Understanding the Derivative via Strogatz

If you've never read Steven Strogatz's book The Joy of x, you should put it on your reading list.  Strogatz, in my opinion, is able to sell and teach the development of mathematics to a general audience--which is no easy task.  He's a brilliant teacher in this book and can be appreciated by both "math people" and "non-math people," educators and non-educators alike.

I have a class set of his books, and I got to put them to use for the first time this week.  I had my calculus students read the beginning of the chapter entitled "Change We Can Believe In."  Strogatz does such a great job explaining the value of a derivative in this chapter.  I gave my students an anticipation guide and explained the value of anticipating where an author is going with the material...before you read the actual material.  I think this is especially true in mathematics:  it took me a looooong time as a student to realize math textbooks could be used for more than just the problem sets.  But, when I did start to fully appreciate math texts for their entire content, I was invested in the material because I would make predictions about the proofs before reading.  If I could get through the proof without the help of the author, wohoo! (rare, but wohoo nonetheless).  If not, I had invested enough time and energy into the problem that, by golly, I was going to figure it now.  Which meant I needed to READ.

I digress.  This wasn't supposed to be a post on the value of this literacy strategy.  But there you have it anyway.

Here's the AG I gave the kids.  They did argue through a few of the statements, which is exactly what I'd hoped for.

The Joy of X-Change We Can Belive in (AG)

Students asked when they would get to read from the book again and where they could their own copy of the book...so I count this as a success.

Derivatives of Exponentials and Logs with Desmos

Here's a Desmos activity I typed up for my calc kids to find the the derivatives of exponential and logarithmic functions.  Sadly...the class set of laptops would not connect to the domain because they hadn't been used all summer, so we did this together as a class, which was not what I wanted, but what can you do?  Instead of having the kids click pause on their own screens, I had them yell "STOP" at me...so it was still entertaining.

All this to say, I don't know if this is good or not since I haven't gotten to test it out on students yet, but here it is.  Use/modify if you'd like!

Derivatives of Exp and Logs Desmos

In the "notice/wonder" section of f(x)=a^x, one student said he noticed that the derivative was proportional to the given function.  This made me a very proud momma and was a perfect segue into finding the derivative when a is different from e.  (We explored f(x)=2^x, f'(x), and g(x)=lna(2^x), and found that a=2).

Desmos also sent me this great online activity.

Power Rule Warm Up

My kids just learned the power rule in calculus.  I love this part of the course...it comes right when students need a little confidence boost after some of the abstract thinking we ask them to do about limits and the formal definition of a derivative.

I put together this warm up for them yesterday to continue practicing:

I'm embarrassed to say, I didn't realize what good functions I had chosen until the kids started working on it.

"Can there be more than one pair?"

"Ummmm...yes, but you should match up the functions so that you use each exactly once."

The puzzle was that, for example, 2x+3 could be the derivative of x^2+3x-7...or it could be the antiderivative of 2 (not that we're using that jargon yet...).  Similarly, 5 could be the derivative of 5x, or it could be the antiderivative of 0.

Again, I would love to say I planned all that in advance, but it was a total accident.  However, I'm not sure I've ever seen kids this into a warm up.  They asked for more like this... 

Mission accomplished.

Unintentionally. 

Answers

So, he wants to be a math teacher

A former student of mine recently wrote to me telling me he was considering pursuing a math education degree.  At first, I was thrilled!  Another math teacher!  How fabulous!

Then, I started wondering, Does this kid really, truly have what it takes?  Does he know what he's getting himself into?

A few comments he made had me thinking that maybe he was pursuing teaching because he really didn't have any better ideas.

So, I took the weekend to come up with what I hope was a fitting response.  Here are chunks of my letter, modified a bit.  The student is actually still set on pursuing an education degree, so...wow!

Dear -----,

I'm thrilled that you're considering education! As a young adult myself, I love to hear that other young adults want to teach. It's an incredible job; I'm certain I will never leave this field.

However, I do want to be realistic with you. There are two things I would recommend thinking very seriously about...

First: the commitment. Teaching--if you want to do it well--is an incredibly time-consuming job. I'm usually at the school ten to eleven hours a day, I work from home during the weekends, and I spend a good deal of my summers researching best practices other teachers are using in their classrooms. As a good teacher, yes, you'll spend time preparing lessons and grading tests, but you'll also be contacting parents, students, principals, and counselors; you'll be writing recommendation letters; you'll be losing sleep over the kids you're particularly worried about. It's never-ending and exhausting. But, again, I wouldn't trade it for the world. What I am saying is this--don't pursue this job unless you're extremely passionate about learning, helping others learn, and loving on your students. If those aren't passions of yours, I don't recommend teaching--you'll be burnt out in a couple years. If, however, you are passionate about helping others grow and are willing to put in the time and energy necessary, I would beg you to please consider teaching. We need all the passionate and loving people we can get.

Second: the money. I used to have a motto: "Study what you love and figure out how to get paid for it later." I can't stand by that motto any longer--especially if a student is taking out large loans to pay for school (i.e., don't take out $100,000 for a job that won't give you the means to pay that back). The reality is, we have to make ends meet. And it's no secret--teachers do not get paid a whole lot. Salaries obviously vary, but I started around $30,000 and will peak--after 25 years on the job and with a Master's degree--at about $50,000 (in today's money). Compare that to my friends who got the same degrees but started at $70,000. On top of this, your friends and family will often say your pay is appropriate because you get off at 3 PM and you have your summers off. As a teacher, you have to be ok with the fact that pretty much all your friends and relatives will be making more money than you. And you have to be ok with the financial sacrifices that may accompany the job.

That's my two cents. I wouldn't recommend pursuing this job unless you know what you're getting yourself into and you're ready for (and hopefully excited about) the challenges that accompany teaching. If, after my warnings, you're still ready to jump in, then please, please do. Future students will greatly benefit from you, and you will get even more out of this job than you put into it.

Please keep me informed.  Much love,

Mrs. Peterson

Getting a little better at math history

I've changed my Mathematician Spotlight routine a bit this year for PreCalculus after seeing what @Fouss did with it in her class (here).

This year, instead of having kids write a report on the mathematician, I give them a quote from the mathematician and three tasks:

1. Rephrase the quote in your own words.
2. Write a paragraph describing why you agree or disagree with the quote.
3. Find three facts on the mathematician; write one on the board.

These papers are waaaaaay more interesting to read than last year's.  After reading the papers last year, I was like, "Ok, I know the guy was born in 1596, for the love of all that is holy, please tell me something I haven't read 43 times already."  With this format, the kids are giving me opinions, which is one of my goals for math class, so I'm enjoying it much more.  Hopefully they are, too.

I change the mathematician once a unit, so this counts as their bonus for the chapter test.

Here's our first one!

I love how they wrote their facts in columns...

Continuity and IVT

So, this isn't anything ground-breaking, but my calc kids responded so well to this one slide, that I thought maybe it's worth sharing.

Continuity has been the topic of discussion the past week.  Even though my kids learn about the Intermediate Value Theorem in PreCalculus, I wanted them to be able to do more with it than just find a couple of y-values.  They could have done that in Algebra 1.  Let's get to some more interesting questions.  So, we worked through a version of these questions.  They hated it.  I loved it.  I will use it again, no doubt.

The next day, I showed them this slide.  Again, it's nothing you can't find elsewhere, but the kids were amazingly into it.

I had the students discuss the questions with their partner before I took a class poll (thumbs up/down).  I would ask a thumbs down student to defend his position, and then a thumbs up student to defend hers.  The kids got into a couple debates, which made me super super happy.  I love it when they argue about math because then I know they're invested in the problem and they're using higher levels of critical thinking.

The most interesting one for us (I think) was the population of the earth.  The fascinating part was that even the students who said it was not everywhere continuous did not come up with the correct reasoning (or, at least the vocal ones didn't).  So, that one's a keeper for future years.

Anyway.  Some seriously good results here.