There are people who have taken Calculus one or more times and still don’t have a feel for what it is they know. They can solve specific problems in context. They have learned which word problems are like which of the completely solved examples in their text. They know how to curve sketch because the rule says that if in this region the curve has a positive first derivative and a negative second derivative then the curve must look like this.

I took Calculus twice. Once while I was in high school and once in college. I could have skipped it the second time but I had an advisor smart enough to recommend otherwise. The first time I learned to do the problems and the second time I had time to find out a bit about what was really going on.

When I started working on HF Calc, I contacted friends who had been teaching Calculus for a while. One of them, Chris Butler, has reminded me that there are people in another camp as well. There are people who understand the theory and principles but can’t solve problems to save their lives. For them the algebra has no meaning. The symbols on the page don’t seem to be related at all to the graphs their calculators produce.

I’m sure the seeds are sown early. Some teachers work better with one group or another.

My nine year old daughter is just beginning to solve algebra problems. When she gets something like

` x + 12 = 30`

she writes

` x = 18 `

. The teacher writes her notes that she will receive no credit for this work without showing the steps. She is to write:

```
x + 12 = 30
x + 12 - 12 = 30 - 12
x = 30
```

For Maggie, this is fine. It doesn’t hurt her to show her work and would have helped the teacher identify what she’s doing wrong in cases that she doesn’t get the right answer.

But there are plenty of people who memorize the steps and don’t * feel* the problem.

The title of this blog is “Extreme Teaching”. It comes from parallels I’ve long felt with Extreme Programming which I talk about in other posts. But you will hear people talk about code and how the programs they are working on push them in a particular direction. The code suggests a simplification. You’ll hear novelists talk about the characters they create in the same way. This character that springs entirely from their own head now seems to have a life of her own.

The same is true in even the simplest of algebra problems. There is a tension in the equation

`x + 12 = 30`

This tension wants to be resolved. The resolution is, what we would call in XP, a refactoring. In refactoring we initially go slowly until we have confidence in what we are doing. So initially we would substract 12 from both sides by explicitly writing it out. It’s how we can see that we truly didn’t change the equality. Later we just “bring the 12 to the other side”.

I’m not sure if people who learn this way are in the first or second group. I think it’s a mix.

I worry about the ones who are convinced that they can not do mathematics. The ones who hate mathematics are a different story for a different day. But those who have a hard time with calculations,are really bothered by

little mistakes the teacher makes on the board.

When I used to teach, I used to keep colored chalk on hand to highlight and annotate the body of notes. If I was working through a calculation and something interesting happened to get us from one step to the next – it might require a dream sequence. In the classroom, the students hear what the teacher is saying and understand what they are doing – but later their notes are missing all of the annotations (unless the teacher specifically wrote them down) and may not be able to reconstruct what was done.

I often taught concepts on more than one level to reach people from both camps. Different people learn differently. I’m certain that Head First Calc will speak to the people in the first camp and I’m working hard to make sure it addresses the needs of those in the second camp.

* A bit more of a ramble than intended. I actually set out to talk about the closest number to zero. It will have to wait for another day. Please add your thoughts below.*

February 22, 2006 at 3:30 pm

One tricky part of learning to ‘think’ calculus is the classroom model. Any examples you learn in a classroom are going to be felt to be fake or specific, just because of the context. If you show rate of change problem in physics, people will learn that particular example. If you make up something more creative, people may think it’s too farfetched.

The things I understood best from physics were when a friend of mine and I would chat about various things completely unrelated to schoolwork, and he would mention an application of calculus to solve that particular problem. At that point, it opens the blinds, so to speak, and shows what that technique is capable of.

If students could work on problems independently of class, trying several ways to find a solution, then learn the calculus way of doing it, then it might help. But that sort of thing wouldn’t really work as part of the calculus learning process proper, because of the contextual issues mentioned above. The trick is to separate the context from the calculus, in order to make its real-world usefulness more obvious.

If I, you know, am making any sense.

February 22, 2006 at 11:44 pm

x + 12 = 30

x + 12 – 12 = 30 – 12

x = 30

Am I missing something? Shouldn’t x = 18?

February 23, 2006 at 6:05 am

[…] Bisher hat das Blog erst 6 Einträge, aber ich hatte schon ein echtes “cool—so hab ich das noch nie gesehen”-Erlebnis beim Lesen: Gleichungen umformen ist eine Art Refactoring. […]

February 26, 2006 at 4:00 pm

Dan –

My excitement for the book you propose is tempered by the news of your loss. In my time as a father I have watched two men very close to me put their children to rest. It was hard enough to sense their pain and grief — I can’t imagine how this affects you. I’ll keep a thought for you and your family as you work through this.

I have thought about helping my own children understand what really goes on in math. I’ve sat down at the kitchen table and tried to say a few things about integral and derivative calculus to my son, and got a huge charge when he started asking questions.

What I don’t have is enough background or practice, however, to take them much further. Most of what I learned about math seemed to be in spite of my college faculty, and I ultimately focussed on other things.

The idea of this book thrills me. I would love to help in any capacity you might need. Please feel free to contact me if you could use an ear, a manuscript reviewer (I’m a decent editor/writer), or what.

Take care.

February 26, 2006 at 9:15 pm

[…] I came across this from an interesting blog on teaching Calculus (the Other Camp): There are people who have taken Calculus one or more times and still don’t have a feel for what it is they know. They can solve specific problems in context. They have learned which word problems are like which of the completely solved examples in their text. They know how to curve sketch because the rule says that if in this region the curve has a positive first derivative and a negative second derivative then the curve must look like this. […]

February 26, 2006 at 10:02 pm

“The teacher writes her notes that she will receive no credit for this work without showing the steps. ”

That statement reminds me of my math teacher only giving me 1/3 marks (where all answers where correct), because she had shown us how to do it with the working.

I had an argument in class with her, due to the test not stating “show all working”. I now see the value, but took exception to her assuming people would follow her steps as that was the only way she had shown.

P.S. I’m absolutely sorry to hear of the lose of your child.

July 20, 2006 at 1:44 pm

I too took calculus twice, and had similar thoughts the second time around. I wonder how many other math educators out there, have simialr experiences?

January 10, 2007 at 7:19 pm

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January 30, 2011 at 12:38 pm

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