Dor Abrahamson

University of California, Berkeley

Ruth M. Janusz

Nichols Middle School, Evanston, IL

Uri Wilensky

Northwestern University

Journal of Statistics Education Volume 14, Number 1 (2006), jse.amstat.org/v14n1/abrahamson.html

Copyright © 2006 by Dor Abrahamson, Ruth M. Janusz and Uri Wilensky, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

**Key Words:** Computers; Education; Mathematics; Sample; Statistics.

This paper overviews our design work and reports on an implementation of this design in a middle-school classroom, focusing
on the combinatorial-analysis activity. Following an overview of the design, we introduce the types of mathematical
problems that are achievement goals for our students, and then we use these problems to explain the principles, structure,
and activities of the combinations tower and some of the computer models. The article continues with a description of the
lessons in Ms. Janusz’s two 6*n* = 21, *n* = 19), interleaved with our reflection
on what we believe students learned in these activities. Next, we propose how combinatorial analysis might help students
understand statistical distributions in populations, for instance why it is that the heights of all 6

Readers interested in the design-research framework that informed this study are referred to Abrahamson and Wilensky (2004c).

In order to help students compare, contrast, and connect the three identified conceptual pillars of the domain—theoretical
probability, empirical probability, and statistics—we designed a new mathematical object as a *bridging tool*
(Abrahamson 2004, Fuson and
Abrahamson 2005) between these pillars. This object is the “9-block,” a 3-by-3 grid in which each square can be either
green or blue (see Figure 1, center, for an example of a 9-block combination).
As we now explain, the 9-block features in different media and different activities as the “math-thematic” bridging tool –
in its different guises, the 9-block functions either as a template for conducting combinatorial analysis
(Figure 2a), a stochastic device generating random outcomes
(Figure 2b), or a sampling tool
(Figure 2c). These multiple roles of the 9-block are expressed through the
following classroom activities. The combinations tower (see Figure 2a) is a
construction of all the 512 different 9-block combinations determined through combinatorial analysis, a mathematical
methodology often applied in the solution of problems in theoretical probability. For the empirical probability part of our
unit, we prepared interactive computer models that generated random 9-blocks (see for example
Figure 2b). For the statistics part of our unit, we developed a computer-based
learning environment in which 9-blocks were samples taken out of a giant population grid (see
Figure 2c, for the computer interface of an individual student who is taking
9-block samples from this population). Our design is for teachers to help students untangle and understand the conceptual
pillars of the domain – theoretical probability, empirical probability, and statistics – by using the 9-block as a basis
for cross-referencing, comparing, and contrasting these pillars.

Figure 1. This diagram illustrates the overall plan of ProbLab, a curricular unit in probability and statistics. The “9-block,” in the center of the figure, is the math-thematic object of this unit that helps students understand and connect theoretical and empirical probability and statistics. This paper focuses on the top-left space of this diagrams - the Combinations Tower that is a combinatorial-analysis classroom project - and discusses its connections to the unit.

Figure 2a | Figure 2b | Figure 2c |

Figure 2. The 9-block ties together our designs for: (a) combinatorial analysis and theoretical probability (on left);
(b) empirical probability (in the center); and (c) statitsics (on the right). On the left, Ms. Janusz and her
6^{th}-grade students stand by the combinations tower constructed from paper-and-crayon 9-blocks that they
created and cut out of blank-grid worksheets. In the middle is a computer generated random 9-block. On the right are
9-block samples from a computer-based population of thousands of squares that are each either green or blue.

There are 512 different combinations of 3-by-3 arrays (“9-blocks”) made up either of green or blue squares (2^{9}).
Of these 512 combinations, there are 126 different combinations with exactly 4 green squares, as derived from the binomial
formula, so there is about one quarter of a chance (126 / 512 = ~0.25) so as to draw a “four-green 9-block” out of a hat
that contains all 512 different combinations. Thus, the combinatorial analysis creates a sample space of all events that
each have the same likelihood of occurring, and by identifying and counting up a subgroup (class of events) within that
space we can anticipate the results of a probability experiment.

Whereas combinatorial analysis and computing probabilities are middle-school learning goals, the mathematical formulas and
procedures for calculating these combinations and likelihoods may be difficult for many middle-school students (see the
scores of U.S.A. Grade 12 students, NCES 2004). For example, only 3% of
12^{th}-grade students solved correctly the following problem (NCES 2004):
“A fair coin is to be tossed three times. What is the probability that 2 heads and 1 tail in any order will come up?” In
designing the combinations tower that is the focus of this paper, our rationale was that students may need more support in
building sense for and connecting between combinatorial analysis and probability experiments (see also
Wilensky (1997)). Using paper, crayons, and glue, students collaboratively
construct a classroom visual representation of the combinatorial sample space of all 9-blocks and in doing so they
“construct” their own meaning for probability (see Papert (1991), for a
pedagogy that relates these two kinds of constructing; see also
Eizenberg and Zaslavsky (2003), on the advantage of collaboration for
learning combinatorics).

Investigating the 9-block offers opportunities for engaging in combinatorial analysis of a compound event. Within the
context of probability experiments, any random 9-block outcome is a compound event, because its identity – its particular
green/blue pattern – is a specific outcome configuration of its nine independent squares. For example, a 9-block with only
one green square in the top-left corner is the compound event configured of nine concurrent independent outcomes: “top-left
square is green AND top-middle square is blue AND top-right square is blue, ..., AND bottom-right square is blue.” Note
that unlike 9 tossed coins, which would land “all over the place,” the inherent spatiality of the 9-block fixates the
location of the independent outcomes and thus creates unique visual identities (patterns) for each of the 512 possible
combinations. (A plausible compound-event probability problem might be the following: “A fair coin is tossed 9 times; What
is the probability that 4 heads and 5 tails in any order will come up?” An analogous problem would be, “9 fair coins are
tossed at the same time. What is the probability that 4 heads and 5 tails in any order will come up?” Understanding the
mathematical analogousness of these two situations – the sequential and the simultaneous – is in and of itself non trivial,
yet we will focus on the latter situation, where outcomes occur concurrently.)
Attending to these unique identities of 9-blocks, such as in the context of constructing their combinatorial
space, may encourage naming and classifying these combinations, e.g., as in terms of *k*-subsets. For instance, a
given 9-block, which may be interpreted as bearing a specific identity, e.g., “the 9-block with a row of three green
squares on top,” may also be interpreted as “one of the possible patterns for exactly three greens,” i.e., as a member of
the *k*-subset “all the three-green 9-blocks.” Appreciating the difference between a unique element in the combinatorial
space and a class of elements is conducive to understanding that whereas each element is equally likely to occur, classes
of elements differ in frequency (compare, for example the 1/512 frequency of the zero-green class to the 9/512 frequency of
the one-green class). In contrast, in the case of 9 tossed coins one cannot readily attend to the individual identities of
each coin, and so it is more difficult to “discover” methods of combinatorial analysis. In sum, the 9-block form is
designed to scaffold student insight into the systematicity and rigor of combinatorial analysis by serving as a template
that helps students initially see, create, and organize the combinatorial space.

We believe that there are educational benefits to the combinations tower activity even if computers are not incorporated into the unit. At the same time, the computer-based components can inform students’ strategies because computer-based probability experiments foreground the difference between rigorous combinatorial analysis and random sampling. Whereas the focus of this article is the combinations tower activity, we discuss the computer work so as to demonstrate some advantages of a mixed-media learning environment (Abrahamson, Blikstein, Lamberty, and Wilensky, 2005).

The combinations tower is a “combinatorial sample space” (see Figure 3). It is a hybrid histogram that shows not only the heights of each column as in regular histograms—it stacks the collection of the combinations themselves as elements in their respective columns (as in a Galton Box). In these columns, the combinations are grouped according to how many green squares are in each combination. A visual comparison between the heights of columns in this histogram may support a sense of the columns’ relative proportion in the sample space and therefore of expected relative frequencies in probability experiments. For instance, the one-green column has 9 permutations and the two-green column has 36 permutations, and this means that a two-green combination is 4 times as likely as a one-green combination. The underlying powerful idea is that combinatorial analysis can help us anticipate the results of an empirical-probability experiment. ProbLab is designed to foster this powerful idea linking theoretical and empirical probability. In addition, the unit links probability and statistics: the combinations tower represents the chances of sampling a 9-block combination with exactly 0, 1, 2, ..., 8, or 9 green squares, respectively, from a matrix “population” with thousands of randomly-distributed squares, where 0.5 of the squares are green and .5 are blue (see Figure 2 and see section 1.4, below).

Figure 3. A combinations tower in the NetLogo probability experiment (the complete tower is on the left, and an enlarged fragment is on the right). The tower is the exhaustive combinatorial sample space of all 512 3-by-3 arrays in which each of the nine squares can be either green or blue.

After 25 trials | After 100 trials | After 200 trials | After 512 trials | After 5120 trials | After 51200 trials |

Figure 4a | Figure 4b | Figure 4c | Figure 4d | Figure 4e | Figure 4f |

Figure 4. On top: In simulating an experiment in probability, a NetLogo interactive computer model generated 9-blocks randomly and plotted their cumulative distribution according to the number of green squares in each. As the simulation runs, the distribution tends in shape towrds the sample space from which random samples are chosen, that is, the combinations tower. On the bottom: This is a screenshot from the teacher’s computer interface that is projected onto the classroom screen. On the left is the NetLogo model that produces an occurences distribution as it runs (empirical probability), and on the right is a picture of the combinations tower produced by another model and resembling the classroom combinations tower that students built (see also Abrahamson (in press)).

a b c

d

Figure 5. Selected features of the S.A.M.P.L.E.R. computer-based learning environment.

The projected histogram shows all student guesses and the classroom mean guess, and this histogram interfaces with the self-indexing green–blue population. Note the small gap (Figure 5d, middle) between the classroom mean guess and the true population index. Because a classroom-full of students takes different samples from the same population, the histogram of collective student input typically approximates a normal distribution and the mean approximates the true value of the target property being measured. The students themselves constitute data points on the plot (“I am the 37” ... “So am I!” ... “Oh no ... who is the 81?!”). So students can reflect both on their individual guesses as compared to their classmates’ guesses and on the classroom guess as compared to the population’s true value of greenness. Such reflection and the discussion it stimulates are designed to foster opportunities for discussing and understanding typical distributions of sample means.

S.A.M.P.L.E.R. can constitute a standalone set of activities, yet the general framework of ProbLab is for students to participate in activities that interleave and juxtapose the statistics component, the theoretical-probability component, and the empirical-probability component. Activities are designed for the 9-block to play a pivotal role in students’ bridging between S.A.M.P.L.E.R. and the other pillars of ProbLab. The 9-block features in S.A.M.P.L.E.R. as samples of size 3 by 3. Students taking 3-by-3 samples from the S.A.M.P.L.E.R. population may construe the greenness of the population in terms of 9-blocks, and this interpretation may help students bridge from statistics to both theoretical and empirical probability, as follows.

Students may bridge between statistics and theoretical-probability by comparing between the S.A.M.P.L.E.R. population and the combinations tower. Specifically, students may construe a S.A.M.P.L.E.R. population as a collage from “the right side” (more green than blue) or “the left side” (more blue than green) of the combinations tower.

Students may bridge between statistics and empirical-probability using 9-block distributions: the act of sampling a 9-block from the S.A.M.P.L.E.R. population is meaningfully related to generating a random 9-block, e.g., as in the 9-Blocks interactive computer model. Both in S.A.M.P.L.E.R. and in the 9-Block model, the user expects to receive a 9-block but does not know which 9-block will appear on the interface. So students may think of the S.A.M.P.L.E.R. population as a collection of many random 9-blocks. This may support students in developing and using sophisticated techniques for evaluating the greenness of the S.A.M.P.L.E.R. population. Specifically, students may attend to each 9-block sample individually, and learn to use histograms so as to record sample values as distributions. Otherwise, students often count up all the green little squares they have exposed and then divide this total by the total number of exposed squares, in order to determine the greenness of the population. Such a strategy, albeit effective for achieving the goal of evaluating the greenness of the population, misses out on a learning opportunity, because it does not make for mathematizing the variety of samples as a distribution—it “collapses” the variation, resulting in an impoverished notion of distribution. Therefore, bridging between probability-and-statistics activities is helpful not only for building a cohesive understanding of the entire unit but also for understanding each of the conceptual pillars of this unit (see also Abrahamson (2006)).

The implementation of S.A.M.P.L.E.R. follows three stages: introduction (server only), student-led sampling and analysis (server only); and collaborative simulation (clients and server). Typically, the first two stages take between half an hour and an hour, depending on student age group. The third stage may take between one and three periods, depending on student engagement and the teacher’s flexibility in “weaving into” the PSA other ProbLab activities, such as NetLogo models, that may challenge students to reason carefully and thus deepen and enrich the discussion.

**Introduction**. The activity begins with the facilitator showing students a population of green and blue squares (the
population is entirely exposed). Students offer their interpretations of what they are seeing. The teacher then asks
students how green the population is, and students discuss the meaning of the question, offer intuitive responses,
reflect on the diversity of responses in their classroom, articulate personal strategies, and develop more rigorous
strategies and suggest how they could be implemented in the computer environment. The teacher facilitates the discussion
by reminding students of mathematical content they had studied in the past that appears relevant to students’ intuitive
strategies. In doing so, the teacher introduces mathematical vocabulary that will help students communicate during the
activity. For instance, a student might say, “It’s too much to count all of the little squares—if only we could just look
at one little place and decide with that,” to which the teacher may respond, “So you want to focus on just a sample of
this entire population of squares—how should we decide what a good sample is that will allow us to make a calculated guess
or predict the greenness of the entire population?”

**Student-led sampling and analysis**. The teacher creates a new population that is not exposed. A student uses the teacher’s
computer, which is functioning as the “server” of the activity, to take a single sample from the population. To determine
the size of this sample and its location in the population grid, the student–leader takes suggestions from classmates,
asking individuals to warrant their suggestions. Once the sample is taken, by clicking with the mouse on a selected point
in the population, students discuss the meaning of this sample in terms of the goal of determining the population’s
greenness. For example, if a 5-by-5 sample has 4 green squares and 21 blue squares, students may want first to describe
it mathematically, e.g., “The ratio is 4 to 25” (correct), and then draw conclusions from this sample, e.g., “There are
16 green squares on the whole screen, because 4/25 is like 16/100” (partially correct). Students then debate over the
location and size of another sample, further discussion ensues based on this new sample, and then more samples are taken.
The teacher encourages students to keep a record of the data and to draw conclusions from the accumulated data. For
instance, let us assume that students have taken ten samples each of 25 squares and have received the following data,
couched in terms of the number of green squares in each sample: 8, 4, 4, 9, 21, 6, 4, 8, 9, 7. What are we to do with
these data? Sum them all up? Decide that the answer is “4,” because “4” occurred more than any other number? Ignore the
“21,” because it does not fit with the others? Calculate the average - 8 - and state that 8% of the population is green?
Perhaps we should conclude that, seeing as the samples are inconsistent, these data are useless? The teacher guides
students towards effective procedures by recording all the ideas and then exposing the population and discussing with
students which procedure appears to be yielding the best results over repeated trials.

**Collaborative simulation**. The teacher creates a new unexposed population and, through the server’s interface, enables
students’ sampling functionalities. Students each take samples. The total number of little squares students may expose
is limited by a “sampling allowance,” for instance a total of 125 squares, that the facilitator sets from the server.
This allowance is “replenished” between rounds. To optimize the gain from their limited sampling allowance, students
each strategize the size and number of their individual samples as well as the location of these samples on the population
grid. Figure 6 illustrates two different strategies students often use. One
student (see Figure 6a) worked in the “few–big” strategy, spending the
allowance mostly on a single location where the student took an 11-by-11 sample (a 121-block). Students who operate thus
often say they are trying to create a reduced picture of the entire population. Some of these students choose to take the
large sample from the center of the population (and not from a corner as in Figure 6a) and say that the center is the most
representative location for the whole population. They also suggest that they can more readily calculate the proportion of
green in their samples if they take just one sample and not many. Another student (see Figure 6b) worked in the
“many–small” strategy, spending the sampling allowance by scattering samples of size 3-by-3 (9-blocks) and 1-by-1
(1-blocks) in a more-or-less uniform pattern across the population. Students operating thus often say they are trying to
cover as much ground as possible, in case there is variance in the population that could not be found through a single
large sample. Also, the “many–small” students are more likely than the “few–big” students to use averaging methods in
analyzing their sampling data. Classroom discussions address individual techniques for maximizing the utility of the
limited sampling resources and for making sense of the data.

Figure 6a | Figure 6b |

Figure 6. Examples of student sampling strategies: “few-big” and “many-small.”

At the end of each round, students use a slider to indicate their guess for the population’s greenness, e.g., 83%, and press a button to input this guess to the server. A histogram that shows all students’ guesses is thus projected on the overhead screen. Often, this histogram approximates a bell shape. The teacher exposes the population and then “organizes” it so that the population’s true value of greenness is evident. Whereas individual students may be up to 20 or more percentile points off mark of the true value, the mean of the histogram—the “class guess”—is often less than 5% away. Moreover, often no student has input the value of the classroom mean guess—this mean is indeed only the guess of the classroom as a whole.

An optional feature of S.A.M.P.L.E.R. is that students begin each round with 100 personal “points.” When students input their guess, they also commit either to their personal guess or to the classroom mean guess. Once students have input their guesses, each student has some points deducted according to the error of the guess they had committed to. For instance, based on her samples, Maggie input “70%” and committed to her personal guess. Assuming the true value of greenness turns out to be 50%, Maggie will lose 20 points. But, assuming that the class’s mean guess is 55%, had Maggie committed to the class guess, she’d have lost only 5 points. The juxtaposition of personal and pooled accuracy often engenders a pivotal moment in the activity: as individuals, students each can view themselves as a single data point on the histogram, but as an aggregate, the classroom embodies a distribution. This identity tug-of-war, “me vs. classroom,” that is stoked by personal stakes in the guessing game and by social dynamics around this game, is designed to provide opportunities for students to ground the ideas of distribution and mean.

Once the classroom guesses have been plotted as a histogram and the true value of greenness has been exposed, volunteer students go up to the front of the classroom, explain the histogram, analyze the accuracy of the classroom guess, and respond to their classmates’ questions. In particular, students share their personal sampling- and data-analysis strategies in a collaborative attempt to improve on the accuracy of the classroom mean guess on a subsequent round.

Following several practice rounds, the facilitator may challenge students by decreasing the sampling allowance so that students each have limited personal information about the population. Some local as well as classroom-level spontaneous conversation may emerge, through which students coordinate their sampling so as to maximize the total exposed area in the population (because it would be redundant to take multiple samples from the same location). If students conclude that it is better, individually, to “go with the group guess,” should the group somehow collaborate to ensure higher accuracy? Some students believe that, once a new population is created and students have taken samples, it is better first to discuss their estimations and then input guesses rather than first to input their guesses and then discuss the distribution. These students argue that by first discussing, the group can decide on a single guess and, thus, ostensibly, achieve higher accuracy. This strategy, which obviates the range and variance of the distribution, affords an opportunity to discuss properties of the distribution and recontextualize the rote procedures of calculating a mean.

With the description of S.A.M.P.L.E.R., we have concluded the introduction of the design of ProbLab. We will now turn to examine data from an implementation of ProbLab in the second author’s two middle-school classrooms.

Figure 7a | Figure 7b | Figure 7c | Figure 7d | Figure 7e |

Figure 7f | Figure 7g | Figure 7h | Figure 7i | Figure 7j |

Figure 7. Variation in individual student work on the first task of creating a green-blue pattern.

Next, we gave students worksheets with thirty-two blank 9-blocks, and asked the students to fill in as many different combinations as they are able. After some individual work (see Figure 8), students came to realize that there are many more possible combinations than they had initially estimated.

Figure 8a | Figure 8b | Figure 8c | Figure 8d |

Figure 8e | Figure 8f | Figure 8g | Figure 8h |

Figure 8. Students work on creating different combinations of the green-blue 9-block. They use personal methods that range and develop from explorative to rigorous.

We showed students a NetLogo model, “9-Blocks,” model that randomly created blue–green 9-blocks in succession (see Figure 9, for a set of screenshots that show a fragment of the interface with the virtual 9-block). Students commented that neither is the computer working according to any method nor is it keeping track of whether or not it is repeating guesses. These observations stimulated students to discuss potential methods for rigorously determining all the different combinations. This discussion also engaged students who had not been methodical in their individual work yet came to appreciate the need for a logical–mathematical procedure (algorithm).

Figure 9. These are fifteen separate screenshots from the NetLogo model Stochastic Patchwork. The model generates such
random combinations successively. The user can control the number of squares in the sample as well as the speed of the
experiment and other parameters. In this particular run there happened to be a single repetition (the 3^{rd},
9^{th}, and the 10^{th} samples).

Many students noticed how the total number of green and blue squares in the 9-block is complementary—a 7-green block is the same as a 2-blue block—and so one need not create both of these but only one of them, because they are essentially the same class of 9-blocks. One strategy students used was to create a combination and then reverse the green and blue colors. This work helped students realize that the number of different combinations with, say, two green squares is the same as the number of combinations with two blue squares. This means that the distribution of combinations itself is symmetrical (as in the combinations tower). Students who had made these discoveries became leaders who presented and explained their ideas to the whole classroom. Figure 10 summarizes some of the students’ insights. The figure shows how green and blue are complementary. The bottom row, “combinations,” represents the number of different 9-blocks in each column. At this point, at the end of the first double period, only three values have been discovered (reading from the bottom-right corner, and moving to the left): 1 combination with 9 green squares (or 0 blue); 9 different combinations each with 8 green squares (or 1 blue), and 36 different combinations each with 7 green squares (or 2 blue). The ‘AM’ and ‘PM’ captions designate which part of the construction work will be done by each of Ms. Janusz’s two classrooms, the morning class and afternoon class. It was our idea that the two classrooms collaborate, and student leaders were glad to organize this collaboration because several of them were personal friends with students in the other classroom.

Figure 10. A table of combinatorial analysis that summarizes students' discoveries on the first day at the end of the double period. Students found that there are 36 different combinations with seven green squares (or two blue squares; see bottom right corner).

Note, in Figure 10, that the PM classroom has found three values for the table – 36, 9, and 1 – whereas the AM classroom has not filled in any. This apparent advantage of the PM group is by and large an artifact of the experimental design. The first two authors were implementing this experimental design for the first time, so the afternoon classroom periodically enjoyed improved facilitation, including better organization in using available learning tools to support classroom collaboration and discussion. As it turned out, this advantage was helpful, because in the AM classroom there were several more advanced students than in the PM classroom.

Several students voluntarily led the classroom work. These were students who typically were more enthusiastic about learning mathematics. After the lesson was over, these students spontaneously approached us to discuss the activities. In the subsequent lesson, these ‘student leaders’ took on the following responsibilities:

- Design the collaborative engineering task, including the distribution of students between construction groups, with a sensitivity to their classmates’ mathematical prowess
- Assign tasks to students and explain these tasks to them
- Circle in the classroom to supervise the production of the combinations
- Manage the construction of the combinations tower

We introduced the combinations tower by challenging students to engineer a display that would help us to compare easily between the subgroups of 9-blocks that they had created (with zero-green, one-green, two-green, etc.). Also, we advised students that their display should communicate to non-participants their discovery of the symmetry of the distribution.

Figure 11a | Figure 11b |

Figure 11. Student achievement on the second day is seen in this figure: on the left is the distribution table with all the values filled in correctly and on the right is the combination tower with three complete columns on each side and four central columns yet to be assembled and built.

The vocabulary that had been developed on the previous day, for instance “anchor” and “mover,” which students had developed for finding combinations with two green squares, was adopted by many students in both classrooms. Also, students elaborated on this vocabulary so as to accommodate the more complex problems they were addressing. Thus, words served as tools for students to communicate in engineering and constructing the combinations tower.

On their third day of working on the combinations tower, Ms. Janusz’s students from the morning and afternoon classes completed the tower. In the last 10 minutes in both of the morning and afternoon classes, we used this tower to help students relate theoretical and empirical probability (see Figure 12). One student, Emma, who had been quite reticent during the unit made the following observation in comparing the combinations tower (theoretical probability) and a NetLogo model that produced random 9-blocks and plotted their cumulative distribution (empirical probability; see also Figure 4).

Figure 12. A student explaining why a probability experiment (on the left) produces a histogram that resembles the representation produced through combinatorial analysis (in the center).

We had asked the classroom why it is that the histogram “grows” to resemble the tower. Specifically, dove-tailing a student comment, we asked why it is that the 4-green column in the probability experiment was taller than other columns. Emma said:

“Maybe because there’s more of that kind of combination. Just basically, because if there’s 512 different combinations, and we know that there’s more [possible combinations] in the middle columns, [then] even though there’re duplicates, there’s still going to be more combinations in the middle columns. [The student is now using a pointer to explain what the class is watching on the screen] Even though these patterns [in the empirical live run, on left] may have duplicates in this [in the combinations tower, center] it’s still counting all the patterns, so it’s going to have the same shape…. It’s going to be the same shape, because it’s basically the same thing. Because in the world there are more patterns of these than there are of the other ones.”

Several of Emma’s classmates expressed similar budding understandings.

Next, to introduce the sampling activity feature, the facilitator used a population that was entirely revealed but for which the greenness value was not disclosed. The facilitator asked the students how one could determine the greenness of the population. Students said that, in principle, one could count up all the green squares in the population and divide this number by the total number of squares in the population. However, students said, there are too many little squares to count, making this strategy unfeasible. Other students suggested that it might be useful to focus on a single area of the population and count up the green squares in it. The facilitator reiterated this idea, calling that area a “sample.” The question on the table then became, “If we could only take a single sample, where should we take it from?” The following transcription illustrates classroom discussion about sampling.

Student 1: It would be better if there were a way to get a random spot. [for the sample]Researcher: A random spot?

St. 1: Yeah, because if you chose somewhere, you might think, “Mmm, this one has a lot of green, let’s do it there.”

Res: But what if

randomlythe computer gives me a place with a lot of green or a lot of blue?St. 1: Well, then that’s what you’ve got to guess on.

St. 2: [You should put the sample] in the middle, a little higher ... it seems a little sort of balanced.

St. 1: But that’s just what I’m saying. If you try to find something balanced, it’s going to be around 50% no matter what.

These students’ exchange reflects a pivotal quandary of statistics—is the sample sufficiently representative of the population, and what measures can we take to ensure that it is? A feature of the design that supported this conversation was that the facilitator could toggle between a view of the whole population and a view of different samples. Thus, students could gauge whether various suggested sample sets were sufficiently representative of the population. Most students did not use proportion-based mathematical vocabulary, possibly because they were not fluent in its application to novel situations. Yet, the visualization features of the learning environment enabled these students to communicate about proportionality qualitatively.

The lesson continued with students working on their individual computers. Students took samples from the population, inputted their guesses to the server, and examined results once the population and its true greenness value were revealed. The teacher worked with individual students as they participated in these activities. In Figure 13a, the teacher is working with one of the students she had listed as high achieving in mathematics. They are comparing the student’s guess for the population’s greenness with the population’s true greenness value. In particular, the student is showing the teacher that she had guessed correctly—the green–blue partition in the population is precisely where the student had indicated it would be. The student explains to the teacher her sampling strategy. In Figure 13b, the teacher is working with one of the students she had listed as low achieving in mathematics. The student had taken samples from the population and had input a guess that did not seem to reflect all the samples he had taken. The teacher is discussing with the student whether it would help for him to consider all samples the in determining the greenness of the population. These classroom data demonstrate both that the S.A.M.P.L.E.R. activity enables immediate feedback to the teacher and helps the teacher elicit a wide range of student difficulties, which she can then address in classroom discussions. Also, these data demonstrate one way that PSA integrate group- and individual work: the framework of the activity is collaborative, but to participate successfully in this collaboration, students must each achieve an understanding of the activity.

Figure 13a | Figure 13b |

Figure 13. During student work, the teacher has opportunities to work with each student.

This is a 4-block. It is empty.

- Make a combinations tower of all the different black-and-white 4 blocks.
- If a computer were making 4-blocks randomly, what is the chance of getting a 4-block with exactly 3 black squares?

Of the students who did not respond correctly, about a half were not careful enough in constructing the 4-block combination
tower, so either they left out or duplicated one or more blocks (see Figure 14,
for examples of constructions that led to correct responses). Bearing in mind the 12^{th}grade students 3% success
on a comparable item (NCES, 2004b), our 6^{th}grade students’ performance of over 50% correct indicates that this
mini-unit may constitute a contribution to helping students build meaning for probability.

Figure 14. Four examples of student constructions of the 4-block combinatorial sample space. Over half of the studnets could build these combination towers and solve correctly a probability question concerning the chance of producing a 4-block with exactly 3 black squares.

Another item asked students whether it is better to commit to one’s own guess or to commit to the group guess. That is, which of these two strategies ensures better long-term results? Students’ answers varied, and they depended on the students’ mathematical ability. High-achieving students preferred going alone, unless they were very unsure of themselves, whereas lower-achieving students preferred to trust the group guess. So the lower-achieving students were those who believed that the compiled guess is a more accurate measure of the statistical data as compared to an individual guess. This finding is somewhat counter-intuitive. One might expect that the higher-achieving students and not the lower-achieving students would be those who gain this mathematical insight. Possibly, the higher-achieving students are those who more often suffered from their classmates’ “wayward guesses,” i.e. off-mark input that resulted from incorrect analysis and not from “extreme” sample. So the accuracy of students’ individual guesses resulted both from a random factor—the specific samples each student exposed—and from a skill factor, students’ individual mathematical competency reflected in their ability to calculate a percentage.

In their written responses, all students referred in one way or another to the distribution and range of the guesses, couching these in terms of ‘left,’ ‘right,’ average, and balancing (“it evens out”). We interpret this finding as indicating that the S.A.M.P.L.E.R. activities created a shared classroom artifact that carried shared meanings, experiences, and vocabulary. Such shared mathematical images could serve as helpful anchors in future classroom discussions.

Yet another item asked students whether one should first input a guess and only then discuss the input or first discuss and then guess. Many students thought that discussing first might either confuse you or bias the group guess—that a wider distribution guaranteed more accuracy of the classroom group guess. We interpret this finding as indicating that students experienced how an aggregation of random outcomes can nevertheless effect higher accuracy than would a “centralized command” (see also Wilensky (1997, 2001); Surowiecki (2004)).

Finally, students varied in what they considered to be a “good guess.” Some students were happy to be several percentage points off the true value, whereas other students were more critical of their guesses (for a more detailed report on students’ spontaneous sampling strategies, see (Abrahamson and Wilensky 2004b).

Our design of curricular material for technology-assisted middle-school mathematics-and-science classrooms is an ongoing project that is constantly informed by implementations in classrooms. Students’ engagement in our activities—their high levels of participation, excitement, and feedback comments—encourage us to improve these activities and research them in more classrooms and with more teachers. One direction that we find promising is using the combinations tower activity so as to help students build meaning for the statistical concept ‘normal distribution’ (bell-shaped curve). We will now explain the rationale of this future work.

We are all familiar with the bell-shaped (‘normal’) curve that characterizes many phenomena in science, biology, social
sciences, evidence-based medicine, etc. For instance, if we were to measure the heights of all 6^{th}-grade female
students in the U.S. and plot these as a histogram, this histogram would be bell shaped. But why do these phenomena fall
into this distribution? How can we make sense of this? (see also Wilensky
(1997), for an analysis and meta-design solutions
for students’ difficulty with this concept). The combinations tower may serve as a clue or ‘model’ for beginning to tackle
this puzzle.

For the sake of clarity, let us assume a grossly simplified scientific model that may then serve as a conceptual basis for understanding real phenomena that are complex. That is, if we accept this model as basically sound, we can then examine each of its specified assumptions so as to evaluate whether, how, and why the model falls short from representing the more complex reality. We would then explore how we may adapt and enrich this model so as to make it more general without losing its basic coherence. Such adaptation will possibly touch upon profound scientific ideas, because the model would give students a handle for articulating their intuitions mathematically.

In this basic mathematical–scientific explanatory model of height distribution in a large population:

- Only nine variable factors, genetic and environmental, contribute to determining a person’s height, such as ‘parents’ height,’ ‘nutrition,’ ‘climate,’ etc. These nine variables map onto the nine squares in the 9-block.
- Each person is randomly assigned values for each of the nine variables.
- Instead of looking at a range of values per variable factor, we will assume that each variable can only be either a ‘positive’ (green) or a ‘negative’ (blue).
- Each variable value has an equal likelihood (0.5 probability) for positive or negative, e.g., ‘nutrition’ could equally be either ‘good’ or ‘bad.’
- Each of the nine factors contributes equally towards determining a person’s height, e.g., ‘good nutrition’ would be as powerful as ‘tall parents.’
- The more positive values a person has, the taller this person is, along some interval scale.
- The variables do not interact. For example, the contribution of a positive value for ‘climate’ is not dependent or affected by the value of any of the other variable properties.
- The range of the distribution of heights is limited both on the minimum and maximum (there is a “cutoff” standard deviation from the mean).

Given the above assumptions, the combinations tower is the sample space of all combinations of the nine variable factors contributing to a person’s height, and these combinations are organized by “height groups” from ‘shortest students’ (on left) to ‘tallest students’ (on right). The leftmost ‘shortest students’ column and the rightmost ‘tallest students’ column each hold a single combination, the no-positive and the all-positive, respectively. In between, there are more combinations that yield a count of 4 or 5 positives as compared to, say, 3 or 6 positives, and there are more combinations that yield a count of 3 or 6 positives, as compared to 2 or 7, etc. Yet note that each of the 512 combinations is equally likely to occur. Thus, the bell curve can be understood as a combinatorial sample space of a cluster of variables that has been detected as contributing to a property of an observable phenomenon. Each variable is independent of the others, but as a cluster of variables that inform a property of a phenomenon, these nine variables are co-dependent, and these co-dependencies create the bell-shaped combinatorial distribution. This is why instances of a phenomenon are often distributed such that there are more “average” incidents. For example, there are more people of average height than there are short people or tall people.

By way of demonstrating how this model may be complexified, consider the dichotomous variable of each square in the generic
model. If we were to increase the space of possible outcomes to 3, the combinations tower would grow to encompass
3^{9} = 19,683 different possibilities. The shape of this tower would be closer to a bell shape. If we were to
modify the relative likelihoods or weight of individual squares or if we were to introduce causal contingencies between
squares within the 9-block, we might affect the shape of the distribution. Implementing this model as a computer-based
interactive simulation would enable us to readily explore the parameter space, tinker with the procedures underlying the
emergent distribution, and receive immediate feedback in the form of mathematical representations. By way of comparing
these simulated experiments to information from scientific and statistics resources, we can evaluate the explanatory power
of our model and iteratively modify the model toward a better fit with the data.

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Dor Abrahamson

Graduate School of Education

University of California

Berkeley, CA 94720-1670

U.S.A.
*dor@berkeley.edu*

Ruth M. Janusz

Nichols Middle School

Evanston, IL 60202

U.S.A.
*JanuszRM@aol.com*

Uri Wilensky

Center for Connected Learning and Computer-Based Modeling

Northwestern University

Evanston, IL 60208-0001

U.S.A.
*uri@northwestern.edu*

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