Mathematical Mondays

Math Rant: Screwy Stats

I have to say, I love me some statistics. Have I collected student scores and done a little analysis? Why, yes, I have. Have I collected and graphed data related to my writing? Oh, wait, you already know I have.

The thing is, I also know the limitations of statistics—what it takes for them to be meaningful, how far you can or can’t take the results. That data I analyze from my students? I use it to give me some direction as a teacher, figuring whether things are improving, whether a particular concept fell through the cracks, etc. Not much more than that.

As we all know, of course, statistics on education can get used for a lot more. I get the need for assessment (in some form) and accountability (in some form), but often when I see articles reporting school success/failure, I wonder if the people involved have the first clue about statistics.

Case in point: I recently saw an online report about the 50 best and 50 worst schools in the state, in reference to percentage of students achieving proficiency on the state’s high-stake testing. It reported results for Language Arts, Science, and Math.

The first thing that struck me was that whether looking at the 50 best or 50 worst, the percent passing math was WAY lower than the other two the majority of the time. That made me scratch my head, so I glanced down at the comments.

Several people noted that AP students didn’t take the state test.

I haven’t had a chance to dig into it yet, but if true, it makes those reported percentages almost meaningless. “We want to see how your school measures up … but we’re not going to count the top students.”

This is why when I see statistics reported, I have next-to-no reaction. Not until I know more about where the numbers are coming from. In broader situations, I ask myself questions like, who was included in the sample? How was the sample selected? How were questions worded?

Be careful when reporting statistics as part of an argument. They may or may not back you up as much as you think. Dig a little deeper to find the whole story.

ETA: Did my own digging-a-little-deeper, and it’s actually worse than I thought. The last math courses to participate in the state test are Algebra I and Geometry. The website was reporting on the results of high schools, many of which around here are only grades 10-12. By that time, even the “average” students are past those levels. So the published results only showed the proficiency of the lowest group. No wonder the math percentages were so much worse than the other subjects (which I believe test higher numbers of students in high school).

Have you run into questionable statistics? Any pet peeves on how you see them reported? Do you find yourself completely confuzzled when facing the numbers?

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My Flavor of Origami

If you’ve read my profile over there on the right, you might have noticed the “origami-folding” part. But when you think of origami, what do you think of?

Paper cranes? I don’t know how to make those.

Or maybe Origami Yoda? I only wish I were that cool.

So what’s up with me saying I’m origami-folding? What can I make?

That’s right! Piles of brightly colored, crinkly parallelograms!

Okay, I’m kidding. I hadn’t assembled them yet. Here’s what they really make.

It’s called a stellated icosahedron. The “stellated” means it’s pointy and star-like. The “icosahedron” means if those pointy parts were flattened down, it’d have twenty faces.

In general, this style is called modular origami. You make a bunch of identical pieces and assemble them. Very geometric.

You can imagine why I like it so much.

It’s also a great way to fill a day of math classes when the timing doesn’t work for a regular math lesson. Like the last day before Christmas break and you just did a chapter or unit test the day before, so you definitely don’t want to start a new chapter.

It also makes the students think their math teacher is pretty cool.

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Dads by the Numbers

Okay, this is kind of a stretch for a Mathematical Monday, but Fathers Day was yesterday, and I’m involving numbers. We’ll pretend it works.

In contemplating Fathers Day, I found myself thinking about the students I’ve taught recently and the various roles fathers have (or haven’t) played in their lives. Here come the numbers.

0 Dads
I’ve had several students raised by single mothers without any father in the picture. Some of them mentioned offhand that the last time they heard from him was years and years ago. I have another student whose father died just last year. Even in the absence of a father, the experience can vary widely.

1 Dad
This is just the standard, average situation, right? For some, yes. Some students have the basic one mom, one dad, still married after all these years. (That’s the situation I come from.) There are others whose parents are divorced, but their dad has stayed just as involved as their mom.

It’s not always so standard, though. I had one student who was raised by her dad because her mom passed away years ago.

2+ Dads
Anyone with half a brain should know that biology isn’t everything. When one of my students mentioned her dad, sometimes she meant her biological dad, but often she meant her step-father. She has a great relationship with him.

When another of my students mentioned his dad, he almost always meant his foster dad. The only time he meant his biological dad was when he talked about filling out paperwork and making sure people included the “Jr.” so his father’s criminal record wouldn’t come up and get mistaken for him.

There are lots of kinds of dads, and they cover the spectrum from amazing to appalling. As a writer, I try to hit on various types and situations. Whatever our situation, we have to be grateful for the good, and grateful for every chance to overcome the bad.

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Covering the Full Spectrum

I seem to talk about balance a lot. (I just ran a search on “balance” and it came up with a dozen posts here on the blog.) It certainly comes up plenty when talking about writing. Balance description with pace. Balance clarity with mystery and intrigue. Adjectives aren’t evil; the overuse and abuse of them is.

Basically, I don’t go in for absolutes on a lot of things. It’s not just in writing, either. It holds in other areas, even when I make a statement that might seem absolute. For instance, I’m sure this won’t come as a surprise:

But does this mean I love math absolutely? That I love all math? That I love every single thing pertaining to math?

Nope.

There are parts I love more, parts I love less, and parts I love not at all. Like what, you ask? Here you go—examples.

Math-Thing I Love a Lot: Being able to break down a complex problem into steps or pieces that logically flow from one to another.

Math-Thing I Love Less: Sketching visuals (graphs, diagrams, etc.) by hand. I can do them pretty well on paper, but I’m a teacher. That means whiteboards. And that means, uh, not so pretty. (Favorite math quote of all-time: “Geometry is the art of correct reasoning from incorrect drawing.”)

Math-Thing I Don’t Love At All: Having to do things the long way when I know there’s a shortcut. That might be more of a math teacher thing, but it came up sometimes when I was a student, too. If a student can prove to me they understand the foundations contained in the long way and can justify their shortcut working consistently, I’ll usually let them use it. But as the teacher, I’m generally stuck with the long way in the early days of teaching a concept.

But here’s the good news about having such a full spectrum even within something I love. I suspect it means even students who generally hate math will have some aspect of it they don’t hate. My job is to find that aspect, because that’s where I can get my foot in the door.

How about you? If you love math, what part of it do you hate? If you hate math, what part of it do you love?

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Even Math Teachers Can Have Math Weaknesses

I’ve taught just about every math subject and topic you can imagine right up through calculus. I’m pretty good at all kinds of math problems, which is nice when you’re expected to help kids make sense of them. But there’s a skill—mathematical in nature—that I’m not so hot at.

Spatial estimation.

How many feet are between me and the car in front of me? Couldn’t tell you.

How many gallons of water fit in my bathtub? No idea.

When it comes to teaching, this isn’t really a problem. I know about measurement. I know how to take measurements. I know a few benchmarks (like a football field) and can easily estimate whether something is more or less than those.

It’s a problem when I’m moving today. Do I have enough boxes? Is everything going to fit in the size of moving truck I’ve rented (plus the two cars coming along)?

I guess I’ll find out.

After packing up my classroom this past week, though, I’m happy to say I’ve at least gotten better at optimizing box space. I got some books packed in an arrangement that was a thing of beauty.

Is there a minor section in your area of expertise where you don’t feel so expert? Has it gotten in your way? How do you work around it?

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Chances Are, We Don’t Understand Chances

Personally, I think probability is one of the most fun math concepts to teach. Break out the dice, the coins, the different-colored marbles, and the spinners. Do a bunch of trials to see how the experimental compares to the theoretical.

Despite the fun, I see a lot of students get all the way to high school without a solid understanding of what probabilities really mean. Take, for example, these two questions:

People often think these are asking the same thing. Our gut instinct for #1 is that if we’ve already gotten an uncommon five heads in a row, surely the chance of getting heads again isn’t that good. But the coin doesn’t know what it landed on before. The situation only has two choices: heads or tails. For that single sixth flip, it has a 50% chance of landing heads just like every other time.

The situation in #2 is completely different. You’re taking all six flips as one situation, so there are a lot more “choices” for the results. All heads, all tails, one tail and five heads (with six different configurations for this one alone), and so on. There is only a 1/64, or a little more than 1.5% chance, of this happening.

The difference in the two is that in #1, the five heads in a row have already happened, and cannot influence the sixth flip.

It’s also good to talk about what makes a game fair or unfair, and why gambling isn’t such a great idea.

The thing about probabilities is that they often make an assumption about all else being equal. The coin or dice being evenly weighted. Every individual outcome (like heads or tails) having an equal chance.

In life, we can’t always make that assumption. That’s where people sometimes confuse “probability” with “statistics.” For example, say we collect some data and find that 2% of writers querying a novel this year will secure representation with an agent. Does that mean any given querying writer this year has a 2% chance of getting an agent?

Not remotely.

Within that pool of querying writers, we can’t say “all things being equal,” because they aren’t. Some of the writers don’t have a clue what they’re doing. (You’ve seen Slushpile Hell, right?) Some aren’t making such egregious mistakes, but just aren’t ready yet. Some just don’t have the right timing with market trends. Some aren’t querying that aggressively, only sending out a few here and there. And then some are at the top of their game, do their homework, and go at it. The percentage of that last group getting representation is probably quite different.

So, strange as it is for a math teacher to say, don’t get caught up in the numbers when it comes to these subjective, highly variable, real life scenarios. Save thoughts of probability for when you’re deciding whether to walk into a casino, or figuring out whether you should take an umbrella when you leave for work.

When it comes to situations where all things aren’t equal, work to make sure you belong to the group that successes draw from. That’s the way to up your chances.

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