A few weeks ago, I’d have deemed this question pointless. Less a real question than a kind of vague hand gesture.
But I just read a great book that shifted my thinking. [EDIT: I forgot to say which book! It’s The New Math: A Political History, by Christopher Phillips.] Now, rather than a vague hand gesture, I see the hard/easy question as a quite precise and pointed gesture. (No, not that gesture.) Specifically, the question points toward another, bigger question:
What is math class really about?
One possibility: Is math education for cultivating a general excellence of thought? For building the capacity to reason logically? For sharpening the mind into an all-purpose tool?
In that case, it should probably be as hard as possible. The mind, like a pencil, needs friction to grow sharper. Not too much friction, or you’ll break the pencil (and the metaphor), but all else being equal, a good challenge is a good thing.
Or, another possibility: is math education for imparting a specific set of useful skills? Is it about data literacy, arithmetical know-how, and algebraic fluency?
In that case, it should be as easy as possible. Break it down into itty-bitty tasks, and practice each one until you have it cold. Math ed ought to be like drivers’ ed: so blindingly simple that any citizen can acquire these basic, necessary skills.
A simplified contrast, to be sure. But for me, it’s a surprising inversion of my usual thinking.
I most relish the challenges of mathematics when I’m in “traditionalist” mode, when I feel real faith in the skills and methods that constitute the historical discipline. (This isn’t my usual operating mode, but hey, I enjoy the chain rule as much as anyone.)
But wait. If manipulating polynomials is truly a human good, shouldn’t I want to make it easy? Shouldn’t Iwant algebraic literacy to become like English literacy, with 99% of kids achieving it?
In the moments when I’m most indulgent of “hard,” I perhaps ought to be pursuing “easy.”
Meanwhile, I’m wariest of math feeling too hard when I’m in “progressive” mode, envisioning a human-centered math education. On these days, I tend to think that math is not about a specific skill set, but expanding our intellectual toolkit and enhancing our ability to reason.
But if this is my priority, why should I be bothered if math is painful or unpleasant? Sure, too much challenge will drive my students away. But if my task is improving minds, then aren’t hard days a good thing?
Of course, “hard” is not one thing—its definition varies student to student, task to task, moment to moment. There’s no “difficulty” dial I can turn. Still, these thoughts leave me with a strange conclusion that I’m struggling to absorb.
When math education is about the math, I should be striving to make it easy. And when math education is about the education, I shouldn’t shy away from making it hard.
I like this train of thought, thanks for sharing. What was the book that (helped) change your thinking?
Same!
Same – also wondering what book you mentioned
Whoops, should have mentioned it in the post! It’s Christopher Phillips’ “The New Math: A Political History.”
The polynomial puzzle looks like good annoying fun. But are we supposed to assume that all the coefficients are real?
Ha, yes! Real coefficients.
How can that be a degree 9 polynomial? As is, it’s at least degree 8. But since you say that all of the real roots are given (you gave us two of them: 2,3) and complex roots come in conjugate pairs then it must have an even degree.
What if one of the real roots has a multiplicity of 2?
Hint: consider the polynomial x^2, which is degree-2, but has only one real root!
Hint: Repeat roots. Got me for a second as well.
One of the two real roots has a multiplicity of 2.
-20 or -40? nice one btw
I got the same answers.
Enjoyed the challenge!
Math should be fun. Sometime that’s hard. Sometimes easy. (But test should never be so hard as to make them unfun.)
What was the name of the book that shifted your thinking?
Several thoughts:
Education should explain what math IS, introduce mathematical concepts, AND teach practical skills. But kids should know there are concepts and ideas that exist even if they don’t thoroughly understand them. They should also get a taste of how math is everywhere.
For kids and adults who want a deeper dive, for whom the above is not enough, there should be more available, at every level. But even calculus is not for everyone.
Pre-pandemic I volunteered an at elementary school, half an hour once a week for a 3rd, 4th and 5th grade class. FINALLY, I’ll be able to do my schtick at another school, starting with 4th grade. I give them a taste of everything. They love the golden rectangle and the fibonacci sequence and the golden ratio and how it’s EVERYWHERE. They love learning about probability. They love learning about the history of counting and why zero is such a special number. They LOVE learning about infinity and that there are some infinities that are bigger than other.
It blows their little minds. I stress again and again that math is the language of the universe (because that’s what I believe). At my last school, I was called the Math Lady. I’m SO glad I’ll be able to work with another school.
Mmm, I’ve also been thinking a lot about this idea you mention: “math class should tell students what math is.” Topic of my next book, actually — I found it took me a solid 200 pages to give a coherent, accessible, illuminating answer to “what is math?”
(At least, I *hope* it’s coherent, accessible, and illuminating! It’s entirely possible it took me a solid 200 pages to *not* answer the question.)
I am a math and comp sci major. I think everyone should be taught basic arithmetic life skills, but not everyone should be taught math thinking, esp. when they may end up not even learning basic arithmetic life skills. That is, math thinking should be an optional add on as many people will not care about this aspect and/or not be able to do it successfully.
I’m not sure I agree. Knowing how to think mathematically is necessary for high school courses such as algebra, which are required in order to graduate. You can’t understand solving for x unless you understand what an equation is, what the goal is (x = #), and why equations are useful. Rote arithmetic isn’t even going to get you a high school diploma by itself.
Comparing studying math to English is an interesting exercise if you take it a little deeper. In English class you study the things you need to get around in the world—grammar, syntax, spelling. The nuts and bolts. Not hard, not especially interesting, requires a lot of practice, and is of practical importance. You also read Shakespeare etc, and analyze and discuss and write about your ideas. This is hard and interesting. Not strictly necessary for getting through life successfully, but it certainly oh trains you to think in a way that makes life more interesting—the “liberal arts” approach. And why shouldn’t we think of math the same way?
Yeah, I think this is right!
Some elements of math education (particularly those taught prior to age 12) feel pretty essential to me. Everyone should understand fractions, proportional thinking, the basics of geometry and measurement, etc.
Other elements (especially those that dominate high school math) make more sense under a liberal arts framework.
But there’s also a tricky third dimension to math education, which is the “STEM professional training” angle. In the case of Shakespeare, this isn’t really a separate dimension, because there’s not much daylight between “liberal arts-style critical thinking” and “the skills you might need for a communication-intensive job as part of the managerial elite.” But in the case of, say, linear algebra, there may be more of a gap; the humanistic goals and the technocratic training goals may not quite coincide.
I don’t see it. I think all brains, if pressed, can go deeper into English/Literature. For whatever reason, I don’t think all brains can go deeper in mathematics, it’s a different way of thinking. We all need to communicate, we all need to speak, and read, and understand. We don’t all need to do much more than arithmetic, and with cell phones everywhere, most people can’t even do that.
Now, gamers, and gamblers are doing more sophisticated math, even if they don’t know it. So…
I think that’s too pessimistic about brains, Debby!
About 200 years ago, global literacy rates were below 15%. It would’ve been very reasonable to think that some brains just can’t get reading. Such an artificial activity, isn’t it? Scanning your eyes along strings of symbols and determining what ideas they encode? In every civilization thus far, literacy had been either (a) nonexistent, or (b) the exclusive purview of a highly educated elite. What kind of crazy dream was *mass* literacy? And what kind of even crazier dream was *universal* literacy?
Fast forward two centuries, and the global literacy rate is north of 85%. In lots of countries it’s around 99%.
Not easy progress, obviously! It took about 15 other history-shattering transitions and changes to get there — public health improvements, social movements, and the greatest streak of economic growth in the history of the species.
Anyway, I have no doubt it is *hard* to teach algebraic fluency to the entire human population! But on the scale of generations, I see no reason to think it is *impossible*.
In fact, massive test results show the opposite. Difficult literature or vocabulary tests have far less frequent high scores than difficult math or science tests.
Interesting question. I don’t think there’s a right answer that applies across the board. It may be that math should be easy at first but should be made progressively harder as students acquire more facility with core concepts. This is suggested by cognitive load theory (discussed in J. Sweller’s article Human Cognitive Architecture), which posits we have limited resources available for learning and can’t learn if the task exceeds them. The idea is to reduce unnecessary load so that cognitive resources don’t become overloaded – i.e., make it easy (in the math context, for example, by using worked examples and partially completed problems), avoid redundancy (present the information in only one way), use isolated interacting elements (learn separate items first and then learn how they are connected) and different modalities. (You did this with the drawings of the cubes, etc.) As the student’s proficiency increases, cognitive load theory suggests you want to increase demand on mental resources (reduce the guidance, have them focus on the whole rather than the sum of its parts, include redundancy and have them reach) – make the student the teacher.
Great post! I love the polynomial question. I wrestled with it a bit and then came up with -20 or -40 depending on whether 2 or 3 is a double-root.
I think you overestimate the literacy of Americans—nowhere near 99% achieve it:
“On average, 79% of U.S. adults nationwide are literate in 2023.
21% of adults in the US are illiterate in 2023.
54% of adults have a literacy below a 6th-grade level (20% are below 5th-grade level).”
https://www.thinkimpact.com/literacy-statistics/
Interesting! There are evidently multiple definitions of literacy at work here — the detailed US-specific data suggests a number around 80%, but the table of international comparisons at the bottom gives 99% for the US.
I suspect the 99% figure uses a very inclusive definition of literacy, by which an at-grade-level 6-year-old would count as literate.
Obviously, it’s very troubling to have adults only able to read at a first grade level. But this is the proper apples-to-apples comparison with the year 1800; back then, fewer than 15% of people met even this meager standard.
It’s a little scary that so few people were literate back in 1800 until you remember that a lot of languages that now have a written form didn’t back then. Cherokee comes to my mind immediately, but I’m sure this was also an issue for a lot of other languages. When your primary language doesn’t have a written form, literacy is irrelevant.
I hadn’t considered that — it’s an interesting point.
That said, the driving force is just that the world of 1800 was, by the standards of 2023, desperately poor. Real median income worldwide was less than $1k per year in today’s dollars, and life expectancy in every country was below 45. Not good conditions for mass education (which as I understand it didn’t really exist yet).
I think a significant portion of the illiterate adults might have cognitive disabilities such as dementia or aphasia. Also, the literacy tests might be in English, which would make it harder for some immigrants to succeed.
I think there’s a crucial distinction to make about _why_ someone thinks a specific math problem is hard. Is it because they tried and couldn’t solve it? Is it because they didn’t know how to start and they feel discomfort with not knowing (which materializes as “this is hard”)? Is it because it’s perceived to have lots of moving parts to consider and they’re overwhelmed by the information? Is it because it uses symbols they don’t know? In my view these are drastically different situations that “hard/easy” doesn’t adequately convey. For example, I think it’s an educator’s duty to make problems hard in their open-endedness, but easy in symbol interpretation.
Yeah, you’re definitely right to push on that undefined term. Though your decomposition isn’t one I’d had in mind, I like it. Sort of breaks down into three phases:
(1) hard to *understand* the question itself, perhaps because you struggle to interpret the symbols;
(2) even if you grasp the question, hard to *get started* working productively, perhaps because you’re uncomfortable with open-ended problems;
(3) even if you’ve made a reasonable start, hard to *actually solve*, perhaps because a correct solution requires you to coordinate many moving parts.
Trying to retrofit my post into this scheme, I believe I’m writing mostly about #3. And I’m thinking less about the absolute hardness level, and more about this two-part distinction: (1) is the hardness under my control, and (2) is it desirable?
If mathematics education is mental exercise for the sake of building general problem-solving skills, then Hardness #3 is (1) largely under my control, because the specific content is just exercise equipment, to be used or set aside as I see fit; and (2) generally desirable, because harder problems will (all else being equal) build greater problem-solving skills. Hence the polynomial puzzle above.
But if mathematics education is about mastering an existing set of disciplinary skills and understandings, then Hardness #3 is (1) largely beyond my control; and (2) generally undesirable. Under this view, I should be trying to make each of these “moving parts” feel as simple and graspable as possible, by weaving in simple, low-effort practice that builds comfort with the objects. Hence the polynomial “translation” exercise above.
Anyway, that’s the distinction I’m drawing. Reading it over now, I’m not sure how convincing I find it myself! I suppose it’s just another dichotomy for my toolkit of dichotomies — one more low-dimensional space into which I can occasionally project my thinking about math.
Reading this while giving a math final. I will be thinking about that last paragraph for a while.
Math is hard. No matter how “good you are at math” it will eventually become hard. And, and that is as it should be. Math pushes logical thought to its extremes.
But, should mathematics education be frustrating? Yes, but within limits. The greatest opportunity for learning is in the midst of a struggle — but it must be an surmountable struggle. Make it too hard, and your students will give up dissatisfied. Satisfaction and enjoyment in learning math comes from facing something difficult and eventually beating it.
The trick lies in the balance
Ummm, as stated the answer to the polynomial question is any real number except zero.
Let P(x) be a polynomial with the solutions or zeroes as described.
Let P(1) = a
Then c*P(x) = c*a for any choice of c. You get the idea from here..
Oops, I missed the leading coefficient = 1 part. I don’t see a way to delete my response.
Interesting question! While rigorous math is essential for STEM fields, perhaps the focus should be on conceptual understanding and practical application. Making it too hard can discourage students. For a fun break, I play Retro bowl during study breaks, it helps me clear my head!
I totally agree that the “hard vs. easy” question boils down to the purpose of math education itself. I used to think math should be challenging to “train” the mind, but your point about accessibility makes total sense. It reminds me of my own frustrating but sometimes rewarding moments playing Eggy Car. Sometimes the “hard” levels are infuriating, but when you finally nail a tricky jump, it’s so satisfying.
Slope Unblocked is perfect for players looking to sharpen their hand-eye coordination and reaction speed. The game demands 100% concentration, as a single mistake can end your record-breaking run in an instant.