Uri Wilensky
Center for Connected Learning & ComputerBased Modeling
Northwestern University
Annenberg Hall 311
2120 Campus Drive
Evanston, IL 60208
uriw@northwestern.edu
8474673818
Concepts and intuitions from probability and statistics pervade many aspects of our "nonmathematical" lives. When we decide whether to pay more for a car that is reported to be "more reliable"; when we choose to safeguard our children by living in a "lowcrime" neighborhood; when we change our diet and exercise habits to lower the risk of heart disease; when we accept or refuse a job with other options pending; all the above actions involve implicit probability judgments. Arguably, no other area of mathematics can be said to apply so widely. No other area is so potentially empowering so useful in making decisions, both intimate and professional, and affords us such opportunities for making sense of phenomena, both natural and social.
Yet, as the research literature (discussed below) has made abundantly clear, students do not take great advantage of these powerful ways of making sense of the world. This paper seeks to give an account of this missed opportunity in terms of the framework of Connected Mathematics (Wilensky, 1993).The paper begins with a sketch of background issues related to learning probability and statistics. The Connected Probability Project (Wilensky, 1993; 1995a; 1995b) is then described and situated in the framework of Connected Mathematics. The notion of "epistemological anxiety" as a primary obstacle to learning probability and statistics is articulated and developed. The paper then describes how the use of computational environments  specifically objectbased parallel modeling languages  can allow learners to successfully address epistemological anxiety in the realm of probability and statistics. The remainder of the paper seeks to instantiate the theory and practice of Connected Mathematics in case studies concerning learning about the concept of normal distribution. These cases illustrate how traditional teaching of the concept of normal distribution which relies on formalism and macro level summary statistics leads to epistemological anxiety. The cases also illustrate how Connected Mathematics engages issues of epistemological anxiety and how it fosters a deeper understanding of normal distributions. This understanding is advanced through learners modeling normal distributions as emergent phenomena.
Probability & Statistics (henceforward, P&S ) research "methods" courses have achieved notoriety in the undergraduate and graduate curriculum as the bane of many a student. Students in the social sciences are usually required to take a statistics course. Though such courses are intended to introduce them to concepts and methods which they can then employ in making sense of real data, they frequently report being able to work the textbook problems but having little idea of "what they mean," or how to apply them to novel situations (see, e.g., Phillips, 1988; Rubin et al, 1989). Students complain about being able to make several different arguments to solve a problem, all of which sound plausible yet yield different answers (Wilensky, 1993). How are they to choose among the plausible alternatives?
This confusion and indecision leads to what I have termed "epistemological anxiety" (Wilensky, 1993) a feeling, often in the background, that one does not comprehend the meanings, purposes, source or legitimacy of the mathematical objects one is manipulating and using.
In their now classic work, Tversky & Kahneman (1982) document the persistent errors and "misconceptions" that people make when making probabilistic "judgments under uncertainty". Among these errors are systematic misconceptions about probabilities of conjunctions, inverse probabilities, updating probabilities in the face of new evidence and "seeing" nonrandom pattern in random data. These systematic errors are repeatable and don't seem to go away even when people have had significant training in probability. This contributes to a widespread belief that humans are incapable of thinking intuitively about probability.
Tversky & Kahneman speculate as to the origin of these systematic biases in people's assessment of likelihoods. They theorize that people's mental resources are too limited to be able to generate probabilistically accurate judgments. Consequently, people are forced to fall back on computationally simpler "heuristics". In other words, we have been "hardwired" not be able to think about probability and must circumvent our natural thinking processes in order to overcome this liability.
The view that humans must circumvent natural thinking processes in the realm of probability has spawned a large literature and has become very influential. For the purposes of this exposition, let us call this view the "accommodationist" view. The effect of this accommodationist view on P&S instruction has been a deference to the mechanical manipulation of the tokens of formal notation as a guard against unreliable intuitions. This lesson for P&S instruction has reached the highest levels of education. In an introductory graduate course in probability and statistics at a major research university, the professor wrote down Bayes theorem for calculating inverse probabilities and then baldly announced:
It is possible to begin with a more conservative set of assumptions then do the accommodationists. We could assume that P&S is an area of mathematics not widely separated from the rest of the discipline. Furthermore, we could posit a broadly developmental view of mathematical intuition which allows for conceptual change. We would then look elsewhere for the source of learners' difficulties with P&S. We would look at the learning methods, tools and cognitive technologies (Pea, 1987) available to the learner. Given the prevalence of this accomodationist view, it would be of great interest if it could be shown that a suitable learning environment could help learners transform their probabilistic biases into effective intuitions and a solid conceptual understanding of P&S. This work is being undertaken through the Connected Probability project.
In the initial year of the project, seventeen in depth interviews about P&S were conducted with learners age fourteen to sixtyfour. Interviews were open ended and most often experienced by the interviewees as extended conversations. The interviewer guided these conversations so that the majority of a list of 23 topics was addressed. The interview topics ranged from attitudes toward situations of uncertainty, to interpretation of newspaper statistics, to the design of studies for collecting desired statistics and to formal probability problems. In most interviews, a computational modeling environment designed for experimenting with P&S was made available to the learners.
An analysis of the interview data revealed persistent confusions on the part of most interviewees about probability distributions. These confusions and attendant anxieties resulted in the interviewees turning off their sensemaking with regard to probability distributions and merely invoking rote formulae when applying probability distributions to real world problems.
In a Connected Mathematics learning environment, the focus is on learnerowned investigative activities followed by reflection. Thus, students are not led through the mathematical "litany" of definition, theorem proof. Mathematical concepts are not simply given by formal definitions. Instead, mathematical concepts are multiply represented (e.g., Kaput, 1987 ; Von Glaserfeld, 1989) and the environment supports multiple styles and ways of knowing (see e.g., Turkle & Papert, 1991). Mathematical intuitions are not assumed to be static, nor are some mathematical concepts assumed to be "abstract" and thus not amenable to intuitive apprehension. Learners are supported in building and developing their mathematical intuitions over a lifetime and, through this construction process, mathematical objects are seen to be more concrete (Wilensky, 1991; 1993) as learning progresses.
Connected Mathematics views mathematics as something human make ã we build tools to operate upon and make sense of our world . In contrast to learning procedures or formalisms first, as in the traditional curriculum, Connected Mathematics calls for making many more connections between mathematics and the world at large as well as between different mathematical domains throughout the learning experience(e.g. Cuoco & Goldenberg, 1995). Empowering technology is central to this effort. Technology is used as a personally expressive medium  to explore areas of mathematics previously inaccessible, to make abstract mathematical concepts concrete, and to create new mathematics In contrast to reform documents such as the NCTM standards, which portrays an "image" of mathematics as essentially a problem solving activity, its vision of mathematics is a more generative one  the central activity being making new mathematics. A culture of design and critique is developed.
Connected Mathematics pays great attention to the affective side of learning and doing mathematics. As such, it pays attention to the role of play, joy wonder and curiosity in the learning of mathematics and to the role of desire for social connection as a motivation for engaging in mathematical activity (see also Thurston, 1992). Difficulties in learning mathematics are also examined from an affective perspective. Connected Mathematics looks critically at the role of shame in mathematical culture  how shame creates, first, anxiety (see e.g., Chipman, Krantz & Silver, 1994; Steele, 1995; Tobias, 1993) and, ultimately, negative mathematical selfimage (Dweck & Leggett, 1988; Steele, 1995) in the individual learner and how it stifles discourse and enforces hierarchy (O'Connor, 1993) in the mathematical community. In contrast, a Connected Mathematics learning environment fosters an atmosphere in which it is safe for mathematical learners to express their partial understandings and values these understandings regardless of their degree of correspondence with the mathematics canon. In doing so, it parts company with the literature on misconceptions which highlights the gulf between expert and novice. Connected Mathematics, instead, stresses the continuity between expert and novice understanding. This continuity obtains both in their explicit conceptions  novice conceptions are not discarded on the road to expertise but rather repurposed, and in their general "messy" character  even for expert mathematicians, only small areas of clarity have been laboriously carved out from the generally messy terrain. (see also Smith, diSessa & Roschelle, 1994).
The sources of this anxiety are multiple. The culture of the mathematics classroom contributes greatly (Hoyles, 1985; Lampert, 1991; Noss, 1988; O'Connor, 1993). Its insistence on unitary definitions for mathematical concepts reinforces the belief that mathematical knowledge is completely disconnected from the rest of the learner's knowledge (Cuoco, Goldenberg & Mark 1995; Minsky, 1987b; Wilensky, 1993). The learner , thus poised at the lip of a yawning chasm separating her previous conceptions from the new knowledge, is understandably anxious. The formal definitions sever the links to the many related conceptions that could serve as bridging mechanisms and could alleviate anxiety. The anxiety is further enhanced by social isolation. The insistence on answers or results, absent of intellectual texture, and invalidation of personal voice (Confrey, in press; Gilligan, 1977) discourages the learner from expressing her partial and incomplete understanding. This, in turn, can create the sense that everyone else is clear about the meaning of these objects and that you alone are confused. The further knowledge that any admission of confusion will be used to rank you in the mathematics hierarchy and judge your mathematical intelligence makes it practically taboo to share your "messy" concepts (Papert, 1972; 1993; Wilensky, 1993) and attendant epistemological anxiety.
Connected Mathematics provides several sources of therapy for epistemological anxiety. Instead of formal definitions, it emphasizes connections to the learner's knowledge that make the transition to new knowledge both safer and more meaningful. By exposing the universality of confusion and messy concepts, it reassures the learner that her predicament is "normal" and shared. This, in turn, leads to the voicing of previously unvoiced concepts and sets the context for the social negotiation of mathematical knowing. This emphasis on making, voicing and sharing takes public the activity of understanding mathematics. These changes in mathematical culture are both supported and made much more realizable by the advent of computational modeling environments.
1) We are living in a time when the meanings of the basic notions of probability theory, the ideas of "randomness", "distribution", and even "probability" itself are still quite controversial. There is, as yet, no consensual agreement on the foundational concepts of probability † they are still being debated by mathematicians and philosophers (see. e.g., Cartwright, 1987; Gigerenzer, 1987; Savage, 1954; Suppes, 1984; Von Mises, 1957 ). There is particular disagreement on the applicability of probability to individual instances. This leads to confusion about the applicability of probability theory to the unique situations in our lives. It raises doubts about its utility in making individual choices which might serve to animate our interest in the subject.
2) When we encounter statistical data (as we commonly do in newspaper articles), it is often detached from its method of collection. If a statistic is left vague and disconnected in this way, we can not operate on it, transform it and compare it with others so as to make it meaningful and concrete.
3) Short term memory limitations make it difficult to collect large amounts of data in "real time". So instead, we construct summaries of the data. These summaries are not analyzable into their constituent elements and thus necessarily remain disconnected from the original data. In this way P&S is at a stage of its development not unlike other areas of mathematics throughout history. These same kinds of memory constraints once made arithmetic very difficult until memory saving technologies (such as zero, place value and generalized decimal notation) were invented (See e.g., Kline, 1972; McLuhan, 1964; Rotman, 1987). A central conjecture of the Connected Probability project is that computational technology should do for probability what HinduArabic notation has done for arithmetic. Computational environments allow large bodies of data to be visualized at once in a small space and large numbers of repetitions to happen in a short time. As a result, short term memory resource limitations can be overcome. The focus can then be on probabilistic reasoning.
4) It is very hard to get feedback as to the adequacy of our probability judgments. If we assess that the probability is 30% that it will rain tomorrow and it does rain, what have we learned? We require many, many such observations in order to calibrate our judgments to the data. Again, memory limitations block us from receiving the necessary feedback, Moreover, there is little opportunity for control because the world is constantly changing and we cannot repeat the experiments so as to get systematic and controlled feedback 5) Seeing our lives as experiments in probability requires seeing our current situation as one of a large collection of similar possible situations. To make this large collection into a "concrete" object of thought (Wilensky, 1991) requires a massive construction job. It requires forging links between our current situations and situations forgotten in the past, situations not yet arisen, and counterfactual situations, (those situations which could have arisen but did not) .
6) A powerful way to think about probability situations is to think of multiple, interacting, distributed entities. Again we cannot keep so many entities in working memory (Case, 1993; Miller, 1956) at once. Consequently, we construct wholes out of the many interacting parts and understand the behavior of the ensemble in terms of its average behavior.
Memory and resource limitations, absent a notation or medium, do contribute to why, until now, many people have not developed robust probabilistic intuitions. We have reached a time where these limitations will no longer hold sway in P&S. The emergence of computational modeling environments promises to provide a holding environment , a symbolizing medium which can express data distributed over space and time. This should enable us to see view dynamic properties of ensembles and to conduct experiments with immediate feedback. It is for this reason that one of the most powerful forms of therapy for epistemological anxiety is access to a computational environment in which the objects whose epistemological status are in doubt can be modeled, experimented with and debugged to the satisfaction of the learner. The computer serves as a setting and symbolizing medium for building mathematical intuitions.
The modeling approach has both cognitive and affective benefits. By building computational models of everyday and scientific phenomena, learners can develop robust mental models of the underlying probability and statistics. The feedback provided by building and then testing the computational model supports the learner in debugging and successively refining (Leron, 1994) the mental model. On the affective side, the modeling environment supports the learner's expression of partial understanding. In a modeling environment, it is clear that partial understandings are the norm  one builds models by, initially, expressing a tentative model in the modeling language, then gradually refining and debugging the model. In this way modeling activities short circuit the "whole cloth" answer orientation of the culture of classroom mathematics (Sfard & Leron, in preparation). Indeed, the activity of model building is so different from normal mathematics for most students that they, often, do not identify themselves to be "doing mathematics". (Feurzeig,1995; Mandinach & Cline, 1994; Thornton & Sokoloff, 1990). Embedded in a suitable social environment, the opportunity to express partial understandings without judgment can go a long way towards alleviating a sense of mathematical shame. Relieved from the necessity of "getting it right" the first time, even learners not usually considered good at mathematics and science can build models that demonstrate a qualitatively greater level of mathematical achievement than is usually found in mathematics classrooms. Computational modeling serves as a powerful therapy for epistemological anxiety.
Probability distributions can be seen as canonical cases of emergent phenomena. They are stable global structures that arise from the interactions of multiple distributed agents. Typically, in statistics classes, the emergent nature of distributions such as the normal distribution (or "bell curve") is hidden. We "learn" about distributions through descriptions of their global characteristics (e.g., mean, standard deviation, variance, skew, moments). This conceals the way these distributions are built up or emerge from individual instances. As a result, the critical connection between the micro level of the phenomenon and the macro level is severed. The learner is given the macro description and formulae and is asked to accept on authority that these macro descriptions are appropriate for such and such a set of phenomena. This supplanting of experimentation with external authority contributes significantly to epistemological anxiety.
"Objectbased" means that each agent is selfcontained: it has its own internal state and communicates with other agents primarily by local channels  agents don't do much action at a distance. The computer language Logo had a single such object  the "turtle". Papert (1980) argued that the power of the turtle to facilitate learning geometry lay in the fact that the child could identify with the turtle  enabling "syntonic" learning. Objectbased parallel modeling languages such as StarLogo afford greater identification with their objects, and thus, in contrast to more procedural languages, foster syntonic learning of emergent phenomena.
Learners can assign micro level rules to simultaneously control thousands of such turtles. Macro phenomena emerge in relation to these micro rules. This makes both forwards and backwards modeling possible  that is 1) exploring the global effects of various sets of local rules. and 2) trying to produce a known global phenomenon through finding appropriate local rules (Wilensky, 1995a).
Such objectbased parallel modeling languages can be powerful environments for doing P&S. One way to think of these agents is as multiple sided irregular dice. At each step in running a model built in one of these languages, thousands of dice are thrown and results visually displayed. This use of object based parallel modeling languages is reminiscent of the technique of Monte Carlo simulations and shares feature with resampling statistics approaches (Diaconis & Bradley 1983; Konold, 1994; Simon et al, 1976; Simon & Bruce, 1991). Unlike these other approaches, in objectbased parallel modeling languages, control of the program is distributed through the agents rather than centrally controlled and the agents themselves are inspectable, visualizable, "concrete" objects.
Even at this basic level of "dice rolling", many experiments in probability and statistics are suggested. Indeed, most of the standard elementary curriculum in P&S can be easily modeled in StarLogo using these "dicerolling" features of the language alone. In a noncomputational setting, experiments with large data sets are perforce constrained to be summarized by mediating global formulae and statistics. Experiments can be done for small data sets and small numbers of trials and is typically confined to several conflips, dicerolls and spinnerspins. But once the scale is increased, the ability to experiment is gone. As discussed earlier, the critical connection between the micro level of the phenomenon and the macro level is severed.
Additionally, in objectbased parallel modeling languages, the "dice" can also interact with each other. As a result, many complex interactions are possible  the results of which are not predictable using standard mathematical models. The ability to experiment with such complex interactions and get visual feedback however, allows learners to make qualitative sense of different possible patterns, classifying metapatterns and noting trends such as feedback cycles, critical densities, clustering, etc. Even though these agents  thought of as dice  are probabilistic components whose individual states cannot be predicted, the emergent properties of the collection of agents can be stable and predictable. This insight is essential to grasping the key notion of probability distribution. For this reason, objectbased parallel modeling languages are excellent constructionist environments for learning probability  the learner is constructing objects using probabilistic components. This situation is analogous to the acquisition of print literacy as it includes both reading and writing. Instead of just "reading" about probability (by being given the summary formulae), the learner "writes" with probability.
As background to the interview, I asked Lynn about her attitudes towards mathematics. Her reply was indicative of the issues that would arise in the body of the interview:
U: So, what was math like for you in school?
L: Well, I was always good at math. But, I didn't really like it.
U: Why was that?
L: Why? I donêt know. I guess I always felt like I was getting away with something, you know, like I was cheating. I could do the problems and I did well on the tests, but I didnêt really know what was going on.
One of the first questions that arose in the body of the interview with Lynn was: What would you estimate is the probability that a woman in the US would be at least 5'5" tall? Here is the text of the ensuing conversation:
L: Well I guess it would be about 1/2.
U: Why do you say that?
L: Well height is normally distributed and I'd say the mean height of women is about 5'5" so half of the women would be taller than 5'5".
U: Why would half the women be taller than the mean?
L: Because the curve of the normal distribution is symmetric around the mean  so half would be below it and half above it.
U: What about the number that are exactly 5'5" tall?
L: Well I guess they could make a difference, but no  they shouldn't matter because they're just one data point and so can be ignored.
U: You can ignore any one data point?
L: Yes, .... because there are infinitely many data points so one doesn't make a difference.
U: But can you make infinitely many height discriminations? How do you measure height?
Here, I'm just trying to probe Lynn's thinking about discrete vs. continuous distributions.
L: Well.... I guess we measure it in inches  so there probably aren't infinitely many data points. I'm somewhat confused because I know height is distributed normally and I know that for normal distributions the probability is 0.5 of being bigger than the mean, but how come you can ignore the bump in the middle? I guess 0.5 is just an estimate, it's approximately 0.5.
U: How do you know height is distributed normally?
L: I don't remember where I first learned it, but lots of the problems in the stats books tell you that.
My question was intended to elicit an explanation of why Lynn believed height was distributed normally in terms of her personal knowledge, but Lynn responded to it as about probing the context of gaining that knowledge and the authority it derives from.
U: Why do you think height is distributed normally?
L: Come again? (sarcastic)
U: Why is it that women's height can be graphed using a normal curve?
L: That's a strange question.
U: Strange?
L: No one's ever asked me that before..... (thinking to herself for a while) I guess there are 2 possible theories: Either it's just a fact about the world, some guy collected a lot of height data and noticed that it fell into a normal shape.....
U: Or?
L: Or maybe it's just a mathematical trick.
U: A trick? How could it be a trick?
L: Well... Maybe some mathematician somewhere just concocted this crazy function, you know, and decided to say that height fit it.
U: You mean...
L: You know the height data could probably be graphed with lots of different functions and the normal curve was just applied to it by this one guy and now everybody has to use his function.
U: So youêre saying that in the one case, it's a fact about the world that height is distributed in a certain way, and in the other case, it's a fact about our descriptions but not about height?
L: Yeah.
U: Well, if you had to commit to one of these theories, which would it be?
L: If I had to choose just one?
U: Yeah.
L: I don't know. That's really interesting. Which theory do I really believe? I guess I've always been uncertain which to believe and it's been there in the background you know, but I don't know. I guess if I had to choose, if I have to choose one, I believe it's a mathematical trick, a mathematician's game. ....What possible reason could there be for height, ....for nature, to follow some weird bizarro function?
The above was a short section of the first probability interview I conducted . Until the last exchange transcribed, I had the feeling that the interview was not very interesting for Lynn. But after that last exchange she got very animated and involved in the interview. This question of the reality vs. "trickiness" of the normal and other distributions occupied her for much of the next few discussions. She came back to it again and again.
At the time, I was a bit surprised by this reaction. Earlier in the interview, when I asked her about infinitely many height discriminations, I think I was trying to steer Lynn into making a distinction between a discrete and continuous distribution. This, I thought, might have resolved her dilemma about ignoring the "bump in the middle". But I was not expecting to hear such basic doubts about the validity of statistics from someone with so much experience studying and using statistical methods .
Lynn's question was, as she says, "in the background" all the time when she thought about statistics. How could it not be? At the same time that she was solving probability and statistics problems for homework, and later on when she actually implemented a study which used statistical tests to evaluate treatments, she was never really sure that the formulae she was using had any explanatory power, any real meaning other than mathematical conventions. Is it any wonder, then, that she was engaged by this question? Up until now, no one had engaged her in discussion about how to interpret the mathematical formulae she was using. She was left feeling that her own work played by the "official rules" but in a way was not accessible to her  and she harbored doubts about its validity. Her questions, while essentially epistemological, go to the heart of the mathematics of probability. Note that, in emphasizing the normal distribution, her courses had led her to see the process of describing height by a curve as a single process with a single agent (possibly the mathematician) controlling it. The idea of the height distribution as being emergent from many interacting probabilistic factors was not in awareness.
Had we tested Lynn on textbook statistics problems, she would have performed flawlessly . Yet, even in a interviewee of such sophisticated background, epistemological anxiety was lurking in the background. Lynn knew that she did not really know what gave validity to the procedures she had so laboriously mastered. Through engaging in this interview, her fundamental confusion is exposed. To an uninformed observer, it might seem that the interview has caused Lynn to regress, confusing her about material she has mastered. But, in fact, by expressing her partial understandings of distributions through asking whether normal distributions are "just a mathematician's trick", Lynn takes a big step towards a more Connected Mathematical understanding of distributions.
In the next interview, we will see how another interviewee, engaged in building a StarLogo model of normal distributions, deals with these same issues.
A: I never really understood normal distributions. I mean I can do the problems, I've even helped Wendy [his wife] with her statistics problems, but what is really going on? Why should height fall into a normal distribution?
As the interview progressed, Alan became intrigued by the question: "why is height normally distributed?" After some prodding by the interviewer to come up with an answer to his question, Alan made a conjecture as to why height is normally distributed:
A: You start out with this one couple, Adam and Eve say, and they're a certain height. No, make this simpler, we just have Adam and he's a certain height. Now let's suppose we have parthenogenesis, (is that the word?) and Adam has kids on his own. And suppose his kids are a bit off from where he is [that is, slightly different heights] due to copying errors or something. Then they sort of form a little bell curve  anyway a binomial distribution. Now suppose they have children and that process continues many generations, then wouldn't you wind up with a normally distributed population?
To explore this conceptual model, Alan decided to write a StarLogo model with me. The model is initialized to have thousands of "rabbits" at the bottom middle of the graphics screen. (Call the initial location of the rabbits the "origin". For reference, a green vertical line is placed at the origin.) Rabbits can hop right or left from the origin on a line at the bottom of the screen. (Call it the hopping line) Each rabbit has a set of possible step sizes which it can hop in one clock "tick". Associated with each step size is a probability  the probability that the rabbit will choose that step size. To make the interface easier for himself and other possible users who might not be comfortable with probabilities (expressed as real numbers between 0 and 1), the probabilities of rabbit steps are expressed as integer relative odds. The rabbits are initialized with the command "setuprabbits" which takes three arguments: the number of rabbits, a list of stepsizes and a list of probabilityratios. So, for example, if one wanted to create 1000 rabbits that could hop to the right 2 units with probability 3/4 and hop to the left 1 unit with probability 1/4, one would type the command
setuprabbits 1000 [2 1] [3 1].
After executing this command, the rabbits would be all piled on top of each other at the middle bottom of the screen. It is then possible to perform experiments and let them hop for a while. At each clock cycle each rabbit picks a step to hop according to the probabilities and hops that distance on the hopping line. Once the rabbits have hopped for one clock tick, they are no longer all in the same place. They are still on the hopping line, but at different x positions on that line. Above each position at which there are rabbits present, we display a yellow column  the height of which is proportional to the number of rabbits at that location. In this way an evolving "living" histogram of the distribution of rabbit locations on the hopping line appears on the screen.
To make this a bit clearer, letês take a simple example:
If we initialize the rabbits with the command setuprabbits 8000 [1 1] [1 1], then 8000 rabbits will appear at the origin. Each rabbit has been initialized so that it can only hop either one unit to the left or one unit to the right with equal probability. If we let them hop one step, then because the probability of moving left is the same as the probability of moving right, approximately half the rabbits will move one unit left and another half will move one unit right. As a result two roughly equally high yellow columns will appear on the screen above locations x=1 and x=1. If we now let the rabbits hop one more step, there will be a tall yellow column in the middle and shorter, roughly equally high, yellow columns to the right and to the left. Continuing this process will lead to histograms of binomial distributions symmetric about the origin. (See figure below).
setuprabbits
8000 [1 1] [1 1] after 10 "hops"

setuprabbits 8000 [1 1] [1 1] after 40 "hops"


As you may recall, the reason Alan and I created this model was to investigate Alan's theory of height distributions. One way to think of the rabbits when they're all at the origin is as a population of individuals all of a standard height. When rabbits take a step they represent new individuals in the population who have deviated slightly from their parents' height. In our simple example the average height of the population does not change and the population heights are symmetric about this average.
When viewing this example, Alan exclaimed: "I see why population heights are symmetrical, the rabbits hop the same distance to the right as to the left". He then typed in the command setuprabbits 8000 [2 1] [1 1]. "Now", he said, "the graph won't be symmetrical, it will skew to the right". But to Alan's surprise, when the rabbits hopped, the histogram remained symmetrical, but the purple line representing the average rabbit location (or average population height) moved to the right.
Alan quickly understood the "bug" in his thinking:
"I see. Rabbits hop farther to the right when they hop but they're still just as likely to hop right as left. So, the distribution will remain symmetric, it will just move to the right one unit on average for each step. To make the distribution asymmetric, you need to change the probabilities not the step sizes."
If children are just as likely to be shorter as they are taller than their parents but when they are taller their difference from their parents is larger, then the population as a whole will slowly get taller. But since we know that height of the population as a whole is symmetrically distributed, children must be equally likely to be taller as they are to be shorter than their parents. I see why normal distributions are so common. Whenever we make a measurement, we're just as likely to make a mistake in one direction as the other, so the resultant distribution of measurements will be normal. Its average will be the true value of the measurement and the spread of the graph will depend on the accuracy of the measurement.
Alan then went on to investigate what would happen to the distributions if he gave the rabbits a much larger "palette" of step sizes. This investigation leads him to more insights about how different micro level rules produce different probability distributions. When I asked Alan to sum up what he has learned from his modeling experience, he expresses satisfaction that he has gained a deeper understanding of binomial distributions and "what kind of animal a distributions is". But, he also says that perhaps he has not yet "entirely understood normal distributions". He speculates that the normal distribution is "the limit of the binomial distribution as "n" approaches infinity  when you fill in the gaps between the bars of the binomial". He worries, however, that he doesn"t yet sufficiently understand the connection between the binomial distribution and the formula he remembers for normal distributions: "how does the integral of e x2/2 come out of this?".
It is important to remember that Alan began the interview with a command of elementary probability and statistics. He was facile with the mathematics of binomial processes and knew the standard parameters and statistics of normal distributions. It was the connection between these two areas of his knowledge that was missing for Alan. School mathematics, like prenineteenth century mathematics keeps these two disciplines separate. Indeed, with limited computational power, it is hard for a school curriculum to make a meaningful connection between the two. Objectbased parallel modeling languages bring the two back together. By enabling Alan to give probabilistic rules to the individuals and see the resulting pattern of the ensemble, the StarLogo modeling language enabled him to make these connections. Though Alan has made significant progress in his understanding of distributions, he is not done. His desire to further connect his new understanding to the formula he learned in school exemplifies his appropriation of the Connected Mathematics approach.
Both Lynn and Alan had strong mathematical backgrounds, had done graduate coursework in mathematics and had excelled. Both of them were very comfortable doing the formal manipulations that constitute the major component of traditional instruction in P&S. Nonetheless, both explicitly expressed considerable discomfort concerning the notion of normal distribution ã the most salient example of the core statistical notion of probability distribution. Regarding the nature of normal distributions, they suffered from epistemological anxiety. The therapy provided in these learning interviews consisted of two main interventions: 1) Validation of the importance of Lynn and Alan's mathematical voice, their personal view of the mathematics and its relation to their own experience. This validation was achieved by explicit avowal at the start of the interview, by openended interviews that followed the interviewee's own thinking and by long drawn out interviews that declared of themselves that the interviewee's conceptions were valuable and valued. 2) The use of an objectbased parallel modeling language, StarLogo, in which to build computational models of probabilistic phenomena and successively refine them. These supports provided a context in which to work through their epistemological anxiety.
Though, clearly, neither Lynn nor Alan were experts in P&S, their sophisticated mathematics background as well as their formal mastery of P&S techniques would qualify them as experts in the eyes of many educators. These "nearexperts" were no less confused about normal distributions than were the novice interviewees. Indeed, instructors of P&S with whom I have shared Lynn's interview have been "appalled at the incoherence of her remarks ", yet, instructors like these, awarded top grades to Lynn in her P&S courses. Interviews such as these cast serious doubt on theories which assert a strong disjunction between novice and expert understanding. Even experts must laboriously map out clarity from a very messy "terrain". By overemphasizing formal operational thought, typical mathematics education allows intellectual accommodation to be an adequate endpoint. In Connected Mathematics, understanding is taken to mean "multiply connected" and not cleanly and uniquely specified. This embraces "messy" conceptions at all levels of expertise in the process of learning. One has never "arrived".
Rather than emphasizing problem solving, Connected Mathematics emphasizes problem posing. In problem solving, the questions are often disembodied and not the learnerês own. Within Connected Mathematics, modeling is both a medium in which learners can formulate their questions in a precise way and a method for building their understanding.
Once Alan posed the problem of the symmetry of the distribution, his reasoning is expressed and refined in interaction with the model he has built. Alan strives to find the connections between his representation of the individual rabbits behaviors and the global pattern that they form. He debugs his model by varying each of the three inputs to the runrabbits procedure, focusing particularly on the lists of steps and ratios. How will variations in the lengths of the lists and the relative magnitudes of the steps and ratios affect this emergent pattern? He alternates his focus from the micro level, the behavior of the individual rabbits to the macro level of the distribution. To better keep track of the change in the global pattern over time, he takes advantage of StarLogoês openended programmability and builds representational aids such as the "origin line", the "hopping line" and colorcoded classes of rabbits. These representations and tools allow him to debug his original conception that the relative size of the steps is the key to an asymmetric distribution and, instead, to see the relative size of the probabilities as the key factor. By building distributions with probabilistic parts and making them work, Alan makes probability ratios as concrete for himself as stepsizes, and sees how distributions are built out of these concrete building blocks. Extending, enhancing and debugging the computational model itself is essential to the activity of modeling. This distinguishes it from model consumption in which learners are left to tweak the inputs of an expert's model.
In the Connected Probability project, probability distributions are viewed as a kind of emergent phenomena. Rather than manipulate the tokens of formalism, learners are invited to negotiate the interaction of the micro and macro levels of probabilistic and statistical phenomena. Through connecting these levels of emergent phenomena, learners understand the "mechanisms" of probability distributions  how they are assembled from their constituent elements. When these interactions are understood, the global patterns become more than just descriptions to be memorized, they are tools to be used.
Binomial probability distributions are elementary examples of emergent phenomena  emerging from the behavior of noninteracting individuals. Other probability distributions emerge from more interactive individual behavior. Many natural phenomena can be modeled as systems whose global behavior are not predictable using noncomputational mathematics. Such phenomena as the shape of a snowflake, the dynamics of coextenisve wolf and sheep populations, the economy of a country  all of these can be modeled as emerging from the interactions of their constituent individuals. Learning to see emergent phenomena  that is learning to describe the phenomena as arising from the interactions of distributed parts  is, in itself, a worthy goal for educators. From this perspective, P&S becomes not an end, but an entry point to the world of complex systems.
In the Connected Probability project, the role of technology is as a medium of personal expression and articulation of ideas. Empowering literacy implies being able to say something of your own, rather than just manipulate the texts of others. While models are ubiquitous in the current learning literature, modeling is not. Modeling calls on learners to say something on their own, to be authors of mathematics. This implies the best use of technology is for learners to be model builders not just model consumers. The StarLogo modeling language has specific features which allow for this articulation:
Modeling at the level of objects: StarLogo allowed Alan to create, visualize and modify individual rabbits. Alan was, thus, able to experiment with individual rabbits whose hopping behavior he could track, control, model with his own body. These features of the language enabled Alan to leverage his knowledge of individual behavior to knowledge of the ensemble ã to connect the micro with the macro.
Parallelism: The ability to control thousands of rabbits at once gave Alan the experimental apparatus he needed to conduct his investigation in "real time". The ability to experiment with collections of rabbits, to enable subpopulations of rabbits to have different behaviors was crucial to his construction of the distribution.
General purpose programmability: The ability to create his own representations (as opposed to manipulating the parameters of a preconceived model) gave Alan the freedom to construct normal distributions in a way that arose from his own nascent understanding of them. It also enabled him to create computationally active tools, such as the "hopping line". These refinements of his computational model provided greater support for his experimentation and, in turn, led to a refinement of his mental model.
In a Connected Mathematics learning environment, learners are supported in actively connecting areas of their knowledge that have, hitherto, remained separate. Computational platforms, in their ability to bring into being new forms of representation and symbolization, can be powerful tools for making these connections. Objectbased parallel modeling provides a new way of coming to understand and articulate understandings of probability and statistics and, in so doing, engage and relieve epistemological anxiety.
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