Good Math Debt

...part two

I was tempted to conclude my previous post with a stern finger-wagging about the perils of math debt¹. But then I thought, “Debt isn’t all bad in regular life…why should it always be bad in math?”

In fact, if you aren’t taking on any debt, you may be making a serious error. I owe the bank hundreds of thousands of dollars right now, and I have no intention of paying the money back any time soon. Here’s my reasoning: 

1. The interest rate is low. I’m paying less than 3% for the loan. In theory, I could take that money and put it in index funds, where money has historically earned much more than 3%. 

2. I get the benefits of an asset (my house) that I can’t afford to buy all at once. My family and I get to live in the house all day every day, but we only have to pay a small fraction of the cost each month. 

3. The loan structure encourages me to invest in a valuable asset (my house). I pay interest each month, which is not ideal, but I also pay off some of the loan’s principal. If I keep making my payments, I’ll eventually own the house.

The same reasoning applies to good math debt. It’s worth taking on debt if the interest rate is low, the underlying asset is valuable, and you commit to paying off the debt over time. 

The truth is, we don’t have the intellectual cash to learn big ideas all at once. I’ll go a step further and argue that we never fully understand important concepts – we start with toy models of them and upgrade to better and better models over time.

Consider the path the Sun traces through the sky. If someone asked me why it does that, I could come up with a decent explanation. I’d say something about the tilt of the Earth, its daily rotation, and its orbit of the Sun, and I think I could show why these elements create the illusion that the Sun is moving through the sky. But I don’t fully understand the geometry. Suppose there were two Suns in our solar system. There is no way I could predict the arcs of those Suns through our sky. Nor could I deduce the tilt, orbit, or rotation of the Earth if I was allowed to observe those Suns’ paths. I have some understanding of what’s happening, and that understanding is useful up to a point, but I don’t *fully* understand it.

The same is true of students who are learning important new math in school. For example, it’s impossible to fully understand linear equations all at once. You have to work with them, see them in action, apply them in lots of different contexts. And while you’re doing that, you may need to use some shortcuts – maybe “rise over run” for slope, or “that number at the end of the equation” for the y-intercept. Of course, these shortcuts will collapse under the pressure of calculus tests, business forecasts, and robot competitions. But they are easy to apply to introductory questions – they get you started without requiring too much up front. And if you spend some mental energy attempting to acquire a better understanding each time you work on a new problem, you’ll, in effect, pay off the principal. You won’t need to pay the price of using cheap tricks. You’ll own a robust mental model, and with this new intellectual asset…you’ll be poised to borrow even more good debt.

I also see this process at work in my understanding of probability. I’m confident within a very narrow range of problem types – permutations, combinations, expected value…basically anything that shows up on the SAT or ACT and a few techniques that have been useful in Mathchops data analysis and fantasy football. But outside of that, I really just have toy models. I can describe the basic ideas behind Bayesian reasoning and Monte Carlo simulations, but I couldn’t apply them rigorously in the real world. And there are many, many other concepts that I’ve barely heard of and know next to nothing about (Markov models?!). 

But the little knowledge I have has been tremendously useful. I’ve been able to help hundreds of students with their tests, design the games and question selection algorithms for Mathchops, and waste countless hours on fantasy football analysis. I’m using shortcuts, procedures, and generally suboptimal methods throughout, but I’m also learning more about probability each time I work on a problem or project. In effect, I’m living in the house while I pay down the principal. Eventually, I’ll own the house. And after that…who knows? Maybe one day I’ll be able to afford Markov models.

1. In my analogy, math “borrowing” occurs when you want to use a concept but don’t fully understand it. You pay “interest” in the form of applying brittle, inflexible procedures, such as “is means equals and of means multiply” when solving percentages, or “When I get to this part, I divide.” Mental energy that might be spent learning a concept is instead spent remembering and executing a series of steps (“paying interest”). If you try to move on to harder concepts while using the same suboptimal procedures, you’ll eventually go bankrupt (have no idea what you’re doing, fail your tests, hate math).