- Email: [email protected]
Student Reasoning when Models and Reality Conflict
5.7 STUDENT REASONING WHEN MODELS AND REALITY CONFLICT Jerry Leg6 California State University Fullerton, USA Abstract 4 p p e r secondary students were provided data from an experiment which contained a systematic error, all related calculations, and several recommendations for modelling the situation. The students were asked to select one of the given recommendations, or to provide one of their own. The physical context (realityl should have required a direct variation as the model, but data analysis (model) suggested a linear relationship with two parameters. In trying to resolve the internal conflict, a wide range of responses and justifications were offered by participants. Clusters of student responses are characterized, and underlying root issues are suggested as explanations.
1. INTRODUCTION
For years, advocates have called for incorporating more modelling activity into the classroom because of its positive effect on students’ understanding of mathematics. Arguments have included providing a framework for introducing applications (Burghes, 1980), developing problem-solving skills (Blum and Niss, 1991), and promoting conceptual understanding (Lanier, 1999). The kind of thinking that takes place in modelling activity has been associated with a range of other processes, including models for how children learn, especially in mathematics (de Lange, 1987), and the development of meta-cognition among students. Modelling experiences have had the effect of engaging the learner, situating them in the practice of mathematics, and improving their disposition toward more realistic interpretation of word problems (Verschaffel & De Corte, 1997). There has been an inexorable march to develop curricular materials (Blum and Niss, 1991; Usiskin, 1997), and a clearly-articulated rationale for using them. Those arguments range from broad goals such as critical competence (preparing students to be hlly functioning members of society) to specific learning outcomes like the acquisition of “knowledge of existing models and applications of mathematics.. .” (Blum and Niss, 1991, p44-45). Are curricular experiences in modelling a necessary condition for developing a modelling disposition among students? Or can students study mathematics and science as separate disciplines, and synthesize that content knowledge in such a way that it can be activated and adapted when asked to critically examine models for a situation? One way to explore the answer to these questions is to take a group of capable students with no prior experience in modelling, and ask them to build a
282
Student Reasoning When Models and Reality Conflict
model using a situation for which the scientific principles and mathematical relationships have been mastered. This paper reports on one such experiment and the responses it generated. 2. BACKGROUND
A comprehensive, college-preparatory school for mathematics and science in the United States was selected to participate in a research study. The school composition had an ethnic and economic diversity which mirrored the community it served. The students were not necessarily gifted or talented in mathematics or science, but were capable and chose to apply to that school because of the concentrated attention in those subject areas. However, the academic success of the school and its students was unquestionably outstanding - top ten ranking in their state, with average scores on standardized tests between the 90thand 80thpercentile nationally in mathematics and science. The entire junior class had just completed a course on pre-calculus, and participated in a research study of how the curriculum used at that school shaped their understanding of linearity and linear functions. One aspect of that investigation focused on whether students could clearly distinguish when to use equations of the form y = mx from those with form y = mx+b. The task described in the next section was one of the questions used to probe for that understanding, and was administered to half of the students (n = 75). The educational program for schools in that region is defined by various content standards, which serve two main purposes: 1) they prescribe the scope of attention for the standardized tests that students must take, and 2) alignment to these standards forms the criterion for elementary and middle school textbook adoption approval. A review of the content standards for science indicate that by grade 8 (prior to high school), students were to know that density is mass per unit volume and know how to calculate the density of substances from measurements of mass and volume. The associated narrative in that document even prescribed that density is calculated by dividing the mass of some quantity of material by its volume. The content standards for mathematics indicate that by grade 7, students should be able to “plot the values of quantities whose ratios are always the same.. .Fit a line to the plot and understand that the slope of the line equals the quantities (sic)” and “solve multistep problems involving rate, average speed, distance and time or a direct variation” (CDE, 1999 p67-68). In the students’ first three years at that high school, all of them completed three additional science courses by the end of their junior year, including a full year of chemistry. Details about the other two courses are not available, but based on the content standards for science, it should have included some physical and earth science but more attention on developing understanding of biological principles. The mathematics curriculum consisted of a sequence of thematic units, usually with an over-riding problem to solve and featuring extensive use of group work, classroom discussion, short-term and long-term problem solving, discovery learning, and attention to developing process skills and higher-order thinking skills, as well as understanding of mathematics content. While none of the thematic units were designed with an emphasis on mathematical modelling, several have contexts in the physical sciences, require students to collect and analyze data, and spiral the mathematical development in increasing complexity.
Leg&
283
3. STUDENT TASK The actual question to which students were asked to respond is the following: In an attempt to determine the density of a particular gauge of copper wire, the following measurements were made: Sample #1 #2 #3 #5 #6 #4 0.28 0.84 1.15 1.79 2.36 2.81 Length (in.) 0.20 0.07 0.11 0.14 0.25 0.29 Mass (8) 0.25 0.13 0.1 1 Linear Density ( g h ) 0.1 1 0.10 0.12 (Assume the measurements were taken correctly.) The average linear density was found to be around 0.14 g h , and the regression equation obtained from the data is approximately: M = 0.089L + 0.040. It appears to be an excellent fit, but you would think that a piece of wire with length 0” would also weigh 0 g. Select one of the following recommendations for modelling this situation, and explain why you choose that particular one. 0 Go with the regression equation, and restrict the use of the model to the range of data in the table. 0 Modify the regression equation somehow so that it “curves” into the origin (0, 0) to match the end behavior. Use a direct variation equation of the form A4 = k.x and begin looking for a source of error to explain why the data doesn’t behave correctly. 0 Some other explanation (provide the explanation, along with your reasons) Figure 1. Question 10 from research study. The intent was to present a conflict between the conceptual understanding that one might apply in modelling the situation (in this case, about proportionality) and the results obtained from doing data analysis on the information. The linear density, average linear density and regression equation calculations were provided to alleviate time constraints and enable students to focus on the model selection. However, students had access to calculators, however, so they could choose to duplicate or verify that work. The given recommendations allowed the participants to ‘accept’ or ‘reject’ certain beliefs, including the power of technology to yield the “right” answer, the ability to use mathematics to “fix” aberrant behaviors among data, and the reasonableness of interpolation as a means of prediction. Students were also invited to use their creativity if they were not satisfied with the choices provided, especially if they felt it appropriate to use a different mathematical relationship. The students were not allowed to discuss the problem with their peers, or the person who administered the assessment, thereby denying a powerful social component that was routinely present in their learning environment. In not allowing students to replicate the experiment, the task was made deliberately more abstract in two ways. First, the data was dissociated from the sense-making that would come from obtaining them by direct measurement. If students did not automatically know the model that should be applied to this situation, they might need to reconsider those numbers in context in order to reason conceptually about the task. Second,
284
Student Reasoning When Models and Reality Conflict
while density was a science subject that the students had encountered, it was less likely that they had studied linear density. Also, students had studied proportionality among quantities and direct variation as a function form, but it was unlikely that the situation that they examined while studying these concepts specifically involved linear density. Also, designing the task to take measurements find the equation which describes the relationship between the data would shft the emphasis from model selection to data analysis; students might report what “is”, rather than what “should be”. 4. RESPONSES
The good news is that students exist who can fuse knowledge about mathematics and science (studied as separate disjoint subjects), apply that synthesis to model a simple situation and critically question the validity of alternative explanations. In the case of these students, the invariant condition was that the mass of the wire should be proportional to the length of the piece under consideration. Several of them stated that condition as assumptions of uniform diameter and constant density. One particularly eloquent response is provided in Figure 2:
*”&, #&j&.bCI!J&/d#O &,pu‘r*‘&+ flu-cbrU, I bavld dp+ sphuq #?, k look 4~ *f, yuAhtn rn (ow P : k - x , a) &tlrrSkrCCA I,, “2‘ dlgf~m Ly k a h d a,. 14 th kG ~t4~l-ttl~j +htr M&/, j& kdLkl h/. rtmh( ””9 hi a s u u r a n~ u
*tbn
I>
1%
Figure 2 . Student response using conceptual understanding. However, only thirteen students (I 7%) identified the third recommendation as the best answer, and while the range of explanations was impressive, the rationale was often suspect. Four gave no explanation, one was unintelligible, one argued that errors are always a possibility and their existence should be considered, and one eliminated the other options and provided that as the reason. Two students argued that the linear density calculations were reasonably constant, especially if the first set of data was treated as an outlier. One student argued that there had to be a source of error, since (0, 0) did not satisfy the regression equation. Finally, one student applied Ockham’s Razor as a means of resolving the conflict, still convinced that the regression equation should work. Twenty-three students (3 1%) selected the first recommendation - that the linear regression equation should be the model used, as long as its domain was restricted to the range of data from which it was obtained. The largest block of students felt that the data determines the equation that is generated, and therefore it seemed reasonable that the equation should represent those numbers. Other arguments that
Lege
285
were invoked by multiple students included the fact that interpolation was considered a “safe” process, that more data was required to warrant rejection of the regression equation, and measurement imprecision might explain why the regression equation did not go through the origin. Two students misinterpreted the phrase “assume measurements were taken correctly” to mean that the recordings were accurate, and two students reported a belief that regression always obtained the right answer. On a lighter note, one student argued that models do not have to reflect reality, and another said that this was a school task where the goal is to complete the task. Nineteen students (25%) thought that the second option was the best. The prevailing thought here was that this recommendation captured both the trend in the actual data and the expected end behavior for the model. Secondary arguments included that (0, 0) should be considered a data point (without arguing for redoing the regression calculation with the extra data point included), that this kind of density might produce a graph which is “locally linear”, and that polynomial curvefitting was seen as the goal - it was okay to add more data points and allow the curve which described the pattern to become more complex. One student argued that the availability of calculators made this recommendation possible, but gave no details on how that would be done, and another included (0, 0) with a graph of linear density versus mass, showing a sketch that looked like a probability density function skewed to the right. None of these students considered defining the equations piecewise to capture the disparate trends in the data, nor did they try other regression models with (0,O) included in the set of data. Fourteen students (19%) of the students suggested another recommendation for modelling the situation, with most of the responses reflecting one of two different types of thinking about the problem. First, most of them were willing to resolve the conflict by restricting the domain, either by not allowing L = 0, or by allowing extrapolation for large pieces of wire only. Those students suggested arguments like a domain restriction necessarily existed anyway for values of L which were negative, or the relationship involved a ratio which would imply that L couldn’t equal 0 anyway! One student claimed that having “no wire” means that it isn’t a “piece of wire”; in other words, physical material contained matter, which could be divisible, but the absence of that physical material violated the conditions of the problem situation. The other recurring theme among these students was that the behavior shown in the regularity among the known data points might change as one considers increasingly small pieces of wire. One recommendation was to collect more data points in the interval 0 < L < 0.28 to validate having the end behavior approach the origin, or to be able to more adequately describe the existing relationship for such small-sized objects. Another student suggested that the precision error would remain constant for the measurements in the data, but the relative size of that error would change as one gets close to zero, which could explain why the “line” needed to “curve” into the origin.
5. CONCLUSIONS It is possible for some students to combine their mathematics and physics knowledge to reason intelligently in a simple yet unfamiliar modelling situation.
286
Student Reasoning When Models and Reality Conflict
However, those students are exceptions when compared to the performance of their classmates, and the conditions at this school were more ideal than what exists typically. In fact, the dearth of effective responses and the types of arguments used by the students should suggest that modelling experiences be a necessary component of a vibrant mathematics program. Perhaps because the task was given in a mathematics class, arguments reflect a mathematical bias. Students believe in the power of an equation to describe data, especially when using a “best-fit’’ one. Domain restrictions resolve mathematical problems like division by zero. Higherdegree polynomials describe complicated patterns in form, even if using them distances you further from reality. There were no clues to suggest that the constant density should be considered, and little evidence that students reasoned conceptually from the fact that the quantities involved should vary proportionally. The only scientific considerations were superficial ones, like interpolation being more accurate than extrapolation, consideration for the effect of measurement imprecision, or the need for more data points. The suspension of sense-making reflects the school environment, where the goal is to complete the task. If any student had reasoned that by starting with the longest piece of wire, one can cut off pieces to create the condition for the other data, that viewpoint would have led to an analysis by finite differences, revealed that the density is constant, and led them to question the calculations that were provided. The blind acceptance of the data as accurately depicting the real situation was exacerbated by the fact that the linear density calculation was provided, which might suggest that the data itself is okay. One might argue that studying statistics, especially if taught well, would instill a “habit of mind” to question information. However, it is unlikely that students would consider three subject areas at the same time when they already have difficulty working in the interface of two domains. Assuming that learning about existing mathematical models is a valid goal for mathematics education, there are reasons why modelling experiences may be a necessary condition for developing such knowledge. Jablonka (1997) cited several misconceptions that may occur in interpreting mathematical models, including the belief that mathematical exactitude should apply, that models must be reliable, and that the structure of the model must be the structure of reality. In this study, many student responses exhibited similar beliefs that would be offset by such experiences - for example, the equation coming out of a calculator should accurately describe the data upon which it is based, and that curve-fitting as a task should not have a “reality” mirror based on physical laws, natural conditions or common sense. A second argument may be found in Niss (1999) when he discusses obstacles produced by the process-object duality. These students understood certain processes - some relationships behave in such a way that the amount of one quantity is always proportional to the amount of the other, and that one can create models for these kinds of relationships in a particular form 0, = mx). Their understanding had not progressed to the point that the form represented all situations in which the amount of one quantity is always proportional to the amount of the other, or else there would have been more students who reasoned like the one given in Figure 2. Those students who re-expressed that equation form to say that the ratio y I x stays constant found it more difficult to argue in favor of that model, since it generated a new problem for them involving division by zero. Models may be a situation in which the
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process-object duality presents an obstacle to learning. In this case, studying proportionality, solving specific problems about direct variation, and expressing and studying fhctions of the form y = mx may not lead to a deep understanding about a model for a specific situation BEING an equation of that form. Finally, in discussing the use of contexts in situated teaching as a means for developing understanding of mathematical concepts, DaPueto and Parenti ( 1999) identified a critical feature as being students’ “understanding of, or confidence with, the process by which the problem is transformed... (p. 7)”. With this set of students in particular, it is likely that they understood the problem transformation, or would have been able to make sense of the situation given time or the opportunity for social discourse. Collectively, it is clear that they lacked the confidence in the process of how the problem was transformed to be able to resolve the inherent conflict that was contained within the problem. And in a manner reminiscent of the conclusions drawn by Lithner (2000), strategy choices are dominated by established experiences - applying methods they know from similar tasks - which can fail when “familiar routines do not work for different reasons (p. I87).”
REFERENCES Blum, W. and Niss, M. (1991) Applied mathematical problem solving, modelling, applications, and links to other subjects - state, trends, and issues in mathematics instruction. Educational Studies in Mathematics, 22,37-68. Burghes, D. (1980) Mathematical modelling: A positive direction for the teaching of applications at school. Educational Studies in Mathematics, 1 1, 1 13-131. California Department of Education. ( 1999) Mathematics framework for California public schools -Kindergarten through grade 12. Sacramento, CA: CDE. DaPueto, C., and Parenti, L. (1999) Contributions and obstacles of contexts in the development of mathematical knowledge. Educational Studies in Mathematics, 39, 1-21. Jablonka, E. (1997) What makes a model effective and useful (or not)? In S. Houston, W. Blum, I. Huntley, and N. Neil1 (eds). Teaching and Zearning mathematical modelling - Innovation, investigation and applications. Chichester: Ellis Honvood, 39-50 Lanier, S. (1999) Students’ understanding of linear modelling in a college mathematical modelling course. Ph.D. dissertation thesis, University of Georgia. Lither, J. (2000) Mathematical reasoning in task solving. Educational Studies in Mathematics, 41, 165-190. Niss, M. (1999) Aspects of the nature and state of research in mathematics education. Educational Studies in Mathematics, 40, 1-24. Usiskin, Z. (1997) Applications in the secondary school mathematics curriculum: A generation of change. American Journal of Education, 106,62-84. Verschaffel, L., and De Corte, E. (1997) Teaching realistic mathematical modelling in the elementary school: A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28, 577-601.
1. INTRODUCTION
For years, advocates have called for incorporating more modelling activity into the classroom because of its positive effect on students’ understanding of mathematics. Arguments have included providing a framework for introducing applications (Burghes, 1980), developing problem-solving skills (Blum and Niss, 1991), and promoting conceptual understanding (Lanier, 1999). The kind of thinking that takes place in modelling activity has been associated with a range of other processes, including models for how children learn, especially in mathematics (de Lange, 1987), and the development of meta-cognition among students. Modelling experiences have had the effect of engaging the learner, situating them in the practice of mathematics, and improving their disposition toward more realistic interpretation of word problems (Verschaffel & De Corte, 1997). There has been an inexorable march to develop curricular materials (Blum and Niss, 1991; Usiskin, 1997), and a clearly-articulated rationale for using them. Those arguments range from broad goals such as critical competence (preparing students to be hlly functioning members of society) to specific learning outcomes like the acquisition of “knowledge of existing models and applications of mathematics.. .” (Blum and Niss, 1991, p44-45). Are curricular experiences in modelling a necessary condition for developing a modelling disposition among students? Or can students study mathematics and science as separate disciplines, and synthesize that content knowledge in such a way that it can be activated and adapted when asked to critically examine models for a situation? One way to explore the answer to these questions is to take a group of capable students with no prior experience in modelling, and ask them to build a
282
Student Reasoning When Models and Reality Conflict
model using a situation for which the scientific principles and mathematical relationships have been mastered. This paper reports on one such experiment and the responses it generated. 2. BACKGROUND
A comprehensive, college-preparatory school for mathematics and science in the United States was selected to participate in a research study. The school composition had an ethnic and economic diversity which mirrored the community it served. The students were not necessarily gifted or talented in mathematics or science, but were capable and chose to apply to that school because of the concentrated attention in those subject areas. However, the academic success of the school and its students was unquestionably outstanding - top ten ranking in their state, with average scores on standardized tests between the 90thand 80thpercentile nationally in mathematics and science. The entire junior class had just completed a course on pre-calculus, and participated in a research study of how the curriculum used at that school shaped their understanding of linearity and linear functions. One aspect of that investigation focused on whether students could clearly distinguish when to use equations of the form y = mx from those with form y = mx+b. The task described in the next section was one of the questions used to probe for that understanding, and was administered to half of the students (n = 75). The educational program for schools in that region is defined by various content standards, which serve two main purposes: 1) they prescribe the scope of attention for the standardized tests that students must take, and 2) alignment to these standards forms the criterion for elementary and middle school textbook adoption approval. A review of the content standards for science indicate that by grade 8 (prior to high school), students were to know that density is mass per unit volume and know how to calculate the density of substances from measurements of mass and volume. The associated narrative in that document even prescribed that density is calculated by dividing the mass of some quantity of material by its volume. The content standards for mathematics indicate that by grade 7, students should be able to “plot the values of quantities whose ratios are always the same.. .Fit a line to the plot and understand that the slope of the line equals the quantities (sic)” and “solve multistep problems involving rate, average speed, distance and time or a direct variation” (CDE, 1999 p67-68). In the students’ first three years at that high school, all of them completed three additional science courses by the end of their junior year, including a full year of chemistry. Details about the other two courses are not available, but based on the content standards for science, it should have included some physical and earth science but more attention on developing understanding of biological principles. The mathematics curriculum consisted of a sequence of thematic units, usually with an over-riding problem to solve and featuring extensive use of group work, classroom discussion, short-term and long-term problem solving, discovery learning, and attention to developing process skills and higher-order thinking skills, as well as understanding of mathematics content. While none of the thematic units were designed with an emphasis on mathematical modelling, several have contexts in the physical sciences, require students to collect and analyze data, and spiral the mathematical development in increasing complexity.
Leg&
283
3. STUDENT TASK The actual question to which students were asked to respond is the following: In an attempt to determine the density of a particular gauge of copper wire, the following measurements were made: Sample #1 #2 #3 #5 #6 #4 0.28 0.84 1.15 1.79 2.36 2.81 Length (in.) 0.20 0.07 0.11 0.14 0.25 0.29 Mass (8) 0.25 0.13 0.1 1 Linear Density ( g h ) 0.1 1 0.10 0.12 (Assume the measurements were taken correctly.) The average linear density was found to be around 0.14 g h , and the regression equation obtained from the data is approximately: M = 0.089L + 0.040. It appears to be an excellent fit, but you would think that a piece of wire with length 0” would also weigh 0 g. Select one of the following recommendations for modelling this situation, and explain why you choose that particular one. 0 Go with the regression equation, and restrict the use of the model to the range of data in the table. 0 Modify the regression equation somehow so that it “curves” into the origin (0, 0) to match the end behavior. Use a direct variation equation of the form A4 = k.x and begin looking for a source of error to explain why the data doesn’t behave correctly. 0 Some other explanation (provide the explanation, along with your reasons) Figure 1. Question 10 from research study. The intent was to present a conflict between the conceptual understanding that one might apply in modelling the situation (in this case, about proportionality) and the results obtained from doing data analysis on the information. The linear density, average linear density and regression equation calculations were provided to alleviate time constraints and enable students to focus on the model selection. However, students had access to calculators, however, so they could choose to duplicate or verify that work. The given recommendations allowed the participants to ‘accept’ or ‘reject’ certain beliefs, including the power of technology to yield the “right” answer, the ability to use mathematics to “fix” aberrant behaviors among data, and the reasonableness of interpolation as a means of prediction. Students were also invited to use their creativity if they were not satisfied with the choices provided, especially if they felt it appropriate to use a different mathematical relationship. The students were not allowed to discuss the problem with their peers, or the person who administered the assessment, thereby denying a powerful social component that was routinely present in their learning environment. In not allowing students to replicate the experiment, the task was made deliberately more abstract in two ways. First, the data was dissociated from the sense-making that would come from obtaining them by direct measurement. If students did not automatically know the model that should be applied to this situation, they might need to reconsider those numbers in context in order to reason conceptually about the task. Second,
284
Student Reasoning When Models and Reality Conflict
while density was a science subject that the students had encountered, it was less likely that they had studied linear density. Also, students had studied proportionality among quantities and direct variation as a function form, but it was unlikely that the situation that they examined while studying these concepts specifically involved linear density. Also, designing the task to take measurements find the equation which describes the relationship between the data would shft the emphasis from model selection to data analysis; students might report what “is”, rather than what “should be”. 4. RESPONSES
The good news is that students exist who can fuse knowledge about mathematics and science (studied as separate disjoint subjects), apply that synthesis to model a simple situation and critically question the validity of alternative explanations. In the case of these students, the invariant condition was that the mass of the wire should be proportional to the length of the piece under consideration. Several of them stated that condition as assumptions of uniform diameter and constant density. One particularly eloquent response is provided in Figure 2:
*”&, #&j&.bCI!J&/d#O &,pu‘r*‘&+ flu-cbrU, I bavld dp+ sphuq #?, k look 4~ *f, yuAhtn rn (ow P : k - x , a) &tlrrSkrCCA I,, “2‘ dlgf~m Ly k a h d a,. 14 th kG ~t4~l-ttl~j +htr M&/, j& kdLkl h/. rtmh( ””9 hi a s u u r a n~ u
*tbn
I>
1%
Figure 2 . Student response using conceptual understanding. However, only thirteen students (I 7%) identified the third recommendation as the best answer, and while the range of explanations was impressive, the rationale was often suspect. Four gave no explanation, one was unintelligible, one argued that errors are always a possibility and their existence should be considered, and one eliminated the other options and provided that as the reason. Two students argued that the linear density calculations were reasonably constant, especially if the first set of data was treated as an outlier. One student argued that there had to be a source of error, since (0, 0) did not satisfy the regression equation. Finally, one student applied Ockham’s Razor as a means of resolving the conflict, still convinced that the regression equation should work. Twenty-three students (3 1%) selected the first recommendation - that the linear regression equation should be the model used, as long as its domain was restricted to the range of data from which it was obtained. The largest block of students felt that the data determines the equation that is generated, and therefore it seemed reasonable that the equation should represent those numbers. Other arguments that
Lege
285
were invoked by multiple students included the fact that interpolation was considered a “safe” process, that more data was required to warrant rejection of the regression equation, and measurement imprecision might explain why the regression equation did not go through the origin. Two students misinterpreted the phrase “assume measurements were taken correctly” to mean that the recordings were accurate, and two students reported a belief that regression always obtained the right answer. On a lighter note, one student argued that models do not have to reflect reality, and another said that this was a school task where the goal is to complete the task. Nineteen students (25%) thought that the second option was the best. The prevailing thought here was that this recommendation captured both the trend in the actual data and the expected end behavior for the model. Secondary arguments included that (0, 0) should be considered a data point (without arguing for redoing the regression calculation with the extra data point included), that this kind of density might produce a graph which is “locally linear”, and that polynomial curvefitting was seen as the goal - it was okay to add more data points and allow the curve which described the pattern to become more complex. One student argued that the availability of calculators made this recommendation possible, but gave no details on how that would be done, and another included (0, 0) with a graph of linear density versus mass, showing a sketch that looked like a probability density function skewed to the right. None of these students considered defining the equations piecewise to capture the disparate trends in the data, nor did they try other regression models with (0,O) included in the set of data. Fourteen students (19%) of the students suggested another recommendation for modelling the situation, with most of the responses reflecting one of two different types of thinking about the problem. First, most of them were willing to resolve the conflict by restricting the domain, either by not allowing L = 0, or by allowing extrapolation for large pieces of wire only. Those students suggested arguments like a domain restriction necessarily existed anyway for values of L which were negative, or the relationship involved a ratio which would imply that L couldn’t equal 0 anyway! One student claimed that having “no wire” means that it isn’t a “piece of wire”; in other words, physical material contained matter, which could be divisible, but the absence of that physical material violated the conditions of the problem situation. The other recurring theme among these students was that the behavior shown in the regularity among the known data points might change as one considers increasingly small pieces of wire. One recommendation was to collect more data points in the interval 0 < L < 0.28 to validate having the end behavior approach the origin, or to be able to more adequately describe the existing relationship for such small-sized objects. Another student suggested that the precision error would remain constant for the measurements in the data, but the relative size of that error would change as one gets close to zero, which could explain why the “line” needed to “curve” into the origin.
5. CONCLUSIONS It is possible for some students to combine their mathematics and physics knowledge to reason intelligently in a simple yet unfamiliar modelling situation.
286
Student Reasoning When Models and Reality Conflict
However, those students are exceptions when compared to the performance of their classmates, and the conditions at this school were more ideal than what exists typically. In fact, the dearth of effective responses and the types of arguments used by the students should suggest that modelling experiences be a necessary component of a vibrant mathematics program. Perhaps because the task was given in a mathematics class, arguments reflect a mathematical bias. Students believe in the power of an equation to describe data, especially when using a “best-fit’’ one. Domain restrictions resolve mathematical problems like division by zero. Higherdegree polynomials describe complicated patterns in form, even if using them distances you further from reality. There were no clues to suggest that the constant density should be considered, and little evidence that students reasoned conceptually from the fact that the quantities involved should vary proportionally. The only scientific considerations were superficial ones, like interpolation being more accurate than extrapolation, consideration for the effect of measurement imprecision, or the need for more data points. The suspension of sense-making reflects the school environment, where the goal is to complete the task. If any student had reasoned that by starting with the longest piece of wire, one can cut off pieces to create the condition for the other data, that viewpoint would have led to an analysis by finite differences, revealed that the density is constant, and led them to question the calculations that were provided. The blind acceptance of the data as accurately depicting the real situation was exacerbated by the fact that the linear density calculation was provided, which might suggest that the data itself is okay. One might argue that studying statistics, especially if taught well, would instill a “habit of mind” to question information. However, it is unlikely that students would consider three subject areas at the same time when they already have difficulty working in the interface of two domains. Assuming that learning about existing mathematical models is a valid goal for mathematics education, there are reasons why modelling experiences may be a necessary condition for developing such knowledge. Jablonka (1997) cited several misconceptions that may occur in interpreting mathematical models, including the belief that mathematical exactitude should apply, that models must be reliable, and that the structure of the model must be the structure of reality. In this study, many student responses exhibited similar beliefs that would be offset by such experiences - for example, the equation coming out of a calculator should accurately describe the data upon which it is based, and that curve-fitting as a task should not have a “reality” mirror based on physical laws, natural conditions or common sense. A second argument may be found in Niss (1999) when he discusses obstacles produced by the process-object duality. These students understood certain processes - some relationships behave in such a way that the amount of one quantity is always proportional to the amount of the other, and that one can create models for these kinds of relationships in a particular form 0, = mx). Their understanding had not progressed to the point that the form represented all situations in which the amount of one quantity is always proportional to the amount of the other, or else there would have been more students who reasoned like the one given in Figure 2. Those students who re-expressed that equation form to say that the ratio y I x stays constant found it more difficult to argue in favor of that model, since it generated a new problem for them involving division by zero. Models may be a situation in which the
Lege
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process-object duality presents an obstacle to learning. In this case, studying proportionality, solving specific problems about direct variation, and expressing and studying fhctions of the form y = mx may not lead to a deep understanding about a model for a specific situation BEING an equation of that form. Finally, in discussing the use of contexts in situated teaching as a means for developing understanding of mathematical concepts, DaPueto and Parenti ( 1999) identified a critical feature as being students’ “understanding of, or confidence with, the process by which the problem is transformed... (p. 7)”. With this set of students in particular, it is likely that they understood the problem transformation, or would have been able to make sense of the situation given time or the opportunity for social discourse. Collectively, it is clear that they lacked the confidence in the process of how the problem was transformed to be able to resolve the inherent conflict that was contained within the problem. And in a manner reminiscent of the conclusions drawn by Lithner (2000), strategy choices are dominated by established experiences - applying methods they know from similar tasks - which can fail when “familiar routines do not work for different reasons (p. I87).”
REFERENCES Blum, W. and Niss, M. (1991) Applied mathematical problem solving, modelling, applications, and links to other subjects - state, trends, and issues in mathematics instruction. Educational Studies in Mathematics, 22,37-68. Burghes, D. (1980) Mathematical modelling: A positive direction for the teaching of applications at school. Educational Studies in Mathematics, 1 1, 1 13-131. California Department of Education. ( 1999) Mathematics framework for California public schools -Kindergarten through grade 12. Sacramento, CA: CDE. DaPueto, C., and Parenti, L. (1999) Contributions and obstacles of contexts in the development of mathematical knowledge. Educational Studies in Mathematics, 39, 1-21. Jablonka, E. (1997) What makes a model effective and useful (or not)? In S. Houston, W. Blum, I. Huntley, and N. Neil1 (eds). Teaching and Zearning mathematical modelling - Innovation, investigation and applications. Chichester: Ellis Honvood, 39-50 Lanier, S. (1999) Students’ understanding of linear modelling in a college mathematical modelling course. Ph.D. dissertation thesis, University of Georgia. Lither, J. (2000) Mathematical reasoning in task solving. Educational Studies in Mathematics, 41, 165-190. Niss, M. (1999) Aspects of the nature and state of research in mathematics education. Educational Studies in Mathematics, 40, 1-24. Usiskin, Z. (1997) Applications in the secondary school mathematics curriculum: A generation of change. American Journal of Education, 106,62-84. Verschaffel, L., and De Corte, E. (1997) Teaching realistic mathematical modelling in the elementary school: A teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28, 577-601.