Content uploaded by Jérôme Proulx
Author content
All content in this area was uploaded by Jérôme Proulx on Nov 10, 2019
Content may be subject to copyright.
PME 32 and PME-NA XXX 2008 4 - 145
STRUCTURAL DETERMINISM AS HINDRANCE TO TEACHERS’
LEARNING: IMPLICATIONS FOR TEACHER EDUCATION
Jérôme Proulx
University of Ottawa
In this paper, I use Maturana and Varela’s (e.g., 1992) theoretical construct of
structural determinism as a lens to better understand and discuss specific events of
teachers’ learning or non-learning in various situations. Through excerpts from the
literature and data from one of my projects, I illustrate teachers’ personal orientations
that guide their potential learning. These interpretations have implications for teacher
educators, who need to become more than facilitators or guides in order to trigger
learning opportunities for/in teachers.
INTRODUCTION – TEACHERS’ LEARNING ISSUES
This paper is partly theoretical, offering a perspective for thinking about mathematics
teachers’ knowledge and learning, and partly practical, using data from the research
literature and my own studies to illustrate and make sense of the points highlighted.
The discussion is framed around an intention to theorize and develop greater
understandings of teachers’ learning. One central aspects of conducting research in
teacher education is to study phenomena of teachers’ teaching and knowledge of how
to teach (what, why and how teachers know and act), that is, to better understand who
they are and what they know as learners of mathematics teaching and as teachers of
mathematics. These understandings can lead to enriched conceptions of teachers’
learning/knowledge which can in turn contribute to the constant endeavour to
improve the teacher education practices that mathematics teachers are immersed in.
One of last year’s conference research forums was “Learning through teaching:
Development of teachers’ knowledge in practice,” by Roza Leikin and Rina Zazkis
(2007) and collaborators, and focused on teachers’ opportunities to learn in and from
their everyday teaching practices. One of the contributors, Martin A. Simon (2007),
offered us a different view on teachers’ learning, talking about the possible
limitations of teachers’ learning from their practices. Using the insightful but catchy
phrase “we see what we understand,” he talked about how teachers can be limited
from their own knowledge in what they can learn from their practices in schools.
Simon’s interesting contributions and explanations reminded me of some related
issues found in the literature on mathematics teacher education.
One came from the study of Grant, Hiebert and Wierne (1998) who offered videos of
reform-oriented mathematics teaching practices to teachers for them to see examples
of such practices, make sense of them, and draw out some principles for their own
practices. It was found that a number of teachers, who saw “mathematics as a series
of procedural rules and hold the teacher responsible for students acquiring these
rules” (p. 233), were not able to see significant differences between these practices
Proulx
4 - 146 PME 32 and PME-NA XXX 2008
and their own and tended to focus on technical aspects of the lesson and the material
used. For Grant et al., this showed how offering videos of innovative practices were
not a panacea for teachers’ professional development since teachers needed to be able
to appreciate what was in the video to see these differences and innovations.
A similar issue arose in Fernandez’s (2005) research report on a collaborative lesson-
study environment for elementary teachers. Playing the role of a facilitator, she
realized after a while that the mathematical tasks teachers were exploring were not as
insightful and provocative to the teachers as she thought they might have been and
noted that “often the exchanges that took place did not push the teachers’ thinking as
far as they could have or sometimes even took them in unproductive directions” (p.
278). Fernandez then wondered if the teachers’ mathematical knowledge was limiting
to some extent what they were able to draw or not draw from the tasks themselves.
As well, my own research report at last year’s conference (Proulx, 2007a) described
secondary teachers’ attempts to make sense of student solutions for a rate of change
question in which conventions came into play. On analysing a solution where a
student had inversed the roles of Δx and Δy in calculating slope, the teachers
concluded that the student “did not understand anything about variation” and that
nothing more could be drawn from such a solution by a teacher – it was just wrong. It
appeared that the teachers were conflating the (arbitrary) order of the rate of change
with its conceptual understanding. Teachers’ own understanding of rate of change
oriented their interpretation of the solution, leading them to not perceive some of the
student possible understandings of variations in a graph. The same happened for the
order in the coordinates of the Cartesian plan, where teachers did not see (x, y) as an
arbitrary convention and interpreted students’ solution in relation to it.
Here, one could point to teachers’ lack of knowledge or inabilities to make sense of
differences and worthwhile mathematics in these studies. However, what we need to
understand more deeply is the rationale for and the mechanisms that are operating in
these situations. The theory of cognition of Maturana and Varela (e.g., 1992),
especially their concept of structural determinism, can shed light on some of these
issues and help make sense of them. Below, I outline aspects of their theory and use it
to interpret the above situations. Then, I use excerpts from my own research to
illustrate additional ways of understanding the phenomenon of teachers’ learning, and
I raise some implications for the role of teacher educators in these situations.
MATURANA & VARELA’S THEORY AND STRUCTURAL DETERMINISM
Maturana and Varela’s (e.g., 1992) theory of cognition is grounded in biological and
evolutionary perspectives on human knowledge and processes of meaning making.
Fundamental to this theory, and rooted in Charles Darwin’s theory of evolution, are
notions of structural coupling and structural determinism. Darwin used the concept
of “fitting” to make sense of the process of survival of species. Hence, for species to
survive, it must continuously adapt to its environment, to fit within it. If not, it would
perish. The concept of fitting is, however, not a static one in which the environment
Proulx
PME 32 and PME-NA XXX 2008 4 - 147
stayed the same and only the species evolved and continued to adapt. Darwin
explained that species and environment co-evolve, and Maturana and Varela added
that they co-adapt to each other, meaning that each influences the other in the course
of evolution. This idea of co-evolution/co-adaptation is key in regard to the origin of
changes or adaptations of the species to its environment. Maturana and Varela call
this structural coupling, as both environment and organism interact with one another
and experience a mutual history of evolutionary changes and transformations. Both
undergo changes in their structure in the process of evolution, which makes them
“adapted” and compatible with each other.
Every ontogeny occurs within an environment […] it will become clear to us that the
interactions (as long as they are recurrent) between [organism] and environment will
consist of reciprocal perturbations. […] The results will be a history of mutual congruent
structural changes as long as the [organism] and its containing environment do not
disintegrate: there will be a structural coupling (1992, p. 75, emphasis in the original).
Here, the environment does not act as a selector, but mainly as a “trigger” for the
species to evolve – as much as species act as “triggers” for the environment to evolve
in return. Maturana and Varela explain that events and changes are occasioned by the
environment, but they are determined by the species’ structure.
Therefore, we have used the expression “to trigger” an effect. In this way we refer to the
fact that the changes that result from the interaction between the living being and its
environment are brought about by the disturbing agent but determined by the structure of
the disturbed system. The same holds true for the environment: the living being is a
source of perturbations and not of instructions (1992, p. 96, emphasis in the original).
Maturana and Varela call this phenomenon structural determinism, meaning that it is
the structure of the organism that allows for changes to occur. These changes are
“triggered” by the interaction of the organism with its environment. They give this
example: A car that hits a tree will be destroyed, whereas this would not happen to an
army tank. The changes do not reside inside of the “trigger” (inside the tree), rather
they come about from the organism interacting with the “trigger.” The “triggers”
from the environment are essential but do not determine the changes. In short:
changes in the organism are dependent on, but not determined by, the environment.
INTERPRETING DATA IN LIGHT OF STRUCTURAL DETERMINISM
If one uses structural determinism to make sense of the learning process, one
understands that the response of the learner is dependent on the environment he/she is
put in (e.g., video watching, mathematical tasks, students answers), but is determined
by the learner’s own way of making sense and interpreting. Thus, the response to a
stimulus is not in the stimulus per se but is in the person that responds to it.
In the case of Grant et al. (1998) study, one could interpret that it is not that the
teachers did not see the differences between the reform-oriented practices presented
in the videos and their own classroom practices, but that for them these were not
present as differences. In order to notice or appreciate these potential differences, the
Proulx
4 - 148 PME 32 and PME-NA XXX 2008
teachers needed to be aware of the possibility of these differences and be able to
understand what they were. The same can be said of the teachers in my study (Proulx,
2007a), who did not distinguish between the usage of mathematical conventions and
mathematical understanding. The teachers needed to be aware of issues of
conventions in these situations to make sense of them in the student solution, which
did not appear to be the case. As well, in Fernandez’s (2005) study, it is not that the
teachers did not see the mathematics in the problem, but that the mathematics that she
saw in it was simply not present for these teachers. Simon’s point that “we see what
we understand” is of relevance to these potential learning situations: teachers’
knowledge, their structure, did not allow them to “see” these distinctions (that were
apparent to the teacher educators presenting the learning opportunities).
In fact, we stating that there are differences in the videos or mathematical aspects in
the tasks offered is also representative of our own “blindness”: we simply do not
realize that it is we who sees them and that someone else could not see these
distinctions. Or, simply, we make the assumption that these properties are present in
and of themselves in the tasks and that these would determine teachers’ reactions –
leading to our conclusion that “Hum! They did not see that.” It is not that they did not
see, but simply that there was nothing for them to see.
This said, as well as our knowledge/structure can lead us to “not see” some aspects,
our knowledge/structure can orient us to focus on other aspects; or, in a sense, to
“only see” some aspects that we are oriented towards. To use Maturana and
Varela’s words, our structure leads us toward ways of understanding the world:
what we understand is a function of our knowledge and is influenced by it. I further
illustrate this idea through reporting on other data excerpts from one of my research
projects.
THE OTHER SIDE OF THE STORY: TEACHERS’ ORIENTATIONS
Recently, I set up a year-long professional development initiative for secondary
mathematics teachers (Proulx, 2007b). The six teachers that participated in the
project wanted to improve and rejuvenate their teaching practices. They felt their
mathematical knowledge was too focused on procedural knowledge and that this
impacted their teaching practices and ways of making sense of mathematics. Some of
them expressed the following: “Why is it that we are not able to solve by reasoning?
It is because we have not been educated to reason in mathematics. Me, I did copy,
paste, repeat, and let’s go. And I had 95% in mathematics!” or “I never understood
why it worked. When students ask me why, I simply say that this is how it is!” Thus,
the in-service sessions were structured around the study of school mathematics
concepts and focused on sense-making and mathematical reasoning, rather than an
application of procedures, for teachers to explore these issues. Through the work on
specific mathematical tasks during the year, interesting characteristics of teachers’
orientations arose. I underline these to illustrate how their knowledge, as structure
determined beings, oriented them to engage in these tasks in particular ways.
Proulx
PME 32 and PME-NA XXX 2008 4 - 149
Engaging in tasks at the procedural level
Through the sessions, teachers often appeared to approach mathematical problems in
a technical fashion. When a problem was offered to them, their initial approach was
often to look for the procedure to apply; almost as a reflex. It appeared as if this was
their natural orientation to engage in mathematical problems. For example, in the
following problem (Figure 1), teachers looked for and attempted to apply the area of
the square formula, even if this approach was quite inefficient here.
Figure 1. Problem on the area of a square.
It appeared as if these teachers’ structure, a structure heavily focused on
mathematical procedures, oriented them to work on problems through procedures.
But, throughout the year, because the in-service program was focusing on aspects
beyond procedures, the teachers became more and more aware of the possibility of
entering differently into problems. Hence, these first attempts, which often resulted in
a blank outcome, had an interesting effect on teachers as they themselves started to
realize that there was something else to understand and work through, and attempted
to probe from a different angle. In fact, the teachers often expressed out loud that it
was their own procedural orientation that had lead them along this way and that they
needed to work at getting away from this orientation. As sessions went on, the
possibility of working differently on problems became more present to them (i.e.,
part of their structure) and they were able to explore these avenues.
Looking for techniques in mathematics
The teachers frequently expressed that their students had difficulties in specific
mathematical domains and that they had been looking for precise mathematical
techniques to communicate them for helping avoid errors and solve problems better.
The issue was that some of these looked-for techniques appeared to not make much
sense. For example, during a session focused on the creation of algebraic equations
from word problems, the teachers continuously tried to find a specific mathematical
technique to easily create the algebraic equation. This technique, for them, would
help to avoid students’ mistakes. Many options were offered as techniques: using
other letters than x or y; underlining key words; creating an intermediary step where
students would need to write down what each unknown represents; writing down the
Findtheareaofthesquare
.......
.......
.......
.......
Proulx
4 - 150 PME 32 and PME-NA XXX 2008
relations in a table; and so on. All of these approaches had at least some value, but
the teachers’ intentions was to find “the one” and to use it as an algorithm they could
present to their students, with precise steps to follow in order to obtain the correct
answer. After a while, they realized that these techniques were insufficient since one
still had to make sense of the problem and the relation between the data in order to
write down the equation and therefore that it could not be reduced to a simple
technique to apply. Here, it appeared that the teachers’ procedural orientation towards
mathematics was leading them to look for even more procedures in mathematics.
Technical reading of the curriculum
This procedural orientation often brought the teachers to interpret the mathematical
topics of the curriculum as expectations for working on techniques, algorithms and
formulas. For example, when we worked on volume of solids, for the teachers this
topic meant giving and demonstrating the volume formulas to students; that is,
seeing volume as being only about its formulas. The same was true for analytical
geometry, understood as a number of formulas of distance, middle points, etc., or
for fractions, seen as a request to learn the addition, subtraction, multiplication and
division algorithms, and so on. As Bauersfeld (1977) explains, some teachers
develop technical “eyes” to read and interpret the notions and topics of the
curriculum. In this case, the curriculum notions are read as requests to work on
procedures. As studies have shown (e.g., Putnam, Heaton, Prawat, & Remillard,
1992; Ross, McDougall & Hogaboam-Gray, 2002), curricular changes alone seldom
affect changes in teaching practices, something that can be explained by teachers’
orientation to reading these topics: teachers read what they understand, their
structure orients their reading.
IMPLICATIONS FOR EDUCATION: DEFINING THE LEARNING SPACE
Given the examples offered above, we might reread Simon’s comment of “we see
what we understand” to also mean “we only see” aspects one is oriented toward. This
is in fact what Maturana and Varela explain: as structure determined beings, our
structure orients the sense we can make of a situation (enabling or orienting). Thus,
tasks, situations or contexts do not possess “the learning” for teachers; rather this
learning is determined by teachers’ own structure as they interact with these tasks and
situations. But, what implications does this have for the educational act or
educational initiatives? If teachers can only see what they already understand, it
could mean that nothing new can be worked on with them since it needs to be already
known by them. If this is so, this would mean that our structure not only determines
what we learn, but also restricts us and stops us from learning. This is clearly not so,
and Maturana offers some explanation of this process, which places an importance on
the outside environment to provoke reactions/learning from the learner.
Maturana (e.g., 1987; Maturana & Mpodozis, 1999) makes a distinction between our
structure and our actions, which he terms “conducts.” The conducts demonstrate, and
are permitted by, one’s structure/knowledge. These conducts are determined by the
Proulx
PME 32 and PME-NA XXX 2008 4 - 151
structure, but are also dependant on the environment in which they are enacted. It is
in the interaction with the environment that one’s conducts arise, in its structural
coupling with it. So, these conducts are coupled with, and embedded in, the
environment within which they are made possible. Our conduct is a product of both.
And, it is in this space that lays the potential of learning and of change, where the
environment acts as a trigger on the learner’s conduct.
By emerging from the coupling of one’s structure and environment, the triggered
conduct is “new” or created from this coupling. Thus, this conduct triggers back in
return. It triggers back one’s structure, by having emerged from a world of
possibilities in the coupling of structure and environment. This emergent possibility,
this conduct, influences (read, trigger) the structure itself and offers it new
possibilities; possibilities for change, for learning. Thus, this generated conduct in
return affects one’s own structure. [As well, in this structural coupling, the conduct
triggers changes in the environment. However, I am not addressing this here.]
This is how our structure evolves, through a continuous interplay between our
structure (that determines possible conducts) and the conduct that emerges from the
interaction/coupling with the environment. Our structure triggers conduct, and this
conduct, by carrying aspects of the environment with it and therefore being un-
thought of and possessing a new character, triggers our structure in return. It is a
circular, never-ending, loop of structural change. One can infer that a new
environment has the potential to trigger new conducts which in return can trigger
changes in one’s structure. Therefore, what is present in the environment is of
fundamental value to generate these new conducts and in return to generate the
structural transformations. The potential for education, for learning, lays here, where
the environment the learner is placed in can trigger some conduct as the learner is in
constant interaction/coupling with this environment.
FINAL REMARKS: SEEING TEACHER EDUCATORS AS TRIGGERS
From this understanding of the learning space, the environment in which the teacher
is put in is of fundamental importance to trigger learning. One needs to see, in the
teacher education environment, the presence of specific tasks as well as the presence
of the teacher educator. The teacher educator acts as a trigger for teachers and has the
opportunity to open new possibilities for teachers, new ways of making sense and of
understanding (however, not implying he or she possess the “truth”). It is by bringing
or throwing “something” into the learning environment that the teacher educator can
create something and potentially trigger teachers’ learning. This, therefore, calls for
the teacher educator to be very active in the educational process, where his or her
actions are to be seen as triggers for teachers’ learning. This view challenges the view
of an educator as mere facilitator or guide (Kieren, 1996). It calls for a teacher
educator that puts oneself and act vigorously in this learning space to trigger and
provoke something in teachers. On this, I conclude by citing one of Fernandez’s
(2005) remarks as to the importance of the role of teacher educators:
Proulx
4 - 152 PME 32 and PME-NA XXX 2008
This learning [of teachers] was no doubt possible because lesson study created a rich
learning environment for these teachers in very much the same way that rich classroom
tasks like those employed in reform classrooms set up opportunities for students to learn.
However, although students learn a lot from working on such tasks, nevertheless a
teacher who can push, solidify, and sometimes redirect their thinking is critical.
Similarly, the teachers described here could have benefited from having a “teacher of
teachers” help them make the most out of their lesson study work. (p. 284)
References
Bauersfeld, H. (1977). Research related to the mathematical learning process. In H. Athen & H.
Kunle (Eds.), Proceeding of the 3rd Inernaional Congresse on Mathematics Education (pp.
231-245). Karlsruhe, Germany: ICME-3.
Fernandez, C. (2005). Lesson study: A means for elementary teachers to develop the knowledge
of mathematics needed for reform-minded teaching? Mathematical Thinking and Learning,
7(4), 265-289.
Grant, T.J., Hiebert, J., & Wierne, D. (1998). Observing and teaching reform-minded lessons:
What do teachers see? Journal of Mathematics Teacher Education, 1, 217-236.
Kieren, T.E. (1995, June). Teaching Mathematics (in-the-Middle): Enactivist view on Learning
and Teaching Mathematics. Paper presented at the Queens/Gage Canadian National
Mathematics Leadership Conference, Queens University, Kingston, Canada.
Leikin, R. & Zazkis, R. (2007). Learning through teaching: Development of teachers’
knowledge in practice. In J.H. Woo et al. (Eds.), Proceeding 31st Conference of the
International Group for the Psychology of Mathematics Education (vol. 1, p. 121-150).
Seoul, Korea: PME.
Maturana, H.R. (1987). Everything is said by an observer. In W.I. Thompson (Ed.), Gaia: A
Way of Knowing (pp. 65-82). New York: Lindisfarne Press.
Maturana, H.R. & Mpodozis, J. (1999). De l’Origine des Espèces par Voie de la Dérive
Naturelle (transl. by L. Vasquez & P. Castella). Lyon: Presses universitaires de Lyon.
Maturana, H.R. & Varela, F.J. (1992). Th Tree of Knowledge (rev. Ed.). Boston: Shambhala.
Proulx, J. (2007a). Addressing the issue of the mathematical knowledge of secondary
mathematics teachers. In J.H. Woo et al. (Eds.), Proceeding 31st Conference of the
International Group for the Psychology of Mathematics Education (vol. 4, p. 89-96). Seoul,
Korea: PME.
Proulx, J. (2007b). (Enlarging) Secondary-Level Mathematics Teachers’ Mathematical
Knowledge: An Investigation of Professional Development. Unpublished doctoral
dissertation, University of Alberta, Canada.
Putnam, R.T., Heaton, R.M., Prawat, R.S., & Remillard, J. (1992). Teaching mathematics for
understanding: Discussing case studies of four fifth-grade teachers. Elementary School
Journal, 93(2), 213-228.
Ross, J.A., McDougall, D., & Hogaboam-Gray, A. (2002). Research on reform in mathematics
education. Alberta Journal of Educational Research, 48(2), 122-138.
Simon, M.A. (2007). Constraints on what teachers can learn from their practice: Teachers
assimilatory schemes. In J.H. Woo et al. (Eds.), Proceeding 31st Conference of the International
Group for the Psychology of Mathematics Education (vol. 1, p. 137-141). Seoul, Korea: PME.
