Some Funny Comments and Questions.

From some of my students

Following below are a few questions and comments made by some of my students in the last year or so. They are all from different students, and they are unrelated to each other. The only thing they have in common is that I found them very funny at the time I heard them, maybe in part because of the student voices or facial expressions when they made them. Anyway, I quote them here in case you find them interesting:

“Are these all the primes there are? Or, is there any prime bigger than 13?”

“How can you stand so much math? My head hurts!”

“You are just fascinated with this stuff, aren’t you?”

“When I see logarithms I just freeze.”

“What is e to the negative x square? I don’t like ‘e’s! ‘e’s always confuse me.”

“Yes, a big number minus a small number we know how to do that, and a small number minus a big number is possible only if it’s money, or something like it.”

“Negative numbers are evil!!!”

“Wait! You just set those two values equal to each other! Why? Are you allowed to do that? How do you know you can do that?”

“Wow! All that math just to get a result equal to zero? That doesn’t make any sense!”

“I hope I will never see you again. I mean, don’t get me wrong; it’s nothing personal. You are a nice guy, and I am grateful for all your help but man, if I pass this test, I am never going to open another math book in my life again, ever!”

“I just feel like school is sucking my life away!”

“So, do you have a real job or you just tutor?”

Negative Numbers

Building up concepts a little bit at a time.

Beyond a certain age, most students can handle negative numbers. Some make mistakes sometimes, like forgetting writing the negative sign in front of the result, or subtracting the numbers when they should add them but in general they have the concept of negative numbers and their operations.
This is different for little children who have not yet been exposed to the subject.
Today I asked one of them:
“How much is 3 minus 5?”
He said:
“That’s impossible!”
I said:
“O.K., well, let’s see.. Have you ever borrowed money?”
I led him through the example of owing five dollars, having three in our possession, paying that amount and ending up owing only two.
Then I pointed out the fact that five minus three equals positive two, while three minus five equals negative two, and I continued:
“So, it is possible to subtract a big number from a small one, and the way we do it is we really just subtract the small number from the big number, and we write the result with a negative sign in front of it.”
Then he summarized his understanding as follows:
“Yes, a big number minus a small number we know how to do that, and a small number minus a big number is possible only if it’s money, or something like it.”
I thought that was really funny but the key point here is in his mind he moved the concept of “small minus big” from “impossible” to “possible only if it’s money.” So now he accepts the possibility of such an operation at least in some cases.
This illustrates another point, that learning most often than not is a gradual process, where we build up concepts a little bit at a time. Students require several exposures to negative numbers and to the rules governing operations with them, before they can feel comfortable handling such operations. These exposures better be gradual, clear, consistent, and such that the student gets a feeling of success about them. Otherwise confusion sets in, and with it the seed of long-term frustration.
I remember the following dialog with another student a few months back, when I asked her:
“So, when we multiply two negative numbers, what is the sign of the result?”
She said:
“Negative numbers are baaad!”
I asked:
“Really? How bad?”
She answered:
“Negative numbers are evil!!!”
I found that comment very funny, I smiled and I said:
“O.K., well, somehow we have to deal with the fact that your teacher for some obscure reason wants you to add and subtract and multiply those evil numbers so, how are we going to do that?”
Then she said:
“Well, maybe they are not always that bad after all.”
Usually it is not easy to discover (let alone clear them) the blockages installed in a student’s mind around a concept by virtue of unsuccessful teaching techniques.
The problem here is that every teaching technique is very effective with some students, while at the same time being totally useless with some others. Given the amount of material in the syllabus, and the limited time available, teachers in the classroom have to go with whatever technique proves useful for the majority of the class, and some students are left behind.

Factored Integers

A memorization exercise

Last week I added a new page to my tutoring web site. The new page’s title is “Factored Integers.” It is a reference page listing about 200 positive integers, completely factored out as products of smaller numbers, including their prime factorization.
The purpose of such a list is for some standardized test takers to do a memorization exercise. The idea is for the student to copy this list, and to write by hand a portion of it every day, anywhere from 20 to 50 numbers a day.
Just writing down the numbers and their factorizations has a cumulative effect in the student’s memory, as long as they do the exercise every day. Standardized test have many problems that can be solved much faster by factoring numbers out than by doing long multiplications and divisions.
Time is the most precious resource in a timed test, so the goal is for the student to have readily available, fresh in their memory, these factorizations, instead of wasting time thinking about what could be a possible factorization, or even worse, going down the path of long multiplications and divisions, because these operations become very time consuming, and prone to errors when the numbers involved are large.
So the best way to solve these problems is by factoring all numbers as much as possible, and simplifying all expressions as much as possible by canceling out any common factors that can be canceled out before getting into any multiplication.
So the value of the memorization exercise resides in the increased awareness of factors the student develops a little bit each day by writing and re-writing the list of factorizations.
The goal is for the student to start thinking about a number’s factors as soon as they see the number in the problem; naturally, automatically, by default, without even thinking about it. See a number, boom! Factor it. The less time you spend on this process at test time, the better. So the time spent at home writing and re-writing the list will pay off on test day.