If we want to be serious about improving education, one of the first things we need to think about is the real and perceived relationship between effort and results. Take a look at the following three cases. To keep the discussion simple, I am using unrealistically idealized curves, but they should illustrate the main principles.

Let's say that the X axis in the graph above represents days in the semester and the Y axis represents some measure of mastery.

Which of these learning would we like to see?

It is easy enough to eliminate C. Obviously we don't want kids getting frustrated by going most of the semester without any progress .

Of the remaining two, most people would probably choose A, but I am going to strongly argue for B. I want to see B because I want the students to expect B in the future. I want them to go into every new topic believing that they probably won't see much if any progress at first but eventually they will see a more-or-less linear relationship between the work they put in and what they get out of it.

The great problem with A is that it reinforces students' most dangerous misconception: that math is something you get or you don't; that understanding either comes quickly or it doesn't come at all. You simply can't motivate people to work hard when they believe that work is futile.

A blog of tips and recommendations for anyone interested in learning or teaching mathematics.

## Wednesday, November 26, 2014

## Wednesday, November 12, 2014

### Fixing Common Core*

* or at least a small part thereof.

Last week I opened up a thread on what's wrong with many of the homework problems appearing under the banner of Common Core (if you click on the link, make sure to check out the comment section). The post included an example that's been bouncing around the internet for a while.

In this post, I'll drill a little deeper into the problem and discuss what the authors were probably trying to do, how they screwed up and what could be done to fix it.

The offending problem reads as follows:

4. Steven solves 7 x 3 using the distributive property. Show an example of what Steven's work might look like below.

I shared this problem with an engineer, a physicist and a statistician. All three had Ph.D.s and extraordinary analytic chops, but none of them got the problem (the engineer came the closest with a tentative "Surely they don't want you to..."). It wasn't that they didn't understand the mathematics; it was that they understood it too well. What threw them was that using the distributive property was such a silly choice in this context. Once the problem was changed slightly so that the approach made sense, all three got it immediately.

I'll discuss that tweak in a minute, but first, let's talk about what the problem was trying to do.

The idea that the creators are going after both here and in many other problems is that there are different ways of stating the same number. For example, we can think of nine simply as nine or as ten minus one or as three square or as 900% or any other number of ways.

Unfortunately, there is a second equally important part to this idea that the authors generally mishandle or leave out completely: we want to find the form of the number that makes the problem easiest.

It's true that 7x3 = 5x3 + 2x3 = 10x3 - 3x3 but there's no good reason to do the problem those ways. The students have very probably learned their multiplication tables by this point and even if they haven't, it's easier just to add 7+7+7.

So, how can we fix this problem?

For starters, we need to think about what is bookable. Some things are simply better explained in person and left out of assignments the students do on their own.

If we do decide that this is a concept we want to cover extensively in homework, we have to make sure that we are clearly and logically explaining what's going on. This is a particularly weak point in many Common Core based materials. Not only do the worksheets have incredibly inadequate explanations; the books are often even worse.

Finally and perhaps most importantly, we need to set up the problems so that the technique we are trying to teach makes sense.

There is no good reason to use the distributive property to multiply 7×3. There is, however, a not bad reason for using the distributive property to multiply 19×3 and a pretty good reason to use the distributive property to multiply 98×3 and a very good reason to use the distributive property to multiply 998×3.

If you try to do these problems in your head, you will probably find that it's much easier to think of them as

3(20 - 1)

3(100 - 2)

3(1000 - 2)

After tweaking, these problems not only illustrate the technique, they put it in an appropriate context.

This is, of course, a single example but it's consistent with what I've been seeing while helping students from various grades.with their math homework. A large number of the problems I see are so bad as to suggest either extreme carelessness or a profound lack of understanding of what the problems are supposed to accomplish.

p.s. I've also been writing on this topic at West Coast Stat Views for awhile now. You can see a collection of education reform posts here or you can just go to the site and search on 'education'.

## Monday, November 10, 2014

### The unbookable lesson

Have you ever seen the original

You can draw an analogy with different educational media. There's a naive view of teaching that's surprisingly popular among 'thought leaders' like Thomas Friedman (and yes, I do have to use quotes whenever I use the phrase 'thought leader'). It reduces instruction to the words and pictures presented to the students. This is analogous to the idea that you could build a completely autonomous plane just by recording everything a pilot did on one flight then attaching servomotors to the controls and having them replicate all of the actions.

Good teaching is always an interactive process, though the interactions may not always be readily apparent to the casual observer. Even when you aren't actively answering questions or soliciting feedback, being an effective instructor means constantly reading your audience. You have to be alert to expressions and body language. If you see nodding and looks of relief, you might want to speed up. If you see wide eyes and slack faces, you probably need to slow down, add a simpler example, or even go back and start the lesson over..

After you do teach for a while, you will find that when you do ask questions or open up the floor, you will already have a remarkably good idea of where the kids are having problems.

If you take away that feedback channel, things become radically different, almost always for the worse. Explanations that seemed perfectly clear to the class in person will seem incomprehensible when presented over a YouTube video. Lessons that worked beautifully in the classroom will leave students confused and angry if done online.

This all ties closely to the concept of bookable instruction, classroom lessons that lends themselves to adaptation to books and other non-interactive media. Most history lessons are fairly bookable; almost no math lessons are.

When it comes to educational technology, the medium is very much a part of the message. Unfortunately, it’s a part that has been all but completely lost amid all of the hype over MOOCs and iPads.

*Flight of the Phoenix*? If not you might want to skip the rest of this paragraph. The plot involves a group of men trying to rebuild a plane that crashed in the middle of the desert. The plan for the reconstruction comes an arrogant engineer who assures them that he has designed many planes in the past. It is only near the end of the project that the rest of the men learn that engineer only designed model planes that had, at best, flown a few hundred yards. Upon learning that their fate depends on someone who, in their words, makes "toy planes," the men are understandably despondent but the engineer argues that it actually requires more skill to design a plane that doesn't have a pilot.You can draw an analogy with different educational media. There's a naive view of teaching that's surprisingly popular among 'thought leaders' like Thomas Friedman (and yes, I do have to use quotes whenever I use the phrase 'thought leader'). It reduces instruction to the words and pictures presented to the students. This is analogous to the idea that you could build a completely autonomous plane just by recording everything a pilot did on one flight then attaching servomotors to the controls and having them replicate all of the actions.

Good teaching is always an interactive process, though the interactions may not always be readily apparent to the casual observer. Even when you aren't actively answering questions or soliciting feedback, being an effective instructor means constantly reading your audience. You have to be alert to expressions and body language. If you see nodding and looks of relief, you might want to speed up. If you see wide eyes and slack faces, you probably need to slow down, add a simpler example, or even go back and start the lesson over..

After you do teach for a while, you will find that when you do ask questions or open up the floor, you will already have a remarkably good idea of where the kids are having problems.

If you take away that feedback channel, things become radically different, almost always for the worse. Explanations that seemed perfectly clear to the class in person will seem incomprehensible when presented over a YouTube video. Lessons that worked beautifully in the classroom will leave students confused and angry if done online.

This all ties closely to the concept of bookable instruction, classroom lessons that lends themselves to adaptation to books and other non-interactive media. Most history lessons are fairly bookable; almost no math lessons are.

When it comes to educational technology, the medium is very much a part of the message. Unfortunately, it’s a part that has been all but completely lost amid all of the hype over MOOCs and iPads.

## Friday, November 7, 2014

### A step-up SAT/GRE problem -- Circles (Rhombus edition)

In an earlier post, we talked about "step-back problems." The idea is that, wherever possible, each problem should be associated with at least one problem that uses similar format and relies on similar concepts but which "steps up" (is more difficult) or "steps down" (is easier).

In that previous post we talked about problems where you had to find the shaded area of a circle. This problem covers similar territory but takes things up a notch.

Circle 1 and Circle 2 both have radius 2. Each passes through the center of the other. Find the area of the rhombus formed by the two points of intersection (A and B) and the centers of each circle (C1 and C2).

Solution after the break.

In that previous post we talked about problems where you had to find the shaded area of a circle. This problem covers similar territory but takes things up a notch.

Circle 1 and Circle 2 both have radius 2. Each passes through the center of the other. Find the area of the rhombus formed by the two points of intersection (A and B) and the centers of each circle (C1 and C2).

Solution after the break.

## Thursday, November 6, 2014

### Deconstructing Common Core

[A quick note: there's been some confusion over over exactly what constitutes Common Core, I might dig deeper into the question at a later date, but for this post, we'll be talking about materials released under the Common Core banner.]

I volunteer a couple of times a week to help a group that tutors kids from urban schools. My role is designated math guy. I go from table to table helping kids with the more challenging homework problems.

Recently, I have noticed a pattern in helping with Common Core problems. First I explained them to the students, then I explained them to the tutors.

That may be the most noticeable difference between the mathematics of Common Core and the new math of the 60s. In the summer of love, an advanced degree in mathematics or engineering was sufficient to understand an elementary school student's homework. These days, the tutors with math backgrounds often find themselves more confused than their less analytic counterparts since what they know about solving the problem seems to have nothing to do with what the assignment asks for.

To follow a Common Core worksheet, you really need to have a little knowledge of the underlying pedagogical theories. Unfortunately, if you have more than a little knowledge, you'll find these worksheets extraordinarily annoying because, to put it bluntly, much of what you see was produced by people who had a very weak grasp of the underlying concepts.

I'm starting a thread called "Fixing Common Core." I'm going to take some problems that are associated with Common Core and try to explain both what the authors were trying to do and how they could have done better with a different approach. The posts almost certainly won't live up to the title but they will, hopefully, shed some light on the topic.

I'll be back early next week with a discussion of the following:

I volunteer a couple of times a week to help a group that tutors kids from urban schools. My role is designated math guy. I go from table to table helping kids with the more challenging homework problems.

Recently, I have noticed a pattern in helping with Common Core problems. First I explained them to the students, then I explained them to the tutors.

That may be the most noticeable difference between the mathematics of Common Core and the new math of the 60s. In the summer of love, an advanced degree in mathematics or engineering was sufficient to understand an elementary school student's homework. These days, the tutors with math backgrounds often find themselves more confused than their less analytic counterparts since what they know about solving the problem seems to have nothing to do with what the assignment asks for.

To follow a Common Core worksheet, you really need to have a little knowledge of the underlying pedagogical theories. Unfortunately, if you have more than a little knowledge, you'll find these worksheets extraordinarily annoying because, to put it bluntly, much of what you see was produced by people who had a very weak grasp of the underlying concepts.

I'm starting a thread called "Fixing Common Core." I'm going to take some problems that are associated with Common Core and try to explain both what the authors were trying to do and how they could have done better with a different approach. The posts almost certainly won't live up to the title but they will, hopefully, shed some light on the topic.

I'll be back early next week with a discussion of the following:

## Sunday, November 2, 2014

### I had a post in mind for this clip...

But I forget what I was going to say...

Still a great scene, though.

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