I'm afraid that this is because academic math is often taught and tested in a way that rewards memorization rather than understanding. Here's Richard Feynman's take on the problem:
I had a similar reaction. I had a lot of reactions, and found the paper interesting.
First, it reminded me of something a stats professor said in grad school: "there are two kinds of mathematicians, those who are good at arithmetic, and those who are not." He was speaking as someone who identified with the latter.
I can't tell if this is something related to this domain of math in particular or something broader. My guess is it's something broader.
I have colleagues (speaking as a professor) who have complained about admitted students who come in with very high grades and test scores, but who can't actually reason independently very well and despair when they are not "told exactly how to respond" on tests and whatnot. You have to be careful because sometimes these complaints hide bad teaching, but I think this is a common sentiment, and I've seen articles written about similar sentiments at other places.
The paper touches on a lot of issues, like applied versus abstract concepts, generalizability of learning, "being a good student" versus actual cognitive ability, learning how to take tests versus learning concepts, the difficulty of measuring cognitive and academic ability, and the fallibility of measuring complex human attributes in general.
Even in lower education, as a student I hated word problems. Partly, I just wanted to be told what equation to solve. In retrospect, though, I think a lot of it was the framing.
It was always presented as some variation of short exposition followed by a question. The question was usually framed as an outside observer asking for some fact about the story.
Think of the classic "A train leaves station A headed west at 6:30 traveling at 30 miles an hour. A second train leaves another station at 7 traveling 50 miles an hour. When do they pass each other?". There's no problem here to solve. Who cares when they pass each other? Why do we care?
Sure, a little exposition helps build up analysis and application skills, but it doesn't actually offer much in the way of engagement.
I was a college math and physics major, and much later taught a college freshman math course that was a level below calculus.
The point of word problems was to recognize a pattern matching one of the topics from the latest chapter, fill in the parameters, and grind through the memorized algorithm. As a student, I liked word problems, but I knew the secret. It was all a game.
What made math come alive for me was proofs. As for applied skills, I developed those in the lab, and making things.
True, looking at real proofs is what changed the game for me.
Before I actually went through 3-4 books on basics of proofs, math felt... almost meaningless , a game of remembering the right thing at the right time.
Saying that as somebody who oscillated between being "good in math" and "top in class" for all 18 years of studying.
To me proofs never where the interesting part of maths - the ideas and intuition which made the proof possible were.
Proofs were a way of formalizing something and, well, making sure the intuition was actually correct, but they were just a tool and not the game itself.
The best math teachers/professors I had were the ones who focused on the ideas .
Yup, and once again, it depends on how we learn. I'm a strongly "learn by doing" kind of person. For instance, I'd get almost nothing out of reading a math book that was full of ideas but no problems or proofs. Doing problems and proofs is how I wrestle with the structure of the subject matter, and internalize the ideas.
In elementary school, I hated word problems because I kept thinking of things that weren't specified which prevented there from being just one right answer. Sure, the car left City A at 60 miles per hour, but what if there's a stop-light? I know there are lots of stoplights, so it must go slower, and you didn't tell me how much slower it would go...
I like to think that I've turned it into an asset when it comes to software. ("We don't know that the first parameter won't be null...")
"Why should the reader care?" I agree, they aren't framed in a way that engages the reader.
How about for a division problem, start with a bag of candy, or if it HAS to be healthy, a bag of cherries.
Or maybe apply it to cooking. Lets use Metric anyway, even after ( https://en.wikipedia.org/wiki/Metrication_in_the_United_Stat... ) and ask questions about a recipe for some food dishes (use real ones! IDK maybe bread, pasta, some pastry stuff...) and ask things like the total expected volume based on the ingredients. How much X there should be if naively adjusted by exactly a factor of 1/2 or 3x etc. Things people might do if a thing was intended for a family of 4 rather than 2, or a group of guests at a holiday.
> There's no problem here to solve. Who cares when they pass each other? Why do we care?
Not trying to “but acktually” you, however, this is more or less how I calculate the optimal time to take a pit stop in a lap-based auto race. I have a little spreadsheet widget that I made to be able to plug the numbers in, but the problem is simply stated:
If old tires decrease my speed, and making a pit stop takes time, when should I stop.
Agreed that elementary school word problems are dumb, though.
The original problem, and the racing one, are both logistics problems, and everything in the world runs on logistics, and people have to be good at it.
If you don't like it or aren't good at doing it even while not liking it, the problem is not the problem.
I don't think you are arguing against the parents position, but for it, while your answers' odd contrarian positioning also exhibits how critical context and caring are to answering questions. Good job, you.
I think you've missed my point. There's no interesting logistics problem in "when do the trains meet?" Yes, the underlying math is useful for all sorts of things, but the word problem doesn't offer any motivation for knowing the answer. It's purely asking as an impartial observer.
Back when I took calculus in high school, my teacher explained how traffic speed cameras used mean value theorem to prove a car exceed the speed limit.
Here, there's an actual problem, actual actors and observers, and a motivation.
It answers the question why is this useful to know, or to be able to answer.
> There's no interesting logistics problem in "when do the trains meet?"
This is trivially resolved with "and the first train is carrying an urgent package for a passenger on the second train. When will the trains meet to deliver the package?".
But on the exam form, that's just extra irrelevant noise.
Applied math in daily life, such as making change, also involves a lot of memorization, it's just that the person doesn't realize they're committing various formulas and equalities to memory.
The problem with proficiency in, e.g., making change, is that it doesn't carry over to higher-order math, logic, and reasoning. Arithmetic as taught in elementary school is attempting to achieve two things at once: proficiency in applied arithmetic, and foundational number theory (e.g. commutativity).
My mother was a waitress and emphasized skills like making change. While she never articulated the rules, I ended up developing many of the mental arithmetic techniques that (I later discovered) Isaac Asimov discussed in Quick and Easy Math: https://archive.org/details/QuickAndEasyMath-English-IsaacAs... But I never developed an appreciation for number theory until it was too late--i.e. after high school. I did get into philosophy and logic during high school, but the connection (theoretical and applied) between the two didn't click until later.
Sadly (or not?), proficiency in mental arithmetic has become much less common even among waiters, clerks, etc, at least where they don't deal in cash directly and without the aid of a register. And professional mathematicians have always humble-bragged about their impoverished mental arithmetic skills. So maybe we should drop the pretense that we're attempting to teach applied mathematics in the early years and admit the purpose is to lay theoretical foundation for higher-order math, applied and theoretical.
I only learned this in adulthood, too. I had to discover and memorize those shortcuts. Now I know that some of the "brilliant" math students I went to high school with had simply learned these skills the rest of us didn't.
That take has been accurate in specific times and places, but it is also blindly repeated in contexts where it's simply not true. All the teachers I encountered in my time in the US public school system were trained in the progressive spirit expressed by Feynman, with a strong bias against rote learning and in favor of a conceptual, understand-based approach. My math education in school both encouraged and rewarded figuring things out, which was good for me because I hated memorization and was always bad at it.
Despite that, the criticism that school rewards memorization and doesn't teach critical thinking is still the only criticism I ever heard about the education I received. It's the standard thing that well-meaning people say.
Which I think is a shame. When virtually every teacher in the system is trained in the progressive approach to education, and most of them sincerely believe in it and do their best to practice it, only to have the entire society turn around and claim that they are actually implementing ideas that nobody has believed in in a century, must be incredibly discouraging.
Yeah, it seems almost as if a lot of people look back at their experiences of U.S. schools in the 90s, and assume that the schools in 2025 must not have changed one iota. While obviously I can't speak for every school district (nor can anyone), many of the criticisms I've heard seem disconnected from the schools I'm familiar with. (Not that they aren't subject to newer criticisms!) Is my personal experience a big outlier, or are people just extrapolating from the past? It makes me worry that school districts will greatly overcorrect in their efforts to ward off the old criticisms.
I went to school in the early 1990s! The progressive approach to education was already orthodoxy when my teachers were trained. It has been around a long, long time and has been the prevailing belief in education for half a century or more.
The situation is almost paradoxical: you have generation after generation of people saying that education needs to be reformed to eliminate rote learning and focus on understanding concepts, and where did they learn this orthodoxy? In school, from their teachers.
I suspect it has something to do with how teenagers experience school. No matter the pedagogical approach, if kids are distracted with their social lives and normal adolescent stuff, they experience any attempt to teach them as dry and rote.
Feynman was correct for science in school, however arithmetic is fundamental and maybe one level above the root of all mathematics. Children should be able to do most of it via mental lookup tables and apply that knowledge on paper. For some reason, they can't.
No, it goes beyond that. There's "arithmetic," the applied usage of addition, multiplication, subtraction, and division to permute numbers, and then there's Arithmetic, the set of theorems and axioms that give rise to that system of applied arithmetic. Memorization only works for the applied part, and children aren't usually taught that there is a system of reasoning behind those rules. Without that, no amount of mathematical dexterity in pushing symbols across a page will help them understand anything past the 100 level, and sometimes not even that.
I also think there's a huge undercurrent of resistance from adults to having children learn that system of reasoning because adults don't understand why it's useful, and in my experience when people don't understand something they dismiss it.
Edit: A nice example of another axiomatic system that might be more approachable is Euclid's Elements, in which five postulates are used to develop a system of geometry using an unmarked straightedge and a collapsible compass that you could, if you were careful, use to build bridges and other large buildings.
Once I got to calc2 and 3. I was so mad. I realized I had spent nearly a decade memorizing things. When I could use calculus to have a factory that made formulas and the rules were on a whole simpler to remember and apply.
I had a lucky experience taking HS calculus the semester before as physics. I saw other students torturing themselves memorizing the formulae from the physics text and even then struggling to apply it to novel problems.
For the most part, knowing basic calc, it was possible to just draw a free body diagram and either integrate or take a derivative to get the answer. Didn't memorize much beyond f=ma and v=IR, for better or worse.
I still firmly believe that physics and calculus should be introduced together to provide a tangible and practical base to understand the mathematical theory.
It's common in European universities to not have "service courses" that the math departments provide to other departments. The other departments teach the math that is part of their field!
Similar here: There were all sorts of volume and area equations I could never remember, then one slow day at work I decided to try and derive the volume of a spehere using what I'd just learned in calculus. After doing so each part of the equation made sense instead of appearing random, and two decades later I still remember it without having to derive it again.
I mean I remember seeing this first-hand student teaching Mathematics 13 years ago in the US. They got to me having never seen any of that stuff, and the curriculum attempted to provide a good education in mathematics. But the staff and the way the whole system is structured is to skip all of that and memorize the single rule you need to know to get through the test. So it's all done by rote and the only time you find out how you've been cheated is when you try to go through Calculus.
And I remember that was how we learned everything when I was a kid, and the teachers chose not to do anything else. I also remember from my math ed curriculum one of the professors joking about the elementary education students complaining about having to learn middle school math from the college perspective. So I think portions of this apply here.
I've also seen carpenters apply trigonometry very effectively to do things like cuts for roofs and stair jacks, so there's certainly a lot of truth to people learning maths by occupation and not in a formal setting, and I think part of it is the formal setting.
Why should they? We have the tables in our pockets at literally all times; doing arithmetic without it might be useful, or a bit faster sometimes, but is hardly an essential skill.
It builds a numeric intuition. When you repeat something enough, it begins to do itself - you gain a subconscious mastery. Think about yourself as you read these words. Imagine if you were looking at the letters and actually trying to sound out each word, consciously thinking about each words meaning, and then finally trying to piece together the meaning. You'd spend 5 minutes reading a sentence or two, and oh God help yo if tere ws a tpyo. Instead it all just flows without you even thinking about it, even when completely butchered.
And that sort of flow is, I think, obtainable for most of anything. But 100% for certain for numbers. Somebody who doesn't gain an intuitive understanding of basic arithmetic will have an extremely uncomfortable relationship with any sort of math, which mostly just means they'll avoid it at all costs, but you can't really. I don't even mean STEM careers, but everything from cooking (especially baking) to construction and generally an overwhelming majority of careers make heavy use of mathematical intuition in ways you might not consider, especially if you're already on good terms with numbers.
that's why montessori math is so impressive. it starts with counting out beads one at a time by the hundreds until they have internalized that. then they get beads on a stiff wire 10 at a time, and repeat the process counting them out up to a 1000. and so forth until eventually they hold in their hands blocks of 1000 beads glued together in a cube, and only after they have internalized that the beads get replaced with more abstract woodblocks and sticks. and all that happens in the first year or so at the age of 3.
It's a bit ironic to be saying this in the context of HN, the tech stack is built up on layers of abstraction that little of us have mastery of.
If we made it so that only people who mastered assembly could be considered "real programmers" we'd get nowhere, certainly not to build modern web applications or video games.
People that have a poor understanding of the stack under them almost invariably produce very poor software.
Video games are an area where in fact good software is still produced, mostly because the people working on the cores of games DO know (at least how to read) assembly language.
Modern web applications on the other hand so basically the same things we were doing with computers 20 years ago but consume 100x the resources to do so.
No "modern web application" comes anywhere near the quality of Word or Excel 2003.
so every time i go shopping i have to type all the prices of what i buy into my phone and also have the calculator connect to my bank account and not only make sure i have enough savings, but also tell me that i am not spending more than my average for weekly groceries? and when doing that i need to make sure to not make any typos because my lack of numeric intuition won't allow me to recognize where i made a mistake. and i also won't be able to tell if an item is overpriced. nor will i recognize a bargain unless it is marked with a big colorful sticker.
I struggle when trying to solve math problems without context. I learned enough trigonometry to pass the final exams in high school, but I didn't REALLY understand it until I took a graduate-level graphics programming class.
Some people enjoy the process of solving equations and math problems. For me, it's a tough process. Unless I have a tangible goal, I struggle to visualize the problem.
Starting with basic algebra, it would be more effective if mathematics were paired with some practical problems. Computer graphics, engineering, construction, finance and the analysis of data would be good areas to do this in because it's exactly where you'd need said math!
I wonder if this is true for all cohorts. There are a lot of children who are just fundamentally not intelligent, and deal with math classes by basically memorizing things and repeating them without real understanding. But for children who are understanding what they’re learning, I would expect academic learning to translate to other things.
I agree. Academic math is taught as a set of rote rules or steps. The focus is not on intuitive understanding. I was taught the usual method of long division and carrying all by rote. Only later on in my academic life did I work out on my own why it works as it does.
This hasn't been true in most of the USA for decades.
A common failure more is for students to forget something and then claim they were never taught it. Arguably the should have been taught it more thoroughly.
i hate when people quote random celebrities as authoritative on any topic, let alone as a counterpoint to actual authorities (google the authors of this study).
Edit: hn is just as anti-intellectual as any other place these days but y'all style yourselves as intelligentsia because your celebrities are special.
I'll repeat: check out the qualifications of the authors of this study and compare them to Feynman's on this subject. Any reasonable person would conclude that comparing them is exactly like comparing Kim Kardashian and Feynman's on QED.
He is celebrity. He is not authoritative about subject of teaching math to kids. He is authoritative about his area of physics. He also is authoritative about writing popular books for physics that demystify physics to adults. But again, not about kids and math.
"you know him because he's known but because you're actually familiar with his work"
I cannot parse your statement, either there are some missing words, or some problem with your English.
Anyway, from my previous comment, you couldn't have any idea about how I got to know Feynman and his work. I haven't mentioned it at all.
FYI, I got acquainted with his work in 1996 when I enrolled into the university. I was studying maths, my dormitory roommate was studying theoretical physics, and he had several Feynman's books that were very interesting to me, though I must admit that sometimes the underlying apparatus was really complicated for an 18 y.o. greenhorn. But the principles were clear enough.
Linus Pauling's authority in chemistry doesn't make his cuckoo theories about Vitamin C any less cuckoo. Feynman may have been an important physicist, but that doesn't make him knowledgeable about education!
And, to be honest, there's a reason why there are memes about physicists' competence in other fields, like https://xkcd.com/793/.
As well as the texts based off his lectures. [1] His ability to teach was completely unreal. Those 'books' dramatically deepened my understanding of physics.
This is not quite accurate. I mean it is, but only in the same sense that a top 5 chessplayer in the world might regularly bemoan, with no irony intended, his inability to play chess well. There's a lot more context to his comments here. [1]
Seriously, if you are at all interested in physics - read the lectures and they will, with 100% certainty, deepen your understanding. Even on the most fundamental topics. For instance my entire worldview around the conversation laws changed thanks to those lectures, which in turn ties directly into the nature of energy.
Probably because a lot of us have read or watched his work and know first hand that it was extremely high quality. It's not like you have to take people's word for it.
The Feynman Lectures on Physics was used as the textbook for Caltech's introductory physics course for nearly two decades, and it is still used in some universities. I learned physics from it and have met many Caltech alumni who used it as their textbook, all of whom felt they learned a great deal more than "intuition" from it. So I am guessing you've never actually tried to learn something from it if you feel that way.
He taught a two-year introductory physics course at Caltech from 1961 to 1964, which gives him some experience with the matter though. He was known as "The Great Explainer", due to his ability to help people understand and more importantly, be inspired by science and the world around them*. His materials from those lectures were converted into "The Feynman Lectures on Physics",
a highly regarded physics textbook. so I wouldn't have chosen education as my example.
In support of 793 however, he didn't do well with bureaucracy so I'd not listen to his advice on how to run something that favored rigorous rule following even when the rules don't make sense**.
My parent comment is especially jarring because Feynman's findings agree with, and propose a mechanism for, the findings of the study. The comment seems to imply that there's some great tension between "arithmetic skills do not transfer between applied and academic mathematics" and "the students had memorized everything, but they didn’t know what anything meant". Or between "These findings highlight the importance of educational curricula that bridge the gap between intuitive and formal maths" and the less academically worded "There, have you got science? No! You have only told what a word means in terms of other words. You haven’t told anything about nature."
Nobody's quoting Feynman "as a counterpoint to actual authorities". Feynman's excerpt provides first-hand testimony from a teacher on the front lines that fully validates what the study found.
Feynman is no random celebrity. In addition to be a renowned physicist, his famous "Feynman Lectures" and his thoughts on pedagogy are similarly legendary.
The Feynman Lectures are great at giving you an intuitive understanding, but is no substitute for the regular curriculum. You don't find many people who read only the Feynman Lectures who can then go on to solve physics problems well. You do find many who read the regular textbooks and who can.
You have to bear in mind that the lectures in The Feynman Lectures on Physics were only one third of an introductory physics course, the other parts being recitation sections (in which homework problems, quizzes and tests were given and discussed), and labs. Lecture attendance was optional - many people prefer reading to listening - but the recitation sections and labs were mandatory, because they were considered much more important. Nobody learns physics from just reading lectures.
In this case, Richard Feynman is just writing about his personal experiences of a well-known phenomenon. https://profkeithdevlin.org/wp-content/uploads/2023/09/lockh... ("Lockhart's Lament") would perhaps be a better reference, but nearly anyone who's been through the education system would be able to tell you this.
And why should I simply assume that "Education Economists"* really know the subject they purport to talk about? Because they are credentialed members of university departments with some label? Because a few of them won some Bank of Sweden award?
Just because a particular department or field of study exists in academia does not magically give them the imprimatur you think it does.
* Btw, I know for a fact that a few of them are not "education economists"
Richard Feynman is famous for being an educator, and he's clearly quite good at it. Who cares if he has no formal training? I reckon he deserves at least a 1.2 on this scale.
It's amazing how deep the celebrity worship goes. No he's famous for being a mathematical physicist (his Nobel is in physics not education). He was actually a very mediocre educator - you can read his own assessments of his success/failure in teaching the "famous" intro courses.
Or you can ask literally any physics major that's actually had to use those books (they are horrible for actually learning from).
I wanted to upvote your other comment because it caught a detail of "how much" that may have slipped past the other commenter's or other reader's minds but...
0. The Kardashians
The distance between 0 and 1 is vast compared to the distance between 1 and 2. Feynman was a professor and also beloved for his ability to bridge across the academic to pragmatic divide that is the subject of this paper.
What is the relevance of this point? No one has linked a Kardashian's take on anything? So who cares if the distance between 0 and 1 is larger than the distance between 1 and 2 - we are only discussing the distance between 1 and 2.
The original comment you responded to made no comparative claims. It simply offered another person's attempt to describe. Feynman is fairly famous but nonetheless an authoritative source relative to most of the population (probably more so than both of us, though I don't know you do have little basis beyond priors [sorry if you have greater credibility than Feynman, I didn't know]). Feynman is less authoritative on the subject than the authors of the article but still... Being well known doesn't remove the authority level that Feynman does have on the topic.
It's not a counterpoint. The Feynman excerpt and the paper support each other.
The paper's abstract ends, "These findings highlight the importance of educational curricula that bridge the gap between intuitive and formal maths."
The Feynman excerpt is about the issues caused by a lack of practica in education and how they should be resolved.
The paper's authors wrote, "These findings call for a maths pedagogy that explicitly addresses these translational challenges through curricula that connect abstract maths symbols and concepts to intuitively meaningful contexts and problems." And provide 2 examples of Randomized Control Trials of math courses in Brazil and India respectively that address the challenges successfully.
Even if you remove Feynman's name, it's still interesting that a Theoretical Physics professor and educator wrote clearly about a very similar issue they encountered over 60 years before the paper in question was published.
Regardless of the people involved, being asked to consider someone’s opinion on a matter is a world apart from claiming they are an authority on the topic.
https://v.cx/2010/04/feynman-brazil-education