Let’s spend some time today talking about mathematics and why. To me, this is one of the most overlooked components of the modern American curriculum, and the stakes are high. What is at risk is not whether our future citizens will be able to calculate tips in their heads… why would anyone think that is important in a world where we walk around with computers in our pockets? No, what is at risk is the intellectual orientation of our democracy: will citizens be able to analyze large datasets well enough to understand outcomes of policy and affect change in a world that is increasingly disparate in net-worth distributions and political power holdings?*
But let’s take a huge step back here. The goal is to further the depth of understanding** of the questions: Why do we teach mathematics? And why do we teach it in the way that we teach it? And why the specific topics that we ‘choose’ to cover?
I know… the first question seems pretty obvious. We teach mathematics because it is a language, and humans think through the language and tools that we use. Through the study of math, we are able to greatly enhance our pattern-recognition abilities – it’s like the equivalent of being able to put on infrared goggles! Yes, it’s true that in a lot of situations out there, the goggles aren’t going to get me any further than if I didn’t have them; on the flip side, though, there are quite a few situations where having them is pretty freaking cool if not absolutely outcome-changing. We could of course bring the metaphor into the real-world in myriad ways, but to name just one: the ability to program an Excel spreadsheet with the mathematics that make up an amortization schedule for a mortgage probably won’t be all that useful when choosing who to vote for in your local School Board Elections… but it could be a complete game-changer when you are deciding how you want to approach choosing and paying off your own personal mortgage to be able to analyze savings over renting, to get away from using corporate calculators (that have business biases built-in), and to recognize whether investing extra monthly money in pre-payments would be better than in an Index Fund like the S&P.
Of course, this is where we come to a conundrum, an impasse in our exploration of why we study math. The example that I just gave is all well and good – I mean, who can argue with receiving a personal benefit from understanding mathematics, right? But what if I were to keep giving examples of ‘mathematics as infrared goggles’? Without a doubt, my examples would soon exit the realm of personal benefits and lead into the realm of professional benefits – ways in which people could not do or be as effective at their jobs without having the pattern-recognition capacities that math gives us. Now, I don’t mean to say that this is a problem… if society is being furthered as a whole by teaching mathematics in school I call that a great thing! But what I do mean to imply is that therein lies a potential answer to the question ‘why do we cover the topics that we cover in mathematics?’ This answer may also lead into an explanation for the second question ‘why do we teach mathematics the way that we do?’
Now, this -ehem- ‘short’ post is not intended to be a treatise on the history of education in America, so I won’t be making it into one. Instead, we’ll be sticking to the surface level here: the reason we cover some of the topics we do has to have a historical basis. That historical basis is at least partially rooted in the idea that one purpose of schools is economic productivity (see the second quote under note *). Ken Robinson has argued that this economic reason for schooling is also related to the way in which schools are set up: with ‘The Industrial Model’ in mind (I’ve argued for other models elsewhere). Certainly some of the topics that we still cover today were originally part of the curriculum for the express purpose of giving industrial workers the skills that would make all of America’s Industry run smoother and progress faster. However, our current mathematics curriculum seems (to me) to arise from cultural dynamics of the 1940’s – 1970’s. Think about the backdrop: World War II giving way to the Cold War, the Space Race, The Vietnam War, etc. Political power and a country’s skill in physical technology was intertwined. Yes, more of the Country’s jobs were transitioning from industrial labor to blue-collar work that must have required skill with numeric manipulation… but I don’t think that was the reason for curriculum at the time. I think the reason was that as a Nation, we were hoping that if even just 1% of students took hold of these foundational physical mathematical topics early on, they would be doing things like programming missile paths or developing The Bomb by the time they were adults. I mean, seriously: consider the topics we cover! Quadratics as a Freshman or Sophomore in high school?! Why? Because quadratics are foundational to projectile motion. I mean, sure, quadratics can be used to illustrate a lot of advanced mathematical concepts at a basic level… but don’t tell me that even 10% of students studying this very topic right now understand why we ‘complete-the-square,’ and much less understand how you could apply a similar creative trick in order to be able to ‘discover’ (re-discover) how to calculate a definite integral by hand, which will then allow them to invent a completely new technique. They don’t understand why – they just do it. Or more likely, they have been taught to memorize the Quadratic Formula and they just use that, without ever understanding where that formula came from (just complete-the-square with a general quadratic with constants a, b, and c).
Now, I don’t mean to say that quadratics (as the example I have chosen – it could have been ‘rational functions’ or … oh don’t even get me started … ‘conic sections’) are only in the curriculum because for 30 years or so our Country was hoping to gain political power through physical technologies. They are also fantastic examples that show us the history of mathematics. The use of quadratics as an illustration of Newtonian Principles of projectile motion is an amazingly awesome unbelievable happy way to illustrate why and how derivatives and integrals are important and tie our world together (e.g. relate an acceleration to a velocity to a distance traveled and vice-versa). To study how Newton was so amazingly perceptive and precocious is a 100% worthwhile endeavor, and makes studying quadratics worth it on its own! But…. suffice it to say that’s not why or how we teach quadratics.
How we teach quadratics is similar to how we teach most maths content, unfortunately, in the United States. We don’t really teach the history of its development, nor do we teach why for most of what we do. Instead, we try to fly through lots of content at a very shallow level in order to “cover it all.” WHY!?!?
I’d like to propose a different way of teaching mathematics. Now, because my first way involves a proposal that is unlikely to be feasible for most math teachers by next year, I’ll only be commenting on it briefly and will also be providing a more extensive secondary way, introduced in this post and finished in the next.
The first alternative I’d like to propose is that many topics – like quadratics – are outdated in today’s world and no longer need to be taught to the level they are currently. For example, quadratics could be taught at a more shallow level than we teach them now in order to introduce students to the idea that not all curves (trends) in the world are linear. After students grasp that on the x-scale of “Amount that Runners Run,” the optimum y-value of “Mile Time” or “Fitness Level” is probably not at either the “0 days a week” side or the “24/7” side of the x-scale, then they don’t need to learn the logistics of completing-the-square. Instead, use ALL of that extra class time taking a deep-dive into one line of study: Data Analysis. That’s what people do at jobs in today’s world, and it just makes sense given that nowadays political power is more tied to the development of 5G than it is to the development of a physical weapon. Furthermore, the study of data analysis will further student’s quadratic skills too – just in a different way. A way that is, I might add, much more appropriate in a world where people don’t actually ever need to calculate a definite integral by hand, so long as they understand what an integral is (I know I already said this, but I will say it again: right now our students understand the exact opposite of that). A fantastic way to begin a Data Analysis course is through the study of Financial Literacy, as most all Data courses I can imagine have to have some ‘base substrate’ on which to base the study.
Now, as I stated previously, this proposal is unlikely for many educators who read this blog. It’s unlikely for me also – I’ll be teaching quadratics, trig functions, etcetera again next year and the year after that. So how are we to structure our mathematics revolution from within the content we have to teach now? Excellent question! Wait for Mathematics and Why, Part II and I’ll describe it in more detail than you care to hear. 🙂
*Of course, I recognize my bias in this statement – some would argue, perfectly reasonably, that what is at stake is the future of our economy. To that I respond that the economy is a terribly important system that certainly holds up our modern way of life. Yet, I respond with my Jefferson-ian bias: “If a nation expects to be ignorant and free, in a state of civilization, it expects what never was and never will be.”
To give a brief example of why this is important, I will refer you to this post.
For further support of ole’ TJ, I’ll revert to my Postman-ian bias: “Thomas Jefferson […] knew what schools were for – to ensure citizens would know when and how to protect their liberty. This is a man who wrote an essay that could have cost him his life,” wrote Postman in The End of Education, “It would not have come easily to the mind of such a man, as it does to political leaders today, that the young should be taught to read exclusively for the purpose of increasing their economic productivity.”
** Notice that I want to further our understanding of the questions. I did not say we will ‘answer’ them, whatever that means.