Sparks Commentary: Some thoughts about math education

“4 out of 3 people struggle with math.”

– A joke of unknown origin

As some of you know, I've been a professional math tutor since 2012. This is the first post where I've really talked much about my job. (I may or may not do others on this subject.)

Disclaimer: My degree was not in math, but in business

I should give a disclaimer to say that my degree was not in math, but in business; and the highest math class that I ever passed was Calculus & Analytic Geometry II. I passed this class by the skin of my teeth, as it turned out, and had no desire to undertake the third (and final) semester of that calculus sequence. This was because it had a lot of 3-D math that I didn't really want to do. More often, I have tended to have some mixed feelings about learning math myself (particularly trigonometry and calculus), and my record at teaching these advanced subjects is somewhat mixed as a result. I have thus tended to stick to teaching the lower levels of math in my job, and have undertaken intermediate levels of math only on rare occasions. (The more advanced classes, like the three semesters of Calculus & Analytic Geometry, are things that I don't even allow myself to go near; although I am glad that we have a number of math tutors at my school, who are able to tackle these subjects.)

3-D rotational math, like the kind I would have had to do in Calculus & Analytic Geometry III, if I had actually taken this class

The balance between liberal arts and “math & science”

In recent years, higher education has put more emphasis on what some refer to as the “STEM” subjects. This is an acronym that stands for Science, Technology, Engineering, and Mathematics. As it turns out, the liberal arts are still doing quite well today, and there is still a much-valued place for teaching them in our school system. But it turns out that they were once a much more prominent part of our general education requirements, and that the focus on “STEM” subjects has sometimes (but not always) come at the expense of the liberal arts. I'm not going to comment on what exactly is the right balance between them, but in general, there is a need for some sort of balance. In my opinion, it would be catastrophic if we lost the great “liberal arts” parts of our cultural heritage; but it would also be catastrophic if we forgot everything that we knew about math and science. Neither extreme would really be beneficial for our society, of course, so we should remember our needs for engineering and technology subjects as well as those of the liberal arts. It is well that we have allowed people to choose their majors in this country, and allowed them to self-select a specialty that is right for them – including selecting whether it is mathematical or not, among other things. And whether or not people go to college may be up to them, but requiring them to learn some basic math is appropriate for most kinds of education, I think – including (and perhaps especially) the K-12 levels. The most useful kinds of math are actually the basic levels (such as addition, subtraction, multiplication, and division); and we would do our students a service to require the majority of them to learn it. It would also be to our benefit to enforce the official punishments that apply to the ones who don't learn it, in my opinion. This would help to bring some accountability back into our schools, making it so that a high school diploma might actually mean something. (Obviously, it doesn't today – but that's a rant for another post.)

Times tables, like the kind that K-12 students often have to memorize

The problems of overusing calculators in the classroom

Most people realize intuitively that calculators have revolutionized how math is done, and made many kinds of math problems far easier to solve. But as it turns out, there are positive and negative effects of using calculators in the classroom. In general, I have tended to adapt my teaching style to the particular instructor's policies on calculator usage, and I'm in no position to insist on teaching any of my students the long way when their instructor doesn't require them to do so. This is particularly true when the teacher doesn't really give a crap about how their students do the material, and allows their students to take shortcuts, even when it's at the expense of their conceptual understanding. Such teachers exist in virtually every school, and my experience is that their students tend to learn very little about the real concepts involved. Although they are often very well-versed in the technological shortcuts, their understanding about what is really going on is often lacking, to say the least. (Although in fairness, I should acknowledge that other math teachers are much better at teaching the concepts, and I don't wish to paint all of them with the same brush. It is the ones who allow too many shortcuts whose teaching methods I object to here.)

TI-83 Plus calculator, the model I used when I was in high school

When should calculators be allowed, and when should they not be?

In my not-so-humble opinion, the best practice is to require the long way when the students are still learning the particular methods, and to only allow the calculator shortcuts when they have a conceptual understanding of what their calculator is really doing. This is not always easy to measure, as it turns out, since a student's passing of the prior classes is not always done as honestly as it should be, regardless of the school at which they took these prior classes. As the next best thing, calculator shortcuts for the basic arithmetic should only be allowed after a certain point, so that the students are allowed to focus their attention (in these advanced classes) on the new material that is being built on these things – rather than on the old material that, in a perfect world, they were already supposed to have learned long before.

TI-84 Plus calculator, a model used by some of my students

The relevance of computers to some (higher) mathematics classes

And a word about the use of computers in math classrooms today: As mentioned earlier, computers have revolutionized how math is being done. Many kinds of math are now possible which were outside the reach of previous generations, because of the invention of computers in the last century (the twentieth century). As you might expect, the typical computer science major will have to take a fair number of math classes today. But as it also turns out, your typical math major has to take at least a few computer science classes as well. In particular, math majors are often required to take some computer programming classes, and sometimes cannot get into (at least some of) their advanced math classes without them. Thus, there is a place for teaching some computer skills to all of our students, and not just for their applications to advanced mathematics. This might seem something of a tangent in a discussion of math education, but I feel that these comments do actually belong here (at least to some degree), and apply our discussion to the realities of the twenty-first century. Teaching students technological skills will be important, of course, even if calculator shortcuts should be limited in certain educational contexts. (I thus don't wish to come across as “anti-technology” in my rant about calculators, but merely wish to limit their application to those situations where they are truly needed.)

Central angle of a circle, like the kind often used by students of geometry

The reasons that people struggle with math vary somewhat from person to person …

As a math tutor, my students are often disproportionately the struggling students. As you might expect, the reasons for their struggling vary a lot from student to student, but a few common themes are often present for many of them. Some of them are smart enough to learn the material, but aren't really motivated enough to do much of the work. Others of them are trying hard to do the work, but don't really have the adequate math software in their brain – at least, not for the level of math that they're currently being required to take. Either problem can be a challenging one for these students, and the combination of these two things can be particularly challenging for the students who are afflicted with them. But not all of my students are in the same boat, of course, as I will show in my next paragraph.

A graph of a particular kind of polynomial function (used in algebra)

Some personal anecdotes on this subject, from my experience as a math tutor

Without naming names, I have some students who struggle with the most basic rules about negative numbers, and think that “a negative and a negative makes a positive,” even when you're adding – mixing up the negative rules for multiplication with those of addition, sometimes with disastrous results. (Unfortunately, some of them just cannot get it, no matter how many times I tell them the rule; and in fairness, it's not always their fault that they can't get it.) Others of them did perfectly fine with the lower levels of math, but might need some occasional help with their more advanced math courses – for example, college algebra or trigonometry. I freely admit that I am not always able to help them with these kinds of advanced classes, but I do my best; and I acknowledge that this type of student is often very capable at math. Very often, their problems only came when they got to these higher levels of math, and a little bit of tutoring may be all that some of them need to make the difference between passing and failing. This kind of student is sometimes easier to help as a result, although I do my best to help every one of them regardless of their difficulty for me. (I always give my students all of the help that their teacher allows me to give them, and I long for those moments when the light bulb really goes on for them. This is one of the most rewarding parts of my job, and makes the other parts of the job worth it for me.)

A right triangle, like the kind commonly used in trigonometry

The emphasis on practical applications in math classes …

In high school, math teachers have seldom known (or cared) about the applications of their own subject, and this was often (but not always) the case with my own math teachers. Some students don't really care, either, and their opinion of math is independent of its perceived usefulness. The ones who hate it will sometimes hate it regardless, and the ones who intrinsically like it will often like it anyway – sometimes becoming the next generation of mathematicians as a result. But a larger number of students would rather know something about what the math can be used for, and that was certainly the case with me. I probably would have enjoyed the math much more if I had known beforehand about its practical applications. When I saw the relevance of statistics and probability to my business major and economics certificate, I was much more motivated. Others could be similarly motivated, if only they knew in advance what it could be used for.

A bell curve, which is commonly used in statistics

… where possible

Obviously, this is not always possible in early math classes, and students sometimes have to take the teacher's word for it to get any assurance that it has uses. And in fairness, the training of some math degrees does not really emphasize the practical applications; and so some very qualified math professors are unable to tell their students what the math can be used for. But where possible, I believe that this may actually be a valuable goal for some students; and might do much to motivate some of them. In particular, word problems might do much in making this kind of math more realistic for these students; and might help the math to “come alive” for them in the process. I realize that students often complain about the difficulty of word problems (and understandably so), but sometimes it is the only way to really show them what the math can be used for. Some math teachers are already doing this, of course, and I greatly admire the ones who do it well. They go the extra mile in helping students to see the application, even when the students are more likely to complain about it in the short term than in the long term.

A system of supply and demand equations, like the kind commonly used in economics

Creating better schools: The ultimate goal

None of these problems can really be solved overnight, of course, but we can still make some strides towards improving our country's schools – and possibly those of other countries, at appropriate times and places. Naturally, we will never be able to create a perfect system, but we may well be able to create a great system. To a large degree, we already have, I think; and we can do it even more as time goes on.

If you liked this post, you might also like:

Some thoughts about economics education

How do the economic laws of supply and demand work?

Why is my stats class so focused on bell curves?

How fractions are used in the United States Constitution

A review of the BBC's “The Story of Maths” (by Marcus du Sautoy)

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Education