## 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.

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## 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.

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## 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.
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## 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.

My first year teaching was wonderful and difficult – I loved the work from day 1 but worked around the clock and on top of that broke my leg mid autumn, missed work 3 weeks, and therefore had a very stressful spring-term.

The second year was supposed to be easier. Then suddenly I was teaching three psychology courses. My head of school agreed to let me off the hook for all extra activities such as sports days etc. Still, I didn’t have time or energy for outside interests or even a social life.

This year, I’m teaching a new math course and have a new syllabus for one psych class. The lessons I meticulously planned last year, and the year before – I no longer find satisfactory; and so I’m still putting in crazy amounts of work. I have learned a few things though that are helping me and may be helpful for someone else:

• It’s important to identify Good Enough. I struggle with this, but seem to have found a good-enough level for math teaching that works and doesn’t take more than 30 minutes to plan. For psych, I still have little idea but it potentially involves powerpoints. Unfortunately, those take time to create. In math, it does help me distinguish between work that needs to be done and work that I want to get done, which makes a lot of difference for my stress levels.
• Another thing I find useful is to give less help to students out of class. I even plan class-time to allow me to help students in class. This is working out very well and I now have much more uninterrupted time between classes.
• What’s proving very useful this year is to think of work as fun. I love this job, and much of what I do no one has asked me to do. This includes attending webinars, organizing meetings, in-depth talks with students and parents, detailed feedback on assessments, and even planning lessons above the good enough level. If I think of it as “working” it has this connotation to it that says it’s somehow wrong to do it outside and above the hours I’m payed for. But if I think of it as stuff I love to do, then that means I’m very lucky to get payed for a lot of the time I spend doing these things. It’s a happier thought, and by thinking it I can choose work and fun, instead of work or fun.
• Finally, when there is insecurity involved (when teaching a new course, or in a new way, or in a new place) it’s sometimes tempting to resolve that insecurity by working long and hard hours. It’s not fun, it’s something we (or at least I) do to escape an unpleasant emotion. But in this case it is often better to just face the insecurities, to feel it and accept it as a natural part of doing something new and caring about how it goes. That way, instead of having the insecurity chase us and catch up with us every time we relax, we habituate to the situation and in the long-term feel less anxious about work. I learned this “trick” from a cognitive-behavioral psychologist and it completely changed my approach to work last year.
There we go. That’s what I do to stay sane. I hope that this will keep me teaching (and loving teaching) for a good many years. And I’d love to hear about what other people have discovered and are using to handle this wonderful, exciting and demanding job.

## 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.
I’d like to find ways of still teaching for understanding, teaching for developing logical thinking and curiosity, but I’m afraid that any attempt to do so will feel threatening to these students, who crave only the safety and ease that direct instruction can provide.

Posted in direct instruction, strategies | 5 Comments

## 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.
First of all, I experimented with powerpoints in math teaching when I was a teacher-in-training. I abandoned it, because it wasn’t interactive enough for my tastes. It still isn’t, and equations are still a bitch in pptx, and so I still don’t use it with my math classes. Instead I use Geogebra occasionally.
However, I teach psych as well as math, and in psych I use powerpoints all the time. So when on Friday I read these posts I was spurred to reconsider my approach to presentations. A few hours later, I had been further enlightened by presentation zen. And ultimately, I’ve tried to put some principles into practice with the powerpoint on research ethics for my psych class this Monday.
Here is the original Friday file:
And here is the redesigned version:

Following Dan’s lead, I’ve also given some thoughts to the handouts, which are structured using a Cornell Notes framework. The student edition will have blank lines instead of text.

Now I know, I haven’t followed the rules.  I’ve included tons of pictures (but Dan, this is psych! how else would you do it?) and ignored the consistency rule for colors (but tried to keep it for fonts and sizes). As with most other of Dan’s ideas, they get me wondering and thinking and and finally doing something completely different from what he recommends. I love it.