On the Use of History in Calculus Education

My colleague is currently struggling a bit with how to introduce and motivate the Fundamental Theorem of Calculus. So I looked in the essay I wrote for my teaching diploma and found a quote by Leibniz, which is useful, and some less than useful "intuitive" arguments based on distance and speed.
While it's worth remembering that I had done virtually no teaching at the time of writing, the essay does still manage to provide some insights into teaching concepts in calculus.

Maybe someone else will find it useful - it's available here.

Current status - a reflection overdue

There is this itch inside me that signals it's time to stop and reflect. Not just over individual lessons, the way I always do, but reeeeaaally think about what's going right and wrong. It feels a bit weird to air these thoughts in public, but I'm hoping for some reactions and comments from people who can relate.

Motivating e

Here's the background: my honor's class has done sequences and series, some basic function stuff (domain and range, composition and inverses) and some descriptive statistics. During the last week, we've also handled exponent rules, equations and functions. My textbook and syllabus are pushing for me to introduce e as "the natural exponent" next week, but how do I, at this stage, motivate that it's "natural"?
Continuous compounding is not a nice fit right now, and we're not even touching calculus until maybe late spring.

What other options are there?

Edit: oh, and I found this, and it's fantastic and I got the whole e-book and it's making me consider compounding after all.

Edit: I decided compounding may work and made this worksheet that students get as homework (for later class discussion).
I'm also giving something very similar to my regular class (seniors) who are doing financial math. Given my previous less-than-perfect (crash and burn) experiences with doing investigations with this class, I'd really like to get this right. It'll be optional, as e is not in their syllabus. Even so - any suggestions will be highly appreciated.

As always, google docs kills equations, so download the documents for best effect.

No time to think - the IB way of examination

This is an example of a recent test (with correct answers attached) given to my senior class. The time limit was 60 minutes.

Here's the problem: while I have some liberty in designing the test, I am preparing the students for a final examination and so far I've found it easiest to choose questions and time limits from previous IB final exams. 

The students, of course, hate the extreme time pressure. I don't like it either. When at university, my exams were 5 hours long and 6 questions large. Sometimes I left after 1 hour, sometimes after 5 hours, and more often than not I was able to use that extra amount of time available to dig deep enough in memory to find what I needed. Sometimes I re-derived formulas and above all, the type of thinking I engaged in during the extra hours was a good learning experience and added to my understanding of the topics. 

So on one hand I'm preparing students for time-pressured examinations, and want to give them practice in such settings. On the other hand, the exams are frustrating, very procedure-oriented, and not especially conducive to learning. At this point I'm welcoming any suggestions on how to proceed. 

Dealing with a job that's both meaningful and fun - without burnout

When reading Sam's touching post about his work recently and in the immediate future, I realized that I too have been struggling with managing the work load and my own attitudes to work. This post will be about some ways I've found that help me deal with being a teacher.

Winding down: an update to what worked and what didn't and dealing with test results

Today I met my seniors for the first time since their disastrous test. I mean Disastrous. I think only 20% passed, and believe me the passing boundaries are low on this thing: way under 50%.  I asked them to consider what they could do differently, and also how I can improve teaching strategies. As I suspected, the students said that they wanted more "traditional" teaching with me explaining new material and them practicing a few problems. Less investigations, less open-endedness, more of me just showing and them repeating. So I did that, on Geometric Sequences, and they loved it. "I understand something for the first time this semester!" was one memorable exclamation from a usually sullen student.

I hate this. On one hand, fulfilling their request will save me 90% of the time I usually put on planning their lessons. I'll have more time for my honors students, many of whom enjoy the challenges they get in class.
On the other hand, this feels like existential suffocation. What's the point of me spoon-feeding the seniors stuff about trig functions, logic and all the other interesting topics we before us, if all they are doing is trying their best to put in least amount of effort to pass the exams? Teaching loses its meaning and joy.

Redesigned

Dan Meyer has had a huge influence on my life since I first discovered his blog mid-July. So far, thanks to him, I have:

• Developed the habit of reading 20 other good blogs
• Started this blog
• Spent way too much of my free time considering (and, currently, rejecting) the WCYDWT strategy of teaching (of course it's great! but who has the time needed to make it happen?)
• Experimented with open-ended questions in mathematics
• Given much more thought to applying for the ph.d. programme next year

This weekend, however, has been devoted to changing an aspect of my teaching to which I had previously given very little thought: my powerpoint presentations. 

Fractions Mnemonic

My students can invert function and tell me for which values of x a geometric series converges, yet they can't do squat with fractions. It's not for lack of "teaching for understanding" - I made sure every step of the way was solid when we did fractions last year. But since then lack of practice and reflection has pushed the understanding and skills far back into awkward recesses of their long-term memory (at best!).

So I've been searching for a good mnemonic, and haven't found anything that covers the whole topic. In high school, I too had trouble remembering the rules and often used the 1/2 fraction to remember. This week, I saw a student use it the same way I used to, without me or anyone else having shown him, and so I decided that together he and I could introduce this method to the rest of the class.

Here is the worksheet I've designed. Google Docs mangles equations, so download the file for best effect. Please offer suggestions for improvement. I'll hand this out to the class on Monday.