School of Mathematics and Statistics
Curtin University of Technology
There has been an increasing level of interest in the use of Computer Algebra Systems in the teaching of undergraduate mathematics courses. To a large extent the spread of use of CAS as teaching aids is based on the assumption that they generate an adequate learning environment in which students could examine their own constructions and understandings about mathematical concepts that they have been exposed to during lectures and tutorials. While such a view concerning the role of CAS in students' understanding of higher mathematics is gaining currency within the mathematics education community, there exists little empirical data to inform us about the nature of this understanding.
The present study was undertaken with the aim of generating data relevant to the above issue by analysing undergraduate students' understandings of concepts and procedures in the area of linear algebra. Students' understandings are analysed within the framework of conceptual entities developed by Harel and Kaput (1991). Analyses of the quality of knowledge activated and used by two students while they attempted to solve a target problem with the help of MAPLE suggest that a) students' understandings are limited in depth and b) students in the main do not fully exploit the MAPLE environment in order to develop higher levels of understandings that are required for the comprehension of more complex concepts, and the analysis of the structure of novel problems. Implications of this study for further research and the independent use of MAPLE by students as a learning tool are discussed
We examine the above issue in the context of teaching concepts in linear algebra. Linear algebra has become one of the most important areas of university mathematics as it empowers students to tackle problems from fields as diverse as engineering, sciences, economics and computing. In the past decade, computer algebra systems (CAS) have been widely used in the study of linear algebra in many departments of mathematics in Australia and elsewhere. The introduction of CAS, such as Maple and Matlab (Crane, 1991) was expected to not only help student perform extended computations rapidly and easily but also develop students' mathematical intuitions and ability to solve problems.
But the question remains as to what is the nature of this mathematics understanding that we expect from our students, and what role does CAS play in promoting that understanding. One way to examine this issue is to consider the instructional effect of CAS on the construction of schemas or similar knowledge structures. Harel and Kaput (1991) in their examination of advanced mathematical understanding interpreted schemas in terms of 'conceptual entities' which are mental objects which allows students not only to understand rules and procedures that are associated with mathematical concepts but also other representations of that concept. Within this framework, Harel and Kaput (1991:83) argued that conceptual entities have three major roles in mathematical understanding: alleviating working memory or processing load, facilitating comprehension of complex concepts and assisting with the focus of attention on appropriate structure in problem solving. This paper is a report of a study in which we investigated students' understanding of trigonometric and linear algebra concepts in light of the above theory.
A set of mathematical problems involving concepts in calculus, analytical geometry and linear algebra was developed. The solutions to these problems were evaluated by lecturers at Curtin University in terms of their prerequisite knowledge and the role of MAPLE (Ellis, Johnson, Lodi and Schwalbe, 1992) in helping students activate these knowledge components. A major consideration in our decision making about the problem was the degree of integration of the discrete knowledge components that are required for solution generation. Results of this analysis led us to use a Tetrahedron Problem as the target problem for this study. The problem was presented to the students on a sheet of paper. Students were required to answer two focus questions: a) find the angles between body diagonals of a cube and b) show why one of these angles is also the angle between the two lines joining the centre of a regular tetrahedron with its vertices.
Students were given the following information to help them construct a tetrahedron. Part of the given information related to certain properties of a square and its relationship to a regular tetrahedron. In addition, students were given three MAPLE commands for the construction of the required figure.
A regular tetrahedron fits inside a cube, with the edges of the tetrahedron equal to the surface diagonals of the cube.Before the problem was given to the students, they were allowed to go through the many options that were available on MAPLE. Both students have had about six months exposure to MAPLE. Students were given 10 minutes to read the problem and discuss it among themselves. At the end of this session, students were asked to talk aloud as they attempted to construct the tetrahedron and answer the focus questions. The entire session was video recorded and later transcribed for analysis.
The four body diagonals join opposite corners and pass through the centre of the cube.
The commands specified a plane in three dimensions.
Four planes defines the tetrahedron
MAPLE commands: (p0; ploygonplot; axes).
After the above initial question, students used the MAPLE commands to draw the required tetrahedron. Having constructed the tetrahedron the students attempted to answer the focus questions. At this stage they drew two-dimensional figures to interpret the three-dimensional figure produced by MAPLE. The construction of these two-dimensional planar images suggests that students were attempting to decompose the MAPLE figure. This activity can be seen as analysing the structure of the figure in order to see how the various planes and diagonals were linked so that they could get to the diagonals in question, and the angles between the diagonals.
Students' actions at deconstructing and constructing the MAPLE figure also involved the activation of prior knowledge about right-angled triangles, the Pythagoras' theorem and trigonometric ratios in the search for the answers. Here, one could see a powerful pedagogical effect of the CAS in helping students access knowledge that they have learnt before and use in the process of problem solving. In other words, MAPLE, by helping the students construct the tetrahedron has in turn freed their mental workspace and facilitated students' search for previous knowledge that is relevant to the solution. It appears that our students had built up conceptual entities of the type that Harel and Kaput (1991) discussed around right-angles, trigonometric ratios and two-dimensional figures. However, this set of knowledge is unlikely to be accessed and exploited during problem solving had MAPLE not been employed in assisting students draw the tetrahedron in the first instance or assist in the visualisation of the diagonals in the tetrahedron.
The role of MAPLE in problem comprehension can be viewed from an alternate angle where the focus is on decomposition of the target problem into subproblems, and attempts to solve the subproblems. Problem decomposition constitutes an important general strategy (Alexander & Judy, 1988) in the search process which was clearly precipitated by the help provided by MAPLE. Again, having liberated students from the time-consuming task of drawing the tetrahedron, students were able to engage in the more productive and demanding phase of problem representation which in this case involved decomposition and reconstruction. Thus, MAPLE can be seen as promoting the use of conceptual entities which Harel et al argued would assist students in focusing attention on the structure of the problem.
While the above moves were important in the search for the solution of the problem, the students did not rotate the tetrahedron by using different values for the axes MAPLE command. We argue that this action would have helped students visualise the diagonals better. The failure to use this command can be seen as an instance where students were not fully exploiting the available options in MAPLE when it can be shown that they in fact 'had' this knowledge. This has implications for the training of students in the use of MAPLE commands.
|Solution Phase||Prior knowledge||MAPLE commands|
|Problem Analysis||Definition of body diagonals|
|Geometrical||> with (linalg) :|
|Vector||> with (plots) :|
Table 1 shows that during both the phases of problem analysis and problem representation students activated a number of important mathematical and related prior knowledge. Problem analysis required students to search their memory for definitions involving body diagonals. The given figure also prompted students to think about knowledge of trigonometry and two dimensional figures such as right triangles. While these were required to analyse the problem, it was knowledge about unit vectors and axes that led the students to think about visualising the figure more dynamically.
The students could have used pencil and paper to sketch the figure but MAPLE provided them with a facility to sketch such figures. This graphing facility of MAPLE has two significant advantages. Firstly, students were able to graph a function much faster than using the pencil and paper method. Secondly, the ease of sketching in MAPLE acts as an incentive to students to consider graph sketching in their repertoire of problem solving strategies, a point that was emphasised by Schoenfeld (1985) in his analysis of mathematical problem solving and the use of appropriate strategies.
The advantages conferred by MAPLE in promoting mathematical knowledge use could be given the following interpretation from a cognitive point of view. Sweller (1988, 1989) in his examination of mental resources involved in human problem solving demonstrated that in order for the solver to attend to the more novel and difficult parts of the problem in question, he must free his mental space. One way to achieve this is to reduce the cognitive load imposed by the less important tasks or aspects of the problem such as graph sketching, thereby permitting the solver to focus his attention and mental resources on the more relevant and demanding part(s) of the problem. This line of argument about information processing during problem solving suggests that MAPLE, by allowing students to sketch a figure rapidly could have facilitated students' efforts in making progress with the problem. While making progress may in itself not be sufficient to the successful solution outcome, MAPLE does encourage knowledge searches which are more critical for that outcome.
During the second phase of the solution attempt students explored the problem further by constructing a representation of the problem using mathematical concepts and procedures from vectors, i.e., the given problem was interpreted via a vector representation. It is possible that students could have initiated the above move without the aid of MAPLE. However, we argue that the processing of the problem using vectors was encouraged by students' prior experience in drawing on this facility of MAPLE. Our past trials with MAPLE has showed that students tend to become familiar with some of the more commonly used procedures in MAPLE. One such procedure involves setting up vectors to examine a class of problems that involve three dimensional figures. Given that the target problem of the present study did contain a solid figure it appears that students' decision to reinterpret the problem from a new angle was promoted by MAPLE. Likewise, MAPLE could also foster students' patterns of reasoning that could lead them onto developing more sophisticated representations of the problem, an action that is supported by the knowledge rich computer environment (Kaput, 1987).
The above situation not only aided in the construction of an alternative representation, but also triggered a further chain of knowledge activation but this time students drew on and evaluated the usefulness of their prior knowledge of trigonometry and right-angled triangles, a process that was observed in a geometry problem-solving attempt by Lawson and Chinnappan (1994). In sum the transformation of the problem has helped students not only view the problem from another perspective but also promote the activation and use of further prior mathematical knowledge which would otherwise have remained dormant.
Crane, M. (1991). The impact of MATLAB on teaching numerical mathematics. In B. Jaworski (Eds.), Technology in Mathematics Teaching, University of Birmingham, 181-188.
Ellis, W., Johnson, E., Lodi, E., & Schwalbe, D. (1992). Maple V Flight Manual. Pacific Grove, Calif: Brooks/Cole Publishing Company.
Harel, G., & Kaput, J. (1991). The role of conceptual entities and their symbols in building advanced matheamtical concepts. In D.Tall (Ed.), Advanced mathematical thinking, (pp 81-94), Dordrecht: Kluwer Academic Publishers.
Harper, D., Wooff, C., & Hodgkinson, D. (1991). A guide to computer algebra systems. NY: Wiley.
Kaput, J. J. (1987). Multiple representations and reasoning with discrete intensive quantities in a computer-based environment. In Proceedings of the Eleventh International Conference of Psychology of Mathematics Education, Montreal, Vol 2, 289-295.
Lawson, M. J., & Chinnappan, M.(1994). Generative activity during geometry problem solving: Comparison of the performance of high-achieving and low-achieving students. Cognition and Instruction, 12 (1), 61-93.
Newell, A. (1990). Unified thoeries of cognition. Cambridge, Mass: Harvard Univeristy Press.
Prawat, R. (1989). Promoting access to knowledge, strategy and disposition in students. Review of Educational Research, 59, 1-42.
Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12, 257-286.
Sweller, J. (1989). Cognitive technology: Some procedures for facilitating learning, problem solving in mathematics and science. Journal of Educational Psychology, 81, 457-466.
|Please cite as: Chinnappan, M. and White, B. (1998). Development of conceptual entities within a learning environment supported by a computer algebra system. In Black, B. and Stanley, N. (Eds), Teaching and Learning in Changing Times, 66-71. Proceedings of the 7th Annual Teaching Learning Forum, The University of Western Australia, February 1998. Perth: UWA. http://lsn.curtin.edu.au/tlf/tlf1998/chinnappan.html|