## Impossible Learning: watch someone trying to learn calculus from scratch

David Wees retweeted this from Jared Cosulich:

It turns out Impossible Learning is a just-started-last-month blog where Jared is trying to learn Calculus and post about his struggles. It’s terrific and you should read it. See him ask the perennial question When Will I Use This?

And I immediately found myself saying “come on, when am I ever going to have to find the limit of this random equation”.

I felt like I was back in High School again.

But seriously, why is this one of the first things I’m directed to learn when I want to know more about Calculus? Why is it so hard for me to find some practical applications of this material? I know there is value in understanding the abstract math, but I’d like to balance that with at least some understanding of how this works practically…

I don’t think this actually counts as a proof, but it definitely made the “Power Rule” click for me a bit more. Basically it’s saying that the derivative of a square (x²) is two lines (2x) and the derivative of a cube (x³) is three squares (3x²).

So for a square to get a tiny bit bigger you need to add on two lines (one to the top and one to a side). Similarly for a cube to get a tiny bit bigger you need to add a square to three sides (e.g. top, right, and front).

### 15 Responses

1. He is obviously in an untenable position. Sounds like wanting to learn to play Chopin and complaining that the book starts with the fundamentals of piano.

So how did Jared get this way? Successful students seem to develop the necessary appreciation for pedagogical sequencing with little effort.

If for some reason you had take up sky diving, would you buy a ticket and jump out of a plane the first thing? I doubt it. Even though you have never taken a class in sky diving, when you do, you’ll have a pretty good instinctive sense of whether it is a “proper” class.

Why does Jared not have these instincts?

Jared doesn’t want to “learn” calculus.

Is it just calculus that he doesn’t want to “learn” or has everything been that way?

Bob Hansen

• The basic questions of What Is This Thing and What Can I Use This Thing For aren’t absurd to be asking (the limit question in particular in the post really is a bit strange to be starting with, there’s lots more natural entry points into limits that I have posted about here). Really I think he’s just picking the wrong exposition; I checked the Khan Academy’s first video on Calculus which is a little scattershot.

2. When you set out to learn something, those are actually absurd questions. They indicate that your pedagogical motives and instincts are faulty. Students who have attained success with learning have an effective familiarity with the “pedagogical formula”. The “X for Dummies” books essentially capitalize on this formula.

At the beginning of the formula you don’t ask these questions, because you couldn’t possibly understand the answer, yet. And you know this after having gone through the process (successfully) several times in the past.

Jared has a very poor familiarity with the pedagogical formula. I only see this when the student has no familiarity with the formula. Once you get the hang of it, you don’t ask questions like that anymore. Even when the student doesn’t like the subject, they follow the formula. When Jared was young, the teachers followed the formula for him. He learned the basic stuff this way. Now he wants to teach something to himself, and can’t. It isn’t hopeless.

These situations are more easily analyzed if you ask “Why?” first. In this case, “Why can’t Jared learn, while others can?” Something makes Jared different. Jared is essentially ignorant of some basic pedagogical habits of mind. And the “What is this for?” questions are merely a symptom of that. A student with familiarity with the pedagogical formula doesn’t ask those questions. They ask questions about the material they just read.

While your point about entry points is a good pedagogical point (did I say point enough?) I don’t see that here. I think because of his very poor familiarity with pedagogy, he didn’t get deep enough to even distinguish whether the treatment was good or bad. If that were the case his questions would be about limits. Not “What is this for?” How the heck would he know what anything is for in calculus, yet? If he had better success in the past, he would know better than to ask questions like that. We know. He just needs to learn how to learn on his own.

Bob Hansen

3. And by the way, not knowing how to learn looks the same as not wanting to learn. If not at first, very soon after.

4. Bob, I don’t know how much of that blog you read, but I disagree with you entirely. I think Jared’s biggest problem was not knowing where to look. He’s taken some great first steps, trying to make sense of the derivative on his own. I’m following his blog, and have given him some suggestions. I’m quite impressed.

I don’t know how you got such a different impression.

• I apologize. My remarks were entirely based on the statements captured here in Jason’s post. I didn’t even look at the blog.:)

After reviewing the blog, I will say…

1. His point does seem to be to learn calculus.
2. He doesn’t know how to.

Trying to “feel” your way through the middle isn’t learning.

But his motive seems sincere. I will try to offer some insight that might stir him to develop his understanding mathematically rather than this serendipitous case by case justification exercise he is going through.

Thanks for alerting me to my narrow analysis.

• If Jared is reading these comments and he’s serious, he might first look at the following as a start (and to decide if he really wants to continue, such as how urbanmythcafe below describes):

W. W. Sawyer, “What is Calculus About?” (1962).

David Berlinski, A Tour of the Calculus (1997).

Silvanus P. Thompson, “Calculus Made Easy” (1910, 1st edition) and (1914, 2nd edition).

http://www.gutenberg.org/files/33283/33283-pdf.pdf

• I just realized that this 1998 reprint of “Calculus Made Easy” has 124 reader reviews at amazon.com:

5. This is an interesting case. No offense to Jared but I could repeat what he is doing here with 5th graders. What is lacking in his exercises (and in 5th graders) is formal thinking skills. Looking at a rectangle and convincing yourself that a+b sort of makes sense as the change in a*b because a *and* b contribute is hardly understanding anything, let alone the derivative of a function, the slope of a function, a function, slope, etc. etc. etc. It also isn’t math. It isn’t even reasoning. It’s jabberwocky. His analysis of the power rule? More jabberwocky. Quiet, I think I hear the constructivists all rolling in their graves on this one.:)

He needs to…

1. Understand the slope of a function (in general).
2. Understand the distance formula.
3. Understand what a limit is.
4. Apply 1-3 to deriving the slope of a function.
5. Recognize that this results in another function which is the derivative of the original function.

Then I would explore how this applies to tangible scenarios.

But he needs to start with the elements. He needs to trust mathematics which, from what I can see, he seems to want no part of. Instead he prefers to invent reasons out of thin air.

When I get a break tomorrow (or Friday) I am going to try to restart him. I think I will start with a solution of a (simple) problem that cannot be solved using just algebra to impress on him why calculus and why it relies on the notion of limit. I am assuming that he knows algebra, otherwise this whole exercise is for not.

6. I read through the calculus learning blog. The author does seem pretty naive about how to approach a mathematical topic. But his “fuzzy analogy” way of tackling calculus is actually pretty reasonable, and works well for almost any other field of study. But not math.

Here is how I taught myself calculus, starting in 2001, 15 years out of high school.
1. Worked my way front to back through a 1915 algebra book.
2. Worked my way front to back through a 1939 analytic geometry book.
3. Worked my way front to back through two “easy” calculus books.
4. Worked my way through a “real” calculus book.

That is what it takes to teach yourself calculus. With enough determination, the calculus learning blog guy may be able to get there. I started with fuzzy stuff too. Eventually, this person will realize that there is a body of mathematical knowledge and of purely mathematical concepts that he needs to deal with.

Euclid said, to Ptolemy, there is no royal road to geometry.

• “Eventually, this person will realize that there is a body of mathematical knowledge and of purely mathematical concepts that he needs to deal with.”

Well, don’t discount the possibility that he actually already knows this (he has taken calculus before) and is simply trying to retrofit the hokum he is posting to that.:) I see that a lot when people don’t understand the learning process, even though they themselves went through it.

7. Well, however it turns out, I have made the offer to see him through calculus. Whether he accepts is up to him.

http://blog.impossiblelearning.com/post/72710585131/a-simple-misconception

8. Robert, I am deeply troubled by your attitude here. You seem to think you know the ‘right’ way to learn. I’ve taught for 25 years, and thought often and carefully about learning. I can’t tell you the ‘right’ way to learn. It depends on the person, what they want to learn, and so much more. Jared is playing around, which is a great way to approach something new. You might benefit by checking out Maria Droujkova’s thoughts on learning. (See moebiusnoodles.com.) Maria teaches calculus concepts, like infinity and rate of change, to young children. (I should have pointed her site out to Jared. He’d like it.)

>Trying to “feel” your way through the middle isn’t learning.

Sure it is. It might not be to your taste, but it is learning. Jared has a philosophy of learning that is very different from yours. (I am inferring some things about his philosophy based on his link to Diablo Valley School.) I’m surprised that you assume that you know more than he does about how he learns best. I don’t think that’s a useful approach. You might like to read Playing Outside: An introduction to the jazz metaphor in mathematics education, by Jim Neyland. (I might like reading it again myself.) Play is an important part of learning. I have loved math circles, and books like Math Girls, because they’ve helped me re-learn how to lay around with math.

I’m curious why you say Jared has taken calculus before. I tried to read everything on his blog, and didn’t see that mentioned.

• “I’m curious why you say Jared has taken calculus before. I tried to read everything on his blog, and didn’t see that mentioned.”

It was in his first post…

“I actually took Calculus in high school, but I don’t remember much of it. I know it has something to do with measuring rates of change, but I’m not really truly confident that I understand what that means exactly.”

And I should be upset, not you. I’m the one that went against my better judgement and instincts, only to realize it was a pathetic ruse all along.

“Jared has a philosophy of learning that is very different from yours. (I am inferring some things about his philosophy based on his link to Diablo Valley School.) I’m surprised that you assume that you know more than he does about how he learns best.”

Oh, I know. I’ve been doing this for 10 years. I am like the James Randi of pseudo mathematics education. Which is why I should have seen this silly ruse all along. No worries.

The reason we do not see eye-to-eye on learning mathematics is because you have turned it into some sort of spiritual experience not to be understood. Our crazy educational system mandates that every student take advanced mathematics and the stress of 80% of those students failing miserably before they even get through algebra causes some to turn to spiritualism. God forbid if that same educational system mandated everyone play the violin. It wouldn’t be long before some teacher, somewhere, in response to the awful sound emanating from many students’ violins, thinks – “Playing the violin must be about something other than music.” They wouldn’t get very far with spiritualism though, because even though we can’t all make music, practically all of us are very sensitive to bad music. That isn’t the case with mathematics. It’s an open door to mystics and frauds. And I am most certainly not applying those labels to you, just the spiritualism one.

Read Jared’s blog again. With an open mind this time (I am repaying the favor you showed me earlier in this exchange). It is a ruse. You can tell by the crazy leaps and word usage. Kind of creepy actually, to someone that studies pedagogy as much as I do. Maybe his next blog will be “Calculus in a Minute”.

Now if you don’t mind, I am going to share the irony of this situation with some colleagues.:)

Bob Hansen

9. […] Impossible Learning: watch someone trying to learn calculus from … http://numberwarrior.wordpress.com/1. His point does seem to be to learn calculus. 2. He doesn't know how to. Trying to “feel” your way through the middle isn't learning. But his motive seems sincere. I will try to offer some insight that might stir him to develop his … […]