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Reza Heidari Ghezeljeh

Shahid Beheshti University, Iran <[email protected]>

Zahra Gooya

Shahid Beheshti University, Iran <[email protected]>

Abstract In this paper, we report some of the preliminary findings of a larger study that its main purpose was to gain more insight about the ways in which students use their conceptual and procedural knowledge in dealing with new mathematical tasks. The study was conducted on the school year 2006-2007, and 124 students of pre-university (grade 12) level of Mathematics and Physics branch participated in that. The first author of this paper was mathematics teacher and taught calculus to some of these students. To design a series of suitable tasks regarding the concepts of derivative of functions as the major source of data collection, we did a thorough review of related literature, as well as using the first authors vast teaching experience of calculus. Later in a pilot study, we administered these tasks on different individuals to make sure that they have potential to engage students and challenge them to use their different kinds of mathematical knowledge. The analysis of the data collected via these tasks helped us to choose 6 tasks for the main study; those that had richer and more diverse contexts as well. For the main study, we gave these tasks to all participants (pre-university students), and then collected data through them. We also used selected interviews and teachers reflective notes for a finer grain analysis. For the analysis of the data, we used Skemp's (1976) theoretical framework of relational vs. instrumental understanding. The study revealed that when students are dealing with a new mathematical task, the context that the task/problem is situated in is indeed an effective factor that could evoke different kinds of mathematical understandings. Keywords: Relational understanding; instrumental understanding; conceptual knowledge; procedural knowledge; context, problem situation; calculus; pre-university/grade 12 students. Background It has been almost three decades that some of the mathematics educators and among them, Richard Skemp have become interested in different kinds of mathematical knowledge that is shaped or presented to students in mathematics classrooms. These researchers have been eager to acquire more insights about the distinct nature of the different kinds of knowledge and the ways in which students obtain and use them when dealing with various mathematical tasks/ problems. Van de Walle (2001) by referring to Hebert and Lindquist (1990) explains that mathematical educators have long noticed that distinguishing between the two kinds of knowledge i.e. Conceptual vs. Procedural knowledge is useful. He continues to say that conceptual knowledge contains logical relationships which is made internally and is present in one's mind as part of a network of ideas, the kind of knowledge that Piaget called it Logical-Mathematical Knowledge. In addition, Hebert and Carpenter (1992) describes that conceptual knowledge is understood because of its nature. On the contrary, procedural knowledge of mathematics is the knowledge of rules and procedures which one uses in performing routine mathematical tasks. These are the kinds of knowledge that Skemp (1976) has identified, and based on which he has proposed his conceptual model. In his model or theory, Skemp introduced a new construct as "relational understanding" that its main ingredients are conceptual and procedural understanding. Skemp (1989) argues that teaching relationally can help students to consider mathematics as an integrative whole and understand the relationships between different concepts in such a totality. In this way, students have a chance to build suitable mental schemas for different mathematical tasks and as a result, they can absorb new things that they learn into these schemas. In short, Skemp believes that conceptual understanding means knowing what to do and how, which is a result of intellectual or meaningful learning. He also refers to procedural or instrumental understanding as knowing the rules without reason, and regards it as a result of habit or rote learning. However, Fischbein and Muzicant (2002) by acknowledging what Skemp has proposed believe that currently, the teaching of mathematics has generally been reduced to instrumental learning and understanding that basically does not require the relational type of mathematics. This kind of situation might even disable students to apply the procedures properly because in such a teaching circumstances, it is possible that students regard mathematical concepts as separate islands. This means that the major thing that is formed in students minds as mathematics might be a number of independent facts and formulas that perhaps they do not even know where to use them and where not. Skemp (1989) and Fischbein and Muzicant (2002) use the distinction between these two kinds of understanding to possibly identify the kind of students' mathematical understanding. In this research we also


used this kind of distinction in order to identify the kinds of understanding that applied by students to solve the problems based on their contexts. The Study By referring to the related literature, the researchers were interested in gaining more insight about the kinds of mathematical knowledge that pre-university/ grade 12 students in Iran- majoring mathematics and physics- use when dealing with tasks/ problems regarding derivative of functions. The major question that shaped the study was that ,,when students face a new mathematical task, how much they rely on their procedural knowledge and how much on their conceptual knowledge? To design the study, the first author of this paper that is an experienced mathematics teacher, taught calculus to two classes of pre-university/ grade 12 students majoring mathematics and physics in one of the central cities of Iran. The classes had all their routines and students had to take the two most important examinations of their life namely; the provincial exams and the more competitive one i.e. the "university entrance examination." For the purpose of this research, on one hand, the teacher/researcher continued to do his teaching duties and be dedicated to finish the syllabus and prepare students for those two important exams. On the other hand, we tried to explore our research question to find out more about the ways in which these two kinds of mathematical knowledge was used by students when facing a new mathematics task/ problem. For this purpose, the content that we chose was the derivative of functions since it plays a major role in pre-university calculus. We then, designed six multi-part questions requiring short answers based on research findings and the first author's teaching experiences. After piloting the questions with a number of volunteers, we made some minor changes specially in wording the tasks/problems and prepared them for the main study. The tasks were designed to be engaging, challenging, rich and with diverse contexts. For the main study, we gave these tasks/ problems to 124 participants (pre-university students) and asked them to answer the questions if they wished. Almost all of the students took the challenge and we used their responses as our major source of data. We also conducted a number of selected interviews and used teachers reflective notes as other sources of data for the purpose of triangulation and for a finer grain analysis. For the analysis of the data, we used Skemp's (1989) theoretical framework of relational vs. instrumental understanding. For this paper, we only discuss the findings based on questions 1 and 2 and the reason for selecting these two questions was that both of them were conceptually similar, and the major difference was their contexts, i.e. question 1 had a physical contexts and question 2 had a mathematical context. Results For the larger study, we used the 6 tasks/ problems mentioned above. However, in this paper we are reporting on the results obtaining from the following two tasks/ problems: Question 1: The given graph represents velocity vs. time for two cars. Assume that the cars start from the same position and are traveling in the same direction. (Source: NCTM, 2000. P. 305.)

1a) State the relationship between the position of car A and that of car B at t = 1 hr. Explain. 1b) State the relationship between the velocity of car A and that of car B at t = 1 hr. Explain. 1c) State the relationship between the acceleration of car A and that of car B at t = 1 hr. Explain. 1d) How are the positions of the two cars related during the time interval between t = 0.75 hr. and t = 1 hr.? (That is, is one car pulling away from the other?) Explain.


Question 2: The graph of ' (not ) is given in the figure. At which of the marked values of x is

(2a) (x) greatest? (2b) (x) least? (2c) ' (x) greatest? (2d) ' (x) least? (2e)''(x) greatest? (2f) '' (x) least? (Source: Harvard's Project, P. 161.) To give an overview of the students responses to these two tasks, we first provide Table 1 and Figure1 and then, present the analysis of responses to each task separately in more details. Table1 Students' performance on answering questions 1 and 2 1 Question 2a 2b 1b a Correct Incorrect No response Total 7 9 2 3 2 2 1 24 4 11 23 90 12 4 11 19 94 12 4 78 29 17 12 4

2c 35 16 73 12 4

2d 35 12 77 12 4

1c 68 32 24 12 4

2e 21 18 85 12 4

2f 13 22 89 12 4

1d 50 19 55 12

Figure1. Overall picture of students performances.


Results of question 1 Students responses to question 1 showed that when they were facing with this mathematical task that was sort of new to their immediate experience, they mostly relied on their instrumental understanding and used their procedural knowledge to solve the problem. In fact, their performances can best be describe by what Davis (1994) called it as using certain ways to answer certain questions. Further, Leinhardt (1994) called it static knowledge, i.e. the knowledge which caused Students to get into trouble just because the statement of the question was slightly different from those that they saw and worked with in their textbooks. For example, in parts (a) and (d), we asked the students to compare the distance traveled by two cars using velocity-time chart. The only difference between these two questions was the time of traveled distance that is, for part (a) was t =1 and for part (d) was t (0.75,1). About one-third of those who correctly answered parts (a), (b) and (c) that were about the traveled distance, velocity and acceleration in t=1 respectively, either did not answer part (d) or their answers were not correct. The difference in students' performances on these two similar questions with different context was interesting for us and we wanted to learn more about it. In doing so, the first author discussed this difference with a number of physics teachers. They explained that in teaching movement at high school physics, more often they consider the conditions of moving object i.e. its location, velocity and accelerator in a certain time rather than in an interval. The physics teachers further explained that if occasionally the condition of a moving object is considered in an interval, usually the beginning of the interval is the origin of coordinates. This discussion explained that why those students were confused by the time interval (0.75, 1) in part (d). This finding showed the importance of context when students were confronted with new tasks. In Leinhardts (1994) terminology, these students mostly used their static knowledge which depends on the context, and we would like to speculate that this kind of knowledge made their learning fragile and unstable. Results of question 2 Although the content of both questions were similar, the students responses to the first one were much better than those on the second one (in some parts, up to seven times as much). It means that although 1(a) is equivalent to 2(a) and 2(b), the number of students who correctly answered 1(a) was 79 while this number for both 2(a) and 2(b) was 11. To follow up on this matter, we interviewed some of the students. Through the analysis of the interview data, we found that the students had a considerable amount of experience in solving problems similar to question 1 by using their procedural knowledge. However, the context of question 2 was unfamiliar to them and they did not have reliable experience of working with such problems in their mathematics classes. In fact, they needed a reasonable relational understanding to realize that question 2 is also entails the concept of function and the two questions were similar in content. However, none of the participants paid attention to this similarity. Even the few students, who correctly answered part (a) of both questions, had used different solution methods without noticing their conceptual similarities. For instance, those who correctly answered part (a) of question 1 applied the rule of area under the curve to justify their answers. But not even a single student used this rule to answer parts (a) or (b) of question 2. All the eleven students who correctly answered both parts (a) and (b) of question 2 had used similar reasons to justify their answers. They explained that because ' is above x-axis, then ' always exist and it is always positive. So, is always ascending in (x1,x6). Therefore, the function has the least and the greatest amount at the beginning and the ending of the interval respectively. In another case, the number of correct answers to part (b) of question 1 was twice as much as the number of correct answers to parts (c) and (d) of question 2. However, we expected similar responses since the common goal of these parts were finding the extreme point of a function by using its graph. Further, questions 1(b), 2(c) and 2(d) that all had the same purpose of using the graph of ' to answer questions about ' were designed to find more about the students graphical understanding of the concept of function. However, the students' answers revealed that the responses to these parts varied according to their contexts. Similarly, in question 1(c), 2(e) and 2(f), the students were asked to answer some questions about '' using the graph of ' and again, the nature of difference in responses were the same as the above. Likewise, the number of correct answers to question 1(c) was three times more that correct answers to 2(e) and five times more than correct answers to 2(f). Therefore, we got more evidence to state that familiarity with the context was an important factor leading students responses to a mathematics task/ problem which means that they mostly relied on their procedural knowledge. We got the same results by analyzing the responses to other parts of this question, and compared them with the equivalent parts of question 1 having different context. To sum up this part, we would like to confidently say that the context evoked the students procedural understanding more than their conceptual / relational understanding.


Concluding Remarks In this research, we were interested in seeing that when students facing a new mathematical task/ problem, how much they rely on their procedural knowledge and how much on their conceptual knowledge? The analysis of the data gathered via different means revealed that when students face new mathematical tasks, they most likely rely on their procedural knowledge. It was interesting to see that a slight change in the statement of the task/ problem or a change in the context was a challenge for students and even sometimes, made them confused. Further to say, students become prone to familiar contexts and more often, the textbooks reinforce this habit of mind. Thus, having fewer and less challenging context leads them to rely more on their procedural knowledge rather than conceptual knowledge. The review of related literature has also support this finding and has indicated that most students in mathematics classes are taught procedurally and in these classes only slight attention has given to the development of relational understanding that requires a great deal of conceptual knowledge as well. In this kind of teaching, students frequently need new procedures or rules for new situations; the instruments that are usually given to them by external helps without enabling them to apply their existing knowledge to a new situation. This situation makes students fragile and incompetent when facing new tasks or even familiar tasks in unfamiliar contexts. This fragility and incompetence is extremely costly and harmful for both students and educational systems. Thus, as an educational implication of this research, we recommend our mathematics teachers to use various contexts for presenting same content. For instance, even when presenting a procedure like the techniques of maximizing and minimizing a function, we could use a range of different contexts such as physics, geometry and economics to make their knowledge more stable and more transmittable to new situations. We would like to close by referring to Wong's (1994) interpretation of learning the rules without reasons or instrumental understanding of mathematics: When we insist on instructing children to follow the exact operational rules without appropriate elaboration on the underlying conceptual framework, it is very much like leading somebody blindfold by the hands through the streets from one place to another several times and then requesting him or her to repeat walking along the same path (p. 37.). References Davis, R. B. (1994). Comments on Thomas Romberg's Chapter. In A. H. Schoenfeld (Ed.) Mathematical Thinking and Problem Solving (pp. 312- 321). Hillsdale, New Jersey: Lawrence Erlbaum Associates, Inc. Fischbein, E. & Muzicant, B. (2002). Richard Skemp and his Conception of Relational and Instrumental Understanding: Open Sentences and Phrases. In D. Tall & M. O. J. Thomas (Eds.) Intelligence, Learning and Understanding Mathematics (pp. 49-78). Flax-ton, Australia: Post Pressed. Hiebert, J. & Carpenter, T.P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan. Hughes-Hallett, D. & et al. (1994). Calculus: Produced by the Consortium based at Harvard and funded by a National Science Foundation Grant. John Wiley & Sons, Inc. Leinhardt, G. (1994). Comments on Thomas Romberg's Chapter. In A. H. Schoenfeld (Ed.) Mathematical Thinking and Problem Solving (pp. 305- 311). Hillsdale, New Jersey: Lawrence Erlbaum Associates, Inc. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: The Author. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics teacher, 77, 2026. Skemp, R. R. (1989). Mathematics in the Primary School. London: Routledge. Van De Walle. & John, A. (2001). Elementary and Middle School Mathematics: Teaching Developmentally. Addison Wesley. Longman Inc. Fourth Edition. Wong, K. M. (1994). Can Mathematical Rules and Procedures be Taught without Conceptual Understanding? Journal of Primary Education, 5(1), 33-41.



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