Wolfram|Alpha

*David M.
Bressoud August, 2009*

In case there is anyone who has missed it, Wolfram|Alpha
is a web site unveiled on May 15 of this year that combines *Mathematica*
with a large variety of databases and some sophisticated language interpretation
to provide a search engine that returns quantitative and mathematical information.
It hit the news in the *Chronicle of Higher Ed* [1]
and Carl Bialik's column in the *Wall Street Journal* [2]
where attention was drawn to its possible effect on the teaching of mathematics.
Maria Andersen, in her blog "Impact
of Wolfram Alpha on Math Ed" [3], argues that
this will be significant. Derek Bruff has already set up a wiki on *Teaching
Undergraduate Math with Wolfram|Alpha *that includes many examples
of the kinds of problems that might be asked in an undergraduate math class
and for which Wolfram|Alpha provides not just a solution, but reveals and
explains each step in the process of getting to the answer, an option that
is revealed when the user clicks on *Show steps*.

While there are many computer algebra systems (CAS)—many of which
are free—that will crank through standard algorithmic procedures such
as solving algebraic equations for exact solutions or integrating complicated
functions, Wolfram|Alpha is significant for its accessibility and ease of
use. It is not only free and and interfaced via the web, it also is quite
good at figuring out what the user wants to find. To **integrate t
* exp( sqrt(t) )**,* Mathematica *requires precise parsing.
This must be entered as **Integrate[t*Exp[Sqrt[t]],t]**. The
commands are case sensitive, and woe to the user who confuses brackets,
braces and parentheses. For Wolfram|Alpha, **integrate t * exp( sqrt(t)
)** suffices.One could also enter **integral t e to the square
root t** to get exactly the same answer. In other words, this is
the first CAS that really is user friendly. Adding the phrase **from
0 to 2** produces the evaluation of the definite integral, both the
exact value and a decimal approximation of up to 800 digits. The language
interpreter is not perfect. Entering **integral te to the squareroot
t**, it interprets **te** as **metric ton **and
integrates a constant called** metric ton** raised to the sqrt(t)
power. Note that one can use **integral** or **integrate**,
**square root** or **squareroot** or **sqrt**,
and the software does a pretty good job of guessing the variable on which
to integrate.

Wolfram|Alpha also provides more than just the answer. Ask it to **integrate
sin x^2**, and you get a solution in terms of the Fresnel integral,
links to information on the Fresnel integral, plots and series expansions
of the integral, and the values of some common definite integrals of sin
x^2.

This returns us to the question that was posed when CAS systems first appeared: If computers can solve these problems so efficiently, why do we drill our students in answering them? The response is still the same: There are important mathematical ideas behind these methods, and showing one knows how to solve these problems is one way of exhibiting working knowledge of these ideas. The existence of CAS does push instructors to be more honest about their use of standard problems executed by memorizing algorithmic procedures. If a student feels that she or he has learned nothing that cannot be pulled directly from Wolfram|Alpha, then the course really has been a waste of time.

Wolfram|Alpha has the potential to interfere with online homework systems
such as WeBWorK. The kinds of questions that WeBWorK is best at asking are
the same questions that Wolfram|Alpha is best at answering. But as I pointed
out in my April column on WeBWorK
[4] *, *the real point of this software is to free the instructor
to spend more time on higher order questions. If students are held accountable
for basic procedural questions via something like a gateway exam [5],
then they shortchange the learning process at their own peril. I can imagine
students using Wolfram|Alpha to check an answer before posting it to WeBWorK,
but WeBWorK itself checks the answer and allows the student to work a similar
problem until he or she gets it right.

I actually see a bigger threat from the online study sites such as Course Hero, Cramster, Hotmath, and Koofers (see [6]) that could interfere with higher order questions . These are for-profit sites. For a monthly fee, students get access to solutions to the problems in most of the standard math texts, the service of "tutors" who too often just provide detailed answers to more complex problems, and—for a growing number of colleges and universities—access to all of a professor's past assignments and exams. The sites vary. Some of them really are designed to provide tutoring help. Hotmath, for example, only provides detailed solutions to the problems for which answers are given in the back of the book and provides a hint before revealing each next step in the solution. But all of these study sites can be abused.

Yet, is any of this really new? In my undergraduate days, many decades ago, fraternities and sororities were noted for maintaining files of past exams, and it was never too hard to find someone who could provide the answers to a homework assignment. The internet simply has made the process easier and open to more students. That is not a trivial change, but it need not be revolutionary.

Telling students to do the even problems, grading their work by just checking the answers, and never challenging students to go beyond the standard algorithmic problems or holding them accountable for explaining their solutions has never been a good approach to teaching. Both Wolfram|Alpha and study sites should simply hasten the demise of this misplaced notion of teaching.

[1] Jeffrey R. Young. 2009. A Calculating Web Site
Could Ignite a New Campus 'Math War.' *The Chronicle of Higher Education*.
June 11.

[2] Carl Bialik. 2009. Sum Help: New Search Engine for
Mathletes. *Wall Street Journal*. June 17. page A14. online.wsj.com/article/SB124516890985419379.html#printMode

[3] Maria Andersen. Impact of Wolfram Alpha on Math Ed. blog posted Friday, May 29th, 2009 at 2:02 pm and available at teachingcollegemath.com/?p=998

[4] David Bressoud. 2009. WeBWorK, *MAA Online,
*April*. *www.maa.org/columns/launchings/launchings_04_09.html

[5] This is a high stakes exam of procedural fluency on questions such as techniques of integration. It is given in a controlled setting without access to technology, the grade is usually recorded as either pass or fail with a high bar for a passing grade, and students are given the opportunity to retake similar exams until a passing grade is achieved.

[6] Anne Marie Chaker. 2009. Do Study Sites Make the Grade?
*The Wall Street Journal. *April 9.

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Illustrative Resources*.

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columns.

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and President of the MAA. You can reach him at bressoud@macalester.edu. This column does not reflect an official position of the MAA.