## Tuesday, March 9, 2010

### Which is harder and why?

Which is harder for you to solve - the first, or the second?  Again, the first or the second?

Which is harder for you to solve - the first, or the second?  Again, the first or the second?

What am I asking you? And why? I'm asking about your perception of the difficulty of two math processes that are taught to elementary school kids. Subtraction in the first case, division in the second.

What makes a problem difficult? We judge them in several ways:
• My examples (subtraction, division) are what you might call negative operations, in that the result (remainder or quotient) is smaller than a number you started with.
• A positive operation (addition and multiplication) gives you a result (sum, product) that is larger than a number you started with.
The negative nature of subtraction or division does not in itself make a problem difficult, even if it might seem psychologically less pleasant than adding. Notice:

I have ten dollars. I am given ten more dollars. I now have twenty dollars. Yay :-)

seems a bit nicer than

You have forty dollars. You have to pay ten in taxes. Now you have thirty dollars. Boo :-(

It's not this negative nature, but the deficit spending component of some problems that makes them harder. You might have to borrow (or carry, or regroup).

Some problems require no borrowing. Let's look at \$6.77 - 2.67

a. 7 minus 7 is 0, 7 minus 6 is 1, 6 minus 2 is 4 = \$4.10

Some problems require a lot of borrowing. Look at \$93.61 - 41.95

a. 1 minus 5 is [Hold it! I need 10 cents. Hey 6 can I borrow from you?] now 11 minus 5 is 6.
b. Now 6 minus 9 [Wait, the 6 is a 5; I need a  dollar. Hey 3 can I borrow from you?] now 15 tens minus 9 tens is 6 tens.
c. The 3 dollars is 2 dollars [due to our previous borrowing], minus 1 dollar leaves 1 dollar.
d. Finally, 90 dollars minus 40 dollars is 50 dollars = \$51.66

Some are low levels of multiplication (6 into 7 goes 1 time with 1 left over) and others a higher level of difficulty (6 into 42 goes 7 times with 0 left over).

These are still easy. Let's go up a notch. More remainders, even in the answer.  More decimal places in the dividend. Decimals in the divisor....

Do you wonder where financiers and politicians learned about borrowing? We taught them in school.