12 ms·

Helping a kid who made a math mistake

79 points by sebg 3y ago

alex_young 3y ago
This advice could be generalized as how to help someone learn something, especially when you know it well already.
bruce511 3y ago
I help adults solve problems in their code. I'm going to try this approach and see if it works in that context. Hopefully it will. Although I guess in some cases the issue is reversed parameters to a function, or whatever, where the underlying knowledge is weak (or faulty). But it's certainly worth a try.
- madaxe_again 3y ago
  It’s universally and self-applicable. Whenever I’m making a plan, or figuring something out, be it engineering or code or physics or whatever, I rubber duck - that is, I explain what I’m doing and why, verbally, in exhaustive detail, to nothing and nobody in particular (for I do not have a rubber duck), and ask questions of myself that a rubber duck would ask. I often catch a snag where I would otherwise have thought my scheme flawless.
nashashmi 3y ago
I agree with the author here for most of what was said. Except I would summarize it as "explore both the problem the student identifies and the solution the student uses" and while doing so they would catch their mistakes. Explore means to drill down and seriously make the student wonder why they're doing the problem that way. And the wonderful thing about exploration is you don't even have to know the subject. You can just keep asking and engaging in a conversation. The student learns. And so do you.
- sitkack 3y ago
  This article and your statement is the best thing I have read all year. This sets up a self correcting feedback loop for the students learning process AND their correctness checking process. It is brilliant! It is like teaching someone to swim, ride a bike or in this case know that what you know is correct.
msrenee 3y ago
Maybe they can take their own advice and proofread their writeup before posting it. I'm still not sure what step c is.
- riffraff 3y ago
  I was deeply confused by "how t" my brain kept trying to interpret it as do/don't -> how/hown't. But still, typos happen.
boomboomsubban 3y ago
Hey, could you walk me through how you chose your section titles?
ggm 3y ago
Use language appropriate to their apparent level of comprehension of the problem. If they objectify the problem visually in a 2D diagram think very hard before asking them to project it into another representative space, or phase change or whatever is your "this is a better way to think about it" Try very hard not to be patronising. Kids struggling with maths can be super sensitive. They can misinterpret a smile as a smirk. If they had the courage to seek help, don't belittle them. For example a phrase as simple as "that's easy" can be a huge burden. Beware misunderstood questions. If you seek to help by asking them questions be mindful if they do not understand the intent of your question, you may add to their burden not reduce it. Some set work requires a specific method be shown to be used. It's fine to show them other methods, but you may have to help them project the answer back into the one they will be assessed in. If they only want debugging, then help them find the bug, the mistranscribed value, the divide by zero, the specific mistake. If they have no insight into the problem then more exploratory help may be useful. Time out can help. Stress hormones don't make maths any easier.
- vsnf 3y ago
  This advice is equally applicable to assisting your relatives with computer problems.
  - ggm 3y ago
    Totally. Or co workers. My best source of advice refrains from pointing out a strongly typed functional language would solve most of my problems and helps the specific python syntactic burden I have right now. But then does go the distance if I engage. After all, programming is just a subset of mathematics anyway..
    - red75prime 3y ago
      > After all, programming is just a subset of mathematics anyway.. It's more of engineering subset. The resulting artefacts are rarely mathematically provable (bugs and Rice theorem).
    - mbork_pl 3y ago
ARandomerDude 3y ago
> If they still don’t catch the mistake, as them to go slower I’ve also found this method very helpful for proofreading.
tzone 3y ago
Funny coincidence, this is also a great way to get ChatGPT to produce correct solutions when it produces incorrect stuff on its first try.
readthenotes1 3y ago
Or ask: How do you check if you got right answers?
- fijiaarone 3y ago
  Yes, it’s very disheartening to see math teachers not understand how to check your work. There are algorithms (and heuristics) that have been known for centuries to verify (or estimate) check most results from arithmetic to calculus. It’s as if programmers were not taught how to do logging or assertions. In math there are two types of errors, not understanding the problem and mistakes. Redoing the work won’t help at all for misunderstanding or lack of understanding and has only a marginal chance of catching the mistake.
pr337h4m 3y ago
Looks like a good use case for LLMs, especially when integrated with a math backend like Wolfram or Desmos
burnished 3y ago
This is pretty common advice that I am starting to disagree with. In my experience (in person math and chemistry tutoring and answering programming questions on public discords) this technique really only works well with some one who is on the verge of understanding and has perhaps made a simple error or already has mastery over the techniques and concepts at play and are learning one entirely new thing. In those cases this technique is great! You can guide them to that moment of epiphany, and it is very rewarding and efficient. Much more commonly it feels like pulling teeth. I think this is because most kinds of misunderstandings and confusions are hard to identify from the inside. For example, when some one learns a new thing that is in contradiction with something else (that they perhaps misremembered, never fully understood, or misunderstood a nuance in a way that was internally consistent) the resulting inconsistency can be very difficult to identify and verbalize. If you can help them identify and resolve it on their own then great- my experience is that when that works it is exhausting and time consuming and that taking a more active role in identifying the conflict is much more productive. In programming it commonly takes people years to learn enough background context that this method becomes effective. I think it is better to be quick with examples and explanations, there is just too much to know to make this sort of highly examined introspection efficient or effective. Other times its simply because the student isn't intrinsically motivated to learn the material and this style of teaching involves a lot of emotional labor in order to be effective. Turns out people shut down a little when you ask them to explain their thought process and they feel they should not tell you frankly they care not one wit for Pythagoras or your damned flag pole .
- zmgsabst 3y ago
  As a math tutor, I’m with you. But I tend to do two steps: 1. I’ll tell you what the right formula/process/etc is — as a hint on the problem. 2. If you still don’t get it, I’ll walk through the right steps with you. I’ve never found it particularly useful to dig into what you have wrong or try to guide you into self-correcting unless the problem is minor (eg, you clearly understand and just need a nudge), you ask, or I’m dealing with someone that has a professional interest and will invest the effort debugging. I can’t imagine how badly I’d hate it if StackOverflow tried to guide me on some Socratic journey every time, rather than just tell me the correct syntax for the magic invocation of pandas that solves my problem. I’d probably just quit.
riffraff 3y ago
I like this advice, but my daughter has been taught to do basic math "intuitively". E.g. for multidigit addition instead of adding up units then carry, they are taught to just add everything up. Learning the carry method only happens later. She can't explain how she does it, and I can't either, it's been a frustrating experience. I hope I can still be useful when she gets to harder stuff that requires more steps :)
patrulek 3y ago
So its like debugging the math.
charlieyu1 3y ago
As a maths teacher I don’t agree much. The problem is that the student would spend so much time on figuring out their mistakes. If you pass the attention span of children, all they feel is that they made a mistake and feel defeated while getting lectured on something they don’t understand anyway. It is not that easy to find a mistake that you made, even I made mistakes that took me long to find out. I’m actually in favour of the traditional way, show them the right way and let them try. It is fine to make mistakes, but don’t put focus on the mistakes. Instead focus on good practices. Asking them to figure out their own mistakes/making explanations should be done very sparsely.
- dzdt 3y ago
  Yes. The biggest hurdle is usually attention span of the child. If the child has the attention to go thru it slowly the article's approach could be great. But for cases where their attention is already shot, walking them thru the steps or demonstration of the steps can be an easier way thru. Yes they are less likely to have a major breakthru in understanding, but it can help them with memorizing the step sequence without exhausting the limited attention.
kazinator 3y ago
Simple editors that don't point out any problems with code formatting or syntax, and compilers with poor diagnostics like "error: this file has a syntax or type error somewhere" ... turn out to be good parents! Who knew?
ajuc 3y ago
Helping people learn something is a lot like debugging software. Usually the problem is that there's an assumption that is wrong. You need to ask questions and check the answers to find out which assumption is wrong. You can even use bisection between input and output to find the wrong assumption in logarithmic time :), but if there aren't that many steps it's less confusing to go step-by-step from input to output. Once you find it - fixing it is easy. Provide simple counterexamples showing why it's wrong, and watch the person solve the problem on their own.
ouid 3y ago
Is this a mistake in a proof or a mistake in a calculation? Let's say that the distinction here is how we establish that an answer is correct. A proof is correct if it typechecks (no really). A computation is correct if it faithfully executes the algorithm. This distinction is more than a little artificial, but in practice it matters a lot.

79 points by sebg 3y ago

This advice could be generalized as how to help someone learn something, especially when you know it well already.

I help adults solve problems in their code. I'm going to try this approach and see if it works in that context. Hopefully it will. Although I guess in some cases the issue is reversed parameters to a function, or whatever, where the underlying knowledge is weak (or faulty). But it's certainly worth a try.

madaxe_again 3y ago
It’s universally and self-applicable. Whenever I’m making a plan, or figuring something out, be it engineering or code or physics or whatever, I rubber duck - that is, I explain what I’m doing and why, verbally, in exhaustive detail, to nothing and nobody in particular (for I do not have a rubber duck), and ask questions of myself that a rubber duck would ask. I often catch a snag where I would otherwise have thought my scheme flawless.

nashashmi 3y ago

I agree with the author here for most of what was said. Except I would summarize it as "explore both the problem the student identifies and the solution the student uses" and while doing so they would catch their mistakes. Explore means to drill down and seriously make the student wonder why they're doing the problem that way. And the wonderful thing about exploration is you don't even have to know the subject. You can just keep asking and engaging in a conversation. The student learns. And so do you.

sitkack 3y ago
This article and your statement is the best thing I have read all year. This sets up a self correcting feedback loop for the students learning process AND their correctness checking process. It is brilliant! It is like teaching someone to swim, ride a bike or in this case know that what you know is correct.

msrenee 3y ago

Maybe they can take their own advice and proofread their writeup before posting it. I'm still not sure what step c is.

riffraff 3y ago
I was deeply confused by "how t" my brain kept trying to interpret it as do/don't -> how/hown't. But still, typos happen.

boomboomsubban 3y ago

Hey, could you walk me through how you chose your section titles?

ggm 3y ago

Use language appropriate to their apparent level of comprehension of the problem. If they objectify the problem visually in a 2D diagram think very hard before asking them to project it into another representative space, or phase change or whatever is your "this is a better way to think about it" Try very hard not to be patronising. Kids struggling with maths can be super sensitive. They can misinterpret a smile as a smirk. If they had the courage to seek help, don't belittle them. For example a phrase as simple as "that's easy" can be a huge burden. Beware misunderstood questions. If you seek to help by asking them questions be mindful if they do not understand the intent of your question, you may add to their burden not reduce it. Some set work requires a specific method be shown to be used. It's fine to show them other methods, but you may have to help them project the answer back into the one they will be assessed in. If they only want debugging, then help them find the bug, the mistranscribed value, the divide by zero, the specific mistake. If they have no insight into the problem then more exploratory help may be useful. Time out can help. Stress hormones don't make maths any easier.

vsnf 3y ago
This advice is equally applicable to assisting your relatives with computer problems.
- ggm 3y ago
  Totally. Or co workers. My best source of advice refrains from pointing out a strongly typed functional language would solve most of my problems and helps the specific python syntactic burden I have right now. But then does go the distance if I engage. After all, programming is just a subset of mathematics anyway..
  - red75prime 3y ago
    > After all, programming is just a subset of mathematics anyway.. It's more of engineering subset. The resulting artefacts are rarely mathematically provable (bugs and Rice theorem).
  - mbork_pl 3y ago

ARandomerDude 3y ago

> If they still don’t catch the mistake, as them to go slower I’ve also found this method very helpful for proofreading.

tzone 3y ago

Funny coincidence, this is also a great way to get ChatGPT to produce correct solutions when it produces incorrect stuff on its first try.

readthenotes1 3y ago

Or ask: How do you check if you got right answers?

fijiaarone 3y ago
Yes, it’s very disheartening to see math teachers not understand how to check your work. There are algorithms (and heuristics) that have been known for centuries to verify (or estimate) check most results from arithmetic to calculus. It’s as if programmers were not taught how to do logging or assertions. In math there are two types of errors, not understanding the problem and mistakes. Redoing the work won’t help at all for misunderstanding or lack of understanding and has only a marginal chance of catching the mistake.

pr337h4m 3y ago

Looks like a good use case for LLMs, especially when integrated with a math backend like Wolfram or Desmos

burnished 3y ago

This is pretty common advice that I am starting to disagree with. In my experience (in person math and chemistry tutoring and answering programming questions on public discords) this technique really only works well with some one who is on the verge of understanding and has perhaps made a simple error or already has mastery over the techniques and concepts at play and are learning one entirely new thing. In those cases this technique is great! You can guide them to that moment of epiphany, and it is very rewarding and efficient. Much more commonly it feels like pulling teeth. I think this is because most kinds of misunderstandings and confusions are hard to identify from the inside. For example, when some one learns a new thing that is in contradiction with something else (that they perhaps misremembered, never fully understood, or misunderstood a nuance in a way that was internally consistent) the resulting inconsistency can be very difficult to identify and verbalize. If you can help them identify and resolve it on their own then great- my experience is that when that works it is exhausting and time consuming and that taking a more active role in identifying the conflict is much more productive. In programming it commonly takes people years to learn enough background context that this method becomes effective. I think it is better to be quick with examples and explanations, there is just too much to know to make this sort of highly examined introspection efficient or effective. Other times its simply because the student isn't intrinsically motivated to learn the material and this style of teaching involves a lot of emotional labor in order to be effective. Turns out people shut down a little when you ask them to explain their thought process and they feel they should not tell you frankly they care not one wit for Pythagoras or your damned flag pole .

zmgsabst 3y ago
As a math tutor, I’m with you. But I tend to do two steps: 1. I’ll tell you what the right formula/process/etc is — as a hint on the problem. 2. If you still don’t get it, I’ll walk through the right steps with you. I’ve never found it particularly useful to dig into what you have wrong or try to guide you into self-correcting unless the problem is minor (eg, you clearly understand and just need a nudge), you ask, or I’m dealing with someone that has a professional interest and will invest the effort debugging. I can’t imagine how badly I’d hate it if StackOverflow tried to guide me on some Socratic journey every time, rather than just tell me the correct syntax for the magic invocation of pandas that solves my problem. I’d probably just quit.

riffraff 3y ago

I like this advice, but my daughter has been taught to do basic math "intuitively". E.g. for multidigit addition instead of adding up units then carry, they are taught to just add everything up. Learning the carry method only happens later. She can't explain how she does it, and I can't either, it's been a frustrating experience. I hope I can still be useful when she gets to harder stuff that requires more steps :)

patrulek 3y ago

So its like debugging the math.

charlieyu1 3y ago

As a maths teacher I don’t agree much. The problem is that the student would spend so much time on figuring out their mistakes. If you pass the attention span of children, all they feel is that they made a mistake and feel defeated while getting lectured on something they don’t understand anyway. It is not that easy to find a mistake that you made, even I made mistakes that took me long to find out. I’m actually in favour of the traditional way, show them the right way and let them try. It is fine to make mistakes, but don’t put focus on the mistakes. Instead focus on good practices. Asking them to figure out their own mistakes/making explanations should be done very sparsely.

dzdt 3y ago
Yes. The biggest hurdle is usually attention span of the child. If the child has the attention to go thru it slowly the article's approach could be great. But for cases where their attention is already shot, walking them thru the steps or demonstration of the steps can be an easier way thru. Yes they are less likely to have a major breakthru in understanding, but it can help them with memorizing the step sequence without exhausting the limited attention.

kazinator 3y ago

Simple editors that don't point out any problems with code formatting or syntax, and compilers with poor diagnostics like "error: this file has a syntax or type error somewhere" ... turn out to be good parents! Who knew?

ajuc 3y ago

Helping people learn something is a lot like debugging software. Usually the problem is that there's an assumption that is wrong. You need to ask questions and check the answers to find out which assumption is wrong. You can even use bisection between input and output to find the wrong assumption in logarithmic time :), but if there aren't that many steps it's less confusing to go step-by-step from input to output. Once you find it - fixing it is easy. Provide simple counterexamples showing why it's wrong, and watch the person solve the problem on their own.

ouid 3y ago

Is this a mistake in a proof or a mistake in a calculation? Let's say that the distinction here is how we establish that an answer is correct. A proof is correct if it typechecks (no really). A computation is correct if it faithfully executes the algorithm. This distinction is more than a little artificial, but in practice it matters a lot.