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AN INTUITIVE APPROACH TO CALCULUS

Mary Barnes (Adapted from an article published in Reflections Selecting this link will take you to an external site. , November, 1995)

Student difficulties in learning calculus.

There has long been concern about calculus in many parts of the world. Student difficulties and misunderstandings have been discussed and documented by numerous researchers. In 1992, at the 7th International Congress on Mathematical Education in Québec, a working group focused on students' difficulties in calculus. Problems in the teaching and learning of calculus have also been discussed in a series of publications by the Mathematical Association of America.

The traditional introduction to calculus depends on students' understanding the idea of a limit. Yet this concept is inherently difficult and causes problems, no matter how it is taught, partly because many students' intuitive ideas are in conflict with the formal definition. Tall & Schwarzenberger (1978), Tall & Vinner (1981), Davis & Vinner (1986) and Cornu (1991) have described many of these problems. They include conceptual problems related to infinite processes, and logical and manipulative difficulties in dealing with a complex definition containing multiple quantifiers.

The definition of the tangent to a curve is another source of difficulty. Vinner (1984) found that students' intuitive ideas conflicted with the formal definition. Many found it particularly hard to visualize a tangent as the limiting case of a secant (Orton, 1983; Ryan, 1992). Since the traditional approach to calculus depends on this, it is clear why so many conceptual misunderstandings arise.

Other problems identified include an inadequate understanding of the concepts of function and variable (Arnold, 1992; Barnes, 1988; White & Mitchelmore, 1992), a lack of awareness that a derivative is a rate of change (White, 1990) and an inability to deal with applications of calculus (Baldino, 1992; White & Mitchelmore, 1992). Many students react to this confusion by ignoring the conceptual aspects of the subject and relying on memorizing rules and procedures. Mundy (1984, p.171) noticed `a disturbing tendency of calculus students to operate at a rote level of procedures and symbol manipulation, not supported by an understanding of the concepts involved'.

Calculus reform

There have recently been moves in many parts of the world to reform the teaching of calculus. In Britain, David Tall pioneered an innovative approach using computer graphics (Tall, Blokland & Kok, 1990). His ideas have been widely adopted and incorporated in a number of recent curriculum projects. In the United States, extensive funding was made available during the late 1980s for `calculus reform', and a great many projects were set up. These mostly dealt with college level calculus, but some also had links with secondary schools.

In Australia, in 1988, the Department of Employment Education and Training funded the Introductory Calculus Project, as part of the Education of Girls in Mathematics and Science Program. This project aimed to encourage the interest and participation of all students in calculus, with a particular focus on girls (Barnes 1991, 1994, in press). The aim was to make calculus accessible, meaningful and enjoyable for girls as well as boys. New ideas were developed out of a variety of contexts especially those familiar to the students concerned, using investigative activities, collaborative work, discussion and reflection. The approach draws on David Tall's ideas, and assumes that students have access to a computer or calculator graphing utility. Other Australian initiatives include the use of graphics calculators in calculus courses at tertiary level (e.g. Boers & Jones, 1992; Britton, 1994).

A variety of approaches have been adopted to improve students' understanding of calculus (Tall, 1992). Investigative projects, or discussion and interaction in small groups, aim to engage students' interest and involve them in active, meaningful learning, but a key aspect of nearly all the reform projects has been the use of graphics calculators, or computers with graphical software, to help students develop a better intuitive understanding. These technological tools enable students to experiment — to look at many examples, use these to discover patterns and make generalizations, and then test their conjectures by drawing further graphs. Among other things, they can:

What if you have few resources?

In many schools, mathematics classes do not have easy or frequent access to computers. Even when a school is well supplied with computers, they are often fully in use for computer studies, and it can be difficult for mathematics classes to arrange opportunity to use them.

What can teachers do in these circumstances? I would like to suggest that there is no need to be restricted to traditional ways of teaching calculus. I do not want to play down the value of good graphics packages, but the new approach can be adapted for use in low-technology classrooms in a variety of ways. It is possible to give students an intuitive feel for the subject without making a great deal of use of high technology. There are, however, a few key points at which graphics technology really helps. At these points in the teaching sequence it is really worth while `moving mountains' to gain access to a computer lab for a lesson or two, or arranging a maths excursion to a nearby university or TAFE college which has appropriate facilities.

The approach I suggest begins with students conducting experiments to collect their own data. They can analyse and graph the data on a computer or graphics calculator, if one is available. Alternatively they can do the task collaboratively, working on paper. This provides opportunities for them to discuss what they are doing, and helps make it meaningful to one another.

Teachers need to plan experiences for students that will help them to make connections between numerical, verbal, symbolic and visual (graphical) representations of concepts such as the derivative of a function. Graphics calculators or computers can be useful for this, but discussions, games, and other collaborative activities are also helpful. The aim is to develop an intuitive feel for the important concepts of calculus. Note that this approach means that formal proofs based on limits are left until much later in the teaching program.

Representing Motion Graphically

The study of motion is a natural starting point for learning calculus because it deals with experiences common to all human beings. Isaac Newton developed calculus as a tool to help solve problems of the motion of astronomical bodies. In today's world, however, some young people may be more familiar with speed and its measurement than others; for example, those who are learning to drive, or those who do a lot of athletics training and frequently monitor their times over different distances. For reasons of equity, we need to ensure that all students have such experiences, reflect on them, and try to describe them and represent them mathematically.

Every child has experience of moving around in the world, walking, running, jumping, riding a bicycle or playing on a swing. We can build on this personal experience by getting students to graph the motion of their own bodies. A measuring tape and a simple timing device are all that is needed to investigate the graphical representation of motion.

The tape, about 50 metres long, is stretched out along the ground in a straight line. One student moves along it, according to a secret set of instructions given by the teacher. Another student is the timekeeper, telling the mover when to start, and then calling out the time every two seconds. Every other member of the class has a card labelled with a time in seconds, on which to record the position of the person moving at the time on the card. You need to have a rehearsal first, so that everyone knows roughly where to stand. Record the data on the second attempt. The procedure can be repeated with several other movers with different instructions. It is best to start with something very simple, like `Walk at a steady pace along the tape'. After that the instructions can become more complex: the mover may be told to walk, jog, or run at different times, to stand still for a while, or to turn round and walk back towards the starting point — there can be as much variety as you like. The final mover may be given an instruction like `Begin by walking very slowly, and then gradually increase your speed until you are running as fast as you can go'.

When the data have been collected, the students return to the classroom and record the data in a table with a row for each mover, and a column for each time. Each student will have noted the position of every mover marked on his or her card, so will be able to fill in one column. When the table is complete, reading across a row gives the position of one particular mover at different times. Graphing this information can be a collaborative effort, with each group of students graphing the motion of a different person. If you have the technology available, you could use a spreadsheet for this, but sharing out the work gets it done by hand quite quickly.

The graphs then form the basis for a discussion. How can you tell from a graph whether the person was moving slowly or fast? How can you tell when they were standing still? How can you tell whether they were moving away from the beginning of the tape or towards it? Can you reconstruct the secret instructions given to each mover?Following this discussion, students can be given other motion graphs and asked to describe the motion. Encourage them to use language such as `away from the starting point', `towards the starting point', `stationary', `faster', `slower', `speeding up' and `slowing down', and to relate these ideas to the gradient of the graph they are talking about. At this stage, too, the concepts of displacement and velocity can be introduced.

Finally, give students descriptions of situations involving motion and ask them, in each case, to graph the distance from the starting point as a function of time. Here are some examples: `I left home and walked to the bus stop. When I was half-way there, I saw the bus coming and began to run. I ran as fast as I could, but I just missed the bus, so I waited at the stop for the next one.' `The cyclist went very slowly up a steep hill, rested for a while at the top, and then went fast down the other side.' In the second of these, the cues `uphill' and `downhill' are traps for the unwary. It is easy to fall into the error of drawing a picture of the terrain (see Swan, 1988) instead of a graph of displacement against time!

For a variation of this activity, use sets of matched cards: one set with the story written in words, another with a displacement–time graph of the motion. Mix up several such pairs and give the whole collection to a group of students to sort. Encourage members of the group to justify to one another why they match the cards in a particular way, and to point out the salient features of the graphs.

In the activities described above, many different aspects of the person are involved in the learning: watching their classmates move — and moving themselves — uses the visual and kinaesthetic senses; talking and writing about what they have been doing uses verbal and logical faculties as well as the imagination. Such activities provide students with concrete referents for thinking and talking about motion. Later on, they will be used to introduce the idea of a rate of change and to motivate the subsequent development of the concept of the derivative.

Sketching gradient functions

The next step is to go back to some of the simpler displacement–time graphs, which consisted of a number of straight-line segments. Ask students, again working in groups, to draw graphs of these situations, showing the velocity as a function of time. Then ask them to do the same thing for the person who began by walking very slowly, and gradually increased speed. This raises the question of what we mean by velocity when the displacement–time graph is a curve. Until now, students have been able to estimate the velocity of a moving object as the gradient of a straight-line section of the displacement–time graph. You will need to spend a little time discussing ways of estimating the velocity in this new situation.

From the table of data collected by the class, students can work out the average velocity of the mover between each pair of consecutive observations. Graphing this gives a good approximation to the velocity–time graph required, as shown in Figure 1. Students quickly appreciate that recording the position more frequently would allow them to make more accurate estimates of the velocity.

These ideas lead to a working definition of instantaneous velocity as the average velocity over a very short interval around the time in question. At this point the class can be told that what they have been doing is actually calculus, and a little of its language can be introduced. There is no need to use the term derivative just yet. It is enough to talk about the rate of change of a function and to point out that velocity is the rate of change of displacement, or position, with respect to time. As a result of their work on motion graphs, students more easily grasp the idea that the rate of change is measured by the gradient of the graph of the function. They should be able to explain that if the graph is a curve, the rate of change can be approximated by taking two points close together on the curve, and finding the gradient of the line joining them. The velocity–time graphs they have been drawing represent the rate-of-change or gradient function of the position function. At this stage a variety of other examples of rates of change can be discussed. For example, students could draw graphs to represent the amount of water stored in a dam as a function of time, and think about how they would interpret the rate of change of this function. Or they could find out how temperature and pressure change as a function of the depth under water, or height above the earth's surface, and again discuss the interpretation of the rate of change.

Once the idea of the gradient function has been introduced, students can learn to draw rough sketches of the gradient function of a given abstract function. There is no need to have an algebraic formula for the function — a graph is sufficient. Students can make rough estimates of the gradient at a number of points along the curve, and use this to sketch the gradient function. They should be encouraged to focus on where the gradient is positive, zero, or negative, and where it is increasing or decreasing. It helps if they work in pairs, and begin by describing aloud to their partners the shape of the original function, noting where it slopes upward and where it slopes downward, where it is getting steeper, and so on. The partner can then translate these properties into properties of the gradient function and use them to draw its graph. This is a difficult exercise because it is easy to confuse properties of the function with those of its derivative but it is worth spending a little time on it. Once students have grasped the ideas involved, they have a very powerful tool that will help them to make sense of other ideas they encounter in calculus.

At this stage, another card-matching activity can help students gain a better understanding of the relationship between a function and its gradient function. You need pairs of cards with the graph of a function and the graph of its gradient function or derivative. Give groups several such pairs mixed together and ask them to sort them into pairs again. Later you can add more cards with verbal descriptions of both the function and its derivative. This helps to familiarize students with language such as maximum, minimum, turning point, point of inflexion and so on.

Discovering differentiation rules

I suggest that students should not be taught rules for differentiation until they have developed a good understanding of what a derivative is, and a familiarity with the relationship between a function and its derivative. This may help to avoid what Ryan (1992, p. 486) has described as `the rush to the rule', where the meaning is ignored or forgotten and students operate on a purely mechanical level, pushing symbols around on paper.

When the time does come to develop differentiation rules, you can work out the gradient function by a collaborative activity, with students working in pairs. Each pair needs a copy of a large, carefully drawn graph (a computer-drawn graph is best). Assign one or two points on the curve to each pair, and ask them to estimate, as accurately as possible, the gradient of the curve at each point. They can do this by selecting another point very close to it on either side, drawing a line through these two points, and extending this line as far as they need to in order to find its gradient. Collect together the results for the whole class and graph the gradient function. It is also fairly easy to convince students of the rules for finding the derivatives by having them draw rough sketches of the gradient functions and then asking if they recognize the graph they have drawn.

At this stage, the use of graphics calculators or computers makes an enormous difference. If you cannot arrange access at your school to graphics calculators or computers running a suitable graphics package, now is the time to plan an excursion. You need software which will graph the derivative of any function you input, and allow students to use `guess, check and improve' techniques to find algebraic formulae for the derivatives. They can then investigate the derivatives of a variety of different functions. I have found that students quickly spot patterns and can easily work out rules for differentiating polynomials and sine and cosine functions. Without a computer or calculator, the process is too laborious.

There are also strategies for investigating other differentiation rules, like the product and chain rules, without computers. These are described in Unit 6 of Investigating Change (Barnes, 1991).

What is a tangent?

The traditional approach to calculus begins with limits and then introduces the idea of a derivative by the question `How can we find the gradient of the tangent to a curve?'. In the approach that I am suggesting, limits are left until the end of the course. The derivative, as we have seen, is introduced as the rate of change of a function. By working with many graphical representations, especially representations of motion, students come to see that the rate of change of a function is the gradient of its graph. At no stage are tangents essential to the development of the ideas.

For completeness, however, tangents need to be discussed. There is no need to do this until after students are familiar and comfortable with gradient functions. You can then introduce tangents by a variation of the `scientific debate' developed by researchers in Grenoble (Legrand, 1992). Draw the graph of a function on the board. The graph should be a curve, and, for the moment, have no awkward points such as cusps or discontinuities. Students will already have intuitive ideas about tangents, probably derived from work on circle geometry, so ask them to draw a tangent to this curve. Then pose the question `How would you explain to someone why this is a tangent?' and for clarification `How would you define the tangent to a curve?' Give the class some time to discuss this in small groups, before inviting suggestions. Ask several different groups to present their ideas, and then let the class discuss them and try to decide on a version that satisfies everyone. As teacher, your role is to ask for clarification or explanation of student statements, and to act as devil's advocate. You may need to steer the class away from definitions such as `A tangent is a line which meets a curve in only one point' by providing examples of graphs for which this would obviously not be satisfactory. It is useful to ask the class whether they think a curve can have a tangent at certain special points, such as a point of inflexion or a cusp. You may also want to ask questions such as `Do you think it should be possible to have more than one tangent to a curve at a point?' and `Do you think it should be possible for the tangent to cross the curve, or must it always lie on the same side of the curve?'. In this way you can ensure that they think about important issues and confront common misconceptions.

You may need to be prepared to accept the definition on which the students agree, even if it is not worded exactly as you might wish. Ideally they will give a definition along the lines of `A tangent to a curve is a line through a point on the curve with a gradient the same as the gradient of the curve at the point.' More thoughtful students may want to add the proviso that if the curve does not have a gradient at a particular point like a cusp, then it does not have a tangent there either. If the definition the class decides on conflicts significantly with the standard definition, you will need at some stage to point out what is the conventionally accepted definition.

Conclusion

The approach described ties the abstract ideas of calculus to the physical reality of students and their classmates walking and running. You can add other experiments to the one described: rolling a ball down an inclined plane and measuring the time to roll different distances; running water at a steady rate into bottles of different shapes and recording the water level at different times; or measuring the temperature of a cup of water at regular intervals as it cools. In each case, students can graph the data, find rates of change, and discuss the physical meaning of the rate of change.

I have tried to explain how to improve the introduction to differential calculus by developing the important ideas in an intuitive way. This needs to be done well before you teach differentiation rules or give a formal definition of the derivative based on limits. Some of these ideas can be communicated much more effectively by using the power of the computer or graphics calculator to process information rapidly and produce powerful visual images.

Nevertheless I hope that I have also shown that practical experiments, graph-drawing, discussion and card-matching games can be used to develop the idea of a derivative in a way that is meaningful and strongly intuitive. This will help to combat the meaningless symbol manipulation which is all that many students experience in traditional calculus courses. Although low-technology approaches do not have the same power as computer-based methods, they can still be very effective in developing an intuitive understanding of the important ideas of calculus.

References

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