“Well,” I said, looking at the student’s math problem, “everything is fine until here: you have (-14/23) + (15/23), which is correct, but you wrote they equaled 18/23.”
“What should it be?”
The student paused. I continued, “What is minus 14 and plus 15?”
She took out her calculator.
“No,” I said. “No calculator. What is minus 14 plus 15?”
She thought for a few seconds. “One”.
“Exactly.”
“I must have pushed the wrong button on the calculator.”
I tried not to cringe. “You don’t need a calculator to add minus 14 and plus 15. By using a calculator, you run the risk of pushing the wrong button and believing what comes out. Use your head. You knew what it was.” The woman nodded.
We tutors sat alongside a wall at desks, and students, who sat at one of many tables, working on math problems, would come to any of us if they wanted help. I spend four hours, sometimes more, two days a week as a volunteer, and I am usually busy the whole time. A fellow tutor about my age commented how we once had main-frame computers (the adjective has now almost disappeared from the language) that took up a whole room, required air conditioning, and needed punch cards to do simple calculations. I can remember learning BASIC, written by John Kemeny, mathematician, teacher, entrepreneur and president of Dartmouth College for 11 years, including the year I graduated. Kemeny was a genius.
In 1969, Wang calculators appeared in a small room in the chemistry department, where I was a major. These were slightly smaller than a laptop keyboard, and we could then do exponentials and natural logs. We used to use log tables to do calculations. Our generation knew the word “anti-log” and the term “9.xxx-10”. If you know what I mean, you are at least 55. We thought these new calculators were incredible. We could put away our slide rules. Imagine, cube roots by pushing buttons! We played with [ln (0)], to see how the calculator handled undefined numbers. It was fun.
Now my TI-83, a rather old calculator, can graph any polynomial I want in any size graphing window I want. It can give me the values of derivatives, intercepts, z-values for proportions, t-tests, Chi Square values, raise numbers to the 100th power or more, which makes interest calculations easy, store numbers, do linear, quadratic, cubic, and exponential regression, has a built in z- and t-table, does binomial calculations, take n th roots of things, and fits in my back pocket.
What it cannot do IS TELL ME WHAT TEST I SHOULD USE. Nor for that matter, can it square any number with 25 digits, all of which are 9, as fast as I. I think I can multiply any pair of two digit numbers as fast. Squaring three digit numbers ending in 5, I can also come close to beating it. But graphing y= [sin x/(x-2)]^2]? No way I can come close. I can do logs to the base 10 of 2,3,4,5,6,8,9, and 10 to 5 decimal places faster, but I can’t do the log of 7 or larger primes. I can’t do them to six or seven places. Nor can I do trig functions other than the simple 30, 45, 60, 90 and multiples. But I can estimate them quite closely, which surprises people, when I say the tangent of 35 degrees is about 0.7. The tangent of 30 is √3/3=0.5773, and the tangent of 45 is 1, so I just interpolate. Many times, that is all I need. I am metric fluent, but I can’t use polar coordinates well. The TI-83 does that and shifts to radians with a couple of key strokes.
These calculators are great. BUT THEY CAN’T THINK. They take your thoughts and input and give the result. They don’t say whether you asked the proper question. They don’t tell you that a quadratic function might fit the data better than a linear function, They will tell that you inputted an incorrect formula but not a wrong number.
Nor will the calculators say if you violated the assumptions with linear regression, something that many people who argue against global climate change, for example, fail to recognize. One cannot simply make a scatterplot, draw a line through the points, or even have a calculator do it, and conclude nothing has changed in the last decade. A calculator can minimize the sum of the squared vertical distances of the points from the line, showing the best “fit”. It can do it faster than I, although it can’t prove why the formula works, and I can. A calculator can show whether the line is going up or down, even how much, and if one asks for more, the p-value for the line, the probability the occurrence is different from a fixed number’s being a chance occurrence. The calculator, however, cannot cannot tell you whether the distance of each of the points from the line follows a normal distribution with mean 0 and the same variance, without some work and judgment from the user. Nor does it tell one whether a point should be an outlier. That also requires judgment.
Calculators have to be used for the right things, just like computers, and one needs to have an idea when the answer makes no sense. If I Google “East Pacific Ridge,” wondering why the Jet Stream Wave #3 has stayed stuck for so long, which is devastating the American West, I might read something that says only warm moisture from south is getting through, because of the radiation in the water at higher latitudes from the 2011 tsunami can’t, which is nonsense, akin to pushing the wrong button. We push wrong buttons sometimes, or we forget to erase something in the calculator. We look at an answer and should realize it makes no sense. That ability has been greatly diminished, and in some places has disappeared, along with my slide rule. I am not a whiz with a calculator. But I have the ability to look at an answer, a graph, say about what things ought to be, and whether the answer makes sense. I agree that Common Core is teaching estimation, but there are multiple ways to estimate, and children need to be taught multiple ways to do math, because they learn differently. I estimate differently from the way Common Core does. I learn differently. Indeed, discovering how an individual learns is one of the most important things education should do.
A 10% increase in the likelihood of a rare disease is not 10% likelihood you will get it.
Decrease in the rate of growth does not mean the total is declining. It is not.
Four different outcomes do not mean they all have equal probability. They usually don’t.
There’s a place for the mean and another place for the median. A person who uses them interchangeably to describe a set of data is either lying or doesn’t know the difference.
Finally, (-14/23) + (15/23) can’t possibly be larger than (15/23).
Tags: MATH AND MIKE, Philosophy
Boreal B[l]og 
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