Teaching concepts and not JUST procedures

Last week, our district hosted some PD based on getting 5th grade and 6th grade math teachers together and collaborating. During the workshop, the organizers emphasized again and again the importance of teaching mathematics on a conceptual level as opposed to just teaching procedures.

Here's an example: if I want to see which fraction is bigger, 13/24 or 7/12, I can use the butterfly method (also known as cross multiplication) to multiply 13 by 12 and 24 by 7 and compare the results. But if I do this, I am not really engaged with the fractions at all. I haven't learned anything new about either fraction, and my answer wouldn't help me put either on a number line. If I at least rewrite the second fraction and realize that 7/12 = 14/24, I have a natural way to determine which fraction is bigger that uses the fractions themselves and not somewhat unconnected whole numbers. Here, the butterfly method is a procedure that may obscure the conceptual understanding that comes from using equivalent fractions.

Well, once I started thinking like this, I started seeing procedures trumping conceptual understanding a lot more. For example, there's a great paper in the NCTM journal Mathematics Teaching in the Middle School called Saving Money Using Proportional Reasoning. It's by Jessica de la Cruz and Sandra Garney and outlines tasks that would make a great project after discussing rates and proportions. Instead of emphasizing cross multiplication, the authors emphasize unit rates and equivalent fractions (which are both hugely important in 6th grade Common Core). Here's a great picture from their article about why cross multiplication can give confusing answers:

What exactly is a dollar-pound?

I was doing some more searching on the potential harmfulness of teaching cross multiplication and found the wonderful site www.nixthetricks.com and the free 83 page book that you can download here. It's written by Tina Cardone, and it's full of tricks that math teachers use, the potential harm those tricks may do, and how you can reteach using conceptual understanding. I particularly loved this flowchart on determining if something is a trick:

How often do you hear "Because ____ said so"?

I'm pretty sure I will switch up how I teach dividing fractions just because I read this book (and hopefully will avoid over-reliance on "Keep-Change-Flip"). If you are looking for deeper understanding by your students, this is a good place to start.