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Article Excerpt
This paper discusses programs that clarify some statistical
ideas often discussed yet poorly understood by students. The programs adopt the approach of demonstrating what is happening, rather than using the computer to do the work for the students (and hide the understanding). The programs demonstrate normal probability plots, overfitting of models and generalized linear models. Although the implementation is in Matlab, any suitable language is appropriate.
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The use of technology in teaching mathematics and statistics has reached the point where it is hard to imagine teaching mathematics and statistics without using computers. The University of Southern Queensland (USQ) has chosen to standardise on using Matlab in its mathematics and statistics courses. First year students learn algebra and calculus by incorporating Matlab as a numerical and graphical tool; they learn the basic language and graphical commands. Matlab is then used in many subsequent courses, including most statistics courses.
Matlab is a computational environment for technical and numerical computing. It has a strong pedigree in robust, stable and efficient numerical algorithms, and has excellent graphical capabilities. The user-interface is typically the command-line, but Matlab has facilities for implementing a graphical user interface, as used for the programs discussed in this paper.
Cretchley, Harman, Ellerton and Fogarty (1999, 2000) analysed the influences on attitudes and learning, finding almost all students responded positively to Matlab for ease of computation and graphing. Using the software as a tool was found to strongly impact the learning strategies adopted and the students' confidence toward mathematics. In addition, "students with high computer-mathematics interaction feel that computers enhance mathematics learning by providing many examples, enable the user to focus on major ideas by reducing mechanical toil ... and find computers helpful in linking algebraic and geometric ideas" (Galbraith, Haines, and Pemberton, 1999, p. 220).
Matlab is used in numerous ways at USQ. Often it is used to actually do the work for the student, saving the student time in fiddly or prone-to-error calculations, or allowing the students to do more complex problems impossible to do "by hand." In higher-level courses, some Matlab programming is expected.
The programs discussed in this paper, however, use the computer to show what is happening, rather than to do a particular task. For example,
It is just about impossible to illustrate Newton's method adequately with freehand sketches using a piece of chalk. A computer printout on a transparency, or even an animation using a PC, can be a great help. Likewise Simpson's rule, the Runge-Kutta method, and other numerical techniques can be best illustrated with a computer and an attractive display. (Krantz, 1998, p. 81)
Dunn and Harman (2000) argue that Matlab can be incorporated into learning in three main ways: Firstly, as a numerical and graphical tool. In this role, the computer enables faster illustration of the applications and better visualisation of complex ideas (such as three dimensional graphical representations). Typical in this approach is to use command line instructions in Matlab to develop the ideas while no programming skills are required; see, for example, Cretchley et al. (1999, 2000).
Secondly, Matlab can be incorporated into the material using appropriate programs then used by the student to facilitate the learning of ideas and concepts. In this approach, the programs are provided to the student for their use or possible modification. Penny and Lindfield (1995) and Borse (1997), for example, have adopted this practice; Colgan (2000) discusses a course built around this approach. A basic knowledge of programming is helpful or necessary in this approach.
Thirdly, Matlab software can be used to show the procedure or concept in a more sophisticated way. This is done by using larger, stand-alone and robust programs with well-designed graphical user interfaces. No programming skills are required in this approach. Of course, these three approaches can effectively complement one another.
In this paper, the emphasis is given to the third approach. A number of statistical and mathematical tools are presented in Matlab, though the language itself is not important. Examples from other areas of statistics can be found in Dunn (1999). Two tools for use in regression, a tool for use in teaching generalized linear models, and details on how to obtain and use the programs are presented in these findings.
TOOLS FOR USE IN REGRESSION
Regression is an important tool for any student of statistics to study and, consequently, it is an essential component of study in most University statistics programs. Two issues often discussed in relation to multiple regression include normal probability plots (also known as rankit plots, Q-Q plots, probability plots, quantile-comparison plots, and normal plots), and overfitting. This paper discusses two Matlab functions that demonstrate important points about both issues. Although Matlab is not primarily a statistics package (it does, however, have a comprehensive Statistics Toolbox), these features make it suitable for use in statistics classes at certain times. We certainly believe that our programs based on Matlab enhance the student's experience within statistics.
It has been argued that the impact of technology "should be especially strong in statistics" (Bratton, 1999, p. 666) since technology plays such an important role in the actual practice of statistics. Bratton (1999) then argues that there are three ways in which proper implementation of technology into statistics can happen: By making some topics unnecessary; by introducing new topics; and by teaching some methods better. In this third category, he especially makes mention of randomness and probability concepts. One of the programs we discuss herein addresses this issue. West and Ogden (1998) discuss a number of Java applets using statistical demonstrations that operate over the World Wide Web, though none cover the same issues as those presented here. The rationale, however, is...
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