Unequal understanding

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.

I really want to knit this.

Finally, a worthwhile project for those two shades of blue mystic alpaca fingering yarn I have in my stash. I'm debating the merits of randomly switching colours versus a pinstripe sort of look or a heavier stripe or a stripe on the body that leaves the sleeves one colour. I don't have enough of either colour to do the whole thing, and I'm not sure I would if I could; I think the two-colour look might work better for the soft feel of this sweater. In any case, I can't imagine a better yarn than the alpaca. It's beautiful against the skin and has a lovely drape to it.