Monday, December 28, 2020

Dividing Fractions

I'm on an email list on which the topic today is a video showing a silly way of doing division.  Literally, it involves drawing columns of dots.  It is no more understandable than the standard division algorithm, and certainly less efficient.  Here's the video:


 Here's what I posted on the maillist:

What I kept thinking while watching that video was "why not just draw 245 dots and circle them in groups of 5?"  It would have been only slightly less efficient but far more understandable.

Finding nifty techniques for math, techniques specifically designed to avoid the standard (and extremely efficient) algorithms, is often a case of mistaking cause for effect.  People like us, who understand math, can often see very interesting patterns or ways of viewing things that make math vibrant. If only everyone saw these things, math would be vibrant for them as well, yes? NO.  We see these things BECAUSE we already grok the underlying math: showing our patterns and techniques doesn't help people LEARN the underlying math.  "If a then b" does not imply "if b then a".  It amazes me how many people in education, how many people who should know better, make that mistake.

The division algorithm is plenty efficient already, and students should learn it.

Now let's shift gears a bit.  Knowing there's an exception that proves the rule, there is one non-standard-algorithm technique that I kinda sorta like.  It is slightly more difficult than the standard algorithm, but it combines some math that students already know with some simple understanding.  It's dividing fractions.

Here's the standard "invert and multiply" algorithm.  

 


It's easy enough to teach why/how this works (and I always did), but it still leaves some students confused.  Some cannot follow the explanation as to why it works, and some forget which fraction to invert.  Practice, though, should cement the process, even if they don't understand why it works.

Some teachers would get all worked up at this point--"understanding it is the whole point", they say.  To which I reply, no it's not.  I doubt most of those teachers could explain the 4 strokes of the internal combustion engine that powers their cars, but that isn't the point of driving.  The car is a tool, not an end in itself--and elementary math is exactly the same.

I won't get into how I demonstrated that the "invert and multiply" rule works, those many years ago when I taught junior high.  What I want to do here is show a slightly different way of doing the same problem.  Yes, it has a more difficult step than the process above does, but the payoff is a little more understanding of what the answer means.

So let's start with the original problem:

If students are learning how to divide fractions, they should already know how to add fractions--and hence, to get a common denominator.  So, let's rewrite the problem using a common denominator:

When students learned how to multiply fractions, they learned to "multiply straight across".  Now that we have a common denominator, it's easy to divide straight across.  No, you don't need a common denominator to divide straight across, but it doesn't buy you anything to do that step without a common denominator (try it if you don't believe me).

Since any number divided by 1 is that number itself, then

Generally speaking, students know that the problem above is asking "how many times does 5/8 go into 2/3 ?"  The number 16/15 doesn't seem to relate to that answer, even though it's correct.  By getting a common denominator, students can better understand the new question, "how many times does 15 (24ths) go into 16 (24ths)?"  Now, the 16/15 makes a little more sense.  Fifteen goes into 16 one time with a 15th left over.

This process has the advantage of using skills students have already learned (common denominators, "multiplying straight across", and dividing by 1) to get an answer to a new type of problem (division of fractions).  It required no new skills or rules, and it allows for an understanding of what the answer means.

If I were teaching division of fractions, this is what I'd teach first.  Then, after students become proficient with this interim process, I'd show them the "shortcut"--the "invert and multiply" process.

This video shows another method but doesn't at all explain why it works, when it's really just a variation of "invert and multiply":


Easy, but just a "plug and chug" with no understanding.

This video shows one of those ohmygawd methods--the process is so tedious that any understanding goes out the window:

Wouldn't 3 3/4 divided by 2 1/2 be unnecessarily difficult using this method?

The standard algorithms are standard because they're the most efficient way of to get answers manually.  The vast majority of the attempts to "make math clearer" fall prey to the point I made on the maillist--they are only clearer to those who can already do the math.

And that doesn't help anyone except the person trying to make a name for him/herself with some "new" method.

7 comments:

  1. "they are only clearer to those who can already do the math.
    And that doesn't help anyone except the person trying to make a name for him/herself with some "new" method."
    Right there, that's the whole problem. It's also what infects Education majors, in exactly the same way.

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  2. Anonymous9:35 AM

    When my daughter was in middle school, we spent an evening trying to help her multiply fractions using some method that involved a box of little squares. DH and I were finally able to reverse engineer the methodology since we actually knew how to multiply fractions.

    Finally, we showed our daughter how to multiply fractions using the standard algorithm. The shock and disbelief on her face as she said, "Why didn't you just show me how to that the first time?"

    Of course, we had no good answer.

    BTW, I have a Ph.D in a stem field. I don't know that i could explain today why you invert and multiply to divide fractions.

    My theory on math education for some time has been that math is difficult for some people, those people are writing the math curriculum, they feel that math should be difficult and confusing for everyone.

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  3. Send me an email, I'd be happy to demonstrate why "invert and multiply" works.

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  4. cthulhu12:19 PM

    “We see these things BECAUSE we already grok the underlying math: showing our patterns and techniques doesn't help people LEARN the underlying math.”

    Exactly.

    I’m an engineer in my 50’s; when I took Calc 1 my first semester of college (concurrently with analytical geometry and Phys 1, not a combination I recommend post hoc, but I didn’t know any better at the time), the highest math I’d had was trig, because that’s the highest math my very-rural-in-a-rural-state high school offered. I was good at memorizing rote procedures, so I got A’s all the way through PDEs a few semesters later, but I had to work a lot more problems in those classes to obtain mastery than I ever did in high school. But a few months after getting good on the rote solving, I found myself starting to understand the why’s of what I was doing, and that understanding deepened with more time and more skills. Our brains didn’t evolve to solve, say, line integrals; we co-opt existing functionality in our brains to learn how to do line integrals, then over time, we co-opt more to truly understand what we’ve learned. (This may not explain people like Newton, Euler, Gauss, Von Neumann, etc., but that’s ok.) But somewhere along the line - the roots are likely in the “new math” that many of my generation were pummeled with - the causality got inverted, as you said.

    I’m all for understanding, but understanding without proficiency is useless, and (unless you’re, say, another Gauss), proficiency only comes with long practice.

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  5. I am kind of amazed I was never shown this in school. Here's the generalized case, where you find a common denominator, then divide across. (I think I have the spacing working.)

     a       c           a·d        b·c              (a·d) / (b·c)          (a·d) / (b·c)              a·d
    ---  ÷ ---   =   ------  ÷ -------     =    -----------------   =   -----------------   =     ------ 
     b       d           b·d       b·d              (b·d) / (b·d)                  1                        b·c

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  6. cthulhu:
    "There is no royal road to geometry." --Euclid

    Auntie Ann: to prove the invert-and-multiply technique, start where you did to the left of the first equal sign and set the answer equal to x. Multiply both sides of the equation by c/d. Now multiply both sides of the equation by d/c. x now equals (a/b)*(d/c).

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  7. My first thought on the dot video is why didn't you do five dots per row. That would have been visually more understandable IMO.

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