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Saturday, December 8, 2012

Do Not Use Baby Talk to Teach Math

Number sense is like a mighty oak rooted in the subconscious. Beginning in infancy, it is little more than a humble acorn. Misconceptions are weeds that also root in the subconscious and stunt the acorn's growth. The language we use to express number sense can nurture the acorn or plant the seeds of misconceptions. The resulting weeds are pulled only with great difficulty.

The baby talk some teachers use to teach addition can plant misconceptions that prevent students from properly developing the concept of mixed numbers. We should never, ever say, “2 and 3 makes 5.” Even a good quality text like Singapore Math talks baby talk, but that is because something was lost in the translation to English. We should say properly, “2 plus 3 equals 5.” Children are perfectly capable of learning correct language, and it saves them the trouble of unlearning it later. After all, we do not expect them to say “2 and 3 makes 5” forever. We expect them to transition to adult math expressions sooner or later.

So what is wrong with “and” anyway?

AND means something mathematically, and it is not “plus.” For example, 2½ does not mean 2 + ½. You do not believe me? How about -2½? Does that mean -2 + ½? Of course not, but that is not obvious to kids. The mixed number -2½ means “minus 2 and ½,” not “minus 2 plus ½.” more technically, it means “minus 2 and minus 1/2” or -(2 + ½). Subtracting a mixed number is often the child's first exposure to the distributive property, however I have never seen a textbook make it clear. Instead, we routinely plant misconceptions and then wonder why kids sometimes have so much difficulty with math.

It is not all that hard to teach either, especially if using money to illustrate. “If I have three dollars, and I spend two and a half dollars, how much do I left left?” I spent 2 dollars AND I spent ½ dollar. A seventh grade teacher mentioned in this blog the difficulty his own students were having.
I saw this post about a week after it appeared, and so I was prepared to prove MY 7th grade pre-algebra students would not make such mistakes. Equation-solving did them in, with this as a solution: -5¼ + 2½ = -3¾. I had previously showed them how illogical such a thing was, and how it didn't make "number sense", yet the method error persisted.

Break it out the way students do, and the thinking error emerges: -5 + ¼ + 2 + ½ = -3 + ¾ = -3¾. Our long custom of misrepresenting “plus” as “and” has led them to the idea that all you have to do is take out the plus sign and shove the fraction up against the whole number. If it is already shoved together, pull it apart, put the plus sign back in, and voila! The problem is solvable.

Because the root of the misconception is in the subconscious, even if they get some number sense training and even understand the training, they will fail to see the error of their thinking, and so the error persists. The teacher will probably have to name this misconception directly and explain to students how they were mis-taught in the past. They may then be able to pull it into their conscious mind and deal with it.

Decimal numbers might help. 37.2 is not “thirty-seven point two.” It is “thirty-seven AND two- tenths.” The function of the decimal point and the meaning of “and” is to differentiate the wholes from the part, whether in decimal numbers or mixed numbers (which brings me to another pet peeve. It is not that we are “converting” from decimal numbers to mixed numbers. Both forms are essentially the same: a whole number with a fraction). The decimal point does NOT mean “perform the operation of addition.”

AND is a mathematical operation called “union.” The performance of AND yields a result similar to addition only when the sets contain entirely discreet members. Otherwise, the result of the AND operation is a smaller number than the result of ADD. It used to be that AND (and OR) could be tough to teach. Nowadays, with Internet searches, lots of kids readily understand that search terms with OR between them will get you a bazillion, mostly useless hits, while search terms with AND between them will get you a smaller number of hits than each search term alone. Set theory using sets of hits makes sense, and a great way to exploit technology such that technology actually increases learning, instead of being the usual monumental distraction.

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