Elementary School Mathematics Education Mysteries

mystery

Since these two problems are really the exact same problem, in two different forms, why not just use “x” to teach it, from the beginning, in elementary school, instead of using the little box? The two symbols have the exact same meaning!

To the possible answer, “We use an ‘x’ for multiplication, instead, so doing this would be confusing,” I have a response: why? Using “x” for multiplication is a bad idea, because then students have to unlearn it later. In algebra, it’s better to write (7)(5) = 35, instead of 7×5 = 35, for obvious reasons — we use “x” as a variable, instead, almost constantly. This wouldn’t be as much trouble for students taking algebra if they had never been taught, in the first place, that “x” means “multiply.” It’s already a letter of the alphabet and a variable, plus it marks spots. It doesn’t need to also mean “multiply.”

Why are we doing things in a way that causes more confusion than is necessary? Should we, as teachers, not try to minimize confusion? We certainly shouldn’t create it, without a good reason for doing so, and these current practices do create it.

These things may not be mysteries to others, but they certainly are to me.

[Note: for those who do not already know, I am a teacher of mathematics. However, I do not have any experience teaching anything at the elementary level. For this particular post, that’s certainly relevant information.]

“How Tall Are You?”

metric system

When I am asked for my height, anywhere — especially at school — I answer the question honestly. I am 1.80 meters tall.

I also live in the USA, one of only three remaining countries (the other two holdouts are Liberia and Myanmar) which have stubbornly refused to adopt the metric system. However, I am every bit as stubborn as other Americans, but, on this issue, I choose to be stubborn in the opposite direction.

It should surprise no one who knows me well that my classroom, whether I am teaching science or mathematics, is, by design, an all-metric zone. After all, like >99% of people, I have ten fingers (assuming thumbs are counted as fingers), ten toes, and almost always use the familiar base-ten number system when counting, measuring, doing arithmetic, or doing actual mathematics. (Doing arithmetic is not the same thing as doing real mathematics, any more than spelling is equivalent to writing.) Using the metric system is consistent with these facts, and using other units is not.

Admittedly, I do sometimes carry this to an extreme, but I do so to make a point. Metric units are simply better than non-metric units. Why should anyone need to memorize the fact that there are 5,280 feet in one mile? It actually embarrasses me that I have that particular conversion-factor memorized. By “extreme,” I mean that I have been known to paint the non-metric side of meter sticks black, simply to make it impossible for students in my classes to confuse inches and centimeters, and prevent them from measuring anything with the incorrect units.

To those who object that American students need to understand non-metric units, I simply point out that there are plenty of other teachers who take care of that. This is, after all, the truth.

Often, after giving my height as 1.80 meters, I am asked to give it in other units. Unless the person asking is a police officer (in, say, a traffic-stop situation), however, I simply refuse to answer with non-metric units. What do I say, instead? “I’m also 180 centimeters tall. Would you like to know my height in kilometers?”

If pressed on this subject in class — and it comes up, because we do lab exercises where the height of people must be measured — I will go exactly this far:  I am willing to tell a curious student that there are 2.54 centimeters in an inch, 12 inches in a foot, and 3.28 feet in a meter. Also, I’m willing to loan calculators to students. Beyond that, if a student of mine really wants to know my height in non-metric units, he or she simply has to solve the problem for themselves — something which has not yet happened. I do not wish to tell anyone my height in feet and inches, for I do not enjoy headaches, and uttering my height, in those units I despise, would certainly give me one. Also, obviously, you won’t find my height, expressed in non-metric units, on my blog, unless someone else leaves it here, in a comment — and I am definitely not asking anyone to do that.

I might, just for fun, at some point, determine my height in cubits. For all I know, a person’s height, measured with their own cubits, might be a near-constant. That would be an interesting thing to investigate, and my students, now that I’ve thought about the question, might find themselves investigating this very issue, next week. The variability of cubits, from one person to another, makes them at least somewhat interesting. It also makes cubits almost completely useless, which explains why they haven’t been used since biblical times, but that’s not the point. One can still learn things while investigating something which is useless, if one is sufficiently clever about it.

Feet and inches, however, are not interesting — at all. They are obsolete, just as cubits are, and they are also . . . offensive. It is not a good thing to insult one’s own brain.

Do Not Drink the Twenty Proof Gasoline!

141008_0000

We’ve all seen labels like this, stuck to gasoline pumps. While filling up my car’s gas tank earlier today, I felt compelled to take a picture of this familiar label — because I suddenly realized that what this small sign actually means is that the alcohol content of the gasoline being sold (in an area where liquor sales are illegal, no less) might be as much as twenty proof.

Twenty proof gasoline. Twenty proof gasoline! One never thinks of it this way, but it is both mathematically and chemically accurate. There are many different alcohols, but the one people drink for purposes of intoxication, and the one found in this gasoline, are the exact same molecule: C2H5OH. I then realized that the people who design these labels are being sneaky with the wording on purpose, for they don’t put “contains alcohol,” or anything like that, on these stickers found on gas pumps all over the place.

The reason for use of the official, less-familiar chemical term “ethanol” then became both obvious, and horrifying, all at once. Gas pumps must be labeled this way because there are people out there who are so incredibly stupid that they would actually drink gasoline if they knew it contained, well, booze.

What’s more, there is an unwritten assumption in play here, and I think (or at least hope) it is a valid one: anyone sufficiently educated to know that “ethanol” and the “the alcohol people drink to get drunk” are synonyms is also, presumably, smart enough to know better than to drink gasoline. Drinking gasoline would, of course, be dangerous in the extreme. Even inhaling gasoline fumes is hazardous, but drinking the stuff would be far worse. Consuming enough of this ethanol-containing gasoline to actually get drunk would, in fact, very likely be fatal, due to the mixture of toxic hydrocarbons present, in addition to the alcohol. The most toxic component of gasoline with which I am familiar is benzene, a potent carcinogen. Benzene is really nasty stuff, if it somehow makes it into a human body.

So, for the record, do not drink the up-to-twenty-proof gasoline — even though that is an accurate way to describe it.

My Australia Story

australia

I once got into a huge argument, as a 7th grade student, in a “talented and gifted” section of Social Studies. The issue:  how many countries are there in the continent of Australia?

The assignment was to choose a continent, and draw a map of it on a full-size posterboard. I had worked for hours on this map, only to get it back, ruined, for the teacher had taken a red ball-point pen, slashed through my line “state and territorial boundaries” in my map’s key, and had written, as a correction, “not states — COUNTRIES.” She also docked points from my grade, but that was a minor issue, to me, compared to her ruining my map. She could have, at least, written her incorrect comment on the back of my map!

When I confronted her about her mistake, she maintained that the political divisions you see above are independent countries. In my opinion, “Northern Territory,” especially, doesn’t sound particularly sovereign, and I said so, but she may not have understood the definition of “sovereign,” for that did not work. Confronted with this absurd situation, I proceeded to grab the “Q” volume of a nearby encyclopedia, and began reading the article about Queensland, loudly enough for the entire class to hear: “Queensland:  one of the states of Australia….” I freely admit that, at the time, my goal was to embarrass and humiliate her right out of the teaching profession — for the benefit of her present and future students. I’ve changed my approach, a lot, since then.

A huge brouhaha ensued, and we ended up taking each other to the assistant principal’s office:  her, to report a disruptive and defiant student; and me, to report an incompetent teacher, who, in my view, at that age, should have been fired on the spot. Dealing with this situation was probably one of the stranger, and more difficult, situations of that assistant principal’s career, for he knew that Australia is both a single country, and a continent — but he could not, for political reasons I did not yet understand, agree with me in front of this teacher. As for me, I was simply incredulous that someone could be a certified social studies teacher, and not know this basic fact about world geography. The whole scenario, to me, was surreal.

The assistant principal handled it well. To the teacher, he said, “You can go back to class — I’ll handle Robert.” He then “handled” me, after she left, in the only way that could have possibly worked:  with an apology, and a polite request to do my best to endure her ignorance until the upcoming end of the year. I respect honesty, was being given a request, not an order, and he had conceded that I was correct. I therefore chose to cooperate — with his polite request.

If he had not taken this approach, I likely would have added him to the list I had, at the time, of people (a mixture of administrators and teachers) whom I was trying to drive out of the education profession, for the benefit of all — but he did the right thing, thus earning my respect.

As for the teacher, I survived the rest of her class, brain intact, and assume she is now retired, this being well over thirty years ago. I’m now in my twentieth year as a teacher, myself, and am pleased to report that average teacher quality has dramatically improved since this fiasco happened. (I wish I could say the same about average administrator quality, but there are, at least, a few competent people working in that field, as well.) During my years of teaching, I haven’t encountered a single teacher who lacked this basic bit of knowledge about world geography. In fact, I count, among my colleagues, many of the smartest people I know.

I am glad, however, that I don’t have to call the teacher in this story a colleague. I simply cannot respect willful, stubborn ignorance, especially in the face of evidence that one is wrong. When one of my students catches me making a mistake, I do the right thing: I thank them, make certain everyone understands the correction, and then we move on with the lesson. That’s what this 7th grade teacher of mine should have done, as well.

On Coming Out of the Closet

I have a strong belief that asking nosy questions is an incredibly rude thing to do, and I find it particularly appalling when person A asks person B to identify his or her sexual orientation. What could be more personal, and private, than what consenting adults do in their own bedrooms? For this reason, I don’t ask questions on this subject, nor do I tell (when asked). To ask would be rude, and to answer such a nosy question would be to encourage rudeness. I don’t wish to do either.

There is exactly one situation I can think of where person A needs such information from person B, and that is if person A wants to have sex with person B. Even then, though, are there not far better ways to begin a flirtatious conversation?

At a school where I used to teach, I co-sponsored a student-initiated GSA, or Gay-Straight Alliance. On the basis of this co-sponsorship, plus my wearing a rainbow-colored wristband, many assumed that I was gay. To make a statement about how such things do not matter, I refused to ever confirm, nor deny, such rumors. On Facebook, I didn’t answer the “interested in” questions, nor did my blog reveal my sexual orientation — until now.

At this time, having considered the ramifications carefully, I’ve decided it is time to come out. I’m teaching at a new school, and am no longer co-sponsoring a GSA. I’ve kept quiet on the subject of my own sexual orientation long enough to make my point. And, now, I’m going to make another point . . . by coming out, here and now.

coming out straight

Yes, world, I admit it . . . I am a raving heterosexual, and one who has finally decided to come out — as a straight person. I’m not ashamed of it; this is part of who I am. There may be some who do not approve of this, but I do not have to care what they think.

In case my point is escaping you, please consider what it is like for gays, lesbians, bisexuals, and others who don’t conform to society’s norms, to do what I just did. Imagine what it would be like, for “heteros,” if social pressure favored homosexuality, rather than heterosexuality. How many “straights,” in such a world, would be willing to risk being ostracized, losing their jobs, being shunned by former friends and family, etc., simply for admitting they have a preference for the other sex, rather than the same one? If you, yourself, are also straight, and lived in a gay-dominated world, would you have the courage to come out as one of “those people,” the “heteros,” with their allegedly-“disgusting” sexual practices?  

It’s ridiculous that anybody has to think about such things. “Live and let live” is not too much to ask — of anyone.

A True Story of a Young Aspie Getting in Trouble with “Show and Tell”

In elementary school, in the 5th grade, I managed to get in trouble for a “show and tell” project. As usual, getting in trouble was not my objective, but it happened anyway. This was decades before I learned I have Asperger’s, but, looking back, none of this would have happened were I not an “Aspie,” as we call ourselves.

This image, which I found here, is very much like the poster I made, by hand, and used for this project:

nuclear chain reaction

That was the “show” part of this “show and tell” project. For the “tell” part, I explained how nuclear chain reactions work, and then explained how nuclear bombs are made. It’s very simple:  you have two slightly sub-critical masses of uranium-235 or plutonium-239, and physically bring them together, so that the total mass exceeds the critical mass. At that point: boom.

The hard part, of course, is actually obtaining the U-235 or Pu-239, for those aren’t things you can simply buy at the local hardware store. Ironically, I did know where to find both uranium and plutonium — at the very same university, about an hour away, where I’d spent far too much time conducting mostly-unsupervised experiments with both elements, along with lots of liquid mercury, before my tenth birthday. (I still suspect that all that radiation may have turned me into a mutant.) However, I also knew that the uranium and plutonium there would not have nearly enough of the correct isotope of either element, making this information irrelevant to my “show and tell” report, and so, for this reason, I did not tell them where to find the uranium and plutonium I had previously used for experiments.

I didn’t figure this out in class that day, since I’m not particularly good at “reading” emotions, facial expressions, and body language, but, apparently, I really upset, and scared, my teacher. This became apparent when she called my mother, and, later, my mother asked me to tell her what I’d done in school that day. Being excited about the “show and tell” presentation I’d given that day, I immediately told my mother all about it. When she told me the teacher had called her, concerned about me explaining to my class how to build atomic bombs, I was confused, since I didn’t understand, at all, why what I had actually said posed any problem. To explain this to my mother, I simply said, “But, Mom, I didn’t tell the class where to actually get the uranium-235 or plutonium-239! I don’t know where to find those isotopes!”

This was enough to convince my mother that I had not, in fact, done anything wrong. She called the teacher back, and simply asked if I had, or had not, included that critical bit of information: where to find the actual fissionable material needed for a nuclear bomb to work. When the teacher replied that I had not done that, my mother’s response was both sensible, and logical:  “Well, then, what’s the problem?”

—–

Postscript, for those who might be worried about the childhood experiments I mentioned above: at around age 40, I asked a physician about my worries regarding early exposure to mercury vapor and radiation. He told me that any problems I might have, as a result of such experiments, would have already showed up by then, and that I could, therefore, stop worrying about this. Thus reassured, I did exactly that.

On “Digging to China”

hole

When I was a little kid, my sister and I dug a big hole, in our front yard, and simply called it “the digging-hole.” It looked a lot like the hole shown above, except for the fact that, during daylight hours, our digging-hole usually included two small, dirt-covered, determined children, armed with plastic shovels. We tried, for years, to dig that hole as deep as possible. My personal goal, of course, was the Earth’s molten core, not India, and certainly not China.

Why do Americans so often talk about digging a hole straight down to China, anyway? Even if the Earth were solid all the way through its interior, digging straight down, from almost anywhere in the contiguous 48 states of the USA, would not put you in China, nor even India (which is, at least, closer to being correct than is China), but at the bottom of the Southern Indian Ocean. Salty water would suddenly rush into your newly-dug tunnel, killing you instantly, as soon as you got close to enough to the other side for the extreme water-pressure there to finish your digging project for you. The only exceptions to this watery doom would be coming out of the tunnel on one of the islands in that ocean, which would require great precision to hit deliberately.

Also, the fact that China and the USA are both Northern-hemisphere nations easily rules China out as the hypothetical “solid-earth” destination for Americans who dig straight down, and all the way through. If you could go through the center of the earth from North of the equator, you’d have to end up South of the equator. Isn’t that obvious? Don’t people look at globes?