Checking Students' Symbolic Math on a Computer
Benjamin Crowell, Natural Science Division, Fullerton College, 321 E. Chapman Ave.,
Fullerton, CA 92832

Conceptual Mathematics
This note describes an open-source computer program that allows students to check theiranswers to symbolic homework problems. Traditional introductory physics courses emphasize
solving a problem to obtain a numerical result, but this skill is only weakly correlated withconceptual understanding.1 We state the laws of physics in the language of algebra, but manystudents find it difficult to connect a purely symbolic equation with its interpretation; to them, anequation is nothing more than an algorithm for numerical computation.
We can build up our students' conceptual-math muscles using activities that develop thelink between an equation and its meaning. In particular, we can assign purely symbolic
homework problems, in which the student is asked to interpret the result. The student finds, forexample, that the acceleration of a cart on an inclined plane is g sin θ. Now why is it that themass of the cart has canceled out? What happens when θ is zero? When θ is 90 degrees? We findfew purely symbolic problems in most introductory texts.2 Why? One reason is that students
resist them: they need help in overcoming an initial feeling that such problems are ``impossible''or ``theoretical.'' To build students' confidence with numerical problems, we give answers in theback of the book, but this approach is not as useful for symbolic problems. A printed numericalanswer provides no information about how it was derived, but given that the acceleration of a
cart on an inclined plane is g sin θ, many students will be able to cook up a derivation that gives
the right result. The danger is not so much that students will misrepresent their understanding asthat they will overestimate it.
The Software
To attack this difficulty, I have written a program called Spotter, which allows students tocheck their symbolic (and numerical) results. The software can be used, copied, and modified
freely, and full documentation is available online.3 The instructor installs the program and ananswer file on a server, and students check their answers through a web browser. Common
incorrect answers can be added to the answer file, along with an appropriate hint for the student.
The idea was suggested to me by another open-source program, LON-CAPA.4
Developing Spotter involved two challenges. First, there is the problem of accepting
mathematical expressions in machine-readable form from naive users. Since we cannot expect
students to learn a programming language for this purpose, Spotter must be able to interpret input
whose form is as close as possible to ordinary mathematical notation, a language with all theinconsistencies and ambiguities common to human languages. For example, the symbol ln may
stand for a natural logarithm, or it may indicate a variable l multiplied by a variable n. Spotter isable to do a relatively good job at resolving these ambiguities because its input is framed within acertain context: the software knows what variables are expected to occur and what units theypossess.

Second, the software must be able to recognize answers that are mathematicallyequivalent. If the instructor has entered a+7b in the answer file, Spotter must accept 7b+a aswell. It does this by choosing random values for the variables and checking whether the twoexpressions give the same result. The random numbers may be unphysical, so there may beresults such as the square root of -1, or the arcsine of 2; these cases are handled by analytic
continuation of all functions into the complex plane. This analytic continuation method makes it
unnecessary for the instructor to specify which values of the variables to sample.
Results
My students like the program's immediate feedback. I have often found it difficult toconvince my students to attend my office hours, and a salutary effect of using Spotter has beenthat they now often come to ask for help in troubleshooting solutions that Spotter has warned
them are wrong.
I was initially surprised to find that some students simply did not take advantage of thesoftware. One of my goals had been to allow students to correct certain commonly occurring
mistakes before their papers arrived on my desk, but some students were still turning in thesewrong answers, which Spotter would have detected. I suspect that this resulted fromprocrastination: students completed their homework too late to be able to consult Spotter. I have
had better results with a new policy, under which students are required to enter their answers tosymbolic problems in Spotter, and they automatically receive credit for them. They write thedetails of their calculations in a notebook, which I collect periodically.

I was concerned that students would try to abuse the system by inputting expressions andsearching for the right answer by trial and error. This has only rarely happened, presumably
because they know they are required to show their work in their notebooks, and also because
most problems would admit an astronomical number of answers, all of them equally plausible in
the absence of any physical understanding. The software also imposes a minimum waiting time
before a student can retry a problem.
Trying, Using, and Installing the Software
Readers who are interested in test-driving the software can do so through a web browser3
without installing any software on their own computers. This can be done anonymously, without
having to log in. There is a demonstration page containing the full text of two simple
mathematical problems. For samples of physics problems, there is also a page that allows users tocheck their answers to the problems in my own textbooks; the full text of the problems is in thebooks, which can be downloaded for free in Adobe Acrobat format. 5
Teachers who want to write their own answer files for use by their students will need toread the online documentation, 3 but the following is a brief summary of the steps involved. I will
be glad to answer e-mail6 from users who need help with installation. First, one downloads andinstalls the Spotter software on a Unix server. (The software is written in the portable Perllanguage, but has not been tested on Windows.) Next, one writes the answer file in a format
called XML, which is similar to the HTML language used for writing web pages. The XML fileis placed on the server, and the students are given a World Wide Web address at which they canaccess it. If the students will be recording their answers online for credit, accounts are set up forthem, and they choose their own passwords.
References
1. Eric Mazur, Peer Instruction: A User's Manual (Prentice Hall, New Jersey, 1997), p. 7.
2. A notable exception is Paul G. Hewitt, Conceptual Physics (Addison Wesley Longman,Reading, MA, 2001), which also has a strong emphasis throughout on connecting equations totheir interpretations.
3. http://www.lightandmatter.com/spotter/spotter.html
4. http://www.lon-capa.org/, described in E. Kashy et al., Amer. J. Phys. 61, 1124 (1993). The
current version of CAPA (now called LON-CAPA) has a facility similar to Spotter's for handlingsymbolic mathematical answers. Another noncommercial piece of software that does similar
things is WeBWorK, http://webwork.math.rochester.edu.
5. http://www.lightandmatter.com/area1.html
6. http://www.lightandmatter.com/area4author.html