Ideas on a Workbook for Calculus

Ideas on a Workbook for Calculus

Ansie Meiring

Department of Mathematics and Applied Mathematics

University of Pretoria

South Africa

Abstract

The fresh approach of Reform Calculus has inspired many lecturers of Calculus all over the world. It has however, as any new system does, brought complications of its own. It requires more reading and more insight from the student than traditional Calculus. This is exactly why the average student experiences difficulties with the new approach. More guidance should be given to assist the student in improving his reading skills in Mathematics and to cultivate the required insight. A workbook is proposed to assist the student in extracting the essential concepts.

1. Introduction

Our method at the University of Pretoria of teaching Calculus to first year university students is probably no different to the way in which it is being taught in many other places all over the world.

The students are presented with a lecture, often excellent, on a certain topic in Calculus and are then expected to go home and read the particular section in the textbook to reinforce what they had heard in class and to gain more insight. In addition to this they are then supposed to attempt the allocated problems on the day’s work. They follow this pattern for the whole week. The problems for this week are then discussed in the tutorial class of the following week where the students have the opportunity to resolve what minor difficulties they may still have with the previous week’s work.

In most institutions however, the reality is far different from the ideal situation described above, especially in South Africa.

Students do get presented with excellent lectures. The groups are big, however, ranging from 100 to 160 and it is a basic principle that there is exponential decay in understanding as the size of the group increases. General and not individual needs are catered for. Adding to this are the vastly different levels of preparedness on entering the first year. Due to educational disparities of the past there is no uniform standard for the final school exam and furthermore there are a number of students that took Additional Maths at school. So on the one hand we have the under prepared students and on the other hand we have a few over prepared students. Although we have tried to divide students into groups according to ability, this is neither politically acceptable nor can it be done in a way where no party feels that it is losing out on something. In short, our lectures are not as effective as we would like to believe them to be.

The student now goes home after a tough day and goes into crisis management mode. What might get him in trouble if it is not done by the following day gets done first. In the unlikely event of having time to spare he might decide to pick up his Maths textbook, but Reform Calculus requires a of lot of reading. He finds the reading too much and too difficult, mostly because he has not been trained to read mathematics and therefore does not do it in a properly directed way. In our case, for the majority of people, it is written in their second language. They lose concentration and fall asleep or become distracted.

Come the following week, Maths becomes a priority on the evening before the tutorial session when the possibility of getting into trouble the following day becomes real. He does not quite remember the study material of the previous week but knows he has got to solve several problems. He reads through a problem and pages through the textbook desperately trying to find something, a formula or an example, that will help him solve the problem. There definitely is no time to study the whole topic now. He is expected to apply concepts, but has never really studied the concepts themselves.

Although the above describes the behaviour of an average type of student, there is the danger of generalising, especially when dealing with large groups. For a good student any system would probably work. On the other side of the scale we have the under prepared students who are far more dependent on the system and the help that it could offer. In our case, we find that most of these students are extremely hard-working. They simply are not capable of fully grasping the concepts in class and cannot resolve their difficulties by reading the textbook. They do, however, really want to learn and waste an awful lot of time in hours of aimless study.

2. The process of studying - ILUO

An idea, which has its origin in Japanese industry, is that the mastering of a basic skill follows a so called ILU-pattern.

Being able to:

• perform a basic skill under instruction earns the worker an I rating.
• perform the skill on his own, to a specified quality level, earns the worker an L rating.
• instruct others on the skill earns the worker a U rating.

Two modifications have been added to this. An O rating can be earned by the worker if he is able to improve on the specified skill. The Japanese have also subdivided the original I rating into two levels, namely that of a i-ratingfollowed by a I-rating. They feel that the first process of learning a new skill happens in two phases. The first phase is acquiring basic knowledge necessary to perform the skill. This will earn the worker a i rating. The second phase is when he assimilates the knowledge to make it his own, that is when he grasps the full implication of the skill. This earns him a I.

The concepts can be applied to the process of mastering Calculus. The learning process is probably somewhat different because is it on a higher cognitive level and there is hardly any physical skill to perform. By allocating slightly different meanings to the various legs we may then apply the ILUO-principle as follows:

The lecture is the student’s introduction to a new topic. A few basic concepts are introduced. If the student has followed what was said in class it would earn him an I rating. It could be said that this is the INITIAL stage of the learning process. The student has by no means grasped the full implications of what he has learned. He hopefully has a basic understanding of the new concepts but has not been confronted by the finer points and has not really been able to make this knowledge his own. It is an unfortunate truth that most lectures, however excellent, serve only as an introduction to a particular topic.

The student should now, ideally, move to the next leg in the process, the L phase. This starts when the student reviews what he has learned in class. He picks up his textbook and reads about the new concepts. If he does this as thoroughly as he should then he is bound to come to a better understanding of the concepts and gain insight. He ideally should be confronted with “What if?” type of questions in his mind which could be resolved by rereading a particular paragraph in the textbook. He should also make notes of the most important concepts and spend some time on the examples. Unfortunately this important phase is also generally most neglected. A good analogy would be to that of baking a loaf of bread. The initialphase, or I phase, is the mixing of ingredients. The lecturer’s knowledge is mixed into the mind of the student. The second phase, the L phase, then is the phase of LEAVENING. The bread is left in a warm place and although nothing is added, the bread rises. This is essential for the next phase. In a similar way the student’s knowledge should be kept active and although no new knowledge is added, his insight should increase by studying the concepts. Because most students find it so difficult to simply read the textbook, they should be guided through this phase. Support material and a good tutor system is advisable here. This phase should add to the student’s confidence level when confronted with problems during the next phase.

The third leg of the process should be entered when the student’s knowledge is mature enough to answer penetrating questions on the concepts in the paragraph. Most students try to solve problems before they’ve mastered the concepts properly, and normally try and use the problems as a departure point in finding the essentials from a paragraph. In other words, if no problem on a particular concept is given, the student could remain unaware of its existence and he only knows enough about any concept to solve a particular problem. The problem solving phase is extremely important because it normally leads to fruitful discussions and therefore deeper understanding. The success of these discussions is largely dependent on the maturity of the students knowledge. This is the U phase, or UTILITY phase. The student should be ready to use his knowledge. He should find pleasure in being confronted by questions and not feel threatened by it as normally happens. Drawing further on the bread baking analogy, this is the baking stage. Heat is added to turn the well-risen dough into a usable loaf of bread. The question is whether the student can stand the heat of problem solving? Unfortunately a lot of students have not gone through the LEAVENING stage, and the result is hardly usable.

Having gone through the three phases mentioned above, the student should be master of the particular topic. This is the level to which we would be happy to elevate our students to. There is a fourth stage, the O phase. This is the phase where the student knows the topic so well that he can actually teach it himself. He could also appreciate different approaches to the same topic. This is the OVERVIEW phase. The student is so familiar with the concepts that he can view it objectively. Not all students reach this level and it is not expected of most students either. In terms of our analogy, it is the phase where the loaf of bread can be shared around and consumed. This is of course the level of understanding that we expect from our tutors. Tutors should assist in helping the student master concepts rather than to solve problems.

How can the system be improved? The one phase that seems to be totally neglected in the present system is the L phase. The student should not be confronted with problems directly after the lecture and before he has gained more insight into the concepts introduced during the lecture. We recommend that a support book is provided with the textbook that guides the student through the textbook. It is somewhat ironical that Instructors Manuals are often readily available to guide the lecturer but the students are not supported enough.

Before discussing the format of such a support book, let us briefly digress and mention two important aspects.

3. Other aspects of interest

3.1 Information Mapping

In South Africa the concept of INFORMATION MAPPING is being introduced in the business and industrial sector. Information Mapping was triggered by the fact that written information is mostly presented in paragraph style with facts obscured and therefore overlooked. When looking for a particular snippet of information, it is often necessary to read attentively through a whole page of script. Information Mapping shows information in bullet lists, tables or often graphical, rather than in paragraph style. In a developing country such as South Africa where reading often poses a problem, there has been a fair amount of success by using this approach.

Now lets face it, Reform type Calculus is most definitely not written in Information Mapping style. Far from it, and surely no-one advocates that it should be done. But that does not mean that the student should not be guided to establish what the essential concepts are when reading, and to write these down as he goes along. In other words, he could do some Information Mapping on a topic himself, in addition to the textbook.

Jumping to pre-school education: One of the principles along which the Montessori schools run is that children learn through their hands. When learning the alphabet they physically touch letters made from sandpaper and arrange them to form words. I believe that this applies to our students as well. Reading a definition is one thing, writing it down is quite another. Read through an example and the solution looks so simple. Change the information slightly, try to do it yourself and it is a different story altogether. You cannot study Mathematics without using pen and paper.

3.2 What do the students say?

In a recent end-of-semester questionnaire the following interesting information was obtained:

How much did the students follow in class?

One third of the students claimed to have followed less than 60% of what was said in class. For the rest, even if they did follow in class, it did not mean that they fully comprehended the concepts.

Did the students read the textbook?

A shocking 49% claimed not to have read the textbook at all, only depending on notes which they scribbled down in class. Only 8% of students spent more than 3 hours per week on reading the textbook and 43% spent less than 3 hours per week on reading the textbook.

Did they find reading the textbook easy?

Again 49% confessed to not finding it easy to read, therefore probably deciding not to read it at all.

What did they mainly study from?

Here 54% claimed to have used mainly their notes as a basis for studying, whereas 34% studied from both the textbook and their notes and only 12% studied mainly fusing the textbook.

How much time did they spend studying in between the lecture and the tutorial session?

A total of 69% of the students started preparing for the tutorial session on the previous day only, emphasising the point of crisis management.

4. What is the solution?

It has become clear that the typical student fails to gain the level of insight that we require of him. Performance on insight questions in test papers prove this. Should we like to continue teaching Reform type Calculus, more support will have to be provided for the students. A workbook is proposed to assist the student in identifying and mastering the concepts in the textbook, and to encourage reading the textbook. A workbook should accompany the textbook and should guide the student through the textbook and better prepare him for problem solving.

Fill-in type of statements form the basis of the book. Students don’t feel quite as threatened when confronted with an incomplete sentence as they do when confronted by a direct question. It also helps the student to read and improves his ability to verbalise concepts.

The basic ingredients for such a book are:

1. Key points. The student is assisted in listing the key points in every paragraph in bullet point style. This guides him to an overall picture of the study matter. Note again, that these are not provided, the student has to find these himself.

2. From the book. Essential detail from the book is extracted by the student himself in a guided way. It is sometimes even necessary to guide the student to a particular sentence in the book and help him extract the concept. In this way the student will find important detail which he might otherwise have overlooked.

3. Checking the concepts. The student is assisted in checking whether he actually grasps the key points in every paragraph. Having extracted the essential concepts from the book, he can now improve his insight.

4. Your contribution. The student gets the opportunity to provide his own examples to illustrate his understanding of the concepts.

5. Checking the samples. This is one of the most important, and often neglected issues. Working through a particular example, then changing some of the data and repeating the example is invaluable. It also provides an opportunity to discuss selected examples in review fashion - what the example tries to show and why it is important.

6. A Diagrammatic view of the chapter. As the student works through a chapter, he is guided to draw up a diagrammatic representation of the structure of the chapter. He should be able to visualise the progress in the chapter. This should prevent the student from getting lost in all the detail and not see the wood for the trees.

7. Problems. Finally, problems are selected for the student to do. Having worked through a particular paragraph he should be ready for, and enjoy, doing the problems.

5. In conclusion

The reform approach to Calculus has been welcomed widely because of its potential of giving the student more insight into the subject. Although our way of presenting Calculus has changed, we have not made it user-friendly enough for the students in the average and below average categories. The proposed workbook is be the next step in the development of our new approach to Calculus. It will improve the student’s Mathematical reading skills and help him lift out essentials and assist him to reach a level of full understanding.