Many decisions benefit from understanding probability, e.g., when a patient must interpret the meaning of a medical test result or when a politician must weigh the costs and benefits of a particular policy. Unfortunately, Tversky and Kahneman demonstrated that everyone, even professional statisticians, suffer from systematic biases in their intuitive judgements of probability. Students make a variety of identifiable mistakes when solving probability problems and even graduate students who plan to teach mathematics retain strong misconceptions.
The challenge is how to help students develop an intuitive grasp of these abstract concepts. We are particularly interested in combinatorics, a branch of probability that deals with the enumeration, combination, and permutation of sets of elements and their mathematical relationships, because it results in a combinatorial explosion: even simple problems result in hundreds of possibilities that cannot be represented simply with physical objects, virtual or otherwise.
The original motivation for this project stemmed from observations of students in a university-level course in combinatorics. Faced with only paper and pencil, many had difficulty developing intuitions about probabilities and suffered from the ‘stereotype threat’ that they are poor in math. We hoped that letting students manipulate concrete objects while simultaneously observing the corresponding changes in deep structure, e.g. a probability tree, would reinforce their intuitions about the underlying mathematical principles. Our goal was to create an engaging and playful environment that avoids excessive mathematical notations and encourages discussion.
Combinatorix offers a novel approach that combines tangible objects with an interactive tabletop to help students explore, solve and understand probability problems. Students rearrange physical tokens to see the effects of various constraints on the problem space; a second screen displays the associated changes in an abstract representation, e.g., a probability tree. Combinatorix supports several input techniques: a camera detects the location of fiducial markers and a wiimote provides the position of multiple infra-red pens. A projector displays additional information around the tangible objects. The interactive surface is 60 x 45 cm. and can accommodate up to four students at the same time.
The underlying application is written in Java and uses the Reactivision engine to detect fiducial makers. Additional libraries, e.g., wrj4P50, communicate with the wiimote. The system is modular and can easily accommodate the creation of additional operators for constraining the sample space.
The current version displays two kinds of information: first, the tabletop interface shows a specific number of placeholders for objects. Letters can be placed on those spots to form a new combination. At the same time, the remaining number of letters for each step is displayed on top of each placeholder. A second screen displays a probability tree reflecting the current state of the problem. Letters can easily be replaced by other elements, including virtual, laser-cut and 3D-printed physical objects. Combinatorix supports up to 10 tangible objects and 20 virtual ones.
A user study revealed that the best way to implement Combinatorix in a classroom is to use the system as a way to prepare students for future learning. We found that students who used this tabletop learning environment before attending a lecture learnt more than students who first attended a lecture and then used Combinatorix.
Thus our findings suggest that educational designers should carefully think about how they want to implement their systems to existing classrooms. Choosing the wrong sequence of activities may impede students’ learning, whereas adopting a constructivist perspective is likely to foster knowledge building.