The marble scooper is an empirical-probability device for sampling a fixed number of marbles out of a vessel containing many marbles of two colors, e.g., a 4-Block marble scooper may draw a sample of 3 green marbles and 1 blue marble out of a box with equal numbers of each color. The scooper is a unique stochastic object—its "urn" mechanism is designed to help learners decouple the stochastic object from its outcomes, its simultaneous sampling problematizes issues of independence, and it's inherent ordering of independent trials scaffolds critical attention to permutations. [read more]

The combinations tower, a theoretical-probability artifact, is the distributed binomial sample space of a stochastic device from the X-Block family, such as a 4-Block, a 2-by-2 grid in which each square can be either of two colors. The 16 unique configurations of the 4-Block are arranged in the tower according to the number of white (green) cells in them, resulting in a 1-4-6-4-1 formation (the corresponding binomial coefficients). This is the anticipated shape of the outcome distribution from experiments with the marble box containing equal numbers of marbles of each color.

The 4-Block Stalagmite NetLogo model is the interactive computer-based embodiment of the 4-Block mathematical object, simulating experiments with the marble box. Students are to bridge empirical and theoretical aspects of probability by constructing empirical distribution as a dynamic "stochastic stretch" of the sample space. Randomly generated 4-Blocks descend into columns according to the number of green squares, converging on a 1-4-6-4-1 stalagmite formation that can be re-sorted and/or colored to visually enhance connections to the sample space. [read more]

The 4-Blocks NetLogo model, too, is an interactive computer-based embodiment of the 4-Block mathematical object, simulating an empirical probability experiment in which the randomness generator is a compound of 4 squares that each can independently be either green or blue. The model helps conceptualize relations among theoretical and empirical aspects of the binomial function: combinatorial analysis (what we can get) and experimentation (what we actually get). A unique within-column stratification feature helps monitor accumulations of specific sample-space outcomes. [read more]

Histo-Blocks is a theoretical-probability NetLogo model for exploring the binomial function. The stochastic object is a "4-Block," a 2-by-2 grid, in which each of the four squares independently can be either green or blue. Through several dynamic and interlinked displays, the model shows connections among the sample space of the stochastic device, the probabilities of independent outcomes when operating this device under different p values, the probability of events, and the expected outcome distribution in experiments with this device. [read more]

Dice Stalagmite is an empirical-probability NetLogo simulation of an experiment with a pair of dice. It constitutes a complementary activity to conducting combinatorial analysis of the sum of a pair of dice. In this simulation, the outcome of each trial is a single roll of two dice. However, this double roll is treated simultaneously, and separately, as two different events: either a pair of dice (one pair per trial) or a single die (two trials at a time), and these two emerging distributions are juxtaposed dynamically. [read more]

The Mathematical Imagery Trainer (MIT) is a mechanical device for inducing a mathematical image for proportional progression, e.g. the progression 2:3, 4:6, 6:9, etc. Using two pulley wheels with 4 and 6 inch diameters, the device raises the user's hands while maintaining a 2:3 ratio of the distance from the two hands to the table. This device is a prototype for a computer-enhanced apparatus that will be used in the Kinemathics project. [view video] [working paper]