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.
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.