Mathematical Mondays

You Gotta Represent

If you’re like me, you probably didn’t learn much about data and statistics when you were in school. There were bits and pieces sprinkled throughout my math textbooks, usually at the end of a chapter, and usually sections that teachers deemed skippable.

Not so anymore. Since just before my teaching career began, data and statistics have been getting a lot more attention in math curricula. One of the last courses I took for my bachelors degree was a stats class where I learned about box-and-whisker plots for the first time. When I started my internship a few months later, I discovered kids were learning about those plots in Pre-Algebra.

It makes sense when you think about it. We have data flying at us every day in the form of survey results, charts, and infographics. It’s important to be able to interpret all that information with a critical eye.

Something getting particular emphasis is the idea of sampling methods and using a representative sample. Say you’re doing a survey on career goals among the student body at your school. You’re not just going to ask the kids in the advanced art class and call it good. Likewise, if you want to know the average height of teenagers, you’re not just going to measure the basketball team.

I got to thinking about this in reference to writing. Specifically, getting critique and feedback. It kind of follows the examples above, plus the reverse. You want to be a little bit narrow (if you’re looking for information relating to teenagers, including grandparents in your sample doesn’t make sense), but also not too narrow.

What determines “narrowness” in this case? There are certain things any reader can point out for you—typos, grammatical errors, things that truly don’t make sense. But for the more subjective “Does this work or not?” questions, you probably don’t want to seek the opinion of someone who doesn’t read or even like your genre. If such a person comes along and gives their opinion anyway, you should see if anything’s valid, but don’t get too carried away with it.

At the same time, you don’t necessarily want all your feedback to come from people who only read exactly the kind of thing you write, who may even write very similarly to you. Like I’ve said before, my critique partners are great because even though we all write YA, their strengths and preferences vary enough from mine to push me out of my comfort zone and make me stretch.

Have you had a representative sample in your beta readers and critique partners? How did you get that perfect selection?

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State of the State Testing

Oh, joy, the time for state-mandated testing is upon us.

For my school, the brunt of it happened last week. We made a patchwork quilt of our schedule so testing would always be in the morning, but the kids wouldn’t miss just their morning classes all week. The kids who had to take it (those in their second or third year of high school) were divvied up into groups and teachers were assigned to administer certain portions of the test.

I only had to miss one class for my test administration. Not bad. But giving these tests to deaf kids is always a big-time drain on the brainpower.

Most of the kids have a testing accommodation on their IEP stating that any test material (other than in the Reading section) can be signed to them. No problem. I handle math stuff in ASL all day.

Except this is totally different.

First, I don’t get to see the test until the day of. Second, some math signs are so iconic, they may give away too much information. So I have to read each question, decide what’s being tested, and determine which words should be spelled rather than signed.

For example (and these examples are completely made-up and unrelated to any I saw in the test), if a test question said, “What is the numerator of the fraction 4/5?” I couldn’t sign “numerator.” Why? The sign for it is one hand held flat like a fraction bar and the other making the N-handshape above it. (Guess what denominator is. Yeah, same thing, D-handshape below.)

Another example is “parallel.” If a question asked, “Which lines are parallel?” I couldn’t use the sign. It’s too visual. On the other hand, if a more complex problem relied on the fact that two lines are parallel and some information needed to be derived from that, I could sign “parallel.”

It makes my head hurt.

It’s a test to gauge mathematical ability, so the accommodation is there to make sure English reading ability doesn’t get in the way of the kids showing what they know. But it’s such a delicate balancing act between that and giving an unfair advantage.

Anyone else have brain-busting balancing acts going on in their lives right now?

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It’s Opposite Day!

Grr, Blogger’s post scheduling failed me again … better late than never?

No, not really. Some of my students think every day is, though, finding it highly amusing to speak in opposites, so I’ve had enough of that.

Math is all about opposites, though. More specifically, inverses, such as inverse operations. Many of the math problems we did in school related to “undoing” something. Pretty much anything we did could be undone. (I hit some advanced math courses in college with operations that were irreversible. Most of you probably don’t want to go there.)

Addition and subtraction. Multiplication and division. Squaring and square-rooting. Even all the way up to trig—sine and arcsine–and calculus—derivatives and integrals.

It’s an accepted fact that mathematicians, at their core, are lazy creatures. So it makes me a little crazy when I have students who don’t grasp the power of the opposite. These two things add up to 64. You need 100, so how many more do you need? Some kids will guess and check, or count up … or add 64 to 100. (This particular case kind of goes back to my earlier rant on subtraction.)

It happens in higher math classes, too, though less frequently (fortunately). In algebra, the idea of “undoing” is huge, so when someone gets to Algebra 2 without catching that division undoes multiplication, I get a little headdesk-y. Then I teach them about it until they understand. The squaring/square-rooting dyad is newer to them, so I make sure to drill it into their heads.

And because I’m on the topic, here’s a little puzzle you can solve with the power of opposites.

I have a mystery number. I divide it by two, subtract 200, square the result, multiply that by ten, and add 52, getting a result of 412. What was my mystery number?

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How Do You Measure Up?

I have a math-teacher confession. (Again.) It’s not something I’m proud of. Not something I like to admit.

I’m not very good at estimating measurements.

Oh, I’m okay at the small stuff, particularly with length. I can say, “This is about two inches,” or even, “That’s about fifteen centimeters.” But if you go much beyond something I can hold in my hands, I’m pretty hopeless.

This drove me nuts in driver’s ed. Rules like, “When parking on the street, you must be X feet from the corner,” were useless for me. Thirty feet, fifty feet, doesn’t matter. I have no mental gauge for a distance like that.

Weights are even worse. Give me something and ask me if it’s closer to five pounds or ten, and I’ll be straight-up guessing. I know the fifty-pound bags of salt are pretty close to the limit of what I can comfortably lug around, so if something else is close to that, my guesstimate will be okay.

You know what this all has in common? Experience.

I can estimate lengths of things smaller than a breadbox because I’ve done a lot of measuring with a 12-inch ruler. I can tell when things are close to that fifty-pound mark because lugging those salt bags down to the basement is a memorable experience. I don’t have a lot of experience measuring and knowing larger distances.

I bet if I played football, I’d have a pretty good feel for five yards vs. ten yards vs. twenty.

Except … I have students who play football and don’t know what a yard is.

*headdesk*

As much as I’m not great with measurement, it’s a much weaker area for many of my students. (Oh, if I could tell you how many times I’ve asked, “How many inches are in a foot?” or even, “How many months in a year?” and gotten blank stares!) Some of it’s a language issue, and some is that it hasn’t been prioritized in their previous years of math education. Mostly, it’s a combination of both.

So, students in some of my math classes will be attacking objects with rulers and yardsticks and tape measures and scales. I will throw lots of questions at them like, “If you were measuring the water to fill up a bathtub, would you use gallons or cups?” And I will hope some of it sinks in.

What mad measurement skills do you have? What areas trip you up? Any tips or tricks? I’d love to hear ’em.

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Gender Wars, Math-Style

This isn’t a war so much as an observation. Not even a highly scientific observation. It’s not based on a fancy statistical study or anything, just my own observations in my classroom over the years. The conclusion isn’t anything like 100%, but the vast majority in my small sample seems to follow the pattern.

Because I’ve taught in the same small school for several years, I’ve often followed the same group of students from Algebra 1 on up, some of them all the way to Calculus. I’ve kept an eye on what students liked and didn’t like, what methods they chose when given a choice, and where their strengths and weaknesses were.

By and large (again, in my relatively small sample), girls prefer the analytical and algebraic. They’d rather have an equation to manipulate and solve, going step by step to isolate the variable. Boys prefer more visual approaches—geometry over algebra, analyzing a graph over an equation. There have been a couple of exceptions, but every year I’ve had more kids split down the expected line.

I’ve found this particularly interesting since these are all deaf and hard-of-hearing students, so you might expect they’d all lean toward the visual approach. Is it something in how males and females are respectively wired that makes us tend to lean toward one or the other? I remember reading things in school about how girls tend to be stronger in verbal-linguistic areas, while boys are stronger in logical-mathematical areas. (Again, these are just tendencies and obviously not true across the board.) Perhaps this is something similar.

Or maybe my students are just strange. 🙂

Do you fall into my expected categories or defy them? Have you noticed other unexpected (non-stereotypical) areas where divisions tend to fall along gender lines?

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Hating Math

Last week I discussed my deep and abiding love for calculus. I understand not everyone feels this way, and in fact, some people don’t even have the slightest hint of positive feelings toward any kind of math at all. If you’re one of those, this post is for you.

I’m not going to tell you it’s wrong to feel that way, or that you have to change your mind. (I will ask that you try to refrain from saying, “Ugh, I hate math!” around children who are still forming their own opinions. Can’t tell you how many times I’ve had a student who tells me, “My mom/dad hates math and thinks it’s stupid and doesn’t think there’s any point to learning it.” Thanks for sabotaging my work, Mom/Dad.)

The people I know who hate math usually fall into a couple different categories. Some hate it because they really struggle with it no matter how hard they try. Sometimes a learning disability is involved. Sometimes nothing’s been diagnosed, but it’s clear their mind just isn’t wired for numbers.

I recently had a student like that. Brilliant artist and some strong writing skills, but math just Would. Not. Click. Bless her, though, she kept trying and was incredibly patient, no matter how many times she had to erase and rework a problem. And the thing is, she did make progress. Not at the same pace as her peers, but she improved, because she didn’t give up. She admitted she didn’t like it at all, but she hung in there.

I think most of the math-haters I know, however, fall into the second category: those who had at least one really bad math teacher, usually at a critical juncture in their math education. This often happens either at fractions in elementary school, or a little later somewhere around pre-algebra/algebra, when things start to get more abstract.

What’s the key to teachers not facilitating the mathematical downfall of their students? I think a big part is recognizing that many students are likely to hit a wall a those junctures, so the teacher needs to be extremely flexible. One way of explaining a tough concept isn’t likely to work for everyone. If a kid isn’t getting it, you have to look for another bridge to get them across.

Even bigger key—don’t make the kid feel stupid for not getting it right away. Kids are good enough at doing that on their own. They don’t need our help.

Many times, I’ve had adults watch me teach or listen to me discuss a lesson and say, “If I’d had a teacher like you, I probably would’ve liked math.”

Best compliment I can receive, but I don’t really mind the hating math. My goal with the haters in my class is for them to hate math a little less. Even if they still hate math, I try to make sure they like the class. Because if they do, their minds stay a little more open, and even if they don’t want to admit it, they learn.

Are you a math-hater? If so, which category do you fall into? Or is there another reason I haven’t accounted for here?

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