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I’ve always found that an indictment of math education — and spent many, many hours discussing it. When teaching addition, workbooks commonly use a box, eg, “[

I’ve always found that an indictment of math education — and spent many, many hours discussing it. When teaching addition, workbooks commonly use a box, eg, “[ ] + 2 = 5” — and first graders have no conceptual problem with this. Somehow, we lose people by the time we’re trying to formalize the same concept in algebra. There’s been many times I’ve written a box around letters in a problem and asked students “what’s in the box labeled x?” Pedagogy is hard.

Buttons840 2y ago
Go from "[ ] + 2 = 5" to writing it "box + 2 = 5--what is box?". Then "b + 2 = 5--what is b?" then "x + 2 = 5--what is x?".
- sublinear 2y ago
  I agree. I think the actual problem is that the student is trying to comprehend what it means for anything to have mathematical value other than explicit numbers. Numbers and letters are taught together, but not as symbols. Letters are taught with sounds and numbers are taught with counting. The notion of a symbol isn't really emphasized much. I would explain it more like after [ ] + 2 = 5 what happens if you need more than one box for a complicated problem? Teach the idea that saying box #3 is equivalent to assigning an arbitrary letter for whatever reason you want, but that people more familiar with math prefer letters because they stand in for words that describe what the number is for. You might want to use 'c' for the number of cats you're trying to figure out. In a room of five animals two are dogs. How many cats? a = 5, c = ?, d = 2 a = c + d so... 5 = c + 2 what is c? Light bulb goes off: "You can do that?" Yes, you can do whatever you want and it's not all about carrying the one or whatever other rote teaching they've been given. They can get creative and be engaged, and then you let them know that actually there are some conventions people like to use for what they're trying to do. They might even believe they've invented a new idea. At least they're having fun.
  - dr_kiszonka 2y ago
    I agree with you. To me, a lot of pre-college math education could be summarized as "In this class I will show you a bunch of abstract problems, a bunch of ways to solve them, and I will test if you have learned them." Learning in these classes is often limited to memorizing a sequence of steps. That's why I would frequently ask "You can do that?" myself when talking to those whom I considered mathematically gifted (math olympiad winners and such). I think they realized that as a problem-solving tool math could be used creatively. I saw it as a largely useless hammer that to work had to be held in a very specific way. I remember connecting sets in, I think, Pascal to what I had learned in school and realizing that all that math was perhaps not as useless as I had thought : - )
- Jensson 2y ago
  That is what math books already do.
  - Viliam1234 2y ago
    Some of them do it better than others.
- Viliam1234 2y ago
  Basically, don't teach the new concept and the new syntax both at the same time. New things should be introduced one by one.
- Izkata 2y ago
  You skipped a step. One of the problems is more obvious with a different operator we learn when in the "box" stage: > Go from "[ ] x 2 = 10" to writing it "box x 2 = 10--what is box?". Then "b x 2 = 10--what is b?" then "x x 2 = 10--what is x?". From memory, we didn't switch from "x" to dot for multiplication until at the exact same time we started using symbols. If we'd done it earlier (or even right from the start) it might not have been as much of a problem.
uwagar 2y ago
back when we was new in programming it was similarly difficult to grok X = X + 1 once we got it, it was a like new world!
- 5- 2y ago
  most likely this very unfortunate misnomer started with fortran, where it was deemed lucrative to point out "how much programs look like mathematic formulas!". not only is this overloading a symbol (equality) with a completely different meaning (assignment), it is also a poor choice typographically, as it represents a directional operation with a directionless symbol. using an arrow for assignment is much better. it's also worth pointing out that unlike most others, logic programming languages (e.g. prolog) have actual variables, not references to mutable or immutable memory cells.
  - uwagar 2y ago
    arrow for assignment is cool, but the backspace key is the only closest arrow-like key on the keyboard but it has a different purpose. plus the arrow key should be laid out such that u dont have to press a SHIFT/CTRL/ALT to produce it. for this reason, i felt C a breath of fresh air cos u could just assign using = instead of what we was doing in pascal which was the horrible := where u had to press SHIFT for the : things like this matter.
    - dr_kiszonka 2y ago
      Right. If we went more in the direction of APL, we would have had way more symbols. https://upload.wikimedia.org/wikipedia/commons/9/94/APL-keybd.svg https://upload.wikimedia.org/wikipedia/commons/9/94/APL-keyb...

Buttons840 2y ago

Go from "[ ] + 2 = 5" to writing it "box + 2 = 5--what is box?". Then "b + 2 = 5--what is b?" then "x + 2 = 5--what is x?".

sublinear 2y ago
I agree. I think the actual problem is that the student is trying to comprehend what it means for anything to have mathematical value other than explicit numbers. Numbers and letters are taught together, but not as symbols. Letters are taught with sounds and numbers are taught with counting. The notion of a symbol isn't really emphasized much. I would explain it more like after [ ] + 2 = 5 what happens if you need more than one box for a complicated problem? Teach the idea that saying box #3 is equivalent to assigning an arbitrary letter for whatever reason you want, but that people more familiar with math prefer letters because they stand in for words that describe what the number is for. You might want to use 'c' for the number of cats you're trying to figure out. In a room of five animals two are dogs. How many cats? a = 5, c = ?, d = 2 a = c + d so... 5 = c + 2 what is c? Light bulb goes off: "You can do that?" Yes, you can do whatever you want and it's not all about carrying the one or whatever other rote teaching they've been given. They can get creative and be engaged, and then you let them know that actually there are some conventions people like to use for what they're trying to do. They might even believe they've invented a new idea. At least they're having fun.
- dr_kiszonka 2y ago
  I agree with you. To me, a lot of pre-college math education could be summarized as "In this class I will show you a bunch of abstract problems, a bunch of ways to solve them, and I will test if you have learned them." Learning in these classes is often limited to memorizing a sequence of steps. That's why I would frequently ask "You can do that?" myself when talking to those whom I considered mathematically gifted (math olympiad winners and such). I think they realized that as a problem-solving tool math could be used creatively. I saw it as a largely useless hammer that to work had to be held in a very specific way. I remember connecting sets in, I think, Pascal to what I had learned in school and realizing that all that math was perhaps not as useless as I had thought : - )
Jensson 2y ago
That is what math books already do.
- Viliam1234 2y ago
  Some of them do it better than others.
Viliam1234 2y ago
Basically, don't teach the new concept and the new syntax both at the same time. New things should be introduced one by one.
Izkata 2y ago
You skipped a step. One of the problems is more obvious with a different operator we learn when in the "box" stage: > Go from "[ ] x 2 = 10" to writing it "box x 2 = 10--what is box?". Then "b x 2 = 10--what is b?" then "x x 2 = 10--what is x?". From memory, we didn't switch from "x" to dot for multiplication until at the exact same time we started using symbols. If we'd done it earlier (or even right from the start) it might not have been as much of a problem.

uwagar 2y ago

back when we was new in programming it was similarly difficult to grok X = X + 1 once we got it, it was a like new world!

5- 2y ago
most likely this very unfortunate misnomer started with fortran, where it was deemed lucrative to point out "how much programs look like mathematic formulas!". not only is this overloading a symbol (equality) with a completely different meaning (assignment), it is also a poor choice typographically, as it represents a directional operation with a directionless symbol. using an arrow for assignment is much better. it's also worth pointing out that unlike most others, logic programming languages (e.g. prolog) have actual variables, not references to mutable or immutable memory cells.
- uwagar 2y ago
  arrow for assignment is cool, but the backspace key is the only closest arrow-like key on the keyboard but it has a different purpose. plus the arrow key should be laid out such that u dont have to press a SHIFT/CTRL/ALT to produce it. for this reason, i felt C a breath of fresh air cos u could just assign using = instead of what we was doing in pascal which was the horrible := where u had to press SHIFT for the : things like this matter.
  - dr_kiszonka 2y ago
    Right. If we went more in the direction of APL, we would have had way more symbols. https://upload.wikimedia.org/wikipedia/commons/9/94/APL-keybd.svg https://upload.wikimedia.org/wikipedia/commons/9/94/APL-keyb...