Gary Rubinstein’s Blog: Is Math the Emperor’s New Subject?
Most people probably suspect that students spend too many hours learning Math in school.
From Kindergarten through twelfth grade, it seems that most students learn some Math pretty much every day. The same can’t be said of other things like social studies or music or art or physical education.
I have two children in public schools. My son just finished 4th grade and my daughter just finished 7th grade. At my son’s school, it seems like he did math for about an hour every day. My daughter is in middle school so she had one period of Math each day, about 50 minutes long. By the time they finish twelfth grade, my children will spend about 2000 hours learning Math in school and maybe another 1000 hours on homework. This will comprise about 12.5% of their learning time.
Though many might worry that studying this much Math is too much of a good thing, it’s a view that they don’t express too loudly. Obviously there must be some people who think this is just the right amount of Math otherwise why would we as a country, and a World really, choose to do this? We are told that Math is important and that Math is useful. If it really is important and useful, you don’t want to be the person arguing against it. But most people don’t know enough about Math to know if it is really as important and useful as we are told. Well I do know a lot about Math, having taught it for 30 years and been a Math major in college before that so I hope my opinions are considered, at least, well informed.
Back in 2013 I wrote one of my most widely read blog posts about this issue. (It was featured on Andrew Sullivan’s dish archive.) Since I tend to be pretty wordy, I’m not sure how many people who start my posts make it to the end, I’ll say right here at the beginning that we absolutely dedicate too much time to studying Math and that time is also not efficiently spent. With a lot of changes we could spend about half as much on Math and simultaneously make the Math that we teach much better.
In that 2013 post I argued that about 40% of the topics that we teach in Math could be cut from the curriculum and they wouldn’t be missed. I also said that beyond 8th grade math should be an elective so students would learn up to what we now call Algebra I and the other things like Geometry, Trigonometry, Algebra II, Precalculus, and even Calculus would not be required (Precalculus and Calculus are not ‘required’ right now, but in a sense they are because students are told that colleges ‘like’ when you take AP Calculus so there is a pressure to take it.)
In 2012 author Andrew Hacker wrote a The New York Times op-ed called ‘Is Algebra Necessary’ went viral and had thousands of comments on it. His book ‘The Math Myth’ came out in 2016.
Hacker makes several points, some are valid and some are not. His most valid point, I think, is that for all the money, time, and other resources that this country spends on Math, we don’t seem to be getting a very large return on that investment. Most adults could not pass a test about the math they learned in high school and many would not be able to pass a test on elementary school math. On the other hand, most adults would easily be able to pass a test that required them to read something that was written on an elementary school level.
But this alone does not mean that something is not worth studying. Maybe adults can’t do well on a Math test right now but if they were to prepare for it maybe the Math is lurking in their subconscious. I took tennis lessons when I was a teenager and I was able to hit the ball back and forth eventually. I can’t do much of a rally right now but I’m sure that if I were to take a few lessons I could get back to being able to do that.
But there is something to the fact that most Math you learn in school is so ‘forgettable.’ Why do we spend so much time on something that can so easily be forgotten? Shouldn’t things we learn in school be ‘mind blowing’ so that you can’t forget them even if you tried?
This is a big problem with the modern Math curriculum. Many of the topics that are in it are very boring. But at least they are useful, right? Well, not really and that’s another big problem. Most of the Math we do in school is not useful in the sense that many people will ever find an opportunity to use most of it. Besides addition, subtraction, multiplication, and division, what Math does an adult really use? Maybe you’re thinking ratios and percentages, but those are just applications of multiplication and division in my mind.
A typical ratio problem is that if 5 widgets cost $200, how much would 7 widgets cost? You don’t really need fractions or ratios to answer this question. If you have a good understanding of division you might think: If 5 widgets cost $200 then 1 widget costs $200/5 = $40. And if 1 widget costs $40 then 7 widgets cost $40*7 = $280. Just division and multiplication.
The same goes with some of the Algebra that is supposed to be useful. If the rent for a retail store is $200 and you make a profit of $4 for every widget sold, how many widgets must you sell to make $300 after paying the rent? Well, yes, this can be set up as 4x-200=300 and then solved 4x-200+200=300+200, 4x=500, 4x/4=500/4, x=125, but do you really need Algebra to do that. Can’t you just use addition and division? You need to bring in a total of $300+$200=$500 to cover the rent and make the profit. But then you have to divide $500 by $4 to see how many widgets you need to sell.
I could do this for almost any ‘practical’ question through Algebra I.
Beyond Algebra I, there’s Geometry, a topic very close to my heart. I leaf through my copy of Euclid’s Elements like it is The Bible. The organization and the development of hundreds of theorems, many that seem like magic but the proofs are undeniable. How amazing is it that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse? I’ve studied over 100 proofs of what I consider to be the best theorem in all of Math, The Pythagorean Theorem. So I love Geometry, but I think some of the best parts of Geometry could be incorporated into middle school Math and to force so many students to take a year long course is wasting resources. I feel the same way about Algebra II, Trigonometry, Precalculus, and Calculus.
Math is a good thing, but there are other good things that are neglected in school so if we were to cut back on the amount of Math that gets taught I would want to see those resources applied to other things. When I think of things that would enhance my life, there are some things I wish I had learned in school. Like how to tie a good knot. I can tie my shoelaces OK but beyond that I can’t do any good knots. I’m not sure in what course that could be taught (there is something called ‘knot theory’ in Math, but it’s a really advanced course and I don’t know that you actually learn to tie a knot in it.) I also wish I knew how to use a power drill. I know some schools have a wood shop elective but I don’t remember considering taking it back when I was in high school. And now I have this towel rack I’m supposed to put up this summer and my building’s handyman is dragging his feet on it and I feel helpless even though I own a drill that I’m scarred to use. Another thing I wish I learned more about in school was gymnastics. My body is falling apart, I’ve been to physical therapy for my back, my wrist, my leg, and my foot over the years. I know that way back in the Greek times, gymnastics was a core subject. I’m not saying I need to be able to do a back flip or anything, but more physical education might have served me well. I guess what I’m saying is that taking away some of the thousands of hours of Math and dedicating it to other subjects is a good idea. We have come a long way from Medieval times where the seven areas of study were grammar, logic, rhetoric, arithmetic, geometry, music, and astronomy.
I mentioned that we lie when we say that Math is important because it is so useful. I recommend an essay called ‘Is Math Necessary‘ by professor Underwood Dudley, he explains better than anyone why it is OK if math is not really useful in the practical sense. I agree with him. I think Math is worth studying because when done correctly it is fun. So something like a Math riddle that has a surprising answer and that generates discussion and that makes students want to answer it, I think that makes Math worthy of learning. Maybe if that is the thing that is good about Math, though, forcing everyone to take it for twelve years is too much.
If you want to see what I’m talking about, I have taught some electives over the years where I am not constrained by the curriculum but can teach whatever topics I want. I teach an elective like this to 9th graders and I used to teach a similar course to 12th graders who did not want to take Calculus. A topic that I loved to teach was different ways to calculate the square root of numbers that don’t have easy square roots. Like the square root of 7. Since 2*2=4 and 3*3=9, the square root of 7 is somewhere between 2 and 3, but how precise can we get it? Over the centuries different cultures have tackled this question in different ways. Their methods were ingenious and with the right teaching, students can figure out the algorithms themselves with some hints or they can analyze the algorithms to see why they work. I spend a few weeks on this. It is something that has no practical value anymore since every calculator has a square root button and for the square root of 7, it probably is good enough to say it is between 2 and 3 for any practical purposes. But I’m not teaching it because it is practical, I’m teaching it because it is thought provoking and it is fun.
So I agree with parts of Hacker’s ‘The Math Myth,’ but he unfortunately undermines his own credibility with many of his proof points. He seems to think that any Math concept that has an intimidating sounding name must be completely ridiculous to teach. On page 99 he critiques a 2008 US department of Ed report because some topics for Math it recommends are “rational expressions and binomial coefficients to quadratic polynomials and logarithmic functions.” But just because something sounds useless and overly complicated does not mean that it is. Having an ugly name has nothing to do with whether something is worth studying or can be interesting and fun to learn. For Hacker sometimes the names of real things are not absurd sounding enough for him so he fuses together different words and invents gibberish that sounds like an actual thing, but really isn’t.
In a section of chapter 8 called ‘Pascal’s Triangles And Pythagorean Triples’ he uses what he thinks is a good example to show how absurdly hard and irrelevant one of the common core standards is:
Here is the tex of the standard from the High School Algebra list:
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
Now I’m not the biggest common core cheerleader, but this is one part that someone who knows about math and math teaching will agree with. One of the things students learn in Algebra is to simplify what are called algebraic expressions. Like, for example, maybe you change x+x+y+y+y into 2x+3y. It is a pretty dry topic and not very inspiring. Sometimes as practice, students are given two different looking expressions and they have to simplify to both to show that the two expressions are equivalent. A simple example might be to show that 3x+x+2y+y = 2x+2x+y+y+y. Since both sides of the = sign simplify to 4x+3y, the two expressions are equivalent and when you do this is is called an algebraic identity. So most examples of algebraic identities are taken out of any meaningful context and they are just an exercise in simplifying expressions. But sometimes, even in the high school level, the checking to see if two expressions are equivalent can be the key step in the proof of something surprising.
So Pythagorean triples are numbers like 3,4,5 that have the special property that 3*3+4*4=5*5. If you are not a math teacher, you would find it challenging to find another three numbers for which this is true. It works for 6,8,10, for example, of if you multiply 3,4, and 5 by whatever number you want. But there are other Pythagorean triples like 5,12, 13 or 8,15,17, or 20,21,29. They are kind of rare and it is one of the mysteries of math to try to find three numbers where this happens. It isn’t the most practical thing there is, it is a problem from a branch of math called ‘number theory’ but it is mysterious and if taught correctly can be something that students find fun — trying to locate these mystical number triplets. So it turns out that if you take any two numbers, call them x and y and make x the larger of the two numbers and you calculate x^2+y^2 and you also calculate x^2-y^2 and 2xy, something surprising happens. Like if I make x=3 and y=2, the first thing becomes 3^2+2^2=13, the second becomes 3^2-2^2=5, and the third becomes 2*3*2=12 and 5, 12, 13 is a Pythagorean Triple, you can check that 5^2+12^2=25+144=169 and that 13^2=169.
So this standard is saying that when possible, rather than just make a dry identity proof that doesn’t prove anything surprising, look for opportunities to make them more meaningful. In this case because both (x2 + y2)2 and (x2 – y2)2 + (2xy)2 both simplify to x4 +2x2y2+ y4 It ‘proves’ that those three expressions will always make a Pythagorean Triple. For sure, having something meaningful to prove with an algebraic identity is superior to just doing an identity for the sake of doing an identity.
Other times he explains a topic that he thinks is absurd for students to learn when someone who really knows about school math knows that the topic is very reasonable. So when he talks about how crazy it is for a student to find the measure of the smallest angle of a 3, 4, 5 right triangle, does he know that it just requires looking up the number 3/4=0.75 on a chart and seeing what row it is in?
Then in chapter 12, Hacker explains what he thinks Math teaching should be by describing a course that he taught. But the creative lessons he seems to think he has invented are things that have been around for decades if not centuries, such as using measurement of string to estimate the value of Pi. Still I agree with his point that Math lessons are better if they start with a thought provoking question and if students get an opportunity to think about the question and make progress toward figuring out the answer though an engaging activity. This isn’t just true for Math, but for any course I think.
Still, I like ‘The Math Myth’ for the basic premise that too many resources are going toward Math education and we are not getting much bang for our buck out of it. He also addresses the requirements in college to take a Math course that serves for some as an insurmountable unnecessary obstacle.
Algebra I is usually taught in 9th grade, nowadays. There is talk of different districts who want to give ‘Algebra To All’ of 8th graders to raise the ‘rigor’ of the curriculum and allow the students to progress further and other districts who want to give ‘Algebra To None’ of the 8th graders since it is not fair to give it to some and not to others. Of course either extreme is wrong. The problem isn’t too many students taking Algebra or not enough taking Algebra. The problem is that some students haven’t mastered the prerequisites for Algebra and others have. I think that if we make Math more fun, more students will learn it and more will want to go further with it.
So you might be wondering if I have such contempt for a lot of math topics, how do I go and teach it for almost 30 years? The answer is that no matter what topic I have to teach, I do everything I can to make it meaningful, interesting, and fun to learn. And a good teacher has this ability to find the thought provoking questions to ask about whatever topic they are supposed to teach. Many of the topics in the modern Math curriculum are not the topics that I think are very interesting but if my job is to teach breaking down logarithmic expressions then I am going to find a way to teach it in a way that is engaging. And I get enthusiastic about this so I doubt my students would even know when I’m teaching a topic that I wish I didn’t have to.
I guess it’s like if I were a musician but I wasn’t good enough to be a famous one with my own original songs. So I get a job as a singer and keyboard player in a wedding band. It is a steady job and pays the bills and I get to do the thing I love, making music. And at the wedding I have to play ‘Celebration’ for the five millionth time and you know what, I’m going to do the best rendition of ‘Celebration’ that I can because for that couple that is getting married, this is their only wedding and they deserve to have an enthusiastic rendition of ‘Celebration’ even if it is not my favorite song in the world. So I see my job as trying to do the best most thought provoking and engaging lesson on whatever the topic is. There is no topic that can’t be taught in an engaging way and it is a challenge that I enjoy to try to make even a really boring topic interesting in the way that I structure my lessons. Some topics are harder to make interesting than other topics. And not every Math teacher is great at animating lifeless topics. I surely don’t always succeed at it. If we got rid of some of the topics that are hard to do this for and only the most interesting topics remained, everyone would enjoy Math more.
Some relevant links you might like:
Underwood Dudley’s ‘Is Math Necessary?’ This is the absolute best.
Paul Lockhart’s ‘A Mathematician’s Lament’ This makes some good points and has a great analogy comparing Math instruction today to an awful music curriculum.
My post ‘The Death of math’ from 2013.
A link to my YouTube playlists. If you want to see what kind of Math I think is worthwhile to learn, see the Math Explorations and Math Research videos. Also you can see my series where I try to explain Math starting at kindergarten and going through trigonometry in 10 hours!
The Mathematical Association of America’s response to Hacker’s ‘Is Algebra Necessary’
November 2, 2021