Unequal understanding

[velvetpage]

American students tend to misunderstand the meaning of the equal sign far more than international counterparts.

The article is decent, and the study says exactly what I'd expect it to say. The idea of balancing numbers on both sides of the equal sign is crucial to algebra, but most students taught procedurally understand the equal sign to mean, "This is where I put my answer."

However, the article attributes the problem to poor textbooks. While this is probably a factor, I'd call it correlation rather than causation, because over-reliance on textbooks for mathematics instruction is symptomatic of the poor teaching that leads to the misunderstanding of the equal sign. Studies have shown that teachers who teach from the textbook most of the time generally rely on the textbook to lay out their plans for them. They'll spend exactly as much time on a topic as the textbook will - even if their students don't yet understand. Any mathematical concept that the textbook is unclear on, the students will be unclear on too, because the teacher is unlikely to address it outside the framework of the textbook.

Solution: get kids as young as grade one working on addition and subtraction sentences that involve balancing equations: 4+2=9- ___, for example. For primary classrooms, have a graphic of a teeter-totter with the equal sign on the fulcrum, and make it clear that the idea is to balance the teeter-totter in the middle. Do this all the way through the primary grades with increasingly complex problems and manipulatives.

By grade five, kids are ready to be introduced to the idea of a variable to take the place of the blank; they're also ready to solve problems by making tables of values that rely on one thing being equal to another: "A spider travels 19 cm every second. How long will it take her to travel the perimeter of a room that is 3m x4m?" One logical starting point is 1 second = 19cm, and the table of values can be built up from there, provided the students know they have to keep counting the number of nineteens in order to balance the equation.

Thanks to ankh_f_n_khonsu for the link.

Comments

[bonne-a-rienne.livejournal.com]

It's always the textbooks' fault. which totally flies for some subjects. you can't teach a second grader history? you've probably got a shitty textbook. (although that's a great thing about where I live, because it's like, 'hop on the el, jase, we're going to go see how our country was formed today for kicks') but MATH? kids can't understand an equals sign, and something has gone grievously wrong in the education process. when jason was in first grade I started giving him basic problems like you mention, largely because he'd figured it out on his own, and thought he'd invented something new.(he was so proud, I had to deflate the bubble slowly) we used fun things in the blank instead of a blank or an x,like banana or fish; but we're weird. He can still do them without blinking (I just wrote down five addition and subtraction problems for him(using x and y), I also asked if he knew what an = meant, and he looked at me like I was soft in the head. They teach kids math SO SLOWLY here, though. Jason doesn't have his times tables down, they didn't start teaching multiplication until the last couple of months of third grade.

[velvetpage.livejournal.com]

That's when multiplication should be taught, honestly. There's a lot of basic concepts that kids need earlier than that, and multiplication gets abstract enough, fast enough, that delaying mastery until fourth grade makes perfect sense. That said, if they're not doing a good job of the other strands of math - geometry, probability, data management, patterning, measurement - then there aren't a whole lot of excuses for doing poorly on the computation as well.

Elizabeth is quite capable of doing the types of problems I mentioned, too. Her teacher gave her some - I loved her teacher when it came to math, she was excellent - and I gave her some more.

When he thinks he's discovered something new like that, don't deflate the bubble. He constructed a piece of knowledge that was new for him, and he should be proud of that. The fact that others have constructed the same knowledge doesn't make his accomplishment any less valid, because everyone has to come to understanding by constructing knowledge exactly the way he did.

[amyura.livejournal.com]

over-reliance on textbooks for mathematics instruction is symptomatic of the poor teaching that leads to the misunderstanding of the equal sign

Amen to that! I always know, upon meeting a new teacher, how bad they are when one of the first things they ask me is what textbook our district uses for a class. Curriculum mapping doesn't always solve the problem either, because many teachers who volunteer to do the mapping (including our department head!) just go through the book and match book topics to the state standards. *headdesk*

What are the odds you could come to Massachusetts and run PD for elementary teachers around here?

[velvetpage.livejournal.com]

I'm still trying to convince the grade two teacher in my school that, no, we should NOT purchase a textbook for her class. It's an uphill battle.

Suggest they get their hands on a couple of Marilyn Burns books, or John van der Walle, or both, and base their programs off of those instead of a textbook. They'll get further.

[kisekileia.livejournal.com]

Half the grade twos aren't even going to be able to READ a textbook.

[velvetpage.livejournal.com]

One of the first things most kids learn about textbooks is that there's no need to read them. Skip the stuff at the top and go straight to the questions you're supposed to do, and you'll be fine! If you have trouble, the teacher will come to point out what you're doing wrong and you still won't need a textbook. In other words, the very act of using a textbook short-circuits the skills kids need to make best use of textbooks.

[kisekileia.livejournal.com]

Ah, okay. I guess I see textbooks a little differently because I'm used to reading them in university, and because I taught myself a couple of years of math from textbooks.

[morpheus0013.livejournal.com]

I never thought about this before. It's fascinating to me.

For my part, now that you've made me consider how I comprehend it, my brain reads the equals sign almost different in those situations. I obviously know what it means in either case, but I feel like it's a different symbol. In my head.

I'm not making much sense. But I can do math, I promise!

[velvetpage.livejournal.com]

No, that makes sense; that's how I processed it, too. I still have to fight with myself to keep myself from using a "running equal sign" when doing strings of math, even though it's bad notation and leads to exactly this kind of misunderstanding if I do it in examples. And yet I've always intuitively understood what it meant in algebraic equations.