Math reform plan: Hammer away at basicsInability to handle fractions is hurting, says presidential panel

updated 1:03 p.m. ET, Thurs., March. 13, 2008

WASHINGTON - Schools could improve students' sluggish math scores by hammering home the basics, such as addition and multiplication, and then increasing the focus on fractions and geometry, a presidential panel recommended Thursday."Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra," the panel, appointed by President Bush two years ago, said in a report.Because success in algebra is linked to higher graduation rates and college enrollment, the panel focused on improving areas that form the foundation for algebra. Average U.S. math scores on a variety of tests drop around middle school, when algebra coursework typically begins. That trend led the panel to focus on what's happening before kids take algebra.A major goal for students should be mastery of fractions, since that is a "severely underdeveloped" area and one that's important to later algebra success, the report states.It goes on to say that other critical topics — such as whole numbers and aspects of geometry and measurement — should be studied in a more in-depth way.

History

• In the 1950’s a large scale stress on national testing began. This caused non-math elementary teachers to focus on students getting test answers rather than proper methodology for solving problems.

History

• Elementary Teachers observed that when dividing fractions if they multiplied the right denominator times the left numerator and the right numerator times the left denominator, from right to left, and then simplify would get a correct answer as illustrated below.

7 ÷ 3

9 5

History

This was the origin of “Cross Multiply”. This method was successful in helping students get a correct test answer. But it does not follow the rules or laws of math.

The rules of math are in the Order of Operations

which says PEMDAS from left to right. Cross multiply only works from right to left.

History

• Then teachers noticed that when adding or subtracting fractions with prime number denominators, they could just multiply the denominators to get the LCD.

• Then to change the numerator, just multiply the right denominator times the left numerator and left denominator times the right numerator. This correctly change the numerators and makes a new application for the term “Cross Multiply”.

History

• Middle School teachers picked up on this term and used it to help students learn to solve proportions, just cross multiply and divide.

3 = x

5 5• So now “Cross Multiply” was taught to add,

subtract, divide, and proportions.

History

• But, the same “Cross Multiply” cannot be used for adding, subtracting, dividing and proportions. Each is a slight variation and none follows the Order of Operations, PEMDAS from left to right.

• In addition, students remember the term “Cross Multiply” and try and use it to multiply fractions, the one time it cannot be used.

History

• So, students were promoted to higher math classes with a technique to solve fractions that does not follow the properties of math and therefore will not work at higher math levels.

• To address this problem, in the 1960’s “New Math” was developed to explain WHY steps were taken. The problem was the teacher needed to be a “Math Person” to understand how to teach this method. So it failed.

History

• Since then different methods of math instruction have been utilized to add and subtract fractions that include such methods to find the LCD as “Try and Figure” what number both denominators will go into evenly. However, “Try and Figure” is not a property or law of math. The student is expected to “Conjure” a number. Conjuring a number is mysticism not mathematics.

History

• Another method tried recently is to take multiples of the denominators and find the first one that is the same. For example the denominators of 2 & 32 4 (6) 8 10 12 14 1618 3 (6) 9 12 15 18 21 2427

Problems in Teaching Methodology

• So, now try the denominators 2a and 3b using this method.

2a 4a 6a 8a 10a 12a 14a 16a 18a 3b 6b 9b 12b 15b 18b 21b 24b 27b

As you can see, this method does not work with this level of problem. Imagine students trying to use this method to find the LCD of x2 – 1 and x2 + 2x + 1. It cannot be done!

How to Fix the Problem

• Any real math education fix must be – mathematically correct– simple for students to learn– simple enough for non-math teachers to be able

to understand as well as teach.

• I developed such a method which is in my book and the fraction section information follows.

Problems in Teaching Methodology

• Take the problem 3 x 2 =

4 9Current method is

3 x 2 = 64 9 36

Then simplify to 3 x 2 = 6 = 14 9 36 6

This method is multiply, then simplify. There is nothing wrong with this method. However, it cannot be used in higher math.

Problems in Teaching Methodology

• For example, if the student is working on a quadratic fraction like this one

(3x2 + 5x + 2) x (4x2 – 9)

(2x2 + 9x + 9) (2x2 -3x -3) they definitely do NOT want to multiply, then

simplify. They want to factor and simplify, then multiply. So factor and simplify, and then multiply is what I use in my method

How is your school doing?

Ask one of your elementary teachers how they would teach students to work the problem on the left and note the steps they take, probably multiply, then simplify. Then ask them to use the same steps to work the problem on the right.

3 x 8 (3x2 + 5x + 2) x (4x2 – 9)

4 9 (2x2 + 9x + 9) (2x2 -x -3)

Solution: Multiply & Divide Fractions

• I give my students the following steps to multiply or divide fractions:1. Make it a fraction. If a mixed number, make it into an improper fraction.

2 3 ÷ 1 1 (3x2 + 5x + 2) x (4x2 – 9)

4 8 (2x2 + 9x + 9) (2x2 -x -3) 11 ÷ 9 Already in fraction form

4 8

Solution: Multiply & Divide Fractions

2. Determine if it is multiply or divide. If multiply go to the next step. If divide inverse, reciprocal, or “flip” the second fraction and only the second fraction.

11 x 8 (3x2 + 5x + 2) x (4x2 – 9)

4 9 (2x2 + 9x + 9) (2x2 -x -3) These are both multiplication. Nothing needs to

be done.

Solution: Multiply & Divide Fractions

3. Factor into prime factors the numerators and denominators and make them one fraction.

11 x 8 (3x2 + 5x + 2) x (4x2 – 9) 4 9 (2x2 + 9x + 9) (2x2 - x - 3) 11•1•2•2•2 (3x+2)(x+1)(2x+3)(2x-3) 2•2•3•3 (2x+3)(x+3)(x+1)(2x-3)

Solution: Multiply & Divide Fractions

Solution: Multiply & Divide Fractions

Solution: Add & Subtract Fractions

The key here is to teach students how to CALCULATE the LCD and then change the numerator. Here are the steps I teach for these two fractions. They could be addition or subtraction.

1 + 2 3 + 2___ 15 9 x2 + 2x + 1 2x2 +x - 1

Solution: Add & Subtract Fractions

1. Make it a fraction.These are both already fractions, so no changes are needed. If mixed numbers, make them improper fractions.

1 + 2 3 + 2___ 15 9 x2 + 2x + 1 2x2 +x - 1

Solution: Add & Subtract Fractions

2. Factoring is the key to all fractions. The LCD is looking for the least common denominator, so factor the denominators.

1 + 2 3 + 2___ 15 9 x2 + 2x + 1 2x2 +x – 1 1 + 2 3 + 2____ 3•5 3•3 (x+1)(x+1) (2x – 1)(x+1)

Solution: Add & Subtract Fractions

2. Circle the factors that are the same in both denominators.

1 + 2 3 + 2____ (3)•5 3•(3) ((x+1))(x+1) (2x – 1)((x+1))

Solution: Add & Subtract Fractions

3. Next, put a box around all the other factors.

1 + 2 3 + 2______ (3)•[5] [3]•(3) ((x+1))[(x+1)] [(2x – 1)]

((x+1))

The purpose is to identify the parts that will be used.

Solution: Add & Subtract Fractions

4. Then multiply the box(es) on the right times the numerator on the left.

1[3] + 2 3[(2x-1)] + 2___ (3)•[5] [3]•(3) ((x+1))[(x+1)] [(2x – 1)]((x+1))

5. Then multiply the box(es) on the left times the numerator on the right.

1[3] + 2[5] 3(2x-1) + 2[(x+1)]___ (3)•[5] [3]•(3) ((x+1))[(x+1)] [(2x – 1)]((x+1))

Solution: Add & Subtract Fractions

6. Find the LCD 1[3] + 2[5] 3[(2x-1)] + 2[(x+1)]__ (3)•[5] [3]•(3) ((x+1))[(x+1)] [(2x – 1)]((x+1)) The LCD is ALWAYS the circle(s) on the left times

ALL box or boxes (3)[5][3] ((x+1))[(x+1)][(2x-1)]

Solution: Add & Subtract Fractions

7. Order of operations, multiply as indicated. 1[3] ± 2[5] 3[(2x-1)] ± 2[(x+1)]___ (3)[5][3] ((x+1))[(x+1)][(2x-1)]

3 ± 10 6x – 3 ± 2x ± 2 45 2x3 + 3x2 - 1

Solution: Add & Subtract Fractions

8. Combine Like terms, if possible, then check to see if it can be simplified.

3 ± 10 6x – 3 ± 2x ± 2 45 2x3 + 3x2 – 1

13 or -7 8x – 1____ or 4x – 5___ 45 45 2x3 + 3x2 – 1 2x3 + 3x2 – 1

These cannot be simplified.

Summary

The Common Core by its very design, is requiring schools to systemize their math teaching methodology from K to 12 grades. My system is the ONLY system, of which I am aware, that can be used and taught from k-12 WITHOUT CHANGE. In addition, it is very visual and when color markers are included in instruction, greatly improves student and teacher understanding.