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[quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous]What in the world is a "brittle understanding of multiplication"

It's really not a hard concept at the basic level.  Sure, eventually there is more to learn, but understanding the basic concept (ie, repetitive addition) isn't a hard one, and the vast majority of 2nd or 3rd graders, and most 5 or 6 year olds, can understand that.  
I can't begin to imagine a problem with a kid memorizing their times tables after a 10 minute explanation of what the x means.  it doesn't mean that they won't learn more and develop a deeper understanding.  But it's crazy to pretend that somehow by having quick recall of multiplication tables that you are hurting their math ability.[/quote]

If your child memorizes the times tables up to 12x12, that's great.  It will help a lot.

AND

At the same time she will also need to understand the concepts enough that she can do 12x17 in her head.  Quickly.

A child who can do the first but not the second is to me an example to me of one who has a "brittle understanding of multiplication."  It's not just about the basic concept of repetitive addition.  It's about understanding the concepts enough to think on your feet and apply strategies to solve problems you haven't seen before or memorized.

A well-educated child needs to do both.  Memorize the tables and have an ability to apply the concepts well beyond the tables.

Along those lines, I also agree with the PP who said that the timing of memorization depends on the child.  Some can memorize them early and still tune in later in class and even have fun playing with the basic concepts and manipulations, even though they can immediately call up the "right answer" from memory.  Others who already have done the memorization feel crazy bored by all the repetitive "process" work and tune out because they already "know the answer" from memory.  Those are the kids who miss out on the deeper learning and struggle to catch up later, even though they seem ahead in third grade.
[/quote]

What is your "quick way" to you of multiplying 12 x 17? I can think of several, but maybe there's a quicker way I haven't thought of that should have learned.[/quote]

New poster but isn't it obviously 12 x 12 + 12 x 5 ?[/quote]

Yes. And this is a easier method than base ten for anyone who has memorized through 12x12. [/quote]

Cool. I'm the one who suggested base 10 above. For whatever reason that's what came to mind for me first. But your way is easy and intuitive, too.

I think that's the whole point, really. Numbers are great. There are so many different ways to get to the same answer, and different paths/strategies will resonate more with different kids.

So we want to do whatever we can to help them manipulate and play with numbers so they can learn many different ways to get to the same end point.  In part so they have different tools in their arsenal, and in part so they go into higher math with confidence that they can break down complicated problems into familiar parts in order to solve them. Shortcuts and memorizing math facts are helpful but ultimately not nearly enough. [/quote]

I find this fascinating, actually. I'm an engineer, so no stranger to math. But I was shocked when we started working on the multiplication tables with DS (rising 3rd grader) over the summer, at his school's request. After we got through the "easy" ones (2, 5, 10), we threw 6x8 at him, just to see what would happen. He thought for a few minutes, counted on his fingers, and announced 48.  Huh?!?!?

We asked him to explain what he did. He told us that 5x8 = 40 (got that by counting by fives, using his fingers to keep track, until he got to 8 5s).  Then another 8 would be 48.

He still uses that method, although he knows more of them by recall now just through repetition. But I love that he has this fallback method to figure things out, and really understands what is going on. I learned the old-school way of sitting down with a huge matrix and running flash cards until I was blue in the face, so I think this way is awesome! [/quote]

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