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.

Fraction Friction

I was speaking with a friend that teaches adults preparing for their GED’s. They were describing the moans and groans heard as the teacher announced they would begin to review fractions. It reminded me how many of my middle grade students hate fractions and love decimals. There is something very reassuring about decimals to students in grades 6-8, and possibly even older.

I have a theory about why fractions are so hated among this crowd. These theories have no research to back them up, so I hope that readers out there will feel free to call me to the mat on this. First lets do a little comparison between fractions and decimals.

	Addition	Subtraction	Multiplication	Division	Comparison
Fractions	☹	☹	☺	☺	☹
Decimals	☺	☺	☹	☹	☺

So the ways in which decimals are better than fractions are the addition, subtraction, and comparison (by which I mean being able to compare the magnitude of numbers). I would say this is at the heart of my middle school student’s love of decimals. These are the operations which students feel most confident with. In fact, I would venture to say that students often don’t really feel the “know” a fraction. If they must give answer they like to give a decimal. Students love to give answers like 3.291098094. At this point the reason for this is almost beyond my grasp.

I will say of the three areas I think that comparison is the most important. I believe that this is because each student has a sort of internal measuring tape. They feel they know a number when they know where it goes on the measuring tape. It can be difficult for a student to place with a degree of certainty 4/13 on such a mental measuring tape.

How do we help students feel more comfortable with fractions? I propose the following steps:

➢ Emphasize fraction’s strengths. They solve division problems easily 3 divided by 23 is 3/23 for example. They can be multiplied easily. A good example is .125 * .75 is 3/32. Division too.
➢ Help student to understand the meaning of fraction in many different ways. As rates, as ratios, as the solution to division problems. As a number with a unit e.g., 3 eighths. Eighths can be thought of as a unit, like feet or inches. This is why we must have a common denominator to add.
➢ Help students find ways to estimate the size of fractions. This may help them place the answer on their mental number line. For example, 4/13 is a little smaller than 4/12 = 1/3 so 4/13 is just a little smaller than one-third.
➢ Help students understand through pictures and a variety of situations why multiplying fractions can lead to smaller numbers and division can lead to bigger ones. This should include a thorough discussion of the meanings of multiplication and division. Too often students just see these as ways to make numbers bigger and smaller.

It really is an interesting question. Why do students dislike fractions so? Fractions predate decimals by a good 600 years. I often wish I could watch a middle school math class prior to Simon Stevin to see what the kids were bitching about then.

Good Children's Book

As part of bedtime tonight I read The Three Silly Billies by Margie Palatini. It is a wonderful retelling of the Three Billy Goats Gruff with appearances by from many of your other favorite story characters (Jack, Little Red Riding Hood, etc.) The conflict is for the heroes to get the 1 dollar needed to pay the toll on the troll bridge. Slowly each character adds a little bit for their carpool.

This could make an excellent literature connection for a K-3 class which is studying the values of various denominations of money, or 2-digit plus 1-digit numbers. Check it out.

Determinants

Determinants are interesting part of mathematics. They are an important measure of a transformation, they form a way to compute the cross product, my mathematical training says that they are connected with character theory. But how do I help student's understand their significance. Determinants have not been around very long, getting started about 1750. If I don't know much about matrices, transformations, or characters, then what are determinants. Until we can answer that question I don't know how to teach determinants. I don't want to just teach an algorithm to compute a number. We must have reasons for it.

Here is a reason that I found while reading mathworld. It is possible to create determinant equations that immediately give equations for lines, circles, parabolas given the right number of points. This emphasizes several of the important properties of determinants: linearity, and the property that if two rows are identical then the determinant has value zero. In fact, it is this last fact that is the key point of this. Let's see some examples. We start with equation for a line. (Bear with me, I don't know how to format the math yet.)

If I want to find the equation of a line through the points (3, 2) and (5, 6) then I can set up the following determinant:

| x  y  1 |
| 3  2  1 | = 0
| 5  6  1 |

First, this is a linear equation by the linearity of the determinant, and if we substitute x=3 and y=2 into this determinant then we get a true equation because of the property that if two rows are identical then the determinant is zero, similarly with (5, 6).

When you expand the determinant you get -4x+2y+8=0. This is not quite standard form, but close enough.

What is more amazing is that this trick works for other types of equations such as parabolas. In the case of the parabola you use a determinant of the form:

| x^2 x y 1 |
| a^2 a b 1 |
| c^2 c d 1 | = 0
| e^2 e f 1 |

where (a, b), (c, d), and (e, f) are points on the parabola.

There is even a version for a circle given three points and on mathworld there is the general case for any conic given 5 points.

I like the way that this approach unifies these different processes and emphasizes the important qualities of the determinant, but I am still left with my big question. What is a determinant? (Clear and Concise please)