Strategies for teaching confusing topics in Mathematics: A case study

Ibrahima Faye, Mohd Yunus Nayan Department of Fundamental and Applied Sciences

Universiti Teknologi PETRONAS 31750,Tronoh, Malaysia

ibrahima_faye@petronas.com.my, yunusna@petronas.com.my

Abstract—Some topics in Mathematics can be confusing for students. Given two very close topics in Mathematics, how should they be taught? Will it better to teach them back to back or will it be better to insert some other topics in between. This paper considers the use of Power Series and Fourier series in solving Differential Equations. Two groups of students were taught the two topics. The first group has the two topics taught back-to-back while the second had one topic (Laplace transform) inserted between the two. The two groups were then assessed with a set of questions combining Power Series and Fourier series. The results showed that the mean of the group that had Power and Fourier series back-to-back is higher compared to the mean for the group that had the insertion of Laplace transform between Power series and Fourier series. The difference is statistically significant as showed through a T-tests.

Keywords—Teaching Method; Mathematics Education;

I. INTRODUCTION

Probably one of the common trends in today’s Education is the measurement of outcomes. After setting the goals, different measurements tools may be set to match the outputs with the inputs. To get to the correct outputs, a few teaching methods have to be implemented. Finding the right method for achieving the goals is a very challenging problem. This is mainly because of the various types of learning styles and capabilities. Some students may be Visual, Auditory or Kinesthetic learners [1]. Many students most commonly consider Mathematics as a difficult topic. Different strategies for teaching Mathematics have been considered. Technology based methods are getting popular among the teachers [2]. A few works show improvement of students understanding when using Technology in Teaching Mathematics [3, 4, 5]. Despite the potential of new technologies to improve teaching outcomes, a few points have to be decided in advance. For example, what should be the right sequence for the different chapters and what should be the weights given to the different chapters?

Designing a proper syllabus is always a complex task. Several disciplines could be involved: Psychology, Sociology, Neuroscience, Computer Science, Philosophy, Teaching Practice at different levels [7, 8, 9]. A teacher may be able to understand the learning style of a single student by experience, after a certain time teaching him. Although it may be still challenging for a single student, it is certainly accessible to a

devoted and attentive teacher. For a batch of students, the challenge is very high. The variability of learning styles among students is probably the first point to consider. The learning style may also vary for a given student, depending on the subject, the topics and even on the lecturer or the teaching method. How each student will find a way to differentiate topics that are close each other?

Given a syllabus with some topics that are close to each other, how to arrange them for an optimal understanding. In the particular case of dealing with similar topics or topics that look alike in Mathematics, the challenge is even higher. Having two close or similar topics, taught back-to-back, will it increase the risk of confusing students, or will it help in clarifying the specificities of each topic? In the other hand, inserting one or more topics between two similar topics, will it help students to digest the first topic before starting the second topic? Will the answers to the previous questions be the same when considering different subgroups of students: High performer, Medium performer and Low performer students?

This paper considers the case of the application of Power Series and Fourier series in solving Differential Equations. Series in general are not among the favorites of many students. Having to deal with two different types of Series may bring confusion to students. The remaining of the paper is organized as it follows. Section II presents the method implemented in this case study. The results and discussions are shown in section III. The conclusion is presented in section IV.

II. METHOD

A. ParticipantsThe participants of this study are students of Universiti

Teknologi PETRONAS (UTP). UTP produces graduates in Civil, Chemical, Electrical and Electronics, Mechanical, and Petroleum Engineering. In their Foundation studies, the students take Calculus 1 and 2. In Calculus 2, the students are taught the basics of Differentiation and Integration. Also, they revisit Sequences and learn Power series. In their first year of Engineering program, the students take Ordinary Differential Equations (ODE). In ODE, they learn mainly how to solve first and second order Ordinary Differential Equations using different tools. They study among others, how to solve ODEs

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using Power series, Laplace transform and how to construct Fourier series of periodic functions. The later will be used for solving Partial Differential Equations (PDE).

In the previous years, some confusions have been noticed among students when comes to dealing with infinite summations. In the particular case of using Power series or constructing Fourier series, the confusion remains since it is on top of dealing with Differential Equations. The participants are taken from different engineering programs. To the question “what is the difference between Power series and Fourier series?” given during a quiz, most students could not answer. A few students pointed the difference in terms of cosine and sine functions present in Fourier series and not in Power series.

B. Experiment Two groups were formed. The first group of students was

taught Power series solutions of ODEs right after higher order ODEs, followed immediately with the construction of Fourier series of periodic functions. Laplace transform and its applications to ODE were taught to the first group after Fourier series. The second group went also through higher order ODE, followed with Power series solutions of ODE. But before studying Fourier series, the group was taught Laplace transform and its applications in ODE.

At the end of the semester, all students were assessed in a special test containing only Power series solutions of ODEs and the construction of Fourier series of periodic functions. The special test consisted in exactly two independent questions. One question involved solving a second order homogeneous linear differential equation using power series. The second question involved sketching the graph of a periodic function and finding its Fourier series. For a balanced comparison, the same number of students is extracted from each group. The distribution of students’ level of performance is also taken as the same in both groups. For each group the same number of Low, Medium and High performer students is considered. The classification of Low (L), Medium (M) and High (H) is based on the students’ results in an intermediate Test paper. The intermediate Test paper was common to all different programs and was about using other methods in solving differential equations, mainly:

- Solving first order ordinary differential equations using Separable, Linear, Exact and non-Exact, and substitution methods.

- Solving second order differential equations sing the methods of undetermined coefficients and variation of parameters.

The Test was marked over 40 marks. A student is considered as low (L), medium (M) or high (H) performer if his Test mark is respectively, less than 20 over 40, between 21 and 26, and above than 35.

III. RESULTS AND DISCUSSIONS

The first part of the analysis consists in looking into the overall performance of the participants from both groups for each topic. Table I shows a sample of marks. For each topic, the mark is over 10. With a T-test on comparison of means, the Null hypothesis is rejected at a confidence level of 99%. The result confirms the general trend that students are more confortable with Power series compared to Fourier series. When asked to choose between Power series and Fourier series for a special session of revision, the students choose almost unanimously Fourier series. The results may be explained by the fact the students studied Sequence and Series in their Foundation, unlike Fourier series which is totally new for them.

TABLE I.

Power vs Fourier series

Participants Category Power Series

Fourier Series

1 H 5 42 H 8 23 H 2 64 H 10 25 H 10 66 H 10 17 H 4 68 H 10 99 H 10 9

10 H 10 611 H 10 412 H 10 613 M 0 614 M 0 715 M 2 216 M 10 617 M 4 218 M 4 219 M 0 420 M 4 221 M 2 222 L 1 423 L 4 324 L 5 225 L 5 526 L 4 227 L 6 428 L 10 429 L 2 530 L 5 0

Mean 5.57 4.03

Considering only the marks for Power series, the two groups did not show any difference. A T-test on the means of the group of students that were taught back-to-back Power series then Fourier series and the group of students that were taught Laplace transform between Power series and Fourier series did not show any statistically significant difference. The test failed to reject the null hypothesis even at 90% significance level. The results go along with the conclusion of the previous section: Power series are not that challenging for students compared to Fourier series.

The difference of the means for the two groups is statistically significant when only the marks on Fourier series

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are considered. The null hypothesis is rejected at more than 95% confidence level since the obtained p-value is approximately . The group that has been taught the two series back-to-back showed a lower mean. This can be explained by the confusion made when the chapter on Fourier series is introduced right after Power series.

The significant difference between the means of the two groups is also observed when considering the overall marks from the special test on series. The marks by category of performance are shown in Table II. The same number of students in each category has been considered. Figure 1 summarizes the average mark par category and per group.

The marks for the test on series are consistent with the categories defined earlier. Indeed in each group the average mark is increasing when going from low to medium, then high performer students. The average mark for the group which had the two topics on series back-to-back is higher than the average for the group which had the Laplace topic inserted between Power and Fourier series. That is valid for the three categories of low, medium and high performer students. The gaps between the averages between the two groups are getting smaller when considering the categories from low to medium then high. That suggests that the low performer students are more affected by the insertion of Laplace transform between Power and Fourier series, followed by medium and then high performer students.

As in the previous paragraphs, T-tests are realized to compare the means from the two groups when considering the levels of the students in terms of low, medium and high performers. In all cases, the results are the same: the null hypothesis on the equality of the means is rejected at more than 95% confidence level. There is difference and it is statistically significant. There are slight differences in the obtained p-values which depend on the category considered. That reflects the decrease of significance from low to high performer students. The p-values are however all very small since they are at the order of .

TABLE II.

Back-to-back versus Insertion Category Back-to-Back Insertion

H 4.5 7H 5 7H 2 9.5 H 8 10 H 6 7H 8 7.5 H 2.5 8H 8 8.5 H 9.5 9.5 H 9.5 10 H 8 10 H 7 10 M 3 9.5 M 3.5 7M 2 7.5 M 8 7.5 M 3 7.5 M 3 7.5 M 2 8M 3 8.5 M 2 9L 2.5 5.5 L 3.5 6L 3.5 6.5 L 5 7.5 L 3 8L 5 8.5 L 7 9L 4 9L 2 10

Mean 4.8 8.2

The results show that there is a significant difference when teaching Power series immediately followed by Fourier series and when inserting Laplace transform between Power and Fourier series. It goes along with our hypothesis that as for spatial proximity, the temporal proximity will help to differentiate similar (or confusing) topics. Putting two objects that look alike in spatial proximity (i.e. side-by-side) will certainly help in differentiating them. In the same way, we hypothesize that having two topics that look alike in temporal proximity (e.g. taught back-to-back) helps in differentiating them.

The significant difference may however be due to the effect of the followings:

- The variability in lecturers: in the study, the two groups were taught by two different lecturers, which may affect the results.

- The variability in marking styles: the special common Test on series was also marked separately by two different lecturers. The variability in marking style may also affect the results.

0

2

4

6

8

10

L M H

Back-To-Back

Insertion

FIGURE 1

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IV. CONCLUSION

This paper presents a case study on how to introduce the chapters Power series and Fourier series in the course of Differential Equations. Based on some confusions noticed in the previous years, two possibilities were considered in this study: introducing Fourier series right after Power series or inserting Laplace transform between the chapters of Power and Fourier series. The results suggested that the first option (back-to-back), creates less confusion among students. The difference between the two options was statistically significant. Although, there may be other factors (variability of lecturers, students, etc.) explaining the difference, the work suggest such strategies may be useful for an efficient delivery of the topics. A few other topics that are more or less confusing may be considered in future work. Having the same lecturer teaching different groups with different strategies will certainly give better results since it will at least suppress the effect of the difference in teaching and marking styles.

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[5] Makar, K., & Confrey, J. (2006). Dynamic statistical software: How are learners using it to conduct data- based investigations? In C. Hoyles, J. Lagrange, L. H. Son, & N. Sinclair (Eds.), Proceedings of the 17th Study Conference of the International Commission on Mathematical Instruction.

[6] Y. Yorozu, M. Hirano, K. Oka, and Y. Tagawa, “Electron spectroscopy studies on magneto-optical media and plastic substrate interface,” IEEE Transl. J. Magn. Japan, vol. 2, pp. 740-741, August 1987 [Digests 9th Annual Conf. Magnetics Japan, p. 301, 1982].

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