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Show HN: Interactive Calculus via LaTeX

ximera.osu.edu

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Re: Show HN: Interactive Calculus via LaTeX

#3

Post by skh »

jovas wrote: post #2

Tools: https://github.com/ximeraproject Source: https://github.com/mooculus Linear algebra: https://ximera.osu.edu/laode/linearAlgebra

I really like the interactive exercises. I teach math at a community college and would like to create interactive exercise sheets for some of my classes. I’m not familiar with Github and poked around a bit but I could not figure out how the answer checking was done. In one problem I did the answer was 6 and I was pleasantly surprised the answer 5+1 was counted as correct. Do you know how this is done?

Re: Show HN: Interactive Calculus via LaTeX

#4

Post by shawn »

skh wrote: post #3
jovas wrote: post #2

Tools: https://github.com/ximeraproject Source: https://github.com/mooculus Linear algebra: https://ximera.osu.edu/laode/linearAlgebra

I really like the interactive exercises. I teach math at a community college and would like to create interactive exercise sheets for some of my classes. I’m not familiar with Github and poked around a bit but I could not figure out how the answer checking was done. In one problem I did the answer was 6 and I was pleasantly surprised the answer 5+1 was counted as correct. Do you know how this is done?

One way to do this is to install Mathematica and then run

  $ wolframscript --code "6 == 5 + 1"
  True
Suppose you know that a function is equal to "(x - 1)(x + 1) - 42". You want to check whether the student has given an equivalent function, like "x^2 - 43":

  $ wolframscript --code "f[x_] := (x - 1)(x + 1) - 42;
  Simplify[f[x]] == Simplify[x^2 - 43]"
  True
But it should also return True for any other form, like "-43 + x^2". And indeed it does:

  $ wolframscript --code "f[x_] := (x - 1)(x + 1) - 42; 
  Simplify[f[x]] == Simplify[-43 + x^2]"
  True
So in general, you can write a script which inserts the student's answer into the right-hand "Simplify[...]" block. If it prints True, they're equivalent.

It's also possible to feed a natural language expression like "the integral of Sin(x) from 0 to pi is equal to 2" into wolfram alpha, and check whether it returns True or False:

http://www.wolframalpha.com/input/?i=the+integral+of+Sin(x)+...

(To be clear, I don't think this is how this project does it. I was just giving a way to do it.)

Re: Show HN: Interactive Calculus via LaTeX

#5

Post by kisonecat »

skh wrote: post #3
jovas wrote: post #2

Tools: https://github.com/ximeraproject Source: https://github.com/mooculus Linear algebra: https://ximera.osu.edu/laode/linearAlgebra

I really like the interactive exercises. I teach math at a community college and would like to create interactive exercise sheets for some of my classes. I’m not familiar with Github and poked around a bit but I could not figure out how the answer checking was done. In one problem I did the answer was 6 and I was pleasantly surprised the answer 5+1 was counted as correct. Do you know how this is done?

I'm one of the developers of Ximera.

Expression equality is handled by https://github.com/kisonecat/math-expressions using a few heuristics, which is necessary because Richardson's theorem says this problem is undecidable. The main trick is to regard the expressions as complex-valued functions, find a point where the value is (nearly) equal, and then check for numerical equality in a neighborhood. (Doing this over the complex plane addresses the problem that many common expressions like sqrt(small number-x^2) have a restricted domain over the reals).

We rely on unit tests from Chris Sangwin (who wrote the textbook _Computer Aided Assessment of Mathematics_) to ensure that we're adjudicating expression equality mostly correctly.

Re: Show HN: Interactive Calculus via LaTeX

#6

Post by kisonecat »

jovas wrote: post #2

Tools: https://github.com/ximeraproject Source: https://github.com/mooculus Linear algebra: https://ximera.osu.edu/laode/linearAlgebra

The biggest trick here is that TeX source like $\answer{2} + 1 = 3$ provides "answer blank + 1 = 3" on the web, but you can do more complicated things like $\sqrt{\answer{4}} = 2$.

The website uses Fraser's differential synchronization algorithm, so a student's work is saved as they do it -- and that happens in real time, so an instructor can "look over their shoulder" and watch a student working in real time. The server is actually a git server, so an author publishes content by performing a git push, and because the server knows the history of the content, the server provides the version of the content on which the student last did work.

Re: Show HN: Interactive Calculus via LaTeX

#7

Post by romwell »

kisonecat wrote: post #5
skh wrote: post #3

Earlier quoted context omitted.

I really like the interactive exercises. I teach math at a community college and would like to create interactive exercise sheets for some of my classes. I’m not familiar with Github and poked around a bit but I could not figure out how the answer checking was done. In one problem I did the answer was 6 and I was pleasantly surprised the answer 5+1 was counted as correct. Do you know how this is done?

I'm one of the developers of Ximera. Expression equality is handled by https://github.com/kisonecat/math-expressions using a few heuristics, which is necessary because Richardson's theorem says this problem is undecidable. The main trick is to regard the expressions as complex-valued functions, find a point where the value is (nearly) equal, and then check for numerical equality in a neighborhood. (Doing this over th…

Interesting! When you assess the equality on a neighborhood, what's the algorithm that decides on the number and location of samples?

For polynomial functions, it's clear how many samples to use; what about others? Do you settle on a magic constant and call it a day?

Re: Show HN: Interactive Calculus via LaTeX

#8

Post by kisonecat »

romwell wrote: post #7
kisonecat wrote: post #5

Earlier quoted context omitted.

I'm one of the developers of Ximera. Expression equality is handled by https://github.com/kisonecat/math-expressions using a few heuristics, which is necessary because Richardson's theorem says this problem is undecidable. The main trick is to regard the expressions as complex-valued functions, find a point where the value is (nearly) equal, and then check for numerical equality in a neighborhood. (Doing this over th…

Interesting! When you assess the equality on a neighborhood, what's the algorithm that decides on the number and location of samples? For polynomial functions, it's clear how many samples to use; what about others? Do you settle on a magic constant and call it a day?

The magic number is currently 100 samples; the perturbation size is "adaptive" in that the perturbation size is changed when the expressions appear to equal but they're also close to zero.

The main goal is to make sure that we pass the tests listed in https://github.com/kisonecat/math-expressions/blob/master/sp...

Re: Show HN: Interactive Calculus via LaTeX

#10

Post by jovas »

pvaldes wrote: post #9

Can't see how is this better than using sweave

Can sweave be integrated with a learning management system?

Ximera allows authors to write online interactive texts, that can serve as the HW system for a class.

The student's work can be directly sent to the grade book via LTI.

Moreover, you can simultaneously produce a PDF that looks like a traditional text.

http://www.lulu.com/shop/the-ohio-state-university-math-depa...

Which nobody needs to buy, since the PDF generating the text is freely available.

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