The curriculum used software that represents manual ..... solution which best represents the situation ("close" to the situation) provides a description of the.
GRAPHS THAT ARE CLOSE TO SITUATIONS: AFFORDANCES AND CONSTRAINS Shoshana Gilead and Michal Yerushalmy OBJECTIVES
Recent research on algebra curricula suggests that a function based algebra curriculum within a computerized environment that allows exploring the multiple representations of functions and the way that bodily activities are connected to temporal graphs, are useful resources for students solving word problems. We study when and how graphs of functions can be a helpful resource for students in such an environment for solving motion word problems. THEORETICAL FRAMEWORK
The deep structure of a word problem Motion word problems such as problem 1 in Figure 1 can be categorized with attention to two sorts of structures: the quantitative structure, which describes the set of given quantities as constants, unknowns, and constraints (Shalin & Bee, 1987), and the situational structure, which describes relationships among physical properties of the elements within a story problem (Hall, Kibler, Wenger, & Truxaw, 1989). Problem 1 A and B are 470 km apart. A truck and a car started traveling at the same time towards each other. The car traveled from A to C at an average speed of 80 km per hour. The truck traveled from B to C at an average speed of 56 km per hour. Both drivers reached C at the same time. a. How long after starting their journey did the two vehicles reach C? b. How far had each of them traveled when they arrived at C?
Figure 1: A typical motion problem In problem 1 for example, the quantitative structure is determined by the given constant speeds (v1=56, v2=80), the unknown amounts of time spent traveling (t1, t2) and the unknown distance traveled by each of the two vehicles (s1, s2). The constraints are the given total distance traveled (s1+s2=476) and that both traveling times are the same (t1=t2). The situational structure is described by two vehicles setting off from the different places at the same time, traveling in opposite directions, and simultaneously arriving at the same destination. Graphs of functions that describe the position over time of motion phenomena (Figure 2) may act as a visual representation of the situational structure as well as of the embodied Source-Path-Goal schema described by Lakoff & Nunez (2000).
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The position versus time graph for problem1 as presented in Figure 2 reflects the situational structure of this problem. It reflects the situational structure of two vehicles setting off at the same time. Both functions "start" at t0=0 from different locations ( F(0), G(0), F(0)≠G(0)), traveling in opposite directions (F(x) has a negative slope, G(x) a positive one), and simultaneously (at tf) arriving at the same destination (G(tf), F(tf) for which G(tf)=F(tf )). p o s it io n r e la t iv e t o A B
F (t)
C
G ( t f) = F ( t f) G (t)
A
tf
t- tim e
Figure 2: The position versus time graph for problem 1 Combining the situational and quantitative structures Once the situational structure is graphed (Figure 3a), the quantitative structure of the problem (Figure 3b) can be integrated (Figure 3c), thus creating a visual model of the combination between the two (Figure 3d). In problem 1, this combination is characterized by the given slopes of both functions (m1=-56, m2=80) and a given point (A(0,0), B(0, 470)) on each function in the graph. P o s it io n R e la t iv e t o A F (t)
v1=56km/hour v2=80km/hour s1+s2=470km t1=t2 hours
s2
G (t) t im e
Figure 3a: A sketch of the situational structure graph in problem 1
P o s itio n R e la tiv e to A s1+ s2= 4 7 0 v 1= 5 6 s1
Figure 3b: The quantitative structure of problem 1
v2= 8 0 tim e t2 t1
Figure 3c: Integrating the quantitative structure into the situational structure graph (v1, v2 are absolute values)
Position relative to A B( , )
m 1 =
m 2 = A( ,)
time
Figure 3d: The combined structure's model of problem1 (as defined by the situational structure and the quantitative structure).
Using the combined model, one can describe the corresponding linear functions symbolically (F(t)=80t, G(t)=470-56t) by the time variable (t) and then solve the problem by solving the equation
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(F(t)=G(t)) or one can accurately graph the two functions and read the values of their intersection point. We defined problems with such a combination between the situational structure and the quantitative structure as graphable problems. Not all motion problems are graphable. Problem 2 (Figure 4a) has the same situational structure as problem 1 (Figure 4b) but a quantitative structure (Figure 4c) that differs in one constraint: s2-s1=given, instead of s2+s1=given. Problem 2: C is located between A and B. A truck and a motorcycle started traveling at the same time. The truck traveled from A to C at an average speed of 60 km per hour. The motorcycle traveled from B to C at an average speed of 42 km per hour. Both drivers arrived at C at the same time. The truck traveled 45 km more then the motorcycle did. a. How long after starting their journey did the two vehicles reach C? b. How far had each of them traveled when they arrived at C?
Figure 4a: An example of a similar constant-rate motion problem, but with a sketchable structure v1=42 km/hour p o s it io n P o s it io n p o s it io n r e la t iv e t o A R e la t iv e t o A =60 km/hour v 2 r e la t iv e t o A s2-s1=45 km B ( ,? ) v1 s1 F (t) t1=t2 hours m 1= F (t)
s2 G (t)
s 2- s 1= 4 5
t im e
Figure 4b: A sketch of the situational structure graph in problem 2
v2
t im e t2 t1
Figure 4c: Figure 4d: Integrating the The quantitative quantitative structure into the structure of situational structure graph problem 2 (v1, v2 are abs values).
G (t) B ’ (0 ,4 5 ) m 2= A ( , )
t im e
Figure 4e: The combined structure's model of problem 2 (as defined by the situational structure and the quantitative structure).
In this case, one of the functions in the graph (F(t) in Figure 4e) cannot be fully specified by the given quantitative structure without solving the problem first since point B on F(t) is not determined. Point B' that is not on F(t) ((0,45) in Figure 4e) is given instead, therefore the situational structure graph can only be sketched and can't be graphed accurately before solving the problem. We call problems with such a combination of the situational structure and the quantitative structure, Sketchable problems. In this case formulating one of the functions in the situational structure graph (F(t) in Figure 4e), requires using letters as variables and as unknowns within the same expression (F(t)=k-42t, where t describes the time variable and k the unknown position value of point B in Figure 4e. Thus, these two types of problems should pose different cognitive challenges to students whose algebra studies are based on visual representations of temporal processes that are described by 3
functions' expressions. In the current research we used graphable and sketchable motion problems as a tool to explore and report on these challenges as well as to study if, how and for what purpose students draw graphs in their solution of a problem in context and if the graph they draw acts as a means to avoid or as a means to support an algebraic solution. We conducted this exploration within the specific context of motion problems to illustrate the delicate connections between the embodied cognition and the use of algebra. THE PARTICIPENTS
The participants in the study were all ninth graders from four different middle schools of a similar socio-economic background. All these schools adopted the 'Visual Math'1 curriculum for learning algebra. This curriculum emphasized the function and its multiple representations as the main concept in developing the algebraic notation. The concept of function was developed mainly through investigating temporal phenomena. The curriculum used software that represents manual mouse motions as a sketch of a temporal graph (Yerushalmy; Shterenberg; 2001). Another software allowed investigating the transition from stories describing temporal phenomena to their graphical sketches of temporal functions (Yerushalmy & Shterenberg, 1993). This established the graphical representation as one that can be sketched directly from a story without necessarily passing through the algebraic representation first. The algebraic notation that was introduced later stressed the algebraic letter mainly as a variable, the algebraic expressions as functions and algebraic equations as a comparison between two functions. HYPOTHESIS
1. Successes in solving graphable problems would be higher than success in solving sketchable problems 2. Students will use graphs in their solution as a visual aid to formulate an algebraic equation. 2.1 Students prefer to sketch their graphs rather than draw them accurately 2.2 Students tend to use a graph that best describes the situational structure of the problem 3. Students will use equations to determine the solution 3.1 Students will more frequently describe equations as a comparison between two functions in different representations rather than a statement of relations between unknowns. 3.2 Students will formulate their equation as a comparison of the functions that correspond to the graphical representation of the situation. 1
The visual Math curriculum is an intensive technology function approach to algebra. It was developed in the Center of Educational Technology, Israel. 4
METHODS AND DATA SOURCES
Based on classification of distance-rate-time problems by both their quantitative structure and their situational structure (Yerushalmy & Gilead 1999) we chose to concentrate on 21 possible combinations of quantitative structures and situational structures. For each of these 21 combinations we created a rate problem. 17 classes of 9th graders who had learned algebra in a function based approach participated in the study. 42 questionnaires were prepared for each class. Each questionnaire included four of the 21 problems. The order of problems was not controlled, and each set of four problems was randomly chosen from the 21. Questionnaires were randomly administered. At least 98 solutions were submitted for each problem. Solutions were submitted with details that allowed careful analysis of the solution method. Although students studied Algebra with the aid of technology they were not allowed to use any graphical software to solve the four problems on their test. For this report we refer to four combinations of two situational structures (
and
in Figure 5)
and two quantitative structures (Q1 and Q2 in Figure 5) combined to create two graphable problems and two sketchable problems (Figure 5). Problems 1,4 are Graphable problems and 2,3 are Sketchable problems. Situational structure Quantitative structure Q1: v1=given, v2=given, t1=t2
Problem 1 - Graphable B ( , )
Problem 3 - Sketchable
m 1=
B(,?)
s1+s2=given
m 2 =
m 2=
A(,)
A ( , )
Q2: v1=given, v2=given, t1=t2 s1-s2=given
Problem 2 - Sketchable B ( ,? )
Problem 4 - Graphable
m 1=
B(,) m 2= A ( , )
m 1 =
m 1 = m 2 =
A( , )
Figure 5: Four combinations involving two quantitative structures and two situational structures
In our analysis, the graphable set of problems and the sketcable set of problems acted as the independent variables. Success in solving these two sets of problems and the characteristics of the 5
solution for each set were the dependent variables. Students' performance was scored as correct or incorrect. A solution was considered correct if a correct mathematical model was presented either obtained graphically by reading the solution from an accurate graph, symbolically by an algebraic equation or a system of equations, or numerically by reading the solution from a table of values or an accurate graph. To explain the differences in success between these two sets of problems we focused on the type of graphical representation that was used in the solution, and if it was sketched or drawn accurately. We also checked the use of Algebra in each solution and the way it was connected to a graph as well as the mathematical model which was used to determine the solution. We used Chi square tests for analyzing the qualitative data we got.
RESULTS
Success rate: Analyzing the performance on the graphable set of problems and the sketcable set, we found significant differences between the two: 90.8% success in solving the graphable problems versus a 41.8% success rate for sketchable problems. This confirms hypothesis 1 regarding the difference in complexity between the two sets. The type of graph: most of the solutions included a situational structure graph (87.2% of the solutions for the graphable problems and 71.4% of the solutions for the sketchable problems), but still with a significant difference between the two. This confirms hypothesis 2.1, that students tend to use a graph that best describes the situational structure of the problem. Most of the solutions included a sketch of the situational structure graph (73% of the solutions for the graphable problems and 71.4% of the solutions for the sketchable problems). This confirms hypothesis 2.2 that students prefer to sketch their graphs rather than draw them accurately. Only a small number of solutions for the graphable problems (14.3%) included an accurate graph of the situational structure. A few solutions which included an attempt to draw such an accurate graph for the sketchable problems (6.1% of the solutions for the sketchable problems) included one accurate line "starting" at the origin while the other line in the situational structure graph, which couldn’t be drawn without finding the solution first, was missing. Use of Algebra: Algebraic expressions of two functions were present in most of the solutions for the graphable (95.9%) and for the sketchable (88.8%) problems, but still with a significant difference between the two. This confirms hypothesis 2 that students will use graphs in their solution as a visual aid to formulate an algebraic equation.
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73.5% of the solutions for the graphable problems included both a situational structure graph and algebraic expressions of two functions that matched the graph. This was not the case for the sketchable problems. Although a situational structure graph was included in 71.4% of the solutions for the sketchable problems; only 27.6% of the solutions included expressions that matched the situational structure graph. These findings regarding the sketchable problems reveal the difficulty that these students have in using a letter to represent an “unknown” value instead of variables. For both graphable problems and sketchable problems, a significant connection was found between the correctness of the solution and the correspondence between the situational structure graph and the algebraic expressions. However this connection behaved differently for each problem type. In the case of graphable problems, 96.5% of the solutions that included a correspondence between a situational structure graph and algebraic expressions resulted in a correct algebraic equation. Whereas among the solutions that lacked such a correspondence only 51.9% resulted in a correct equation. In the case of sketchable problems, an inverse relationship was observed. Only a negligible percentage (1.9%) of solutions with a correspondence between a situational structure graph and algebraic expressions resulted in a correct algebraic solution, whereas among the solutions in which a correspondence was not observed, 51.4% contained a correct algebraic solution. The model used for determining the solution: Algebraic equations were the most popular model for determining the solutions (90.8% for the graphable problems and 65.3% of the solutions for sketchable problems) this confirms hypothesis 3 that students use equations to determine the solution. Although 14.3% of the solutions for the graphable problems included an accurate situational structure graph, students did not provide a solution by reading the numerical solution from the graph or its corresponding table of values. Only in 4.6% of all the solutions an accurate graph acted as the only model for determining the solution of the problem and only in 4.6% of the solutions a table of values acted as the only model. In the case of the graphable problems, 70.9% included a correct equation in one variable ( F(x)= G(x)) presenting the comparison of the two functions (F(x) and G(x)) in the situational structure graph. This confirms hypothesis 3.1 and 3.2 that students tend to formulate their equation as a comparison of the functions in their situational structure graph. Such a common solution for the graphable problems is presented in Figure 6. This solution for problem 1 includes a sketch of a situational structure graph (the change in position over time of each vehicle relative to point B), two algebraic expressions that match the functions in the graph (g(x)=56x and f(x)= 476-80x) and an equation (476-80x=56x) that represents the comparison of these two functions. 7
Figure 6: A common solution for a graphable problem (problem1) Formulating an equation with "variables" and "unknowns" for the sketchable problems and then writing another equation as a statement about unknowns, was found to be a very complicated task for the students in this study. In fact, only one student succeeded in presenting a correct algebraic model that was based on functions matching a situational structure graph (see Figure 7 for the solution of problem 3). This supports hypothesis 3.1 that students tend to view equations as comparison between functions rather than a statement of relations between unknowns.
Figure 7: The single correct algebraic solution for a sketchable problem (problem 3) with an equation comparing the two functions in the situational structure graph 8
This solution includes a sketch of the situational structure graph and algebraic expressions where x is the time variable and y is the unknown distance AB. The comparison “72x=50x+y” of these two function, allows the student to express their meeting time in terms of y (x=y/22). The second equation (33.3y+3.3y-y=427) states relations between unknown lengths: AB=3.3y, BC=3.3y-y and the total lengths of the two is 427. This student seemed to overcome the main obstacle that most of the students faced in solving sketchable problems; he managed to use algebraic manipulations in order to shift from viewing expressions as processes described by functions to viewing expressions as unknown quantities. This allowed him to formulate an equation that describes relations of quantities instead of functions. Many of the students found it difficult to use a letter representing an “unknown”. Though some overcame this problem, they faced difficulties in pursuing the solution that required formulating an equation as a relation between “unknowns”. Such a common difficulty can be observed in the example in Figure 8.
Figure 8: A solution for a sketchable problem (problem 2) with an equation as a comparison of the two functions in the situational structure graph.
Here the student succeeded to express algebraically both functions in the situational structure graph (60x and -42x+y) using the letter y to present the unknown distance. Though she succeeded in formulating an equation that corresponds to the comparison of these two functions (-42x+y=60x), she was not able to pursue the solution further. 9
All but one of the correct solutions for the sketchable problems were algebraic solutions, e.g. the solution for problem 3 in Figure 9, disconnected from the situational structure graph.
Figure 9: A false attempt to solve a sketchable problem (problem 3) by formulating two functions that match the situational structure graph, followed by a solution ignoring the graph
This student also succeeded in using the letter y to represent the unknown distance. However, probably because he couldn’t see how to pursue the solution, he dropped the idea of constructing a solution which corresponds to the situational structure graph and formulated a correct equation (50x+72x=427) which does not describe the functions in his sketch. Most of the students, however, could not formulate an equation that does not directly describe the situational structure graph. A common example for this difficulty can be viewed in the solution for problem 3 shown in Figure 10. Here the student succeeded to first formulate a correct equation (60x=42x+45) representing the relation of lengths. In this equation the expression 60x can be read as describing both the function that passes through the origin in the graph and the unknown distance AC, while 42x+45 describes only the unknown length BC but not the other function in the graph.. Since the student probably was not aware of the difference between the two meanings of an expression and since 42x+45 does not correspond to the decreasing function in the graph, he replaced it with the function -42x+60. In this way he obtained a decreasing function but this resulted in altering his correct equation (60x=42x+45) into an erroneous equation (60x=-42x+45).
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Figure 10: An attempt to solve a sketchable problem (problem 2) using an equation representing the comparison of the two functions in the situational structure graph, leading to an erroneous equation
DISCUSSION
This study allows an insight into how students that learned for three years function based algebra and who used technology to draw different graphs of functions for a situation, tend to use such graphs for solving problems in context when the technology is not available. The study allows an insight into their capability and preference in drawing graphs, the way the graphs are used to solve problems in context and the benefits and obstacles they experience while using them. 1) Sketchable problems are more difficult to solve than graphable problems. 2) The graphical representation of functions acts as the base of a solution attempt to solve problems in context. 3) Students prefer using a graphical representation that best describes the situational structure of the problem 4) Students prefer to sketch their graph rather than drawing it accurately. 5) Students tend to use in their solution attempts algebraic expressions of functions that correspond with the situational structure graph they sketched.
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6) In the case of graphable problems, creating algebraic expressions corresponding with the situational structure graph usually led to the formulation of a correct equation that represented the comparison of the functions in the graph. This common solution strategy for solving problems in context, explains the high success rate in solving the graphable problems and the low success rate in solving the sketchable problems. Previous studies (Hall et al, 1989) showed that experts tended to include in their solution attempts for problems in context a visual representation to describe the situational structure of the problem. The findings in this study indicate that students in a function based environment acted almost in the same way. However, instead of inventing such a representation, they sketched graphs of functions to represent visually the situational structure of the problem. In such graphs the horizontal axis represented the elapsed time from the time that the objects started moving and the vertical axis represented location relative to the starting point of one of the moving objects. The preference to draw such a graph over others can be explained by the its ability to visually capture the embodied Source Path Goal motion schema (Lakoff &Nunes, 2000). The students in this study demonstrate a different skill level in drawing graphs than those mentioned in other studies. Janvier ( 1978, 1981a) argues that students tend to draw a graph point by point and are not able to visualize the graphic representation of a situation without drawing its graph accurately. Herskovics (1989) considered this to be a cognitive obstacle. However, in this study, the students' ability and confidence in sketching the process of motion as an entity instead of creating it point by point indicates that we can overcome this obstacle by pedagogy. The main argument and concern among math educators is that function based approaches, by using multiple representation of functions, encourage the use of non-algebraic tools instead of developing algebraic thinking (Kieran,1994; Filloy & Shuterland, 1996; Shutherland & Pozzi, 1995) and that the improved performance in solving problems in context is mainly related to the ability of students to escape from using algebra in their solution and solve such problems only numerically or graphically. Our findings suggest that this is not a necessary outcome of a function based approach. The students in our study presented a different way of using graphs in their solution process. The fact that most of the solution processes for the graphable problems included a sketch of a situational structure graph as well as algebraic expressions that matched the graph indicates that in general the graph did not act as a tool for solving the algebraic equation but rather as an aid for constructing it. This provides evidence that students may develop an appreciation for algebraic tools over the numerical ones and use the other representations of the function mainly to support the formulation of the algebraic solution rather than as a way to escape it.
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The high success rate in solving the graphable problems over the sketchable problems exposes the gains as well as the limitation of function based approaches. The emphasis within such an approach on a letter presenting variables and on an equation presenting the comparison of functions (in the graphical representation) helped in formulating a correct equation for certain problems ( the graphable) and helped to overcome difficulties that students experience in solving such problems algebraically by the equation unknown strand. However this emphasis on the algebraic letter and equations seemed to prevent the formulation of a correct equation in the case of sketchable problems. Thus, while known complexities disappeared and students were successful in solving graphable problems by sketching graphs as a visual aid, sketchable problems present a challenge we have yet to meet when using the Visual Math approach.
EDUCATIONAL IMPORTANCE OF THE STUDY
Embodied cognition has an important role in solving motion problems. Students in a functionsbased algebra educational environment tend to use graphs of functions as a visual representation of their Source-Path-Goal scheme of motion phenomena. Students seem to prefer sketching graphs to graphing them accurately, and these graphs mainly support formulating an algebraic solution rather than as a way to determine the solution via graphic means only, avoiding the use of algebra. This seemed to work well in the case of Graphable problems, in which creating an algebraic solution which best represents the situation ("close" to the situation) provides a description of the situation algebraically with letters interpreted as variables and equations as comparisons of the functions describing the situation. However, in the case of the Sketchable problems creating such a solution requires a broader interpretation of the algebraic letter not only as a variable but also as an unknown and a broader view of the equation not only as a comparison of functions but also as a statement of equality between unknown numbers. The study suggests that within a functions-based curriculum, motion problems may become interesting modeling tasks where different mathematical tools are used to describe the situation. The graphical tools seem to have the capacity to facilitate using algebraic representations instead of being a way of avoiding them. Although graphable problems suit this best, the sketchable problems shouldn’t be ignored in a functions-based curriculum. These problems may act in such curricula as a channel for starting an interesting conversation regarding the different meanings of algebraic letters, expressions and equations, to facilitate a deeper understanding of algebra.
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REFERENCES
Filloy, E.& Sutherland, R. (1996) Designing curricula for teaching and learning algebra. In Bishop A. J. et al (eds.) International handbokk of mathematics education 1, 139-160. Dordrecht, Kluer Academic Publisher. Hall, R., Kibler, D., Wenger, E., Truxsaw, C. (1989). Exploring the episodic structure of Algebra story problem solving. Cognition and Instruction, 6, 223-283. Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. in S.Wagner, C. Kieran (Eds.) Research Issues in the Learning and Teaching of Algebra. NCTM & Erlbaum Associates. Janvier, C. (1978). The teaching of graphs: A language approach. Proceedings of the thirtieth conference of commission internationale pour l' etude et l' amelioration de l' enseignement des mathematiques. Santiago, Spain. Janvier, C. (1981a). Difficulties related to the concept of variable presented graphically. In C. Comiti & G. Vergnaud (Eds.) Proceedings of the fifyh international Conference for the Psychology of Mathematics Educayion (pp. 189-192). Grenoble, France: Laboratoire I.M.A. Kieran C. (1994). A functional approach to the introduction of algebra: some pros and cons, In da Ponte, J. P. & Matos, J. F. (Eds.). Proceedings of the 18th International Conference for Psychology of mathemaics education , 1, 157-175 Lakoff G. and Nunez R.E. (2000). Where mathematics comes from. How the embodied mind brings mathematics into being, Basic Books, NY. Sutherland, R. & Pozzi, S. (1995). The Changing Mathematical Background of Undergraduate Engineers, The Engineering Council, London. Yerushalmy, M. & Gilead, S. (1999). Structures of constant rate word problems: A functional approach analysis. Educational Studies in Mathematics, 39, 185-203. Yerushalmy, M & Shterenberg, B. (1993) The Function Sketcher (in Hebrew, English and Arabic) for Center of Educational Technology, Ramat-Aviv. Yerushalmy, M & Shterenberg, B. Move On! (2001) Motion Laboratory (software in Hebrew and English) for the Center of Educational Technology, Ramat-Aviv. http://www.cet.ac.il/mathinternational/more12.asp
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