What makes teaching math special
Related:
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Designing good math curriculum for elementary and high schools requires one to have two kinds of expertise: deep understanding of math, and lot of experience teaching kids. Having just one of them is not enough. People who have both are rare (and many of them do not have the ambition to design a curriculum).
Being a math professor at university is not enough, now matter how high-status that job might be. University professors are used to teaching adults, and often have little patience for kids. Their frequent mistake is to jump from specific examples to abstract generalizations too quickly (that is, if they bother to provide specific examples at all). You can expect an adult student to try to figure it out on their own time; to read a book, or ask classmates. You can’t do the same with a small child.
(Also, university professors are selected for their research skills, not teaching skills.)
University professors and other professional mathematicians suffer from the “curse of knowledge”. So many things are obvious to them than they have a problem to empathize with someone who knows nothing of that. Also, the way we remember things is that we make mental connections with the other things we already know. The professor may have too many connections available to realize that the child has none of them yet.
The kids learning from the curriculum designed by university professors will feel overwhelmed and stupid. Most of them will grow up hating math.
On the other hand, many humanities-oriented people with strong opinions on how schools should be organized and how kids should be brought up, suck at math. More importantly, they do not realize how math is profoundly different from other school subjects, and will try to shoehorn mathematical education to the way they would teach e.g. humanities. As a result, the kids may not learn actual mathematics at all.
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How specifically is math different?
First, math is not about real-world objects. It is often inspired by them, but that’s not the same thing. For example, natural numbers proceed up to… almost infinity… regardless of whether our universe is actually finite or infinite. Real numbers have infinite number of decimal places, whether that makes sense from the perspective of physics or not. The Euclidean space is perfectly flat, even if our universe it not. No one ever wrote all the possible sequences of numbers from 1 to 100, but we know how many they would be.
If you want to learn e.g. about Africa, I guess the best way would be to go there, spend 20 years living in various countries, talking to people of various ethnic and social groups. But if you can’t do that… well, reading a few books about Africa, memorizing the names of the countries and their capitals, knowing how to find them on the map… technically also qualifies as “learning about Africa”. This is what most people (outside of Africa) do.
You cannot learn math by second-hand experience alone. Imagine someone who skimmed the Wikipedia article about quadratic equations, watched a YouTube channel about the history of people who invented quadratic equations, is really passionate about the importance of quadratic equations for world peace and ecology, but… cannot solve a single quadratic equation, not even the simplest one… you probably wouldn’t qualify this kind of knowledge as “learning quadratic equations”.
The quadratic equation is a mental object, waiting for you somewhere in the Platonic realm, where you can find it, touch it, explore it from different angles, play with it, learn to live with it. Only this intimate experience qualifies as actually learning quadratic equations. (Even if you are completely ignorant about who invented them in what century. That is an interesting trivia, but ultimately irrelevant.)
Second, math is not about social conventions and historical accidents. We can decide (individually, or collectively as a civilization) somewhat arbitrarily which mathematical objects we pay attention to, but once we made that choice, the rest is given and we need to figure it out. Once you decide to use natural numbers, 2+2 is already 4, even if no one knows it yet.
The only way to learn that Paris is the capital of France, is to obtain that information from a credible source. The facts that Paris or France exist are historical accidents; they may not exist in parallel universes. The fact that Paris is the capital is also a historical accident; given different history, it could have been some other city. Even the fact that we have the concept of “capital city” is a social convention.
The only way to learn how “cheese” is spelled, is to obtain that information from a credible source. Yes, it comes from Latin “caseus”, but that just passes the buck, and also doesn’t explain why the spelling has changed this way rather than some other way. Or why by a historical accident English didn’t import some other nation’s word for cheese—perhaps there was no sufficiently impressive cheese empire nearby, but in a parallel world that could have happened.
This is why I wouldn’t trust a geography teacher, or an English teacher, to design a math curriculum. They would probably unconsciously import the assumptions valid in their subjects, which would be wrong for math. I would expect them to teach mathematical theorems as arbitrary rules to be memorized, rather than show why they are inevitable given the nature of the mathematical object.
By the way, did you memorize or did you derive it? If you forgot the equation, could you re-derive it without the help of internet? I am asking because this was the point when my personal nullius in verba hit the wall at high school. I despaired: “oh no, here comes the point when I stopped understanding math; from now on I will only be able to memorize it, just like everyone else does”. Luckily, it turned out that the following math lessons made sense again, and a few years later I re-derived that equation. And maybe this specific equation is not so important in a larger picture, but generally, memorizing instead of deriving is a bad habit. If you memorize too many things, you will forget them soon after you stop using them actively. Also, each memorized theorem locks the door to something else; if you can’t solve the quadratic equation (and you have internalized that as “not necessary”), you are not going to solve the cubic equation either. How many other doors have you already locked?
(I am not commenting on subjects like physics, because I am not an expert there. It seems to me that physics has a lot of real-life knowledge and a lot of mathematical models, so designing a good physics curriculum would probably be even more difficult. Or maybe easier, because you could choose one or the other for elementary school, and leave the rest for the older students? I honestly don’t know. Anyway, university professors of physics will probably have the same problems as university professors of mathematics.)
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So, here comes the love-hate relationship with constructivism.
On one hand, some constructivist approach in education is necessary, because learning math is what constructivism in the sense of Piaget and Vygotsky was about, probably more so than other subjects. You internalize the mathematical objects. Merely looking at them (memorizing the definitions and theorems) is not enough; you need to digest them (explore the mathematical objects in different contexts, find their relations to other mathematical objects you know, explore different cases, explore the what-ifs).
Notice how LLMs suck at math, by the way. IT industry has created a humanities polymath before it could create a machine with 3-rd grade math knowledge, LOL. One of the reasons is that mathematics is not about knowing the facts, but using the tools.
On the other hand, constructivism seems hopelessly infested by the plague that calls itself “radical constructivism”, or more frequently just “constructivism”. The kind where people believe that 2+2 is a mere social convention, and might equal to anything else, if different people invented math instead. The current discourse seems to focus on the false dilemma between something called personal constructivism and social constructivism, where the former seems to assume that kids can derive all knowledge without any help from outside, and the latter seems to assume that all conclusions are arbitrary and therefore the kids can only learn them by memorizing. (Or they should be brave enough to reject social conventions and choose their own truth, because there is no such thing as truth anyway.) I keep saying “seems” because I don’t really understand most of this, and the more I find out, the more it makes me depressed. Asking GPT-3.5 to summarize their position, radical constructivism:
rejects the idea of an objective reality that exists independently of the observer
reality is considered to be a mental construct
In education, the emphasis is on fostering an awareness of the subjective nature of knowledge. It suggests that educators should recognize and respect individual interpretations and avoid imposing a single, objective understanding.
Ok, fuck that. You definitely do not want those guys to design a math curriculum. (Or anything else.) But it seems inevitable that they would insert themselves into the debate. It seems also inevitable that you will be associated with them once people sense that you have something to do with constructivism (even in its Piagetian/Vygotskian form).
But, you know, constructivism (Piagetian/Vygotskian) is basically this:
learning by actively solving problems, thinking and talking about them
using specific tools (think: Montessori) that help us understand abstract ideas
students create mental models of the world, and try to fit the knowledge there
students can figure out things within small inferential distance on their own
so the textbooks should be arranged accordingly
learning is supposed to be a pleasant experience
And by negating these things, you get:
students learn by memorizing abstract statements they do not really understand
and as a consequence, they are stressed and feel stupid
And I think most people who are good at math would agree that it should be like the former, and most people who hate math would agree that their lessons at school actually were more like the latter.
So we just cannot ignore this topic.
As of now, the state of the art of debating math education seems to consist of mutual yelling between math professors who say “the abstractions that I work with in my daily job are the real, high-status math, and if the kids don’t get it, it’s their fault, not mine; I don’t mind if most of them grow up hating math, as long as a few talented students get it anyway (frankly, usually from sources other than school), and those can become my PhD students”, and those who should properly be called New-Age Math activists who say “dude, math isn’t even real, so how could you teach it?”.
We need to have a different kind of talk.
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Let’s address the false dilemma between “personal” and “social” more closely. Math as such, is impersonal. (You could solve equations alone on Mars.) But the process of learning math is inevitably social. (A child couldn’t learn math alone on Mars, without teachers, without textbooks.) You learn about most mathematical concepts because other people care about them. Only a few people invent their own math, and even that only comes after they learned the existing one.
Furthermore, some aspects of how-humans-do-math are grounded in our social experience. Consider the notion of “proof”. Now that you have sufficiently internalized it, it can be a solitary activity of checking something thoroughly. But the psychological base this was built on is “words that would convince another human”. First you learn to prove things to others, and only after you get good at it, you can reliably prove them to yourself even when no one is around. (More generally, as Vygotsky would say, first you learn to talk, then you learn to think.) You cannot reinvent the math without the social experience of doing math together. And if you are doing things together, but those things aren’t math, you cannot reinvent it either. But if you learn the math together with others, you can continue to explore more of it alone. And the results you derive—if your math was correct—will be the same as the results anyone else derives (no matter their race or gender; sorry for dragging politics into this, but other people already do so).
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Perhaps the idea of “learning math” is itself a bit confused. There is no such thing as knowing all math—even Terence Tao doesn’t know everything about math, no offense meant. Once we accept that everyone’s knowledge of math is limited, we can talk about the trade-offs such as “knowing more things, superficially” and “knowing fewer things, deeply”. (Though we can agree that knowing very little, superficially, is bad.)
This is in contrast with e.g. language education, where there are natural Schelling points; for example, once you learn to speak Spanish as well as a native Spanish speaker, we might consider your learning of Spanish complete. (Even if there are still some obscure words you don’t know, and you could still study etymology of everything, etc.) Reading comprehension, similarly: once you can read a story and you understand it completely, you are done.
There are no such natural limits in math. Categories of “elementary-school math” and “high-school math” are arbitrary. Should you learn the basics of calculus at high school? I guess, if your country teaches it that way, you will probably say “yes”, and if your country doesn’t teach it that way, you will feel that calculus is clearly university math. We can agree that addition and multiplication belong to elementary schools, but at the boundaries it gets blurry.
I am saying this because no matter what curriculum you design, there will always be someone complaining that you could have added an extra topic or two, if you only didn’t waste so much time practicing the trivial topics. And therefore, your curriculum sucks.
But of course, if you comply with that, people will notice that your students learn a lot, but they also forget a lot, quickly. And they are stressed, because there is so much information, and so little time to understand it. Therefore, your curriculum sucks, too.
Saying “we could test the mathematical knowledge” just passes the buck. Does your test include a lot of superficial knowledge from diverse parts of math? Or does it torture you with complicated examples from a specific part of math, and ignores everything else? (Who decided which part of math, and why?) This is how the tests can drive the curriculum.
And of course there is correlation between math knowledge and success at test. The problem is that at the extremes the tails come apart. Kids that are better at multiplication will be better at the tests. But the kids who score maximum at the tests and the kids who win math olympiads, are probably two distinct groups. I am saying this as a former kid who did well at math olympiads, but often got B’s at high school, because of some stupid numeric mistake. The opposite example, I imagine, is some poor kid whose parents insist he or she must drill dozen textbook exercises before breakfast, so the kid always gets the A’s but will probably never have a single original thought about math.
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Okay, time to conclude this rant. Is there a conclusion to be made?
I think if you want to fix math education, you need to start with lower grades first. Because the higher grades are constrained by what knowledge the kids already come with. So you should fix the elementary-school math first, then the high-school math, and maybe then the university math.
You should be looking for people who:
have a PhD in math, or won a math olympiad, and
were teaching (1) math (2) at an elementary school (3) for the last ten years, and (4) their pupils don’t hate them—all four points are non-negotiable
You probably won’t find many of them. They might all fit into one room, which is exactly what you should do next.
Then you should pay them 10 years of generous salary to produce a curriculum and write model textbooks. You need both of that. (If you let someone else write the textbook, the priors say that the textbook will probably suck, and then everyone will blame the curriculum authors. And you, for organizing this whole mess.) They should probably also write model tests.
The system should have a lot of slack:
the curriculum should only cover 80% of the available time, so that the teachers can make their own decisions how to use the remaining 20% (whether it means learning something outside the curriculum, or repeating the topics where the students struggle most)
the first half of the first year at high school (and wherever else your school system has lots of students moving from one institution to another) should be spent practicing the stuff the students were supposed to learn during the previous grades (because chances are many of them actually didn’t)
tests should be written in such way that learning 80% of the curriculum gives you 100% at the test
Everyone else needs to shut up. There will be many university professors who never taught little kids who will provide “useful suggestions” (such as making the math more abstract, or not wasting time giving too many specific examples). There will be many self-proclaimed educational experts who will provide “useful suggestions” (such as memorizing more, and reminding the kids that math is just a social convention invented by the evil white men). These should not be allowed to disturb the people in the room.
The authors should be provided books written by Piaget and Vygotsky (and Montessori, Papert, etc.), but ultimately, the decision is on them. If something doesn’t feel right, just ignore it. Only use the books for inspiration.
Of course, you need to test the lessons with actual kids.
If it seems like the kids are having too much fun, you are going in the right direction. Keep going. (As long as it is still math; but given the selection of the people in the room this should not be a concern.)
You do not need to hurry. Build your house on solid foundations. Time is not wasted by making sure that students actually understand the lesson before moving on to the next one.
At the end, you win if kids following these lessons will understand the math and love it.
And then, of course, the new curriculum and textbooks should be tested on a randomly selected sample of all schools.
Oh, and don’t expect too much praise. Most people will complain anyway, either because this is not the math education as they know it, or because this is not the math education as they would have designed it. Expect to get a lot of complaints, often mutually contradictory. (Your new curriculum is simultaneously too easy and too hard. It either only works for the gifted kids and not for the average ones, or vice versa. It is simultaneously an irresponsible experiment and “just common sense, nothing special”.)
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Some ideas were taken from Ladislav Kvasz’s Princípy genetického konštruktivizmu (Principles of genetic constructivism). The article is in Slovak; as far as I know there is no English translation. All mistakes and exaggerations are my own anyway.
A related article from 2005 arguing that it’s important to teach elementary school teachers more mathematics in order for their teaching to improve: https://www.ams.org/notices/200502/fea-kenschaft.pdf.
This is perfect. Thank you!
I don’t think this is literally true—the subculture of the mathematically savvy people seems quite self-sustaining, especially because, as the author has previously correctly noted, they often ignore the school system and teach their kids mathematics at home (at least until the kids get sufficiently solid fundamentals so they can continue learning from books and videos).
There are also some great educational tools such as Khan Academy etc., where the problem is on a meta level: you need to be told about the tool, and you need the motivation to use it. (There is probably also a lot of educational snake oil out there.) The problem with these tools, from the perspective of the article, is that they teach math, but they don’t teach teaching math. They are great for the students, but the teachers need something different to do their jobs well.
Continual remedial programs for math teachers, definitely a good idea! (I assume they would be voluntary, which means the participants would be motivated.) Just maybe change the name to something like “math teachers giving each other advice”; anyone can join, anyone is free to give or not give a lecture, always invite one or two experts. Because you also want feedback from the teachers, what works for them and what does not, and also encourage them to try their own things.
(Congratulations for surprising me, by the way. I didn’t expect to read an article that has “racial equity” in title that I would agree with so much.)
I’m glad you liked the article! As for your one disagreement with the quoted passage, I think the all in the phrase “all university mathematicians” is key to her point. Mathematicians at prestigious private universities and well-known state flagship universities are indeed able to teach university-level mathematics due to the adequate supply of mathematically-prepared students that remains despite bad k-12 math education. But a large percentage of university mathematicians are at less prestigious institutions where very few students major in math and most of the demand for math courses comes from students who didn’t have a good math experience in k-12 schooling (and weren’t taught math at home by mathematically savvy parents) but need to satisfy a gen ed requirement.
Have you run into this problem actually happening or it just seems likely?
Just seems likely.
I have an experience with general population that their attitude to math problems is generally: “stop explaining, just tell me the rule so that I can memorize it”, not realizing the problems this predictably causes later. But that is general population, not geography or English teachers specifically.
(And I wouldn’t trust a typical math teacher to design a math curriculum either.)
Upvoted because it’s a good intro discussion to a problem that I am personally involved with (as a maths teacher). But my personal experience is that what makes a good maths curriculum is much more complicated than that. In particular I’m pretty certain now that different students have such wildly different needs that any attempt at (universal) standardisation is doomed to fail (of course some curriculum are still better than others...).
I think this is a key point. Even the best possible curriculum, if it has to work for all students at the same rate, is not going to work well. What I really want (both for my past-self as a student, and my present self as a teacher of university mathematics) is to be able to tailor the learning rate to individual students and individual topics (for student me, this would have meant ‘go very fast for geometry and rather slowly for combinatorics’). And while we’re at it, can we also customise the learning styles (some students like to read, some like to sit in class, some to work in groups, etc)?
This is technologically more feasible than it was a decade ago, but seems far from common.
One of my favorite math moments was actually a teacher explaining that some aspects of math are a social construct. Specifically, I was confused about why x^0 = 1 and not 0. There’s a lot of good reasons for this, but rather than starting with those, he pointed out that you could define it that way if you wanted to, and mathemeticians have just agreed on the 1 definition (and then he went into how the 1 definition is much more useful).
I remember getting the same answer to the same question at school and being quite surprised.
If I was asked the question now I would show the interested student this pattern, and ask them what makes sense for the last term.
x^3 = 1*x*x*x
x^2 = 1*x*x
x^1 = 1*x
x^0 = ?
(if they are still unconvinced invite them to approach 0 from the other side)
x^(-3) = ((1/x)/x)/x [up to brackets = 1/x/x/x]
x^(-2) = (1/x)/x [1/x/x]
x^(-1) = 1/x
In some ways the negative powers (if the student already knows those) are cleaner, because the implicit factor of 1 is made explicit in fractions. Although the need for brackets makes the pattern less elegant.
I think the main reason the x^0 = 0 definition is awful is that it means you can no longer rely on the fact that
x^n * x^m = x^(n+m)
(because for n=0 the left hand side is zero, but the right hand side is x^m, which may or may not be zero). Although I think that (depending on the age of the student) this might not be the best explanation.
Well, when you put it like this, the conclusion feels irresistible!
But it also feels like an artificial move. Not something the student would come up with, unless they already understand it.
I would probably go with:
x^3 / x = x^2
x^2 / x = x^1
x^1 / x = …complete the pattern, then contemplate on what “x^1 / x” is
I do exactly what you describe with my students, but sadly with extremely limited results.
Yeah I think showing the pattern is helpful, but I like the version where instead of showing the (under dispute) 1 term, you show the division pattern, like in this video https://www.youtube.com/watch?v=X32dce7_D48
Essentially the argument is that x^(n-1) = x^n / x applies in every other case, so why not apply it for x^0 also?
For me it was also helpful to point out that you could define x^0 = 0 (and in fact, some fields leave 0^0 undefined), but it would cause all kinds of other problems you’d have to account for and make arithmetic a lot harder and less useful.
Technically, you could define 2+2=5, but that would cause a lot of problems everywhere. :D
I agree that sometimes the definitions are motivated by “how do you intend to use that, and which definition will make your job easier?”
But I think that situations with two meaningfully different answers are rare. I imagine there is often either one good answer, or one good answer that brings some problems along (such as introducing a new type of number) so some people choose to leave it undefined. For example:
1⁄0 is either undefined, or we introduce some notion of “infinity”
0⁄0 is probably better left undefined, or some kind of “error” value
square root of −1 is either undefined, or we introduce complex numbers
a sum of zero numbers is = 0
a product of zero numbers is = 1
a union of zero sets is = empty
an intersection of zero sets is either undefined, or the entire universe (if such thing exists)
0! = 1
x^0 = 1 …maybe unless x = 0, in which case one might reasonable argue also for 0
parallel lines either do not intersect, or they “intersect at infinity”
infinity minus one is either undefined, or we introduce surreal numbers
Maybe I just selected examples that support my point. Feel free to add examples to contrary.
Seems to me that 0^0 is the only case where one could successfully argue for both “0^0=1” and “0^0=0″ depending on whether you see it as a limit for “x^0” or a limit for “0^x”. (Even here I feel that the former choice is somehow better, because it also extends to negative x.)
(In set theory, there is a disagreement about which axioms to use, and it seems to be resolved as “agree to disagree” but I am not an expert.)
The one I like is that for cardinal (counting) numbers, x^y counts the number of functions from a set of y elements to one with x elements. This is very foundational to how powers are defined in the first place, and in many ways even more foundational than addition and multiplication.
If y is empty (regardless of x), then there is exactly one such: the empty function. So x^0 = 1 for all cardinal numbers x including x=0 and all the natural numbers, but also all the infinite cardinalities as well.
From there you can extend to integers, rational numbers, reals, and so on.
Of course, from a pedagogical point of view it may be hard to explain why the “empty function” is actually a function.
Just to check, did you here mean 0^0 ?
It’s been a while since I did much math, but I thought that was the one that counterintuitively equals 1. Whereas 0^1=1 just seems like it would create an unwelcome exception to the x^1=x rule.
Er yeah, I’ll edit. Thanks!
I don’t think there is a clear problem statement anywhere about what is wrong with math (or any other) education, or what “good enough” looks like. Almost nobody writing about this really accepts the differing abilities, motivations, and social support (parents and friends) of the students. There CAN BE no single solution—the variance is just too great.
Personally, I focus on getting the most out of the top potential learners, which DOES lead to research-ey theory of education, as it is there to identify and encourage future researchers and advanced thinkers. This portion of education leaves MANY students behind, because that’s not what they need, and not how they interact with educators. I think there needs to be another tranche of “smart, well-supported, non-academically focused” students, getting enough math to inform their daily life and help develop a numerate mindset. I don’t know how to move the less-well-supported and less-well-equipped (regardless of reason) into the well-supported group.
This is one of the problems that can’t be discussed very well in public, as it’s currently outside of mainstream Overton window to admit that there is a pretty large cognitive and motivational variance among humans. In fact, I feel it’s important to state that I don’t think this variance is necessarily ingrained or permanent across generations or across an individual’s lifetime. But I do think it’s real.
I have a strong intuition that we are far from the Pareto frontier. For most kids, math is suffering, and at the end they do not remember almost anything. Maybe we could teach them more, maybe we couldn’t… but if it’s the latter, then at least we could make them suffer less.
There is probably a lot of signaling involved. By making kids needlessly suffer, we signal that we care about their future well-being. By suffering, kids signal their diligence. Maybe the math is made difficult on purpose, because difficult math more clearly separates the more talented kids from the less talented ones.
I agree that people should pay way more attention to the differences in IQ. I also think that people often use it as a lazy excuse. (“Hey, most kids don’t understand your lecture.” “Not my problem, large cognitive variance.”—conveniently ignores the fact that the kids who didn’t have a problem with the lecture were the ones who already knew all of that from some other source.)
Remember the “camel has two humps” paper about programming aptitude? At first, it seemed that only human nature denialists could disagree with the harsh truth. And then it didn’t replicate. What I am trying to say is that although ignoring the cognitive variance is popular, it is also popular to overcompensate in the opposite direction and blame basically everything on genetics, even when something is clearly wrong.
(For example, some people have told me that teachers didn’t explain to them that “fractions” and “division” are the same thing, only written differently. You can’t blame that on a difference in IQ, because those people then successfully learned skills such as adding fractions, they just never clearly understood what they were doing; they just operated the numbers mechanically according to the rules they learned.)
(This summer I had a lecture for a group of people about “how I would explain complex numbers to an 8 years old child”. After the lecture, several math teachers thanked me, saying that this was the first time they actually understood the concept. I am currently rewriting it as a blog post… that, and some research related to that, was actually what triggered me to write this post.)
Or, you know, people say how much they learned from some educational YouTube videos. Okay then, why don’t we make some of those videos a part of school? I mean, completely literally, why couldn’t we have each week two hours of “video watching lessons”? Perhaps make it elective, like you can either watch a math video from 3Blue1Brown or a history video by Sabaton (and then of course you can watch the other one at home, if you want to). Just one example of what seems to be a low-hanging fruit. Or when kids learn numbers from 1 to 10, why don’t we let them spend a lesson or two solving easy Sudoku?
Oh, quite! I tried to make that clear in my comment, but maybe it’s not possible to point it out without the assumption that one is on the complete opposite end. Sad, really—the multiple dimensions of ability, willingness, context, and possible improvements are too complicated to use a one-factor model.
“conveniently ignores the fact that the kids who didn’t have a problem with the lecture were the ones who already knew all of that from some other source.”
This is definitely not true in general and probably a rare case. N=1 of course, but I never had problems with maths lectures (or any other lectures) and I never was in the situation of knowing all of the maths before the lecture (I usually knew history and physics lessons in advance though). And it’s the same thing with my current students : even the best ones are clearly unfamiliar with the material I cover.
I think the lesswrong crowd has in general a very unusual experience with both school and maths, even compared to the average gifted maths student. Beware of the typical mind fallacy.
This argument is in no small part covered in
https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician’s_Lament.pdf
which is also available in 5-times-the-page-count-and-costs-$10.
The problem undergirding the problem you’re talking about is not just that nobody’s decided to “put the smart people who know math and can teach effectively in a room and let them write the curriculum.” As a matter of fact, both New Math and the Common Core involved people with at least all but point 2, and the premise that elementary school teachers are best qualified to undertake this project is a flawed one (if it’s a necessity, then Lockhart may be the most famous exemplar adjacent to your goals, and reading his essay or book should take priority over trying to theorize or recruit other similar specialists.
This is culture-dependent; e.g. in (some parts of Russian grading culture) making a numeric mistake might be counted similarly to other gaps in the reasoning. An attitude of mistake is a mistake is quite prevalent in grading for university admissions (in at least one of the few places they remain supplementing the standardized tests). Unsure how prevalent that was in olympiad grading, I never was competitive there (although did encounter this outlook a bit in my small stints of grading olympiad tasks).
[There is usually a bit of leniency if miscalculation happened to be in the last few calculations to be done. Otherwise, you are out of luck.]
In this culture poor calculation skills are punished, and you basically have to have them at a certain level in order to stay competitive. Another example of this phenomenon is competitive/olympiad programming—grading is done automatically, your solution has to pass the tests. It does not matter that you understand the idea (which is most of the time the difficult part); the implementation has to be correct.
[Depending on the competition: some give partial points for working on parts of the test set. Reiterating: this is a culture some popular competitions subscribe to and which I’ve had quite a bit of exposure to.]
Sidetrack: a coapproach to the mistake is a mistake is it doesn’t matter how you got the answer if it is correct. This is difficult to implement in grading because this makes cheating significantly easier, but has led to some of the most fun solutions I’ve had to math/programming problems. I vehemently despise the “you have to complete the task in exactly this way” problems.
There’s a big gap between “you have to complete the task in exactly this way” and “mistake is a mistake, only the end result count”.
I routinely gives full marks if the student made a small computation mistake but the reasoning is correct. My colleagues tend to be less lenient but follow the same principle. I always give full grade to correct reasoning even if it is not the method seen in class (but I quite insistently warn my students that they should not complain if they make mistakes using a different method).
The advantage of “a mistake is a mistake” approach is that it makes it much cheaper to evaluate. Imagine having to grade tests from 100 students—how much time would it take if you only check the results, and how much time if you also check the process.
It also depends on how much the teacher’s verdict is final, or how much the students are allowed to argue. Imagine that out of those 100 students, at least twenty want to debate you individually on whether their processes were only 60% correct or 80% correct (because a partial point here and a partial point there might together make a difference in their final grade). It would drive me crazy. Also, it would be unfair against the less “litigious” students.
A mistake of the same magnitude can have a different overall impact, depending on where you make it. If at the last step of long computation you accidentally do 20+50 instead of 20-50, but everything else was correct, it is obvious that you would have solved the problem correctly. But if you made exactly the same mistake at the beginning of the problem… it could have sent you on a completely different track. Like, maybe you needed to calculate a square root of that, and then you solve a task with real numbers, while all your classmates were solving a task with complex numbers. If the point of the test was to verify your knowledge of complex numbers, then this completely failed the purpose. And yet, it was the same mistake.
I think my preferred option (assuming “veil of ignorance” where I can either be the student or the teacher) would be: a mistake is a mistake, but you can take the test again (with slightly different questions).
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At math olympiad, both your process and your answer matter. The process matters in the sense that it has to be mathematically sound (but it can be completely different from what other people would use). If the answer is wrong because of some stupid mistake but the process is generally sound, you get partial points; so I think a typo in the last step would still get you 5 out of 6 points.
(You also submit your working notes, which is like potential extra evidence that can be used in your favor. This is optional—if the problem is so clear to you that you immediately start writing the official answer without any previous note, of course you will not be penalized for that. It’s more like… if the way you wrote something in the official answer is unclear, people will check your working notes for possible evidence how you actually meant it. You will not be penalized for also trying approaches that didn’t work in the working notes; that is a part of what they are for.)
However, if you are competitive, you aim for 6 out of 6 points, so we are kinda back to “everything must be perfect” again? I guess the difference is that if you get most of the problems 100% right and one or two of them are “generally okay, but made a stupid mistake that only had a local impact”, you might still win.
The other advantage of “a mistake is a mistake” is that it matches the real world. When solving problems because you care about the world, rather than because of a score, the process matters for repeatability, communication, and trust of others, but the answer is all that matters in terms of impact.
Most emphasis on rote learning leads to school students in not being able to sustain interest. Sadly, one of the subjects not taught in school is “learning how to learn” such as →
Barbara Oakley’s “learning how to learn”—https://www.coursera.org/learn/learning-how-to-learn
Richard Hamming’s “learning to learn”—https://jamesclear.com/great-speeches/learning-to-learn-by-richard-hamming