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Marsha Ratzel on Teaching Math

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The Great Fire Wall of China

As my regular readers know, I am writing from China these days, and have been doing so four years so far. Sometimes the blog becomes inaccessible to me, making it impossible to post regularly. In fact, starting in late September 2014, China began interfering with many Google-owned entities of which Blogspot is one. If the blog seems to go dark for a while, please know I will be back as soon as I can get in again. I am sometimes blocked for many weeks at a time. I hope to have a new post up soon if I can gain access. Thank you for your understanding and loyalty.

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Sunday, December 16, 2012

We Teach Our Students to Misbehave...

… and then complain when they do exactly as we expect. A certain student came home from school reporting on a substitute teacher the class did not like. She began her report by asking me, “Doesn't a teacher have to let a student go to the bathroom if it is an emergency?” I did not answer; I asked what happened. The class in question took place after just after lunch. A girl came in, saw there was a sub, and began “dancing” near the classroom door. The teacher asked her to sit down. She continued to loudly make a scene. The teacher continued to ask her to sit down. The girl never did sit down, and continued to interrupt as the teacher tried to finish taking role. Then the teacher let her go.

After she left, several members of the class began asking the teacher out loud, “Why didn't you let her go? You have to let people go to the bathroom. What if it's an emergency?” The teacher explained that he would have let her go a lot sooner if she had just sat done when she was asked and allowed him to finish taking attendance.

Typical of many classes with a sub, there was a video which the teacher showed using the TV monitor. The students demanded out loud that the teacher put the video on the powerpoint projector instead of the TV monitor. The teacher refused. The students insisted their regular teacher always puts videos on the powerpoint projector. The teacher still refused. (an aside: students are really spoiled by all the technology available in classrooms these days. When I started teaching, we handed out purple, smelly mimeographs and showed filmstrips on a reel-to-reel. Some people reading this post may not have any idea what I am talking about. LOL)

My little friend came home complaining about what a mean teacher the substitute was. I guess she expected me to commiserate, and she was thoroughly astonished that I had an entirely different take on the incident. I told her the fault was with the students, not the the sub. First, the girl created a public scene when she could have walked respectfully to the teacher and quietly made her request. But no. She engaged in melodramatic, loud theatrics and essentially set a trap for the sub. She probably did not have to go to the bathroom at all.

Second, the students thought it was okay to question the teacher's response out loud, but worse, the teacher thought he had to answer their objections. Third, the students whined about the powerpoint projector. I told my young friend that no sub with a speck of common sense will do anything just because the students say the teacher does it. That is exactly the way to guarantee a whole period of one piece of nonsense after another. My little friend thought that perhaps her teacher forgot to write the part about the powerpoint projector in the lesson plans she left. Maybe so, I said, but the class will just have to do without, and they should never have been disrespectful to the sub about it.

She countered, “If we like the sub, we behave.” Wrong answer. Students behave because they are expected to behave whether they like the sub or not. Students have not the power, responsibility or authority to decide that they will behave “if we like the sub.” Liking the sub is irrelevant. Shame on our society for even giving such a wrong-headed notion any positive attention.

Then my little friend asked, “But what if the sub doesn't like kids?” What was she really saying, that kids have the right to punish a sub they decide does not like them? Wrong again. It is irrelevant whether “the sub likes kids” or not. Students are expected to behave, period. The problem in our society is that students did not get these wrong-headed ideas from nowhere. They have been socialized to them their whole lives. As a society, we do not really expect students to behave, the obligatory first day “expectations” lectures notwithstanding. In fact, I would suggest if we are still giving these first-day lectures to students older than about ten, both students and teachers have already conceded that students are expected to misbehave, regardless of our actual words. Furthermore, maybe we need to think a lot more deeply about what we mean by “student-centered.” In addition, there is tremendous pressure on teachers to avoid sending unruly students to the office.

As a junior high and high school teacher, I also gave the first-day lecture. I told the students I was doing so because I knew they were expecting one. I told them they had heard all the same expectations every first day since kindergarten, and now that they are secondary students, they get to hear the same expectations multiple times in one day. I listed the expectations anyway “on the off-chance there is even one person here who has not heard them,” and I explained the consequences of misbehavior. Faddish and wrong implementation of “democratic discipline” models leads to specious student “empowerment.” (Oh, I do hate buzzwords).

Up until this point, the students have basically tuned out what they have already judged to be merely a stricter sounding version of the usual first-day yadayada. But then, they all perk up when I say, “Here's the catch. There are no warnings. You guys are way too old for childish warnings. And I don't do second chances, and I do not negotiate.” About the third day, a student (usually a boy) will test me. I apply the consequence immediately and shut down the inevitable attempt to negotiate. Normally, I have no more problems during the year, because the thing is, students actually know how to behave. They just need teachers who genuinely expect them to. It is the Pygmalion Effect. When I had laryngitis while teaching in a school for "troubled" (read: disruptive) students, I learned that the students really do know how to behave. I learned to raise my expectations instead of my voice.

Japanese and Chinese students have a reputation for being well-behaved. I directly observed that overall, the expectation that students will behave is a Japanese societal given that does not require an annual review. Interestingly, a study found that “Chinese teachers appear less punitive and aggressive than do those in Israel or Australia and more inclusive and supportive of students’ voices,” and this in a country stereotyped to be just the opposite.

If you doubt that we encourage the very misbehavior we decry, check out this actual example from curriculum purporting to teach “critical thinking.” Really gotta watch for that “invisible curriculum.”

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.