Who Are We to Question Why, Just Invert and Multiply!

At the end of last week's class, our instructor posed an interesting question...  She was talking about dividing fractions and, of course, we all know the rule to divide fractions is "invert and multiply".  That is, you keep the first fraction the same, invert (or flip) the second one, and change it from division to multiplication.  Like this:

(https://www.coolmath4kids.com/math-help/fractions/dividing-fractions)

But then she posed the question..."WHY?"  Why do we invert and multiply?  And none of us knew the answer right off the bat!  We were likely all thinking, "Because that's how it's done."  I mean, as the title of the blog says, "Who are we to question why, just invert and multiply!" Seriously though,    speaking for myself, I have never even really thought about why we invert and multiply to divide fractions.  So I decided to do some research and find out.  There is a wonderful page I found at Mike's Math Club (http://www.mikesmathclub.org/div_fractions.pdf) which explains it perfectly.  The images below are from that PDF.

Firstly, it's worth the question... If we can multiply fractions by going straight across, why can't we just divide fractions using the same method.  And the truth is...you can!  In fact, if the fractions have a common denominator, then it's a very efficient method!  For example:

And even if there isn't a common denominator, if the two numerators and two denominators divide nicely, then it's still faster to just divide straight across:

So, now then, what happens if the two numerators and two denominators don't divide nicely?  Well, you have two options:

1.  You could manipulate them to have common denominators, then use the method above.

2.  You could still divide them straight across using the following method.  Oh, and by the way, an "identity element" is something that, for a particular operation (like multiplication, division, etc), returns exactly the same answer that was there to begin with.  So the identity element for addition is 0; for multiplication, it's 1.

So there you have it! 

(http://www.funnyism.com/i/funnypics/when-you-finally-understand-a-mathematical-concept)

I truly love finding out the "why" of the simple math tricks we were taught in high school.  I actually felt quite excited when I realized that you totally can divide fractions straight across just like with multiplication!  Honestly, I'd just never really tried - when you're taught to "invert and multiply", or, as my math teacher said, "Keep Times Flip", it's just what you do.  But now I know why it works...and so do you!

Is the "New" Way of Teaching Math Really Better?

This weekend I attended a math conference as part of my "teaching math" course.  There were some really interesting things I learned, particularly regarding technology in the classroom, but there were also some things I really didn't like regarding how to teach math.

This post may end up being a big rant, but sometimes a good rant gets everyone thinking.  I want people to think about the way some teachers are suggesting that math should be taught, because I think it's just not working.  Allow me to explain...

At this conference were two very competent math teachers who had some very innovative ideas on how to teach math.  Unfortunately, I really just didn't agree with them.

What I want to focus on here is the concept of giving problems to students without giving them the tools to solve them just to see if students can innovate ways to solve the problems.  Here is an example that was given to us at the conference:

Mrs. Lin walks into the Sweets Emporium and buys 3 candies and 4 chocolates.  It costs her 26 cents.  You walk into the same store and buy 7 candies and 2 chocolates. It costs you 24 cents.  What is the cost of the candy and the chocolates?

This problem is intended to be given to students who are not yet well versed in the method of using algebra to solve this type of problem.  They are supposed to be given the candies and a bunch of pennies so they can figure out the answer visually.  This is all well and good, but as far as I'm concerned it's a complete waste of time.  Sure, we could give the students half an hour to mess around with the pennies until they come up with the correct answer.  OR, we could give them a 10 minute lesson on algebra, give them the tools they need to solve the problem, and then let them have it at - they'd be done in half the time!  The students who prefer algebra will solve it that way, and, to be honest, the students who aren't strong in algebra will solve it visually anyway.

They are giving students problems without first giving them the mathematical tools to solve them!  This is completely backwards as far as I'm concerned.

The thing is, if math had been taught to me like they're suggesting to teach it to students now (ie - backwards), I think I would have hated math...  I, the person who loves math, would have hated it.  And that's sad to think about.

It has been proposed to me that maybe I don't like this way of learning math because I wasn't taught that way, but I think that's just not true.  I can easily think of two examples of how I was taught in high school that could have been improved.

1.  In English we had to read Shakespeare.  I wasn't a big fan of the old-style writing and so I found it difficult to follow along.  Someone in one of my university classes the other day suggested an amazing method of using a diagram on the board to join different characters together visually with pictures and words to help people remember who's who.  I think this would have helped me immensely and would have been a much better way than just having us read the book out loud.

2.  In French classes in high school, we spent a huge amount of time on grammar and vocabulary, but very little time on conversation.  This, I believe, was a mistake.  We should have spent way more time on conversational French as this is what is needed to be able to speak the language well.  I think 50/50 would have been a good proportion.

So, there you go.  It's not just how you were taught that influences what you think works well and what doesn't when it comes to teaching.

The other big thing is, this way of teaching math is not how real life works.  When you get hired for a job, they don't sit you down, give you a problem and tell you, "Now we're not going to give you the tools yet, we're going to see how well you do on your own," then come back a half-hour later to see how you're doing with your problem solving and then give you the tools.  NO, they will give you the tools, equip you as best as they can, then let you do your job (time is money!).

Some may say that this method of having students problem solve without the tools may help in everyday life. But I beg to differ.  If you have a loose doorknob, you're not going to MacGyver a solution.  You're going to either Google it, or you're going to get yourself to Home Depot to ask an expert what to do.  And what will they say?  Not, "Why don't you go home and ponder it for a while.." No, they will lead you to the tools necessary to fix it!

Anyway, I'll end this rant now.  To be honest, I know these teachers are very good at what they do, and they wouldn't be teaching us this method if it didn't work for their own students.  However, for the reasons mentioned above, I must, respectfully, disagree.