Tuesday, April 29, 2014

Common Core and Fractions




By Professor Doom

     “See that big thing with the hairy mane? Don’t go near that, it will eat you!”
--advice from adult to child on the Serengeti, to avoid lions. This is really all the explanation a child needs, at least at first. A discussion of biology, the need for the lions to eat, their caloric intake, the size and weaponry of lions, and their hunting habits, while interesting, distracts from the key thing the child needs to know. If he follows the adult-provided guideline, he’ll live long enough to learn the other things.

      It’s time to talk about Common Core. I grant that this mostly affects primary and secondary schools, but what affects the schools will eventually reverberate into higher education…not to mention that much of so-called higher education is a fraud, merely re-teaching the material already given in schools.

     Before I can address the problems that are most evident in Common Core, I want to talk about “adding fractions.” I imagine a wave of fear just passed through some of my readers at the mere thought of “fractions.” A great number of my students are terrified of fractions, to the point that the class can completely shut down if I put a fraction on the board.
      For all I know, in the public schools, around 3rd grade or so, the students are all lined up and a fraction comes in and touches each student, inappropriately.

     That’s a joke, but the point is students are trained into freaking out at the sight of a fraction. The reason for this is the schools, in an effort to “explain the theory” of fractions, buries the student in so much crap that they lose track of what the theory is for: to be able to add fractions.

     Let’s go over all you need to know about how to add fractions. I’m sorry to start with fractions, because I know many readers will simply shut down. That’s entirely my point: many readers only know fractions from the incredibly and stupidly complicated method taught in schools, and I’m going to show a simple way to do it. I want to compare two techniques, the “easy” way, and the way taught in public schools. Both assume the student knows the basic times tables, and perhaps a little about division.

The easy way:

1) If the denominators (the numbers on bottom) are the same, you just add the numerators (the numbers on top), and leave the denominator alone…then you’re done.
2/5  +  7/5   =  9/5  (no need to do anything more)
Sometimes you’ll need to simplify:
1/6  +  2/ 6  = 3/6, but 3/6 simplifies into 1 / 2, since “3” is a common factor of the numerator and denominator. So, 1/6 + 2/6 = 1 / 2.

2) If the denominators are different, it’s a little harder.
Multiply the first fraction (top and bottom) by the denominator of the second fraction, and don’t simplify.
Multiply the second fraction in the same way, by multiplying top and bottom by the denominator of the first fraction.
Now that the denominators are the same, add the numerators, and simplify as before. Here’s an example:
1/3 + 1/4   (note: denominators different)
Multiply 1 / 3 by 4 / 4 (i.e., multiply both numbers by 4), to get 4 / 12
Multiply 1 / 4 by 3 / 3, to get 3 / 12
Now add:
4 / 12  +  3 / 12   (the denominators are the same)
7/12   (add the numerators).

     Now, the above is a very simple “sledgehammer” technique, guaranteed to work every time. The only issue with the technique is sometimes you have to simplify the fractions (by eliminating common factors), but conceptually, “sometimes you need to simplify” is still far easier than the theoretical methods taught in school (which, still, sometimes need to be simplified).

     I emphasize: above, half a page of text, is all you need to know to add fractions. I’ve tutored dozens of “special ed” students that had no idea how to add fractions after YEARS of public school.

     I show these “special” students the above technique, and in a matter of minutes they’ve mastered adding fractions.  It requires no intuition, or knowledge beyond the times tables; you use the numbers that are right in front of you.

     Why do many (most?) kids coming out of school approach fractions with fear and awe? Because the schools take a heavy theoretical approach, one the kids get browbeaten with starting around the 3th grade…they’re never shown any guideline that’s as easy to follow as “stay away from the lions.” Instead, they’re taught such a ridiculously overcomplicated method that, while mathematically more sound, is just unreasonable to inflict on an 8 year old.
    
     The simple method for fractions really highlights what Common Core will do to our children. The overcomplicated methods will create a generation not just terrified of fractions, but afraid even of addition of whole numbers.

     I encourage the reader to practice adding fractions using the “sledgehammer” method above (with another example below), to better appreciate how sad it is that more than half of high school graduates have trouble adding fractions:

Example:   1/3 + 2/5   =   (5/5) * (1/3) + (3/3) * (2/5) = 5/15 + 6/15 = 11/15

Now try:
1 /2 + 1 / 4
2/3 + 1/5
3/7 + 1/3

    Now, for an adult, three problems is usually enough to master a basic skill. Children are, generally, slower. Did your child learn to “pick up the laundry” after only being told three times? How about “take out the garbage”? How quickly did he learn to tie his shoes? 

     Common Core, to judge by the worksheets I’ve seen, seldom gives the child even three chances to learn the skill.

     I know I’m losing some readers by talking about fractions first, but if you thought fractions were hard, try to learn the above method, and see how simple it is. I want the reader to be angry about being trained into hating fractions, so that the reader can better appreciate what Common Core will be doing to his children, not just with fractions, but with basic addition and subtraction.

     Next time, we’ll go over how students are taught to add fractions in the public schools, and then start on Common Core.