Doing math multiple ways

On gifted forums, sometimes parents complain that the teacher says the kids have to do something X way, but DC gets the right answer doing it a different way.  So why should they have to do it X way when Y way is obviously working?

It’s kind of reminiscent of the argument that elementary schools no longer need to teach math because we have calculators now.

I disagree with that sentiment.  It’s important to do math multiple different ways.  There’s value in learning a different way to get the same answer.  You get a better understanding of how numbers (and later, symbols for numbers) are put together.  That leads to more accurate math, better estimates, faster calculations even without a calculator or pencil, and a greater knowledge of the possibilities of what can be done.

Even if we have computers that can do calculus, it’s still important to know how calculus works, because you know what is possible, you have ideas about what to try for things… and that’s even ignoring that math just makes you smarter.

DC1’s school just switched from Saxon math to Chicago math, but we’re doing Singapore math at home.  I’m glad ze’s learning the traditional computational methods at school (and we practice them in hir Brainquest workbook during summer and on the weekends), but I love love love that Singapore math looks at the same things in a different way.  For example, we just hit multiplication of 2 or 3 digits by a 1 digit number.  The traditional method ze’ll learn in school (and practice in brainquest) is to start with problems that don’t require any carrying.  Probably lots of x2 and x3 simple problems (23 x 2 = ?, 12 x 3 = ?), in order to cement the idea of multiplying the ones digit and then the 10s digit (and then the 100s digit another day).  Eventually they’ll introduce the concept of carrying (23 x 4 = ?).  (Then next year, the mechanics of double digit multiplication.)

The Singapore method, instead starts with some pictures.  It says, you remember when you learned multiplication how that was like having 3 rows of 4 balls?  And 3 * 4 = 12?  Well, what if, instead of each ball being worth one, that each ball is worth 10.  So you have 3 rows of 4 (10) balls.  (In pictures this is more obvious than in words.)  They’ve done the 10 ball representation previously with place value and with skip counting and x10s, so they’ve seen this idea before multiple times.  So 3 * 40 = 12 tens, and they know that 12 tens = 120.  Then they move on to 3 * 400 with the same pictorial representation.  Finally they finish up with 6 sample problems:  5*9, 5*90, 5*900, 9*5, 9*50, 9*500.  These last problems are set up in a way such that there’s pattern matching insights there for students who are good at getting insights from pattern matching, but it isn’t forced on kids who aren’t.  (At this point DC1 asked if 50*90 = 9*500 and 5*900.)  The next day moves on to 2 and 3 digit times 1 digit without carrying, but teaches it using these insights with the distributive property (13* 2 = 10*2 + 3*2), and this is not the first time they’ve seen the distributive property either– they’ve worked a lot with it with addition.  By the time Singapore math kids get to algebra a lot of tricky algebra concepts should seem pretty obvious.

I believe there’s value to being able to do math with both of these techniques.  They each provide different insights to how numbers are put together.  They each have different numerical problems for which they are the faster and easier method of solution.  In addition, the standard US method tends to be easiest when one has a pencil handy, whereas Singapore math is often best for mental math.  It isn’t that one technique is better than the other (though I confess that Singapore is more beautiful and I can see the sneaky ways it’s introducing higher level math while working with simple numeric problems, something beautiful in itself).

Being able to use multiple methods is even more valuable, however, than the sum of being able to use two individual methods.  Because of the insight given by seeing two different ways to solve the same problem, I would argue that the value of learning a second method isn’t even multiplicative, but instead exponential (or maybe factorial…)  Each new way provides a deeper insight into the magnificent world of numbers.

And, with that pattern matching turned on… if there are multiple ways to get to the right answer in math, maybe there’s multiple ways to get to a solution in other kinds of problems too.  If everyone had that particular insight, then maybe government policy wouldn’t be quite so messed up (a long shot, perhaps).

Do you think there’s a benefit to learning different ways to get the same answer?