The Information and Communication Technologies as Cognitive Tools in the Teaching and Learning of the Probability and Statistics Santiago INZUNSA
[email protected] Facultad de Informática, Universidad Autónoma de Sinaloa Culiacán Sinaloa Mexico Guadalupe QUINTERO
[email protected] Facultad de Informática, Universidad Autónoma de Sinaloa Culiacán Sinaloa Mexico ABSTRACT In the present article results of a research with undergraduate students about confidence intervals and their relationship with sampling distributions in a simulation environment are reported. The results show that power of cognitive tool of the computer and the design of suitable activities had an important effect so that students developed correct meanings of concepts like influence of the sample size in sample variability and margin of error in the intervals, but not in the wideness of the intervals. Keywords: Computers, Probability, Statistics.
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1. INTRODUCTION The development of the computer technology has revolutionized the practice of statistics [1], [2]. Nevertheless, the application of the computers in statistics education has left behind in respect to practical approaches. The recent incorporation of probability and statistics in the curriculum of middle and high school, and the insufficient research about the learning problems in the area are the main causes [3]. Nevertheless, in the last fifteen years, many statistical educators have developed research projects with computers based technology (computers, graphical calculators, Internet) in
the teaching and learning of probability and statistics. The results are emerging and orient us towards the appropriate ways to use this technology, however, also show us the difficulties that students face when this technology is applied. These results have had impact in the curriculum of school mathematics around the world, where computers in probability and statistics classes have been incorporated. An example frequently described in the literature, are the Curricular and Evaluation Standards for School Mathematics [4] and the Principles and Standards for School Mathematics [5] in the United States. Also, new developments of educational software especially designed for statistics learning have been developed. Recent examples of this tools are: Fathom [6], Probability Explorer [7], TinkerPlots [8]. 2. THE COMPUTERS AS COGNITIVE TOOLS Mathematical educators have the tendency of considers computers as “amplifiers of mind” [9]. From this perspective, the computers are considered tools that help us to accomplish tasks in a much faster way, without a qualitative increase in comparison of what we can do without them. Another approach consists of considering computers as cognitive tools, with potential to bring about structural changes in the system of the student’s cognitive through a
of
the
A cognitive technology “is any medium that helps trascend the limitations of the mind in thinking, learning, and problem-solving activities” [9]. On the other hand, to Jonassen [10] “cognitive tools are both mental and computational devices that support, guide, and extend the thinking process of their users”. Recent developments of software for the teaching of probability and statistics, incorporate dynamics environments, abundant representations and are highly interactive, in such a way a computer can become an important cognitive tool, since “the dynamic and interactive media provided by to computer software make gaining an intuitive understanding interrelationships among graphic, equational, and pictoral representations more accessible to the software user” [9]. Among the important functions that a software tool must incorporate to work like cognitive tool in mathematics education, is the mathematical exploration. An exploration facet that is suggested in many concepts of probability and statistics is the computer simulation. Dörfler [11] -supported in the ideas of Pea [9] -, propose a conceptual framework about the use of the computer in mathematics education and identifies some ways in which a computer tool can generate changes in students’ mental activities: 1. Shift the activity to higher cognitive level. 2. Change of objects of the activity. 3. It focuses the activity on transformation and analysis of representations. 4. It supports the situated cognition mode of thinking and problem solving. In the present work we will use this framework of reference and we will analyze the implications in the concept of sampling distributions and probability distributions. We will as well provide some empirical results of a
study with a group of university students. 3. THE SIMULATION IN THE TEACHING OF SAMPLING DISTRIBUTIONS The probability distributions and sampling distributions occupies an important place in the curriculum of high school and first grade university statistical courses. Until now, formulas and tables of probability, with emphasis in the calculation of probabilities, have predominated in its teaching processes. From this perspective, many students successfully develop the necessary calculations to solve the problems; however frecuently they don’t´understand the concepts and the process involve in it. With the development of more flexible software tools and with greater potential in educational terms, many researchers (e.g. [12], [13]) have turned around the glance towards the computer simulation as an education resource that can help the understanding of these concepts. Computer simulation approach show diverse advantages with respect to the theoretical or complemented approach, such as the very little mathematical background required to be implemented. In order to have an idea of the simulation potential in the probability distributions we showed the convergence of the binomial distribution to normal distribution (fig. 1). Also, the superposition of the frequency distribution of the results of a random experiment developed by simulation is presented here. And the expected results according to the theoretical model of probability (fig 2). Func tion Plot
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y = b in o m ia lPr o b a b ility ( r o u n d ( x ) , r o u n d ( n ) , p ) y = n o r m a lDe n s ity ( x , n • p , n •p (1 − p ) )
Fig. 1: Convergence of the binomial distribution to normal distribution
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Fig. 2: Comparison between frecuency distribution and probability distribution With this fact can be justified in visual terms to use the normal distribution to calculate probabilities of a binomial experiment. To demonstrate the previous thing requires very sophisticated mathematical tools like the central limit theorem. Of such form, the use of computer tools offers the possibility of teaching concepts to a greater level of depth clearly indicating that without computers tools, some students would not be able to understand. 3.1 An Example of a Teaching Situation Based on Computer Simulation The purpose of presenting the following example is to show the way in which can be used the simulation in the teaching of sampling distributions. The example comprises of an activity that was developed in a study with the main purpose to explore the potential of the simulation (with software Fathom) to introduce the sampling distributions with students with null or few mathematics antecedents. The study was carried out with a group of 24 students from the Public Politics School of the Autonomous University of Sinaloa (Mexico). The study was conducted when an introductory course on Probability and Statistics was taken by the students. Before of each simulation sessions, the students solved problems by means of probability formulas and tables. Each of these topics was accompanied by a theoretical explanation with computer simulation by the teacher. Three activities were designed (1.5 hours each one) that involved sampling distributions of a
proportion, and they were linked with the confidence intervals. Each of the students had its own computer to solve the activities. At the end of the activities a questionnaire was provided to evaluate its understanding and some students were interviewed. A learning process was designed where the participants’ students themselves built a conceptual meaning based on the solved problems. Details of the last activity are showed next: A survey made in the Faculty of International Studies and Publics Politics, with a sample random of 20 students, indicates that 6 are smokers, that is to say, 30%. a) It generates sampling distributions with 1000 samples of size 20, 50 and 100 respectively. b) It calculates the mean and the standard error of each distribution c) It determines a confidence interval of 95% for each distribution and its respective margin of error. The first sampling distribution (n = 20) constructed by one of the students is next: Collection 1 fumadores =
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The amplifying power of the computer tool, allowed the student to simulate the extraction of a great amount of samples of a population, to
calculate in each sample the amount and the proportion of cases of interest and to graph them of immediate form, obtaining similar results to which would be obtained if the sampling were made in fact. A manual simulation of the previous situation is evidently impractical in the classroom. The power of cognitive tool of the software allowed having three different simultaneous representations, bound to each other, from the same situation (numerical, symbolic and graphical). Such form allowed, by pressing a key or modifying some parameters, the immediate generation of 1000 more samples and to visualize the changes in the representations. At the end of the activity, the students obtained three sampling distributions for sizes of sample (20, 50 and 100) in the same screen, which allowed them to visualize patterns of behavior in the different representations and to establish conjectures about the form in which the sample size can affect the behavior of the sampling distributions, the dimensions of the confidence interval and the margin of error. In the course of the activities the majority of the students demonstrated an adequate conception of the sampling variability and the expected value of the proportion in one sample. This was due to the activity of the students with the software in which they simulated a repetitive extraction of samples and observed that the results, if they varied from one sample to the other, frequently they corresponded to an interval close to the parameter. For example, once the simulation activities were concluded, in the context of a problem that signalled that in the Mexican population approximately 30% of obese children (binomial population), a student (Karina) was asked to answer what the results of the simulation in the Fathom cases table meant to her. (See figure 2): R: What does the first number in the table mean? K: That in the first sample there are 39 obese children, 33 in the second. R: How many obese children would you expect in a sample of 100?
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Fig. 2: Simulation Developer by Karina K: 30 R: But there is a sample of 21. K: The result is not always the same, they vary. R: Could you tell me a variation interval? K: Approximately from 25 to 35. Another student (Rosario) in the same context answered as follows: R: Could you explain the meaning of each of the numbers that appear in the table? Ro: In case 1 there are 20 obese children from the 100 children in the population. R: What do the 100 numbers represent? Ro: 100 samples R: Is it normal that the sample results vary? Ro: Of course, you do not always get the same. R: What would be the variation interval? Ro: From 24% to 34% approximately. We can see that both students have constructed a correct meaning of sampling variability since they identified that the results vary from one sample to another and they defined an acceptable variation interval around the parameter. This is due to the availability of a chart with a great grouping of samples and their graphic representation. In other words, the representational elements of the software have an important impact in the discovery of properties and the construction of the sampling variability concept. On the other hand, the sample size influences in sampling distribution properties and its relationship with intervals, the activities were
designed so that students simulated sampling distributions for sample sizes of 20, 50 and 100. Simultaneous histograms of distributions were used so that students identified behaviour patterns easily. Once more the software representations contributed in an important way so that the majority of the students constructed a correct meaning of the effect of sample size in the behaviour of sampling distributions. The interviews and the final survey demonstrate the results. For example in the final interview with a student (Adalberto) answered the following: Do you think there is a relationship between the mean of each sample and the mean of the population? A: Yes because in both the 100 sample size and in the 200 sample size there is no variation and they are near 0.30. That indicates that the 30% data about obese children of the population is important. R: In the standard deviation? In the sample of 100 we have that it’s worth 0.04 as opposed to the sample of 200 that is 0.03. We can see that the biggest sample has a minor standard deviation. R: Now construct a distribution with sample size of 500 and compare the three distributions. A: Practically the mean is the same, it equals to 0.30, or 30%, and dispersions lower, now they are worth 0.02. R: Which of the three is more precise? A: The one with a sample size of 500 is more precise because the standard deviation is smaller and the mean is closer, actually is equal to the population proportion. R: If we continue incrementing the sample size, what do you think will happen with the center of the sampling distribution? A: It would be the same, it does not vary. It would be equal to 0.30 and the standard deviation will continue to decrease. R: If we generalize. A: For diverse sample sizes, we see that the mean is equal (it does not vary a lot) but as the population is bigger the dispersion is smaller.
We observe that Adalberto understands the effect of the sample size in the center and variability of the sampling distributions and he adequately relates it with precision About the relationship between sampling distributions and intervals, students were asked to make calculations in a theoretical manner and by simulation. The result of the simulation was very near and in many cases almost equals to the theoretical results. Thus, the simulation besides of allowing the exploration of the involved concepts and to help understanding the relation among them, can be a tool that provides results sufficiently precise to solve problems. For example, a student obtained the following results: Size of sample 20 50 100
Simulation results Interval 09% - 51% 17% - 43% 21% - 39%
Error 21% 13% 9%
Theoretical results Interval Error 10% - 50% 20% 17% - 43% 13% 21% - 39% 9%
In the construction of the sampling distributions process, the students made an intensive use of representations (cases tables, charts, formulas and descriptive tables), to explore the relationship between the sample size and the behaviour of distributions and intervals (see figure 1). This contributed to that many students showed an adequate comprehension of the effect of the sample size in sampling variability and the margin of error of intervals. However, in a final questionnaire, 17 students manage to identify that statistics are variables and 15 of them identify that this variation was in concordance with a distribution formpredictable pattern. While, 21 students indicated that the statistics serve to estimate parameters. Twenty two of the students indicated that the variability of the sampling distribution diminished as the sample size increased and 18 indicated that the average distribution is not altered, and only three of the students identified that the mean of the sampling distribution is equal to the population parameter.
In an item that contained the graph of three sampling distributions, 19 students assigned the correct way of the sample sizes taking care of the variability and 17 correctly identify the average of the population from which the samples come. Respect to the sample size effect in the margin error of a confidence interval, all the students indicated that this diminishes it as increases the sample size, but 15 of them incorrectly indicated that the wide of the interval increases. Some students had difficulties to interpret the simulation results when these are concentrated in the graph. They saw the sample distribution, when in fact it was a set of samples. This difficulty has been identified in a previous study with another type of students [13]. 4. CONCLUSIONS The computer simulation can become an important cognitive tool, mainly when software allows that the student constructs important parts of the process and provides with dynamic and interactive representations that allow students to relate the concepts each other. In the case of the sampling distributions the students managed to understand important properties of intuitive way and could calculate intervals of confidence with a greater sense, than simply using the formulas and the tables of probability. Nevertheless we considered, that suitable activities design focused to the concepts of interest, is crucial for the students to develop a better understanding of sampling distributions. 5. REFERENCES [1] Moore, D. S. (1990). Uncertainty. In L. Steen (Ed.). On the shoulders of giants: New approaches to numeracy. pp. 95137. USA: National Academy Press. [2] Thisted, R. A. & Velleman, P. F. (1992). Computers and Modern Statistics. In D. Hoaglin & D.S. Moore. Perspectives on Contemporary Statistics. pp. 41-53. Mathematical Association of America. MMA Notes, Number, 21. [3] Biehler, R. (1991). Computers in probability education. In R. Kapadia y M. Borovcnik
(Eds.). Chance Encounters: probability in education. A review of research and pedagogical perspectives. pp. 169-212. Dordrecht: Kluwer. [4] NCTM (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA. Author. [5] NCTM (2000). Principles and Standards for School Mathematics. Reston, VA. Author. [6] Finzer, W., Erickson, T. & Binker, J. (2002). Fathom Dynamic Statistics Software. Key Curriculum Press Technologies. [7] Sthol, H. (2002). Probability Explorer Software [8] Konold C. & Miller, C. (2005). TinkerPlots Software. Key Curriculum Press Technologies. [9] Pea, R. (1987). Cognitive Technologies for Mathematics Education. In A. Schoenfeld (Ed.). Cognitive Science and Mathematics Education. pp. 89-122. Lawrence Erlbaum Associates Publishers. [10]Jonaseen, D. H. (1994). Technology as cognitive tools: learners as designers. Department of Instructional Technology. University of Georgia. Available Online: http://itech1.coe.uga.edu/itforum/paper1/pap er1.html [11]Dörfler, W. (1993). Computer Use and Views of the Mind. In Ch. Keitel & K. Ruthven (Eds.). Learning from Computers: Mathematics Education and Technology. pp. 159-186. Springer Verlag. [12]DelMas, R., Garfield, J. & Chance, B. (1999). Exploring the Role of Computer Simulations in Developing Understanding of Sampling Distributions. Paper presented at Annual Meeting of AERA (American Educational Research Asociation). [13]Sanchez, E. & Inzunsa, S. (2006). Meaning´s construction about sampling distributions in a dynamic statistics environment. In A. Roosman & B. Chance (Eds.). Proceedings of the 7th. International Conference on Teaching Statistics. IASE-ISI. Salvador, Brazil.
