Manipulatives for Everyone

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Many years ago, it was discovered that far too many elementary students were being taught mathematics by giving them rules.  Some teachers motivated the rules with examples (See?  It really works!), others just insisted that the kids learn the rules, and use them.

Presently, it was decided by at least some authorities (and all backed up with research), that it was better for a child to learn a method slowly, using materials that worked like money, for instance, than to simply teach the child rules of addition.

In the case of place value, the big idea in representing numbers, this works as follows.  You first explain that all numbers must be represented by dollars, dimes, and pennies.  Ten pennies are worth one dime, and ten dimes are worth a dollar.  Okay; that certainly is a rule, but it is a definition, from the real world, so it isn't too illogical.

The next step is to convert every number of one, two, or three digits, into the least possible number of dollars, dimes and pennies; e.g. 123 = 1 dollar, 2 dimes, and 3 pennies.

Now you teach addition. 123 + 456 = 5 dollars, 7 dimes, and 9 pennies.

Next you teach carrying, which is merely trading 10 pennies for a dime, etc.

Base-Ten Blocks.  Units are represented
by the little cubes (yellow, in this set),
Tens by the rods (green, in this set),
and Hundreds by the slabs (blue, here),
and thousands by the big (red) block.

Teachers who are adept at this can diagnose which part of the procedure a kid has failed to get.  For instance, if a student gives a response to the problem 99+43=1312, you can tell that there is difficulty with the concept of carrying, but the actual addition is proceeding moderately well.

People outside the teaching profession take the concept of "Carrying" for granted.  A student taught with Base Ten Blocks, which is the classroom equivalent of money, knows exactly what carrying is.  It is more important for an elementary student to understand how what we call place-value notation works (an Arab invention, we believe), than to be able to use rules fast.

Calculators

Some "experts" are of the opinion that if kids can use calculators from the start, they will figure it all out someday.  I don't think this is either true or false; the big question is whether it will actually serve the child better in the long run.

What got me started on this harangue is a recent blog criticizing how students in some states were being tested for the so-called Common Core.  Some of the questions were undeniably bad, but a large part of the criticism was based on the critic completely missing the point of the question.  Unfortunately, some of the teachers inventing creative questions were also a little off the mark.

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