# Category Archives: Problems (teaching)

## 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.)

The title does not mean what you likely think it means. I cannot read children’s minds.

But I am reading Children’s Minds.

Michael Doyle recommended it to me.

Well, that’s not quite right. He called me the successor to Margaret Donaldson (author of the book in question). I had never heard of her. I consulted Amazon and now possess a first American printing of the 1978 book.

There is lots of interesting stuff in it. Good, thoughtful critique of Piaget; a lovely read.

I have been enjoying and identifying but not really seeing Michael’s point.

And then this.

To Western adults, and especially to Western adult linguists, languages are formal systems. A formal system can be manipulated in a formal way. It is an easy but dangerous move from this to the conclusion that it is also learned in a formal way.

Replace language with math and you pretty much have everything I’ve been saying on this blog.

I cannot say that I’ll live up to Ms. Donaldson’s legacy, but Michael Doyle is astonishingly good.

## What did you learn?

One thing Malke Rosenfeld and I agreed on over breakfast the other day is that the question, What did you learn? makes us uncomfortable. Weird, right? We are teachers and find both answering and asking this question makes us uncomfortable.

I have many reasons for not liking the question: that it implies the process has ended; that when I ask it of my students, they may be inclined to say what they think I want to hear; that it doesn’t invite further questions; on and on.

Being asked this question in Malke’s (fabulous) workshop* led me to something new, though.

New to me, anyway.

This coming school year, I will characterize learning—for myself and for students—in the following way.

Learning is having new questions to ask.

If I have learned something, it is because I can ask questions that I previously could not. Some examples…

### example 1: Algebra II

Reading Nicholson Baker’s article on Algebra II in Harper’s [behind pay wall; also available at your local library. And seriously, a Harper's subscription is like \$15 a year.] recently, I didn’t learn anything. Much of what he had to say about the course and the way students experience it is pretty familiar and the tone resonated with many of my feelings. But when I read Jose Vilson’s response to it, I had questions. Jose writes,

If someone said, “Let’s end compulsory higher-order math tomorrow,” and the fallout happens across racial, gender, class lines, then I could be convinced that this was a step towards reform.

I wondered whether I would view Algebra II differently if I were a man (or woman) of color. I wondered yet again about the place and effect of developmental math and College Algebra on the economically and culturally diverse population of my community college. I have new questions to ask, so I learned something from my colleague Mr. Vilson that I didn’t learn from Mr. Baker.

And you are reading Jose Vilson’s blog on a regular basis, right? If not, now would be a good time to start.

### Example 2: Percussive Dance

• the relationship between variable and attribute,
• the importance of decomposing things by their attributes and paying attention to one of these attributes at a time, and whether that is a fundamental characteristic of mathematical activity,
• whether a characteristic of a novice is an inability to distinguish noise from pattern,
• how children’s experiences with sameness in their non-mathematical lives informs and constrains their ability to work with sameness in mathematics,
• whether I was taking seriously my responsibility and opportunity to use physical classroom space for student learning, and
• what kinds of equivalence relations we could use in Malke’s percussive dance work, and whether we can form a group from the resulting elements, together with composition (my hunch is yes and that the resulting group is non-Abelian, but I haven’t worked out the details).

Now you should watch Malke in action. I’ll be surprised if this 3-minute video doesn’t give you some new questions to ask.

### Conclusion

See, in math classes asking questions is usually a sign that you have not learned.

“Any questions?” is a signal to students to speak up if they don’t get what has just been explained.

We have it all backwards.

It shouldn’t be, “What questions do you have?” [I hope you have none so that I can tell myself you learned something.]

It should be, “What new questions can you ask?” [I hope you have some because otherwise our work is having no effect on your mind.]

*Asked by someone who is not Malke, for the record.