## 9-Block Stalagmite

9-Block Stalagmite is one of several interactive models for probability and statistics authored in the NetLogo modeling-and-simulation environment. The model is part of ProbLab, a curricular unit designed to enrich student understanding of the domain. The online unit package will include a suite of models, student worksheets, and a teacher guide. Below is an applet of 9-Block Stalagmite. You can interact with this model by changing the slider values and switch settings and then pressing Setup and Go to run this model under different settings. You may want to slow down the model in order to get to know it -- use the ADJUST SPEED slider that is on the top-left corner of the graphics window. For more details, please see the model itself in the NetLogo library. Note that this model is still under development and is yet to undergo our rigorous checkout procedure. Also, the size of the model below has been reduced from its original size so as to fit your browser window.

CM ProbLab: 9-Block Stalagmite -- bridging theoretical and empirical probability
Don't see nothin'?

Gist

Samples are generated randomly and then drop down into a column that reflects the number of green squares in the sample. For instance, a sample with exactly 4 green squares (so 5 blue squares) will "travel" to the column that has a "4" at the bottom, and then slide down the chute. You can control the size of the sample, and therefore of the combinatorial sample space of all possible samples of that size, with the SIDE slider. For instance, a setting of 'SIDE = 3' will give you samples of size 3-by-3, that is, a sample containing 9 little squares. We call this a "9 block."

Try running the model under the two possible settings of the KEEP-REPEATS switch -- 'Off' and 'On.' When the switch is on, samples will stack up even if they are repeats of samples beneath them in that column. When the switch is off, a repeat sample will vanish the moment it lands on the top sample in its column. You can toggle between building the combinatorial space per se (keep-repeats? off) and the experiment (keep-repeats? on) of randomly generating samples and watching how the distribution roughly replicates the shape of the combinatorial space. Note that in the interest of viewing convenience, the graphics for this applet have been set to a size that cannot accommodate the entire sample space for SIDE that is bigger than 2. If you have downloaded NetLogo, you can adjust the size of the graphics window in this model.

9-Block Stalagmite is a rich model that can serve as a kick off for conversations pertaining to key ideas in the domain. For instance, for settings of 'probability = 50%,' 'SIDE = 2' (4-block samples), 'KEEP-REPEATS? Off,' and 'STOP-AT-ALL-FOUND? On,' the model will run until it has found all 16 permutations of the combinatorial space. But...

• Riddle #1: How many random samples should the model take, on average, until it finds all the permutations? Let's begin by considering the case of a 50% chance of being green.

• Riddle #2: What is the range of probabilities at which a 2-green block is the most prevalent in a randomly-generated 9-block?

Perhaps the most interesting relation to explore in 9-Block Stalagmite is between the setting of 'probability' and the mode in the histogram. Why is it that the mode is in the column around p*N'? Consider the case of 'SIDE = 2,' that is, the 4-block. It is pretty obvious that for a setting of probability-to-be-target-color = 0, we'll get a mode of 0 green -- every square will always be blue, and so every single 4-block will be completely blue. Likewise, we'd be surprised if a probability-to-be-target-color = 100 did not give a mode of 9 green -- all 4-blocks are completely green. But what about probabilities in between 0 and 100%? That's where it gets fuzzy. But fascinating, too. Setup the model with the SIDE at 2 and the 'probability' at 70%. Press Setup and then Go. Note that the %-target-color tends to 70%. Also note that more sample combinations collect up in the column of 3 green patches than any other column. So '3' is the mode. We have found it interesting and perhaps somewhat confusing to shift between thinking about the 'mode' and thinking about the 'mean.' For instance, if you set the PROBABILITY to 83%, you might think that since 83% of 4 is 3.32 then the mode should still be '3.' Is it? What is going on here? So for some probability setting, the 3-column will most often get to the top before other columns. For other probability settings, the 4-column wins. Try to determine the probability ranges in which each column "rules." Is this trivial? (See Riddle #2, above)

We believe that this phenomenon is only ostensibly obvious. The compelling match between .70 chance at the micro level of each square and .70 at the macro level (see the %-target-color monitor) may hide the inherent complexity of the situation. The micro-to-macro congruence indexed by the near overlap between parameter settings and experimental outcomes deceivingly suggests a certain closure. That is, the micro-macro congruence may appear too trivial a connection to be stated, let alone explored. Yet, it is this micro-to-macro phenomenology that we wish to problematize. Such is the double-edged sword of computer-based models (see Abrahamson, Berland, Shapiro, Unterman, & Wilensky, 2004): For learners who are not fluent with designing computer-based experiments and authoring and modifying code, computer environments and, specifically, computer-authored and run simulations, implicitly signify an ineluctable authority. Moreover, the input-output fit lends a 'makes sense' feeling that dis-invites further inquiry. At best, the learner might remain with a superficial insight into a relation, but never begin to ask hard questions concerning the why and how that underlie this relation. We are currently working on various activities both to evoke student interest in this central phenomenon of probability and to help students understand -- both at an intuitive and formal level -- why/how this phenomenon obtains.

[last updated July 8, 2005]