Moving over to Blogger

My previous blog host went off line. I think Blogger.com will offer a much longer lasting service. Anyway, I am re-posting here the backup copies of my old posts, so I apologize for the inconvenience if you already saw this material. I expect to catch up pretty soon.

Negative Zero

Yes, it is a real number.

Every once in a while, I find students who show surprise or disbelief when they first encounter the concept of -0 (negative zero).
Let's say they are solving some equation, and close to the end it reads like x = - (a - b), where a and b represent two numbers known to be equal by virtue of the conditions set at the beginning of the problem.
So, in this example, the next step would be to write x = - 0.
My observation here is that, some students in this situation freeze, turn their head towards me, with a strange look in their face, and go: "There is no negative zero, is there?"
Usually I reply: "Why not?"
And they go: "But ..., what is it?"
I say: "It's zero."
Then they say: "Oh! Really? Just that, zero? Are they the same?"
And I say: "Yes, they are the same thing."
And they go: "O.K."
They seem to suddenly realize that the concept makes sense and it's not really that big of a deal.
I mean, what else could it be? What else could negative zero be if it wasn't equal to zero? There is no other option.
Actually, being equal to its own negative, is a defining feature that uniquely identifies zero.
Zero is the only number equal to its own negative. If you find any number x for which x = - x holds true, then you know x must be zero.
However, the momentary puzzlement, surprise and disbelief some students show when confronting this concept for the first time, is quite natural.
Remember it took centuries for Western civilization to come in contact with the concept of zero, and to fully adopt it as part of the number family. At first it was not considered a "true" number, but only an artificial placeholder used in the representation of "true" numbers.
Not only zero had difficulties being accepted as a number, but also the number One went through a period in Greek history when it was considered more like a philosophical, psychological, or even a religious concept, not a plain mathematical entity.

Another Important Key to Problem-Solving

Write down ~everything~

Very often I see students struggling with confusion when they try to solve a word problem all in their head.
It is so common, it's amazing. First of all, nobody says you have to produce the answer by just looking at the problem.
This is what happens, they read the problem, they understand the first sentence, or the first few sentences, and they are already asking themselves: "How do I solve this?" "What do I do with these numbers?" "What formula do I apply?" "What operation am I supposed to perform?"
They are obsessed with the idea of taking action steps. This is the first obstacle.
As soon as they come up with an idea about what to do, they start doing it, they start performing the operations, all in their head.
Then they get a partial result, and they immediately jump with that result into the next operation, without writing down anything. It's just unbelievable!
When I see them doing this, I tell them: "You are using your mind as a calculator and as a piece of paper at the same time. Don't waste energy like that."
For the average person, the mind can be much more effective as a calculator than as a piece of paper. The short-term memory that stores numerical results from previous calculations is very volatile.
When you try to use your head to do the operations and to remember the results at the same time, you are headed for trouble and confusion.
Let's say you make a mistake. If you write down all the steps of your calculation, and don't erase anything, you are much more likely to catch your own mistake when you go back and check your steps.
If you don't write down anything, you won't even remember what operations you performed, let alone catch a mistake.
So, write down everything, not only the partial results from each operation, but the whole calculation.
Write down not only the calculations you perform, but the ideas that made you perform those calculations.
Write down everything, every single step, all of it, your ideas, your examples, the formulas you are going to use, everything.
You will be amazed how easy the process becomes when you create this habit of writing down everything as soon as you think of it, and not erasing anything.
Even with mistakes, in the end it works better just to mark them with a red circle, and rewrite the correct expression somewhere else in the page, instead of erasing them. Many times mistakes are useful for reference.
As a general rule, the more you write, the better. The more you write, the less stress you put on your mind and the easier the process becomes.

One Key to Problem-Solving

Not "How To?" but "What do we have here?"

I always emphasize this to my students. When first facing a math problem, especially a word problem, do not try to get the answer right away. This is an unrealistic expectation. The answer will come as the result of properly developing all the relevant, detailed information contained in the problem. Pay attention to the wording. Create a clear mental image of the situation the problem is describing. Sketch a graph, a table, or a drawing to represent the situation. Avoid jumping into pre-packaged, memorized formulas after having read the problem only once.