Teaching Arithmetic
By Samuel Blumenfeld
Printed in Practical Homeschooling #42, 2001.
In my last column I discussed the genius of our place-value arithmetic system, how it consists of only ten abstract symbols - we call them numerals - with which we can perform any arithmetic function.
Counting
Arithmetic is a counting system, used to keep track of quantity. Thus, the first task is to teach your child to count. You can use pennies, for in the future your child, like everyone else, will be using arithmetic mainly for keeping track of money. Show your child that the numeral 2 stands for two units, or pennies; 3 for three pennies; etc. Show how a nickel stands for five pennies. Thus the child sees the numerals and the quantities they stand for and can articulate this information. One uses all the senses in learning to count. And let your child count anything that can be counted: days, weeks, years, birthday candles, marbles, pages in a book.
Show the child the number at the bottom of each page so that he or she learns the convenience of using numbers. Incidentally, numerals are merely quantity names. The word five, for example, means the sum of five units and position five in a sequence.
The Four Operations
Once your child has learned to count at least up to 50 or 100, you can begin to teach the four basic arithmetic functions: addition, subtraction, multiplication, and division, starting with simple addition facts. Each fact should be demonstrated with concrete items, so that the child sees the truth of the fact. Again, using pennies, demonstrate the truth of 5 + 6 = 11 by having the child count the five pennies at the left plus the six pennies at the right so that he sees that they add up to eleven. Do this with all the arithmetic facts, making a table of these facts, which the child will be required to memorize. Teach the facts in each column from top to bottom. Once they are learned you can mix them up to see how well the child has memorized them.
As we have pointed out, arithmetic is a counting system. In addition we count forward. In subtraction we count backward. In multiplication we count forward in multiples. In division we count backward in multiples.
Place Value
In order to demonstrate place value, you might construct a Hindu counting board with its columns denoting ones, tens, hundreds, thousands, etc. You might place pebbles, or pennies, or peanuts in each column so that the student can see how place value is derived from the counting board. Have the student write the quantities in each column on a piece of paper starting from the far right and proceeding to the left. Doing enough of these exercises will leave an indelible idea in the student's mind of what place value is and where it came from.
Memorization Works!
Rote memorization is the easiest and fastest way to master the arithmetic facts. You can make your own arithmetic tables, starting of course with the addition table. Have your child read each column in the table with the answers visible. Thus your child sees the fact and says it at the same time. To test how well your child is memorizing the facts, cover the answers with a slip of paper. Whenever the child hesitates indicating that he or she doesn't know the answer, don't tell the child to "figure it out," but quickly show the answer. And keep showing the answer until the child tells you not to show it because he or she has memorized it.
If we tell a child to "figure it out," he will merely resort to counting by ones. If that becomes a habit, it becomes an obstacle to developing a good arithmetic memory. The reason why you should show the answer is because that is the only way to create permanent memorized knowledge. For example, supposing you were in a play and had to memorize many lines. Supposing you forgot some of the lines. Would the director tell you to "figure it out?" No, you would see and reread the written lines so that by repetition you would remember them.
The simple fact is that our place-value, ten-symbol arithmetic system is a memory system. It requires memorization in order to be able to use arithmetic with efficiency.
How about calculators? Why have your child labor over memorization when the calculator can provide the answers? The trouble with using calculators instead of memorization is that the student won't know when he or she has made an error. The student will simply accept whatever number comes up on the calculator.
Thus, memorization must come first. Can memorization be made easy and enjoyable? Yes, if you tell your child of the powerful benefits of memorization. Besides, some of the facts involving ones, twos, fives, and tens are easy to memorize. Concentrate on those that are not as easy: threes, fours, sixes, sevens, eights, and nines. What could be easier than seeing the answer when one has forgotten a fact? In short, rote memorization is a technique, which makes best use of the mind's remarkable ability to remember what it sees and hears.
debh9876
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| I appreciate the reminder that at the end of the day, it is necessary to memorize basic math facts. However, I also believe it is critical when teaching math that kids understand conceptually not just memorize procedures.
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| Thu Jun 10, 2010 9:30 AM |
dkocur
Fort Worth |
Great article! And DebH, you make a good point as well.
There's a big difference between memorizing the math facts and memorizing procedures. Memorizing math facts is essential. Memorizing procedures is not only a waste of time, it prevents the student from learning. A close friend of mine struggled with math. I tried to help her when she was in high school, but all she knew to do was find an example problem and follow the same procedure. She didn't have a clue what she was doing.
Sadly, this has caused many teachers to over-react and stick to a "no memorization whatsoever" policy, which is the other extreme. Like I said, memorizing the math facts is essential. Studies have repeatedly shown that student's that attain "automaticity" (near instant recall of math facts) fair far better when they get to algebra and higher level math than those that don't.
Memorizing math facts doesn't have to be a chore either. Kids love computers, and there are a few computer programs like mine that do a really good job of harnessing that enthusiasm to get the kiddos to learn their math facts.
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| Thu Oct 21, 2010 5:20 PM |
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