X·Y Points

I’m annoyed with education in our culture. I guess it’s the Olympics: these wonderful athletes from around the world causing such universal feelings of joy with their amazing performances.  These performances are the product of natural physical gifts and much mindless drilling.  What could be more monotonous than swimming back and forth in the same indoor pool day-in and day-out?  Think of the number of laps that Michael Phelps swam in preparation for this year’s Olympics.  Good that he had a coach who understood the importance of training and not an educator  who promoted “self-esteem” over performance.

Consider the practice routines of the divers, the gymnasts, the runners, the tennis players, and all the others: careers built on “Drill and Kill”.  Yet they sure seem alive while they’re performing.  It’s sure thrilling to watch them.  Maybe we should change the slogan to “Drill and Thrill”.  Maybe a thrill is a more likely outcome of drilling than death.

Of course it is.  In learning math practicing the basic facts until they are mastered forms a foundation for the later thrills of solving difficult problems or finding new mathematical truths.

The two dimensional addition table is compact: the addition symbol appears once, the numbers added once each, and the equal sign not at all. To find the sum, however, you go on a little journey: from the top left corner advance to the right to the column of the first number, take a 90 degree turn down to the second number’s row. At the intersection of this column and row you find the numerals for the sum.

With the sequentially ordered list of problems, you simply scan the list until you find the desired equation. The sum is to the right of the equal sign.

Both addition tables offer the learner an orderly and complete picture of the facts to be mastered. Either may serve as a reference during the process of memorization. Once this process is finished and instant recall of any of these facts is acquired,  the student will no longer need the external table: it is now etched on the mind.

Multiplication is a form of addition. A product is the result of adding the same number a certain number of times.Here too we have the single digit facts to master. Their orderly presentation in a table or list corresponds to that of the addition facts just described, so that finding the product matches the method to find the sum.

The two dimensional times table, however, provides an additional representation of the product that is invaluable to the visual learner. The abstract symbol of the product is found where the column and the row of the two factors intersect, but also the area of the rectangle thus formed is 56 square units, which you can see! When you find the product of 7 times 8 you see the symbol 56 but if you consider the rectangle formed by this motion, you see the 7 columns of 8 unit squares that add up - - repetitive addition - - to 56 unit squares, the area of this 7×8 rectangle.

Multiplication Table

The addition table lacks this. The sum of 9 + 9 is 18, yet its rectangle contains 81 square units.  This is a significant shortcoming.  It does not add up visually!

Did anyone’s grade school math teacher address this?  Please comment about your experiences with using these addition and times tables.

Arithmetic begins with counting, which in its simplist form is the repetitive adding of the number one.  The next important operation is adding two single digit numbers.  The number of possible ways to add the two single digit number totals 81 and these can be best represented in the form of a table.  There are two traditional forms of addition tables. First is the following listing of the possible operations:
 1+1 = 2
 1+2 = 3
 1+3 = 4
  …
  9+9 = 18
This presentation is complete and orderly, but very redundant: notice the repetition of the “+” and “=” signs and the repetition of the digits.  A more efficient table is the two dimensional array that follows:

Here’s the picture.

 
It will take a multitude of postings to explain it.