An instructional challenge through problem solving for physics teacher candidates

This paper reports on the analysis of a problem solving discourse involving two cohorts of pre-service physics teachers in a University that recruits people with a minimum of a bachelor’s degree in physics or related degree, such as engineering, for a one-year Bachelor of Education degree program. Lately, in the Canadian context, there has been an increase in the number of pre-service physics teachers with bachelor’s and even master’s level degrees in physics or engineering entering the teacher education program. This is regarded as a good thing since these are some of the previously top science students who had joined other professions because of their excellence in physics. In other words, teacher education programs in Canada are now preparing some of the top academic performers to become teachers.

This background has made it challenging for teacher education programs to critically examine the one-year teacher preparation model, as there is a widely held view that content knowledge equates to pedagogical knowledge. What defenders of this position may not appreciate is that teaching is a discipline that has rules grounded in research and scholarship. Those who join teacher education programs expect to be prepared to teach effectively. Effective teaching in part involves the ability to develop grade-level-appropriate instructional strategies. These strategies vary from one discipline to another. Physics embodies a problem-solving character that uses mathematics knowledge as a tool. But caution needs to be taken at the teacher preparation level to help pre-service teachers link their own content knowledge with pedagogical strategies appropriate for their students. Our pre-service physics teachers are highly qualified individuals who have more than enough content knowledge to teach high school students. However, subject content knowledge does not equate to pedagogical content knowledge (Shulman, 1987) or a teacher’s ability to develop effective instructional strategies.

Planning for instruction, including selection of resource materials and instructional strategies is very important for effective teaching. However, one-year teacher preparation programs are short in duration, and hence, run the risk of mechanistic apprenticeship of pre-service teachers. At the University of British Columbia, our teacher candidates spend a short time (38.5 hours) on physics methods out of the 12 months spent in the Teacher Education Program. But to investigate the teacher candidates' ability to develop grade-level-appropriate problem solving strategies, at the end of each physics methods course, we have given our pre-service physics teachers a physics problem to solve as homework. We require them to use as many methods as possible. Our teacher candidates are specifically challenged to generate at least one solution procedure that employs methods the majority of high school (grade 11 and 12) students can easily understand or in which they have some knowledge or experience.

In this paper, we use one particular problem as an exemplar because of the various approaches it attracted from two cohorts of 11 and 16 pre-service physics teachers in two consecutive academic years, Y1 and Y2, respectively. Though limited to the two cohorts and this one problem, the investigation is based on our belief that through participation in a classroom discourse, pre-service teachers are provided opportunity for reciprocal learning. Moreover, engaging in this activity assisted teacher candidates' learning through socialization, allowed modeling of pedagogy, and thus elicited the teacher candidates' content knowledge including their mental stock of problem solving methods. Data for the study comes from detailed record keeping of the discussions and the exercises during the two cohorts' physics methods classes.

Group discussion discourse was employed to model what we considered to be an appropriate strategy for instructing mixed ability or diverse classes as well as offering opportunity for the teacher candidates' socialization, which is an important step to instilling into the candidates a sense and appreciation of the power of group learning. As part of our pedagogical approach to this physics methods course, we hoped that group discussion of the various problem solving approaches that were generated amongst our pre-service teachers could be provocative for their thinking about methods that they could use with their future students. Through our pedagogical method, we hope that the majority of students, and not just the few high achievers (gifted), will be “taught". It is not uncommon, for example, to find a particular problem, such as the one used in this investigation, attracting different approaches to its solution from among subject-content-rich persons (e.g., pre-service teachers). The question this raises is whether the strategies employed are appropriate to the level of the students. In particular, given that physics problem solving employs mathematics, the question is whether the mathematics employed to develop solutions is within the grasp of the students, a majority of whom do not have broadly-based background knowledge or are not “gifted". It is now widely acknowledged that building on students' prior knowledge to develop solutions to physics problems can be very fruitful in terms of student learning, thus directing us to attend to how our pre-service physics teacher candidates are socialized to consider and utilize their students' prior knowledge.

Teacher Learning Through Socialization

According to Zeichner and Gore (1990), teacher socialization is the process whereby an individual becomes a participating member of the society of teachers. As applied to this paper, teacher candidate socialization involves being a participating member of a learning group. What this means is that in belonging to the group one participates in all activities of the group including learning from other group members and through a process of reciprocal teaching (Palincsar & Brown, 1984). What is learned in such a setting can profoundly influence any future actions in activities similar to what was experienced in the group.

In other words, socialization in this paper is seen as a construction process whereby attitudes, beliefs and ways of doing things are influenced during group discussion, hence our view that we were enhancing group learning through a socialization process. An earlier study investigating teacher candidates' perceptions of the status of Physics 12 revealed that such a construction process occurred when they were high school students (Nashon & Nielsen, 2007). Teacher candidates' perspectives were in large part shaped by the experiences they had while in high school or during their undergraduate programs. Right from their experience as elementary, high school or university students the teacher candidates were immersed in the culture of teaching and learning (Lortie, 1975).

Brousseau, Book and Byers (1989) see the effects of a “teaching culture" in shaping a teacher’s education beliefs as spanning school contexts. In their study of teachers, Brousseau et al. conclude that the number of years that a teacher has worked in this capacity significantly affected or influenced their beliefs on teaching. At the beginning of their career then, it is important to help teacher candidates gain an awareness of their own beliefs about teachers alongside development of appropriate pedagogical models for use in teaching. In our view, effective pedagogy in teacher education also involves modeling learning through socialization mechanisms such as working in groups.

We do not hesitate to add that the teaching experience modeled for the teacher candidates is likely to have profound influence on their future instructional practice. Teaching culture is conveyed and experienced differently by teacher candidates as they attend classrooms with so many instructors, and this varies to some extent depending on the subject area (Burden, 1990). In the same vein, Behar-Horestein, Pajares and George (1996) consider teaching beliefs as affecting students' learning behaviour. With this background, it can be argued that instructional models used in teacher education programs are likely to influence the teacher candidates' teaching behaviour when they become teachers, and this behaviour is further influenced by their own prior knowledge.

The Role of Prior Knowledge

Development of grade-level-appropriate problem solving strategies or approaches as much as is feasibly possible should, among other things, draw from students' prior knowledge. As Kelly (1955) has said: “All thinking is based, in part, on prior convictions. A complete philosophical or scientific system attempts to make all... [this] prior knowledge explicit" (p.6). What Kelly seems to be suggesting is that prior knowledge plays a prominent role in human attempts to interpret experience. Further, understanding new concepts involves the reconstruction of incoming information in terms of prior knowledge held by the individual, and prior knowledge can be replaced or reconstructed. In the same vein, Novak and Gowin (1984) echo this view by describing the distinction between meaningful and rote learning:

To learn meaningfully, individuals must choose to relate new knowledge to relevant concepts and propositions they already know. In rote learning... new knowledge may be acquired simply by verbatim memorization and arbitrarily incorporated into a person’s knowledge structure without interacting with what is already there (p. 7).

Consistent with the above citation, Bodner (1986) quotes Ausubel on prior knowledge: “If I had to reduce all of educational psychology to just one principle I would say this: The most important single factor influencing learning is what the learner already knows" (Bodner, p. 877).

We see a trend of scholars and researchers all trying to underscore the role that students' prior knowledge plays in learning. Already this paper underscores this condition of learning. Ausubel (1968) delineates meaningful learning from the rest saying: “It is apparent... that insofar as meaningful learning outcomes in the classroom are concerned, the learners' cognitive structures constitute the most crucial and variable determinants of potential meaningfulness" (p. 40). This way of acquiring knowledge is what has come to be known as constructivism. Theories that guide this method of learning subscribe to the view that knowledge is constructed and not just merely added. Hodson (1998) sheds more light on what constructivist theories of learning are all about. Learning is about the process of eliciting, clarifying and constructing new ideas, all of which take place in the mind of the learner. It seems therefore in the interest of good pedagogy to elicit students' prior knowledge, which very likely will influence the understanding of or meanings accorded to new concepts or experiences (Ausubel, 1963, 1968; Bodner, 1986, Kelly, 1955; Novak & Gowin, 1984).

The power of prior knowledge in influencing conceptual understanding or practice has been underscored in contemporary research and literature in science education. We have even seen this evidently reflected in successful analogies - those that employ the use of ‘knowns' to explain ‘unknowns' (Nashon, 2000, 2001, 2004a, 2004b). The process of building on prior knowledge involves a reconstruction of the already possessed knowledge systems (accommodation), where the existing knowledge is inadequate to explain new encounters/phenomena or filling gaps (assimilation) within the existing knowledge systems (Posner, Strike, Hewson, & Gertzog, 1982). This creates a cognitive conflict where the learner is challenged to fit the incoming information into the existing knowledge (Gunstone, 1992). But the question remains whether all students' prior knowledge is “acceptable" to the scientific cannons of physics. Some ‘knowledge' could be counter-physics, or even, “un-physics."

It seems to make good pedagogical sense for the physics instructor to identify counter physics ideas and target them by providing experiences in which such “un-physics" ideas get challenged. In fact, according to Hodson (1998), “Secure conceptual understanding is the ‘trigger" for changing the language and for making progress towards more sophisticated understanding" (p. 24). Thus, in the current paper, apart from illuminating the challenges pre-service teachers confront in developing grade-level-appropriate problem solving strategies, which is a skill we believe must be deliberately taught and exemplified, we aimed to reiterate the importance of prior knowledge in learning. And given the fact that mathematics is a tool of physics (Von Weizsacker & Juilfs, 1957), there is strong evidence to suggest that physics instructors should utilize students' relevant prior knowledge in explaining solutions to intended physics problem solving tasks. This requires a great deal of preparation and thinking, which is consistent with developing teachers' pedagogical content knowledge [PCK] through their professional training and practice.

Through the process of becoming qualified teachers, the problem solving strategies modeled in this study would then be part of the teachers' PCK (Shulman, 1986), including their understandings of the connections between physics and mathematics. In a recently concluded study about the status of Physics 12 in British Columbia, the physics teacher and teaching styles were prominently mentioned as impacting students' decisions about Physics 12 (Nashon, 2005). In the same vein Blanton (2003) and Kumagai (1998) indicated that quite often science teachers conform to instructional models they were exposed to as high school students. In retrospect, this, in part, constitutes the kind of pedagogical knowledge that is constructed or modeled in the context of teacher candidates' problem solving tasks that could shape their view of the nature of problem solving in physics.

Connection Between Mathematics and Physics

According to Weizsacker and Juilfs (1957), “The tool of conceptual thought in physics is mathematics, for physics treats the relations measured, which is numerically determined, magnitudes" (p. 11). The connection between mathematics and science (physics included), is further expressed by Kline (1980): “Science must seek mathematical description rather than physical explanation. Moreover, the basic principles must be derived from experiments and induction experiments" (p.51). This is the principle on which Newton and Galileo operated, and in which contemporary thought in physics still resides. The importance of mathematics in physics classes is evident and is perceived so by instructors of physics. Curriculum materials portray a similar image: it is virtually inconceivable to have a page in a physics textbook ending without a single equation or other form of mathematical expression.

In some cases physics is synonymous with mathematics. In short, mathematics is important as far as physics is concerned, but with due respect to other forms of knowledge domains. But, as far as physics is concerned, mathematics constitutes a large portion of its language. What is troubling though, is the fact that some instructors of physics seem to recognize this importance and yet never make deliberate effort to sharpen their physics students' mathematical knowledge needed for the moment - a moment when the mathematical knowledge appropriate to the teaching of the intended physics concept is required.

Arguments exist about whether or not physics can be taught without the use of mathematics (e.g. Tao, 2001). Nonetheless, it is almost impossible to imagine complete physics knowledge without its quantitative aspects (Nashon, 2006). In other words, it appears almost a given fact that the physics knowledge domain is constructed through both qualitative (involving observation and description) and quantitative (involving measurements and calculations) methods.

The current investigation centred on quantitative methods of problem solving, since these employ mathematics as a tool for use in the process of physics knowledge construction. The challenge is how to teach high school students, many of whom may not be as proficient at using this tool of physics as we would want to assume (Basson, 2002; Nashon, 2005, 2006).

Many problem solving tasks in physics are characterized by the use of equations and other forms of formulae. In our view, students coming to physics classes where instructions utilize knowledge of equations they already know experience minimum obscurity of the intended physics concepts by the mathematics. Conversely, if too much new information is to be learned concurrently or over too short a period of time, students may experience cognitive overload. Of course, we might partly appreciate this, in a metacognitive sense (Gunstone, 1992; Nashon & Anderson, 2004), as we consider fundamental issues such as those underscored by Sherin (2001):

What does it mean to understand an equation? The use of formal expressions in physics is not first, a matter of rigorous and routinised applications of principles, followed by the formal manipulations to obtain an answer. Rather, successful students learn to understand what equations say in a fundamental sense; they have a feel for expressions, and this guides their work (p. 479).

Because the majority of high school students lack a depth of mathematical competency, understanding of physics tends to be obscured by the students' attempt to understand the mathematics, which is used to develop the logical arguments that bring about understanding of the intended physics concept(s), which Ausubel (1963, 1968) describes as meaningful leaning. The lack of mathematical competency could lead to an over-emphasis on qualitative methods by physics teachers. But, such an approach would necessarily be limited in scope, since certain aspects of physics explanations are rooted in and explicated through the mathematics knowledge domain (Nashon, 2006). It is for this reason that the current paper reports on an investigation aimed at determining pre-service teachers' ability to appropriately employ problem solving methods that utilize mathematical tools that are suitable to the grade level of students they are teaching.

There is a question as to whether physics instructors discern what prior mathematics knowledge their physics students possess so as to apply it to the intended physics concept. This paper underscores the idea of sensitizing pre-service physics teachers during their initial teacher preparation as to the need to always build on students' prior knowledge. Therefore, as an attempt to underscore the importance of this aspect of pedagogy, in the current investigation, we used a problem solving task that drew from the kinds of knowledge that the majority of high school students already possess, and which is open to a variety of problem solving approaches. Through this pedagogical approach, we examined our pre-service teachers' ability to use mathematically modeled strategies that were appropriate and relevant to grade 11 and 12 physics teaching. The intention was to sensitize the pre-service physics teachers to the fact that, although all possible approaches could lead to the same answer, some high school students may not understand all of the approaches. Hence the questions for the current study: What problem solving strategies does a physics problem challenge elicit from teacher candidates with content-rich physics backgrounds? Which of the strategies are appropriate to grade 11/12 levels and employ grade-level-appropriate mathematics? What pedagogical implications does this experience offer?