ENHANCING THE TEACHING OF PROBABILITY AND PROBABILITY DISTRIBUTIONS IN UPPER SECONDARY

USING COMPUTER GRAPHICS

Tema: Pengintegrasian ICT dalam Pengajaran dan Pembelajaran (P&P)

Rasimah Aripin Pusat Pengajian Statistik

Fakulti Teknologi Maklumat dan Sains Kuantitatif Universiti Teknologi MARA

Shah Alam, Selangor, Malaysia drasimah@salam.uitm.edu.my

Aidawati Zainan Abidin Pusat Pengajian Statistik

Fakulti Teknologi Maklumat dan Sains Kuantitatif Universiti Teknologi MARA

Shah Alam, Selangor, Malaysia aida@tmsk.uitm.edu.my

Abstract

Probability theory has been applied in many disciplines such as business, life sciences, humanities, education, agriculture, and the social sciences. Currently, basic probability concepts are taught at the secondary level in the Additional Mathematics subject for the Sijil Pelajaran Malaysia (SPM) examination. However, the focus is more on the computational aspect rather than on building the underlying principles of probability and probability distributions. Furthermore, since examination questions on this topic are not compulsory, very often they are neglected by both students and teachers. As a result, students are not able to build their reasoning and judgment skills acquired through the principles of probability at an early age. In order to enhance the interest in learning probability, new approach need to be introduced. Many studies have found that the use of simulation, animation and graphics visualization are effective for teaching difficult and abstract concepts commonly found in mathematics and statistics, such as probability. This paper presents the use of computer graphics to enhance the teaching of basic probability and probability distribution concepts. These graphics can be developed from widely available open source software. The teaching materials proposed can be extended to enhance the teaching of other abstract mathematical concepts such as functions, differentiation and integration. Keywords Probability, probability distributions, computer graphics 1. INTRODUCTION

Statistics and probability provides the faculty to develop reasoning and judgment skills. It is now widely recognized that the foundations for statistical reasoning should be built in the earliest years of schooling rather than being reserved for high school or university studies (NCTM, 2000). Statistics has already been established as a vital part of the secondary school mathematics curriculum in many

countries (Chacko, 2002; Accrombessy, 2006; Parsian & Rejali, 2008). However, inadequate attention has been paid to teaching practices. There is substantial evidence of poor understanding and insufficient preparation to teach statistical concepts among mathematics teachers assigned to teach statistics and probability (Serradó and Azcárate, 2006). Most are likely to have weak understanding of the statistical concepts they are expected to teach and as a result, tend to focus on the computational procedure rather than on conceptual understanding. Problem arises when secondary school leavers proceed to pursue tertiary education where most academic programmes require at least one semester of statistics and probability course. They also need the analytical competency when conducting research projects required at the various stages of their studies. Results from a study by Albert (2003) indicate that students were generally confused about the classical, frequency, and subjective notions of probabilities. Although the students were able to solve routine probability problems, they were not competent in other situations. Many methods have been proposed. to enhance the effectiveness in the teaching of difficult and abstract concepts commonly found in mathematics and statistics Among them are using simulation and visualisation with computer graphics. This paper proposes the use of computer graphics to enhance the teaching of basic probability and probability distribution topics at the secondary school level. The use of these materials can be extended to teach other abstract mathematical concepts such as functions, differentiation and integration which have vast applications in statistics and probability. Many of these graphics courseware are readily available online or can be developed from widely available open-source software. 2. STATISTICS AND PROBABILITY AT THE SPM LEVEL

Statistics is one of the five components of the core package in the Sijil Pelajaran Malaysia (SPM) Additional Mathematics syllabus (Ministry of Education, 2006). Probability and probability distributions are two of the four topics under this component. In the SPM examination, Paper 1 consists of between 2-4 questions under this component, usually on combination, permutation and probability of events, worth a total of between 10-16 marks. Section A of Paper 2 requires students to answer only routine questions on frequency distribution and descriptive statistics such as mean and standard deviation, while Section B has one question on Binomial and Normal probabilities. Table 1 summarises the contents of the SPM Additional Mathematics Paper 2, 2007. The marks allocated to questions on statistics and probabilities are very minimal compared with the other components. For example, students may ignore the probability topics totally and this will not affect their chances of getting good grades. Initial investigations also reveal that topics such as probability and probability distributions, being among the last topics on the syllabus, are often received minimal attention or even totally left out. Further review of previous years’ examination papers reveals that questions on statistics and probability focus on routine tasks, which encourages rote learning. The questions are very similar from year to year which could be due to the limited content covered in the statistics component. As suggested by Curcio (1987), even at the secondary level, examination questions should go beyond the data by testing on the underlying concepts. Limited content is partly due to the lack of appropriate courseware to teach probability as it is not a simple concept to illustrate in a traditional classroom. As a result, it is not surprising if teachers and students did not see the importance of statistics at the secondary level when in reality, it is required in higher education to build the foundation for analytical competency in research undertakings.

Table 1: Components of the Additional Mathematics Paper 2, Sijil Pelajaran Malaysia 2007

Section Requirement Questions on Statistics and Probability Marks

A Answer all 6 questions Question 5: Complete a frequency distribution, compute the inter-quartile range

6

B Answer 4 out of 5 questions

Question 11: Binomial and normal probabilities

10

C Answer 2 out of 5 questions

None 0

3. USING GRAPHICS FOR TEACHING PROBABILITY AND PROBABILITY

DISTRIBUTIONS

Computer environments have the potential to extend children’s experiences and foster a rich exploratory approach to the study of early mathematical concepts (Drier, 2000; Schneider, 2007). There are quite a number of free coursewares for probability and probability distributions available for use. Among them are ESBPDF Analysis, Probability Explorer, Adaptive Statistics Tutor and e-Methodology. Numerous open source software can also be used to develop interactive teaching modules appropriate for the statistics component of the SPM Additional Mathematics syllabus. The graphical capabilities of such software enable complex concepts to be illustrated in classrooms, to enhance the understanding needed before performing manual calculations required for examinations. This section presents some illustrations of graphical capabilities and probability computations provided by ESBPDF Analysis and an open source software, R. ESBPDF ANALYSIS ESBPDF Analysis provides everything needed for using discrete (Binomial, Hypergeometric and Poisson) and continuous probability distributions (for example, Normal, Exponential and Exponential) in a single application. Other useful features are binomial coefficients, factorials, permutation; standard normal table and critical t values. For a small fee of less than RM100, ESBPDF Analysis is ideal for those who wish to understand probability and probability distributions better. Figure 1 shows the Analysis window for selecting the distribution of interest, setting up the parameters of the distribution and selecting the probability region to be computed. The resulting probability is provided on the same window. The corresponding region under the graph can be viewed in the Graph window (Figure 2). This interface allows a student to visualize the shapes of various distributions, in particular, binomial and normal distributions, with given sets of parameters. This is especially useful for exploring more examples on probability computations under various probability distributions, which may not be previously possible.

Figure 1: Setting up the parameters for Binomial(8,0.5) and Prob(3≤X≤6)

Figure 2: Graph for P( 3≤X≤6) when X is Binomial(8,0.5) An example of a probability computation for the normal distribution is shown in Figure 3 while the corresponding area under the curve is in Figure 4. The illustration is for the computation of the probability that a random normal variate from a N(130, 16) distribution is between 114 and 150. This problem is one of the questions on probability in Paper 2 of the SPM examination Additional Mathematics paper, 2007. Using this feature, a student will be able to visualise different areas under the curve corresponding to given events of interest.

Figure 3: Analysis window for computing probability under the normal distribution

While ESBPDF is sufficient for simple computations of probabilities, it is not sufficient for demonstrating underlying concepts in probability and probability distributions. For instance, it lacks the capability to compare the same distribution with different shapes or different distributions, such as the t-distribution and the normal distribution on the same graph. In addition, the features are quite rigid with no options for modifications or adding new features to the existing graphics.

Figure 4: Analysis window for the computation of normal probability

R for Probability and Probability Distributions R is an open source software for statistics that provides an environment for data analysis and graphics. With R codes, one can create sophisticated graphs, carry out statistical analyses, and run simulations. R is also a programming language with an extensive set of built-in functions that allows the user to extend the language and write their own code to build customized applications. Advanced

users can even incorporate functions written in other languages, such as C, C++, and Fortran. There are several reasons that make R an excellent choice of statistical software for an introductory statistical course to teach a range of topics from descriptive statistics to statistical modeling. First, R is free and available on the Web and thus can be used on home computers. Second, R is a powerful, widely-used software with new codes actively being shared among users on the web. Third, even though it is not the simplest statistical software, the basics are easy to learn. R provides excellent choice of graphics parameters, which includes 657 different shades and colors to choose from to make your graphics presentation appealing to the users, especially young learners of abstract concepts such as probability. Figure 5 demonstrates the relationships between parameters and shapes of normal distributions produced using R codes. These relationships are not easily demonstrated without the aid of such graphics. A sample of codes written in the interactive mode of the software is shown in Figure 6 while the resulting graphics are in Figure 7. Other than generating graphics for teaching probabilities under normal and binomial distributions, R also provides excellent graphics for demonstrating data presentations for introductory statistics course such as pie chart, box plot, stem-and leaf and line plots as shown in Figure 8.

Figure 5: Comparing the shapes of three normal distributions using R

Figure 6: R codes for selected probabilities under Normal(130,16)

Figure 7: Probabilities under Normal(130, 16) distribution

Figure 8: Probabilities under Binomial(8,0.5) distribution

Figure 9: Graphics for data presentation 4. CONCLUSIONS There are advantages when using graphics software such as R to teach abstract concepts including probability and probability distributions. Plotting and exploring various distributions with different combinations of parameters help to internalize the concepts of probability under the various distributions and parameters. Traditional classroom teaching of probability and probability distributions are greatly constrained to simple distributions and computational aspects only. In view of this, a project is being undertaken to develop the courseware to cater for the statistics component of the SPM Additional Mathematics paper. Other than graphics, animations and simulations are to be incorporated in the courseware which should be appealing even to pupils in primary schools to learn probability.

REFERENCES Accrombessy, F.D. (2006). An Evaluation Study of the Process of Reform of Statistics Teaching at the

Secondary Level in Benin: Assessment and Perspectives. Proceedings of ICOTS-7.

Albert, J.H. (2003). College Students' Conceptions of Probability. The American Statistician, Feb. 2003.

Chacko, I. (2002). An Analysis of Secondary School Examination Syllabus, Public Examination Test Items and Statistical Literacy Levels of Students. Proceedings of ICOTS- 6.

Christou, N. (2008). Enhancing the Teaching of Statistics: Portfolio Theory, an Application of Statistics in Finance. Journal of Statistics Education, www.amstat.org/publications/jse/v16n3/christou.html.

Curcio, F.R. (1987). Comprehension of mathematical relationships expressed in graphs. Journal for Research in Mathematics Education, 18, 382-393.

Drier, H. S. (2000). The Probability Explorer: A Research-based Microworld to Enhance Children's Intuitive Understandings of Chance and Data. Focus on Learning Problems in Mathematics 22(3-4), 165-178.

Mavrotheris, M.M. (2006). Improving the Teaching of Statistics in Early Grades Through Technology-

Enhanced Learning Environments. Proceedings of the 7th

International Conference on Teaching Statistics.

Ministry of Education Malaysia (2006). Intergrated Curriculum for Secondary Schools.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author

Parsian, A. and Rejali, A. (2008). A Report on Preparing Mathematics Teachers to Teach Statistics in High School. Proceedings of the IASE Round Table Conference

Schneider,K., (2007). Visualizing Statistical Inference Using SAS®. Proceedings of SAS Global Forum 2007

Serradó, A. and Azcárate, P. (2006) Analyzing Teacher Resistance to Teaching Probability in Compulsory Education. Proceedings of ICOTS -7.