Gary Rubenstein has been writing a series of posts on the question of whether the math curriculum is useful. Some parts of it are indeed useful, others not so much. In this post, he describes the “useless” topics.
He writes:
I’d estimate that about 15% to 20% of school time in K-12 is spent on math. Elementary and middle schools often have their students do 90 minutes of math a day. And it is common for students to take a math class every year throughout high school.
In my last post I listed a meager six math topics that I consider ‘useful’ and by that I mean that those math skills are really needed by adult consumers and also, to some degree, in a lot of professions. And if you believe me about this and you think that any math that is not useful should not be taught in school you might wonder how much time should be dedicated to those topics throughout a students schooling. Now I’m not saying that I think that we should cut all topics besides these few but if I had to answer how long it could take to teach those, I’d say that we could do it in about 1/3 the amount of time. Math would be a thing like music, art, or physical education.
It’s still an interesting thing to think about, though, because it gets to the fundamental question of ‘what is the purpose of learning math?’ or ‘what is the purpose of learning anything for that matter?’ or ‘what makes this thing better to learn than that thing?.’ I will eventually provide my opinions on these questions.
But before we cut 2/3 of the time that we dedicate to math, we should take a look at what sorts of things would we be depriving the students of and whether there would be negative side effects of these discarded topics.
In Part 2, I mentioned a topic that I said was not ‘useful’ of finding the prime factorization of composite numbers. While it is true that hardly anyone in their adult lives are ever asked to break 555 into 5*3*37, maybe the ‘use’ of this skill is not so direct. The ‘use’ of some ‘useless’ topics is that they are prerequisite skills to more complicated topics in future years and those more complicated topics might be ‘useful’ in some science applications. So some ‘useless’ topics might have some utility as scaffolding to other topics.
Another reason that something like factoring has more ‘use’ than it at first seemed is that prime numbers are really important in more advanced math. They are the building blocks of all other numbers. Maybe someone who loves factoring eventually becomes a math major and they use advanced factoring to create a new cryptography method based on it.
Open the link and keep reading.

“I’d say that we could do it in about 1/3 the amount of time. Math would be a thing like music, art, or physical education.”
I hope you’re not suggesting the time allotted for music, art and PE is appropriate! Didn’t you mean to say “I’d say that excess math time would be better used to increase music, art and PE time”?
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“So some ‘useless’ topics might have some utility as scaffolding to other topics…Another reason that something like factoring has more ‘use’ than it at first seemed is that prime numbers are … the building blocks of all other numbers. Maybe someone who loves factoring eventually becomes a math major… ”
Important point, and perhaps such topics and outcomes would be facilitated by removing the pressure of grades for those topics. Those parts of the curriculum could be explored in a grade-free, exploratory math lab format.
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In the post, Rubenstein discusses topics like radians and a particular geometry theorem. Mate had a lot of good things to suggest about it. everyone should read the original post and the responses. Thanks, Gary
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I coulda been a contender
I coulda been somebody
(who stopped the useless
topics taught in classes)
I coulda/woulda/shoulda
but didn’t…
Damn choice schools
Damn wrong wingers
Damn voters
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We do what we can. What’s the alternative? Do nothing?
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And I highly recommend reading Diane’s Slaying Goliath, which is replete with stories of rank-and-file educators making a difference, changing things. No coulda/woulda/should there.
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Disclosure: I’m not an education professional, & I was good in math through HS & my freshman year college, when I was an engineering major, then switched to music & never looked back.
As a musician, I’ve found that math & music are very closely related: Math is the inherently abstract science, & music is the inherently abstract art. The origins of Western math & music both trace back not just to one time period, region, or culture, but one person: Pythagoras.
Since school, I’ve heard the laments of highly intelligent, well-educated people who’d had difficulty with math, pointing out that in their adult lives, they’d never once had to factor a quadratic equation, compute the volume of a sphere, or utilize a logarithm; asking why they had to be put through that torture in school.
While I sympathize with their frustration, I wonder if, despite their struggles, that math may have been more useful & applicable to their lives than they realize. Through its process of establishing a series of steps to determine a precise value, math develops our brain function into a particular mode of reasoning. These processes are analogous to all critical thinking, regardless of the subject matter. Due to its abstract nature, math focuses on these processes at their most basic level, in a manner designed to be transferable to a variety of concrete applications — not just physics, but also literature, art, & human relations.
Is it possible those neurological functions developed through study of math are used unconsciously by most people every day in the normal course of their lives, in both professional & personal lives? I’m not a neurologist & can’t cite any studies exploring this, nor can I demonstrate that current math curricula are the most effective, efficient way to achieve this result, but I do wonder if math study may be more relevant to our lives than we realize.
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Lenny—just gonna take a devil’s advocate position here. Family of musicians. Hubby math major/ engineer; me totally artsy with weak math but strong interest/ practice in music. Sons artsy, strong music, strong computer chops [unlike me]— but not great in math. [We all have excellent pitch.]
We line up along the typical can or cannot easily read music. I am easily: no longer do piano, but can read complex classical choral scores no sweat. The men are very strong in musical structure, multiple instruments, lotta bands; sight-reading did not come easily but gradually improved, always an effort (unless chord charts/ single-instrument band charts). Millennial sons teach musical instruments for a living.
So what is this thing about reading scores easily or not, & how does it relate to math ability? Hubby only one strong in math, yet reads no more easily than artsy [but computer-choppy] sons. Me, weak math student, dumb on computer, strong musical-score-reader.
I have to wonder whether the score-reading facility isn’t related to other abilities (not math). My main strength is in foreign languages. For me there is an easy audio/pitch-speech [or sung note] connection which is tied into the symbols/ alphabetic representation…
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Hi Bethree — You make interesting points. As I said, I’m not an educator, mathematician, or scientist, so anything I say is just my (?educated?) guess about what seems to make sense.
Both math & music encompass multiple modes of perception: conceptual (numbers); visual (physical structure; distance); tactile (rulers & other measuring devices); aural (vibrations, rhythm). Both disciplines include representations & applications. Individuals may have varying degrees of innate inclination toward these various areas.
Some people may have an affinity for, or difficulty with, expression or interpretation within any of these areas. There are musicians who write scores without touching an instrument, & those who don’t read but can reproduce an entire piece after hearing it only once. (These are extreme examples, but they’re more frequently observed in lesser degrees.) There are mathematicians who gravitate toward primarily theoretical (Stephen Hawking reportedly did the complex calculations for his astronomical theories in his head), & those who excel in graphing or plotting topological connections. As a mathematician advances to higher levels, they’re likely to specialize in something that concentrates more in a particular area.
Musicians, obviously, are strong in the aural area. You, in addition, seem to be strong in the visual area. A musical score is a highly stylized graph representing a set of pitches & timbres occurring over time. You can see this graph & hear internally the music it represents. Your family members who are less skilled in reading, but excel in performance, may be stronger in the tactile area: To produce a note, in what shape or position should your fingers/limbs/mouth be? Just how much force should be applied, at what angle? They don’t calculate this; they feel it.
My interpretation of your descriptions of your family’s different musical inclinations & applications is maybe there are contrasting affinities for the different mathematical areas related to music. This is a really engaging area to study. Thank you for suggesting it!
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Fascinating, thought-provoking reply, Lenny– thanks!
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A I would argue that the radian is a much more natural” (intuitive) measure of an angle than the degree.
I’m sure all math teachers understand the following, but how many actually explain the radian to their students this way?
One radian is just the angle you get when you “wrap” a string with length equal to the circle’s radius around part of the circle. One radian is the “angular distance” around the circle corresponding to an arc with length of one radius.
If it was explained that one radian is quite literally the “radi(us) an(gle)” it would probably make a lot more sense to most people.
On the other hand, the degree has no natural mathematical relationship with the circle, which if anything is actually more confusing.
Radians might be confusing in practice but not because there is anything inherently confusing about radians but instead because people are generally exposed to degrees first and then simply expected to know (ie, memorize) the conversion.
But I think the biggest issue here is the fact that people are expected to learn more than one system of (in this case angular) measurement. Most engineering and science actually uses radians rather than degrees and the only common use of degrees i can think of is on a compass, but unless a person is a sailor, they probably don’t need to know about that.
The same “multiple systems of measurement” issue rears its ugly head in a much more full blown (even more confusing) manner with the English/metric issue.
We were told decades ago in the US that America intends to switch to metric but in the meantime, you need to learn both systems. Well, the “meantime” has turned into over half a century with no end in sight. And to this day, every mechanic has to have 2 complete sets of wrenches often for different bolts on the same car and there is even a famous example of s NASA space probe that burned up because one team was using Metric and the other English and the two teams did not interconvert their calculations.
And of course, students in the US have to learn both systems. It’s just goofy.
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Yes, radians are a more organic way of measuring angles. Radian measure relates the angle to the radius of the circle, which is the crux of it. Also radians have no units. They are pure number and very useful in functions.
We have 360 degrees because way back in the days of Babylon it was believed that a year was 360 days in length and hence 1 degree was one day’s worth of travel in the night sky. Cool stuff.
Radians are for more higher math usage whereas degrees are less abstract and useful for more tangible applications.
I love your insights. You have a great mind.
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I think you are right to a degree.
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I’ll take what I can get.
I’d rather be right to a degree than to just 0.0174 radians
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Right to a Degree
Right to a degree
But wrong to three five nine
Unless the angles be
Counterclockwise twined
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I can see your radiance.
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“Rad to high Degree”
I’m really rad
To high degree
And never grad
At all, you see
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I’m really rad
To high degree
I’m Two-Pi-R’d
A circle, me
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SDP– Well you get to try it out on this math-dumb person. When you say “angle,” I picture the straight line each end of the arc traces to the center. But you may actually be talking about the literal distance traversed along the arc. Maybe I need a map! Words don’t do it. How does a degree have no mathematical relationship to the circle? Gotta explain it to me better, can’t make head nor tails of your post.
Not saying you aren’t a good teach IRL, SDP, and I have to admit, talk about “multiple systems of measurement” is intriguing. Wish I could sit in on your classes.
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When it’s said a degree has no mathematical relationship to a circle, it means that unlike a radian, there’s nothing about the structure of a circle that indicates one degree must be 1/360 of the way around it. There are possible reasons for the use of 360 as the number of degrees < https://www.quickanddirtytips.com/articles/why-does-a-circle-have-360-degrees/>, but they’re all convenient applications of elements of the environment or culture of the societies who used that measure. For instance, the use of a 360-day year in early times meant that one degree was the angular distance the earth moved around the sun each day, but that applied only to the particular circle of the earth’s orbit. It’s not intrinsically related to the construct of a generic circle. That makes it relatively arbitrary compared to the radian.
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Thank you!
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A great series, Gary. I read it all and left comments on the first one that Diane reposted. Well done. I wish everyone were as thoughtful about their practice as you, though I know that a lot of teachers are. I despair, however, of seeing any real change despite the record of utter failure here. Why? Well, every beast of the field has an opinion on this subject, almost all of them kneejerk, and so we will most likely continue doing what we have been doing despite the gargantuan failure and waste of it. Inertia because it’s impossible to get the stakeholders to agree on the necessary changes. How do we change that? Or will people 50 years from now, shortly before the AIs take over from us, still be reading Lockhart’s Lament and saying, yup. He nailed it. And then doing nothing.
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Click to access LockhartsLament.pdf
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Useless”
Useless math
And useless lit
Useless graph
And useless writ
Useless biz
And useless buzz
Useless is
What useless does
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I’m agreeing that math helps with logical thinking and should be taught. HOWEVER, right now it is taught to the exclusion of other very important things, such as civics and basic world knowledge, which ARE used every day. Could some hyper-advanced math be cut in order to focus on civic engagement and world knowledge? Perhaps.
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Math and civics could be combined eg, by teaching about ranked choice voting and the electoral college.
There are also lots of ways that math can be and is applied to understanding the world and its people.
Maybe what is needed is a more interdisciplinary approach.
How much of that is even done between math and science departments in k-12?
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I NEVER gave any credence to the, “Oh, it isn’t useful but helps to build thinking ability” argument. The same one was used for teaching classical Latin to all kids. LOL.
It’s a bs argument. MUCH better things could be done to that end (building neural pathways for thinking). But that’s a very long conversation that I am not going to get into.
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My basic argument is, if you want to teaching procedures for thinking carefully, then teach procedures for thinking carefully. LOL. Don’t teach Latin declensions and operations on matrices in hopes that kids will build thinking skills from those. That’s like setting your house on fire so you can make S’mores.
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Bob– Um, wait a minute. My measly 2 yrs of hischool Latin helped me understand the Eng lang better, and helped immeasurably in studying Fr, Span, Ital. Even German, because I’d already gained that sense of waiting to end of sentence for the verb (not to mention declensions).
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I did not say that Latin was worthless. I had the Latin intensive reading course in college, and I have used this all my life. But would you really want to go back to requiring Latin of all K-12 students? I don’t think so.
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Well, that shoots down my theory! (or should I say “theory”?😀) Seemed like a good idea at the time. Since I’m not any kind of educator/mathematician/scientist, I have to defer to your judgment.
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I’m not entirely sure that you are wrong, Mr. Rothbart. Studying this stuff probably does grow neural pathways for thinking, but I suspect that that can be done more directly and effectively for people generally.
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OK, Bob — thanks for the encouragement! As I sad, it’s pure speculation on my part. Even if my concept is plausible, there’s certainly no reason to believe Rhee aren’t other, & possibly more efficient, ways to accomplish the same sort of neural development.
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Oops— “Rhee” = “there.” Gotta love that spellcheck.
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The essential problem with school math education is that we think of it as a list of topics. I haven’t read the full three-part series and maybe he comes to this same conclusion. The individual skills we teach are not the point, and we’re not primarily training people to be better consumers. In Geometry one day, we rotated a regular pentagon around its center to discover some facts about the interior angles. It turns out that each central angle has 72 degrees and each interior angle has 108 degrees. Does any of that matter? In math we like to have the students generalize their results. In this case, we would generalize to a regular polygon with an arbitrary number of sides n. Does the formula for interior angle measures matter?
I don’t think any of those things will be important, but the process of finding out is math reasoning. We’ve been trapped by the need for uniform, machine-scorable tests that compare students and sort them into a rank order. And the same with teachers. In math education, we wants students to know, understand, and be able to do. Recollection of knowledge is easy to test, as is ability to perform a task. We trick ourselves into thinking those two things indicate “understanding” and simply don’t worry about that part, but it’s the most important.
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Ravi– Bravo! “the process of finding out is math reasoning.” I was terrible in math [except geom] K12, but it was this “process of finding out” that intrigued me, and keeps me interested in the subject to this day.
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Well, Gary’s list of ignorable topics needs to be thought over, imo. For example, he writes
“#5 Radian measure”
Well, measuring angles in degrees is a completely arbitrary decision, and as sound as measuring length in inches, feet, yard, miles: there is better, namely the metric system. Measuring angles in radian has a geometric meaning; it’s the length of the arc, corresponding to the angle, on the circle of radius one. Yeah, pi appears scary for most kids, but it’s because it’s not taught well; on the other hand, there is no way around pi since it’s everywhere where circle is.
Gary then writes ” The extra layer of confusion prevents students from being able to understand the more important concepts in trigonometry that are in the course.” But the reason trigonometry needs to be learnt to some degree is for students to understand periodic motions (such as springs’ or clocks’ motions) , waves, and there degree is inappropriate.
We already talked about the necessity of understanding the exponential function, and periodic functions are closely related to them. They are the most important functions, not quadratic functions or polynomials.
The connection between using degrees or radian measure is that wherever we see 360, we replace it by 2pi. Both look arbitrary, hence need to be memorized, but one is completely artificial, the other one is organic to math and nature.
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The other one on the blacklist I would argue against is the greatest common factor and least common multiple, since they are also related to periodic motions and processes.
What I certainly put on the list is solving endless equations, whether they involve linear, quadratic or trigonometric functions, square roots. Also forget about all those countless identities in trigonometry, and forget about trig functions except sin, cos, tan.
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Newton loved Trig
Isaac Newton loved his trig
Kind of like he loved his wig
Both inclined to make him
scratch
On his giant noggen, natch
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Ours is not to reason why
Just to add and multiply
Integrate and find a root
Factor out the x to boot
Ours is not to reason why
Ours is but to 2pi i
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