After reading Clements & Sarama's (2004) Learning Trajectories in Mathematics Education a few days ago, I wanted to go back to the origins of learning trajectories: a 1995 paper from Martin Simon that explored how mathematics can and should be taught differently with a constructivist mindset. Simon is a professor of math education at NYU and has a history of researching how students and their teachers come to understand their mathematical knowledge.
From the start, I immediately appreciated two strengths of this article: Simon's clear writing and the relatively straightforward description of constructivism he offers. When you're still trying to sort out what constructivism is and is not (like me and many classroom teachers), it's a whole lot easier to parse this article from 1995 than, say, a heavily theoretical, mid-2000s piece by Jim Greeno. Recognizing that there are multiple (and often subtly different) ways to describe constructivism, Simon lays out his interpretation like this:
Constructivism derives from a philosophical position that we as human beings have no access to an objective reality, that is, a reality independent of our way of knowing it. Rather, we construct our knowledge of our world from our perceptions and experiences, which are themselves mediated through our previous knowlege. Learning is the process by which human beings adapt to their experiential world. (p. 115)
So when we have an idea that "works," meaning it does what we need it to do to make sense of our experiences, then we've constructed knowledge. This can be a tough sell to the mathematically-minded who say things like, "I didn't construct two plus two equals four. That's an objective fact." In response, the constructivist would disagree about having access to objective reality, but acknowledge that we (almost?) universally construct the knowledge that 2+2=4 because we experience no evidence suggesting otherwise (what Simon and others refer to as disequilibrium).
There's also a theoretical debate about the construction of knowledge as an individual, cognitive process versus a social process. This is an interesting debate, to be sure, and it has been pushing the leading edges of theories for learning mathematics for about the past twenty years. If you're wearing a theoretician hat, then you care deeply about how this debate might be won. But if you're wearing a researcher hat (like Simon is here), then you use both theories to help you gain whatever insights they might afford you. Simon (crediting work by Cobb, Yackel, and Wood) calls this coordination of psychological and sociological approaches "social constructivism," and compares it to how a physicist can better explain the nature of light by considering it both a particle and a wave.
Simon takes pains in this article to separate the theory of constructivism from a notion of "constructivist teaching." It's a mistake that I've seen and heard many times, and it's important to understand the differences and nuances. Simon states:
As I stated above, constructivism, as an epistemological theory, does not define a particular way of teaching. It describes knowledge development whether or not there is a teacher present or teaching is going on. ... There is no simple function that maps teaching methodology onto constructivist principles. A constructivist epistemology does not determine the appropriateness or inappropriateness of teaching strategies. ... The commonly used misnomer, "constructivist teaching," [suggests that] constructivism offers one set notion of how to teach. The question of whether teaching is "constructivist" is not a useful one and diverts attention from the more important question of how effective it is. From a theoretical perspective, the question that needs attention is, In what ways can constructivism contribute to the development of useful theoretical frameworks for mathematics pedagogy? (p. 117)
Using this perspective and a lot of theoretical support from work done in the early 1990s, Simon sets out to explore "the ongoing and inherent challenge to integrate the teacher's goals and direction for learning with the trajectory of students' mathematical thinking and learning" (p. 121, emphasis in original). Unlike a traditional perspective, where the pedagogical focus tended towards chopping mathematical content into manageable pieces to be demonstrated and practiced, Simon wished to focus on student understanding and a plan for mathematical tasks that improved that understanding.
I won't describe Simon's teaching experiment in great detail (after all, it was data-rich enough for Simon to publish multiple papers), but it involved a group of preservice elementary teachers and a set of tasks designed to elicit understandings about how multiplication related to the simple area formula A = l x w. Simon knew that his students had no trouble multiplying or using the formula. That wasn't the problem. Instead, he gave them this task:
Determine how many rectangles, of the size and shape of the rectangle that you were given, could fit on the top surface of your table. Rectangles cannot be overlapped, cannot be cut, nor can they overlap the edges of the table. Be prepared to describe to the class how you solved this problem. (p. 123)
Students used their rectangle (I'm imagining an index card) to measure the length and width of their table. A few groups questioned whether the rectangle should maintain its orientation, or if the long edge should always align with the edge of the table. This launched a class discussion and Simon pushed students to explain how they found the area without defaulting to "I used the formula." Some students talked about rows and columns, some talked about counting rectangles, but comments about "overlapping" rectangles suggested misunderstanding was still apparent. Compounding the problem was that these students were not accustomed to provide this level of justification.
Simon tried varying the task to elicit better student explanations. Some students seemed to get it while others still struggled or remained silent. (The transcript excerpts in Simon's paper are very valuable here, if you can get a copy of it.) Simon began to worry that students were actually misunderstanding things about area, not just how multiplication relates to rectangle area, so he assigned a second task about finding the area of an irregular shape. This was less of a problem for students, so Simon returned to the "turned rectangle" problem and tried another activity with measuring tables, both with rectangular cards and also with sticks. Some students were stuck in their thinking that the area unit must be the size and shape of the card, while others began to see how using the long edge of the card for length and width of the table created new, square units not shaped like the card.
All these classes were observed by researchers who took notes and videotaped the classroom activity. Simon also kept a journal of his reflections after each lesson and planning session. Following the teaching experiment, Simon analyzed his role as the decision maker in the classroom activities. First, he had hypothesized that his students would otherwise be satisfied with knowing and using a formula for area, but had probably never explored why the formula worked. This hypothesis was based on prior experience with similar students, prior research, and pretesting. Simon carefully thought out what he thought would happen in his initial activity, saying this thinking
provides an example of the reflexive relationship between the teacher's design of activities and consideration of the thinking that students might engage in as they participate in those activities. The consideration of the learning goal, the learning activities, and the thinking and learning in which students might engage make up the hypothetical learning trajectory, a key part of the Mathematical Learning Cycle described in the next section. (p. 133)
The "Mathematical Learning Cycle" Simon, in a simplified way, suggests how a teacher's knowledge can be used to create a hypothetical learning trajectory (containing a learning goal, a plan for learning activities, and a hypothesis about the learning process), and how assessment of student knowledge gives the teacher new and better knowledge upon which to refine the hypothetical learning trajectory. (I can't help but wonder if Simon once thought he'd be remembered for "learning cycles," not "learning trajectories.") The trajectory as planned is always hypothetical because it is just the teacher's prediction and the true trajectory cannot be known in advance. Modification of the trajectory happens as the teacher increases his/her knowledge about what students understand, which can be during a planning session between lessons or on-the-fly during a classroom activity. Of course, the more knowledge a teacher has in advance -- about their students, about the mathematical content, and about theories of learning that content -- the better the hypothetical learning trajectory can be. Sometimes we don't have all the information we want, says Simon:
As a teacher, I often do not have a well-developed map of the mathematical conceptual area in which I am engaging my students; that is, I may not have fully articulated for myself (or found in the literature) the specific connections that constitute understanding or the nature of development of understanding in that area. ... Thus, in such cases, my operational definition of understanding is the ability to overcome these particular difficulties; I may not have unpacked the difficulties in order to understand the conceptual issues that are implicated. Thus, even if I do not have a thorough knowledge of what constitutes mathematical understanding in a particular domain, having a rich set of problem situations that challenge students and having knowledge of conceptual difficulties that they typically encounter provide me with an approximation that lets me be reasonably effective in promoting learning in the absence of more elaborated knowledge. (This is not to suggest that the more elaborated understanding would not be more powerful.) (p. 139)
In his summary, Simon reiterates some major themes:
- Student understanding is prioritized in the design of instruction
- Teachers learn as students learn
- Planning instruction includes the creation of a hypothetical learning trajectory
- Because of #2, teachers need to constantly revise #3
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205