The Chain Rule was harder than I thought

Last week I taught the Chain Rule to both my Calculus I and my Business Calc class.  Let me rephrase that.  Last week I was supposed to teach the Chain Rule to both my classes.  I’m pretty sure I didn’t quite get there.  At least not yet.  I was really looking forward to teaching it, because it shows up everywhere.  Also, I never remember thinking the Chain Rule was a particularly hard concept.  But maybe I’m romanticizing my beginning calculus experience.

This is my first go at teaching either class, so I’m sticking pretty close to what the books say.  I figure the authors are the experts on both the subject and the audience, so it’s a good starting point.  Both books teach the Chain Rule quite differently, so I was excited to try both and compare and contrast.

Unfortunately, one thing I found across the board was that many of my students don’t have a firm grasp on composition functions. Sure, they can compute fog, but ask them to go the other way–to decompose a function–and all of a sudden at least half of them look at you like you’ve asked them to please go swim across the Atlantic Ocean.  It was a frustrating moment as a teacher because I couldn’t find a way to explain decompositions without using the typical vague words like “inside” and “outside” functions. I tried saying that the “inside” function is what has parenthesis around it, or the expression you could put parenthesis around without changing anything.  Yeah…that works for functions like

f(x)=sin(3x−5)

And

f(x)=4x2+2−−−−−−

But when we got to

=e
they told me the inside function was e.

Not e to some power. Just e.

FAIL.

On me, not my students.

Note to self: learn how to teach the decomposition of functions.

The Chain Rule via Leibniz notation did go a bit better. We talked about how if a company that produces video games wants to know how much it is making per minute, it could take how much it makes per game sold and multiply that by how many games it sells per minute:

Similarly, if y=f(u) changes 1/2 as fast as u, and u=g(x) changes 3 times as fast as x, then we can conclude that y changes 1/2 times 3, or 1.5, times as fast as x:

y=(3x2−4)2

Or…

Another thing I tried that I stole from the Business Calc book was beginning with a “guess” for the derivative of a function such as

f(x)=(x3+5)2

For the “guess” for f'(x) we applied the Power Rule to the “inside” function and got

[latef′(x)=2(x3+5)

Then we found the actual derivative for f(x) by expanding it and using the Sum Rule.  We found that the derivative was the same as our guess but multiplied by 3x^2!

I thought this would be a great “AH-HA!” moment.  Alas.  It was not.

I just got a bunch of, “So, what was the guess for?”  “What’s the final answer?”  “How would you enter that into MyMathLab?”

Sigh.

I think the “guess” thing really could have been powerful.  I just need to ponder how to present it better.

So, that’s the Chain Rule.  Some things worked.  Some didn’t.  Most didn’t.  But this is one of the amazing parts of teaching mathematics–learning how others learn math.

y′=2(3x2−4

Derivative Cards

Two weeks down of summer classes, six to go.

These past two weeks have reminded me how hard it is to teach a class for the first time.  I continually feel like I’m coming up short because I compare my performance in my summer classes to my performance in a typical 16-week class that I’ve taught many times.  I know that’s not a fair judgment, but I still do it.  I also know you can’t learn how to teach a course well if you don’t ever teach it that first time.  Still, I feel all my energy is spent just trying to get half-way decent lectures ready, to keep up with homework questions, and to write tests.  I don’t have much time or energy to provide learning experiences outside of the typical lecture.  And I hate that.

That said, I’m learning more about the teaching and learning of calculus every day.  And I think that’s pretty priceless.  Also, I think I’ve been given a set of unusually patient and gracious students this summer.  I get thanked about every other day for doing what I do.  And for someone who needs pats on the back, that is the biggest reward I could get.

That’s my vent.  Now for some calculus…

Every semester I have my students write an introduction about themselves.  In addition to hobbies and life goals, I ask them to tell me why they’re in the class, their math background, and their current feelings towards mathematics.  In doing so this semester, I learned that many of my students have had calculus before.  So, before we ever talked about the Power Rule, I would ask something like, “How can we find the derivative y=3x-2?”  I would, of course, get an eager, “Well, I learned how to find derivatives another way, and you just take the exponent and multiply it by 3 and then reduce the exponent by one…so the derivative is 3.”  Everyone loves the Power Rule.

“You’re absolutely right, and we’ll talk about the Power Rule soon.  But can you think of the geometric definition of a derivative and tell me what 3 corresponds to in the linear function?”

Apparently, we still need to work on writing “lim as h approaches 0.”

All this to say, I wanted to introduce the Power Rule differently, somehow.

I split the class into eight groups and gave each a “Derivative Card.”  On the top it had “Find f ‘(x) when f(x)=…”  I used four basic functions:  f(x)=x^2, x^3, 1/x, 1/x^2.  Each function had its own color.  When the students were done, I asked them to find the other group with the same color and see if they got the same answer.  I loved this because the students were doing all the work.  We then created a table on the board using what the groups just found.  I started with f(x)=x (which I didn’t give to any group).  The table looked something like this:

It’s nothing new, but the students came up with it themselves, which is the great part.  They were able to generalize the rule no problem too.  Which makes my heart very happy.

Average Rate of Change: Burgers

Summer classes started this week, which means I’ve started my two calculus classes (wohoo!).  One is essentially a survey course and the other is Calc I.  I’m enjoying it a lot, but I’ve also been continually up to my eyeballs in lesson plans, homework, and tests.

And it’s only the first week.

In any case, as I was preparing for these classes last month, I came across this wonderful introduction to rates of change using (wait for it) a burger-eating competition.

So the first day of Calc I, I asked the class, “What do you think of when I say, ‘How fast’?”

I got some good answers, like speed and rate.  I also got a suspicious answer:  average rate of change…way to steal my thunder.

We talked a bit about the importance of units when it comes to speed and then I asked them to consider speed in another setting–the speed of a person consuming an exorbitant amount of burgers:

 

And now for the questions…

• How many burgers were consumed in total?
• What does that mean that this man/woman’s average speed was (in burgers per hour and burgers per minute)?
• Is this speed constant?  How can you tell?
• During what 10-minute interval was his/her speed the greatest?  How can you tell?
• Find the average rate of change (or speed) in burgers/minute between 50 and 60 minutes.
• Between 30 and 60 minutes.

So it’s not revolutionary or anything, but it was a nice way to introduce calculus and ease everyone’s nerves a little (including my own).

“Exponentially Better”

Dear residents in my little area of the country,

I keep hearing you use the word exponential in reference to a drastic change from one examined point in time to another.  For example, “I slept exponentially better last night than I did the night before.”  Or, “My coffee tastes exponentially worse at this Starbucks as opposed to the one on Main Street.”

I usually just smile and nod.  As if I understand what you’re saying.

But, I really don’t.

How do you know the change is exponential when you’re only comparing two points?  How do you know it’s not linear, perhaps with a steep slope?

To illustrate my point, let’s say on Day 1 you rate your sleep as a 2.  On day 2, you rate your sleep as a 4. (I have no idea what these numbers indicate, but you must be able to rate your sleep on some kind of scale if you can make a statement like the aforementioned one.)  Well, then, maybe your sleep pattern, indeed, is following an exponential trend line:

An exponential function: y=2^x

Or…you could have trend lines such as:

A linear function: y=2x

A quadratic function: y=(2/3)x^2+(4/3)

A logarithmic function: y=2+2.88539ln(x)

And these only represent a few of the possibilities.

So, residents of the great Midwest:  I urge you–be more creative in your comparisons.  Don’t assume you need to use the phrase “exponentially better.”  Nay.  Why not try something like, “quadratically better,” or “logarithmically better”?

Sincerely,

The Picky One

Introverts in the Classroom

I recently watched a TED talk on introverts by Susan Cain.  It was fascinating.  I related to a lot of it.  In essence, Cain says that our society often looks down on introverts.  She recalls when she went to summer camp as a young girl, expecting it to be a time to escape into the imaginary worlds created by her books, while being surrounded by many other girls her age.  Obviously, she was sorely disappointed.  One girl asked her, “Why are you being so mellow?”  As if finding your energy by being alone–as opposed to being with others–was a flaw.

Cain encourages us to let introverts be introverts, because that’s when they do their best work.  And for the most part, I agree with that.

In terms of education, Cain argues against continual group work and “pod” seating arrangements. In her closing, Susan Cain says, “Stop the madness for constant group work.  Just stop it.”  And receives a flood applause for this statement.

I get a lot of what she’s saying, I really do.  And I don’t think she’s saying, “Stop all group work,” because she concedes that most great works require collaboration of some kind.  But, as a teacher, (and may I add, as an introverted teacher) something didn’t settle quite right with me.  Maybe it’s just hard for me to take advice from people who (I assume) have never taught in the classroom setting.

So, I’d love to hear thoughts on the following video.  As I said, I think it’s excellent.  I’m just trying to digest what it means for me as an educator.

When you have to give him a B

My students did incredible on the common College Algebra final that all CA students on our campus have to take.  The median for my classes was a high B.  I think that’s pretty darn good considering their instructor didn’t even know what was going to be on the test.

Here’s what kills me.  Even though I had some students do exceptionally well on the final, and–in fact–rather well on tests all semester, I gave out fewer A’s than I would have liked.  For example, I had a student whose lowest unit test grade (out of 5) was a 94.  And the kid got a B in the class.

WHY, BOY, WHY??

Because, to do well in my class you have to do more than do well on the tests.  We have homework grades and we have study guide grades (and next semester, we gonna have attendance grades, too, let me tell ya).

Because I’m such a young teacher I have adjusted my grading scale every semester, hoping that one day I’ll reach something I like, while still abiding by whatever rules the college has already set up.  For example, when I was a TA, I had to enforce a pretty strict attendance policy.  I had to email every student that missed a class, tell him what he missed, and give him his current attendance count.  Once a student missed more than four classes, I was to drop him.  This was really a lot of work.

So, when I finished grad school, and could structure my courses to my own liking, I went to the other extreme:  no attendance policy.  I thought, “Why make kids come to class if they already know this stuff?”  This really applied to my students, too, as they are concurrent high school seniors who have recently seen every thing College Algebra covers.

And then it hit me.

We are a class.  As in a group.  A team.  We learn from each other, and when someone doesn’t show, the group is robbed from the opportunity of learning from that student.  I am robbed from the opportunity of discovering something new about mathematics or about the learning and teaching of mathematics.

Furthermore, I don’t want to send the message of, “I don’t care if you come to class or not.”  Because I do care!  Immensely.

Hence, next semester there will be an attendance policy.  Because I want to teach my students more than mathematics.  I want to teach them life lessons, such as we learn from each other.

And that’s why I take other grades into account, too.  Like homework.  I get that not every student needs to do it (but, let’s face it–for math, most college students do).  But I want to show them that sometimes in life you have to do things you don’t want to do.  And the same rules apply to everyone–whether you’re a math superstar or not.

So, that’s how someone with a near-perfect test average got a B.  Because he didn’t grasp the fact he needed to follow instructions along with everyone else, because the instructions were set up for the benefit of us a group, as a community.

It hurt me a little to submit that B for one of the brightest students I’ve ever had.

I’m still learning.  Hopefully I always will be…

My thoughts on College Algebra

I’m 24 and have taught College Algebra 9 times.

I realize that’s not some kind of record.  But it is a lot.

When I first started teaching it, I was so excited:  How could anyone possibly find any of this boring?!

I’m finally starting to understand my students’ frustrations.  Don’t get me wrong, I still love math, and I love teaching College Algebra.  But I’m starting to wonder if it’s really accomplishing what it set out to do.

I know this isn’t the case everywhere, but in the part of the country where I live (the great Midwest), College Algebra is the peak of most college students’ mathematical career.  Any previous or remedial work is a build-up to help students succeed in College Algebra.

But what is College Algebra?

It was designed to help students succeed in calculus.

The problem is that only one in ten students enrolled in a College Algebra class will end up taking a full-length calculus sequence.

So what’s the point in preparing students for calculus if very few of them will end up taking it?

I believe College Algebra needs to be revamped (and kudos to the schools who are already working on this).  I’m not one to jump on the “Math education has to be applicable to real-world situations” bandwagon; although I’m not at all opposed to this approach.  What I am opposed to is giving pure math a bad rap.  Pure math can be just as fun as applied math.  The thing is, in a typical College Algebra course, we teach very little of the fun stuff in pure math.  Call me crazy, but I’d love to see a beginning college course that showcases the best of the world of pure mathematics.  I truly believe every college student can do a bit of abstract algebra, a little number theory, a piece of combinatorics, and a snippet of analysis/calculus.

How encouraging would it be for a student to finish her final mathematics course and be able to say, “I can do calculus!”  As opposed to what several of our current students end up saying:  “I barely passed a class that I had already taken in high school.”

I currently teach high school seniors who are taking College Algebra for both high school and college credit.  I teach them nothing new.  Nothing.  I’ve compared the topics I have to cover to the topics taught in their Algebra 2 curriculum and they’re identical.  Do we really want to send the message of This is the end-all of math!  Functions and equations.  That’s pretty much all math is.  Specifically, rational, exponential, and logarithmic functions and equations.  The end.  In fact, we think this is so important, we’re going to teach it to you again!  And again!

That’s really not the message I want to be sending.  But what are students to think if that’s what every single math class they’ve taken (both in high school and in college) focuses on?

At the Conference to Improve College Algebra in 2002, Arnold Packer said the following:

Many would skip College Algebra if they did not have to pass it to get the degree they need to enter their chosen career field. Enrollment in CA tends to fall dramatically when colleges make quantitative reasoning or intermediate algebra the requirement. Finally, a few years after finishing the course, getting their degree, and starting their professional life, they cannot recall anything they learned. Or, equivalently, they have never used anything they learned in College Algebra.

All of this is unfortunate and related. Mathematics courses that seem hard, boring, and irrelevant prior to College Algebra establish the expectation that College Algebra will be more of the same. Moreover, the course – as conventionally taught – does nothing but confirm the foreboding.

That’s the sad truth.  But I think things will change.  And I hope to be part of that change in some small way.

This fall I will start an adjunct position at my alma mater, where I will be teaching a Survey of Mathematics course.  My hope (albeit lofty) is to showcase some of the best and most interesting topics of pure mathematics.  We’ll see how it goes.

In conclusion to this rant, a final plea…

To high school teachers:  Encourage all your students to take their required college math classes ASAP (i.e., discourage them from waiting until their senior year of college).  I’ve taught too many amazing individuals that were–and probably always will be–one class short of a bachelor’s degree:  College Algebra.  Oftentimes this is due to a prolonged break in their mathematics education.  Also, encourage your bright students to test out of College Algebra (CLEP it if allowed) and move on to Calculus or take some other math elective, lest we produce a generation that thinks polynomial functions are the epitome of mathematics.

To College Algebra teachers:  Let’s try to make this class as engaging as possible.  It will be more enjoyable for both ourselves and our students if we challenge ourselves in this way.  Also, I try to remember:  my students have seen all of this before.  That doesn’t mean they remember it all, but they have seen it.  Thus, according to the law of diminishing marginal utility, we’ll have to work harder to sell it the second (or third) time around.  (That’s right, my husband’s a CPA.  I know my economics talk.  Sort of.)

To college administrators:  Decide who really needs to take this class and who might benefit from a different kind of mathematics course.

Overheard in the Math Lab

In addition to teaching at the college, I also tutor at our Math Lab.  This job is both rewarding and frustrating.  Some days I leave very proud of my work; other days I leave very disappointed by my lack of grace and patience.  But one thing that’s almost always a given:  I will hear or see something interesting.

Below are some of the things I’ve heard or seen in the Math Lab recently.  Let me be clear, my intention is not to make fun of any student.  Think of this as a “Kids Say the Darndest Things:  College Edition.”
                   
Student:  If physics makes sense to you, then chemistry won’t. Just like if you like geometry, you can’t do algebra.

Apparently liking geometry and doing algebra are now mutually exclusive.
                    
Student coming up to the tutor table:  So, are you good at math?
                    
A problem a student worked:

I just love the third line.

                    
Student to a man with his PhD in mathematics:  Are you a math tutor?
Professor:  I try to be.
                    
Me:  What’s zero minus zero?
Student:  2.
(Me:  frantically trying to figure out how to salvage that answer)
                    

 

And how does one plot a y-intercept at (-infinity, infinity)?

Me:  What does x^2 divided by x simplify to?
Calculus student:  One-half!
Me:  Maybe let’s try that again.
                    
Some of my favorite repeats:

“So, can you show me an easier way to do that?”

“Can you just write it and I watch?”

“I’ve only missed like one week of classes…”

We’re getting closer with recursive definitions

Last semester was the first time I used a College Algebra curriculum that taught sequences and series.  The first section of this chapter was a nice little intro to sequences and series, just to get students used to notation.  I thought I did such a good job explaining sequences that are defined recursively.  Until I saw the test.

Not.  So.  Hot.

Then, as I helped students in our Math Lab, I realized something–recursive definitions are not that obvious to students.

I think what happened here was a classic case of it’s-so-obvious-to-the-teacher-she-automatically-thinks-it’s-obvious-to-everyone-else.  We’ve all had teachers like this.  My absolute favorite prof from grad school loved the phrase, “Oh, this is kindergarten stuff!”  Which usually had one of two effects on me:  (1) Ahhhh!!  This is NOT kindergarten stuff!  I just spent the majority of my weekend trying to figure this out!  (2)  Where in the world did you go to kindergarten?  Remind me to send my kids there.

But I digress.

What hit me was that when I see something like:

an=an−1+an−2

I automatically think, “If I want to find a certain term, I need to sum up the two previous terms.”  Furthermore, I know that

=an+an−1

means the same thing as the previous equation.

On the other hand, when my students saw a recursive definition, I’m pretty sure they thought, “WTF.  Skip it.”

So, this semester I paid much more attention to these types of sequences.  The very first thing I did regarding recursive definitions was show a slide with this at the top:

I asked students to fill in the blanks and then asked them three questions:

1. What do the dot, dot, dots mean?
2. What do we call the term before a_n?
3. What about the term after a_n?

Maybe this is an obvious starting point, but it was such a revelation to me.

We then did some examples with

an=an−1+an−2 

I had them tell me what they thought it meant (with a lot of guidance from questions like, “a_(n-1) is related to a_n how?”).  Then we wrote a few equations in symbols and in words.  For example for,

a4=a3+a2

an=an−1+an−2

I made them write “The fourth term is equal to the third term plus the second term.”  And so on.

Then came

an+1=an+an−1

which everyone was convinced was a totally new problem (darn you, indices!).  But, once we did the same examples (finding a_4, etc.), I think/hope all minds were changed.

After working some specific examples, where initial values were given, I gave them an exit ticket of something like:

List the first five terms of the sequence defined by:
a_1 is the number of boys in the room; a_2 is the number of girls;

an=an−1+2an−2

Finj I think about 80% of students got it with zero help from me.  Not perfect, but I’ll take it this time around!

Conic Sections: Parabolas

Much like with circles, I need some major help in the area of teaching parabolas (through the lens of conic sections).  Maybe I’m just not cut out to teach geometry.  It’s quite possible.

In any case, here are a couple of things that I found/made that did work nicely:

1. This graphing paper from MathEdPage.org is awesome.  It gives a very nice low-tech option for discovering the geometric definition of a parabola.  I split the class into groups of two or three and gave each group a sheet of this graphing paper with one of the lines darkened (which is to be the directrix).  I told the students to plot seven or so points that are equidistant from the point in the middle and the darkened line.  We found a couple together, and then they were good to go for the rest.  When I gave them the following definition, they were able to fill in the  blanks no problem:
2. I made a little graph with sliders that shows what happens to a parabola in the form x^2=4py where you change p.  It wasn’t a ton of work, but I’m still pretty proud.  Plus, I continue to absolutely adore Demos graphing calculator at abettercalculator.com.  I also love their new “Projector Mode” under Settings.

So, those are two things that worked.  The 5-10 minute intro.  But once we got to working examples, I wasn’t too pleased.  I feel like I jump all over the place when I work these problems.  “What’s the vertex?!  How do we find the focus from there?  And the directrix?”  I think students get it during class, but that’s with me asking all the right questions at all the right times.  Ideas for making them do more of the work?

I got to borrow one of these from my college. I really want one. Unfortunately they’re a little pricey.