TITLE: Using Programming to Learn Math, Even at the University
AUTHOR: Eugene Wallingford
DATE: October 17, 2016  4:10 PM
DESC: 
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BODY:
 There is an open thread on the SIGCSE mailing list called "Forget
 the language wars, try the math wars".  Faculty are discussing
 how to justify math requirements on a CS major, especially for
 students who "just want to be programmers".  Some people argue
 that math requirements are a barrier to recruiting students who
 can succeed in computer science, in particular calculus.

 Somewhere along the line,
<A HREF="http://www.horstmann.com">
   Cay Horstmann</A>
 wrote a couple of things I agree with.  First, he said that he
 didn't want to defend the calculus requirement because most
 calculus courses do not teach students how to think "but how to
 follow recipes".  I have long had this complaint about calculus,
 especially as it's taught in most US high schools and universities.
 Then he wrote something more positive:

<BLOCKQUOTE><EM>
 What I would like to see is teaching math alongside with
 programming.  None of my students were able to tell me what
 sine and cosine were good for, but when they had to program
 a dial [in a user interface], they said "oh".
</EM></BLOCKQUOTE>

<BLOCKQUOTE><EM>
 Couldn't that "oh" have come earlier in their lives?  Why
 don't students do programming in middle school math?  I am
 not talking large programs--just a few lines, so that they
 can build models and intuition.
</EM></BLOCKQUOTE>

 I agree wholeheartedly.  And even if students do not have such
 experiences in their K-12 math classes, the least we could do
 help them have that "oh" experience earlier in their university
 studies.

 My colleagues and I have been discussing our Discrete Structures
 course now for a few weeks, including expected outcomes, its
 role as a prerequisite to other courses, and how we teach it.  I
 have suggested that one of the best ways to learn discrete math
 is to connect it with programs.  At our university, students
 have taken at least one semester of programming (currently, in
 Python) before they take Discrete.  We should use that to our
 advantage!

 A program can help make an abstract idea concrete.  When learning
 about set operations, why do only paper-and-pencil exercises when
 you can use simple Python expressions in the REPL?  Yes, adding
 programming to the mix creates new issues to deal with, but if
 designed well, such instruction could both improve students'
 understanding of discrete structures -- as Horstmann says, helping
 them build models and intuition -- and give students more practice
 writing simple programs.  An ancillary benefit might be to help
 students see that computer scientists can use computation to help
 them learn new things, thus preparing for habits that can extend
 to wider settings.

 Unfortunately, the most popular Discrete Structures textbooks
 don't help much.  They do try to use CS-centric examples, but
 they don't seem otherwise to use the fact that students are CS
 majors.  I don't really blame them.  They are writing for a market
 in which students study many different languages in CS 1, so they
 can't (and shouldn't) assume any particular programming language
 background.  Even worse, the Discrete Structures course appears
 at different places throughout the CS curriculum, which means
 that textbooks can't assume even any particular non-language CS
 experience.

 Returning to Horstmann's suggestion to augment math instruction
 with programming in K-12, there is, of course, a strong movement
 nationally to teach computer science in high school.  My state
 has been disappointingly slow to get on board, but we are finally
 seeing action.  But most of the focus in this nationwide movement
 is on teaching CS <EM>qua</EM> CS, with less interest in emphasis
 on integrating CS into math and other core courses.

 For this reason, let us again take a moment to thank the people
 behind
<A HREF="http://www.bootstrapworld.org">
   the Bootstrap project</A>
 for leading the charge in this regard, helping teachers use
 programming in Racket to teach algebra and other core topics.
 They are even evaluating the efficacy of the work and showing
 that the curriculum <EM>works</EM>.  This may not surprise us in
 CS, but empirical evidence of success is essential if we hope to
 get teacher prep programs and state boards of education to take
 the idea seriously.
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