# Category Archives: Wolfram|Alpha

## Conrad Wolfram’s vision for mathematics education

A partial answer to the questions I brought up in the last post about what authentic mathematics consists of, and how we get students to learn it genuinely, might be found in this TED talk by Conrad Wolfram called “Teaching kids real math with computers”. It’s 17 minutes long, but take some time to watch the whole thing:

Profound stuff. Are we looking at the future of mathematics education in utero here?

Wolfram, Inc. has just rolled out its newest creation: Wolfram|Alpha Widgets. These are small “apps” that execute a single W|A query using user input, without actually loading the W|A website. In just the last few days since W|A widgets have been around, hundreds of them have been made, from widgets that find anagrams to widgets that calculate comparative economic data between two states to widgets that take derivatives. Each widget also comes with the option to customize, share among social media applications (21 different services are represented), or embedded in popular blogging and wiki services such as WordPress and Mediawiki. (Sadly, there’s no WordPress.com embedding yet.) Take a look through the gallery at what’s been done.

What’s really exciting here is that you don’t need any programming knowledge to create a widget. You start with a basic W|A query, then highlight the specific search terms you want to turn into user-defined variables, and the graphical tools on the website do the work. In other words, if you can perform a W|A query, you can make a widget out of it in short order and then share it with the world via social media or embedding on a blog or wiki.

There’s a lot of potential here for use in teaching and learning:

• The ability for anybody, with or without programming skill, to create widgets from simple W|A queries opens the door for creative technology projects for students at almost any level. An instructor could assign a project in which students simply have to create a widget that does something useful for the class, for example to generate a comparison of two stocks in an economics class (though that’s already been done) or generate a contour map of a two-variable function in a multivariable calculus class. Students work in teams to create the widget and then post on a class blog or wiki.
• Instructors can easily add a W|A widget to a homework or writing assignment for easy generation of data from user-defined sources. For example, a standard exercise in precalculus and science is to determine when a sample of a radioactive substance is reaches a certain mass, given its half-life. In textbooks, we have to stick with one element and its half-life. But an instructor could now create a widget where the student enters in the name of an element or selects it from the list, and the widget spits out the half-life of that element. The instructor can alter the problem to say, “Pick your favorite radioactive element and use the widget to find its half-life. How long until 10mg of that element decays to 8mg?”

I’m very excited about the shallow learning curve of these widgets and the consequent potential for students to make and play with these things as creative components of a class. Here’s a screencast on how to make a widget, in which I do a complete walk-through of the creation process.

What are some other ways you could see Wolfram|Alpha widgets being used effectively in a course?

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## Partying like it’s 1995

Yesterday at the ASEE conference, I attended mostly sessions run by the Liberal Education Division. Today I gravitated toward the Mathematics Division, which is sort of an MAA-within-the-ASEE. In fact, I recognized several faces from past MAA meetings. I would like to say that the outcome of attending these talks has been all positive. Unfortunately it’s not. I should probably explain.

The general impression from the talks I attended is that the discussions, arguments, and crises that the engineering math community is dealing with are exactly the ones that the college mathematics community in general, and the MAA in particular, were having — in 1995. Back then, mathematics instructors were asking questions such as:

• Now that there’s relatively inexpensive technology that will do things like plot graphs and take derivatives, what are we supposed to teach now?
• Won’t all that technology make our students dumb?
• Won’t the calculus reform movement dumb down our curricula with all this nonsense about graphs and multiple representations and so on?
• How can you seriously call a person a mathematician/engineer if they can’t [insert calculation here] by hand? What if they are in a situation where they don’t have access to technology?

And yet, I actually heard all of these questions almost verbatim from mathematics and engineering professors this morning, multiple times. (To the great credit of the speaker who was asked the last question, he replied: If an engineer is ever in a situation where he is without access even to a calculator, we have a lot bigger problems on our hands than just bad education. TRUTH.)  I was having serious third-year-of-grad-school flashbacks.

These questions aren’t bad; they’re moot. In 1995, mathematical technology was just at the level of expense, accessibility, and functionality that these questions needed to be dealt with. Students could conceivably purchase calculators for a couple hundred dollars that were small enough to slip into a backpack and could calculate $\int \ln(\cos x) \, dx$ symbolically. Should we ban the technology, embrace it, regulate it, or what? Should we change what and how we teach? The technology could be controlled, so the question was whether we should, and to what extent, and these were important questions at the time.

But that was 1995. What the TI-92 could do in 1995 can now be done in 2010 using Wolfram|Alpha at no expense, using any device with an internet connection, and with functionality that is already vast and expands every other week. (This says nothing about W|A’s use of natural language input.) The technology is all around us; students are using it; there is no argument against expense, accessibility, or functionality that can reasonably be made. It’s going to affect what our students do and how they accept what we present to them regardless of what we think about it.  So I’d suggest that questions such as What do students need to know how to do in an engineering calculus course? and How do we ensure they can do those things? are better questions for now (and might even have been better then).

Some of the conceptions of innovative teaching and learning strategies I saw also seemed stuck in 1995. I won’t name names or give specific descriptions in order not to offend people who probably simply don’t know the full scope of what’s gone on in mathematics education in the last 15 years. (Although I must call out the one talk that highlighted the use of MS Excel, and claimed that there were no other tools available for hands-on work in mathematics. Augh! You should know better than that!)

I will simply say that people who concern themselves with the mathematical preparation of engineers simply must look around them and get up to speed with what is happening in technology, in the cultures and lives of our students, and in what we know now about student learning that we didn’t know then. Read some seminal MAA articles about active learning. Talk to other people. Read some blogs. Something! We can’t stay stuck in time forever.

## ICTCM underway

It’s a beautiful day here on the shores of Lake Michigan as the ICTCM gets underway. It’s a busy day and — to my never-ending annoyance — there is no wireless internet in the hotel. So I won’t be blogging/tweeting as much as I’d like. But here’s my schedule for the day.

• 9:30 – Exhibits and final preparations for my 11:30 talk.
• 10:30 – “Developing Online Video Lectures for Online and Hybrid Algebra Courses”, talk by Scott Franklin of Natural Blogarithms.
• 11:10 – “Conjecturing with GeoGebra Animations”, talk by Garry Johns and Tom Zerger.
• 11:30 – My talk on using spreadsheets, Winplot, and Wolfram|Alpha|Alpha in a liberal arts calculus class, with my colleague Justin Gash.
• 12:30 – My “solo” talk on teaching MATLAB to a general audience.
• 12:50 – “Programming for Understanding: A Case Study in Linear Algebra”, talk by Daniel Jordan.
• 1:30 – “Over a Decade of of WeBWorK Use in Calculus and Precalculus in a Mathematics Department”, session by Mako Haruta.
• 2:30 – Exhibit time.
• 3:00 – “Student Projects that Assess Mathematical Critical-Thinking Skills”, session by David Graser.
• 5:00 – “Visualizing Mathematics Concepts with User Interfaces in Maple and MATLAB”, session by David Szurley and William Richardson.

But first, breakfast and (especially) coffee.

## Courses and “something extra”

Some of the most valuable courses I took while I was in school were so because, in addition to learning a specific body of content (and having it taught well), I picked up something extra along the way that turned out to be just as cool or valuable as the course material itself. Examples:

• I was a psychology major at the beginning of my undergraduate years and made it into the senior-level experiment design course as a sophomore. In that course I learned how to use SPSS (on an Apple IIe!). That was an “extra” that I really enjoyed, perhaps moreso than the experiment I designed. (I wish I still knew how to use it.)
• In my graduate school differential geometry class (I think that was in 1995), we used Mathematica to plot torus knots and study their curvature and torsion. Learning Mathematica and how to use it for mathematical investigations were the “something extra” that I took from the course. Sadly, the extras have outlived my knowledge of differential geometry. (Sorry, Dr. Ratcliffe.)
• In the second semester of my graduate school intro abstract algebra class, my prof gave us an assignment to write a computer program to calculate information about certain kinds of rings. This was a small assignment in a class full of big ideas, but I had to go back and re-learn my Pascal in order to write the program, and the idea of writing computer programs to do algebra was a great “extra” that again has stuck with me.

Today I really like to build in an “extra”, usually having something to do with technology, into every course I teach. In calculus, my students learn Winplot, Excel, and Wolfram|Alpha as part of the course. In linear algebra this year I am introducing just enough MATLAB to be dangerous. I use Geometers Sketchpad in my upper-level geometry class, and one former student became so enamored with the software that he started using it for everything, and is now considered the go-to technology person in the school where he teaches. In an independent study I am doing with one of my students on finite fields, I’m having him learn SAGE and do some programming with it. These “extras” often provide an element of fun and applicability to the material, which might be considered dry or monotonous if it’s the only thing you do in the class.

What kinds of “extras” were standouts for you in your coursework? If you’re a teacher, what kinds of “extras” are you using, or would you like to use, in your classes?

## Unexpected kudos

Last night I received this email from my colleague Dan Callon, who is at the Joint Mathematics Meetings in San Francisco:

Robert,
I went to a session at the national joint meetings tonight on Wolfram|Alpha, sponsored by the MAA Special Interest Group on Mathematics Instruction Using the Web, with speaker Bruce Yoshiwara of Los Angeles’ Pierce College.  He cited the blog of the best-known expert (outside of Wolfram itself) in the country on using Wolfram|Alpha in education: Robert Talbert.  Congratulations!
Dan

I would have to rank Maria Andersen way above myself both in terms of her expertise with W|A and in terms of how well-known she is, but still, I’m honored by Prof. Yoshiwara’s mention. And I’ll keep trying to crank out relevant posts about Wolfram|Alpha in the future.

Filed under Casting Out Nines, Technology, Wolfram|Alpha

## Wolfram|Alpha as a self-verification tool

Last week, I wrote about structuring class time to get students to self-verify their work. This means using tools, experiences, other people, and their own intelligence to gauge the validity of a solution or answer without uncritical reference an external authority — and being deliberate about it while teaching, resisting the urge to answer the many “Is this right?” questions that students will ask.

Among the many tools available to students for this purpose is Wolfram|Alpha, which has been blogged about extensively. (See also my YouTube video, “Wolfram|Alpha for Calculus Students”.) W|A’s ability to accept natural-language queries for calculations and other information and produce multiple representations of all information it has that is related to the query — and the fact that it’s free and readily accessible on the web — makes it perhaps the most powerful self-verification tool ever invented.

For example, suppose a student were trying to calculate the derivative of $y = \frac{e^x}{x^2 + 1}$. Students might forget the Quotient Rule and instead try to take the derivative of both top and bottom of the fraction, giving:

$y' = \frac{e^x}{2x}$

Then, if they’re conscientious students, they’ll ask “Is this right?” What I suggest is: What does Wolfram|Alpha say? If we type in derivative of e^x/(x^2+1) into W|A, we get:

The derivative W|A gets is clearly nowhere near the derivative we got,  so one of us is wrong… and it’s probably not W|A. Even if we got the initial derivative right in an unsimplified form, the probability of a simplification error is pretty high here thanks to all the algebra; we can check our work in different ways by looking at the alternate form and at the graphs. (Is my derivative always nonnegative? Does it have a root at 0? If I graph my result on a calculator or Winplot, does it look like the plot W|A is giving me? And so on.)

But how is this better than just having a very sophisticated “back of the book”, another authority figure whose correctness we don’t question and whose answers we use as the norm? The answer lies in the  “Show steps” link at the right corner of the result. Click on it, and we get the sort of disclosure that oracles, including backs of books, don’t usually provide:

Every step is generated in complete detail. Some of the details have to be parsed out (especially the first line about using the Quotient Rule), but nothing is hidden. This makes W|A much more like an interactive solutions manual than just the back of the book, and the ability given to the student to verify the correctness of the computer-generated solution is what makes W|A much more than just an oracle whose results we take on faith.

Using W|A as a self-checking tool also trains students to think in the right sort of way about reading — and preparing — mathematical solutions. Namely, the solution consists of a chain of steps, each of which is verifiable and, above all, simple. “Differentiate the sum term by term”; “The derivative of 1 is zero”. When students use W|A to check a solution, they can sit down with that solution and then go line by line, asking themselves (or having me ask them) “Do you understand THIS step? Do you understand THE NEXT step?” and so on. They begin to see that mathematical solutions may be complex when taken in totality but are ultimately made of simple things when taken down to the atomic level.

The very fact that solutions even have an “atomic level” and consist of irreducible simple steps chained together in a logical flow is a profound idea for a lot of students, and if they learn this and forget all their calculus, I’ll still feel like they had a successful experience in my class. For this reason alone teachers everywhere — particularly at the high school level, where mechanical fluency is perhaps more prominent than at the college level — ought to be making W|A a fixture of their instructional strategies.