RESEARCH
THE MATHEMATICS CLASSROOM
A Rich Academic Setting for Teachers’ Professional Development
TENOCH ESAÚ CEDILLO ÁVALO
Tenoch Esaú Cedillo Ávalos is a professor/researcher in the academic area of “Teaching and Learning in Mathematics,
Science and Art” at Universidad Pedagógica Nacional, Carretera al Ajusco Number 24, Colonia Héroes de Padierna, Tlalpan,
Mexico DF, CP 14200. E-MAIL: [email protected]
Abstract:
This article reports on an evaluation of the results of a development program for secondary school mathematics
teachers. Twenty classroom sessions with students in the second year of secondary school were prepared and
videotaped. During the sessions, crucial topics on the curriculum were addressed and the students were given a
series of non-routine problem situations to solve. The solutions required elementary mathematics and creativity,
rather than advanced knowledge. Subsequently, ten sessions with teachers were organized and filmed. After
solving the same problems as the students, the teachers watched the videos of the student sessions. The teachers
then discussed prepared questions to guide their reflections on their practice, based on the problem-solving they
and the students had done. The analysis suggests that relevant changes occurred in the teachers’ conceptions of
teaching and learning mathematics, along with improvement in their knowledge of mathematics.
Keywords: mathematics, teaching and learning strategies, secondary education, Mexico.
Introduction
One of the central goals of this study was to show the degree that current theories on teaching and
learning mathematics can be put into practice. Attaining such a goal requires going beyond the
elaboration of novel materials and the proposal of sensible recommendations for teaching and learning
mathematics at school; in addition, we believe that an essential factor is demonstrating, through data
obtained in real classroom situations, the role that teachers must play so that their students learn how to
construct conceptual networks based on solid mathematical knowledge. We frequently hear that
students need to construct their own knowledge, that they should not be receiving knowledge
previously “digested” by teachers, and that teachers should modify the roles they play in the classroom.
The problem that we decided to address required us to create an atmosphere of learning, with
didactic strategies that would indicate how to reach our objectives. To do so, we adopted the strategy of
encouraging students to generate knowledge in a manner similar to mathematicians who are producing
new knowledge. In brief, we assume that in the process of constructing a new result, mathematicians
first address an unsolved problem, then explore the relations that exist in the problem, analyze
particular cases, formulate conjectures to attempt to generalize their observations, and lastly, search for
irrefutable arguments that sustain their conjectures.
Using that strategy as a frame of reference, we designed a series of learning activities based on
solving nontrivial problems that would represent a challenge for students while motivating their
intellectual curiosity. We wanted to make them “feel” capable of overcoming difficulties if they made a
concentrated effort, supported by appropriate teacher guidance.
During the past thirty years, research on learning mathematics has emphasized the need to generate
forms of teaching, paradigms for teacher education, study plans for basic education, and new evaluation
procedures (Kilpatrick, 1992). The results of that research have led to a reformulation of the
mathematics curriculum, which has presented new demands for teaching work. Theoretical postures
based on social constructivism have also had an impact on educational programs. Such theories
conceive knowledge as a product of the intellectual work of communities of creative individuals—the
idea behind courses and materials that suggest teachers abandon their role as transmitters of concepts,
basic facts, and skills, to become tutors in developing their students’ mathematical thinking (Cobb,
Word and Yackel, 1990). The new educational paradigms demand that teachers make a significant
change in their mathematical knowledge and in their conceptions of teaching and learning mathematics.
Such theoretical postures are based on the premise that students enter the classroom with their own
ideas, and that teachers must be prepared to provide new experiences that induce students to compile
and analyze information—information that will help student to confirm or refute the conjectures they
generate through an intuitive approach to the formal knowledge of mathematics.
Therefore, teachers in their daily practice must show that they are convinced that students are not
“containers waiting to be filled”, and that they consider students to be intellectually creative individuals
capable of asking nontrivial questions, solving problems, and constructing theories and reasonable
knowledge. Satisfying such requirements requires teachers to eliminate—from the textbook as well as
from their own person—their roles as intellectual authorities in the classroom; they must deposit that
authority in rigorous arguments that they produce along with their students (Thompson, 1992). Studies
have focused on the role that teachers play in educational improvement (Linares and Krainer, 2006). A
specific case for the past fifteen years in Mexico has been the reform of the secondary school
mathematics curriculum, which has required teachers to develop didactic focuses and teaching methods
centered on student learning.
This article reports on work that we have carried out in Mexico with classroom teachers in
secondary schools during the past four years. We begin the article with a discussion of our previous
experience with teachers, which was an important influence on our selection of the method and
instruments used. In the second section, we describe and discuss the main elements of the program of
professional development that provide the context for carrying out the research reported here: the
questions that guide the work, the individual profiles of the project participants, and the project and
teaching/learning activities undertaken. The third section addresses the findings we have obtained to
date. Lastly, we present a series of indications based on the work we carried out, along with the studies
we believe should be done in the future.
Background
Our initial work with teachers was part of a professional development program carried out for four
years in one hundred Mexican secondary schools (Cedillo, 2003). From the beginning of the project,
our data confirmed that one of the greatest challenges would be for the teachers to have a genuine
feeling that important reasons were behind the need to change (Cobb and Mc Clain, 2001; Cedillo and
Kieran, 2003). The information compiled during the training sessions showed that the factor of most
influence in convincing the teachers to participate actively in the project was the opportunity to observe
what they called “unexpected mathematical achievement in students”—events which occurred during
the classes conducted by the project instructors (Cedillo, 2003).
The teaching style adopted by the instructors consisted of orienting classroom events to the
students’ reasoning. On adopting this principle we assumed that learning is a construction process that
is shared socially by the students and teacher. The process requires the teacher to adopt a role that we
describe as a multiple chess game, in which the teacher is the expert who simultaneously plays against
thirty players who can talk to each other before making a move. The expert makes the first move
(proposes the activity) and must be prepared to receive up to thirty different responses; in his second
move, the expert (teacher) must give specific responses that imply a challenge for each player, and so
on. An important difference between a conventional chess game and the version we use as a metaphor
is that in our game, the teacher must ensure that over time, the students win a fair match.
This metaphor implies viewing mathematics teaching as an active confrontation between students
and mathematical challenges. To adopt such a posture, the students’ and teacher’s previous knowledge
must be considered:
[…] It is unproductive to ignore recent knowledge and the fresh ideas of students; it is also unproductive to
ignore the knowledge produced by generations of mathematicians. Therefore, teaching mathematics is a
situation that presents a continual dilemma for teachers: on one hand, they need to begin where the students
are, and on the other hand, they try to help students to develop an understanding of concepts that are part of
a body of historically constructed mathematical knowledge (Krainer, 2004:87).
The empirical evidence we compiled in previous studies indicates that teachers attach almost no value
to successful teaching experiments made at other locations, in school contexts different from their own.
This finding agrees with the reports of Cobb and Mc Clain (2001). Our data show, however, that
teachers attach high value to the experience of holding meetings with their peers in other schools. We
found that if they learn that other teachers are successful in having their students learn, then they are
more likely to modify their practices and increase their mathematical knowledge. The data we collected
indicate that the process of having teachers change is quite lengthy. Initial evidence of change was
observed after one year of ongoing work, and the most significant transformations appeared after three
years (Cedillo, 2003).
The Current Study
The research we present was carried out during the implementation of the program called “Enseñanza
de las Matemáticas” (“Teaching Mathematics”). The program was based on the hypothesis that an
increase in the quality of teaching can have positive effects on students’ learning. Another premise of
the program was the assumption that discovering the students’ forms of mathematical reasoning would
provide the teacher with a more solid foundation for instruction and professional development
(Carpenter and Fennema, 1989). The program was financed by the Inter-American Development Bank
and supported by Mexico’s Secretariat of Public Education, the Latin American Institute of
Educational Communication, and the National Pedagogical University.
The results we report here refer to the effects of the methodological strategy employed in the
professional development program; the data collection method we used was based on a critical
observation of the sessions in which teachers were given opportunities to compare their form of
reasoning with the reasoning of their students. The participating teachers and students were asked to
solve the same mathematical problems, in a separate form. Further detail is provided below.
Research Questions
The following questions guided the study:
•
•
If teachers compare their forms of reasoning with those of their students, what are the effects on
the conceptions of learning and teaching mathematics?
If teachers observe the classes in which students solve the same mathematical problems that the
teachers have already addressed, what are the effects on the teachers’ mathematical knowledge?
Method
The basis of working with the students was a didactic strategy based on conducting class activities by
making adjustments along the way, following the students’ forms of reasoning. In the teachers’ work
sessions, the teachers were asked to focus their attention on the students’ progress in order to orient
their reflections on their teaching practice. We expected that such an approach would eventually give
rise to favorable changes in teachers’ mathematical and pedagogical knowledge.
This didactic model was used during twenty class sessions with students enrolled in the second year
of public secondary school; all sessions were videotaped to permit usage at as many locations as
necessary during the following phases of the program of professional development. The workshops
with the teachers were also filmed; the videos were the main source of data for the research project. We
believe that these materials will be a valuable resource for later studies involving classroom teachers and
as didactic material for teacher educators.
Classroom Sessions
To determine the mathematical content of the classroom sessions, a review was made of the research
literature and the official mathematics curriculum in Latin America. The information obtained from this
analysis supported our selection of the topics in the mathematics curriculums under study; some topics
are characterized by the difficulties they represent for teaching, and others by their relevance in the
further study of mathematics.
Another necessary consideration was the financial restrictions and the assigned time for the project.
As a result, we selected the curriculum areas and topics described below:
Arithmetic
Algebra
Geometry
Multiples and divisors
Numerical patterns and
generalizations
Measurement and similar
triangles
Greatest common divisor
Games and algebraic regularities
Areas and the Pythagorean
theorem
Least common multiple
First-degree equations
Measurement and
trigonometric ratios
Reading and construction of
Cartesian product graphs
Two fifty-minute sessions were devoted to dealing with each topic. The classroom sessions were held in
two public secondary schools in Mexico City that voluntarily participated in the project. The directors
of each school allowed us to participate in the school provided the project did not disturb daily
activities; thus we were required to work with groups during their normal class schedule. This
unexpected situation was very valuable for the project because it gave natural encouragement to the
spontaneity of students’ work. Another advantage of the situation is that the students who took part in
the classes were not selected in any form by the school’s director or teachers, or by the team
responsible for the project.
The population consisted of the school groups available during the work sessions, which were
programmed as a function of the schedules of the technical team that professionally recorded the
sessions. The teachers who conducted the classroom sessions are not part of the school’s faculty. Three
researchers from the academic team in charge of the project played the role as teacher. Their
participation was necessary in order to carry out a pilot phase, during which our methods were refined
at intensive meetings with the project leaders. The pilot stage was completed in schools that did not
participate in the main study.
The classroom sessions followed the students’ responses to a series of articulated problems, and the
teacher’s actions were guided by the previously described metaphor of the multiple chess. This
metaphor implies a method similar to the teaching cycle proposed by Simon (1995) and to the concept
of instructional sequences developed by Cobb and McClain (2001). Simon describes the teaching cycle
as a process in which:
[…] At all times the teacher has a pedagogical agenda and therefore a feeling of direction. The agenda,
however, is subject to continual modification in the act of teaching […] This manner of acting in the
classroom includes a purpose and a permanent attitude of openness to the possibilities offered by students’
reactions to the proposed activities (Cobb and McClain, 2001:215).
We wish to point out that this didactic posture in no way suggests that activities are carried out without
clear objectives; for the duration of the class, the teacher must have a main goal in mind and the means
to attain it.
Another methodological aspect taken into account was the selection of mathematical problems,
which needed to be sufficiently complicated for the teachers as well as for the students. We selected
problems that would allow us to approach the topics from the curriculum in unconventional ways.
For example, in the case of arithmetic, the topic of divisibility was chosen. According to the
documentary review we carried out, the starting point for addressing this topic in Latin America is a
formal definition of divisors, multiples and prime numbers, followed by a series of exercises to
reinforce students’ understanding of the definitions. To present an alternate approach, we decided to
address the topic of multiples and divisors by using a set of questions that would encourage students to
do the following: to face situations that would challenge their mathematical curiosity and intellectual
capacity and lead them to “discover” the relations among the involved concepts; to propose
generalizations based on the observation of numerical regularities; and lastly, to express and justify
these generalizations by using the algebraic code. It is important to mention that the students were
allowed to work in cooperative form in small groups if they so desired.
Proposed Mathematical Problems
Described below are the problems we used for each topic. We shall address some in detail in order to
express our didactic approach with greater precision. The other problems are presented in a more
general manner.
Multiples and Divisors
The basis for addressing these topics was the responses the students gave to the following questions
and activities:
•
•
•
•
•
Can you find numbers that have exactly two divisors? In the next four minutes make as long a list as
possible of those numbers. The challenge is for your list to include only numbers that meet that condition.
Can you find numbers that have exactly three divisors? Can you propose a rule that allows us to construct
several numbers that have exactly three divisors? Is there more than one rule that allows us to do this?
Can you find numbers that have exactly four divisors? Can you propose a rule that allows us to construct
several numbers that have exactly four divisors? Is there more than one rule that allows us to do this?
Can you find numbers that have exactly n divisors? Can you show us a rule that allows us to construct
several numbers that have exactly n divisors? Is there more than one rule that allows us to do this?
Can you find a natural number other than 1 that you cannot factor by using exclusively prime numbers as
factors?
Greatest Common Divisor
This concept was addressed by means of a well-known problem that consists of three containers (A, B
and C); no container is graduated, and the only facts known are the capacities of individual containers A
and B. The capacity of C is greater than the other two added together. The problem is to find a general
rule that will allow us to determine the amounts in whole liters that can be obtained by pouring liquid
from container A to B, based on the knowledge of the capacity of each. We used this problem so that
the students could recreate the concept of the greatest common divisor, and we encouraged them to
find numerical regularities that would eventually allow them to propose general solutions. Each team of
students was supplied with pitchers and water that could be used as considered necessary. The problem
was extended to permit an intuitive approach to solving Diophantine equations. The students were
asked the following questions in the sequence listed below.
•
•
•
•
•
You have three pitchers, A, B and C. Pitcher A holds three liters, and pitcher B holds 5 liters. Pitcher C can
be used to hold the amount of liquid you want or to pour liquid to pitcher A or B. Can you obtain 4 liters
by pouring liquid from pitcher A to pitcher B? Can you find a way to record the sequence of the
movements you make from one pitcher to another to convince us that your answer is correct?
Take into consideration the same conditions as in the above problem, but now pitcher A holds 2 liters and
pitcher B holds 4 liters. Can you obtain 1 liter by pouring liquid from pitcher A to pitcher B? Can you find a
way to record the sequence of the movements you make from one pitcher to another to convince us that
your answer is correct?
Now pitcher A holds 6 liters and pitcher B holds 9 liters. Can you obtain 1 liter by pouring liquid from one
pitcher to another? Can you find a way to record the sequence of the movements you make from one
pitcher to another to convince us that your answer is correct? Can you make a list with the different
amounts of liters you can obtain by pouring liquid from the 6-liter pitcher to the 9-liter pitcher? Can you
propose a method to know the numbers of whole liters you can obtain by pouring liquid from the 6-liter
pitcher to the 9-liter pitcher?
Now pitcher A holds 7 liters and pitcher B holds 10 liters. Can you obtain 1 liter by pouring liquid from one
pitcher to another? Can you find a way to record the sequence of the movements you make from one
pitcher to another to convince us that your answer is correct? Can you make a list with the different
amounts of liters you can obtain by pouring liquid from the 7-liter pitcher to the 10-liter pitcher? Can you
propose a method to know the numbers of whole liters you can obtain by pouring liquid from the 7-liter
pitcher to the 10-liter pitcher?
Look carefully at the lists you made to show the different amounts of liters you can obtain by pouring
liquid from one pitcher to another. Do you see any regularity in these numbers? Could you find a general
strategy to discover if you can obtain a certain amount of liquid if you know how much each pitcher holds?
Least Common Multiple
This topic was addressed by using the well-known problem of gears. The version we proposed was to
consider two gears like those shown in Figure 1. The number of teeth on each gear can be different.
The question posed to the students was to find the least number of turns each gear would make before
coinciding again at the point where they began to turn. This problem was extended to the case of more
than two gears and to problems of the type of “A friend of mine bought apples and oranges. He paid 5
pesos for each apple and 3 pesos for each orange. He paid the same amount for the apples and the
oranges. What is the least amount of each fruit he bought?”
FIGURE 1
Gear Problem
Giro
GIRAS
A
Pre-algebra and Algebra
We included four topics: two induce the need to employ the algebraic code to express and justify
generalizations (numerical patterns, games and algebraic regularities). The sessions on numerical
patterns and generalization consisted of the students’ finding a function to comply with the numerical
sequence suggested by a diagram like that of Figure 2.
FIGURE 2
Problem of Cubes
To introduce the activity, the students were asked questions like the following: How many cubes will
the fourth figure have? How many cubes will the tenth figure have? If a figure in this sequence has 225
cubes, what is its place in the sequence? Can you express in algebraic language the rule you used to
answer the above questions?
Other problems used in this section were the type of “think of a number”. For example: Think of a
whole number between 0 and 10. Add 10 to the number you thought of and remember the result. Now
subtract 10 from the number you thought of and remember the result. Add the two results. Can I guess
the final result you obtained? It’s 20! Am I correct? How do I know? Can I guess the number you are
thinking of if it is greater than 10? Can I guess the number you are thinking of if it is less than 0? Can I
guess it if it is not a whole number? Why?
The sessions on games and algebraic regularities were addressed by means of a problem involving
Hanoi towers (figure 3). The goal was for students to find the rule that governs this game: function f(n)
= 2n-1. To solve the problem, the students were given materials that could be manipulated and software
that simulates the movements of disks from one tower to another. Students were allowed to use either
or both of these resources.
FIGURE 3
Problem of Hanoi Towers
The game began by asking the students to move three disks from one tower. The rules of the game
stipulate that a disk can never be on top of a smaller disk, and that the best strategy uses the fewest
movements. The students were asked the following questions: What is the minimum number of
movements needed to pass the three disks from one tower to the other? If the disks were on tower A,
toward which tower would you make the first movement? The difficulty of the game increased as the
number of disks increased.
The topic of first-degree equations was addressed by asking the students to find “the lost number”
in a given equation. We consider it important to indicate that the students had received no instruction
on the conventional methods of solving equations when the session was held. The activity’s difficulty
was increased gradually until reaching the equations that contained parentheses and slashes as signs of
grouping. Another type of activity used in this session was to ask the students to write a problem that
could be solved by a given equation and vice versa.
The topic of reading and constructing Cartesian product graphs was addressed by asking the
students to create stories that corresponded to the information presented in graphs of time and
position and vice versa (such as the graph of Figure 4). Another type of activity used in these sessions
was to find the rule of correspondence of a function based on the information given on a linear graph.
FIGURE 4
Time-position Graph
Automóvil
Distancia (Km)
500
400
300
200
100
0
08:00
09:00
10:00
11:00
12:00
Tiempo (horas)
Geometry
13:00
14:00
The geometry sessions were developed around the concept of measurement. The central notions
addressed were the similarity of triangles, the Pythagorean theorem, and trigonometric ratios in right
triangles as tools of measurement. Since these topics are not included in the study plan for the second
year, we were sure that the students would first see them in the project sessions.
The students were introduced to these topics by using a research strategy that encouraged the
students to discover the central notions involved, in a form similar to that employed in a natural science
laboratory. To support this type of activity, the classroom was equipped with usable geoplanes,
software that simulates geoplanes, and software with dynamic geometry and scientific calculators. For
example, to address the topic of trigonometric ratios, the students were asked to draw a right triangle,
given the measurements of its inside angles; then they calculated the quotients by looking at the lengths
of the triangle’s sides and comparing the results with their classmates. The dynamic geometry software
allowed them to carry out this process quickly and explore their conjectures by using as many individual
cases as they found necessary to formulate their conclusions.
The use of scientific calculators allowed the students to confirm the results they had obtained and to
learn the technical names of the trigonometric ratios they were using. Subsequently, they were asked to
give mathematical arguments to explain the numerical regularities they had observed. To finalize the
session, the students looked at problems that could be solved by measuring the sides or angles of a
right triangle. A similar method was used to study the topics of similar triangles and the Pythagorean
theorem.
Workshops with Teachers
Ten five-hour workshops for teachers were held once a month from February to November, 2005.
Classroom mathematics teachers participated in voluntary form and allowed the sessions to be
videotaped. As we mentioned above, the same activities and problems that had been used with the
students were carried out with the teachers. In the workshops, the teachers did mathematical
explorations based on written instructions, and made the decision to work in small teams.
The workshops were structured as follows: The teachers had one hour to address the problems
proposed for that session. The following two hours were assigned to watching the videos of the two
student sessions; before seeing the videos, the teachers were asked if they believed second-year students
would be able to solve the problems the teachers had just seen. If the answer was yes, then they were
asked to respond to what degree, how, and why. The final two hours were used for the teachers to
discuss what they did in the context of what they had seen in the videos. To guide their attention as
they watched the videos, the teachers were asked to take notes on the following points:
•
•
•
•
•
Unexpected strategies the students used
Factors of most influence in the students’ success in solving the problems
Factors that caused student confusion or the production of incorrect answers
Forms of student reasoning that differed from reasoning the teacher would have used to face the
same problem
Other classroom events considered important
The teachers who participated in the workshops work in public secondary schools in Mexico City. The
teacher with the most experience in this group had taught mathematics for twenty years, and the
teacher with the least experience had taught for two years. Two of these teachers worked in the schools
where the student sessions were held. This fact added a factor of credibility to the classroom events
observed in the videos.
Results
Once we had analyzed the data from the videos and the activity worksheets filled out by the teachers,
we found that in general, their remarks showed interrelated reflections on the mathematical content, the
pedagogical content, and student skills. Taking into consideration these restrictions, we divided the
results into the following sections: a) the teachers’ expectations regarding the students’ abilities, and b)
their reflections on their knowledge of mathematics and their teaching practice.
The Teachers’ Expectations
Having both the teachers and students work on the same problems gave us important information on
the teachers’ expectations in terms of student skills. The compiled data indicate that the teachers center
their evaluation of student abilities to learn on their own skills as teachers. As mentioned above, before
the teachers saw the video of the student sessions, they were asked if they believed the students would
be able to solve the problems. The first time the teachers answered this question, they emphasized that
they did not believe the students could solve them, especially the problems that required formulating a
generalization and producing sound arguments for acceptance or rejection. For example, in the case of
the topic of divisibility, the teachers sustained that for the students to reach such an ambitious goal,
they would first have to be taught the rules of divisibility, a method to obtain prime numbers, and the
way to use algebra to work with generalized numbers. The teachers did not believe the students could
feasibly answer questions requiring algebra because their knowledge in this respect was still incipient.
They stated that they were sure their opinion would be confirmed by the videos and that the only
variable would be the degree the videos would confirm their hypothesis.
Once the teachers saw how students were able to work with the problems by using rudimentary
mathematical tools, they began to try to determine why their initial perception of student abilities was
so distant from what they were able to do. For example, the teachers were surprised that the students
could analyze the factors of a given number by using the algebraic code. On discussing this matter, the
teachers referred to cases like that of the student who used the blackboard to explain the discovery: “If
we say that p is a primer number… p4 has exactly five divisors, because p4÷1=p4; p4÷p=p3; p4÷p2=p2;
p4÷p3=p and p4÷p4=1…” The student continued his explanation: “First we tried many numbers like this
and it worked. Then a person on the team did it with letters and we realized it was easier with letters
than with numbers.”
This type of evidence led the teachers to search for plausible explanations; they reached the
conclusion that the way the instructor guided the arithmetic reasoning gave the students selfconfidence: “This made the students start to produce ideas, like analyzing individual cases that
ultimately let them see possible generalizations”. One of the teachers emphasized the following:
[…] You see, the teacher [the project instructor] never told any student no. He was always patient […] If a
student said something wrong, the teacher asked the group if they agreed with the student’s response […] He
acted the same way even if the student was giving a brilliant idea. In this way, the teacher was giving them
many opportunities to correct or confirm their responses on their own […] This teaching attitude gives
students an opportunity to learn more […] In this manner, the students are willing to provide arguments to
reject or accept solutions.
The teachers concluded that they had believed the students would not be able to solve the problems
because they were basing their opinion on their own teaching experience and resources. They stated
that they were sure the didactic sequences they used were good, and that they believed that if the
students approached the problems in the way we proposed, it would be almost impossible for them to
solve them successfully. The following extract from a teacher’s remarks illustrates the above:
The way we teach is what made us think that the students would not be able to give the correct responses to
such complex questions […] When I teach these topics, I start by giving the students the necessary procedures
and definitions and concepts about the content of what I want them to do […] Then I give them a good
number of examples to try to reaffirm their understanding. The rules of divisibility are needed in order for
them to become good at finding factors, so I teach them these rules […] When you said that you would give
the students these exercises the same way you did with us, we found it very difficult to believe that the
students would be able to do it well.
We expected that the teachers’ opinions about the students’ abilities would change after the first
workshop session, but they did not. The change we observed was that the teachers took more time to
express what they believed would occur in class with students. For example, in the class about the
“problem with the pitchers”, they doubted the students could provide the solution. We pressured the
teachers to give a more precise response, and they finally agreed that:
[…] We did not believe the students would be able to solve the problem since they knew very little about the
maximum common divisor, and the activity required not only handling the definition of this concept, but also
keeping a clear log of the changes in liquid from one pitcher to another, and finding relations between the
numbers involved in the problem.
Once again, the students’ achievements were in strong contrast with the teachers’ predictions.
Near the end of the study, nine of the twenty-five teachers who participated in the workshops
reported that they had tried to use a different teaching method in an attempt to emulate what they had
learned in the workshops. One of the main obstacles they mentioned was that “preparing and giving a
class like the project instructor took too much time”; another difficulty they mentioned was “the
required mastery of mathematical content in order to respond to the students’ questions and remarks
and help the students strengthen their mathematical thinking […] Over time, after giving the same class
various times, we’ll be able to do it.”
The teachers who did not implement a new teaching method stated that both the project
instructor’s teaching and the students’ progress were positive, but that a focus like the one we proposed
was not viable since their school organizations were not ready for this sort of program. The teachers’
argument was that they lacked confidence in their own knowledge to be able to speak with the director
or to explain to the supervisor how this focus could be used to cover the topics on the program and
reach the goals of the official curriculum. Our data suggested that such a lack of confidence is related to
weaknesses in the teachers’ pedagogical and mathematical knowledge, as shown by the following
extract:
I’m impressed with the students’ achievement; I find many favorable points in the way the instructors
conducted the class, and how they handled the content and adjusted their remarks along the way according to
the students’ answers. I don’t believe I could do it. The school where I work is not as well organized as the
schools where the videos were taken. I know those schools well. My students do not behave like the students
we saw on the videos. It is very difficult to keep them focused. I have said before that I liked very much the
classes we saw on the videos, particularly the one with the Hanoi towers. I tried it with my students. That day
the director of the school entered my classroom unexpectedly and stayed to observe the class. She commented
on my work with the inspector and they ordered me not to do it again. I was told to limit my work to the
topics included on the curriculum. I’m sure I’ll try again in the future. For now I need time to improve and
learn more about mathematics and teaching mathematics […] One thing is the game with the Hanoi towers
and something entirely different is untangling the mathematical concepts that the game involves. This is
necessary in order to prepare a class that will allow the students to play the game mathematically. This requires
the teacher to have, in addition to a good knowledge of mathematics, notable skill for teaching. The problem I
see is that my workload is very heavy, as it is for many of my colleagues. Nonetheless, I’ll try to get organized
to do this and I’ll volunteer again to participate in the type of programs as often as I have the opportunity.
Teachers’ Reflections on Their Practice
The remarks the teachers made during the workshops suggest that the students’ achievements were an
important factor in motivating them to reflect critically on their teaching practice and on their
pedagogical and mathematical knowledge. This finding confirms what we hoped to accomplish when
we designed the current study. As mentioned above, on concluding the workshop sessions (one year of
work), some of the participating teachers attempted to implement different teaching methods; these
attempts were a result of the workshop discussions and analyses of the teaching focus. Even teachers
who stated they had no intention to change, mentioned that having observed the students’
achievements motivated them to review their teaching.
The compiled data suggest that the main factor that motivated teachers to try a different teaching
method was having observed the change in student attitudes as students were given opportunities to
explore, make mistakes, and move forward and backward, with the self-confidence produced by having
the support of a teacher to help them. One of the teachers referred to the students’ attitudes as follows:
I believe that the students’ favorable attitudes are closely related to their mathematical performance […] We
saw how the students gained self-confidence as the group accepted their ideas […] It was impressive to see
how they participated enthusiastically by proposing ideas and discussing heatedly with their classmates and the
teacher. The students we observed in the videos looked happy to be in the mathematics class […] I ask myself
why my students almost never look like that. I’m impressed with how far student learning can go when the
teacher guides them as we saw in the videos […] I would like my students to behave like that and I am
convinced that a lot depends on me.
The teachers’ reactions indicate that at the beginning, their attention was centered largely on the
students’ responses, but that they soon stopped focusing exclusively on responses and began to analyze
how the students had reasoned to reach those responses; this process led them to conclude that the
way the instructors conducted the classes was an important point of discussion. In many sessions, the
teachers tried to find plausible reasons to explain why the young people could generate mathematical
solutions for the problems they were given. The teachers emphasized in their discussions that the
instructor tried not to give the students answers; in contrast, he would respond with another question,
“trying to make the students find the error for themselves”. Our analysis suggests that the teachers
became aware of these events because they do not teach like the instructor in the videos.
The point the teachers discussed most was their opportunity to implement the teaching focus they
had observed in the videos. They reached the conclusion that the students who took part in the videos
were no different from their own students. This type of discussions allowed the teachers to conclude
that a teacher’s pedagogical skills are not the only factor that allows classroom activities to be fruitful;
they noticed that pedagogical skills are evident in other important factors, such as the materials the
teacher uses for an activity, the criteria for selecting or designing problems, and the way a class is
planned is structured (which can help the teacher to anticipate student responses). This idea is
illustrated by a teacher’s comment:
After discussing these topics, I realized the importance of preparing the class with care. I realized that
planning a class consists of combining my pedagogical and mathematical knowledge to help students learn
more meaningfully […] What I want to say is that planning a class requires the teacher to translate
mathematical knowledge in a way that can create a context of teaching and learning.
Final Considerations
Our work was influenced by other studies out in teacher development programs that have assumed that
solving sufficiently difficult mathematical problems can have positive effects on teachers’ pedagogical
and mathematical knowledge (Zaslavsky, Chapman and Leikin, 2003; Murray, Olivier and Human,
1995; Schifter, 1993). The results of our study emphasize the importance of this type of mathematical
problems as an instrument for teachers to improve their teaching practice. Our data also confirm
previous findings on solving non-routine mathematical problems as a means to encourage teachers
reflect on their practice and eventually to improve. An important source for our study was the research
carried out by Carpenter and Fennema (1989), whose work assumes that familiarity with students’
mathematical thinking will provide teachers with a more solid basis for their teaching work and
professional development.
A supposition in the current study is that teachers must learn mathematics in the same manner that
they are expected to teach the subject. Since our study was directed to classroom teachers, we had to
use a method that would allow us to give teachers an opportunity to learn mathematics in the same
manner that we expected them to teach mathematics. Working with teachers who were currently giving
classes was a challenge for us. Putting them into the position of comparing their knowledge with their
students’ knowledge allowed them to analyze their teaching critically in the framework of how they
were teaching. The results of this study confirm the results reported by other studies: that a relevant
component in teachers’ professional development is their mathematical knowledge. The facet our data
suggest in this respect is that teachers are more likely to participate actively in the process of
strengthening their mathematical knowledge if it is approached adequately within their teaching
practice. Our data ratify the need for sound mathematical knowledge so that teachers will be capable of
designing or returning to activities that give their students opportunities to learn mathematics in a more
meaningful manner. We wish to emphasize that this study indicates that if we want to strengthen
teachers’ mathematical and pedagogical knowledge, we must motivate them by involving them in
didactic proposals whose results are reflected in the quality of their students’ learning.
The results of this study confirm that the change in teachers’ conceptions, practice, and knowledge
is not an event, but the result of a long process. After a year of working with classroom teachers we
have evidence of only incipient changes in their practice and conceptions. This experience indicates that
the monthly work meetings provided a favorable setting for teachers to learn from each other; it also
emphasizes the pertinence of continuing to promote the creation of professional teaching communities:
an enormous challenge in Mexico since our teachers usually work in isolation.
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Article Received: March 14, 2007
Ruling: July 13, 2007
Second Version: July 31, 2007
Accepted: August 16, 2007