Small Math

The idea of "small math" has been bopping around in my head for about two months now. What is "small Math"? Technically "small math" is a neurological syndrome named mathminuitis. I affects roughly 75% of all students ages 8-21. Symptoms include

* A desire to learn the minimum amount of math necessary to pass the next test, period.

* A belief that roughly 70% of the material studied in math course is superfluous, and can be more easily solved by cross-multiplying (whatever that means).

* A belief that all math is a series of fun mnemonic devices such as: FOIL, and "Don't ask why, just flip the second and multiply".

So far, there is no known course of treatment. Though many believe that by focusing on understanding what math is truly about, by including greater and greater amounts of applications within the mathematics curriculum, we can begin to stop the spread of this syndrome.

The most worriesome part to the trend for me is the number of math teachers that seem to show signs of coming down with mathminuitis. These teachers believe that the by shrinking the curriculum, by focusing on rote algorithms that have cute sayings attached we can create some worthwhile learning clearly are suffering from mathminuitis.

Determinants are calling me...

Good Story about a class, and my continuing fascination with determinants.

I go to class today, and it is Algebra 2, which I am never really sure how to approach. We are doing a chapter on lines. Now I had all this students last year for Algebra I and it is an honors course, so they are all pretty strong. I come in to class and put the following question on the board:

Determine if the points (-2, 1) (0, 3) and (7, 12) are collinear

And do you know what all the students asked (Go ahead test yourself, see if you can put yourselves in the mind of an 8th grader)

I hope you guessed "What is collinear?" Cause that would be right.

Now we are at the tony private school, so the smartboard is still on from the last class. I login in and go to wolfram.mathworld.com and look up collinear. I let the students read the first couple lines, and now they are off and running. Get lots of good answers/approaches. Some students find the equation of line using two points and check the third point in the equation to see if it works. Some calculate the slope two different ways and see if they are equal. Some calculate the equation two different ways and see if they are equal. It is all very good.

While the students are working I can help but see the following information just a little bit lower down on the mathworld entry for collinear (and this is not a direct quote)

You can check if three points are collinear if with determinants. For points (x_1, y_1), (x_2, y_2), and (x_3, y_3) create the determinant.

|  x_1  y_1  1 |
|  x_2  y_2  1 |
|  x_3  y_3  1 |

if the determinant equals zero then the three points are collinear.

I had totally forgotten about this, even though I had written a post about determinants two months ago.

Now I am standing in front of my class while they work the collinearity problem wondering, should I go for it. Should I show them determinants right now? Should I use this moment to introduce this strange monster of mathematics? It then occurs to me that we can use this to find equations of lines as well, and that in the next chapter I am going to introduce Cramer's Rule. Cramer's Rule is nothing but determinants, and so I go for it. The kids really liked it, and it is a most amazing thing that the series of calculations does this.

The way to find the equation of the line is to take two points and create the determinant:

|  x    y    1 |
|  x_1  y_1  1 |=0
|  x_2  y_2  1 |

this works because you are looking for all the points that are collinear with the other two.

Another way to look at it is the fact that the determinant

|  x_1  y_1  1 |
|  x_2  y_2  1 |*.5
|  x_3  y_3  1 |

Gives the area of a triangle with those coordinates therefore the three points are collinear if and only if the three points define a triangle with zero area.

So the class went well, and they really had an appreciation for determinants. They liked the way this got them pretty close to standard form. They may even remember it. I showed them two by two determinants as well, we will see if they recognize it when it comes up in about two weeks.

So, after another foray with determinants, I left pondering. I wonder "What is a determinant?" What is it's core? It does so many things: linearity, area, invertibility, etc. I just don't know.

Delta/Epsilon in HS

I was tutoring a AP calculus BC student today that goes to a different private high school. I saw in her notes there the definition of the limit as it regards the definition of the definite integral. It was complete with delta and epsilon, and of course mesh size and all the rigmarole. I am teaching Calc BC at my high school and we had gone over such stuff this week, and I didn't mention this topic at all.

I know why I did this. The short list is (and in no particular order):

* Not on the test so why burden the students
* Delta/Epsilon definitely not the test at all
* I have previously tried to explain Delta/Epsilon arguments to people and found that every method/analogy I have is ultimately more complicated than the actual argument. People clear agree that Delta/Epsilon is complicated, so why try to make what you can't make simpler simpler.

The only thing I have found that helps explain Delta/Epsilon proof is the following diagram:



This image is meant display that the function maps from the real numbers to the real numbers. What we are trying to do is show that a for a every little region around what anticipate will be the limit, it is possible to find a region of the domain that maps into the chosen section of the range.

This might be what Alfred S. Posamentier is talking about in this op-ed.

Three 3's Competition

It was parent's night at the school that I teach at this week. There is generally a lot of waiting around for the teachers. So based on the brainteaser over at Text Savvy I decided to give my colleagues a little competition. Just as at Text Savvy the rules were:

1. Write a mathematical expression that evaluates to 9.
2. Use exactly three 3's and no other numerals.
3. Use no plus signs.

The idea was to be the "Most Creative", probably as easy to judge as "learning" so why not. I have to say that I was tremendously impressed with the results. With in the first half hour I had the following entries:





I really thought they were great. Some quibbling could be made with the sin(pi) and the units in the first, but they were creative.

Of course, word spread and the chemistry teacher at the school, who drove home with me on Wednesday came up with the following three in the car, which I think are wonderful too.



All in all a great success.