This is a report about two students and two teachers in a high school physics class at an urban high school. One of the teachers is a highly experienced, competent teacher who has been working with students for over two decades. The other teacher is the author of this report. I have taught one lesson at the time of this writing. The intent of this report is to examine the effectiveness of the instruction that occurred during three consecutive days in the classroom by interviewing two students about the content that was presented to them.
Due to unfortunate circumstances, the student interviews were necessarily rescheduled such that the following events occurred: On Monday (hereafter labeled day 0), I taught an introductory lesson on the reflecting properties of concave mirrors. On Tuesday (day 1) and Wednesday (day 2), the classroom teacher (hereafter labeled CT) continued instruction. Interviews were conducted on day 1 and day 2. Unfortunately, my own instruction weighs heavily into the students' initial understanding of the concepts -- and we will see this manifest itself in the interviews. Hence, this report is perhaps biased by these events. At the same time, completing this assignment has been an invaluable experience for me since I can scrutinize the students' understanding and misconceptions and can then attempt to alter the teaching of this content in subsequent years in order to correct such misconceptions.
The structure of this report will be mainly chronological. I will present a day by day analysis of the teaching that occurred and the student interviews that followed. A final summarization of the analysis will appear at the end of this report along with concluding reflections. By no means does this report contain all information in the interviews. While each interview was very interesting and revealing, space does not permit such a detailed discourse. The content that this report will discuss is typical of a senior level class in high school. I will presume that the reader of this report has a basic understanding of the "law of reflection", which states that the angle of incidence is equal to the angle of reflection when a light ray strikes a surface. For those readers who need further information about this principle, I would urge them to examine carefully Appendix II before proceeding.
Typical of teachers' education programs are evaluations by the supporting university. On day 0, my teaching was evaluated by a university supervisor. The lesson plan for this day was to introduce spherical concave mirrors and the real images that result from them. Appendix III contains photocopies from the textbook from which I was to draw content and the lesson plan that I used. Also included are the projection screen slides and handouts that the students received.
I began the lesson with a demonstration that illustrated how (concave) bathroom magnifying mirrors could be used to project images of candles upon a screen. Following the demonstration, I discussed and demonstrated the physical principle that causes such images. The critical physical law that governs the production of these images is the law of reflection. A light source, such as a candle, radiates light evenly in all directions. If we isolate a single ray of light, and follow it to the mirror, it will obey the law of reflection. The reflected rays recombine at another location to form a real image. I demonstrated this principle by first reviewing the law of reflection for a planar mirror, and then applying this law to spherical concave mirrors. I urged the students to understand that the law of reflection, was the governing principle since it is a fundamental structure of the subject of physics and according to Bruner, "understanding the fundamentals makes a subject more comprehensible."
Using the law of reflection to find an image can be tedious -- as it involves calculating the incident angle, and then drawing the reflected ray such that it makes the same angle with the normal line. This process must be repeated several times for different incident rays. To reduce the number of rays that must be drawn, physicists have developed the "ray-tracing-method." In this method, only three critical rays are drawn from the object to the mirror in order to find the image. These rays must obey the law of reflection -- but they are chosen such that they are easy to remember (please see p.321 of the textbook). The textbook mentions that these rays must obey the law of reflection, but stresses the ray-tracing method.
Since the law of reflection is extremely important in physics, I repeatedly demonstrated to the class how this works in union with the ray-tracing-method. There is a danger in using the ray-tracing-method: the fundamental law of reflection can be lost in the process of memorizing 'which rays are supposed to do what' while applying the ray-tracing-method. After students practiced using the ray-tracing-method, we derived the mirror equation as a class experiment in accordance with Schwab's idea that syntactic knowledge of a discipline must be taught (the experimental process) as well as the substantive structure (the mirror equation). The mirror equation allows one to compute the location of the reflected image if one knows beforehand the location of the object and the radius of curvature of the mirror. The textbook simply presents the formula to the students without a formal derivation. I believe in Bruner's ideology -- that any subject can be taught to students at an intellectually honest level, so I opted for an experiment in which the students would have a role in developing this equation. By doing this, I was hoping to meet one of Bruner's ideals:
"Various people who have worked on curricula in science and mathematics have urged that it is possible to present the fundamental structure of a discipline in such a way as to preserve some of the exciting sequences that lead a student to discover for himself. [p.20]"
The experiment worked well, as the students were able to derive the correct expression from data that they had created while practicing the ray-tracing-method. There is one final important point that I would like to make about this lesson. The class arrived at the following expression: 1/f = 1/object distance + 1/image distance. I would like to draw the reader's attention to the fact that 1/f appears on the left-hand side of the equation. The book presents the equation in reverse order (see page 322 of the textbook). The order does not change the meaning of the equation -- but it will be an important detail to remember for the interview process.
My day in the limelight was complete. I was relieved to have the CT at the front of the class so that I could focus my efforts on completing this report. The CT began class by drawing the students' attention to a carefully constructed diagram that he created on the chalk-board of a concave mirror that used the ray-tracing-method to locate the image. During the entire class, the law of reflection was never mentioned -- but rather the ray-tracing-method was underscored.
He then presented the magnification formula to the students as it appears in the textbook. The magnification formula is used to determine the size of the image if the size of the object and the location of the object and image are known. As part of the homework that I created for day 0's lesson, the students will derive this equation using simple geometrical figures. I was relieved that he merely presented the equation to them. He then proceeded to work through a few examples of this equation's use. Lastly, he asked the students to generalize their findings for concave spherical mirrors by asking them to consider the five possible locations of the object relative to important physical characteristics of the mirror. He asked them to locate the image and the relative size of the image for each case. Mostly, the lesson was a review of the material that I had covered on day 0. With twenty minutes remaining in the class, the CT allowed the students time to work on their assignments.
Student A (hereafter labeled by 'SA') is described as a very bright, quiet student by his CT. However, he doesn't always put his best effort into his work. I initiated the interview by asking SA what today's lesson was about. He admitted that he wasn't paying attention for most of it --but still conveyed the essence of the lesson accurately to me.
For all of the interviews, my focus of concern was centered on testing the students' understanding of the law of reflection. Hence, I asked SA if he could remember the fundamental law that governs the way that images are formed. Instead of describing the law of reflection, he stated the mirror equation in the same form that we had derived in the class -- not the manner in which the book had presented it. This was very revealing. For a student who usually puts little effort into homework, he did a remarkable job of remembering a "trivial" formula. The fact that he recalled it in the form that contains 1/f on the left hand side indicates to me that durable ownership of this equation resulted due to the experiment that he participated in --further strengthening Bruner's claim that active construction of one's own knowledge is essential.
Although this diversion was interesting, it showed me that he remembered the equation, rather than the fundamental principle involved. I prompted him further by placing an illustration of a plane mirror in front of him and drew an incident ray. I asked him to draw the emerging ray and explain what his reasoning was. He immediately utilized the law of reflection to draw the correct reflecting ray -- but couldn't remember the phrase "law of reflection". His use of the words "angle of incidence" and "angle of reflection" suggested a complete understanding of the law applied to this mirror. Although it might have been nice to have him remember the name of such an important law, I was happy that he could apply the principle. .
The next stage of the interview utilized Vygotsky's zone of proximal development. SA was very capable of constructing images from planar mirrors and concave mirrors. Both of these genres had been discussed in class. But what about convex mirrors? Could he locate the image that a convex mirror generates even though he hadn't studied these previously? If one grasps the fundamental law of reflection, then one should be able to construct such an image. .
I drew an example of a convex mirror (figure AI-1) and asked him if he could locate the image for this type of mirror. He began to search for the focal point of the mirror, indicating to me that he was going to apply the ray-tracing-method rather than the law of reflection. When he couldn't find one, I asked him to consider the law of reflection. Immediately, he began constructing tangent lines to the circular arc and drew the appropriate rays. The fact that he needed some "adult" supervision indicated that his knowledge of these mirrors was within his zone of proximal development. But more importantly, SA used an alternative method for using the law of reflection. He used tangent lines to construct his rays rather than using normal lines that radiate outward from the center of the circular arc. I had taught the class the latter method during day 0, but SA was able to apply a more general version of the law of reflection. This clearly demonstrated that he had an excellent understanding of the use of this law. The use of such diagnostic problems is a valuable tool for understanding the students' cognitive structures according to Goetz, et al. We will see in the next interview that they can clearly reveal misconceptions that students may hold.
Student B (hereafter labeled as SB) is described by his CT as "not as bright as SA", but possibly works harder. Indeed this was the case. When I interviewed SB, he had already completed the assignment before the class started. During class on day 1, SB commented that he had tried to use a concave spoon to project an image of a flame at home. It didn't work for him and he asked the CT for an explanation. It appears to me that this student is highly motivated to learn the material since he went as far as attempting a similar experiment during his own time. His interview reveals a that he has a deep interest in physics, especially since he has been noticing how it manifests itself in everyday objects. .
When I asked SB what today's lesson was about, he did superb job of recounting the day's activities. Several times during the response he used the word "reflected". Although SB has gained ownership of the word "reflected", during the problem solving section, we notice misconceptions about the application of this term. When I asked him what the fundamental law of the past days' lessons was, he plainly stated, "the law of reflection." I presented the same problem to SB regarding the planar mirror and asked him to apply the law of reflection. He first constructed the normal line, whereas he should have first drawn the incoming ray and then he should have constructed the normal line at the point of intersection. He later discovered his error and presented an accurate representation of the law of reflection for planar surfaces. .
I then posed SB with the identical convex mirror problem that I had given SA. Unfortunately, I had made a slip of the tongue and told him to apply the law of reflection. Despite this, he determined a focal point (on the wrong side of the mirror) and fully completed the ray-tracing-method in order to find an image. His use of the ray-tracing-method would have been correct if this mirror had been a concave mirror and if the focal point would have been correctly determined. .
I then asked him to construct a normal line at one of the points of intersection and asked him to verify that the law of reflection was intact. He did so, and realized that the law of reflection was not upheld. He then went about applying the law of reflection using the method that I had taught in class to construct the appropriate rays. The fact that SB immediately began using the ray-tracing-method even though I had stated to use the law of reflection indicates to me that this student isn't fully aware of the fundamental law that governs mirror reflections. The ray-tracing-methods were in direct violation of the law of reflection, yet the student never made an accuracy check using both methods. This is cause for concern and reflects the manner in which the student constructs knowledge from the instruction. SB is the type of student who may be skilled at applying algorithms without understanding why the algorithms work and lacks the syntactic knowledge to verify the correctness of such algorithms when applied to different situations. During the final summary, I shall discuss measures that might be taken to prevent such blind applications of these algorithms.
Day 2 was rather disappointing in the quantity of actual instruction that occurred. The CT began class by presenting the solutions to all of the homework problems that the students had been assigned in the textbook. These solutions were prepared in advance and simply placed on the overhead projector. Future plans for upcoming laboratory experiments were discussed briefly, and then the CT showed a video that pertained to lenses, a topic that the students would be studying soon. The video was interesting, but was superficially informational, rather than presenting the principles of physics. When I was in high school, we called such films "fillers", as it usually meant the instructor had something else to do that day. Such was the case in this class. I had prepared some "back-up" interview questions in the event that this would happen. Thank goodness for foresight!
On day 2, I had to teach a lesson at the university for a methods class, hence I needed to allow for transportation time back to the university. For logistical reasons, both students were interviewed simultaneously on this day. This was unfortunate, but unavoidable since the interviews were necessarily rescheduled. After some preliminary questions that might reveal information about the students' motivation and mathematical background, I continued to pursue the issue of the law of reflection. .
Interestingly, the law of reflection can be applied to physical objects (such as pool balls) as well as to light. This has been, and continues to be, a puzzle that keeps many physicists' paychecks coming each month. I asked both students if they play pool very often. Both of them replied, "yes". Good -- this will make the interview "culturally relevant" .
I presented each student with his own diagram that resembled a traditional pool table and a single ball on the table (see figure AI-2). I asked each student what they would do to shoot the ball (without a cue ball) such that it banks off of the top rail (by line xx' in the diagram) and lands in the side pocket on the left hand side. SB began to explain how he would go about this, including a description of the spin that he would put on the ball. I realized that these guys were experts in the game, as I hadn't anticipated this much detail. Hence, I asked them to keep it simple. .
I asked them where the ball would go if I hit it "straight on", directly against the banking rail (such that it made a 90 degree angle with it). SB correctly responded that it would come straight back. I then asked them what would happen if I hit it at a slight angle. SB again responded correctly that it would come back "at the exact angle". He proceeded to say, "it's like, it's like, the law of reflection".
This is exactly what I had wanted them to see. Pool balls act like light rays -- they both obey the law of reflection. But did they really make this connection? That is, now that they know how to apply the law to concave spherical mirrors, would they be able to carry the same principles over to physical particles? Read on. .
The students knew that they were to apply the law of reflection to a pool ball striking the banking rail -- so I asked them to locate the point on the banking rail that they should aim for so that the ball reflects off of the wall and lands in the side pocket. Both students went to work on the problem. After a few minutes of work, they showed me their results. Their results did not match each other's. While SA's answer was somewhat more correct (in the sense that it was closer to the actual point), SB's answer was dramatically wrong. .
I asked each student to explain how they did their problem. The students both said something about using the diagonals on the squares of the graph paper on which the problems were drawn. This completely surprised me -- as I have no idea how to explain this solution. Perhaps it was a remnant from a lesson earlier in the school year which I had not seen. It astonished me that both students would do this, yet each student applied this idea in his own manner. Before I told them that their answers were incorrect, I asked them to look at their solutions in light of the law of reflection. That I would have to remind them to use such error- checking procedures indicates to me that they lack the some of the syntactic knowledge of physics: those canons of evidence that allow one to verify truth in the discipline. After this prompt, they both realized that their solutions were incorrect. .
I set them to their task again, hoping that they might find the correct solution now that they firmly knew that the law of reflection had to be satisfied. After several minutes of silence, both admitted that they couldn't do the problem. In a way, I'm not surprised. This problem essentially involves applying the law of reflection in reverse. But I wanted to pose this problem to the students to ensure that they really did know the 'ins and outs' of the law of reflection. .
I showed them how the problem was to be done and received an emphatic "Wow!" from SB. Remember, SB is the student who looks for applications of the physics he has learned. I provided him with an application that will sharpen his pool skills and he was grateful and impressed that physics is hard at work on his pool table. SB is the type of student who would benefit greatly from instruction that is taught in the spirit of the Monk and Finkel article: carefully designed learning environments that allow for self-discovery. .
Lastly, I gave the students a "Goofy Pool Table" exercise. Figure AI-3 illustrates a "regular" pool table with a semi-circular top. The concave side of the circle is used in this new pool game. I asked each student how I should shoot the ball so that it would land in the new hole that I created. The answer is very simple: it is a direct application of the ray-tracing-method or the law of reflection -- whichever the student decides to use will yield the correct answer. After a little bit of work, SA arrived at the correct answer, but he remained silent as SB continued working. SB had constructed all sorts of rays on the diagram, but wasn't making much progress. He finally gave up. I gave him the solution and he replied something like, "Do you mean to tell me that if I just hit it straight, it'd go in that hole? So I made it harder than it was."
My analysis of these incidents is as follows: SA truly had a firm grasp of the fundamental concept of the law of reflection, and could also implement the ray-tracing-method to situations that extended beyond the mirror and light scenarios. I believe that he had a solid conceptual foundation of this all-important law. SB, on the other hand could apply the ray-tracing-method only to identical situations (such as the concave mirror problems) yet could not extend his understanding of the law of reflection to other, similar circumstances. Perhaps this indicates that he really doesn't grasp this law in its fullness.
It is interesting that SA, who couldn't remember the name of the law, could apply the law better than SB, who apparently knew the rules of the law and its name. This may indicate a difference in the substantive and syntactic knowledge base of the two students. While each type of knowledge is important, the syntactic knowledge base is generally more valuable for applying physics principles to new problems. SB has a greater challenge before him if he continues to study physics at the college level. When SB is presented with equations, he may remember them as individual elements rather than simplify them into broad concepts. SA has the ability to simplify all of the mirror genres and the pool table problems into "law of reflection" problems, whereas SB must remember how to do each type of problem separately.
These differences may be due to the type of instruction each student has received. I have only known these students for a few months, and know nothing of their former educational background, so I cannot substantiate this claim. But there is an important point to be made: this material can (and should) be presented so that the fundamental structure (the law of reflection) is accentuated and well understood by all students. It may be helpful to teach a unit on the law of reflection by itself. Within that unit one could discuss mirrors, light, pool tables, and pool balls and the other applications of this law. Instead, the students at this high school learn about the reflecting properties of mirrors in the unit on light. Several fundamental concepts of physics must be well understood to proceed through this unit. It's a difficult decision to weigh. Do we teach in terms of broad physical laws such as the law of reflection, and the law of energy conservation, or do we instead teach in seemingly logical units such as "light" and "motion" that encompass many different fundamental laws in each unit? I don't have the answer.
Perhaps, however, we could look at Bruner's spiral curriculum. Since the law of reflection is also a fundamental topic for mathematics classes, it may serve the students best to learn this very carefully in their math classes -- all the while applying it to physical problems such as those discussed in this report. Then, if we do decide to teach physics in logical units such as "light", the students will already have the concept of the law of reflection firmly embedded within their schema. We simply revisit this law on another part of the spiraling helix. These are problems that cannot be solved by a single teacher. The entire curriculum must be adapted, with cooperation among all disciplines. This won't be an easy change to make, but would certainly improve the likelihood that the important fundamentals of each discipline are well understood by our students.
You were right. I did feel like cursing you at times during this assignment, but those were mainly the times when I had to schedule interviews. After my grandfather passed away and I missed my first set of interviews, the cursing became even stronger. But once I conducted the interviews, I was extremely excited about writing up this report. I found some amazing stuff in the heads of my students. In a way I wish that the interviews would have come at a time when the CT was doing more teaching -- but I also found out some important things about my lesson plan. Revisions are certainly in the makings. I wholly agree with Bruner's idea of teaching the fundamental principles. SA is living proof that if you know these, you can go a long way in physics. Perhaps I am at fault for spending a disproportionate amount of time teaching the ray-tracing-method to my students, but I was under mandates from my CT to do so. In my own physics class of the (hopefully) near future, I will certainly stress the fundamentals more than the mechanized algorithms.
All in all, I believe that the interviews went very well. I have only covered a portion of what we discussed in them, and the rest of the material is equally interesting! (At least in my opinion). I don't want to sound like I'm "brown-nosing", but I did enjoy completing this assignment - and again, you were right: "Thank you!" Lastly, I'm sorry about the length of this report. It's hard to contain a project to X number of pages when you enjoy working on it.
p.s. Noticeably absent from this report is any of Norman's iceberg metaphor. Please don't take this personally, but I don't find his metaphor useful or interesting. It oversimplifies the task of explaining how we are to examine students' schemas and how we are to view the effectiveness of instruction. It's just my opinion, but I have my right to express it!