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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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)

Test Corrections Pro/Con

Pro:
➢ Students must have a way to be successful in the class beyond the test.
o What if the student is not a good test taker?
o What if the student is going through some huge personal stress, a bad day?
➢ Teachers must have a way of documenting improvement.
➢ Test Corrections can be done in a quick and efficient framework. With out too much disruption to the class as a whole.
Con
➢ Re-working one problem gives less assurance in general of mastery of a topic. There is a benefit to the “Test” structure. The randomness of the questions requires a greater level of mastery.
➢ A student’s ability to do math should be the primary reflection of the grade. The mastery of the subject is primarily calculated by tests.
o When a parent sees that their child has received a B in a class, there is a reasonable expectation that they have achieved some level of competency with the material.
➢ There is an ability to cheat the system. Tutors, other students, even over utilizing the teacher.
Unknown:
➢ There seems to be little in the way of research in the effectiveness of corrections.