Math Made Easy: Thoughts on the Common Core
I’m a tutor for my day job. As a tutor, I encounter a variety of students, from grade levels K12th, all at different levels of expertise. Some are far behind their grade level, some right at it, and others ahead of it. But I find myself encountering commonalities amongst all of them, which are all a result of the way math is taught in American schools. Part of this is that, almost to a student, they hate math. And even if they don’t like math, there are certain parts of it they universally hate, like fractions, and other parts that are just tedious and slow, and they, to a student, fail to make certain connections that are necessary not only in higher mathematics, but practical realworld applications thereof.
After observation, it seems the primary reason for this is that math is taught in a segmented, rote manner. Students are taught one way and only one way to approach each situation, and are essentially handed the method, but never taught how it works or why it works. So, like good little robots, they can completely copy it when they see it presented exactly as it was previously, but when they encounter a situation that’s not quite the way they were taught, my students find themselves a t a complete loss. Further, the segmentation of the way math is taught – in other words, it appears that secondary school teachers will teach one topic, then move on to another, then another, never fully connecting the various topics, so essentially they’re teaching one type of screwdriver, then a type of wrench, then a type of hammer, but never showing the students that they can use all of the tools together.
The Common Core is supposed to address that, suggesting (I say suggest because the Common Core is not a requirement, but many schools do adopt it) that teachers present mathematics in a way that applies it to real world situations. But, having seen the homework and exams that many of my students bring home, and addressed their questions concerning it, the majority of the teachers doing this seem not to know *how* to apply the mathematics to the real world, or at least, don’t know how to present it to their students in a way that demonstrates how to make the decisions they need to manipulate the data presented them – it seems that those teachers are flailing about in an ocean they don’t know how to tread, and their education degrees never taught them how to do so.
So this leads to two issues that must be addressed if American students are going to catch up with their European and Asian counterparts in mathematics (we are sadly lagging behind, ranked 33rd in the world in Mathematics according to the 2021 Program for International Student Assessment survey (http://www.oecd.org/pisa/keyfindings/pisa2012results.htm).
First, we need to stop teaching by rote. While this does not preclude requiring students to perform many problems of the same operation (after all, the only way to learn math is to do math), what it DOES mean, is that, when teaching operations that have multiple methods for completion, actually present those multiple methods, and communicate to the students that any of those methods work, accomplish the same thing (and show them HOW they accomplish the same thing), and allow the students to pick what works for them. I repeatedly say the following to my students: “Math is a tool. And as a tool, we want the math to work for us, not for us to work for it.” When we teach by rote, and teach only one method to do a certain operation or series of operations, we are forcing the student to work for the mathematics, and not the other way around. We must present it more like a toolbox, where the student gets to pick the most appropriate (and for them easy to use) tool for the situation. I’ve watched, on more than one occasion, my students struggle with a problem, forcing it into a mold of the one way they’ve been shown, and taking inordinate amounts of time to complete a problem that, after letting them do it the slow way (I usually do this at least once to get the point across to them), I will then show them a much faster, alternate way of doing it, and they continue with the problem set and complete the remainder of it in half the time it took them to do one single problem.
Second, we need to connect the various segments of mathematics. It is no good teaching one part of the Pythagorean Theorem to students, then, a year or so later, present another portion of it, but never telling them that this new portion is indeed also part of the Pythagorean theorem, or even connecting the various elements of said theorem (which I also see being *not* done almost universally across the board). So when we show them trigonometric functions, we must also demonstrate to them from where they arise, and connect them to the most basic function in the Pythagorean Theorem, the a^2 + b^2 = c^2, which I rarely see done at the secondary school level. Similarly, with fractions, while they have them pounded into their heads, I rarely see the further step taken of demonstrating to the students just *why* those fractions are being pounded into them, or *how* they can be taken advantage of to make their mathematics easier.
Thirdly, let’s arm our teachers with the ability to actually present mathematics in the realworld applicable methods that the Common Core suggests. How to build realworld based problems, how to present those problems to the students so as not to overwhelm them, and how to help the students learn the critical thinking methods they need to tackle those problems, understand the data that’s presented them, and make the choices of operations necessary to build their way to the solution of those problems. For example, I had a 9th grader, taking classes through an online high school, who was presented a problem involving car tires on a geometry test. She was to calculate the height of a tire based off the tire label. For those of you who don’t know (as I didn’t before she showed me this problem, nor did she), automobile tires are labelled in the following manner (example): P215/65R15. To someone not familiar with car mechanics, this is absolute gibberish, and a horrible thing to throw at a student in the middle of a test, especially a student who is just learning the basics of geometry and units conversion. How to read the label: The first number gives the width of the tire (the treads side), in millimeters. The second number gives the aspect ratio of the tire, or how much taller the sidewall is than the tire is wide on the treads. The last number tells you the diameter of the rim, in inches. Personally, I think this is a godawful labelling system. You have three different types of measures in one label, when you could easily use one unit for all three numbers. WHY give an aspect ratio when you could just give the measure, in millimeters, of the sidewall? Why give the measure of the rim in inches, when the width of the tire is given in millimeters? Be consistent! Anyway, to solve the problem, she first had to make sense of this gobbledygook, then figure out the fact that she needed to either convert the millimeters of the width to inches, or the inches of the rim to millimeters to get her units consistent, calculate the height of the sidewall using the width of the tire and the aspect ratio, then add twice that to the diameter of the rim. The girl was completely lost when she came to me, and almost in tears because she didn’t even know where to start, and the teacher wasn’t helpful to her at all. So I first had to teach myself that system, having to look it up online because it was nowhere in her lesson, then teach it to her, then teach her the multiple steps she needed to finish the problem. This demonstrates that the teacher had NO clue in how to connect the mathematics to the real world in a way that the students could comprehend and handle. And this example is by no means an isolated incident – I’ve encountered many others. That’s something we, as a country, need to fix. Mind you, I'm not trying to bash teachers here. I think they're every bit as much victims of an inadequate system as the students, and just need help in learning how to make that connection and presentation.
So, in my contribution to this end, I’m going to start in parallel to my Physics Made Easy articles, a series on mathematics, Math Made Easy. After this introductory article, I’m going to tackle multiplication, and will demonstrate an easier, faster method than the traditional multiplication method. See you in the next article!
