Category Archives: Problems (teaching)

Brief thoughts on being ready for calculus

A smart friend (whose permission I have not asked) read an article of mine that will be published in Mathematics Teaching in the Middle School sometime soon. The article is based on my NCTM talk last spring, titled “They’ll Need It for Calculus”.

This friend asked by email:

For clarification: are you arguing that the sorts of problems that you point to will help students better understand calculus, or that these sorts of problems will help students do better in their calculus classes?

I was pretty sure that you were making the first argument, but not the second.

My reply, which I stand by, is this:

That these two things are different from each other is a pretty damning critique of the whole affair, is it not?

You know what will help them do well in their calculus classes? Memorizing about 20 of these:

The Fundamental Theorem of Calculus

I have taught Griffin (9 years old) the Fundamental Theorem of Calculus.

That is…

$\frac{d}{dx} \int_{a}^{x} f(t)dt=f(x)$

Details and discussion coming soon.

In the meantime, see Kristin’s related post.

Full disclosure. Griffin was paid a sum of \$0.25 for his performance.

Questions as evidence of learning

I have argued that learning is having new questions to ask.

Here are a few questions that have surfaced in the early weeks of the semester. These are all student questions in College Algebra.

(1) Can it still be a variable if it only has one value?

This was asked by a student as we were sorting out whether $y=2$ counts as a function, and whether it counts as a one-to-one function.

(2) How do you solve $x=|y|$ for $y$?

This was asked by a student as were considering the relationships among functionsinverses and inverse functions.

(3) Is the inverse of a circle an inside-out circle?

See, we were using a set of equations, considering x as the domain and y as the range. We were asking whether each equation—so viewed—is a function and whether it is one-to-one.

Then we were switching domain and range (i.e. swapping x and y) and asking the same questions about this new equation. Bonus question was to solve each of the new equations for y.

One of our equations was $x^{2}+y^{2}=1$. Swap $x$ and $y$ and get back the same thing. Thus, a circle (as a relation) is its own inverse. Which fact I had never considered.

But my purpose here is to check in on the progress I am making in fostering and noticing student questions as evidence of learning.

Life’s dashboard [#NYTEdtech]

Quick!

Think of something complicated that all of the competent adults in your life are equally good at.

Having trouble?

Consider the following possibilities.

• Parallel parking
• Folding maps
• Making risotto
• Growing tomatoes from seed
• Doing laundry
• Consoling friends

So what would your life’s dashboard look like?

Three stages of mastery in the new Khan Academy dashboard for teachers. Students are organized into rows; content in columns.

Is your goal for every adult in your life to master each of these skills? Is it OK for the adults in your life to attain some familiarity with each and to improve throughout their lifetime? Or must the dashboard be solid blue?

Additional question: How would you behave differently if life’s dashboard were available on your mobile device or desktop computer?

Much of the rhetoric at the New York Times Schools for Tomorrow Conference this past week was based on individualization. The mantra here is alluring.

We have been treating time as fixed and mastery as variable. We need to flip that so that everyone attains mastery and the time they take to do it is variable.

This was a much retweeted component of Sal Khan’s keynote address (see it at 12:56 in this video).

Watch live streaming video from nytschoolsfortomorrow at livestream.com

Instead of holding fixed how long you have to learn something and the variable is how well you learn it, do it the other way around. What’s fixed is every student should learn; we should all get to 100%, or 99% on basic exponents before moving on to the negative. And the variable should be how long we have to learn it and when we learn it.

The larger idea of which this is a part is competency-based education.

Perhaps the principle here is too broad for meaningful debate, but I do think the assumption is worth questioning. My Life’s Dashboard thinking is one way of doing that.

Another would be to state some explicit areas for concern. One is equity. We can imagine students cycling endlessly through arithmetic content deemed foundational, and never being given access to (say) algebra.

Another area for concern is the power that is given to those who create the knowledge map. A careful look at the KA knowledge map, for instance, reveals that the prerequisite knowledge for adding decimals consists of addition and subtraction skills together with additive whole number and negative number relationships.

No knowledge of fractions is necessary; no knowledge of the multiplication and division relationships underlying place value, decimals and fractions is necessary.

These assumptions about how people learn decimals are flawed, and they are known to be flawed. But powerful people are creating flawed knowledge maps, which then form the basis of the appealing fixed mastery, flexible time meme.

I have written multiple times about Cathy Fosnot‘s idea of the landscape of learning. This is a useful metaphor that conflicts in some important ways with Khan Academy’s more linear knowledge map metaphor (and at 9:21 in the video).

So I get how appealing this flexible time/fixed mastery thing is. I understand its allure. And the idea that we can summarize this information for teachers in a tidy array? Also appealing.

But it just isn’t that simple.

Mr. Khan? You got some ‘splainin to do!

Oh dear.

No.

No.

No, no, no.

No.

I know we went over this. I know we did.

Seriously, Mr. Khan. Gimme a ring next time, OK? I’ll talk you through it. I promise.

I’m easy to find.

(“Thanks” to Frank Noschese for alerting me.)