By Professor Doom

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 3

^{rd}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.

*--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.

Maybe a century back adding fractions could have been taught this way (i.e., taking the LCM). But these days the students just don't have practice at calculating LCMs and HCFs (highest common factors); count yourself lucky if they know the multiplication tables. Your way is the easiest.

ReplyDeleteThe old computational skills have gone out the window. Even school teachers these days usually cannot calculate square roots by hand.

I think the Archdruid has it correct here:

"Computer-free mathematics. Until recently, it didn’t take a computer to crunch the numbers needed to build a bridge, navigate a ship, balance profits against losses, or do any of ten thousand other basic or not-so-basic mathematical operations; slide rules, nomographs, tables of logarithms, or the art of double-entry bookkeeping did the job. In the future, after computers stop being economically viable to maintain and replace, those same tasks will still need to be done, but the knowledge of how to do them without a computer is at high risk of being lost. If that knowledge can be gotten back into circulation and kept viable as the computer age winds down, a great many tasks that will need to be done in the deindustrial future will be much less problematic."

http://thearchdruidreport.blogspot.com/2014/01/seven-sustainable-technologies.html

In industry, we had the saying "If it works, don't fix it." The traditional methods of teaching arithmetic, such as multiplication tables and handling fractions as described in the previous post, worked. It put men on the moon. It built ships, bridges, and aircraft. It created whole new industries such as electronics and computing. It runs cities and countries.

ReplyDeleteEducationists, however, insist that they must "improve" things by meddling with what has been proven to be effective. I shudder to think what would have happened to me had I been taught fractions in the method you just described.

It certainly explains why many of my students couldn't do basic arithmetic and they were recent high school graduates.

I thought about mentioning GCF ("Greatest"), but decided tossing that in there would be piling on confusion onto what is already a ridiculously complicated algorithm.

ReplyDeleteI'm not sure I agree with the archdruid on many things; the whole "oil is melted dinosaurs" theory has siginficant issues. I really don't see humans forgetting bookkeeping skills, and I promise you, even if computers shut down tomorrow, and every single textbook burned, we'd have some rebuilt log tables within a short period of time...assuming nobody would just copy

If it's such a fear, I guess someone could start printing books on plates of steel or something? A total industrial collapse is possible (I politely concede), but "Mad Max" is just not as likely for the world as real-world Somalia. I'm just not willing to believe humanity is that incapable on every level.

Shouldn't that more complicated but elegant way be left for when students start taking algebra, where they'll need to know what a greatest common factor is in order to factorise polynomials and simplify rational expressions?

ReplyDeleteAlso, about simpler ways of handling fractions, is it frowned on everywhere (not just in schools) to multiply fractions by going straight across (multiplying the numerators and denominators) then simplifying the resulting product fraction? I've found this way to be easier for me, even though it does take longer. Oddly, I'm more likely to mess up when cross-cancelling! This is been an issue teachers have seen me deal with for many years now. They do not like seeing say 3/4 * 6/21 = (3 * 6)/(4 * 21) = 18/84 = 9/42 = 3/14.

That's as good a way as any to multiply fractions, but students have been trained to think "multiplication is harder than addition". They're so blown away by addition of fractions that they figure the reason so little time is spent on multiplying that it must be impossible, and few ever learn it, at least to judge by what I see on papers.

ReplyDeleteKnowing stuff like prime numbers & what numbers are highly composite (like 12) can be very handy. Also seeing stuff like a number being a perfect square, cube, or half/quarter of something or double/treble/5x/10x something. I am glad that when I was attending middle school in the mid-1990s, we still did things that 'old style' way you describe. I can't imagine how they teach decimal division these days. Do they teach short division in schools along with long? Now doing square root simplification by hand I never learnt. I would like to learn it though.

DeleteSquare roots by hand isn't much fun, but Method 2 of Wikihow explains it clearly. Google "square root by hand method" to get there.

DeleteMost public schools, like college, have 'two track' system. So, it's possible some students in a school learn long division, while others learn how to make shoebox displays.