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.


  1. Is this still a relevant skill? Maybe it's just like having to use a fountain pen to write assignments and exams in script. It's so outdated! I can add fractions, for example, but I'm one of those who had to use a fountain pen.

  2. Well, is spelling a relevant skill? I mean, there's always spellcheck. How about knowing the meanings of words? I mean, you could always just look up words in the dictionary. While these skills seems about as irrelevant as working with fractions, imagine trying to read a book or write a message while having to look. up. every. single. word.

    It's almost impossible to get a narrative that way. Similarly, in math, if I put a fraction up on the board, pure chaos ensues in the lower level classes, making it impossible to address many concepts. Next time I'll explain why.

    But as far as relevance, like every skill, relevance is in the eye of the beholder, and the marketplace. Tossing a football 80 yards in the air doesn't seem to be relevant to most people's lives, but I hear tell some folks with that skill make many millions a year...

    It's also funny you mention about writing in script. There was a major court case, and the high school graduate witness hurt the case when it was shown she couldn't read script.

  3. To be fair, the letter in question was quite illegible. There are a few words I couldn't understand or that I had to guess based on the context. I did have the experience of not being able to understand a book in a foreign language and eventually giving up only to find it much easier years later, when I finally read the whole book.

  4. On another point of conversation: it looks as though The Atlantic Monthly has been reading your confessions. This writer discusses much of the difficulty Professor Doom points out with the multiple bad ideas of depending on adjunct profs as a contingent workforce. She has some interesting data pointing to higher dropout rates for college students with multiple adjuncts teaching classes. She also makes a point that I hope Professor Doom can run with: we're facing the loss of a generation of scholarship because of the way administrations prioritize the nomadic adjuncts.


    Looking forward to your inside opinion on the veracity of this article.

  5. That's a nice overview article, but the fatal flaw is at the end, where there's an assumption made about the integrity of administration (lower rankings? Every administration I've seen has sold out as fast as possible, every time). The final line, "accreditors could change this game overnight" sure does sound like something right out of my post, but the article neglects to mention that accreditation is a fraud, as I've shown many times. It needs to be fixed first, before any other "change" to higher education will be relevant. Right now, even diploma mills are fully accredited; admin can do whatever they want under the current accreditation scheme, that's the simple fact.

    But, I sure hope the article helps adjuncts...I just don't see it happening until accreditation is fixed, and "willingness to hurt children for personal profit" is taken out of the administrator's job description.

    I lean against unions in general, but I see no other way adjuncts will even have a chance in this system. It used to be, your graduate degree WAS your union card, but Educationists destroyed that idea, watering down graduate degrees to the point of irrelevancy.

    Perhaps I will address this article eventually, but I have three more essays on CC to post.

  6. Fractions is where things start going wrong in the urban schools. Even with your method, the children have to know how to multiply numbers -- and teaching the multiplication tables today is considered passe ("we've got calculators now"). The lack of drill question is also a big problem. Modern math teaching emphasizes concepts but de-emphasizes memorization (of the multiplication tables, for example) and drill problems (which are essential to acquiring real skill). To compound the error, the emphasis on concepts goes into lengthy and utterly redundant explanations of why something is so rather than how to solve concrete problems. Even mathematicians and physicists generally don't work this way -- they try to solve concrete problems, stumble across some heuristic and ad hoc way of solving them, and only subsequently try to polish them, generalize them, and explain *why* they work (with varying degrees of accuracy and plausibility).

  7. Back in 1958, in third grade, we memorized the multiplication and division tables. We were tested daily, briefly, too less than ten minutes in speed tests.

    I even recited the tables while skipping to the jump rope!

    During my life making prototypes for IBM and Texas Instruments and building houses, being able to do math in my head was great!