# An unexpected lesson from technology

This discovery happened a few years ago, but I’ve just started ‘blogging, so I guess it’s time to share this for the “first” time.

I forget whether my calculus class at the time was using the first version of the TI-Nspire CAS or if we were still on the TI-89, but I had planned a very brief introduction to the CAS syntax for computing symbolic derivatives, but my 5-minute introduction in the first week of introducing algebraic rules of derivatives ended up with my students discovering antiderivative rules simply because they had technology tools which allowed them to explore beyond where their teacher had intended them to go.

They had absolutely no problem computing algebraic derivatives of power functions, so the following example was used not to demonstrate the power of CAS, but to give easily confirmed outputs.  I asked them for the derivative of $x^5$, and their CAS gave the top line of the image below.

(In case there are readers who are TI-Nspire CAS users who don’t know the shortcut for computing higher order derivatives, use the left arrow to place the cursor after the dx in the “denominator” of $\frac{d}{dx}$ and press the carot (^) key.  Then type the integer of the derivative you want.)

I wanted my students to compute the 2nd and 3rd derivatives and confirm the power rule which they did with the following screen.

That was the extent of what I wanted at the time–to establish that a CAS could quickly and easily confirm algebraic results whether or not a “teacher” was present.  Students could create as many practice problems as were appropriate for themselves and get their solutions confirmed immediately by a non-judgmental expert.  Of course, one of my students began to explore in ways my “trained” mind had long ago learned not to do.  In my earlier days of CAS, I had forgotten the unboundedness of mathematical exploration.

Shortly after my syntax 5-minutes had passed and I had confirmed everyone could handle it, a young man called me to his desk to show me the following.

He understood what a 1st or 2nd derivative was, but what in the world was a negative 1st derivative?  Rather than answering, I posed it to the class who pondered a few moments before recognizing that “underivatives” (as they called them in that moment) of power functions added one to the current exponent before dividing by the new exponent.  They had discovered and explained (at least algebraically) antiderivatives long before I had intended.  Technology actually inspired and extended my students’ learning!

Then I asked them what the CAS would give if we asked it for a 0th derivative.  It was another great technology-inspired discussion.

I really need to explore more about the connections between mathematics as a language and the parallel language of technology.

### 2 responses to “An unexpected lesson from technology”

1. Interesting. FYI—Mathematica refuses to accept the idea of $\frac{d^{-1}}{dx^{-1}}$. It requires the the “power” to be a non-negative integer. Which makes me curious—what happens if you give the CAS $\frac{d^{1/2}}{dx^{1/2}}$?

2. chrisharrow

Hadn’t thought of $\frac{d^{1/2}}{dx^{1/2}}$ as the “exponent” represents the number of derivatives taken of the given function, requiring the exponent to be an integer under my new understanding (or a natural number under the old). I checked the Nspire CAS to be sure and got the expected “domain error.”

Before doing so, I wondered what 1/2 a derivative could possibly mean and couldn’t come up with anything.

Then again, I had never considered what (1/2)! could possibly mean until I was introduced to the gamma function.