Last time I covered the “sledgehammer” method of adding fractions, a method that is very simple and anyone can master in a few minutes.
That’s not the method taught in schools, however. Let’s go over what you need to know to be able to add fractions, using the public school method, and note that it takes weeks for a student to
almost never master all this:
1) You need to know about prime numbers. Prime numbers are those numbers which only have themselves and 1 as factors. So, 3 is a prime number (the only positive whole numbers that go into 3 are 1 and 3). 10 is not a prime number (since 5 and 2 are factors of 10). This is introduced around the 3rd grade; if the student doesn’t know this, the rest falls apart and the student will never “get” fractions the public school way.
2) You need to be able to “factor” a number into a product of primes. So, you need to be able to take 20, and write it as 2 * 2 * 5. This method is called “prime factorization”, so the prime factorization of 20 is 2 * 2 * 5. As before, if the student misses this part of class, it’s over.
3) When you look at two numbers, you need to be able to find the Least Common Multiple (LCM). For example, the LCM of 3 and 4 is 12. One way to do this is the look at the prime factorization of 3, and 4, and multiply the non-overlapping prime factors. So, to get that 12, I multiplied 3 and 2 * 2 (the prime factorization of 12). It can be a little more involved than this, but I’m taking a simple example just so I can go through the entire public school process of adding fractions in a sane amount of time. A student that doesn’t know LCM will never get past this part.
4) Now, to add fractions with the same denominator, use the same steps I mentioned above (add the numerator, leave the denominator alone).
5) If the denominators are different, well, this is where even my teacher in public school always got scared, but here goes: first, look at the prime factorizations of the different denominators, then find the LCM by comparing the prime factorizations.
As before, I use the example of 1/3 + 1 / 4.
6) The LCM of my first example is 12 (by looking at the prime factorizations of 3 and 4, by using steps 1, 2, and 3). Now, compare the prime factorization of the LCM to the prime factorization of the first denominator. Multiply the first fraction, top and bottom, by whatever prime factors are not in the prime factorization of the first denominator. So, in our example, I’d multiply top and bottom of the first fraction by 2*2 (i.e., 4, since the prime factorization of the LCM is 3 * 2 * 2, and the prime factorization of 3 is just 3). Then repeat the process with the second fraction, by comparing the LCM to the prime factorization to the second denominator.
7) After you’ve performed the prime factorizations and comparisons and multiplications in steps 2-6, you get fractions with the same denominator. Now, just add the numerators and leave the denominator alone.
8) Simplify if necessary.
If you can’t remember the easy way to do this, go back to the previous post, and see how much simpler it was to calculate 1/3 + 1 / 4, to get an idea how ridiculously overcomplicated the above method, used in the public school, is.
Now, the public school method is, absolutely, more mathematically graceful, especially when dealing with unusual fractions…it also overwhelms the students with the amount of material necessary to just add a couple of numbers together. In many classes I’ve taught, including calculus, the students expect the entire discussion to come to a complete halt the moment a fraction comes on the board, because they’re used to having to access insane amounts of material to deal with that fraction.
If a student is weak on any aspect of steps 1 through 7 (and all he has to do is miss a day of school for that), he is forever under the impression that adding fractions is a ridiculously hard concept meant only for “math whizzes.” Since those steps all rely on some underlying theory that few 8 year olds can follow, that’s most students.
The method covered in the previous essay? A little clumsy, but it’s simple, relies only upon basic multiplication, takes a few minutes at most to cover in complete detail…and allows the class to do a problem with fractions in it without everything coming to a halt. A student that missed that day can catch up in minutes, not days.
Since schools use the ridiculously overcomplicated method, most students go through the system with nothing more than fear of fractions.
This is my biggest problem with Common Core—the underlying theory may be fine, but covering all that theory when all a child needs is “here’s how to do it” means that any child that doesn’t follow the theory will simply walk away from math thinking it’s all incredibly complicated, instead of learning the basic skills that, when he is older, will make understanding the theory much easier.
Now that, I hope, I’ve taught the reader a simple way to add fractions, and reviewed the stupidly complicated way of public school, the reader will have an easier time appreciating my claims that added complexity (even with more theoretical grace) will make things much, much, worse for our children.
There’s another reason kids won’t gain skills in Common Core: lack of practice.
“6 hours a day, 6 days a week.”
–when in training, this is how much Michael Phelps would practice swimming. You’d think he’d know all there is to know, but still, he practiced like this. Practice the basic techniques of swimming, over and over and over again. Bottom line, that’s how you get good at a skill…repetitive practice. This level of repetition got him 57 gold medals in various competitions. Why don’t the schools know the obvious?
To judge by the Common Core assignments I’ve seen, simple practice of basic skills is also missing. I’m not saying our kids need to practice 6 hours a day, 6 days a week, but even 15 minutes of practice isn’t going to be part of Common Core for most skills.
This is going to be disastrous.
Next time, I’ll address some specific examples from Common Core showing what I mean.