Inclusive Math: A Classic Bayes Theorem Example

Last week, news broke about how Florida education officials banned 54 math textbooks for including "woke content", which many rightfully criticized as nonsensical. Many of the cited examples of issue appeared as context motivating specific math exercises, not that the math problems themselves were "woke". In fact, a common argument I saw among people was "how could math itself be woke? Isn't it objective?"

I was pretty surprised. Not just at how ridiculous some of these examples were (one of the most commonly cited examples was simply the mention of the word "racist" and the Implicit Association Test), but at the fact that examples I thought would have been banned were pretty much ignored. All of this has been an exercise in seeing how much content these folks can get away with censoring and how much fake hubbub they can generate to anger a misinformed public. It's frustrating, to say the least.

As an educator who does apply critical race theory to my work (and not the fake version of CRT that Republicans have made up), I'll save my disdain for the right's harmful attempts at censorship in a different post. For this post, I wanted to focus on the fact that there are math examples that can be more or less inclusive based on one's beliefs about the world. I don't think that math is objective – math is a system that humans have created as a language to understand our world, and it's directly influenced by culture (and our philosophical or physical beliefs about reality). An easy example to point to is the fact that there is no zero in Roman numerals, because their system of math didn't call for it. It didn't mean that they didn't understand the concept of 0, just that it wasn't used in their system of math; it wasn't until the introduction of the Hindu-Arabic numeral system that we use today was introduced to Europe that we began commonly using 0 as a mathematical numeral that could be used in mathematical operations. Throughout history, as we needed more maths to better describe the world around us, we adopted more maths – Newton needed to invent calculus to work with his physics of instantaneous change, and we realized with quantum mechanics that our traditional propositional logic wasn't sufficient to describe the phenomenon that we now began to observe. Math is shaped by us: our culture and our needs.

The math example I want to discuss in this post isn't very profound. I'll leave the philosophy of math question of "Is math invented or discovered?" for a different time. What I will be talking about, though, is how our conception of the world can influence the set-up of a specific math problem and lead us to different answers.

Context for this math example

This is a math problem that falls within the domain of probability. Specifically, it deals with conditional probability, which you might encounter in an Algebra II or Pre-Calculus class in high school. At the college level, this example is often used to help motivate Bayes' Theorem, which gives us a useful equation to deal with conditional probability.

I encountered this example in teaching Bayes' Theorem as part of an introductory course on artificial intelligence (AI). In the AI world, Bayes' Theorem is central to a variety of algorithms, with one of the most commonly taught being the Naive Bayes Classifier. (The classic example that we apply Naive Bayes to is if we want to sort new mail as being spam or not spam. This algorithm, which uses Bayes' Theorem, helps us do this classification task in a simple, or naive way. This classification task isn't what I want to talk about, but I wanted to provide it here in case you wanted a brief idea of a real-world application of this algorithm.)

All of this to say: whether you're in a high school math course, a college probability course, or an introductory AI course, you'll likely run into conditional probability, and this particular example is likely to show up. The way it's typically discussed isn't very inclusive, and teachers should be aware of this.

So what's the example?

Here's how the math problem is usually worded:

Many of you probably understand the issue with this math problem already.

Typically, the way this problem is solved is that you assume there are only two genders. I mean, we can take a step back and look at an easier math problem that runs into the same issue:

This is a screenshot that I took from a solution sheet that I found on the Internet (I don't know what the class actually is, and I'm not trying to put heat on the person who wrote this). As a non-binary individual, this was a math problem that I found frustrating to answer when I encountered it in college. The way it's worded, it's exclusive to non-binary genders, and though many people might read it as an innocuous math problem, to me and many others it reads as a belief statement or assertion about gender identity, which is a hot topic today that horrible individuals around the world have been loud and wrong about. When I saw the math textbooks in Florida being banned, I thought it was because of math problems like this and proposed solutions to make it more inclusive.

For anyone who's curious, we can take a look at the steps to solving the problem to understand where the issue arises. When asking what the probability is that both are girls, you need to know what the probability is that one child is a girl. The common answer is 1/2. "There is a 50% chance that a person is a boy, and a 50% chance that a person is a girl." Except that's not true, even when you look just at the existence of intersex individuals. For the purposes of our math classes, we make a generalization that's by definition exclusive. In order to answer this question with a solution of 1/4, you first need to accept a view of the world that assumes P(boy) = 1/2 and P(girl) = 1/2. And for all the hubbub that heteronormative people make about how their math textbooks are too political, this right here is an example that reminds us that math has always been political.

Even a math problem as seemingly innocuous as this has political and social components. When a non-binary individual reads in the solution "Hopefully you said 1/4", it's a bit of a slap in the face: "Hopefully you only think there are two genders."

Part of my work as an educator and curriculum developer is to create inclusive content for my students, and these two examples and their solutions are far from inclusive. It's not just a math problem. It's a commitment to a belief that excludes your non-binary students and invalidates their existence. Sure, it might not be that deep – some of us probably won't care and just finish the math problem anyways. But it still makes a difference when we see a teacher who does care and does validate our existence.

It's not just about the politics of a math example and the optics of it at an ideological level – it's about the real impact of these problems to real students.

Possible Solutions

In running into this example, there are two solutions that I have adopted in the past:

1. Just don't use this example. There are other examples of conditional probability that you can use, such as coin flips. (Notice that in this case, students always ask "But what if the coin lands on its side?" and teachers have to say "Assume that doesn't happen" – how come we're comfortable having students raise issues with our assumptions here, but not when gender is involved?)
2. Reword this example. "A family has two children. Assuming each child identifies as either boy or girl, what's the probability that both are girls." This simple rewording changes the context and assumptions of the question so that it's not strictly assuming a gender binary. It's not perfect, for a variety of reasons. How come we're only considering binary gendered individuals in the example? is the question that comes immediately to mind. And of course, the queerphobic crowd would lose their minds at even the sight of the phrase "identifies as", even if the identities described are binary gendered. But for that reason, I do think there's value in pursuing this solution. Instead of avoiding the conversation altogether by talking about something like coin flips, directly addressing this example gives you space to talk about things like gender identity in the classroom. It allows you to affirm the non-binary students in your class, while being intentional about how assumptions in math matter. Especially when I'm teaching an introduction to AI course, conversations like these about gender identity are crucial, because so often computer scientists will develop algorithms based off statistics, based off harmful and invalid assumptions that were never challenged in the classroom. This gives an opportunity to stop and point them out. That's why I like this example so much.