# Tag Archives: listening to students

## Khan’s kindness

Say what you will about Sal Khan (and I have certainly said a lot), but he communicates a tremendous amount of patience with his students.

I watched his video on “Basic Addition” the other day.

He begins with the assumption that the viewer has absolutely no equipment for finding the sum 1+1.

This bears repeating. He assumes absolutely no knowledge of the meaning of the addition symbol in the expression 1+1. None.

As he does so, Khan is patient, supportive and encouraging. He does not condescend and he even apologizes for the word basic in the title of the video-worrying that his viewer may be put off by the term.

When I think of the culture of many math classrooms, in which students don’t ask questions out of fear of looking stupid, or in which instructors use words such as trivial and obvious without apology or concern for the effect these words can have on learners, I get a glimpse of what people find so appealing about Khan’s videos.

Khan gives permission to not know. He reassures the viewer that it’s OK to still be figuring things out. And of course he is happy to repeat what he just said as many times as the viewer likes. Just stop and rewind. The calm, patient demeanor never changes.

The field could learn from Khan’s kindness.

I expect a few folks will stop by this blog after reading my recent critique (together with Michael Paul Goldenberg) of Khan Academy. That critique was based on Sal Khan’s lack of knowledge of common student misconceptions, as evidenced in his videos. It was also based on the fact that he seems not to care.

Forthwith, some more reading on the topic, Teachers need to know about their students’ ideas.

And feel free to argue with me (read the comments-you’ll see that you’re not the first!)

But before you do, please read my post on ground rules. I adhered to these in my Washington Post piece. You need to adhere to them here. It’s how we’ll learn together.

Division of fractions. (Contrast with Khan’s treatment of the matter. Of course our work has different audiences; but I argue that his teaching ought to reflect having though about the issues I wrote about. Does it? Discuss.)

• Ways to think about the sum of the angle measures of a polygon. (Again, contrast with Khan’s treatment.)

• Some thoughts on designing tasks from which students can learn.

• Problem-solving and understanding; notes on their relative importance in teacher preparation.

• A post in which I predicted my own students’ struggles-only partially correctly-and discussed with commenters afterwards.

• A high-concept, mathematically sophisticated way of saying Holy crap! I get why my students struggle with logarithms!

You might also enjoy my ongoing series on Talking Math with Your Kids.

A common theme in critiquing Khan’s critics is to ask, “Why don’t you go ahead and make your own videos?”

This has some merit. But it’s not the fact that Khan’s making videos that I find troublesome. A more apt retort would be, “Why don’t you go ahead and make your own multi-million dollar website?” The answer to that should be obvious.

For me, on the video front, I have. I am still pretty open-minded and curious about what video can do well. I think it can provoke (examples here, here and here). And I think it can provide decent explanations and demonstrations. I do not think it can be the primary instructional medium for a quality math course. And yet, I am ready to be persuaded.

Finally, I noticed that Karim Kai took some heat for a perceived (but fully disclosed) conflict of interest, in that he founded Mathalicious. Concerns in his case are unwarranted in my view. But be that as it may, I want to make clear that while I have written for Connected Mathematics, I have zero financial interest in the venture and my formal relationship with the project has ended. I neither speak for, nor profit from Connected Mathematics.

## The lawnmower problem

David Peterson (@calcdave on Twitter) posted this lawnmower video to 101qs recently. It was love at first sight.

I needed a polar-coordinates-based assignment for my Calculus 2 students, so I pounced on it. The question they have been working on is, How long will it take to mow the lawn?

I read their work today. The following are some quotes from their writing.

“Establishing the polar function was difficult at first, until I thought about it as just a plain linear function.”

“I tried going on the treadmill to see what a comfortable walking speed for mowing would be.”

“Sorry for making this 13 pages. I really got into it.”

“Sometimes math needs a little touch up; this is when Photoshop is there to save the day.”

“The real real-world problem is how to convince your wife to upgrade mowers.”

“Rather than dealing with negatives and reciprocals, this paper will assume the lawnmower ‘un-mows’ the lawn from inside to out.”

“After realizing that the point on the outer ede of a circle has to cover more linear distance than a point near the center, angular velocity seems like it might have some flaws.”

I see in these excerpts students making mathematical connections that result from their struggles with the problem. I see them posing and refining mathematical models based on correspondence to the real world. I see them looking at this small slice of the world through a mathematical lens.

I am so proud of them.

NOTE: In original post, I did not know who had posted the video to 101qs. David Cox came through for me on Twitter. Credit given in revised post.

## Why study math?

As usual, I was the house rabble-rouser at my institution today. Someone sent around a link to an article (subscription required, unfortunately) titled “Why study math?”

Reading the piece from the perspective of my inner middle-schooler, I was unimpressed. It felt to me like a rehashing of the usual vague unsubstantiated claims about transferring problem-solving skills and learning to reason. And also this:

Learning math develops stick-to-it-ness, defined as dogged perseverance or resolute tenacity, and develops perseverance, resilience, persistence, and patience. Students have opportunities to develop their work ethic in my math class by not making excuses, not blaming others, and not giving up easily.

Um, teacher? Do we have even one shred of evidence to support these claims?

So I wrote up my own reasons for studying math. Here they are:

I have two reasons people should study math.

First is that there is a set of very practical quantitative and spatial skills that are necessary for informed participation in society. Access to these skills ought to be both a civil right and an obligation.

The second is that there are many bodies of knowledge that we have agreed as a society are important; to be educated means knowing and having experienced certain things in the arts and sciences. In this way we pass on our culture.

I see these reasons as being quite different from more generalized claims about reasoning and problem-solving skills. An important part of the difference is that my reasons invite conversation and debate about exactly what mathematics we should teach.
If the practical skills are a major reason we impose math on students, we need to inspect the curriculum pretty closely to make sure we’re teaching the right ones. In a technologically advanced society, the long division algorithm for multi-digit decimals is pretty hard to defend from this perspective, for instance (to say nothing of polynomial long division!)

And if passing along mathematical knowledge and ways of thinking are culturally important, we ought to design curricula that give students experiences with mathematical ways of knowing. I would argue that our standard curriculum K-12 and through calculus does a pretty poor job of this.

But if we appeal vaguely to reasoning skills and stick-to-it-iveness, there is no further conversation. We tell students, “Take our word for it-studying this will make you better at that, and you’re gonna need that. So study this now and don’t ask any further questions.”

And while I don’t know everything about how to teach critical thinking, I know that when we talk to students this way-either explicitly or implicitly-we devalue it.

## More than you bargained for

This semester has been out of control since January. One of many consequences is that I haven’t had time to write about my teaching. There’s a serious backlog of stuff to document; no time to type it up.

Today’s post is a bit of a mix.

Nat Banting wrote this week about a lovely moment when a student looked at a task from a new perspective.

The look on my face must have been priceless, because she started to laugh. The scene went on for quite a while. Slowly but surely, every student had approached the desk to see what was up. The student beamed as she explained…

It got me thinking about similar episodes in my classroom; those moments when I take the time to ask instead of tell, and when my students’ ideas blow me away. The moments when my students teach me some mathematics.

Calculus 2 has had many of these moments this semester. We were studying approximate integration (about which, much more in future posts). We had a motto, “When you cannot integrate, you must approximate“. We had approximated with rectangles and trapezoids. We had built up the formulas for these methods based on students’ intuitions. And then it was time to deal with Simpson’s Rule.

If it’s been a while since you studied such things (or indeed if you never have), the basic idea is in the picture below:

We’re trying to find the area enclosed between the function above, the x-axis below, x=-1 on the left and x=1 on the right. Those rectangles give a pretty good approximation of that area. Each rectangle has a bit of extra area (above the function) and leaves out a bit of area; those roughly compensate for each other and the result is a good estimate.

The reason it’s not a perfect measure is that we are using straight lines (apologies to Chris Lusto) to approximate a curvy function.

So I asked my students what a reasonable solution to this problem would be. What curvy functions should we use to approximate f(x)? We all agreed that it would be desirable for these functions to have nice calculus properties, since that’s why we’re approximating in the first place (that original function doesn’t submit to the standard set of techniques for finding this area exactly).

We were building towards Simpson’s Rule, which is based on parabolas. Use parabolas as tops on those rectangles instead of horizontal line segments and you can get a really nice fit. Plus, parabolas are easy to integrate. Plus if you do a ton of complicated algebra, you can find a really nice formula so that you don’t even have to bother integrating (the pedagogical benefits of this are debatable; the calculational benefits are massive and undeniable).

So I asked. Not just rhetorically. I asked and I listened to their answers. They wanted to use sine (or cosine). And they wanted to use exponentials.

Of course. These are the things with the simplest antiderivatives. Polynomials get more complicated when we integrate. Sine, cosine and $e^{x}$ pretty much stay the same so they’re easy to evaluate. It’s brilliant, right? The algebra won’t work out nicely, and you won’t end up with a clean and tidy rule. But who cares? We’re trying to learn some Calculus here; some ways of thinking mathematically about relationships among functions.

I made the evaluation of this integral by a sine-based version of Simpson’s Rule into an A assignment. Several students are working on that right now.

So much more to report. It’s been a productive semester. I’ll get on that in just a few more weeks.

## Talking math with your kids week summarized

A research project at San Diego State University (IMAP) has carefully studied what future and practicing teachers believe about the teaching and learning of mathematics. They have been working on the assumption that there are productive beliefs in this area that will help teachers communicate with their students, and design meaningful learning experiences for them, and that there are destructive beliefs that will constrain the kinds of opportunities children have to learn in these teachers’ classrooms.

These beliefs are just as relevant in the home environment as they are in school. To the extent that parents believe some of the following things, and to the extent that they act on these beliefs, parents will be able to support their children’s mathematics learning in powerful ways:

1. Mathematics is a web of interrelated concepts and procedures (school mathematics should be too).
2. One’s knowledge of how to apply mathematical procedures does not necessarily go with understanding of the underlying concepts. That is, students or adults may know a procedure they do not understand.
3. The ways children think about mathematics are generally different from the ways adults would expect them to think about mathematics. For example, real-world contexts support children’s initial thinking whereas symbols do not.
4. Children can solve problems in novel ways before being taught how to solve such problems. Children in primary grades generally understand more mathematics and have more flexible solution strategies than their teachers, or even their parents, expect.

Each of these is a productive belief. The conversations in my Talking Math with Your Kids series demonstrate what it might mean to act on each of these beliefs in conversations with children.

Belief number four may be the most important. Throughout these conversations you can see me asking questions that I am not at all sure my children know how to answer. You can see me asking them very genuinely how they found an answer. Of course I have hunches about what they are thinking, but I know enough not to assume I’m right. I know enough to ask and to listen to their answers.

## Not really ready for fractions

Talking Math with Your Kids week continues.

I worry about how much fraction work we do in early elementary grades, before lots of kids are really ready.

Some evidence…

Griffin (seven) has been doing his second-grade math homework, which is to find $\frac{1}{4}$ of 4, and then $\frac{1}{4}$ of 8, and then $\frac{1}{4}$ of 20, etc. After several of these, he is to fill in the blanks: $\frac{1}{4}$ of ___ is ___. He begins with $\frac{1}{4}$ of 4 is 1.

Me: You already did that one; it was the first problem. You have to do a new one.

Griffin (seven years old): But I don’t know what else to do.

Me: Any other number that you haven’t already done is fine.

G: Hmmph.

I have to believe that some extended time thinking about sharing situations would be a much better use of Griffin’s homework time than expressing the results of this sharing abstractly as $\frac{1}{4}$ of a discrete quantity.