The Success of Substitutes...

...is the school administration's responsibility. Fact is, if first-year teachers have to sink or swim, the waters are even more dangerous for subs. However, administrators are not there with the life-saving buoy for their first years,and they are even less there for subs. So subs need to help themselves.

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.”

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.

True, Authentic, Real Life Math Problem of an Eighth Grader

A certain student had recently missed much of the first quarter, so her band teacher did not count those weeks when figuring the credit for the weekly practice logs. Therefore, the student had only five weeks worth of practice log grades for the quarter whereas her classmates had ten weeks worth. She got 100% for each of the first three weeks she was back. When her report card came, she was shocked that she had a C. She thought she had an A in the bag.

Looking back at her practice log grades, she saw 100, 100, 100, 70, 0. “I forgot to turn in last week's practice log,” she explained. BUT, she knew how to figure averages, and when she did, she got 74%. “Ah, so there's the C,” she said.

Then she asked me, “How many minutes will I have to practice to bring my grade back to an A by next week?” True to form, I irritated her by telling her to figure it out. A real life math problem was staring her in the face. She had been wondering if there was such a thing as a real life math problem. “How do I do that?” she wailed. I told her to think about what she already knows and what she needs to know.

Her train of reasoning: One more week means I will have six total weeks of practice log grades. To average 100%, I will need 600 total minutes. I have 370 minutes. So I need 230 more minutes. I will need to practice 230 minutes this week. (An aside: I wonder if my teacher will give credit for so many minutes in one week). My practice log is due on Friday, so I have six days to practice 230 minutes.

So far, so good, but then her reasoning began to go awry. She divided 230 by 60, and got 5.5 on a calculator. She did not question the result. We need to teach students to determine the neighborhood of the result before doing any actual computation. I do not like to call this process “estimation,” because almost all kids have reduced estimation to mere rounding, and nothing more. Most kids tolerate estimation lessons at school, but basically tune them out because they have been socialized to value answer-getting techniques. Estimation does not, in their minds, yield “answers.”

(Now I have to explain that during this whole process, I was busy with my own work, so I was only seeing pieces intermittently, as she showed them to me. She showed me the calculator with the 5.5 in the display, which at this point was all I knew. I reconstructed her train of reasoning later from her comments).

I asked, “What does 5.5 mean?” She said, “5 hours and 50 minutes.” Remember, this student has all As in math, but as I have explained before, much of math in schools is misnamed. It is really non-math, but since schools call it math, students believe it is math, and if they get good grades in non-math, they believe they are good at math.

I probed, “How did you get that?” She looked at me like, well duh, isn't it obvious and said a little too loudly, “5.5 is 5 hours and 50 minutes.” Then turning away, she poked something into her calculator.

“How do I round this?” she asked. The display showed 0.9166666.

“You have asked the question wrong. No one can answer your question the way you asked it. You need to specify what place you want to round it to.”

“The thousandth's place. So 0.917.”

“That's right. But what are you counting?”

She pondered a moment and wrote 0.92.

“And what is that?” “Minutes,” she said, and wrote 92.

“How did you get that?”

“I need minutes, so I moved the decimal point.”

“'I moved the decimal point' is never a mathematical explanation for anything. You need to give a mathematical reason for the math you do. What did you do to get 0.92 in the first place?”

“I divided 5.5 by 6 to get the number of minutes I need to practice everyday. 0.92 minutes doesn't make sense so I need to move the decimal to get a number that makes sense.” (With this kind of reasoning, is it any wonder our students are so poor at math? And if they use the same faulty reasoning for any of life's other problems, no wonder decision-making ability is also poor. When they become adults, they are easily scammed by poor reasoning that sounds good to them).

She has three main problems:

1. Using disembodied numbers

Teachers have allowed her and her classmates to disembody numbers since first grade. What I mean is students have been trained to compute with only the numbers and attach the units to the result later. When students do that, they attach the unit they want, not the unit their computation produces. What she should have done is written 5.5 hours = 0.92 hours/day. Her unit was “hours/day.” However, since she was looking for minutes, she did the math the way so many students (and adults) do: 5.5/6 = 0.92 minutes.

2. Mixing bases

She did not realize that decimals numbers are base 10, and clock numbers are NOT base 10. I set up some place value columns for decimal numbers, and another set of columns for clock numbers. Then we did some counting so that she could see how numbers end up in the columns they do. First, we counted decimally, that is, in base ten. Then we counted time. As our paper time clock ticked over 59 in the minutes column to 1 in the hour column and 0 in the minutes column, she exclaimed, “Oh, base 60, like the Incas.” She could tell me that 0.5 = 50/100 = 50%, but still insisted that 5.5 hours = 5 hours and 50 minutes. She realized that she was looking for 50% of 60 minutes, but insisted she should divide 50% by 60. Eventually, understanding dawned. She realized that since 50% means half, then half of an hour is 30 minutes, so 5.5 hours means 5 hours 30 minutes. (My own work had come to a complete standstill long before). “So 'of' means multiply, right?”

3. Misunderstanding “Decimal Number”

She thinks, like so many kids do, that a decimal number is a number with a decimal point. Just take out the decimal point and presto, changeo, it is not a decimal number anymore. What else do we expect when we teach kids tricks,shortcuts and blind procedures,and call this strange conglomerate "math?"

In quite East Asian style, we had spent over an hour on this one problem. Eventually, she determined that (leaving aside the original calculator error), she actually had gotten her answer way back at 0.92 hours/day. She realized that the math had “spoken” to her if she had only thought about it correctly. What the math said was that she would need to practice a little less than an hour a day. She never noticed the calculator error, and I did not point it out.

Epilogue:

She practiced 60 minutes (in 30 minute increments) three days in a row. Then it occurred to her that if she practiced 60 minutes per day for 6 days, her total would be 360 minutes, not the 240 minutes she was expecting. She has not practiced for two days, but plans to practice 60 minutes on the sixth day. She got a real-life lesson in checking the math by plugging the solution back into the original problem, a step her teacher requires, but she resents as a time waster. We talked about that maybe her teacher really does have some wisdom in her requirements. She also admitted that her goal is to do the minimum necessary to secure an A. Excellence and doing one's best is just adult yadayada. At least her bar is set at A.

Tricks and Shortcuts vs. Mathematics

The issue is not whether algebra should be taught in the eighth grade or later. The issue is not whether local schools should be able to make their own textbook adoption decisions. The issue is about how easily states make big changes based on flimsy research which asks the wrong questions, only to backtrack later because solutions that solve the wrong problem do not work. California reverted to phonics in 1995 after abandoning it for a faulty implementation of whole language based on research that answered some questions, but not the questions that matter.

The emphasis on algebra in the eighth grade is misplaced when even students with good math grades enter algebra weak in math concepts. I am working with an A student now who is solving for x in problems involving mixed numbers. She wrote these "computations:" 2 + ¼ = 2¼, 3 + (- ¾) = 3-¾, and -2 + ½ = -2½. Do you see the pattern? In her mind, numbers are disembodied entities with no real meaning. She thinks all she has to do is take out the plus sign and push the fraction up against the whole number.

These silly errors happen in an education system where children have been taught tricks and shortcuts since first grade. The problem is teachers call tricks and shortcuts "math," and when children do well on a test of tricks and shortcuts, they learn their good grade is proof they understand math. Actually the grade proves only that they can reliably implement tricks and shortcuts.

I have worked with children who have terrible math anxiety because they do not do well with the tricks and shortcuts. Some part of their mind has rejected the tricks and shortcuts as not making sense, so "math" does not make sense. If they ever get a chance to acquire true number sense, then they find out they are good at math after all.

Sometimes we reward unthinking compliance (as when kids memorize the tricks and shortcuts) and punish the thinkers for whom the tricks and shortcuts do not make mathematical sense.

American Education is NOT Failing....

...In fact, it accomplishes its hidden curriculum perfectly, according to Danjo1987, who hits several out of the park his first day up to bat on EdWeek forums. hllnwlz wound up the pitch and effectively makes many of the same points.

If you have been following this blog, you know that last year I am the "parent" (from the school's point of view) for a particular child who is now an eighth grader. My kids are grown; still it has been instructive to observe her schoolwork and communicate with a school as a savvy teacher/parent. I have been reminded once again that schools really do not like interacting with savvy parents. When schools say they want parent involvement, what they usually mean is they want parents to bake cupcakes once in a while and make sure the student does the homework everyday. More than that, and you are stigmatized as a "helicopter parent."

Overall, the girl's teachers seem to be competent;a couple strike me as excellent. There is one teacher I simply cannot fathom. On the midterm progress report, this teacher gave this straight-A student a citizenship grade of "N" for "excessive absences" during a medical leave. Upon her return to class, she took a "diagnostic" test and got a "D." This is the student's only grade for the class, and the grade teacher put on the progress report. (The other teachers gave her "I" for incomplete).

For the past five weeks, apparently this teacher has done nothing gradable in class. At the close of the term last Friday, there was only one grade in the online system the school uses: that "D." The student's grade on the report card? C-. I am about to intervene.

Meanwhile, in her other classes, she often brings home homework that astound me with the easiness and triviality of it. I see the kind of homework I used to get as a second or third grader. For example, she has to write a little essay about a short story they read in English class. The first assignment is to analyze the writing prompt, write down the verbs that tell what the student is to do, etc. In eighth grade? And the requirements for the regular notebook checks are beyond ridiculous, but the school feels if they do not force the students to organize, none of them will. Apparently, they did not learn how to collect and organize their work in elementary school. The only reason she writes both her first and last name on papers is because I insisted she write a complete heading on each paper whether the teacher required it or not. "But I am the only one with my name," she complained. Does not matter.

What I see is a disjointed and inconsistent system characterized by low expectations, even as the adults give inordinate emphasis to test scores.

Wrong Questions About Spreadsheet Math

Spreadsheets are a ubiquitous and necessary tool these days. Students need to learn spreadsheet math.

"Our children still spend hundreds of hours perfecting their ability to add, subtract, multiply, and divide fractions. And the pinnacle of math for most of our K-12 students remains the ability to solve quadratic equations. When was the last time you used any of these skills? When did you last multiply two three-digit numbers together on paper, add two improper fractions with unlike denominators, or solve a quadratic equation?"

These are the questions people asked when it came to calculator use, and they are still the wrong questions. Spreadsheet math will not replace the ability to actually understand math any more than calculators did.

When the National Council of Teachers of Mathematics (NCTM) recommended calculators for even the youngest students, they rhapsodized about about how calculators would revolutionize math teaching, using the same sort of language that idealizes the potential of spreadsheet math.

"By teaching our children spreadsheet math we enable them to solve ...fascinating problems, problems without a single right answer, problems that can be explored, problems that get our children thinking "out of the box."

And that was exactly the wrong-headed pie-in-the-sky rationale for recommending calculators. It sounds great but does not work in practice. The problem with math instruction is not whether we should be using calculators or spreadsheets. The problem is the lack of skilled math teachers. The problem is the continued reliance on teaching tricks and shortcuts instead of math. Like calculators, spreadsheets have a similar tendency to replace thinking.

Beginning in 2001, I researched the calculator fallacy extensively culminating in a 78-page report in 2010. Briefly, I found that the research NCTM insisted supported the use of calculator in the early grades did not exist.

I agree that students need to learn spreadsheets, but not as a substitute for learning math. Since our elementary teachers lack an ability to teach math for understanding, abundant experience with mechanical processes, though far from ideal, is pretty much the only way kids learn to tell an unreasonable answer from a reasonable one, and even then they are not very good at it.

Just last week, a friend's eighth grade daughter (A+ in math per last progress report) was sure that if $27.50 could buy 10 lbs of hamburger, then $55.00 would buy over 150 lbs because "I followed all the steps correctly." When I told her that obviously she had not, she argued that even the calculator agreed with her, so I must be the wrong one. Just yesterday she insisted that -3 + ½ = -3½ (by analogy to 2 + ½ = 2½). In her mind, all you have to do is get rid of the plus sign and shove the fraction up against the whole number. When these kinds of misconceptions plague even good students, no wonder students who are not as “good” have math anxiety. Deep down, the anxiety is related to an unspoken and unspeakable suspicion that math makes no sense. They are right. When math is turned into a system of tricks and shortcuts, it makes no sense.