Throwing the Bones and Building Their Models

In this blog post, BCcampus 2020–2021 Research Fellow Rob-Roy Douglas of Northern Lights College discusses his experience incorporating Indigenous approaches to learning into an online statistics course.

I have always used experiential learning in my statistics courses. In face-to-face classes, this could be as simple as bringing some tape measures or a roll of coins to the classroom (students can – and do — learn a lot about probability from coin flipping). I still do this online, both in demonstrations and by getting students to measure things themselves and flip their own coins. For this project I decided to introduce the students to a more advanced approach to probability and Indigenous culture: rolling the bones.

Games of chance have a long and interesting history among Indigenous peoples, and some of the most interesting are played with talus bones. Traditionally called astragalus bones, these are small tetrahedral bones found in the shins of a wide variety of animals. First Nations used astragalus bones from deer, bison, elk, moose, and even wolves both to play games of chance and to divine the future, for instance to predict the success of a hunt. Astragalus bones have been used by both Cree and Inuit shamans in their rituals. And until quite recently, children used them to play simple games of chance.

Each bone has four sides it can fall on: a narrow flat side, a narrow hollow side, and broad concave and convex sides. Each has a different probability of turning upward, and we can model those probabilities. You can be sure that Indigenous gamblers did too, at least informally, by observing which sides tended to fall upward over many throws of the bones.

Astragali are also used by virtually every traditional society in the world for similar purposes. Even the old-fashioned children’s game of jumping jacks was traditionally played with astragalus bones. Among Indigenous peoples, Europeans, and Asians, dice are essentially the descendants of astragalus bones – but with some important differences.

Dice are regular and reliable (as long as they are honest). We can build a model for a die that applies to all honest dice. Astragalus bones are less predictable — but more interesting for it. The outcomes of throwing the bones are much more variable, so the skill of the person throwing them and their understanding of the odds are much more important.

I introduce students to the bones as a way of illustrating probability models. We can model the probability of which way an astragalus bone will fall in the same way we can build a model for how a die will fall. A die is simple, since each side has exactly the same probability of falling upward. The model for a single die looks like this:

What does this mean? It means there is an equal chance for any one side of the die to land face-up: one chance in six. When we add up all the probabilities of each face, we get all the possible outcomes of a throw of a die – which adds up to a probability of one, or 100 per cent of all the possible ways the die can fall.

But astragalus bones have only four sides (they cannot stand on their ends on a flat surface), and the probabilities for each side vary – both among bones from different species of animal and even between different individual animals. The best model for throwing an astragalus bone looks something like this:

Why does this model have only four sides but probabilities of one in 10 or four in 10 for individual sides? Because the probabilities of each side falling upward are very different. There isn’t one chance in four for the narrow flat side to come up but only one chance in 10. So if we threw the bone 10 times, we would expect most of the time that one of those throws (maybe) would come up narrow flat (and one narrow hollow, but four broad concave and four broad convex).

Because this is a model, the total of all the probabilities still adds up to one.

Bones are interesting because they are traditional; they have enormous cultural importance to Indigenous peoples in the Americas and elsewhere in the world. They are also hard to model. They help illustrate how probability models work and how Indigenous peoples worked with probability on a practical level – by understanding how to calculate the odds of throwing bones, whether in a game or to predict the future.

I integrate the above information into a lecture in which I discuss the history of chance, the evolution of gambling, and the emergence of dice. First Nations peoples developed artificial dice as successors to astragalus bones before contact with Europeans. I have students model throwing dice and then throwing astragali. Then I show how probability models can be applied to the bones and why a model for an astragalus bone is different from one for a modern, plastic die.

Integrating Indigenous and experiential learning into an online statistics course is challenging, especially given how much material we cram into a term. I feel strongly that it is as much the approach to learning as the content of the material that counts. Traditional Indigenous learning was through narrative storytelling and experience. I also feel strongly that this is the approach to learning that actually works for most students.

This research is supported by the BCcampus Research Fellows Program, which provides B.C. post-secondary educators and students with funding to conduct small-scale research on teaching and learning as well as explore evidence-based teaching practices that focus on student success and learning.