6 ms·

I'm jealous of kids these days... homework would've been so much easier with this. You could always use a calculator but the whole 'show your own working' catc

I'm jealous of kids these days... homework would've been so much easier with this. You could always use a calculator but the whole 'show your own working' catch meant you had to do it all manually. Not any more!

maxerickson 10y ago
Hopefully the kids of tomorrow end up getting less homework because of things like this. Consider a system that combines practice and assessment. It could individualize both, reducing the need to force students that have mastered a concept to do repetitive practice. It might be a big challenge to get such a thing to work well, but let's not look back at our schooling as an anchor for what students today must do.
- samfisher83 10y ago
  A lot of things in life you get good by doing over and over. Some people are born with innate gifts, but a lot of people have to put in work to get where they are.
  - maxerickson 10y ago
    Indeed. But if you can do assessment on an ongoing basis, you can eliminate the work for the students that have already mastered a given concept and individually focus the work of other students, so that they aren't wasting their time trying to solve problems that they lack the conceptual foundation to solve (math is often like this, missing 2 abstractions needed to solve a problem undermines the utility of the practice, it will be basically impossible). I'm not suggesting that it would remove the need for practice. I'm suggesting that it could be used to make the practice more effective, for students that are doing well and for students that have fallen behind.
- jerf 10y ago
  I once designed myself a system for teaching math online. In addition to using an engine like what they're showing here, my plan was to start from more-or-less the beginning, but also track and categorize the errors a given student makes over time. Errors are not generally made randomly, there's a pattern to them. Then, we could allow the student to "solve" an equation the way they really should, by skipping over two or three steps at a time, but when we see them do something wrong, we can use our table of "errors the student is most likely to make" to explore the space of possible errors between step 4 and step 5, and give focused feedback about what they did wrong. Using that table there's really only a couple hundred possibilities; if that fails we can always ask the student themselves to break it down more tightly. Presumably if someone were making a business out of this, there would be someone on the lookout for errors the computer can't figure out to add what rules they can to the system. (Though there will always be an irreducible residue of incomprehensible error.) Teaching a student math would then be about reducing each of these errors to zero over time. You'd have the computer custom create problems that hold a constant probability of the student making an error at some point during solving it; say 20% or so. Then as the student demonstrates mastery, you naturally make the problems more complex as you have to put in more steps to make the probability of error go that high. Instead of a klunky, chunky "ok now we learn this and you blindly practice it, now you learn this and you blindly practice it, and we hope at the end you've learned everything we taught", you would in theory get a naturally progressive, customized difficulty curve that keeps the student continuously engaged with being about 80% correct, but always progressing forward. This approach also naturally ensures that just because we're covering the quadratic equation this week does not mean you get to forget fractions; once you've seen the simple stuff with integers, we're naturally going to fold fractions back in to the problems, for instance. There's some elaborations on the theme after that, such as pre-examining the generated problems to ensure that the most likely mistakes are all distinguishable by producing different error output. But I don't have time to do this. I'm at least reasonably confident it would work, though.
  - nooron 10y ago
    Diagknowstics is working on this in Ann Arbor. Not sure how far along they are, but it's a very bright team.
    - jerf 10y ago
      Interesting. I'm literally 15 minutes' walk from them right now. I wish them well; I'd love to have something like this for my own kids in the 3-4 years it'll take for it to be useful for them.
  - Pulcinella 10y ago
    As a teacher I would kill for something like this, even if it's entirely paper-based instead of digitized. As you said, creating something like this is incredibly time intensive. There is a reason lots of free and paid teaching resources are basically worksheets, a textbook, and an answer key for "documentation." Anything more complicated is so much work. Even if you knew exactly what you wanted to do and how you wanted to implement it, you still have "beta test" it with students, determine the space of possible errors and misconceptions[1], make corrections, etc. I've thought about creating something like this for science, but I "only" teach the same lesson 4-5 times a day. Anything I learn, any corrections I want to make have to wait until next year when I teach that lesson again. Then if those changes don't work or I realize I need to take different changes, I have to wait another year to make them. Not exactly rapid iteration. [1]Not quite as easy as you might think. Students usually can't explain their misunderstanding very well or at all and you probably understand the material too well and it's hard to understand how to solve the problem any way but the correct way.
- omaranto 10y ago
  I hope the kids of tomorrow get better homework, not necessarily less of it (although in some places they definitely should also get less of it).
trendia 10y ago
I remember feeling pretty awesome because my TI-89 could solve most integrals and even some differential equations. Towards the end of my education, Wolfram Alpha came out and would not only solve those equations, but also show step-by-step solutions. Although the solver could be used for cheating, I think it could also be beneficial. Previously, a student who reached a point where they couldn't understand a problem would have trouble finding the right information to understand the solution (e.g. in Diff EQ there are many "tricks" that are necessary to understand in order to solve an equation and which are not immediately obvious). However, with Wolfram Alpha and the like, the student can work through that problem and understand how to solve it.
- cmrx64 10y ago
  So much this! I had a really time learning calculus, due to these "tricks". Wolfram Alpha's step-by-steps helped me so much. I could have went and harassed a TA, and I still did that occasionally, but I could largely do it on my own with the computer's help.
- mighty_atomic_c 10y ago
  Mathematica was my tool of choice. I could plot, Manipulate and visualize things. That was very powerful for learning quickly and getting a "feel" for things. The ability to check if my work was equivalent to the initial and final equations meant I could catch mistakes when I was stumped and trace through my work. I could see where I messed up. When practicing for exams, I would make mental notes of common errors or properties I forgot, and write a portion of "misc idiot lapses" on part of my crib sheet.
ibmthrowaway271 10y ago
> You could always use a calculator but the whole 'show your own working' catch meant you had to do it all manually. Not any more! You could often cheat (for equation rearrangement questions) if you knew the answer by simply working backwards towards the question, this is often easier than going from problem to solution but still provides all of the steps along the way. I remember one maths teacher hinting at this trick, especially to understand the derivation of the quadratic formula:- ax^2 + bx + c = 0 by starting with:- x = ( -b +/- sqrt( b^2 - 4ac ) ) / 2a and working backwards.
- Asooka 10y ago
  There is also "solving by assuming the negative". It's a technique for problems like "Given X, prove Y", where you assume "Y is false" and use it to prove "X is false", but X is true, therefore Y has to be true. Sometimes it's easier to do it in that direction than X->Y.
  - Tyr42 10y ago
    Contrapositive is the fancy word for that technique. A -> B iff not(B) -> not(A)
SilasX 10y ago
Well, you could always shift to "find the erroneous step in this person's attempt to show their work."

maxerickson 10y ago

Hopefully the kids of tomorrow end up getting less homework because of things like this. Consider a system that combines practice and assessment. It could individualize both, reducing the need to force students that have mastered a concept to do repetitive practice. It might be a big challenge to get such a thing to work well, but let's not look back at our schooling as an anchor for what students today must do.

samfisher83 10y ago
A lot of things in life you get good by doing over and over. Some people are born with innate gifts, but a lot of people have to put in work to get where they are.
- maxerickson 10y ago
  Indeed. But if you can do assessment on an ongoing basis, you can eliminate the work for the students that have already mastered a given concept and individually focus the work of other students, so that they aren't wasting their time trying to solve problems that they lack the conceptual foundation to solve (math is often like this, missing 2 abstractions needed to solve a problem undermines the utility of the practice, it will be basically impossible). I'm not suggesting that it would remove the need for practice. I'm suggesting that it could be used to make the practice more effective, for students that are doing well and for students that have fallen behind.
jerf 10y ago
I once designed myself a system for teaching math online. In addition to using an engine like what they're showing here, my plan was to start from more-or-less the beginning, but also track and categorize the errors a given student makes over time. Errors are not generally made randomly, there's a pattern to them. Then, we could allow the student to "solve" an equation the way they really should, by skipping over two or three steps at a time, but when we see them do something wrong, we can use our table of "errors the student is most likely to make" to explore the space of possible errors between step 4 and step 5, and give focused feedback about what they did wrong. Using that table there's really only a couple hundred possibilities; if that fails we can always ask the student themselves to break it down more tightly. Presumably if someone were making a business out of this, there would be someone on the lookout for errors the computer can't figure out to add what rules they can to the system. (Though there will always be an irreducible residue of incomprehensible error.) Teaching a student math would then be about reducing each of these errors to zero over time. You'd have the computer custom create problems that hold a constant probability of the student making an error at some point during solving it; say 20% or so. Then as the student demonstrates mastery, you naturally make the problems more complex as you have to put in more steps to make the probability of error go that high. Instead of a klunky, chunky "ok now we learn this and you blindly practice it, now you learn this and you blindly practice it, and we hope at the end you've learned everything we taught", you would in theory get a naturally progressive, customized difficulty curve that keeps the student continuously engaged with being about 80% correct, but always progressing forward. This approach also naturally ensures that just because we're covering the quadratic equation this week does not mean you get to forget fractions; once you've seen the simple stuff with integers, we're naturally going to fold fractions back in to the problems, for instance. There's some elaborations on the theme after that, such as pre-examining the generated problems to ensure that the most likely mistakes are all distinguishable by producing different error output. But I don't have time to do this. I'm at least reasonably confident it would work, though.
- nooron 10y ago
  Diagknowstics is working on this in Ann Arbor. Not sure how far along they are, but it's a very bright team.
  - jerf 10y ago
    Interesting. I'm literally 15 minutes' walk from them right now. I wish them well; I'd love to have something like this for my own kids in the 3-4 years it'll take for it to be useful for them.
- Pulcinella 10y ago
  As a teacher I would kill for something like this, even if it's entirely paper-based instead of digitized. As you said, creating something like this is incredibly time intensive. There is a reason lots of free and paid teaching resources are basically worksheets, a textbook, and an answer key for "documentation." Anything more complicated is so much work. Even if you knew exactly what you wanted to do and how you wanted to implement it, you still have "beta test" it with students, determine the space of possible errors and misconceptions[1], make corrections, etc. I've thought about creating something like this for science, but I "only" teach the same lesson 4-5 times a day. Anything I learn, any corrections I want to make have to wait until next year when I teach that lesson again. Then if those changes don't work or I realize I need to take different changes, I have to wait another year to make them. Not exactly rapid iteration. [1]Not quite as easy as you might think. Students usually can't explain their misunderstanding very well or at all and you probably understand the material too well and it's hard to understand how to solve the problem any way but the correct way.
omaranto 10y ago
I hope the kids of tomorrow get better homework, not necessarily less of it (although in some places they definitely should also get less of it).

trendia 10y ago

I remember feeling pretty awesome because my TI-89 could solve most integrals and even some differential equations. Towards the end of my education, Wolfram Alpha came out and would not only solve those equations, but also show step-by-step solutions. Although the solver could be used for cheating, I think it could also be beneficial. Previously, a student who reached a point where they couldn't understand a problem would have trouble finding the right information to understand the solution (e.g. in Diff EQ there are many "tricks" that are necessary to understand in order to solve an equation and which are not immediately obvious). However, with Wolfram Alpha and the like, the student can work through that problem and understand how to solve it.

cmrx64 10y ago
So much this! I had a really time learning calculus, due to these "tricks". Wolfram Alpha's step-by-steps helped me so much. I could have went and harassed a TA, and I still did that occasionally, but I could largely do it on my own with the computer's help.
mighty_atomic_c 10y ago
Mathematica was my tool of choice. I could plot, Manipulate and visualize things. That was very powerful for learning quickly and getting a "feel" for things. The ability to check if my work was equivalent to the initial and final equations meant I could catch mistakes when I was stumped and trace through my work. I could see where I messed up. When practicing for exams, I would make mental notes of common errors or properties I forgot, and write a portion of "misc idiot lapses" on part of my crib sheet.

ibmthrowaway271 10y ago

> You could always use a calculator but the whole 'show your own working' catch meant you had to do it all manually. Not any more! You could often cheat (for equation rearrangement questions) if you knew the answer by simply working backwards towards the question, this is often easier than going from problem to solution but still provides all of the steps along the way. I remember one maths teacher hinting at this trick, especially to understand the derivation of the quadratic formula:- ax^2 + bx + c = 0 by starting with:- x = ( -b +/- sqrt( b^2 - 4ac ) ) / 2a and working backwards.

Asooka 10y ago
There is also "solving by assuming the negative". It's a technique for problems like "Given X, prove Y", where you assume "Y is false" and use it to prove "X is false", but X is true, therefore Y has to be true. Sometimes it's easier to do it in that direction than X->Y.
- Tyr42 10y ago
  Contrapositive is the fancy word for that technique. A -> B iff not(B) -> not(A)

SilasX 10y ago

Well, you could always shift to "find the erroneous step in this person's attempt to show their work."