study. It shows that in the two countries the reference to different theoretical
frameworks results in putting emphasis on different but complementary aspects of
teaching.
INTRODUCTION
In different countries, mathematics teachers’ education and research programs in
math education exhibit differences both in principles and ways to implement them.
There are also different research traditions or paradigms to frame mathematics
education projects, but all of them recognize that mathematical problems or tasks
play an important role in fostering the development of teachers and students’
mathematical knowledge. Contrasting different systems and approaches, in this study
French and Mexican, may be of interest to identify features, common goals, and
differences associated with their traditions in education. Our research interest is to
analyze and document both the processes put into play by pre-service teachers while
working directly with problems and later the way they use these problems in their
teaching practices.
Specifically, the purpose of our common project is to investigate ways in which preservice and in-service teachers work on a series of problems in order to prepare,
organize, schedule and implement their lesson plans. Moreover, the development of
technologies in class is quickly changing teaching practices and it appears necessary
to take it into account. During their interaction with the tasks, pre-service teachers
were encouraged to use computational tools.
In the present contribution, we will focus on the initial step of the project in which a
common problem was used as a teachers’ training tool in both systems of education.
Our main interest was to see the implementation in contexts in which “solving a
problem” has not necessarily the same meaning. Introduced by Polya (1945), problem
solving constitutes the traditional framework for teachers’ training in Mexico. In
France, the traditional framework for teachers’ training relies much more on
Brousseau’s Theory of Didactical Situation (TDS, 1998).
Taking into account the differences in the referent frameworks in France and in
Mexico, we chose the following mathematical problem in both countries for teachers’
training to compare how it was used. Other problems were discussed, but not chosen
by the French team because of the difficulty to use them in secondary teaching.
The task: A square piece of paper ABCD with side l, has a
white front side and a blue back side. Corner A is folded
over point A' on diagonal AC. Where should point A' be
located on this diagonal (or: how far is A' from the folding
line) in order to have the total visible area half blue and
half white? (Based on Carlson & Bloom, 2005, p.71).
D
A
A’
C
B
For each country, we expose briefly the main features of the theoretical framework
used and describe and analyze teachers’ training scenarios based on this problem. We
start with teachers’ training in Mexico City, then with teachers’ training in Blois
(France). Furthermore we analyze the main elements of the comparison.
TEACHERS’ TRAINING SESSIONS IN MEXICO
Problem solving
Mathematical problem solving perspectives have framed and oriented the
development of multiple research programs and supported curriculum proposals in
mathematics education in the last four decades (Schoenfeld, 1985, NCTM, 2000). In
general, a problem-solving route to learn mathematics relies on an inquiry or
inquisitive approach to deal with mathematical ideas or problems (Santos-Trigo,
2007); however, there are multiple factors around its identity and practical
applications. “The patterns that form a problem-solving identity are complex,
involving varied motivational patterns, affective reactions and cognitive and social
engagement in different circumstances both within a given task and across tasks”
(Lesh & Zawojewski, 2007, p. 776).
In teachers’ education, it is important to address issues regarding the nature of
problems, the rationale for using them, and the goals that are intended to be achieved
during the problem solving sessions. Similarly, it is also important to elaborate on
ways to organize and develop the problem solving sessions to be held with preservice teachers. For that, it is important to discuss the dynamic of the problem
solving scenarios in which teachers and students will work on the problems.
The context
The problem was given on a sheet of paper to a group of 14 pre-service teachers.
Afterwards, they were asked to read and make sense of the problem individually, and
later they worked on the problem in pairs. Then, each pair had the opportunity to
present their approaches to the problems to the others. At this stage, all the
participants and the instructor could ask for concept explanation or clarification. At
the end of the session, the instructor encouraged the participants to contrast the
different models used to solve the problems. In general terms, the sessions’ dynamics
fostered the participants’ inquiring approach to the problems. For example, when the
participants introduced an idea or a problem representation, the instructor questioned
them and encouraged them to reflect on their ideas and other related concepts.
Similarly, when the participants ran out of ideas the instructor either oriented the
discussion (through questions) or asked other participants for suggestions to be
considered. The aim was to encourage all the participants to formulate and then go
deeper into the questions. In this perspective, the organization of the sessions is
consistent with the activities recommended by Mason and Johnston-Wilder (2006) to
have students participate during problem-solving discussions. In particular, the
authors identify four ways to organize students’ participation in developing and
discussing their mathematical knowledge:
Individual work allows learners to review, consolidate, and develop their facility, as well
as to reconstruct for themselves.
Work in pairs allows learners to try out ideas on each other before offering them to a
wider group; it also provides an opportunity for learners to consider something that has
happened or been said, and to generate more ideas about this [the problem] than an
individual is likely to produce when working alone.
Work in small groups allows a multitude of ideas to be generated, and also allows a large
task to be split up amongst several people; with discipline, small groups can provide a
forum for discussing ideas, modifying conjectures, and coming to a consensus with
supporting reasons and justifications.
Collective and plenary work allows everyone to hear about novel ideas and approaches,
and to see teachers or peers displaying their mathematical thinking (p. 52, italics in the
original).
It is important to mention that the pre-service teachers were taking a problem-solving
course when they worked on the task. The aim of this course was to use different
computational tools in problem solving activities. The participants worked on the task
during two sessions (2 hours each) held at the end of the semester. The goal of this
report is not to analyze individual approaches to the task but to focus on
characterizing mathematical behaviors exhibited by the group of participants as a
whole. The solution process is structured into episodes which distinguish key
principles associated with the finding of solutions (Santos-Trigo & Camacho; 2009).
First Episode: An overarching inquisitive principle
The core aspect in this approach is to conceptualize and examine the contents of the
problem (definition, theorem, etc.) in terms of questions or dilemmas that need to be
explored. An overarching principle that permeates the entire problem solving process
is that teachers and students should transform the problem statement into a set of
meaningful questions to be examined. Thus, the questions addressed by the group
included: What are the relevant properties of a square? What does it mean to fold a
vertex over to a point on the diagonal? And also questions to examine properties of
the involved figures: What properties does the folding line hold? What figures are
formed in blue and white colors? What happens to both areas when point A’ is close
to point A or close to point C? What type of triangle is formed (blue region) when
point A is reflected with respect to the folding line to determine A’? How can we
calculate the areas of the blue and white regions? In general, the participants
observed that for different positions of point A’ (i.e. when A’ is moved along the
diagonal) the area of one region increases while the other decreases. Then, they
agreed that there would be a position for point A’ where both areas were equal. At
this stage, two main approaches to solve the task appeared: one focusing on
representing the task algebraically and the other relying on constructing a dynamic
model of the problem.
Second Episode: An exploration principle
The figure provided in the problem statement (some labels were added) was used by
some participants to represent the triangle area as x2/2 and the other as l2−x2, which
led them to solve the equation l2−x2 = x2/2 in order to find the position of point A’.
The dynamic approach carried out by some participants required thinking of the
problem in terms of mathematical properties and software commands. This approach
allowed them to visualize and quantify the area variation of both regions as a result of
moving a point on the diagonal.
During the plenary discussion, the participants acknowledged that the two approaches
complement each other because they gave them an opportunity for examining the
task from various perspectives. While the algebraic method gives a general solution
to the problem, the dynamic approach offers not only the possibility of exploring
visually the area variation of both regions; but also to graphically interpret the
solution achieved algebraically (intersection point of curves).
Third Episode: The principle of extension and generalization
The participants asked whether the methods used to compare the area variations of
the square could be extended to the case of a rectangle. They recognized that in the
process of solving a problem or understanding a mathematical concept or idea it is
always important to reflect on the scope of the solution or applications of that
concept. In this context, the method used to explore the square case dynamically was
adjusted to deal with the rectangle case.
Fourth Episode: Reflection Principle
Participants generally recognized that a problem solving approach for learning
mathematics involves the construction of mathematical representations of concepts,
situations or problems, in order to find and explore mathematical relations. In this
process, any mathematical statement (problem, definition, statement, content) is
conceptualized as a starting point for teachers, students, or problem solvers in general
to search for different ways of solving and extending a problem. In this perspective,
the use of technological tools offered the participants the opportunity to explore some
parameters of the task from varied angles and as a consequence, they could relate
different ways of reasoning about the problem. For example, the construction of a
dynamic model requires from the problem solver a functional approach without
defining explicitly a function which is necessary with an algebraic approach.
However, both models complement each other in terms of making sense, visualizing,
and generalizing a set of relations emerging during the solution process.
Conclusion
In this approach, teachers tend to guide their practices through problem solving
principles that lead them to search and explore diverse ways to solve the task. This
particular process takes a long time for the participants to get appropriated and
exhibited in their regular practices. However, they clearly recognized it as powerful
approach to develop a deep comprehension of mathematical concepts and solving
problems. Thus, the emphasis in teachers’ education is to conceptualize the problem
solving process as an opportunity for them to look for variouss ways to approach the
tasks and to search for relations that emerge as a result of using different tools.
TWO TEACHERS’ TRAINING SESSIONS IN FRANCE
Problem situation, open problem and research narrative
The notion of “problem situation” appeared in France in the 1980s in Brousseau’s
TDS, which is based on a socio-constructivist conception of learning. A problem
situation is a learning situation aiming at fostering the acquisition of a new
knowledge by the students. Its setting up implies identifying previously erroneous or
weak conceptions among the students by analysing their errors. On this basis the
teacher conceives of and sets up a situation presenting some specific features,
namely: 1) be relevant for the cognitive objective aimed at; 2) have a meaning for the
student; 3) allow him/her to begin the search for a solution; 4) be rich (in terms of
mathematical and heuristic contents); 5) be possibly formulated within several
conceptual “frames” (Douady, 1986).
The notion of “open problem” was introduced at about the same time (Arsac et
al.1988, Arsac & Mante, 2007). In comparison with the problem situation, the aim of
an open problem is methodological rather than cognitive. The students are induced to
implement processes of a scientific type, i.e. experimenting, formulate conjectures,
test them and validate them. The problem must belong to a conceptual domain in
which students are somewhat familiar with, the wording (statement) has to be short
and induce neither a solution nor a solving method.
The French official curricula for junior high school (BOEN 2008) integrated recently
− though without naming them − open problem and problem situation, which refer to
two complementary sides of mathematical work:
– in the case of an open problem the question is to find a genuine, personal solution,
with one’s own means, the general solution being out of reach;
– in the case of a problem situation the question is, starting from a specific problem,
to elaborate a more general knowledge (concept, process…) which is intended to
be institutionalised, socially acknowledged and mastered by all.
The notion of “research narrative”, which is explicitly linked with those of open
problem and problem situation, appeared in France some twenty years ago [Bonafé et
al. 2002]. It involves asking the student to write an account of the thought processes
he/she has undertaken in order to solve a given problem, pointing out his/her ideas,
successes, failures, etc. The features of the problem are the same as for an open
problem, but it has often several questions and the student must be able to start a
research, test his/her results and validate them. And, if possible, different solutions
need to be considered.
Implementation in high school teachers’ training in Blois
Sessions on the folded square problem were organized with two groups of pre-service
teachers: a “standard group” included 18 trainees in their first year of professional
training and a “special group” included 6 trainees who had not been appointed to a
permanent post at the end of the normal training year.
Besides this different context, the tasks given to the standard and special groups are
quite the same. A material bi-colored square folded along its diagonal is used to
explain the problem. The trainees have to compare both areas using different tools:
software, spreadsheet, calculator, paper and pencil. They were asked to write a
research narrative of their exploration and solution of the problem with the specific
tool used and then they to prepare a classroom session for their own students, to make
explicit their mathematical aims and the material they would use with their students.
The standard group
The described session is a 3 hour lecture, part of a course focused on theoretical
issues aiming at initiating the trainees to Brousseau's theory and the pedagogical use
of open problems. They were also initiated to pedagogical uses of Dynamic
Geometry Software (DGS) and spreadsheet. They generally mastered very well DGS
but not spreadsheet, which is not a common tool for mathematics students at
university. No particular mathematical framework was privileged and students had to
use some specific artefacts to solve the problem. The 18 trainees were split into 5
teams: 2 teams worked with Geogebra, 1 team with a spreadsheet, 1 team with a
hand-held calculator and 1 team with paper and pencil. When they had completed
work, all the teams presented it to the others. After the presentations, they were asked
to prepare a classroom session for their secondary students.
Team with DGS. The task is very open and dependent on knowledge about the
software. Normally, the problem must be handled a bit more in the geometric frame
than with other tools. Students had indeed a very good knowledge of the software and
they immediately solved the problem by linking it to the curriculum for grade 9. We
can follow the methods from the narratives:
- Drawing the folded square, linked with the area of both parts. The dragging point
is point A' on the diagonal.
- Drawing two curves in a Cartesian system (one representing the area of the blue
triangle and the other the area of the white hexagon). The link between the point
and the area is made by using the track mode (not locus). The intersection point is
reached by approximation.
Both teams indicated the objective of exploring the problem as an example of
modelling. They retained that modelling and experimental approach in official
documents must rely on the computer. These teams wanted to support students by
giving them a rich environment favouring an exploration of the problem. They gave
the same problem with the same instruction as that received during the course (for
them the use of digital tools is not a problem for the high school students). In a
second stage, they intended to work on the difference between exact and approximate
values: the exact value controls the different results depending on the square size.
Team with calculator. The team focused on an approximate resolution of the
equation of two curves and followed this method: free hand drawing modelling the
situation, where x is a side of the right angle in the folded triangle; observation that it
is impossible to find an exact geometrical construction of the point equalling the two
areas (hence they decided to approximate the result with a calculator); plotting curves
representing the areas in a particular case; approximating the result with a table of
values and using the calculator zoom; solving algebraically a quadratic equation to
get the exact value.
This team’s objectives were using the calculator and writing out equations. They
therefore used the activity to improve their students’ knowledge of calculator. They
retained the instrument for the course, using it to work graphically on the variation of
two functions and the intersection points of curves.
Team with spreadsheet. This team started the research with a bi-colour paper and
modelling. Students chose the height of the blue triangle (AA’/2) as a variable. After
discussion, they agreed to make the values vary on a range equivalent to the diagonal.
Then they used a spreadsheet to increment the variable with a given step and
calculated the areas of the blue triangle and white hexagon, and their ratio. An
approximate value was reached when this ratio was 1.
To determine objectives for the class, members of the team discussed sharply and did
not agree, some wishing to continue with the spreadsheet, others finding no interest in
this activity with the spreadsheet.
Team paper-and-pencil. This team changed its name into Paper-and-Brain Team.
They immediately solved the problem by using two general equations.
Only junior high school teachers participated in this team (grades 7 and 8) and they
concluded that the activity was too difficult for their students. They made a proposal
for grade 10 students based on the resolution of a quadratic equation. They remained
in a paper-and-pencil environment and did not plan to use software (the institutional
pressure for using technologies is weaker on junior than on senior high school
teachers). They spent time writing instructions for the students with no reference to
an actual folding.
The special group
Each trainee received by mail an envelope containing a square sheet of paper and the
task they had to perform. The trainees were isolated from each other and the
exchanges took place only during the two sessions. They had to undertake a research
of the different possible solving strategies, choosing two environments (paper and
pencil; calculator; DGS; spreadsheet) and to write a research narrative. They also had
to write a complete course for one of their secondary classes and really teach this
course. They had to write down their work and present it to the group (the first
session was mainly devoted to mathematical research and the second to the lesson).
All trainees used a paper-and-pencil environment, which is not surprising. Calculator
was only used by one trainee (approximate values were obtained with GeoGebra by
visual adjustment of the areas, sometimes with the slider tool, in order to get better
values); spreadsheets were not used at all. All trainees used the GeoGebra software
for modelling the folding; no other dynamic geometry software was used.
Only the two trainees who taught in senior high school used the software for solving
the problem by considering intersections of function graphs. It seems that the syllabus
plays a great role for these two trainees, since the senior high school curriculum
emphasises functions and graphs. This link between teacher’s research and topics he
teaches at school also appeared with a grade 8 teacher who focused his research on
geometrical proofs, which are of great importance at grade 8.
Unsurprisingly, the tasks given to their students confirm this last point, since teachers
care a lot about the subjects they have to teach. In spite of this obvious fact, a detail
has to be mentioned: only the two senior high school teachers proposed a comparison
of the areas to their students; all of the others changed the question into “when are the
areas equal?”. It is a fact that the comparison of functions by means of graphic
strategies is an important subject at grade 10.
Conclusion
In both French groups, the grade, or the teaching institution, in which a teacher
teaches seems to influence not only the transposition of the problem for the students,
but also his/her research when he/she solves a problem for himself. And this could be
observed whether trainees knew that they had to set up a course for their students
(special group) or not (standard group).
The software chosen in the special group was still GeoGebra, one could think that
there was in fact no choice of the software. This is linked with software currently
used: the standard group is richer for DGS and especially GeoGebra. We explain this
in this way: GeoGebra is a free software, both a dynamic and analytic geometry
software which can even replace a calculator, and for that reason the training and
school institutions possibly put the stress on it. But this does not explain why
spreadsheet was left aside (maybe because of a lack of teachers’ competences?).
We also noted the importance of the artefact used during the session on the trainees’
school planning: they stayed very close to the experience they lived during the
training. This argues for repeating this kind of training sessions but changing the
tools used by the teams.
COMPARISON OF FRENCH AND MEXICAN SESSIONS
There are a lot of similarities between teachers’ training in France and Mexico.
Obviously, we can first mention structural organization similarities: number of
trainees in a course, duration of sessions, trainees teaching in a real class. There are
also some pedagogical organization similarities, such as sharing courses into
individual work, work by groups (even by pairs) and plenary sessions. These
similarities show that the way to plan teachers’ training is quite equivalent from a
pedagogical point of view.
The emphasis on the use of new technologies is another similarity. Obviously it relies
on didactics and the interest for teaching mathematics, but we could also see an
external pressure of society and more specifically curriculum demands. Nevertheless,
spreadsheet seems to be left out in France which is not possible in Mexico because of
the problem solving framework which demands to explore all solving possibilities.
The main difference is clearly the role of real secondary classes. In France it appears
that the level at which the trainees teach has a great influence on their behaviours
and, in teachers’ training, teaching at secondary level is one of the main aims
(virtually for the standard group and really for the special group). In Mexico, real
classes seem to be out of the training since there is no reference to them in problem
solving sessions. Obviously this is not the case because one of the aims of the
Mexican problem solving sessions is that teachers would be able to do the same
activities with their students. But this is not explicitly worked, contrary to French
teachers’ training.
Another difference could be related to the difference in didactic frameworks
reference. In France the trainees’ work began with a material square whereas in
Mexico the situation was modelled under the form of a geometrical diagram.
CONCLUSION
These examples from two different countries with different cultures and traditions
about math education show that in each of them the stress is put on different points:
In Mexico solving strategies and associated scenarios or episodes, in France the
influence of tools and implementation of the problem situation in a classroom. Both
aspects are important for the training of math teachers: On the one hand they must be
prepared to understand, assess and support any solution proposed by their students,
and on the other side they need to identify the contribution of the situation to the
students’ knowledge and consider how to bring it into play in a real class. We believe
that the development of such cross studies can have an influence on the training of
pre- and in-service teachers and finally help them to develop a more holistic view of
teaching.
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