Homeschool Encouragement


Thursday, February 9, 2012

When Math Manipulatives are too Abstract

Peter was rolling slowly (but successfully) through Right Start book B. 

011016Patterns are easy, Ma.

Then he hit a brick wall and it had zeros written all over it.

It was the thousands wall.

I started thinking about this and I realized that I shouldn’t have been at all surprised.

Did you get thousands when you were 6?

I have a feeling I didn’t either.  I think mastery-based math has its limits.

Peter is my super concrete kid.  It was only a few months ago that he started being able to add numbers together on paper instead of just pictures of objects or actual objects.

Now he realizes that 3+3, 2+2+2, 2+4, and 4+2 all equal 6.

Making that abstract jump from adding together 2 pictures of 3 balls to adding together 3 and 3 is an amazing feat.

It’s a developmental milestone.

So, yeah, he’s doing great.

But I saw the wheels start to come off when RS introduced hundreds.  And not just the idea of hundreds, but counting by hundreds.  And adding hundreds.

Of course, RS uses a lot of manipulatives, so kids can see the math, right.

One hundred in RS is represented by a complete abacus (which has 100 beads) or by one of these abacus tiles (there are other cards and place value cards we use, too):



There’s 100 beads on each tile, but you can see at a glance that there are 10 groups of 10 (100).  Easy, right?  Um, maybe.  But let’s look at that a little more closely. 

I’ll add that RS discourages counting.  So, when your child figures a math problem on the abacus, for instance, he wouldn’t count 4 beads and then count on 2 more beads to get 4+2, he would move over 4 beads as a group and then move over 2 more as a group and come to recognize it as the same as 5+1 which is self-evidently 6 since it’s one more than 5 (it’s actually not self-evident, but I’m not going to get into a philosophical discussion at this point, but I think that you can start to see what I’m getting at).

The point being that the child becomes accustomed to recognizing numbers on sight, including 10s and eventually 100s.  Once it’s been demonstrated that all the beads on the abacus equal 100, he’s supposed to come to know it.  And accept it.

Even if we accept that, how is one thousand represented?  With 10 of those abacus tiles.  Or as 10x100.

And it’s self evident, right, if we count by 100s?

But let’s step back a minute.  What if the whole concept of 1000s and even 100s is too abstract in and of itself.  Sure, we adults used to adding and subtracting big numbers when we balance our checkbooks (or faint at the country’s debt), but do we really have a firm grasp of what 100 or 1000 means?

When was the last time you counted 100 of anything?  And how often are you right when you try to estimate the number of candies in a jar for one of those guessing games (you know the kind I’m talking about, don’t you?)? 

The fact is, beyond about 10 or so of anything, it gets hard (really hard) for the average person to estimate at a glance how much of something there is, especially if they aren’t lined up nice and neat in a row.

Can you imagine what 100 M&Ms would look like?  How many do you think you get in a single serve bag?

Now, imagine that you are in 1st grade and can’t quite count all the way to 100 without making a mistake.

But you’re using abacus tiles to add hundreds and even thousands together?


What does it all mean?

The fact is that, to Peter, adding 2000+5000 is just adding 2+5 and putting 3 zeros at the end.  He doesn’t get the concept of 1000s.

And that shouldn’t really be surprising.  It’s simply too abstract.  The manipulatives are abstract (no one’s going to count all the beads on those tiles, are they?). 

And so, we put aside Right Start.  Again.  Perhaps I will be selling the curriculum in the near future.

For now, Peter will be working with smaller numbers, skip counting, adding multiple numbers, and so on.  There will be time enough for 1000s in the future.

Ever have to scrap your math program?

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From the Edge of Order to the Pit of Chaos?
Raising Boys:  A Look at Our Past