Monthly Archives: June 2013

Clever math

Here are what I think are three clever uses of math by students.

This was a gift certificate I received from a student last month from a graduating senior.

1

In my last week of classes at my former school in May, 2013, my entire Honors Precalculus class showed up wearing these shirts designed by one of my students, M.  The back listed all of the students and a lovely “We will miss you.”  As much as I liked the use of a polar function, I loved that M opted not for the simplest possible version of the equation (r=2-2sin(\theta )), but for a rotation–a perfect use of my transformations theme for the course.

2

Now for a throwback.  When I was a graduate student and TA at Syracuse from 1989-1990, one of my fellow grad students designed this shirt for all of the math and math ed students.  I don’t remember who designed it, but I’ve always loved this shirt.

3

Calculus Derivative Rules

Over the past few days I’ve been rethinking my sequencing of introducing derivative rules for the next time I teach calculus.  The impetus for this was an approach I encountered in a Coursera MOOC in Calculus I’m taking this summer to see how a professor would run a Taylor Series-centered calculus class.

Historically, I’ve introduced my high school calculus classes to the product and quotient rules before turing to the chain rule.  I’m now convinced the chain rule should be first because of how beautifully it sets up the other two.

Why the chain rule should be first

Assuming you know the chain rule, check out these derivations of the product and quotient rules.  For each of these, g_1 and g_2 can be any differentiable functions of x.

PRODUCT RULE:  Let P(x)=g_1(x) \cdot g_2(x).  Applying a logarithm gives,

ln(P)=ln \left( g_1 \cdot g_2 \right) = ln(g_1)+ln(g_2).

Now differentiate and rearrange.

\displaystyle \frac{P'}{P} = \frac{g_1'}{g_1}+\frac{g_2'}{g_2}
\displaystyle P' = P \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)
\displaystyle P' = g_1 \cdot g_2 \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)
P' = g_1' \cdot g_2+g_1 \cdot g_2'

QUOTIENT RULE:  Let Q(x)=\displaystyle \frac{g_1(x)}{g_2(x)}.  As before, apply a logarithm, differentiate, and rearrange.

\displaystyle ln(Q)=ln \left( \frac{g_1}{g_2} \right) = ln(g_1)-ln(g_2)
\displaystyle \frac{Q'}{Q} = \frac{g_1'}{g_1}-\frac{g_2'}{g_2}
\displaystyle Q' = Q \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)
\displaystyle Q' = \frac{g_1}{g_2} \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)
\displaystyle Q' = \frac{g_1'}{g_2}-\frac{g_1 \cdot g_2'}{\left( g_2 \right)^2} = \frac{g_1'g_2-g_1g_2'}{\left( g_2 \right)^2}

The exact same procedure creates both rules. (I should have seen this long ago.)

Proposed sequencing

I’ve always emphasized the Chain Rule as the critical algebra manipulation rule for calculus students, but this approach makes it the only rule required.  That completely fits into my overall teaching philosophy:  learn a limited set of central ideas and use them as often as possible.  With this, I’ll still introduce power, exponential, sine, and cosine derivative rules first, but then I’ll follow with the chain rule.  After that, I think everything else required for high school calculus will be a variation on what is already known.  That’s a lovely bit of simplification.

I need to rethink my course sequencing, but I think it’ll be worth it.

Enhancing Creativity at Home

I had a great conversation last month with a Westminster colleague, Kay, as school was winding down.  While from completely different academic departments, our daughters are nearly the same age, and so we share lots of parenting ideas and stories.  Here’s some thoughts we developed to use with our kids this summer to enhance creative thinking and enhanced abilities in many areas.

Creativity in Literature:

Our daughters LOVE to read, so our first challenge to our girls would be for them to find a good stopping place 20-40 pages before the end of each book they read. Then we encourage them to describe how they think the story might end and why they think their hypothesis is possible.  If they see multiple ways it could end, they could describe alternates.  They should WRITE their predictions to get their minds to commit and more actively engage.  This need not be anything formal–but it can be if they like.  We just want the girls to think a bit more deeply about what they’re reading and engage.  Use what they know has already happened to draw conclusions about what might have happened.

After their informal (or formal) writing, read the remainder of the book and compare their prediction(s) to the author’s chosen end.  How close did they come to the author’s conclusion?  What information did the author use to end the book that they didn’t use?  Was the author’s conclusion reasonable? Was theirs?

There’s not any particular right or wrong here for us beyond getting our kids to think about what they’re reading and to engage the process.  We also hope they will become more attentive to details in their reading.

Cooking:

Allow the kids to help plan meals; help them understand the daily processes for planning nutritious meals.  Encourage them to participate in cooking, especially for anything you cook from scratch.  What choices do you make and why?

For cooking, talk to them about what the different ingredients do for the resulting dish.  Have them make predictions about what will happen when new ingredients are added or the collected ingredients are prepared.  To the best of your ability, explain what each step does and why it’s important to the final product.

Compare the predictions to the final results.  When some part inevitably turn out differently than predicted, learn why.  Take pictures along the way for you or your child to use when comparing hypotheses and outcomes.  When you come back to a recipe on another occasion, think about how it turned out last time and plan a change or an improvement.  Knowing what your child wants to happen, can she adapt the preparatory steps to accomplish that change?

In the end, this really isn’t all that different from the reading suggestions.  Engage, observe, explore, make predictions, and compare expected and actual results.  All along, use data to explain why you believe your claims are justified.

Conclusion:

This is about where Kay & I ran out of time to chat that day, but we ended with the realization that what we were describing was exactly the scientific process, embedded in arguably “non-science” settings.  As I’ve mused over this for the past few weeks, I’ve realized that this process can be applied almost anywhere:

  • athletics (how do you get a better result? Why?),
  • mathematics (what kind of answer will I get? Can there be more than one? What will it look like?)
  • gardening (how do I get a particular plant to perform a specific way? Is that even possible with that plant?),
  • computer programming (getting a computer to do precisely what you ask of it), and so on.

Engage your children or your students. Get them to hypothesize and justify using data.  Teach them (and yourself) to be more alert to patterns and clues about past and future behavior.  Perhaps most importantly, determine if other outcomes are possible and what it would take to get there.  Envision something she or he hasn’t seen or done before and figure out what is needed to make it happen.  Then … make it happen.