They know the math, but the words get in the way.

Abstract

This study sought to better understand instructional models that could be expected to improve student understanding of graphs of kinematic variables (distance, velocity and acceleration). The effect of using CBL-instruments and cooperative group structure (alone and in concert) was examined for repairing students' misconceptions. Misconceptions were determined using Nemirovsky and Rubin's (1992) definitions for cues that indicate students' misconceptions. Laboratory activities utilized in the various instructional settings were created that incorporated strategies developed by Kykstra, D. I., Boyle, C. Fl, Monarch, I. A. (1992) to promote conceptual change. Results of the study support previous research that even though students understand the requisite mathematical concepts, they still have misconceptions concerning the interpretations concerning the interpretation of mathematical terms in a physical setting. The most problematic misconception was indicated by students' use of Linguistic Cues; students interpreted mathematical terms using "common language" interpretations, not mathematical interpretations. Students continued to use "common language" interpretations even when confronted with physical situations using CBL-tools that contradicted their (incorrect) conclusions. Students needed to not only be confronted by their misconception, but needed the confrontation to be confirmed by the teacher or they maintained that their interpretation was correct and that the physical evidence was wrong.

**********

  "You can not apply mathematics as long as words still becloud
  reality."
  (Herman Weyl 1885-1955)

Mathematicians and physicists believe that when people communicate mathematics using algebraic symbols, communication is precise and unambiguous. However, when applying the symbols of mathematics many students would agree with Wehl that there is a great deal of ambiguity. For example, many students have difficulty articulating their understanding of the relationship between a function, its derivative, and its graph. One of the principle applications of these concepts is with problems involving distance, velocity and acceleration of a moving object (kinematic variables).

Virtually all students come to the classroom with some personal experience with kinematics. The desire to build conceptual understanding of functions, graphs and the physical phenomena that they relate to, using knowledge that students already have, is consistent with widely accepted constructivist principles. Clement states, "We assume that it is desirable to be able to ground new material in that portions of the student's intuition which is in agreement with accepted theory. When this is possible, it should help students to understand and believe physical principles at a 'make sense' level instead of only at a more formal one (1989, p. 1)." Unfortunately, students' personal understanding of kinematic variables may be incomplete or erroneous. Students' difficulties are often grounded in knowledge based on their personal experiences (Monk, 1990; Nemirovsky, J.R., Monk, S., 1992) Further, many students continue to have difficulty interpreting graphs of kinematic variables even following instruction in mathematics and in physics courses (Beichner, 1994; McDermott, L.C., Rosequist, M. L., Van Zee, E. H., 1986). Students recognize that the slope of a velocity graph is acceleration, but fail to reflect on the physical interpretation of negative acceleration, and whether the interpretation is different when velocity is negative rather than positive.

The traditional model of instruction for mathematics and physics courses has been a lecture/homework format, with lectures concentrating on the algebraic interpretation of variables. This traditional format may not be effective for developing understanding of graphs of kinematic variables: "Teachers cannot simply tell students what the graphs' appearance should be. It is apparent from the testing results that this traditional style of instruction does not work well for imparting knowledge of kinematic graphs (Beichner, 1994, p. 755)." Cooperative group structures and activities that utilize a Microcomputer Based Laboratory (MBL, or less costly and cumbersome CBL instruments) show promise for dealing with this problem. There is evidence that cooperative group structure can be effective in improving achievement for more difficult tasks requiring analysis and other problem solving skills (Slavin, 1980; Dees, 1991). The interpretation of graphs of kinematic variables is not a simple computational problem, but involves understanding several concepts. Further, cooperative groups provide the opportunity for student discourse which may assist students' understanding with kinematic graphs (Beichner, 1994; Nemirovsky, J. R., Monk, S., 1994; Monk, 1994; Dykstra et al., 1992).

New technologies have been shown to produce significant opportunities for teachers and students to engage in experiencing mathematical models that were impossible 30 years ago (English, 2002; Mariotti, 2002). In particular, the use of MBL instruments has been shown to produce significant opportunities for teachers and students to engage in experiencing mathematical models that were impossible 30 years ago (English, 2002; Mariotti, 2002). In particular, the use of MBL instruments has been shown to improve student understanding of kinematic graphs (Thornton, 1987; Rosenquist, 1986; Kykstra et al., 1992). Dykstra found that the MBL gave students the opportunity to be confronted by discrepancies in conceptions. Based on his research, he set forth the following strategies for conceptual change:

1. Use and develop trust in tools that extend the senses. It is preferable to use a phenomenon (a) ..., or (b) a phenomenon whose outcome students feel confident predicting but whose outcome differs with their predictions.

2. Have students predict the outcome or explain the phenomenon.

3. Focus on inducing disequilibration by having students test their predictions or explanations.

4. Establish a "town meeting" to discuss, develop and test new ideas in order to resolve perceived discrepancies and differences in explanations (Dykstra et al., 1992, p. 642).

Using Dykstra's strategies, activities were developed for cooperative group structure and CBL environments (alone and in concert) with the goal of better understanding the dynamics by which these models could be expected to improve student understanding of graphs of kinematic variables. Specifically, the goal was to answer the following questions:

1. What difficulties with interpreting graphs of kinematic variables do students bring to the integral calculus classroom and what misconceptions (or lack of conception) are indicated by these difficulties?

2. What is the relative effectiveness of the traditional, cooperative group, and CBL models of instruction for building conceptions, repairing misconceptions and removing difficulties with interpretation of graphs of kinematic variables? In particular, are certain types of difficulties more readily removed by one of these instructional models?

The Study

The student took place at a medium-sized, public university on the west coast of the United States. The sample was drawn from four recitation sections of integral calculus, Winter Term, with identical teaching personnel. There were 121 students enrolled in the four recitations, 98 of whom took both the pretest and post-test. Of those 98 students, 86 were in attendance on the day of treatment.

Students were given a 14-item pretest at the beginning of the term and an identical post-test six weeks later. The post-test followed lectures on kinematic variables that included a demonstration by the instructor using a CBL device, and a laboratory activity on kinematic variables. The pretest and post-test for this study were adapted from the Test of Understanding Graphs in Kinematics (TUG-K) developed by Beichner (1994). Beichner reported that the 21 item TUG-K had Kuder-Richardson reliability coefficient of 0.83. Beichner established content validity through examination of the items by 15 science educators including high school, community college, four year college and university faculty. The TUG-K was designed to test for seven objectives that relate to students' difficulties with kinematic graphs, three questions for each of the seven objectives as given in Table 1.

Due to time constraints on the administration of the test in this study, it was necessary to reduce the number of items from 21 questions to 14. Questions were selected from the TUG-K on the basis of the point biserial coefficient, similarity to other questions, and diversity in question format. For example, TUG-K items 4 and 20 are very similar in both give a velocity graph and ask the student to compute change in distance for a specific time interval. Thus, only one of the items were included on the test for this study (item 4). This 14-item instrument included TUG-K problems #1, #4, #6-#11, #13, #15, #17, #18, #19, #21 and had a Kuder Richardson-21 reliability coefficient of 0.74.

The tests were analyzed by examining each student's incorrect responses. Some (not all) incorrect responses were associated with one of six particular misconceptions (or lack of conception). For example, in problem 11 in Figure 1, response (B) could be found by dividing the height of the graph by 3 which could indicate that although the student had difficulty with this problem, he may have had a conception for velocity as the slope of a position graph. Response (E) gives the height of the graph. This response, in conjunction with other similar incorrect responses, may indicate that the student has a lack of conception for velocity as the slope of a position graph.

[FIGURE 1 OMITTED]

Some responses indicated that the student completely lacked the conception that:

1. Velocity is the slope of a position graph.

2. Acceleration is the slope of a velocity graph.

3. Area under a velocity graph represents displacement and/or area under an acceleration graph represents velocity.

The above concepts are just particular ways of thinking of the quantities identified. For example, velocity is the slope of a position graph, but it is also the rate of change of position. However, students who have taken calculus and physics should understand all of these concepts. Clearly, the most difficult of these concepts is the notion of area under a graph. Students look at the area under a graph as something representing "area," that is, square units, and area is always positive. However, when we look at a graph of constant, negative acceleration, then each "square" between the acceleration graph and the x-axis represents units of acceleration x time = meters/[seconds.sup.2] x seconds, which is just meters/second = velocity. The concept that "area" or squares can represent something the students have thought of as linear, such as velocity, is difficulty. Moreover, students have difficulty understanding that the "negative" value is added to an initial velocity (that may be positive or negative) and this determines whether the object is speeding up (negative initial velocity) or slowing down (positive initial velocity).

Three types of misconceptions were indicated by students' use of cues indicating particular types of resemblances between the graph of a function, its derivative and anti-derivative. These cues were defined by Nemirovsky and Rubin (1992) and Monk (1990):

1. Syntactic cues are distinguished by the fact that they are based on graphical features, unrelated to the student's knowledge of motion. For example, given the position functions of two objects, the student draws the graphs of the velocity functions for the two objects as a geometric transformation of position.

2. Linguistic cues are ambiguities of language that support resemblances between a function and its derivative or a function and its indefinite integra. Words such as more and less, or up and down can have ambiguous meanngs. For example, it is true that less velocity for car A than for car B means less distance traveled for car A. But, if a single car A is considered, it is not true that less velocity now than earlier implies that the car has traveled less distance now than before.

3. Iconic cues are distinguished by the student responding with graph that has the same shape as the path of motion. For example, consider the physical situation of a bicycle traveling over a hill. When asked to draw a speed vs. time graph, students may simple draw a hill.

Researchers (Monk, 1990; Nemirovsky et al, 1992) have found that students' use of these cues may be supported by other conceptions they have of physical situations, or may be supported by their personal experience, but their concepts and experience do not generalize to the interpretation of the graph at hand.

For example, Figure 2 gives an example of an item on the pre/post-test. The correct response is (d). A student response of (c) would indicate that the student used the Linguistic cue that "constant" velocity is where the graph is constant and "slows down" is where the graph is decreasing. A student response of (a) or (e) would indicate that the student used Iconic cues, that is, that the graph will be a "picture" of the physical situation described. A student response of (b), although incorrect, was not associated with a particular misconception or lack of conception.

A student was diagnosed with a particular misconception or lack of conception if she gave all the incorrect responses that indicated this misconception or lack of conception. In each classification--three classifications for lack of conception and three classifications for an indicated misconception--there were two or three questions on the pre and post-test that indicated the classification.

[FIGURE 2 OMITTED]

The laboratory activities on kinematics were developed for four different instructional environments: working either individually or in groups using algebraic tools; working either individually or in groups using CBL tools. There were four problems on all activity sheets; the first three problems were common to all four working environments (see appendix), Problem 4 was different for each environment as described below.

Students in the Group environments were assigned to a group of three during the first recitation class. Assigning students to groups of three were based one experience that indicated that groups of size three were optimal for promoting student interaction in comparison to groups of size two or four. However, due to typical turnover of students in this course, it was expected that group size would fluctuate between two and four. The student's assignment to a particular group was based on students' ability as determined by their grade in the previous differential calculus course. Each group was mixed-ability with a narrow-range, i.e., all groups had students that were a mix of high-ability and medium ability only, or medium-ability and low-ability only. Webb (1991) found, "All students in these groups tended to be active participants, with questions eliciting help more frequently than in mixed-ability groups with a wider range of ability" (p. 379). Due to typical fluctuations, the configuration of the cooperative groups by the fifth week of class was nine three-member groups and five four-member groups.

Students in the CBL/Group environment created the physical situation they described in Problems 1, 2, and 3 (see Appendix), and then used the CBL tools to collect and produce graphs on their calculators in Problem 4 (Figure 3).

Students in the CBL/Individual environment responded to the instructor's questions regarding the physical situation they had described in Problems 1, 2 and 3, and the instructor then produced graphs, using the CBL tools, for situations described by the students. Problem 4 in Figure 3 was modified for the students in this environment to the following:

4. Your instructor will use the CBL system to monitor situations similar to those you described in 1.c), 2.c), and 3.c). Sketch the acceleration, velocity, and position graphs produced by the CBL system in each case and explain any differences you see between the graphs you predicted and the CBL graphs.

Students in the Algebraic environment used symbolic tools instead of CBL-tools for Problem 4. The only difference between the two Algebraic environments was whether students worked in groups or individually. Problem 4 in Figure 2 was modified for the students in the algebraic groups to the following:

4. a) The motion of an object is described by an acceleration and an initial velocity and position as follows:

a(t) = -0.5 m/[s.sup.2], for 0[less than or equal to]t[less than or equal to]3 v(0) = 4 m/s s(0) = 0 m

[FIGURE 3 OMITTED]

Find expressions in terms of t for the velocity v(t) and the position s(t). Sketch their graphs and explain any differences you see between the graphs you predicted in Problem 1.

4. b) The motion of an object is described by an acceleration and an initial velocity and position as follows:

a(t) = -0.5 m/[s.sup.2], for 0[less than or equal to]t[less than or equal to]3 v(0) = 2 m/s s(0) = 0 m

Find expressions in terms of t for the velocity v(t) and the position s(t). Sketch their graphs and explain any differences you see between the graphs you predicted in Problem 2.

4. c) The motion of an object is described by an acceleration and an initial velocity and position as follows:

a(t) = 0.5 m/[s.sup.2], for 0[less than or equal to]t[less than or equal to]0.5,

a(t) = -1.25 m/[s.sup.2], for 0.5[less than or equal to]t[less than or equal to]2,

a(t) = 0.5 m/[s.sup.2], for 2[less than or equal to]t[less than or equal to]3

v(0) - 2 m/s s(0) = 0 m

Find expressions in terms of t for the velocity v(t) and the position s(t). Sketch their graphs and explain any differences you see between the graphs you predicted in Problem 3.

The laboratory activities were developed to promote conceptual change using Dykstra's strategies (Dykstra et al., 1992). The professor in the lecture section gave a demonstration using the CBL the week prior to the recitation in this study. This entailed a lecture on the motion, velocity and acceleration of a pendulum followed by a demonstration of the graphs of these variables using a CBL and a pendulum set up in the front of the lecture hall. The motion of the pendulum was monitored by the CBL and the graphs were visible to the students via a view screen on an overhead projector. This was done to develop students' trust in the CBL as a tool. Conceptual change was promoted by: using a phenomenon (motion) whose outcome the students felt confident predicting; giving students the opportunity to make predictions (Problems 1, 2, and 3); to then to test their predictions (using the CBL) and explanations (Problem 4); and it was planned that they would conclude the laboratory activity with a whole class discussion led by the instructor.

In the recitation sections where students worked in groups of three or four, students were videotaped while they were working on the laboratory activities. The researcher supervised the video and audio-taping of the groups (this had been the practice for 4 weeks)--the researcher did not interact with students.

Three groups were videotaped in the class using CBL equipment (Groups A, B, and C) and three using algebraic tools (Groups D, E, and F). Videotapes were used to corroborate a student's demonstrated lack of conception or misconception as evidenced by the pre- and post-test. Videotapes were analyzed for verbal evidence, as demonstrated by the students, of one (or more) of the six particular misconceptions or lack of conception.

Results

The pretest indicated some difficulties in all areas (see table 2).

The most difficulty indicated was concerned with area under the curve. This difficulty would be expected since the concept of area under the curve is not introduced, or introduced only briefly, in differential calculus so many students had little or no instruction on this topic. However, it is a concern that after 6 weeks of instruction, 29% of the students still responded to all three questions on the post-test, concerning area under the curve, in such a manner as to indicate they had no conception of what the area under the curve meant.

Of significant concern was that many students had a misconception indicated by their use of Linguistic cues (30% of students). Virtually all students had received some instruction on the related topics in a differential calculus course and some had received instruction in a physics course as well. The concern was magnified when analysis of the post-test indicated that many students still had this misconception (16% of students) following additional instruction and the laboratory activities. Further, although there was improvement in this area, the improvement was not as great as student improvement in other areas.

Videotapes further substantiated that students (83% of those observed) had the misconception that "negative acceleration always implies an object is slowing down" as indicated by their use of this Linguistic cue. Students further articulated their misconception that an object with negative acceleration is always slowing down by stating that the velocity of such an object would have to approach 0.

In each of the three groups in the CBL section (Groups A, B and C), the group members proceeded quickly through problem (1) on the activity. Each of these groups agreed upon the misconception that negative acceleration means an object is slowing down (the students are confident in their prediction, which will differ from the results obtained with the CBL). The students in this section could ask the instructor for assistance in completing the activity (either mathematical assistance or assistance with the technology). The groups did not ask the instructor for assistance when the results they got with the CBL did not reflect their predictions. The following is a short segment of the discourse from Group C. This

particular segment represents a portion of the time they spent working on Problem 2. The group's members agreed that an object with negative acceleration must be decelerating and will eventually come to a stop. The mathematical misconception is that students do not seem to understand that when an object has velocity changing from -2 to -2.5 or -3, the object is not coming to a stop (velocity 0).

Max: It's decelerating.

Nick: I guess we start it negative (due to negative velocity), I guess that's what they want. I don't think it'll come back, I think it'll just keep going down. (This is correct due to negative acceleration.)

John: Will it stop?

Nick: Well, no, if it's going in a negative direction it won't come to a stop (correct).

Max: Well it's just for three seconds.

John: Well if it's got negative acceleration it's got to come to a stop sometime (incorrect)

Nick: Well I don't know if they're considering ...

Max: You don't know how fast it's going in the first place.

John: This has negative velocity of 2 m/s.

Max: Plus the negative acceleration, so it's -2.5.

Nick: That's what I'm saying. So it's accelerating like slower in a way. It will go from -2.5 to -3.

This group continued to maintain that a graph in Problem 2 was of an object going in a negative direction and slowing down (velocity approaching 0); even though Max and Nick clearly stated that the velocity is going from -2 to -2.5 to -3 (which does not approach 0). Their answer to Problem 2.c was "The car is moving backwards and slowing down." When they created the situation in Problem 4-2.c to correspond with their graphs in Problem 2, they monitored a toy car moving towards the motion detector (negative velocity, moving backwards), and slowing down (which would be positive acceleration, not negative). The motion of the toy car they monitored would produce a negative velocity graph and a positive acceleration graph. However, Figure 4 (this groups's completed work) shows a different set of graphs they claimed to have created with the CBL-instruments monitoring this motion. The acceleration and velocity graphs did not reflect the situation this group monitored with the CBL-tools. This group was typical of other groups in that when confronted with graphs different from those they predicted, they chose to simply draw the graphs they wanted, and ignored the graphs on their calculator.

Group B had the same difficulties and also drew graphs for Problem 4 that were not reflective of the situation they monitored with the CBL equipment. When the graphs created with the CBL equipment did not match their predictions, the following discourse took place:

Sara: Are we supposed to get the graphs to match exactly?

Todd: We can always draw the graph like we got it exactly.

In each of the three groups in the algebraic section (Groups D, E and F), the group members displayed the misconception that negative acceleration occurs only when an object is showing down and so the velocity must eventually be 0. In Group E the mathematical misconception was that the vertical coordinate of the velocity graph must eventually be 0. They articulated their knowledge concerning the slope of the velocity graph, thus contradicting this misconception. However, the confrontation did not alter their misconception until they discussed the problem with the instructor. The following is a brief example of the discourse from Group E while working on Problem 2:

Jane: Okay. The velocity is negative so it's just going to go like this (drawing a velocity curve that starts at (0, -2) and ends at (3, 0) which is incorrect).

[FIGURE 4 OMITTED]

Sam: I don't think velocity would be increasing.

Jane: It's not. It's decreasing--or I mean it's getting to zero (Linguistic cue, negative means decreasing which implies approaching 0).

Ian: So once it gets here it'd be zero.

Jane: Actually no. The acceleration is negative so the slope should be negative on the velocity graph. (This is correct; recall her graph has positive slope.)

Sam: I think it's the exact same as last time. It's just down to negative two. (Correct, Sam is saying it looks just like the graph in Problem 1 but starts at -2 m/s instead of 4 m/s.)

Jane: It's the same as last time, going down at negative two--it starts at negative two and goes down (correct).

Jane: Like this? (drawing a graph that starts at (0, -2) and is decreasing.) But isn't it slowing down at a rate of negative 0.5?

Sam: It was last time too.

Jane: Shouldn't it be? So why would the velocity be getting greater? Oh it's getting less. But that zero though. I'm confused about that.

Ian: It kind of doesn't make sense. It seems kinda weird.

Jane: The velocity is never going to get to zero? That doesn't make sense (erases the decreasing graph and redraws the increasing velocity graph).

Sam: The velocity is negative which means it's going backwards.

Ian: The velocity is just getting more negative.

Sam: The velocity is negative then it is slowing down and it's slowing down even more causes the acceleration is negative and so ...

Ian: It's just decelerating.

Jane: If the velocity is negative it doesn't mean it's slowing down, it means it's moving towards the motion detector. Acceleration is negative it means it's slowing down (Linguistic cue), and if it's positive it's speeding up. And it's slowing down ...

Sam: If the velocity is negative it means it's moving towards ...

Jane: Yeah.

Ian: Yeah.

Jane: But the acceleration is negative so it should be a negative slope. I don't understand this.

This group got help from the instructor. They expressed their inability to reconcile a car that is slowing down with a velocity that does not approach zero. The instructor asked them to describe the velocity and acceleration for a car moving towards the motion detector with increasing speed. The group realized their misconception, and were then able to move on to the next problem.

Only one of the six groups was observed to move quickly through Problem 2 (Group D). This observation supported the results of the group members' pretests which indicated that only one of them had any misconceptions or lack of conception and that one group member's pretest only indicated a lack of conception concerning area under a curve.

In the video observations of the students working in groups, only those students in the algebraic section sought help from the instructor when confronted with conflicting information (mathematical knowledge versus Linguistic cue). Students in the Group/CBL section only sought help from the instructor on their use of the CBL equipment.

Students in the Individual/CBL section observed trials run by the instructor using the CBL equipment. The students did not question if the graphs produced by the CBL instruments were correct, even though the graphs were not what the students had predicted, and a classroom discussion followed.

In all four instructional settings the instructor had planned a "whole class" discussion to follow the laboratory activities in keeping with Kykstra's strategy to promote conceptual change (Kykstra, 1992). In an interview with the researcher following all the classes, the instructor stated that in the Group/CBL Section she felt she was, "talking to the air." Observation of videos indicated that during her discussion almost all students were still working with the CBL instruments and/or drawing their graphs and paying no attention to the instructor. In the other three instructional settings a "whole class" discussion did take place and it was observed that most students at least paid attention to the discussion and many participated.

Results from the post-test suggested that many students did not repair misconceptions associated with their use of Linguistic cue. In particular, there was no reduction in the percentage of students in this category in the Group/CBL section. There was significant reduction in the percentage of students in this category in the other three sections. Thus, the data and observations indicated repairing this misconception may not only be aided by students being confronted with contradictory information (mathematical knowledge or a physical situation), but that an explanation by the instructor is also necessary. Only where students sought and received help from the instructor, or participated in classroom discussions were they able to come to an understanding that negative acceleration could mean an object had increasing velocity, specially, when the object also had negative velocity. This is not to suggest that having students work in groups on activities utilizing CBL's to confront their misconceptions is a poor teaching strategy. What the evidence suggests is that in Dykstra's model for conceptual change, the fourth component, "Establish a "town meeting" to discuss, develop and test new ideas in order to resolve perceived discrepancies and differences in explanations" is vital to promoting conceptual change in students.

Discussion

One purpose of this study was to better understand instructional models that could be expected to improve student understanding of graphs of kinematic variables. Observation of the videotapes showed that most students understood very well the mathematical relationship between a function and its graph, as well as the function's rate of change, and the slope of the graph. However, students still had a considerable amount of difficulty interpreting graphs that represented a physical situation.

It was found that students had the most difficulty when mathematical terms were words commonly used in the English language. They interpreted words, principally 'negative,' as it is commonly used to mean 'less' or 'decreasing' which was not correct in the given mathematical context. Students came to an incorrect conclusion by using the common interpretation of a word; they then came to a different (correct) conclusion based on their mathematical knowledge; but the (incorrect) conclusion based on common language is the one they invariably believed to be correct. Further, when confronted with the actual physical situation (using CBL tools), which again indicated their conclusion was incorrect; they still chose the interpretation based on common usage of words.

Thus study supports results of prior research in related topics. Nemirovsky and Rubin (1992), as well as Monk (1990, 1994) found that students' difficulties interpreting graphs of kinematic variables were not necessarily due to simple confusion when reading the axes or a lack of understanding of the meaning of the slope of a graph. As in this study, what they did find was that students have conceptions of kinematic variables based on their personal experience and that the students use these conceptions incorrectly to solve mathematics and physics problems.

The results also support the work of Dykstra (1992). His strategy to promote conceptual change included four components which were employed in this study. In the study environment where the fourth component (a "town meeting") did not occur in the form of a classroom discussion, very little conceptual change took place. This was particularly true where students' misconceptions were based on the "common language" usage (Linguistic cue) or mathematical terms.

To overcome their difficulties with language, students must be given the opportunity to see the conflict. Students need the opportunity to combine their knowledge of mathematical concepts with their use of language and practical experience. However, once they see the conflict, they need to be guided to the correct resolution by someone whose knowledge they trust.

Prior research found that students' difficulties with graphs of kinematic variables were often grounded in the students' knowledge based on their personal experiences. This study found that the most problematic of these difficulties were misconceptions supported by the common-language interpretation of mathematical terms. This raises the question of whether student use of Linguistic cue is the basis for continued difficulties (following instruction) in other content areas as well.

References

Beichner, R.J. (1994). Testing student interpretation of kinematic graphs. American Journal of Physics, 62(8), 750-762.

Clement, J. (1989). Not all preconceptions are misconceptions: Finding "Anchoring Conceptions" for grounding instruction on students' intuitions. Paper presented at the Annual Meeting of the American Educational Research Association at San Francisco, March, 1989.

Dees, R.L. (1991). The role of cooperative learning in increasing problem-solving ability in a college remedial course. Journal for Research in Mathematics Education, 22(5), 409-421.

Dykstra, D.I., Boyle, C.F., & Monarch, I.A. (1992). Studying conceptual change in learning physics. Science Education, 75(6), 615-652.

English, L. (2002). Priority themes and issues in international research in mathematics education. Handbook of International Research in Mathematics Education, 3-16.

Mariotti, M.A. (2002). The influence of technological advances on students' mathematics learning. Handbook of International Research in Mathematics Education, 695-724.

McDermott, L.C., Rosenquist, M.L., & van Zee, E.H. (1986). Student difficulties in connecting graphs and physics: Examples from kinematics. American Journal of Physics, 55(6), 503-513.

Monk, S. (1990). Students' understanding of a function given by a physical model. Paper presented at the conference on the learning and teaching of the concept of function at Purdue University, October, 1990.

Monk, S. (1994). How students and scientists change their minds. MAA invited address, Joint Mathematics Meeting, Cincinnati, Ohio, January, 1994.

Nemirovsky, R., & Monk, S. (1994). The case of Dan: Student construction of a functional situation through visual attributes. Research in Collegiate Mathematics Education, 4, 139-168.

Nemirovsky, R., & Rubin, A. (1992). Students' tendency to assume resemblances between a function and its derivative. TERC working paper.

Rosenquist, M.L., & McDermott, L.C. (1986). A conceptual approach to teaching kinematics. American Journal of Physics, 55(5), 407-415.

Slavin, R.E. (1980). Cooperative learning. Review of Education Research, 50(2), 315-342.

Thornton, R.K. (1987). Tools for scientific thinking--microcomputer-based laboratories for physics teaching. Physics Education, 22, 230-238.

Webb, N.M. (1991). Task-related verbal interaction and mathematics learning in small groups. Journal for Research in Mathematics Education, 22(5), 366-389.

Appendix

[FIGURE A1 OMITTED]

[FIGURE A2 OMITTED]

[FIGURE A3 OMITTED]

Dr. Patricia Hale

California State Polytechnic University, Pomona 

Table 1. Objectives of the TUG-K

Given                           The student will            Item number

1. Position-Time Graph          Determine Velocity           5, 13, 17
2. Velocity-Time Graph          Determine Acceleraion        2, 6, 7
3. Velocity-Time Graph          Determine Displacement       4, 18, 20
4. Acceleration-Time Graph      Determine Change in          1, 10, 16
                                  Velocity
5. A Kinematics Graph           Select Another              11, 14, 15
                                  Corresponding Graph
6. A Kinematics Graph           Select Textual Description   3, 8, 21
7. Textural Motion Description  Select Corresponding Graph   9, 12, 19

Table 2: Results from the Pre and Post-test

Misconception/                                  Cooperative
Lack of                            Cooperative  Group/       Individual/
Conception                  Total  Group/CBL    Algebraic    CBL
                 n =        86     15           16           27

Lack of
Conception:
1. Velocity as   Pretest     9%    20%           0%          11%
   slope of      Post-test   2%     7%           0%           0%
   position
   graph
2. Acceleration  Pretest    17%    33%           6%          22%
   as slope of   Post-test   5%     7%           0%           7%
   velocity
   graph
3. Area under a  Pretest    64%    60%          44%          70%
   graph         Post-test  29%    53%          13%          30%

Misconception
indicated by:
1. Syntactic     Pretest     6%     7%           0%          11%
   cue           Post-test   2%     7%           0%           0%
2. Linguistic    Pretest    30%    33%           6%          37%
   cue           Post-test  16%    33%           0%          19%
3. Iconic cue    Pretest    17%    13%          13%          30%
                 Post-test   8%     7%          13%           7%

Misconception/
Lack of
Conception                  Individual/Algebraic
                 n =        28

Lack of
Conception:
1. Velocity as   Pretest     7%
   slope of      Post-test   4%
   position
   graph
2. Acceleration  Pretest    11%
   as slope of   Post-test   4%
   velocity
   graph
3. Area under a  Pretest    71%
   graph         Post-test  25%

Misconception
indicated by:
1. Syntactic     Pretest     4%
   cue           Post-test   4%
2. Linguistic    Pretest    36%
   cue           Post-test  14%
3. Iconic cue    Pretest    11%
                 Post-test   7%