Ways of Thinking, Not School Subjects
“Subjects” like math and history are less subjects and more ways of thinking.

But how will they learn math, or history, or science?

Like others involved in Self-Directed Education, I hear this a lot. It is a good question that has taken me a while to formulate a response to. Something about the question itself has always struck me as off, and I’ve finally put my finger on it.

I think the concern arises because we wrongly think of things like math, history, and science as subjects. But they are not. They are ways of thinking, and once we treat them that way, the answer to the question becomes more obvious: self-directed learners learn things like math and history when they encounter things they want to do that require those ways of thinking.

Here’s what I mean. We tend to think of math as a subject of study, a body of knowledge that, to learn, we must deliberately sit down and study. (First, you get math instruction, then you can do math stuff.) But we’d do better to think of math as a skill: addition, division, and algebra are really just ways to manipulate numbers, not discrete objects of study. (First, you do stuff that requires the manipulation of numbers, and you learn math in the course of doing those things.)

When we think of it that way, it should become clear that we can’t get very far in life without needing to think mathematically, without the ability to manipulate numbers. Cooking, counting money, shopping, measuring, planning the day’s schedule, even video games (which are often point-driven) are difficult to impossible without being able to work with numbers.1

flour and milk
Photo by pfctdayelise

People learn math when they are confronted with activities they want or need to do that require the manipulation of numbers. Those activities are all around us. Think of a child who is learning to cook. It could be that they get to a point (maybe, wanting to double a recipe that requires 2/3 cup of flour), realize they don’t know how to manipulate the numbers, and that causes them to ask others (or do an internet search) to help them learn the math. It could also be that they are learning with folks who already know how to cook and they learn by watching others perform those operations. Either way, the learning of math is incidental to, but necessary for, learning to cook.

History is another example where the mistake is thinking about it as a subject – a collection of knowledge – rather than a skill. But what is history except for the ability to tell a story about how something has progressed over time? People get into history when they encounter something they want to explain in a historical way. Someone may be into cars and wonder how cars have changed from fifty years ago, and this may lead to the question of when cars were invented, why, or how? The same can be asked of computers, Barbie dolls, or anything that has a history at all. (In fact, I recently binge-watched a Netflix show called The Toys That Made Us, each episode about the history of an iconic toy. Not the kind of history school calls history, but history nonetheless!)

Of course, looking at history or math this way still doesn’t guarantee that learners will learn the facts about the American revolution or how to do algebra as written equations. Should learners be made to learn these types of things that maybe experience alone might not to teach?

I think not. Let’s address the math first. School math just doesn’t look like the kind of math we generally use in the “real world.” This is illustrated in a book-length study by anthropologist Jean Lave. She followed people around (with permission!) to see how they solved real world math problems – things like figuring out how people find the best buys in supermarkets or calculate points in Weight Watchers meetings. What she found was that how people solve math problems in daily life was so different from school math that if she presented the same problem they just solved but on a worksheet, they often couldn’t do it. Here is Lave’s conclusion, in academic jargon: “The rules for transforming problems in school lessons, learned as formulae, mainly by rote, seem very different from the ubiquitous and successful transformation of problems in the supermarket.” Translation: math as a skill of manipulating numbers, a real-life method of problem-solving, just looks different (and more “real world”) than math as the ability to do math worksheets.

What about ensuring students learn about the French Revolution, who all the American Presidents were, or the qualifications of a Senator? I’ll be honest. On more than a few occasions, I’ve completely forgotten historical facts I am sure I learned in school, only to look them up when I needed them. I’d argue that the reason we learn history in school should not be so that we remember historical facts, but so that we become familiar with how to think historically (how to read historical documents, construct and think about historical accounts of things, etc).

One problem with the way history, science, and math are taught in school is that they are usually taught less as ways of thinking and more as bodies of knowledge. To illustrate, psychologist Susan Engel described an instance she observed in a middle school science classroom. The students were doing a lab where they had to work through instructions using lab materials. One group veered off task and started playing – literally, experimenting – with the lab materials on their own. Engel reports that the teacher kindly told her students to get back on task, that she’d “give [them] time to experiment at recess. This is time for science.” For this teacher, science is a body of knowledge to be given to students based on scripted labs, not the process of experimenting!

As another example, several mathematics education professors wrote an article that describes the type of “mathematical habits of mind” that math instruction should be providing, but isn’t. Like Jean Lave, they observed that, “for generations, high school students have studied something in school that has been called mathematics, but has very little to do with the way mathematics is created or applied outside of school.” How should schools fix the problem? Rather than treating math as a body of knowledge to be mastered, they recommended focus on mathematical “habits of mind.” “Such a curriculum lets students in on the process of creating, Inventing, conjecturing, and experimenting; it lets them experience what goes on behind the study door before new results are polished and presented.”

The problem is that even though many schools try to do this – treat these areas as ways of thinking rather than subjects – conventional schools aren’t well designed to do this. It is very hard to teach and test for things like creativity or the ability to conjecture and experiment. (Tests can ask students questions that try to get at these things, but generally, there is an expected “correct answer,” so teacher and student quickly learn to focus on learning those).

Self-Directed Education, on the other hand, confronts learners with real problems – problems that involve mathematical, historical, and scientific thinking – that allow learners to acquire and hone these ways of thinking. One learns to think mathematically less by doing worksheets and math games than by confronting and puzzling through real problems, like how to double a recipe that calls for 2/3 of a cup of flour. And one learns how to think historically less by listening to studying and remembering facts than when one gets curious enough about something to wonder and figure out how it got to be that way.

When we worry how self-directed learners will learn math, history, or science, we are often thinking about these things as subjects that need deliberate study rather than as ways of thinking. As ways of thinking, students will learn science, math, or history when, in the course of living life, they have occasion to need those ways of thinking.

[1] To see this, try to go through a day being conscious of all the times you manipulate numbers. If your experience is like mine, you will discover many instances you’d otherwise have been unconscious of. Yesterday, for instance, I had to take some cold medicine; the directions told me to take two pills every so many hours, not to exceed x number of pills per day. I also listened to a podcast whose duration was over an hour long, but knew that I was scheduled to do a particular thing in less than one hour; I had to think about how much of the podcast I could listen to before my scheduled activity. To many adults, these encounters with math are subconscious, but the more we become conscious of these mathematical ways of thinking in our daily lives, the more obvious it should be that kids truly will encounter math in many places.