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Asia-Pacific Forum
on Science Learning and Teaching, Volume 10, Issue 1, Article 1
(June, 2009) |
Highlighting key points of the group discussion, our analysis raises five issues: 1) the problem attracted nine strategies/approaches (two non- calculus, seven calculus); 2) non-calculus approaches utilized math that could be considered familiar to grades 11 and 12 students; 3) consistent with a constructivist view, our teacher candidates' prior knowledge appeared to have strongly influenced their conformity or lack of conformity with the nature of desired outcome of the task; 4) our teacher candidates' solutions seem to convey the impression that the math and physics concepts imbedded in the problem are interwoven; and, 5) the relevance or irrelevance of mathematical noise in a solution depends on the extent of obscurity or clarity of the intended physics concepts.
Further, it is important that groups be trained to be critical of their own learning or practice. Also, on the one hand, majority of the groups (with calculus solutions) did not reflect on their solutions as they only thought in terms of calculus. On the other hand as a class they learned from different group presentations, including solutions that conformed to the expectation of the problem solving task. The group discussions were the place where this sort of learning could happen.
Problem Solving Strategies and Prior Knowledge
The study’s results, though not necessarily applicable to every pre-service physics teacher or to high school physics teachers in general, nonetheless highlight the need to make the match between grade-levels and problem solving strategies a key part of our pre-service physics methods curriculum, and further, the need to underscore the role students' prior mathematics knowledge plays in their understanding of the knowledge intended to be conveyed through physics problem solving tasks. We modeled this pedagogical approach with the assigned problem solving task and provided opportunity for prior pedagogical content knowledge to be elaborated within the group discussions.
Although questions could be raised about whether or not the assigned problem was suitable for a grade 11 or 12 classroom, it nonetheless served the purpose of the investigation, which was to determine the teacher candidates' ability to generate problem solving strategies appropriate to grades 11 and 12. That is, we deliberately chose this problem because of the challenge it offered to our teacher candidates. We wanted to influence our teacher candidates to always endeavour to build upon what their students already know. It was our view and experience that using a grade 11/12 level problem did not provoke the kind of discussion and challenge that was needed to reflect real life experiences. More particularly, content-rich individuals (e.g. physics teachers) quite often tend to underestimate the difficulty their students experience when what is being taught does not relate to what is already known. Moreover, our goal of provoking thought, engagement and offering real challenge would have been undermined by using a problem at a grade 11/12 level of difficulty.
The problem solving task evoked the teacher candidates' prior mathematics content knowledge and the extent to which they could apply it to solve the problem. In other words, the problem or task helped elicit the teacher candidates' pedagogical content knowledge, which according to Shulman (1986) refers to, “the ways of representing and formulating the subject that makes it comprehensible to others" (p. 9). However, this paper adopts an elaborated version in which PCK includes teachers' interpretations and transformations of subject matter knowledge in the context of facilitating student learning. Simply put, Shulman (1987) considers pedagogical content knowledge as the category most likely to “distinguish the understanding of the content specialist from that of the pedagogue" (p. 4). Pedagogical content knowledge thus depends on complex interactions between discipline knowledge, pedagogic knowledge, and the teacher’s experiences in teaching that knowledge (Cochran & Jones, 1998; Tobin, 1998; Tobin, Tippins & Galland, 1994). We believe that these interactions need to be explored and reflected upon during pre-service teacher education so that teacher candidates' developing pedagogy includes eliciting and building upon their students' prior knowledge.
Candidates' Pedagogical Content Knowledge (PCK)
Many of the teacher candidates could not generate problem solving approaches that conformed to the challenge of the assigned problem solving task, but the task most likely evoked an awareness of what they did or did not know. This interpretation falls within the realm of constructivist theories of learning. In other words, the understandings the candidates developed are according to Kelly (1955),
Ways of constructing the world. They are what enables…[one] to chart a course of behavior, explicitly formulated or implicitly acted out, verbally expressed or utterly inarticulate, consistent with other courses of behavior or inconsistent with them, intellectually reasoned or vegetatively sensed (p. 9).
It should be clear that the prior knowledge physics students possess at the time new concepts are being taught (whether through problem solving tasks or some other means, such as experimentation) is a major factor in determining the ease with which they understand the new concepts and the pace at which the teacher covers the intended content. Problem solving, as we have already argued, uses mathematics in the modeling of solutions to problem solving tasks. Absent, inadequate or poorly understood prior mathematics knowledge might necessarily inhibit a student’s understanding of the intended physics concepts. Use of students' prior mathematical knowledge will most likely enhance their understanding of the intended physics concepts. Teaching or activating the mathematics knowledge will then allow the students to concentrate on the physics concepts since the mathematical skills will be familiar. We see this linkage as an important objective for instructional planning: helping students to build substantive understanding across subject areas, freely utilizing different ways of knowing to deepen and broaden conceptual knowledge structures. This is the pedagogy that was being modeled through the problem solving task, and physics instructors should where possible, utilize math content with which the students are already familiar, thereby reducing cognitive overload. Also it is important to minimize unnecessary mathematics noise, even when it is familiar to the students.
If the purpose of a lesson is to teach physics concepts, it makes good pedagogical sense to minimize any impediments, such as “mathematics noise" (Johnstone & Wham, 1982), to students' understanding since there is always a high possibility that employing complicated mathematics in physics problem solving tasks will obscure understanding of the intended physics concepts. Otherwise, those who concurrently succeed in understanding both the mathematics and relevant physics are likely to be in the minority. Several problem solving strategies developed by our teacher candidates contained mathematics noise. These strategies can expose students to “cognitive overload." Although our teacher candidates were asked to generate strategies that used mathematics familiar to grades 11 and 12 students (which calculus-based strategies C1 through C7 are not), C1 through C3 and C7 have less mathematics noise than C4 through C6. It is even more overloading if the noise involves unfamiliar mathematics. However, if the noise is familiar physics, then in a way such noise might serve a useful purpose in some cases, for example, remediation. In order to minimize cognitive overload or mathematical noise, it might be helpful to the students if physics teachers provided remediation on the necessary mathematics knowledge and skills that may be required for particular physics units or topics prior to launching into the teaching of the physics concepts. Furthermore, since it is widely understood and also acknowledged in this paper that mathematics is a tool of physics, we see it as imperative to impress upon physics instructors the need to utilize students' relevant prior knowledge in explaining or working out solutions to physics problems. This might require a great deal of preparation and thinking - such as what was experienced by the teacher candidates during the problem solving task in this study. In fact, it might be easier for students to understand physics concepts conveyed through mathematics that the majority of students already possess. Realization of this fact and using it to develop instructional strategies that build upon appropriate mathematics content requires deliberate attention and modeling at the teacher preparation level. A further aspect of our teacher candidates' preparation involved their learning through group socialization.
Teacher Learning and Socialization
The kinds of experiences that our teacher candidates encountered during the physics problem solving activities are what this paper considers to be consequences of learning group socialization. Although individual teacher candidates initially generated the solution strategies in this study, they in turn shared them in groups and presented to class what they had agreed upon to be strategies represented within the group. In the process of sharing their strategies, there were knowledge exchanges between group members. In a way this was modeling teacher-learning communities (Lave & Wenger, 1991).
In a similar way, it is hoped that by engaging and wrestling with tasks that challenged them to generate strategies that are appropriate to grades 11 and 12, the teacher candidates experienced and constructed a feel for the student learning experience. The data in Table 1 are about individual as well as group products (problem solving strategies). Moreover, these data were generated within a group learning context, which in many respects involved sharing ideas, having a common experience and appreciating the challenges involved in planning appropriate grade-level strategies. This made our teacher candidates part a socialization process, hence the idea of teacher candidate socialization.
As reported in this study, problem solving strategies developed by individuals were shared in groups and presented to the whole class as group presentations. We believe our task as teacher educators is to help our teacher candidates build and rebuild what they already know about the work of teachers (Feiman-Nemser & Floden, 1986), and this is only possible for preservice teachers when they are put in situations where pedagogy is modeled. We also believe that such modeling is more effective if it involves a socialization process such as working in groups where they share information, experience, challenges and thoughts.
In this paper, it is argued that teacher candidates construct their images or perceptions of teaching in a particular discipline through direct or indirect socialization. The use of the term direct here is intended to mean a deliberate effort made by teacher educators to have the pre-service teachers adopt, practise and accept certain ways of experiencing a subject. Correspondingly, indirect refers to an unknowing on the part of the teacher educator, whereby the teacher candidates, by experiencing the teacher educator’s way of doing things, can come to believe that that is the way things are done or work in that subject. We could further hope that our teacher candidates have thus reconstructed their own images of teaching and that they will teach how they were taught to teach (Blanton, 2003; Goodlad, 1984; Nashon, 2005). The problem solving task in the current study was intended to do just that: model pedagogy where the would-be teachers will elicit their own students' prior knowledge of what they intend to teach using strategies similar to what was used in this study including the use of a problem to evoke as well as challenge existing knowledge. Further, through capturing this knowledge in records such as written assignments, individuals have a frame from which to share their understanding in a group context or social setting. The particular problem used as a context for this type of assignment must also be amenable to multiple interpretations or approaches in order to generate constructive argumentation, for example, such that each person has opportunity to explain their problem solving approach. Hansen (1995) argued that this was specifically the case with technology education students who had come from a variety of business and industrial backgrounds to their teacher education programs, through which were conveyed differing notions of what it is to be a technology teacher, models that may have been inconsistent with those in use in the teacher education program. The teacher education program then, needed to challenge these various perspectives in order that the candidates could be socialized into the culture of teaching.
In this study there is a high possibility that the candidates were driven by the desire to solve the problem and not necessarily to teach the problem. Thus, we argue that the problem was not just aimed at challenging their desire to solve the problem, but also to orient them toward a desire to generate solution strategies appropriate for grades 11 and 12 students. In other words, they were challenged to not merely solve the problem but to engage in a pedagogical process of deciding and generating solution strategies that are appropriate for the grade levels they will teach. But we wish to acknowledge the fact that even the non-calculus strategies involved very subtle non-standard, non-mathematical solutions or reasoning that could be quite challenging for grade 11 and 12 students. Also, our teacher candidates, as we already pointed out, could be classified as experts and it is widely acknowledged that experts tend to quickly characterize problems as being of particular types (Goldman, Petrosino & Cognition and Technology Group, 1999). Thus, once the majority of teacher candidates classified the problem in this study as requiring calculus, they found it difficult to reason beyond this categorization and return to the needs and abilities of grade 11 and 12 students. Through the assigned problem, the study aimed at to raise our teacher candidates' awareness of this difficulty. The fact that the challenge raised our teacher candidates' awareness is to us very important in terms of pedagogy. Of course there are some teacher candidates who were in a way oblivious to the kind of math appropriate to grade 11 and 12 levels and how the problem could be solved without knowledge of calculus. But, it was through the group discussions and class presentations that we used as pedagogical tools in our physics methods course to draw attention to our teacher candidates' use of their own prior knowledge and the mismatch with the levels of both mathematics and physics knowledge among their high school students. We see this as an important strategy in teacher education.
The pre-service teachers' views of school physics appear to have been influenced by how they understood physics and its teaching. In addition to what a number of studies (e.g. Dweck & Bempechat, 1983; Fisher et al, 1978) have revealed about orientations to teaching influencing teacher decisions and actions, Bunting (1984) proffers that assuming a variance between teacher beliefs and teacher behaviours, knowledge of the content of beliefs becomes an important first step in the identification of variables within the educational context which mediate between the thinking and practice of teachers.
