4 ms·

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:

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.

zdragnar 2y ago
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.
- singleshot_ 2y ago
  > 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.
  - zdragnar 2y ago
    See, you're providing a reason to care that doesn't exist in the word problem. That's fully supporting what I would have wanted in school.
    - Brian_K_White 2y ago
      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 have no sympathy for this complaint.
      - simtel20 2y ago
        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.
      - zdragnar 2y ago
        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.
        quesera 2y ago
        > 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.
- mjevans 2y ago
  "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_States#20th_century 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.
- Terr_ 2y ago
  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...")
- analog31 2y ago
  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.
  - vkazanov 2y ago
    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.
    - Agingcoder 2y ago
      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 .
      - vkazanov 2y ago
        Well, i don't disagree with you! in my book proofs are formalized intuition.
      - analog31 2y ago
        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.

zdragnar 2y ago

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.

singleshot_ 2y ago
> 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.
- zdragnar 2y ago
  See, you're providing a reason to care that doesn't exist in the word problem. That's fully supporting what I would have wanted in school.
  - Brian_K_White 2y ago
    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 have no sympathy for this complaint.
    - simtel20 2y ago
      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.
    - zdragnar 2y ago
      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.
      - quesera 2y ago
        > 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.
mjevans 2y ago
"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_States#20th_century 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.
Terr_ 2y ago
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...")
analog31 2y ago
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.
- vkazanov 2y ago
  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.
  - Agingcoder 2y ago
    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 .
    - vkazanov 2y ago
      Well, i don't disagree with you! in my book proofs are formalized intuition.
    - analog31 2y ago
      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.