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:
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.
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.
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.
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 have no sympathy for this complaint.
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.
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.
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.
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...")
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.
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.
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 .
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.
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.
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.
"My phone is just rebooting" the pilot replied.
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.
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.
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.
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.
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.
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.
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!
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.
When I was taught long division I was told how it worked, although it is fairly obvious if you think about it for a few seconds anyway.
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.
Let's see
1. The personal experiences of a guy with no formal training in pedagogy or education
2. A research paper in nature written by expert education economists
Hmmmmmmmm
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.
I should, perhaps, have used:
0. Average person
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).
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"
I'm glad we've arrived at Fox News level takes. At least we can all admit what we are here.
Unfortunately it destroyed your argument.
Hint: I might need to google the authors of the study, I don't need to google Richard Feynman...
My friend that is the textbook definition of celebrity status.
To use such word with regard to one of the most talented and innovative physicists of the 20th century debases the entire conversation.
If my eyes rolled back any further in my head I could look into the past.
But you've proven your own point: you know him because he's known but because you're actually familiar with his work.
So again: textbook celebrity worship masquerading as intellectualism.
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.
And, to be honest, there's a reason why there are memes about physicists' competence in other fields, like https://xkcd.com/793/.
I'm very aware of the "spherical cow" joke about physicists, but this isn't a "random celebrity".
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.
[1] - https://www.feynmanlectures.caltech.edu/info/popular_misconc...
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**.
* https://www.commoncraft.com/honor-richard-feynman-great-expl...
** https://laurakornish.com/2019/04/no-joking-matter-feynman-on...
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.
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.