Part II - Significant Findings of the StudyStudent Views on Learning and Physics

From the first day the participants arrived with a desire to learn about physics and throughout the project generally looked forward to being in class. They were motivated by a variety of factors, including personal interest and knowing that Physics 30 was a prerequisite for certain career choices. At no time during the study did I deliberately have to spend time motivating them to engage in learning physics.

The students had some perceptions about how they learned and the role that teachers had in that process. For the most part they were receptive learners at the beginning of the study and were not comfortable being actively involved in their own learning. They expected me to provide instruction in a direct manner, either through notes or by directly answering their questions. Their expectation was that the notes and answers would explain the physics concepts and that they would develop understanding from these explanations. Initially they were uncomfortable when the instructional strategy did not fit their receptive style of learning.

When the students became comfortable with being actively involved in their learning, they were incredibly active. Although there were only nine students in the class, interactions between participants were numerous. Conversations erupted spontaneously among students and between students and teacher. They discussed ideas with each other and openly argued about interpretations and meanings of definitions, concepts and problems. The high frequency of interactions about physics was a revelation to me. The participants were more talkative during their attempts to understand physics concepts than I had expected. All students were involved in these interactions. Without the videotape records the large number of these learning interactions would not have been fully appreciated.

The participants believed that natural ability was an important factor in their own and others success or lack of success in physics. They believed that some of them had more ability than others to do physics. On occasion individuals assumed that they did not have the ability to use the mathematics which they believed were required in physics. While a range of ability did exist among members of this group of students, they were not always correct in their assessment of their own ability or of other individuals. Their main tool for assessing ability was the marks that they received on tests and assignments. They undoubtedly viewed these marks/grades as a measure of ability as well as an indication of achievement.

Their understanding of the structure of physics knowledge caused them some problems. They perceived that there were knacks or tricks to doing questions and searched for a system of steps to follow that would lead them to correct answers. The participants wanted an all-purpose algorithm (set of steps to follow in solving a problem) which they could use to solve all types of questions. Rather than viewing theory as underlying principle, they appeared to view scientific theories as algorithms which could be used to answer problems. Theories were thought of as providing understanding of phenomena in the world by answering their questions. This interpretation is consistent with their reliance on experiential knowledge to understand phenomena and with their lack of development of conceptual knowledge.

Their use of algorithms was non-discriminatory; that is, they picked an algorithm that matched the variables in the problem. They demonstrated, for example, considerable difficulty answering questions involving kinetic energy or momentum because both concepts depend on mass and velocity. They looked for a formula that contained mass and velocity and solved the resulting equation, rather than basing their analysis of a problem on underlying physics principles. Since both kinetic energy and momentum formulas contain an m and a v, they often chose the incorrect formula for solving a problem. Once they employed an algorithm and determined an answer, they did not reflect on whether the answer was reasonable or not.

The participants were aware of difficulties and inconsistencies in their knowledge construction. When a topic was presented and not understood, they asked questions in an attempt to reduce or eliminate their confusion. Initially they were somewhat reticent to talk about their confusion; however, as they became more at ease, they openly discussed their confusion with me and the other students. For example, when learning about momentum they knew that they did not understand the concept in spite of having done a laboratory activity which was designed to explore momentum. Although achieving good results on the activity they knew that they did not understand momentum.

They appreciated the non-judgmental atmosphere of the classroom, including my own reactions to their struggles and the reactions of their classmates. There were very few cases of students “putting each other down” during the months of the study. In reviewing the tapes and tran******s no cases were identified where students had to be reminded not to discourage each other. They had trouble describing their thinking and problem solving as they learned but without a supportive classroom environment they would never have attempted to do so.

The participants employed a variety of strategies to construct physics concepts but did not appear to attack knowledge construction in a planned or coordinated manner; that is, I do not think they had identified principles of learning that they applied to the process. While other examples occurred the most clearly documented case illustrating different strategies of knowledge construction were exhibited during the discussion on momentum. Four different individual attempts at constructing the concept were evident in a twenty minute discussion. The students clearly learned about momentum differently, even though they had shared common classroom experiences.

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Students’ Experiential and Conceptual Knowledge

In this report experiential knowledge refers to knowledge that students bring to class as a result of their life experiences. This knowledge includes all experiences they have had during their lives and the thinking that they have done to organize their knowledge to help them operate in their world. On the other hand conceptual knowledge is theoretical in nature. This knowledge is formed in the mind as a result of reflection about experiences and generally has principles that can be used to explain a number of experiences. Science requires both types of knowledge, but the conceptual knowledge is the abstract part of science which serves to organize knowledge using laws and theories.

The participants relied on their experiential knowledge to an enormous extent when learning physics. They had not developed conceptual (theoretical) knowledge that was useful in physics and did not seem to understand the process of using conceptual knowledge to explain and understand natural phenomena. Although they used mathematical formulas in calculations, they did not understand the process of representation that has been used to create the formula. When the mathematics became more complex, they did not trust a model to provide interpretations of situations; for example, when asked to calculate the change in velocity of a car which went around a corner at constant speed, they did not think that there had been a change in the velocity, and did not understand what the calculated answer meant. The mathematical model was not useful in assisting them to understand the situation. A second example of the lack of use of conceptual knowledge was displayed when they solved a force board problem near the end of the project. Even those participants who used vector mathematics properly were unable to state an adequate reason for using vector mathematics in their solution. Their choice was determined by intuition and previous examples rather than realizing that vector mathematics are required to represent the properties of forces.

The participants did not reflect to any extent on the application of physics principles in their everyday experience. They had not thought about the action of curling rocks or the place of numbers in science until asked to do so during the study. Initially I was concerned that my interaction was not skilled enough to reach the limits of their reflection about such concerns; however, I no longer think this to be the case. The life experiences of the participants had not created any need to think about the nature of knowledge or what it meant to learn. Generally the participants seemed to be positivist in their view of the world and believed scientific laws were discovered in nature. Their belief was that physics concepts really existed in nature, rather than being constructed by humans to organize and understand their world. On occasion they talked as if they had a constructivist view of the world, but they did not understand the ramifications that such a viewpoint had for learning science and other subjects.

The participants worked well with laboratory apparatus. On several occasions they demonstrated their ability to operate laboratory equipment skillfully. They understood what the equipment was meant to do and the measurements that they were supposed to make during the experimentation; however, this understanding of the operation of laboratory apparatus did not appear to translate into understanding at a conceptual level. This outcome was disappointing because a traditional argument for the use of laboratory activities in all science classes had been to provide concrete examples of concepts that are being studied. These concrete examples were expected to help students develop more understanding of the concepts involved.

On one occasion students were asked to lift a heavy bucket of sand using two ropes and pulling them at various angles. This activity illustrated their lack of understanding of the process of mathematical representation of concepts identified in everyday situations. They were able to describe the relationship between the angle of the rope and the force needed to lift the bucket with considerable accuracy in a qualitative manner; however, they made no headway in representing that relationship using vector mathematics. The ability to perform this difficult representation process was never demonstrated during this activity. Their struggle with conceptual knowledge, its use and development, was ongoing throughout the study.

When asked what concepts were easiest to learn, the participants listed those that they could visualize or identify in their everyday experiences. They were unable to visualize concepts which they considered to be difficult to learn, and wanted me to provide “hands-on” activities and practical examples of abstract concepts. They thought that if they could understand how a concept was used in their experience, then they would be able to understand the concept in physics. In spite of this belief the students did not demonstrate very much development of conceptual knowledge as a result of the hands-on activities.

Although the participants at times exhibited some characteristics of meaningful learning, more extensive probing of their understanding revealed that they had mainly achieved rote learning. The learning that occurred as the result of the momentum cart lab was a good example of this type of mimicry. The manner is which they manipulated the equipment and lab reports submitted indicated they understood some aspects of momentum. Their lack of understanding was identified only when they tried to answer questions which probed their conceptual development. The instructional sequence produced the expected results but these did not accurately indicate the level of student understanding. The assessment items used during this activity, student observation and submitted report, did not correlate well to their conceptual understanding of momentum.

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Student Alternative Conceptions and Communication

As was expected the students showed confusion over the use of words which are used in everyday language but have separate and distinct meanings in physics. Separation of vector addition and subtraction from algebraic addition and subtraction was difficult for the participants who, at times, used the algebraic operations when using vectors. Equally apparent was their confusion between the termsbalancedandequalwhen using vector components. They wrote equations for relationships between components which indicated that they thoughtbalancedmeantequal.They looked only at the magnitude of the component vectors and did not consider the directions of the components as significant.

Confusion over the meaning of pairs of words provide strong evidence for the necessity of clear communication between teacher and students. While this need is always assumed by teachers, the apparent insignificance of an item that can cause a breakdown in communication can not be underestimated. Confusion can arise over seemingly minor points resulting in learning blockages which either produce faulty knowledge construction, or block it altogether. The most important feature documented in this research is that many of these causes of confusion are not identified in the classroom as instruction occurs. Students may be aware of some blockages, especially those that stop learning completely, but are unaware of others because knowledge construction continues but in a wrong direction leading to some form of alternative conception.

Student transfer of knowledge about vector mathematics between mathematics classes and physics was almost nonexistent. No student in this group brought sufficient understanding of vector mathematics to be of practical use in physics. Some had learned algorithms for addition and subtraction of vectors but could not recall them completely. They brought only a poorly developed concept of what a vector is, and no one came with conceptual understanding of what it meant to add or subtract vectors. Teaching the rudiments of vector mathematics in physics classes is likely to continue for the foreseeable future, at least until a different approach is used in teaching these concepts in mathematics classes.

This inquiry has reinforced my understanding of the value of using student questions and comments to build models of their knowledge construction and conceptual development. Students ask questions and make comments on the basis of what they think they understand about a concept. The structure of their knowledge is indirectly revealed in the way that they phase their questions. By using their questions and asking others I was able to explore their knowledge development. Student responses on tests and quizzes, and work at the board provided additional sources of data for development of these models of student knowledge construction. For teachers to develop such models of student knowledge construction, interactions among students and teacher have to occur openly. In a classroom which is highly teacher-centred, this type of model development is not possible, because students do not have opportunities to talk about their developing concepts with the teacher or each other.

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Mathematical Representation of Concepts

The detailed analysis of student thinking made in this research confirmed that the participants did not understand the process of mathematical representation of physics concepts well enough to apply the process to new concepts and situations. Because of this research and my teaching experience I think the majority of secondary school physics students do not have a fundamental understanding of this important process. Students learn to manipulate formulae that are part of the course curricula but, in general, do not understand the relationship between the formulae and the concepts which the variables in that formula represent. Students treat physics formulae as algebraic expressions to be manipulated mathematically rather than representations of certain quantities identified in nature. To some extent I have fostered this attitude in my students by providing algorithms for problem solving and clues in the questions to help students choose the correct pathway to the solution. In this study I attempted to provide the participants with a different view of this relationship by looking at the place of numbers and mathematics in science, but the results of the study are strong evidence that this change was not enough to create understanding of the representation process.

The enhancement of student understanding of the process of mathematical representation cannot be achieved in grade 12 physics classes alone. Students need experience with the principles of mathematical representation much earlier in their formal education than the last year of secondary school. Courses in math and science taken before grade 12 physics will have to begin to develop these skills and understanding. The curricula of those courses will have to be restructured to provide students with primary experiences constructing mathematical representations, rather than observing them as secondary experiences from a teacher or text book. Classroom experiences could be formulated so that their successful solution is dependent on students developing mathematical representation for the concepts under investigation. Computer software and graphing calculators have potential to provide simulations of this process and to perform those mathematical manipulations in which students tend to get bogged down. The use of computers and the Internet in assisting students in constructing physics concepts is an area for further research.

Post-secondary educators would also benefit from students who had a better understanding of the process of mathematical representation. In post-secondary science courses the representation process is essentially the same; however, more complex mathematics, such as statistics and calculus, are required to represent the relationships between concepts with accuracy. Students entering subject areas such as biology, chemistry, ecology and economics, as well as, physics would benefit from a more complete understanding of the use of mathematics in representation.

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Recognizing Direction as a Characteristic

I reflected for some time about the students’ inability to identify direction as a significant characteristic of certain physics concepts before gaining even a hint of insight. In their lives most experiences and problems did not require the awareness of direction that is needed in physics. All participants had driven cars which would seem to be an experience requiring some knowledge of direction; however, closer examination reveals that this was not so. When beginning a trip a driver must start out in a particular direction, but after choosing the correct road few navigational skills are required to arrive at a destination. Students tended to see direction as a means of relating positions on the earth and not a characteristic of certain concepts in physics. Their conception of direction was not the same as that of practising physicists.

The participants memorized algorithms to solve problems that involved direction and used clues in the problems to identify which algorithm to apply. These clues were normally present in the questions as part of the written de******ion. Educators have assumed that successful problem solving of this nature would lead to the development of understanding as experience was gained. Over the years this approach appeared to be an effective way to teach students because they successfully answered problems. This research study has helped to show that this instructional approach did not produce the depth of understanding which was traditionally thought to have been created.

This lack of identifying the importance of direction in physics concepts adds to the inability of students to understand applications of vector mathematics in physics. Without identifying direction as a fundamental characteristic of certain quantities students cannot be expected to see any reason to use vector mathematics in solving problems; and, vector mathematics will make little sense to them until they are able to understand why direction must be part of some mathematical representations. The participants did not benefit a great deal from separate instruction about vector mathematics in geometry-trigonometry classes as was shown by their unanimous surprise that vector mathematics could be used to represent anything in physics. None of the students in this research had developed sufficient understanding of vector mathematics in their mathematics classes to be able to make use of that knowledge in our physics class.

In a sense the confusion is the result of students’ alternative conceptions of direction. Students have a conception of direction in their vocabulary and use this meaning in the physics environment. Their meaning is based more on using direction to describe the location of some object or destination with respect to some fixed point. For example, a car is located to the left of the doorway, or Canada is north of the United States. They do not understand the concept in the manner that is required for success in physics in that they do not associate concepts such force and velocity with having direction. Without a more appropriate comprehension of direction as physicists use it, grade 12 physics students will continue to struggle with the use of vector mathematics. Conceptual development strategies, as described earlier, may produce some of the reconstruction required for successful problem solving involving direction, but more study and research are required before a practical classroom solution can be developed.

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Applying Vector Mathematics in Physics

The students did not understand the process of mathematical representation to any great extent, nor did they understand that direction is a fundamental characteristic of many physics concepts. These two factors combined to make the use of vector mathematics even more difficult for the participants and most grade 12 physics students. The participants lacked a perception of any need for vectors or vector mathematics. They did not have a sense of why they had been taught about vectors in other courses nor could they describe any practical applications when we talked about vectors early in the study. Some were able to perform addition and subtraction using algorithms but they did not exhibit understanding of the mathematical principles involved in these processes. This deficiency was illustrated when they drew vector diagrams to help with adding and subtracting vectors as part of solving problems. Most students did not view these diagrams as aids which showed a resultant vector; rather, they saw the diagrams as separate problems which made the problems more complicated.

The research results show that the difficulties experienced by students learning to apply vector mathematics are very complex. Three elements of the struggle have been described, mathematical representation, alternative conceptions of direction, and not understanding the function of vector mathematics. A simple solution to this problem does not exist because of the complexity of the learning processes that have to be achieved by students. The three elements must be dealt with together and successful resolution can not be achieved in one five-month semester in grade 12 physics. Solutions to student difficulties in applying vector mathematics in physics have ramifications for science and mathematics courses at earlier grade levels. Students must be assisted on three fronts: first to understand the representation process; second, to develop a different conception of direction; third, to develop an understanding of the purpose of representing certain concepts with vectors. Resolution will take considerable time and innovation to create instructional strategies and experiences to accomplish these goals.

I have discussed the three elements separately but any solution will have to incorporate their interdependent nature. While the concerns described are fundamentally cognitive in nature, they must be addressed in curricula to some extent because curricula largely determine what is taught in science classrooms. Resolution will have to start much earlier in science and mathematics education. Elementary and middle years science teachers will have to begin to provide experiences that develop student understanding of these ideas and relationships. Students need opportunities to test their own knowledge in real-life experiences and then to reconstruct it in light of them. Most science teachers do not have the arsenal of instructional strategies and experience necessary to create these experiences for students because the type of instruction that I am advocating had not been used to any extent in science education. This result will also have ramifications for teacher education programs.

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Part III - Considerations and Recommendations Abstract concepts, such as momentum, energy and entropy, are currently a part of the Physics 30 curriculum and are likely to remain so. The nature of these concepts cannot be changed, but instruction can be modified to assist students in achieving something closer to meaningful learning instead of simply memorizing formulae and definitions. Different instructional strategies can be developed to facilitate student construction of conceptual knowledge. Learning about this group of concepts will never be a simple matter, but students are likely to develop better understanding if they learn more about the structure of physics knowledge and the process of mathematical representation than is currently expected in secondary school curricula.

Secondary physics curriculum guides usually describe student learning in terms of outcomes or objectives, but do not provide guidance to teachers in promoting student conceptual development. Physics and other science curricula are not designed to have students explore the relationship between science concepts and mathematics, or the process of mathematical modeling. Science and mathematics courses are developed with little attempt to coordinate *******/topics in mutual support. Some mathematics courses may be required as prerequisites of physics courses, but students can not be assumed to understand the process of mathematical representation used in science as a result of those mathematics courses.

One objective of physics curricula is to have students understand the nature of science knowledge and the processes of science/physics (Saskatchewan Education, 1992). To achieve this goal student learning should be meaningful and new knowledge should be connected to what the learner already knows. This research indicates that reducing the amount of ******* in physics curricula, making ******* more relevant and meaningful for students, and increasing connections between mathematics and physics would be constructive changes in assisting students to achieve meaningful learning within secondary school physics.

Table of *******sAmount of ******* in Secondary School Physics

Throughout my teaching career and this research project I have been aware that students did not develop the depth of understanding of physics concepts for which I was aiming. Because of pressure to cover the ******* in the grade 12 physics curriculum, additional time was not spent helping students develop a conceptual knowledge base to inform their experiential knowledge. I have found it impossible to help students develop conceptual physics knowledge in the time allotted; however, the length of time spent on a given concept is not the only issue. Alternative teaching strategies and learning experiences must be developed to increase learning success. Using the same instructional strategy for a longer period to time will not increase student knowledge development. When curricula are designed with coverage of concepts as a major driving force, the pressure to move on to the next topic or unit dominates teacher decision making. Until a change in curriculum focus is made, the pressure to “cover the course” can not be ignored by teachers. Good pedagogy should direct teachers to ensure an adequate level of student understanding before moving onto a new concept or unit; however, good pedagogy is rarely the driving force in these decisions because of the overwhelming pressure to cover the *******.

If students can answer problems and “do the math,” then they are assumed to understand the function of mathematics and mathematical representation in physics. This research has illustrated the inaccuracy of that assumption. These participants did not demonstrate understanding of the process of mathematical representation even when instruction was designed to enhance it. Little time is allotted to examine this relationship in most physics, science and mathematics courses. Without making the connections between physics concepts and fundamental processes of physics, students can not achieve adequate understanding. New curricula in physics have to reduce the number of physics concepts explored and allow students more time to develop understanding of the processes and relationships in physics. If changes are made only in grade 12 physics, then success is unlikely. To ensure better exploration of the mathematics/science relationship changes to the curricula of other science and mathematics courses taken prior to Physics 30 are required. Most student knowledge of the place of mathematics in science and physics was obtained in the courses taken previously. Changes in teaching science and mathematics in earlier grades could provide the background for application of mathematical models necessary to physics. The rush to move on and cover the ******* in physics might be alleviated if understanding of the processes was learned earlier in students’ formal education.