How can we teach people to understand what multiplication means?

Just as there are ways of working with your brain to learn to walk, sing, recite poetry, or work a puzzle, there are patterns of thinking for understanding mathematical concepts. 

The article: Promoting conceptual understanding of multiplication (Tzur, et al) describes a way to build the ability to think this way.  It describes the importance of "double counting" -- figuring out an increasing quantity *while being aware* of how many equal groups are accumulating. (I'm thinking it's a bit like learning to hear the tenors while singing alto. Some people just do it -- but many more of us *can learn it.*)   

It's not instantaneous.   These six sessions build from concrete, kinesthetic manipulation of cubes to thinking through the counting processes, and then translating the processes to diagrams, and then translating that into equations. Yes, some of us had opportunities that facilitated figuring that out; I ferociously love that these folks have designed a sequence to facilitate getting the right answers **with** the deeper understanding and not falling into "as long as they get the answers, they're succeeding!" thinking.  

It  reminded me of letterbox lessons for learning how graphemes map to form words and how phrases map into sentences with their connected meanings.  Teaching and practicing these processes helps internalize deep understanding and fluency. 

The link to the article is above but ... maybe you want to watch a quick video instead! Pardon my primitive graphics -- 

Part One -- Introduction

Part Two -- First two sessions 

Part Three -- Sessions four through six.