Aiming for variable understanding: Paul White of the Australian Catholic University and Michael Mitchelmore of Macquarie University find that tertiary students' poor performance in calculus often reflects a lack of understanding of much simpler concepts.

White, Paul ; Mitchelmore, Michael

Some research into first year level tertiary students' understanding of calculus (White & Mitchelmore, 1992) revealed results as relevant to the junior years of high school as to the senior years. Basically, we found that many students were unable to use variables when they had to represent some actual, changing quantity and that their notions of functions were very hazy. As a result, even though they were all competent at factorizing, expanding, solving and manipulating symbols in general, most could not cope with simple applied calculus problems.

We want to highlight here some of the deficiencies that we found in students' concepts of a variable and to consider associated implications for the teaching of variables and functions in the secondary school.

The calculus research

The research focused on forty first-year, full-time students enrolled in mathematics at a New South Wales university. A prerequisite for the course was a satisfactory result in the final high school examination for a mathematics course which contained a large component of calculus. It should be noted that none of the students had finished in the top 10% in that examination; the students were in most cases 'average'. Students responded to four calculus problems set in context and to similar problems presented in symbolic form, and were also interviewed extensively.

For example, one test item asked for the instantaneous rate of change for a cube whose sides were shrinking uniformly at 2 em/sec when the volume of the cube was 64 [cm.sup.3]. The symbolic form of this item asked for dV/dt when V = 64, given that V = [x.sup.2] and dx/dt = -2. All symbolic items led to exactly the same manipulations as the corresponding contextual item, but did not require translating rate information into appropriate derivatives.

Another item asked for the area of the largest rectangle with its base on the x-axis and upper vertices on the curve y = 12 - [x.sup.2], the situation shown in the Figure 1. This item became in symbolic form 'Find A when dA/dx = 0, given that A = 2x (12 - [x.sup.2])'. Using the two forms allowed for separate analyses of students' concepts and their ability to perform routine procedures.

It really came as no surprise that many students could not symbolize information and that, even when they realised a derivative was required, they often chose the incorrect one. However, other errors indicated that the biggest obstacle to the successful application of calculus was an under-developed concept of a variable--a concept which is a focus of secondary mathematics. Let us look at some of these results in more detail.

The nature of a variable

How do students conceive of variables? Two examples illustrate typical misconceptions:

1. In the shrinking cube example, some students who realized dV/dt was required wrote "V = [x.sup.3], V = 3[x.sup.2]". No appreciation of the role of the independent variable was evident.

2. Many others correctly interpreted the item and wrote "dV/dt = dV/dx. dx/ dt = -6[x.sup.2] ", but could not finish, or gave two answers: "-6[x.sup.2] for V = [x.sup.3]" and "0 for V = 64". They apparently saw V = [x.sup.3] and V = 64 as totally separate cases. The idea of 64 as a specific value for a varying V was not understood at all. Comments like: "The V = [x.sup.3] and the V = 64 at the same time confused me. I didn't know which one to use", and "V = 64, how can I differentiate the V ?" were common. Such students also happily used constants and variables interchangeably and found no conflict in using a constant as an independent variable.

The manipulation focus

Closer analysis showed that not thinking about whether letters were representing changing or constant values was in fact just part of a deeper problem. The basic problem was that many students focussed on the visible symbols that were available and then searched among the procedures which were known to them for one which used these symbols. No thought was given to what the symbols meant, or to what relationships existed between variables. The error of giving dV/dt as 3[x.sup.2] because V = [x.sup.3] is a case in point. Many other examples of this 'manipulation focus' occurred in all items. The general attitude of manipulation focus students can be summed up by the one who said: "I might try to differentiate the V." Why? "Because I can see it."

The x, y syndrome

A special form of the manipulation focus occurred when procedures were chosen on the basis of the presence of the letters x and y. In the above example on the maximum area of the rectangle under the parabola, instead of using an expression for the area of the rectangle based on the x and y coordinates of a general point, many students inappropriately solved dy/dx = 0. When asked why, they claimed that "it was the rule". Their view of the rule for maximising using calculus was based totally on the symbol dy/dx rather than on the concept of the rate of change of a function. The following comment summarised the type of item with which most students felt comfortable: "It was easy because it was all laid out it front of you with a lot of information in xs".

Defining variables

There was a small group of students who were successful in solving the contextual items and who also made virtually no manipulation errors. Analysis showed that a key factor to their success was the ability to assess the information presented in an item and then to define appropriate variables. The act of defining indicated an overview of the situation and invariably led to linking the variables to form a function which was ready for the relevant calculus manipulation. These select few show the value of concentrating on meaning and not just manipulation.

Implications for school mathematics

If a general concept of a variable is to be developed, then the focus must be on what the letters mean rather than on how they can be manipulated. If manipulation of visible symbols is the only context in which variables are placed, then errors like those which have been described will occur with devastating frequency. Hence, in teaching about variables there needs to be an emphasis on such ideas as these:

* variables have a context apart from the symbolic;

* one variable often changes in relation to another;

* the difference between dependent (input) and independent (output) variables;

* the role of constant values in variable expressions;

* when substitution of values is appropriate;

* the idea of a constant function;

* situations which involve many variables, not always just x and y. Let us see how this might be done. Many articles have been written about introducing variables via geometric patterns. One example is a line of squares, where the number of matches (m) depends on the number of squares (n) (see Figure 2). The research described here certainly supports such an approach because, from the outset, variables have a concrete meaning and not just a symbolic one. Student actions on variables take into consideration how changing one variable (such as the number of squares) affects the other (the number of matches). Even abstract questions such as 'Is a + 1 = 2a?' can be approached in terms of variable concepts rather than in a purely symbolic context. Initially, students may simply try out different values of a to find when the statement is true; later, they may use manipulations to solve the equation. At all stages, however, the answer is "They are not usually the same, but they are when [alpha] = 1" rather than a straight "No".

Variables can only really take on a general meaning when they are closely linked to functions. Number patterns are essentially functions, but there are many other areas of school mathematics where we could take a functional approach. Most area and volume work (in fact, everything which involves formulae) deals with functions, but this opportunity is rarely exploited. For example, in finding the hypotenuse of a right angle triangle using c = [square root of (([a.sup.2] + [b.sup.2]))], we can discuss how a, b and c are variables, what they mean and which are the independent variables (in this case two of them) and which is the dependent variable. Approaching formulae this way can transform the idea of a function to a central theme of mathematics, a position it both deserves and needs.

Problem solving and the investigation of real life situations also play a part in taking a more functional view of mathematics. One especially valuable experience is the interpretation of graphical representations of life-like situations such as those presented in Swan (1989).

Our suggestions so far have had a qualitative touchless emphasis on manipulation, more emphasis on variables in a functional context. There needs to be a quantitative change too. By this we mean the inclusion of problems which involve more complex situations. When situations that involve more than two variables are investigated, students have to base their choice of variables and constants on what they mean, not on what they look like.

One important change is not to always have x as the independent and y as the dependent variable.

Conclusion

The argument has been made that to succeed in calculus in senior secondary school and eventually at a tertiary level, by the end of junior secondary school students need to have a well-developed concept of a variable, to be comfortable working with functions and to be able to use algebraic notation in simple problem solving. Unfortunately, 'word problems' are often put into the too hard basket in junior secondary school; this understandable reaction only makes the chance of developing the required understanding seem remote--even if variable concepts are developed in other ways, such as by using graphs. However, it may well be that we are asking too much from our students. Research by Kichemann (1981) suggests that very few students understand variables at the conceptual level required for learning calculus.

Some hard decisions need to taken about what mathematics is studied at school. We can see three possible alternatives:

1. Encourage more junior secondary students to study a mathematics course which includes far less algebra and functions, taking up these ideas in the senior secondary school and then only with the more able students;

2. Defer algebra and functions for all students until later in the junior secondary school, so that students have more time to develop the concepts; or

3. Expose larger numbers of junior secondary students to algebra and functions taught with an emphasis on concepts, so that students who proceed to the more abstract courses in the secondary school will have a firmer foundation.

Much further research and discussion is needed before we can say which of these might be best policy, that is, the policy which satisfies the needs of all our students.

Published in Vol. 49, No. 4, 1993, Paul Scott (Ed.)

References

Kichemann, D. E. (1981). Algebra. In K. M. Hart (Ed.), Children's understanding of mathematics (pp. 11-16, 102-119). London: John Murray.

Swan, M. (1989). The language of Junctions and graphs. Nottingham, UK: The Shell Centre for Mathematical Education.

White, P. & Mitchelmore, M. C. (1992). Abstract thinking in rates of change and derivative. In B. Southwell, B. Perry & K. Owens (Eds), Proceedings of the fifteenth annual conference of the mathematics education research group of Australasia, (pp. 574-581). Sydney: MERGA.

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Please note: Some tables or figures were omitted from this article.