One nice example of what people did was making change. A cash register added up the total--but did not tell you how to make change. Instead of subtracting people counted up. If the total was 78 cents and I gave the clerk a dollar, the clerk would give me two pennies and then two dimes, counting up 79 cents, 80 cents, 90 cents, and finally one dollar. The actual process of counting out change in this way requires no more effort than it would require today to count out 22 cents after a modern cash register did the subtraction and told us what change to give.
Not only is there less need for traditional arithmetic today--there are also fewer situations in the real world where these skills are reinforced. We cannot expect the students of today to be as strong in this area as students were in the past. But there still is a place for traditional arithmetic.
Suppose two movie tickets and a $3 bag of popcorn costs $13. How much does a movie ticket cost? Without the popcorn you pay $10 for two movie tickets. Dividing by 2 we get $5 as the price of one movie ticket.
This solution is as clear as it is because we can do the arithmetic easily in our heads. If we needed a calculator to find 13 - 3 and 10/2 it would interrupt the flow of the argument. That would be fine if we already understood the process. But if we had never seen a solution like this before, that added distraction would make things harder for us.
One benefit to traditional arithmetic background is this instant access to certain simple basic facts which can be incorporated into when a certain type of problem is first being learned. This allows us to place the greatest possible focus on the algebraic parts of the problem with minimum distraction from the arithmetic part.
A second benefit comes from the fact that algebra in many ways generalizes what we do in arithmetic. Familiarity working with numbers expressed in powers of 10 provides a background that makes it easier to understand working with polynomials expressed in powers of x or some other variable.
Carrying and borrowing are major problems for many people working with whole number arithmetic. These problems are as bad as they are because we rush students into doing problems that require carrying and borrowing, and we do not treat the subject in a sufficiently flexible manner. This is particularly true today when students do not have the mastery of basic facts that students once had.
What we need to do is work with problems that allow students to place nearly all their focus on carrying or borrowing with as little distraction as possible. This can be done if we add or subtract relatively small two digit numbers ending in either 5 or 0. The addition or subtraction in the ones place is very simple. Addition or subtraction in the tens place is fairly simple. Students can place a maximum of focus on the carrying or borrowing part until they become familiar with the process. With problems this simple, students can learn to carry or borrow without ever writing down little carry or borrow numbers. The process is learned with problems simple enough that writing such numbers is not necessary--and once the process is learned in a simple setting such as the one I described, slightly more complicated settings can be considered, such as two digit numbers ending in 2, 4, 6, 8, or 0. It is easier to never learn to write these little numbers than it is to first learn to use them, and then learn to stop using them.
But in a age of calculators, mental math may be more important than paper and pencil techniques. There are different methods for carrying and borrowing in your head. When I add 365 and 366 in my head, I don't start with the ones place--I start with the hundreds place, 3 hundred and 3 hundred make 6 hundred. Sixty and sixty make one hundred twenty for a total of 7 hundred twenty, And finally 5 and 6 make 11 bringing the total to 731.
Subtraction is similar. To find 82 minus 38 I start with eighty minus thirty and I get fifty. Then I try to do 2 - 8 but I am 6 short so I take 6 away from fifty to get 44. Even if our accuracy is less than perfect, being able to visualize simple arithmetic problems in this fashion can help follow algebraic examples as long as the arithmetic isn't too messy.
Another area of difficulty for many students is working with fractions. The rules for adding fractions can be made clearer if we write out the denominators in words. We cannot directly add 1 half and 1 fourth because we would be adding different things--apples and oranges. But if we re-express 1 half as 2 fourths, then we can add 2 fourths and 1 fourth and clearly the answer is 3 fourths.
If we are asked to add two unlike fractions, how can we re-express the problem in a way where we are no longer trying to add different things? How can we make it into a problem where we have like fractions? We can start working with halves, quarters, and eighths--which are the fraction we usually encounter. In this case, the largest denominator will always work--and there are just a few conversions we need to learn. Once we become comfortable with the need to re-express a problem using like fractions, we can worry about the situation where doing so is more of a challenge.
Too many people are misinformed by teachers and textbooks that tell them they need to find the lowest common denominator--and then they are given a complicated procedure involving factor trees and prime factorization they are supposed to use to find the necessary lowest common denominator.
You don't really need to find a lowest common denominator. Any common denominator will work--and you can always find a common denominator by multiplying the two given denominators together. Lower denominators are usually easier to work with--but experience will usually help you find a lower common denominator when there is one, and it is not really that important if you don't use the lowest possible common denominator.
In conclusion I would like to mention one neat little trick that you can do when you subtract mixed numbers. What is 5 and one third minus 1 and two thirds? 5 minus 1 is 4. One third minus two thirds leaves us short one third. This leaves us one third short so we wind up with one third less than 4 or 3 and two thirds.
No comments:
Post a Comment