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.

The Scrap Heap of Mathematics

My nomination for the scrap heap of mathematics, the rational root theorem. It is a lovely pile of math that is collecting all of the math that is no longer relevant, no longer valued to be a part of the K-12 experience. Upon inspection of the heap we find: calculating square-roots by hand, manipulating a slide rule, and if not on the heap lying right next to it balled up like a piece of used kleenex, long division.

I remember learning the rational root theorem in my Algebra 2 class in high school. I loved it. I wanted to factor everything I could, and I needed something that would help me factor higher degree polynomials. Today, I would hopefully have a great understanding of the connections between zeros of a polynomial and the roots. I would graph the polynomial and use that to find the roots. And moreover, I don't think that the proof/reasoning behind the theorem is so enlightening that students will understand mathematics less if we never speak of it again. The reasoning, I believe, boils down to the fact of divisibility of the lead term and constant term. Students get this by factoring quadratics.

Am I missing something important here, or is this a slam dunk onto the scrap heap of mathematics?

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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.

Aha Moment, trapezoid and the series

Math is wonderful. Especially, when it all connects. Last spring I was teaching arithmetic series to my honors 8th grade class, and I made a connection that I had never made before. For those who don't know a arithmetic series is the sum of (generally) a lot of numbers that increase by steady amounts, such as 5+9+13+...101. The goal for our us is to find the sum with out actually doing all the dirty work of adding up all the numbers (easy for Gauss, but hard for many of us). There is a formula for this which is to add the first and the last (5 and 101 in my example), multiply by the number of numbers (25 in my case, harder to figure out than it sounds; the famous fence-post problem) and divide by 2. For those of you that look cool formulas it looks something like this:



The standard way of seeing this is with some pictures of bar graphs each bar representing one summand. The sum that we are trying to find is like adding up all the lengths represented by this bar graph



One nice way to see the formula is to take a copy of the graph and place it on top of the first but running backward from last to first as in this picture



Then each bar has the same height, the sum of the first and last, and we can multiply by the number of bars to find the total. We doubled what we are trying to find so to find the sum of the red columns I take that answer and divide by 2.

How do trapezoids fit in?

We if you look carefully, you will see the that the original picture is pretty much a trapezoid.



How do you find the area of this trapezoid? You take the length of the parallel sides, in this case 5 and 101. Multiply by the height between those sides, I guess this would be how many bars there are 25, and divide by 2. And in fact to proof that this formula works for trapezoids is exactly the same. Double the trapezoid to create a parallelogram that you already know how to find the area of.

So arithmetic series are isomorphic to trapezoids. Who knew?

Math as Metaphor

Tony at Pencils Down has an interesting autobiographical post about the importance on metaphor in mathematics education. Both how it is essential and how it can be a hindrance to learning. It reminds me of the opening section of How Students Learn by the National Research Council. There they use the story Fish is Fish to show how all learning is based on previous experience.

I am currently trying to write a short piece for ASCD about the "The Value of Mathematics". All this analogy, metaphor stuff fits in nicely with the point that I will be trying to make. If ASCD doesn't want it, I will post it here.

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