where the quality of predictions ha

where the quality of predictions has financial consequences. Thus political or financial problems intenningle with sampling issues. This is even more the case in the program Prime Time from the software Sampling: Probability and Prediction.
A tool-like program for studying sampling variation is Stat Lab. It is much more comprehensive than the above mentioned programs. The focus is on laboratory experiments which enable the students to study the statistical concepts of an introductory statistics course by observing how they vary under repeated sampling.
Structure in random sequences
Some programs do not focus on the sampling distribution but on the structure of a sequence of random data. Can we distinguish a sequence of numbers emerging from tossing coins from another which somebody has just written down on a sheet of paper? Randomness is highly structured compared to haphazardness. The assumption of independent stochastical trials has many empirical consequences such as the number of pairs, triples etc. in a sequence. Elementary knowledge about these phenomena is important for appreciating random numbers and their generators. It is also well known that students have difficulties with such features of randomness. Computers can provide empirical background for the necessary conceptual development.
One approach consists in simulating the throw of a coin and plotting the sequence of results with two different symbols on a computer screen. Changing the basic probability p of a head will have well known visual effects. Green has developed a collection of programs for attacking this problem (Probability and Statistics Programs). We will briefly look at the program ClA. (Cointosser Invention Analyzer), which seems to be the most instructive and convincing of the collection.
In addition to displaying data visually, C.I A. includes methods to analyze data. Its basic idea consists of concealing nine 'coin machines' in the computer which produce sequences of binary data. Each machine is either a fair coin tosser or represents some kind of deviation from the model of equal chance and independence. Such deviations are: the probability p differs from 1/2; after head another head is more likely; after head a tail is more likely; heads mostly come fIrst, tails at end; a repeated pattern; a pattern is randomly chosen. Several commands for graphical and numerical analysis of the data are provided. Data can be grouped (group size 2 - 10) and the distribution plotted or tabulated. The number of runs as well as the distribution of run length can be calculated.
In contrast to many other programs, the simulated data are built up and presented on the
screen and can then be analyzed from several points of view. In this rich and moderately complex environment, students can experiment with the machines, make interactive data analyses, model or describe their behaviour, apply and develop theoretical knowledge about binomial distributions or about runs, decide whether a machine seems to be fair or not, etc. According to the accompanying teaching material, a major aim is that students learn to discriminate between a fair coin tosser and an unfair one. However, the environment can be also used for slightly different pedagogical purposes. A basic option is the possibility of repeating an experiment if the user is not sure how to judge a machine. Obviously, there will be individual differences in judgement which can stimulate discussion. Such experience in an ideal chance environment may also be used to provide valuable experience for understanding statistical tests and applying them reasonably. Textbooks on statistical inference only give tests as being definitive and do not explain that though uncertainty is inevitable it may be reduced by further experiments.
Another option is to input data from the keyboard, for instance from real coin tossing experiments or from students' attempts to generate random sequences by hand. This option for combining simulation with real data analysis, however, is only modestly supported; real data cannot be stored and retrieved from discs. The accompanying material discusses the problem of relating the experience in the learning environment to analyzing structures in sequences of real data and interpreting deviations from the ideal pattern in terms of the real situation. Such experience would be a further step in understanding randomness in the sense Feller (1968) describes:
"In testing randomness, the problem is to decide whether a given observation is
attributable to chance or whether a search for assignable causes is indicated ... "
Related to runs he discusses examples
" ... counting runs of boys and girls in a classroom might disclose the mixing to
be better or worse than random. Improbable arrangements give clues to assign-
able causes; an excess of runs points to intentional mixing, a paucity of runs to
intentional clustering." (p.42)
In summary, experience in an abstract chance environments is valuable and difficult to achieve without a computer. But using such knowledge and experience as a reference and tool for exploring real data and systems in a second step is the ultimate goal of understanding randomness.
A simulation and modelling tool as companion of the curriculum
The Computer Based Curriculum for Probability and Statistics project aims at software that can accompany a curriculum where the empirical side of probability is an integrated feature (Konold et al., 1989). This comprehensive tool provides simulation data for the above mentioned problems and for some more which have been identified by research. The central idea is that students express their beliefs about the probability of events. The software allows one to defme a model and sample from it to test and change these beliefs. As the student activity is central, this software is quite different from others which merely demonstrate something by simulation.
Some of the problems of the pilot curriculum material are the following (Konold, 1991). In Coin Flipping, students are asked to empirically explore the probability of patterns like HHTHT, or the probability of H as compared to T after a series of H's. This is directed towards overcoming the gambler's fallacy and the problems with the representativeness heuristics (see Chapter 7). In Random vs Mixed Up, students can compare coin flipping to artificial data; in Coincidences, a box model for the birthday problem is defined and can be analyzed with regard to coincidences.
The software tool Probability Simulator provides a well organized system of commands which enable the student to analyze diverse simulated data. For instance, it is possible to search for specified patterns in a sequence of numbers and count the number of occurrences, to count the (conditional) frequency of outcomes after a certain pattern, to draw a sample until a certain event has occurred for the first time etc. The system of commands is adapted to such problems and therefore different from those commands which are usually available with standard statistical software.
Games and strategy
In computer assisted instruction, games are often related to a topic rather superficially. Obviously, this is different in probability where they provide a major source for the development of probability theory and are still an important domain of application.
Moreover, the statistician's role has been defined as 'playing games against nature'.
Games of chance have often been exploited as a method of teaching and leaming. Can computer-implemented games offer additional opportunities for learning? O'Shea and Self (1983) express a "desperate need for experimental studies of their [computer games] educational effectiveness. "
Existing studies support the sceptics and are consistent with the critical evaluation of the disparity of computer experience that Bauersfeld (1984) has elaborated.
"In spite of motivationally attractive 'packaging', the microcomputer games used
in this study were not very effective at teaching probability and estimation. Giv-
en the proven success of non-computer games, this result was somewhat surpris-
ing and raises the possibility that students may not process information presented
in a computer environment in the same way that they process information in a
non-computer environment. Much more study is needed of appropriate instruc-
tional uses of computers so that teachers know how to exploit the best features
of such environments." (Bright, 1985, p.522)
In the following, a brief review of some pieces of software and related ideas on games is given. Subgame is part of Micros in the Mathematics Classroom. Digits are randomly generated and the students have to decide the place they should be put in the subtraction for a maximal result. Some empirical research on the use of this software from the ITMA Project is reported by Fraser et al. (1988, pp.330). The influence of the teacher and the changing of classroom roles are important message

Kết quả (Anh) 2:

where the quality of predictions has Financial Consequences. Financial problems Political thì or intenningle with sampling issues. This is the case in the more even level program from the software Prime Time Sampling: Probability and Prediction.
A tool-like program for studying Stat Lab is sampling variation. It is much more the coal Comprehensive Mentioned above programs. The focus is on laboratory Experiments Students to study the mà enable the statistical concepts of an Introductory course by Observing how chúng statistics vary under repeated sampling.
Structure print random sequences
Some programs do not focus on the sampling distribution but on the structure of a sequence of random data. Distinguish a sequence of can we Emerging from tossing coins numbers from another mà somebody has just ghi down on a sheet of paper? Randomness is highly structured to haphazardness sánh. The assumption of independent stochastical has many empirical Trials Consequences như số pairs, Triple etc. in a sequence. Elementary knowledge about phenomena is important, for appreciating These random numbers and ask for their generators. It is well known mà cũng Difficulties with such 'Students have features of randomness. Computers can cung empirical background for the conceptual development cần.
One approach Consists print the throw of a coin Simulating and plotting the sequence of results with two symbols on a computer screen khác. Changing the basic probability p of a head will have well known visual effects. Green has Developed a collection of programs for attacking this problem (Probability and Statistics Programs). We will briefly look at the program CLA. (Cointosser Invention Analyzer), mà Seems to be The Most instructive and convincing of the collection.
In addition to Displaying data visually, includes methods to analyze A. CI data. Its basic idea of concealing Consists nine 'coin machines' in the computer sequences of binary data mà tạo. Each machine is hoặc a fair coin tosser or some kind of deviation là from the model of equal chance and independence. Such deviations are: the probability p Diff from 1/2; another head after head is more Likely; head after a tail is more Likely; First come mostly heads, tails at end; a repeated pattern; a pattern is randomly Chosen. Several graphical and numerical analysis for commands of the data are provided. Data can be quả (group size 2-10) and the distribution plotted or tabulated. The number of runs as well as the distribution of tremor length can be calculated.
In contrast to many other programs, the simulated data are built up and hiển on the
screen and can then be analyzed from vài points of view. In this rich and moderately complex environment, Students can experiment with the machines, make interactive data analyzes, models or ask for their behavior describe, apply and develop theoretical knowledge about binomial Distributions or about runs, decide nếu a machine Seems to be fair or not, etc . Theo Accompanying teaching material, a major aim is mà Students learn to discriminate the between a fair and an unfair one coin tosser. Tuy nhiên, the environment can be used for slightly khác pedagogical cũng Purposes. A basic option is the Possibility of repeating an experiment if the user is not sure how to judge a machine. Obviously, there will be the individual print Hiệu can stimulate discussion mà Judgement. Such an ideal printing experience sewing cũng environment dùng chance to cung Valuable experience for understanding and Applying statistical tests added reasonably. Textbooks only give tests on statistical inference as being definitive and do not explain có uncertainty is Inevitable though it lẽ reduced by Further Experiments.
Another option is to input data from the keyboard, for instance from real coin tossing Experiments or from Students' attempts to generate random sequences by hand. This option Combining simulation with real data for analysis, Tuy nhiên, is only modestly supported; real data stored and đã lấy không from discs. The Accompanying material discusses the problem of relating the experience in the learning environment to print sequences of real structures Analyzing and interpreting data deviations from the ideal pattern print real terms of the situation. Such experience would be a step Further understanding randomness in the sense print Feller (1968) diễn tả:
"In testing randomness, the problem is to decide nếu a given observation is
attributable to chance or assignable nếu a search for Causes is indicated ... "
Related to runs he discusses examples
"... counting runs of Boys and Girls in a classroom might, disclose the mixing to
be better or worse than random. improbable Arrangements give clues to assign-
khả Causes; an excess of runs points to intentional mixing, a paucity of runs to
intentional clustering. " (P.42)
In summary, an abstract print experience is Valuable chance environments and Difficult to Achieve without a computer. But using such 'knowledge and experience as a reference and tool for exploring real data and systems in a second step is the ultimate goal of understanding randomness.
A simulation and modeling tool as companion of the curriculum
The Computer Based Curriculum for Probability and Statistics at the AIMS project That software can accompany a curriculum where the empirical side is an integrated feature of probability (Konold et al., 1989). This Comprehensive simulation tool provides data for the above Mentioned problems and for some more mà được Identified by research. The central idea is có có Students express beliefs about the probability of events. The software cho phép one to defme a model and sample from it to test and change những beliefs. As the student activity is central, this software is quite khác others mà merely something proven and demonstrated by simulation.
Some of the problems of the pilot curriculum material are sau (Konold, 1991). In Coin Flipping, Students are to empirically explore the probability Asked of patterns like HHTHT, or the probability of H as sánh to T after a series of H's. This is directed gambler's fallacy Towards Overcoming the problems with the representativeness and the heuristics (see Chapter 7). In Random vs Mixed Up, coin flipping Students can compare to artificial data; Coincidences print, a box model for the birthday problem is defined and can be analyzed with Regard to coincidences.
The software tool provides a well Organized Probability Simulator system of commands enable the student to analyze mà DIVERSE simulated data. For instance, it is im possible to search for patterns in a sequence of rõ numbers and count the number of lần xảy ra, to count the (conditional) frequency of Outcomes after A Certain pattern, to draw a sample off until the event has occurred A Certain for the first time etc. The system of commands is adapted to such 'problems' and therefore khác những mà commands available with standard statistical software Thường.
Games and strategy
In computer assisted instruction, games are often Do related to a topic rather superficially. Obviously, this is khác print probability where chúng cung is a major source for the development of probability theory and are still an important, the domain of application.
Moreover, the statistician's role Đã defined as 'playing games Against Nature'.
Games of chance have often Do được Exploited as a method of teaching and leaming. Can computer-games offer additional thực Opportunities for learning? O'Shea and Self (1983) express a "desperate need for experimental studies of chúng [computer games] educational effectiveness."
Existing studies support the sceptics and are Consistent with the critical evaluation of the disparity of computer experience mà Bauersfeld (1984) has elaborated.
"In spite of motivationally attractive 'packaging', the microcomputer games used
in this study were not very effective at teaching probability and Estimation. Giv-
en Proven success of the non-computer games, this result was somewhat surpris-
ing and raises Students sew the Possibility mà hiển not process information
in a computer in the same way mà environment chúng process information in a
non-computer environment. Much more study is needed of the appropriate instruc-
tional dùng of computers than how to exploit mà teachers know the best features
of such 'environments. " (Bright, 1985, p.522)
In the drop down, a brief review of some pieces of software and related ideas on games is given. Subgame is part of Micros in the Mathematics Classroom. Digits are randomly generated and the Students have to decide the place chúng nên put in the subtraction for a maximal result. Some empirical research on the use of this software from the ITMA Project is Reported by Fraser et al. (1988, pp.330). The Influence of the teacher and the classroom roles are important, thay of message