Measurement: invented, inexact, and indirect

This is another topic I want to write about. I did speak about it a few years ago at Asilomar, but it’s still kinda half-baked and is worth revisiting here because of how it fits with what I want to do in class this year. It will be interesting to look back in May and see what its role was. So, here we go:

Like many math educators, I used to dismiss the “measurement” strand. I thought of it as the weak sister of the NCTM content standards, nowhere near the importance of geometry or algebra, or even the late lamented discrete math. But I have seen the light, and now it’s one of my favorites. Not for how NCTM represents it, but for the juicy stuff that got left out.

I like the “rule of three” slogan of the title: Invented, inexact, and indirect. Oddly, I have trouble remembering it, and I created it. This suggests that something is wrong—but for now, let’s proceed as if it were perfect.

Continue reading Measurement: invented, inexact, and indirect


SBG: The Search for Standards Continues

Yesterday I came across a great resource from missCalcul8: an SBG wiki for noobs. (Thanks to yet another blog, The Space Between the Numbers, by Breedeen Murray, for the pointer.) It includes how-tos from some of the luminaries in this field, plus, joy of joys, actual lists of standards so that we can imagine what they’re really talking about.  (She has also just posted a number of frightening skills lists on her own blog.)

For me, well, none of them are in statistics yet, but maybe that’s a place where I can contribute when I make that list.

So I tried to get started. One place to look for statistics standards is in the GAISE materials. That’s Guidelines for Assessment and Instruction in Statistics Education, put out by the American Statistical Association (ASA) and designed to elaborate on the NCTM Standards. These guidelines come in two downloadable pdf books, one for pre-college (that’s us!) and another for postsecondary. In our book, they define three levels, named A, B, and C. These do not correspond to elementary, middle, and secondary; many high-school students (not to mention adults, not to mention me) have not fully mastered the ideas in levels A and B.

Continue reading SBG: The Search for Standards Continues

Tyranny of the Center

Tyranny of the Center: a favorite phrase of mine that I keep threatening to write about. Here is a first and brief stab, inspired by my having recently used it in a comment on ThinkThankThunk.

In elementary statistics, you learn about measures of center, especially mean, median, and mode. These are important values; they stand in for the whole set of data and make it easier to deal with, especially when we make comparisons. Are we heavier now than we were 30 years ago? You bet: the average (i.e., mean) weight has gone up. Would you rather live in Shady Glen than Vulture Gulch? Sure, but the median home price is a lot higher.

We often forget, however, that the mean or median, although useful in many ways, does not necessarily reflect individual cases. You could very well find a cheap home in Shady Glen or a skinny person in 2010. Nevertheless, it is true that on average we’re fatter now—so when we picture the situation, we tend to think that everyone is fatter.

One of my goals is to immunize my students against this tendency to assume that all the individuals in a data set are just like some center value; I think it is a good habit of mind to try to look at the whole distribution whenever possible. Let’s look at a couple situations so you can see why I care so much.

Continue reading Tyranny of the Center

What I Did On My Summer Vacation

I am not quite done with jet lag. I think I’m trying to hang onto being away, it was so delicious.

I love travel. In my other life as a freelance science-and-math educator, I attended conferences and traveled a lot. Now the schedule does not permit it much, so I guard my opportunities carefully. Summer is the big chance.

So: ICOTS. The International Conference On Teaching Statistics. Held every four years. 2010 was in Ljubljana. (Huh? The capital of Slovenia. Oh. You mean Slovakia? No, Slovenia. Just to the right of the top of Italy. Under Austria. Part of the former Yugoslavia. Forests. Mountains. Caves. Castles. Gelato. Tied USA 2-2 in the World Cup even though we maybe shoulda won.)

There is a great deal to say about this conference, but I am both jet-lagged and still daunted by Dan Meyer’s incredibly thoughtful posts about NCTM. But it’s worth some overarching observations: Continue reading What I Did On My Summer Vacation

Reality interferes with my planning

I’ve been away for three weeks, sans computer—more on that anon—spending much vacation-and-conference time mulling over what I want to do in this class and fantasizing about The First Day.

Yesterday we returned. I logged in. The list was available; I could see what kids will be in my perfect little class. There are 18 cherubs, juniors and seniors, a few of whom I had in class a couple years ago, most of whom I don’t really know at all. And I know that they will be great, that they are smart, that I can communicate my love of the subject and infect them.

But now, instead of an idealized, pristine version of a progressive student-centered, SBG stats class, my vision had actual students in it. And somehow my ideals started sliding out from under me. I could imagine giving fun, engaging lectures instead of designing explorations; awarding points for showing up and doing homework instead of for mastery of standards; dealing with deadlines and extensions; and generally succumbing to the quick and easy path, sliding off the razor’s edge in the direction of being a stand-and-deliver math teacher.

It’s not the kids. They’re great. But they’re real, and reality somehow sucks me towards what’s comfortable.

Fortunately, I have help. One good source of backbone is in the repeated rants at ThinkThankThunk. I once thought that hammering on the same anvil over and over was bad form, an indication of being enslaved to one good idea. But I was wrong. I appreciate Shawn’s willingness to remind us newbies why we decided to think about doing things differently. And I confess that one of the main reasons I put that link in this post is so that I can find it again when I find myself going over to the dark side.

Another place to find clarity, or at least reality, is at f(t), where in the post of that link, Kate Nowak reminds us how messy it all is. There is no razor’s edge, no clean, perfect educational slam-dunk; we deal with human beings every single time, and that is both a burden and a privilege.

Still. I like being in Fantasyland, where standards-based grading works beautifully from day one, where the students who have been badly treated by math in their past realize that they really can look at the world quantitatively; where they connect the math they thought was meaningless to the real world; and where these students design their own projects and critique one another’s work fairly but kindly, building classwide self-esteem while insisting on an appropriate, deepening level of rigor. Ha.

I guess I know that despite this being, after-all, a best-case scenario, it won’t be perfect. I won’t be. The kids won’t be. But we’ll get parts of all that, and a lot more that I can’t predict, because of the particular alchemy of these 18 kids.

I am so scared.

SBG: One Dollop of Fear

I should also record what I’m afraid of. Here’s one that keeps coming back:

Suppose I really do try standards-based grading (SBG). I’ve been reading and lurking. It sounds really attractive. But in order to have SBG, you need S. You need a list of standards (or objectives or outcomes, whatever).

There are lists of these things for statistics, but they don’t list everything that I value. So I’m trying to figure out how to write that list. The problem is like—you know when you’ve been dreaming, and you wake up, and for just an instant it’s all there? But the moment you start telling about it, two things happen: the thing you’re not talking about dissolves, and then what you remember is the words, not the underlying ideas?

That’s what I’m afraid will happen about these other, less effable things that I care about.

At somebody’s suggestion (probably Meg’s) I have been keeping a document that’s a brain-dump list of whatever ideas I come up with, and it is purposely unordered. It’s allowed to be redundant. I don’t have to organize it. And so far, it has gone pretty well. I think the guts of a good list are in it.

In addition, I know that I’ll have ideas in the class and in collaboration with the class, and they’ll let me add or modify these standards on the fly.

What kinds of things am I thinking of? besides, you know, content, I want to include stuff we might call “habits of mind” and the like, such as:

Data Goggles. When appropriate, the student spontaneously looks for the data in a situation and does something useful with it (e.g., make a display).

House of Mirrors. The student consciously and explicitly uses multiple perspectives (graphical, tabular, model, formula, etc) to get insight into a situation through its data.

Could these be standards? By writing these down, do I lose the thousand other wisps of aspirations for my students? Will kids point-grub to get high marks on these? Experienced SBG-ers, any advice welcome.

We’re Idiots!

A great quote:

People will often make the case, “We can’t be that stupid, or we would have been evolutionarily wiped out as a species a long time ago.”  I don’t agree. I find myself saying, “Well, no.  Gee, all you need to do is be far enough along to be able to get three square meals or to solve the calorie problem long enough so that you can reproduce.  And then, that’s it.  You don’t need a lot of smarts.  You don’t have to do tensor calculus.  You don’t have to do quantum physics to be able to survive to the point where you can reproduce.”  One could argue that evolution suggests we’re not idiots, but I would say, “Well, no. Evolution just makes sure we’re not blithering idiots. But, we could be idiots in a lot of different ways and still make it through the day.”

—David Dunning

From an op-ed by Errol Morris in the New York Times