I’m investigation possible models for learning multiplication. Below there are 6 possible models. I’d like to know if you know of more models. I’ve found 12 different models for understanding/doing multiplication over at Natural Math, but have not found explanations of all of these that I can easily share.

Interestingly enough, many of these videos define multiplication at the beginning of the video with no sense that there are other possible definitions.

I’m detecting a difference between Natural Math’s models and your list of six items. The lattice method strikes me as more of an algorithm than a model. Same with binary multiplication, The key difference is that an algorithm is a method to get the answer, where a model is a situation where multiplication is used. Some items can appear under both categories, such as repeated addition.

If the different algorithms presented here are used to multiply, and students look for reasons why they work, they become models for understanding multiplication. Even the standard multiplication algorithm has some use as a multiplication model as well.

I’ve spent a lot of time in the last 20 years thinking about multiplication and talking to kids about it. My quest to demystify multiplication started when I was leaving an elementary afterschool program where I volunteered. A third grader said to me, ” I just don’t get those times things.” I asked him to explain, and he said, “Like 2 times 4 or 3 times 3.” I asked him what 2 four times was and he said 8. I asked what 3 three times was and he said 9. I turned the words around, and he knew just what to do. He could multiply. He just didn’t understand the vocabulary.
Last year I decided to make a multiplication game with a 100s chart, and thought I would indicate numbers that were NOT products of multiplying 1 through 10 by making them gray. I started to change the font color on number after number and discovered that there are only 42 possible products and that nearly half of them are one-digit or multiples of 10. How many hours of frustration do kids suffer trying to learn so few answers? Here’s a link to the 100s chart – http://www.suedowning.blogspot.com/2012/03/is-that-all-there-are.html
After being in classrooms where multiplication is taught with little sayings, I determined meaning not memorization is the way to go. Examples: You have to be 16 to drive a 4×4 or I ate (8) and ate (8) until I was sick on the floor (64). When it is time to learn division, many students in these classrooms are clueless.
Less instruction and more discovery (CGI math) can lead to great reasoning and understanding. http://www.suedowning.blogspot.com/2012/03/marvelous-math-play-date.html
I would teach multiplication by starting with the answers and letting kids discover the facts by creating groups of objects and playing games. I have had great success with these two free activities – http://www.teacherspayteachers.com/Product/Meaningful-Multiplication-Two-Free-Activities-to-Demystify-Multiplication

Thank you for your insights Sue. I have to admit, I have never taught multiplication before, so I post my stuff up here to get feedback from people who have, so thank you SO much.

My observation is that all of mathematics is like this. You need a conceptual framework to hang the concepts on so that when your memory of a procedure fails, you have the ability to reconstruct that procedure as needed. Learning how to learn, in a math context.

Thanks so much for your reply, David. My goal in life is to help students discover the joy of playing math so they develop a conceptual framework that allows them to construct the procedures in the first place. Learning just for the fun of it. What could be better! ;^)

I’m detecting a difference between Natural Math’s models and your list of six items. The lattice method strikes me as more of an algorithm than a model. Same with binary multiplication, The key difference is that an algorithm is a method to get the answer, where a model is a situation where multiplication is used. Some items can appear under both categories, such as repeated addition.

If the different algorithms presented here are used to multiply, and students look for reasons why they work, they become models for understanding multiplication. Even the standard multiplication algorithm has some use as a multiplication model as well.

There is a great article by Brent Davis on the different conceptions of multiplication.

Is 1 a Prime Number?

http://brentdaviscalgary.appspot.com/articles.htm

I’ve spent a lot of time in the last 20 years thinking about multiplication and talking to kids about it. My quest to demystify multiplication started when I was leaving an elementary afterschool program where I volunteered. A third grader said to me, ” I just don’t get those times things.” I asked him to explain, and he said, “Like 2 times 4 or 3 times 3.” I asked him what 2 four times was and he said 8. I asked what 3 three times was and he said 9. I turned the words around, and he knew just what to do. He could multiply. He just didn’t understand the vocabulary.

Last year I decided to make a multiplication game with a 100s chart, and thought I would indicate numbers that were NOT products of multiplying 1 through 10 by making them gray. I started to change the font color on number after number and discovered that there are only 42 possible products and that nearly half of them are one-digit or multiples of 10. How many hours of frustration do kids suffer trying to learn so few answers? Here’s a link to the 100s chart – http://www.suedowning.blogspot.com/2012/03/is-that-all-there-are.html

After being in classrooms where multiplication is taught with little sayings, I determined meaning not memorization is the way to go. Examples: You have to be 16 to drive a 4×4 or I ate (8) and ate (8) until I was sick on the floor (64). When it is time to learn division, many students in these classrooms are clueless.

Less instruction and more discovery (CGI math) can lead to great reasoning and understanding. http://www.suedowning.blogspot.com/2012/03/marvelous-math-play-date.html

I would teach multiplication by starting with the answers and letting kids discover the facts by creating groups of objects and playing games. I have had great success with these two free activities – http://www.teacherspayteachers.com/Product/Meaningful-Multiplication-Two-Free-Activities-to-Demystify-Multiplication

Thank you for your insights Sue. I have to admit, I have never taught multiplication before, so I post my stuff up here to get feedback from people who have, so thank you SO much.

My observation is that all of mathematics is like this. You need a conceptual framework to hang the concepts on so that when your memory of a procedure fails, you have the ability to reconstruct that procedure as needed. Learning how to learn, in a math context.

Thanks so much for your reply, David. My goal in life is to help students discover the joy of playing math so they develop a conceptual framework that allows them to construct the procedures in the first place. Learning just for the fun of it. What could be better! ;^)