What Mathematics Do Students Need To Remember?

I puzzle over this question almost daily as I prepare to teach my classes. I’m torn. When I took mathematics classes in college, computers were not ubiquitous, smartphones were still the stuff of science fiction, and the Internet was … well, it was around, it’s just that it wasn’t the easiest thing to use nor did the masses know of its existence. This would be the late 1980’s.

Now flash forward to today, 2015. I have computers for everyone in my classroom, plus some students bring their own computers, most people have a smartphone, and the Internet is revolutionizing just about every single industry. Education will be no exception.

And this brings me to my original question: What mathematics do students need to remember?

Take factoring for example. Do students need to understand what factors are and how they can be useful in analyzing graphs?

In my opinion, yes. To me, that’s really the important mathematics I want my students to remember. I don’t think we need to continue spending several days teaching all the different factoring techniques like has been done for decades and probably much longer. I just believe it’s no longer necessary.

Today we have tools like WolframAlpha, a seemingly countless number of smartphone apps, and computer software that will do such menial tasks as factor.

I used to say to my Beginning Algebra students that it was highly unlikely they would ever factor out in the “real world”. That wasn’t the point of learning it. Instead it was learning how to break things apart and put them back together that was the benefit of learning how to factor. This kept the complaining down to a minimum, but really, my explanation was bogus.

The point of factoring, which has been embedded into the curriculum for decades or longer before I even began studying mathematics, is because we needed to know how to do that particular skill in order to move forward with more advanced mathematics. There was nothing else that could do it for us. We NEEDED to do the factoring.

So, factoring techniques have been taught ever since, and students spend weeks learning this “skill” with most not really understanding what factors are nor how they can be useful in analyzing graphs or really anything important about factors at all. For some, like myself, it’s a fun puzzle to try and figure out. For others, it’s a never-ending maze of confusion and frustration.

Now I’m not saying the reason we should scale back the teaching of factoring is because it may cause confusion and frustration. Hardly. Most learning happens by working through the confusion and frustration. Instead, I’m saying the actual act of factoring is no longer a necessary skill. Technology has made this skill obsolete.

Yes, I know I’m in the minority, here, but please bear with me.

It used to be a very important skill to find the square root of two by hand, or to look up logarithmic values in a table and then be able to interpolate other values. At one time, these skills were necessary in order to move forward with more science and mathematics. These skills became obsolete when inexpensive hand held calculators became available. Now, thankfully, these skills are no longer taught.

However, the concept of the square root of two is still taught. It’s just that now when we need that particular value, we use technology to give us that value so we can then use it appropriately in the context of our problem. Finding the square root of two itself is not the problem. It used to be. Oh, how boring that must’ve been.

I see the same thing with factoring, the properties of logarithms, calculating derivatives, etc. The list goes on.

So, now I have to ask the question: Does there really need to be a three, or more, course sequence to get students ready for college level mathematics? I’m thinking not.

Think about what’s taught in those pre-college level mathematics courses. You know, the one’s titled Prealgebra, Elementary Algebra, and Intermediate Algebra. Most of what’s taught in those classes are obsolete skills like solving equations, factoring, and finding the equation of a line. These things are still taught because at one point all those skills were necessary in order to move forward.

We couldn’t add rational expressions very easily without factoring to find the least common denominator.

We couldn’t figure out the time it would take for $1000 to grow to $1500 if it was compounded monthly at an annual interest rate of 4% without using the properties of logarithms and knowing how to solve an equation.

We couldn’t figure out a perpendicular bisector without knowing how to find the equation of a line.

But today, technology can do all of the above more efficiently and without making a sign error during the calculation.

I’ve often told my classes that the calculator is only as smart as the person pushing the buttons. If you enter incorrect data, then you will not get the correct solution. You need to know if the results make sense.

To me, that’s an important skill. Learning how to slow down, enter the correct data into the appropriate tool, and interpret the results. I especially want them to know enough to question if the results make sense or not in the context of the problem.

So, I’m no longer sure there needs to be so many math classes. And this doesn’t end with pre-college level classes. In Precalculus, Calculus, Differential Equations, etc. there are several skills that have been made obsolete by technology.

Is it possible instead to have fewer total math classes but they all incorporate learning how to use technology to do the menial skills tasks, while at the same time learning what the given results mean and their relationships to graphs and applications? Could that then introduce the beauty of mathematics to more people?

I don’t know.

I hope so.