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!
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!
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.
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.
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.
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.
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.
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.
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
x = ( -b +/- sqrt( b^2 - 4ac ) ) / 2a
A -> B iff not(B) -> not(A)