Reflections

Somehow summer classes have come and gone.  I know students take summer classes to learn information fast.  I hope that they really do learn.  I do know that I've learned so much more than I thought was possible, so I wanted to take some time to write what I thought went well this semester and what didn't go so well.  This way, if I get to teach calculus again, I can come back and remember what needs to be stressed more.

What I plan to keep:

• Bowman Dickson's method of introducing inflection points through the spread of an epidemic (inflection via infection)!  This worked especially well for my Business Calc students who are not quite as enthralled with pure mathematics as...some.
• Making students find the general antiderivatives for functions and then having them create a chart on the board.  Once a student found an antiderivative, s/he would pass the marker on to another student to find the next antiderivative on the chart.  This was inspired by Prof. Burdette from an article by Faculty Focus.
• Derivative cards.

What I need to change:

• MUCH better review of composite functions before we get too deep into calculus.  The Chain Rule and u-substitutions are just too hard to teach if we can't recognize composite functions as such.
• Critical points.  What are they and why are they important?  I didn't a good enough job emphasizing how important the locations are where the derivative is zero or undefined.  I assumed the geometric representation of critical values would be clear.  Never assume.
• More emphasis on the fact that integration is related to antidifferentiation.  Specifically, Leibniz's elongated s combined with a differential (dx, for example) is the COMMAND to antidifferentiate.  Once you do the command, the elongated s and dx disappear.  Notation, notation, notation.  It's important!  [What worked in private tutoring:  the integral sign is like the capital letter and the differential is like the period at the end of a sentence.  You need both; and combined they tell you to find the antiderivative.]

What I plan to focus on more with my algebra and pre-calc students to get them ready for calculus:

• Compositions.  I want to start early with this idea.  For example, what does √ mean?  Well, nothing, unless it has an argument.  Similarly, sin, ln, and ( )^2 are all meaningless without an input. 
• Exponents, baby.  My Calc I students seem to be fine on this, but my Business Calc students were way behind (in general).  I need students to know that  can be rewritten as x^(-1/4).  And I need them to be able to recognize this quickly.  Furthermore,  as crazy as this sounds, I need them to be able to add and subtract fractions so we can milk the Power Rule for all it's worth.  Please, for the love of all that is good and holy, don't pull out your calculator to figure out what (1/2)-1 is.  Please.
• I want to pound in the idea that the slope of a horizontal line is zero.  
• How do we find the change in a quantity? (Subtraction.)  Seems simple, but it's something I need to emphasize more.
• Difference quotients.  Less emphasis on computing a bunch of them, more emphasis on how it's just slope.  I don't want my students to memorize a formula, I want the formula to flow out of an understanding for how to mathematically write "How fast does y change with respect to x?"

:(

This may be old news to some, but I just found this on YouTube.  It hurts my heart a little.

Ok.  More than a little.

Leibniz v. Newton

I gave my calculus students the following question on their final: "Name the two inventors of calculus.  Which do you like better and why?"

I really didn't know what answers to expect, but I found myself thoroughly enjoying their responses, and thought you might, too...

Newton.  Honestly, I feel that his work was more useful.  Plus, he did much more that I know of other than calculus.

If I had to choose, I like Newton because of his wide range of scientific contributions.  Also, to be honest, his name is easier to pronounce!

I like Newton better because he came up with the laws of motion and his name is easier to pronounce.

I like Newton better because he also came up with his physics laws.

I like Newton better because he has a lot more to do with science in my opinion and I like science a lot.

Leibniz all the way.  Integral signs are fun to draw, and he's overshadowed by Newton too often.

Newton because he's more well known.

Leibniz notation, while more "foreign" is much more straightforward.  Newton, though is considered to be one of the greatest minds in all of history (according to Neil deGrasse Tyson).  I appreciate both their mathematic contributions equally.

I like Leibniz's notation better.

I like Leibniz better because he has a cooler name.

Leibniz because I like his notation better.

I like Newton better because I can spell and pronounce his name. :)

I like Leibniz better because technically he invented calculus earlier than Newton.

Leibniz. His notation was easier for me to understand.

Leibniz--his method of calculation is easier for me to understand (Leibniz notation).

Question for the blogosphere--whom do you like better, and why?

More thoughts on the Chain Rule

I posted about how teaching the Chain Rule was a lot harder than I thought it would be.  I'm still convinced this is at least partially due my students' lack of understanding/recognizing a composite function.  For example, just the other day we needed to simplify the expression



And a student (one of my top students, I might add) suggested we "divide out an ln."

Hold up.

Let's ignore the fact that dividing by any number other than 1 would change the expression.

Dividing by ln?  So...somehow there's not a connection that ln is meaningless without an argument.  "Dividing by ln" is akin to "dividing by √ " or "dividing by cos."  An empty square root or an empty cosine doesn't have any kind of value, and really doesn't mean a thing.

I was further disturbed when I gave my Business Calculus students a function like



and was told that in order to find the derivative, we should use the Product Rule.

Wha...?

Do students view ln as some sort of constant?  Like e?  It seems maybe so if the function above is thought of as a product and if we can indeed "divide out ln."

I decided we needed to revisit the Chain Rule.

I started by showing a slide that had a composite function at the top and four expressions beneath it such as:

I asked my students to tell me why we needed to use the Chain Rule and then to identify the derivative of the outside function (holding the inside) and the derivative of the inside function.  The next slide highlighted the former in red and the latter in blue:

We did several of these.  Then we concluded with other types of functions to test if they knew when to use what rules.

I think I will start with this type of presentation the next time I teach the Chain Rule.  Giving the students a limited amount of options to start out with seemed to worked fairly nicely.

There are still definitely some issues.  But I now have a better understanding of what needs to be emphasized  in terms of composite functions.  I will try to make them more densely populated in my algebra and pre-calculus classes from now on.

If you're interested, here's the slideshow we worked through:

Test 3 Review

M&Ms and the Population of Afghanistan

In Business Calc, we're currently studying exponential growth and decay.  I'm rather excited about this since it's something we study in College Algebra, too, and I feel--because it's material I've taught before--that I can expand a bit.  I'm learning that it's really, really hard to expand (i.e., go beyond an absolutely dazzling lecture *cough*) when I'm teaching a class for the first time.  I sort of feel like I did my first semester as a TA:  I just hope I don't screw something up too terribly. But--you gotta start somewhere, right?!

Anyway.  Back to exponential growth/decay.  In College Algebra, when we study exponential functions, I have my students model the decay of an M&M population.  I had planned to do this with my Business Calc class as well.  Then Bowman Dickson posted places to find awesome data, which made me want to use data the UN has on the world's populations instead.

The question:  How to relate M&M's to population growth or decay?

The answer:  I'm not entirely sure.  Here's what we did though...

We started out with the M&M project as in College Algebra.  Each team found the exponential regression (in the form y=ab^x) and the r^2 value for their data.  We talked about the meaning of a and b.  Then I asked them to convert their regressions to the form P(t)=P_0e^(kt), which turned out to be very close to the trendlines Excel found (yay!).  We talked about what k would mean if it this were a real population and how it's related to the derivative.

Now the challenge:  I asked them to do the same types of calculations for an actual population, using data from the UN.  They were on their own for this project, which may or may not have been a great idea.  Below is what they had to go off of.  I focused mainly on finding the exponential regression on a TI as well as understanding growth/decay rates.  But there's much, much more to do here (Bowman does a week-long project!).

And here's the project!

Population Growth or Decay Instructions

A New Chapter

The past three years I've taught math at the introductory college level.  And I've loved it.  The last year of this three-year gig has been spent teaching concurrent high school juniors and seniors College Algebra.  I had the privilege of teaching some amazing high school students classes sponsored by the local community college.  I loved everything about my job.  So much so, that I distinctly remember telling my husband a couple months ago that if the high school I taught at offered me a job, I would seriously consider switching from college to high school teaching.

And they did!

So I accepted!

I know there are a few things I'm giving up by taking a break from the college scene (there's a reason I went straight for a Master's in math), but there's so much I'm looking forward to.

What I can't wait for

•  Quadruple the amount of time spent with students.  Give or take.  I've felt that it's really challenging to make an impact on my students when I only get them for three hours a week, for sixteen weeks.  Once you factor in holidays and test days, I get to talk to them about forty hours.  That's the amount of time most people spend with their co-workers in one week.  I'm not saying you can't make a difference as a college instructor; I know many, many people who do.  I'm just saying, I'm excited to be with the students more.
• The ability to cover material more thoroughly.  College classes go by so fast.  I'm excited to be able to take my time and, hopefully, not feel as rushed.
• Collaboration.  Having taught at this school already, I know the types of teachers I'll be working with.  They're amazing, and I can't wait to learn from them.  As a college instructor, I've really felt alone in my teaching.  I don't think teaching is supposed to be a solo project.  Now it won't be.
• Concerts, games, talent shows, etc.  I'm ready to be a part of school that has amazing school spirit!

What I'll miss

• Not having to teach until 9 AM.
• FERPA.
• The ability to fail a student and not have it questioned.

All in all, I feel I'm gaining much more than I'm losing.  I'm really thrilled to be teaching at the high school level; although, I'm definitely scared out of my mind.  I've said from Day One that I'm extremely passionate about seeing high schools and colleges have conversations about how to make education a more continuous process from grades 12 to 13.  I'm hoping that gaining high school teaching experience will allow me to someday be a voice for programs that enable high school students to start their college or technical careers early.

So, starting in August, I'll be teaching Algebra 2 and Pre-Calc!  I'll be staying on as an adjunct at the community college, but, rest assured, I'm taking this fall semester off!

Calc Teachers--I need you!

My summer calculus classes are coming to a close (two weeks left--gah!).  For both finals (Business Calc and Calc I), I'm planning on having a computational part and a conceptual/theoretical/write-it-out-in-words part.   I'm working on questions for the latter part, and I need your help.  What other major (mostly differential) calculus concepts would you add?  Are any of the questions unclear?  All help is greatly appreciated!

PS:  These questions are not written in any specific order. 

1. Describe what it means for a function to be continuous at a point x=c.  There should be something about a limit in your answer.
2. A function f attains both an absolute minimum and an absolute maximum on an interval as long as two conditions are met, as described by the Extreme Value Theorem.  (1) Name the two conditions.  (2) Sketch a situation in which the conditions are not met and in which the function does not attain either an absolute minimum or maximum (or both).
3. A function can attain absolute extrema at endpoints and at critical values.  What's a critical value?
4. For a function to be differentiable at a point, it must be _______________ and _____________ at that point.  Give an example of a function (in function notation, not a sketch) that is not differentiable at at least one point.  State that point.
5. Which is true:  For a function to be differentiable, it also must be continuous; For a function to be continuous, it also must be differentiable.
6. Which is true:  For a function to be integrable, it also must be continuous; For a function to be continuous, it also must be integrable.
7. If f '  is positive over an open interval, what can you tell me about f on this interval?
8. If f '' is negative over an open interval, what can you tell me about f on this interval?
9. In your own words, describe what a derivative is and what it can be used for.
10. In your own words, describe what an integral is and what it can be used for.
11. What two words did we say summarize differential calculus?
12. In words, the Mean Value Theorem says that if we look at an interval [a,b], then there is at least one point in the interval (call it c) such that the __________________ rate of change between a and b is equal to the __________________ rate of change at c.
13. Name the two inventors of calculus.  Which one do you like better?
14. Describe what each of the three parts of the following statement mean as they relate to area:

∫f(x)dx