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THE MATH WARS - 1960's REVISITED
So the National Science Foundation expects to contribute $20,000,000 to Milwaukee to improve (?) math instruction. The catch is that there is vast disagreement on how math should be taught. The same types of intellectual battles were being fought forty years ago when most of the public schools cast aside old-fashioned arithmetic and adopted what was called, 'new math'. At that time I accepted the challenge of teaching math in the elementary grades at the two-year-old Academy of Basic Education, now called Brookfield Academy in Brookfield Wisconsin.
After talking to some professors and checking out various textbooks and systems I decided old-fashioned arithmetic made the most sense. I selected some not-so-new textbooks called the Strayer Upton series published by the American Book Company. We had the kids learning their addition and subtraction facts, reciting multiplication and division tables, and working hundreds of regular and word problems to prove they knew what they were doing.
As the school grew, more teachers were hired, and some shared the chore of teaching arithmetic. One young teacher, fresh out of college, decided my methods were abominable. He did not discuss the matter with me, but composed a memo which he sent to the headmaster and circulated among the other teachers. I considered the issue important enough and sufficiently well-stated to require a response. I believe the issue is just as important today as it was forty years ago. Here is the young teacher's memo, and following it, my response:
The advent of the lockstep army has made drills of all types, including mathematics, quite popular; but even this unwarranted popularity has not made these drills less dull or less useless. A soldier is able to memorize the physical and mental necessities of the lock step, and a child is equally able to memorize his tables and flash cards. The chance that either of them will ever act independently, i.e. think, in their respective concerns is decreased many fold by so doing.
Not only does drill require no thought on the part of either the student or the teacher, it is dull and, with very few exceptions, leads to disinterest in or dislike for math. The exceptions are the children who can learn under any circumstance, and the children who have a special penchant for memorization. This is a great waste, because any child can be taught the fundamentals of math, in my opinion. The key to teaching these fundamentals is not drill; the key is understanding. Without understanding, these drills mean nothing. It does not make any difference that an eight year old child can do 100 addition combinations in three minutes. It does make a difference that he understands what he is doing and can use it for something besides making 100s on speed tests. Even a parrot can learn to say that 2 plus 2 is 4.
How does one encourage understanding in a youngster? And how is this different from drill? I do not attempt to prescribe methods which will work in every case or even in a majority of cases; let that be quite clear in the beginning. Since the goal, as I see it anyway, of teaching addition, say, is its application in summing columns of figures or solving problems, early introduction of these applications should be a boon. By the time a student has figured the totals on 250 columns of figures, he is bound to have a good notion of the sum of 12 and 5, by merely having repeated it 30 or so times. Epstean's Law (people will always take what appears to them the shortest way home) will see to that. It is easier to remember that 7 and 5 are 12 than it is to count it out on one's fingers thirty times.
Counting is another useful method of developing mathematical acumen. Counting forwards, backwards, by 2s, 3s, 8s, or 37s not only gets the sense of the order of math across to those who can manage the counting (very few can't), but it adds variety to a class, and can be used to sharpen talents already developed in these areas. If speed with combinations is desired, a little oral work or a brief competition every day (say 20 minutes) stressing quickness will show surprising results with no memorization required by the teacher.
I have suggested three possibilities--they by no means exhaust the avenues open to the imaginative teacher. That is half my point. Other methods do exist. The other half of my point is that other methods exist which should achieve better results than does the drill method.
As I said before, I don't presume to have all the answers, and this is not a memo to propound answers. I have been concerned about the lack of interest in mathematics and lack of mathematical understanding I have observed in many of our students, and have spent some time trying to understand why this is so. You have read what I have thus far come up with. I offer it merely as an opinion, which I do not wish to argue about. I do not suggest that anybody incorporate these suggestions in his math class. I do suggest he might have better results if he did.
Whenever anyone states an opinion it is an invitation to an argument. If he states an opinion and claims to want no argument, he is merely saying, "Here's my opinion. I'm not interested in yours." I'm certain George did not intend this, so have decided to accept the invitation instead of the gag. Besides, I always feel I have paid someone a compliment when I consent to argue with him. Although I'll admit not everyone accepts it as such.
I've never been in the army, so the fact I require my arithmetic students to memorize simple addition, subtraction, multiplication and division combinations is not due to a love of drill or a love of army-type discipline. The students memorize because they need to have these facts instantly at hand if they are to be successful when they get to working with large numbers. In every case I have observed where the student is doing poorly in arithmetic, he is also a plodder who has to spend much of his mental energy trying to figure simple totals, products, etc...
It is no great task to have a child understand why 2 plus 2 is 4; 3 minus 1 is 2; 5 times 5 equals 25. There are lots of ways to do this, and most children see almost immediately that these things make sense. There's no particular advantage in agonizing for 'deeper understanding' when there isn't anything very deep about it. Few children need or want any more than to have the knowledge confirmed within themselves.
Are we creating parrots because we ask children to memorize the facts? No! Teach children to say 2 plus 3 equals 8; 1 plus 2 equals 4; 2 times 8 equals 7. If you succeed in this, you have created parrots. When you teach 2 plus 2 equals 4, children's minds and eyes check the results. They are not just memorizing, for they can see that this is true in many demonstrable ways.
An arithmetic which did not demand lockstep approval, if you please to put it that way, of 2 plus 2 equals 4; 8 plus 1 equals 9; 2 times 8 equals 16, etc. would be completely useless for business or scientific purposes. While it may not necessarily be true that the facts of arithmetic are unassailable eternal truth; it is true, nevertheless, that arithmetic, like language, is a way of understanding and communicating. If we cannot agree on certain basic premises, we might as well forget about the whole business.
I would like my students to learn the basic facts so well that conscious thought is no longer exhausted on them. No sense wasting a child's imagination and independence on such trivialities.
Choosing the method that is used to entice the students to learn the elementary facts is an individual matter--to each his own. Drills, competition, problems, counting, demonstration--most good teachers will use all to some extent, and it does quite often require at least some originality to keep the students on their toes.
In my experience it has not been the children who have drilled and learned the facts who hate math, but the ones who have not. They, therefore, find every simple problem requires tremendous effort; and more often than not answers are wrong because of a lack of ready control of the facts.
Perhaps Epstean's Law is correct, and we all take what appears to be the easy way; but to 99% of the children, unless they are taught otherwise, counting on their fingers appears to them to be the easy way. I suggest there is another law which states the fact that people always look to short range advantage unless they are taught it is to their greater advantage to take the long view.
I may have been walking in my sleep, but I must confess I have observed no widespread hatred or boredom with math at our school. In a great many cases I have seen outstanding ability and great interest. I do not claim we have made math lovers of all students. I don't care how you teach the subject, there will always be a fair percentage who merely tolerate it because it is required. It should not be our goal to make great mathematicians out of all students; or for that matter great scientists or linguists. Let's give them as much as we can in every way, and let the students find out for themselves where their interest will eventually be directed.
This is my opinion. If you want to fight about it, welcome to the ring. There's nothing I enjoy more than a good mental duel with someone who is willing to go at it without pulling his punches; and who is sufficiently charitable to allow me to do the same.
NOTE: This was the last I heard from anyone about cutting out the 3-minute drills for learning arithmetic facts. Thankfully, no foundations with donations were trying to bribe any of us to change to what was called 'new math.'
Computers have made a difference since the 60's, but understanding arithmetic still requires more of a foundation than learning how to push the right buttons.
� 2003 Erica Carle - All Rights Reserved
Erica Carle is an independent researcher and writer. She has a B.S. degree from the University of Wisconsin. She has been involved in radio and television writing and production, and has also taught math and composition at the private school her children attended in Brookfield, Wisconsin. For ten years she wrote a weekly column, "Truth In Education" for WISCONSIN REPORT, and served as Education Editor for that publication.
Her books are GIVE US THE YOUNG--$5 Plus $2.00 P&H WHY THINGS ARE THE WAY THEY ARE--$16 PLUS $4.00 P&H BOTH BOOKS -- $25 Total. A loose leaf collection of quotes titled, SIX GENERATIONS TO SERFDOM is also available--$15 Plus $2.00 P&H. Mailing address: Erica Carle; PO Box 261; Elm Grove, WI 53122.
"Are we creating parrots because we ask children to memorize the facts?"