The Story of an Experiment is another interesting story from Perla (see previous post), which she posted to support her contention that children should not be taught arithmetic, except as connected with real life experiences, until they are at least ten years old. I write about it here, not because I agree with her, since I most emphatically do not, but because the story nonetheless makes some excellent points.
In the early 1930s the superintendent of schools for Manchester, New Hampshire tried an experiment—several experiments, actually. Essentially, he abandoned math as it was taught in the elementary schools, and concentrated instead on language and logic: reading, reasoning, and speaking, with arithmetic introduced only as it came up in the course of the rest of the studies. The results, as he reports them, were spectacular, with the "new curriculum" students far exceeding the abilities of their traditionally-taught age-mates, even in mathematical reasoning, and where they were behind (in basic arithmetic manipulation) they caught up quickly when formal study was introduced in sixth grade.I don't believe for a minute that the secret of their success was in what the students didn't learn, but what they did. Reasoning is a skill, and those who have been taught to think logically can be expected to out-perform those who have not. Before considering that the early teaching of mathematics will "dull and almost chloroform the child's reasoning faculties," one would have to see the accomplishments of those who learned both reasoning and the multiplication tables.
What I suspect is going on here, besides the laudable emphasis on logic and language, and the deliberate connections made between learning and real life, is that the old curriculum was teaching math as if mathematics comprised rote arithmetic skills alone, and these were taught—most likely by teachers with no understanding of the math themselves—with no context, no grounding in real life. Thus the children learned nothing about math, but only how to plug numbers into formulas, and that poorly.
Real learning does not paralyze the brain, but fear does. The errors described in this article are of the same kind and degree as those made by the bright fifth graders John Holt describes in How Children Fail, which he attributes primarily to the fear that comes of seeing math as irrational and incomprehensible—just a collection of magic formulas into which one casts numbers and hopes for the best. In my own experience, I remember being taught what seemed to be a dizzying array of disconnected formulas for finding surface areas and volumes of various solids, none of which would stick in my mind even long enough to make it through a test. Only later, when I made the monumental discovery that math actually could be expected to make sense, did I discover that one could derive most of the formulas with a little thought; then the fear went away and I could even remember most of them. :) Similar fears assailed me in the conversion of temperatures between Fahrenheit and Celsius; I could never keep the formulas straight. Again, a little logic (and the knowledge of a few data points, such as 32°F = 0°C) fixed that problem—but I had been taught not to think, but to apply formulas blindly.
So I'm sure the superintendent's experiment helped the children's mathematical reasoning, not by keeping them away from harmful knowledge, but giving them reasoning skills and by taking away the fear engendered by long hours spent in rote learning. That doesn't mean rote learning is bad, especially at an age when children memorize things as a matter of course. Can it possibly be more harmful to memorize subtraction facts or historical dates than Star Wars characters and commercial jingles? It does mean that memorization should be done in short, frequent doses—that's more effective, as well as more enjoyable—perhaps a few minutes every day, and should be only one small part of the total mathematical experience.
Jonathan is memorizing Scripture; not just short verses, but entire psalms, and when his mother gets around to posting his recitation of Psalm 1 I'll include a link. :) What's more, he's learning the King James Version, full of words and verb forms no longer in his everyday language experience. Not that I have any complaint, because whatever one might say about the KJV's comprehensibility and the accuracy of some of the translation, it's still the most poetic version we have. If he were merely memorizing words it would be of limited value, though even "just words" would establish a foundation he could build on as he matured. But he's not merely memorizing words; he's learning what they mean, and of course it's all in the context of a family life in which the truths are worked out in practical, daily living. In this situation, I find it hard to imagine the learning is harming his developing brain. :)
Frequency, consistency, and context are powerful keys to learning, whatever the subject. So are delight and the absence of fear. If I could teach elementary-age students only arithmetic or logic, I would choose logic, but that's a false dichotomy.
Take time to read the article, it really is enlightening. Note also the shadier side of the superintendent's experiment. Not wanting to deal with complaints from his more articulate, educated parents, he chose for most of his experimental classrooms those "located in districts where not one parent in ten spoke English as his mother tongue. I sent home a notice to the parents and told them about the experiment that we were going to try, and asked any of them who objected to it to speak to me about it. I had no protests."Hello Linda:
Just to clarification,
I never said that children must do not learn arithmetic before age ten. Please, read my post again.
What I did said is that in the first ten years is better to teach math in a practical way, as the suggestion of An Easy Start in Arithmetic by Ruth Beechick, or Math on the Level. Maybe if you check those books, you will agree with me.
I did say that workbooks (math drill in paper) are not the only (and are not the best, better avoid them) way to learn math, those are modern ways that had being probe not to be very efficient in the early ages.
Real life experience, solve real life math problems in the daily life, reasoning, puzzles, table games, counting money, learn to use measurement instruments, etc. are better ways to learn math than drills in paper that usually are disconnected of the reality.
Again, if you read my suggestion (basically Mrs. Beechick’ s book ) , I think you will agree with me
Now, if your postion is: "workbook are the best way to learn math before age ten", well in this case, yes disagree :) , because my posture is about workbooks,not about arithmetic or math in general.
Bye, bye
Hi, Perla! Thanks for commenting. I can tell from your work with Octavio that you are not opposed to teaching math as a subject, and have changed my first sentence to one which I hope you will find more accurate.
The superintendent, however, was more extreme, and I especially quarrel with his contention that any formal instruction in arithmetic should be delayed till at least sixth grade. Certainly I do not believe that workbooks are the best way to learn math -- before age ten or at any other age. But I do believe that teaching logic and giving a good, foundational base in practical experience should not preclude the rote learning of arithmetic facts. Scales are not music, but knowing the scales upside down, backwards, and standing on his head in a cold shower allows the musician to concentrate on making music. Phonics isn't reading, but knowing the rules so well you no longer have to think about them facilitates true reading fluency. Knowing arithmetic facts won't make one a mathematician, but can smooth the way for higher thinking. Workbooks, computer games, flash cards, educational songs, and even skip counting while bouncing on a trampoline (as our children did) are all useful tools; the problem, I believe, comes when they are allowed to take up too much time, or become an end in themselves.
I agree wholeheartedly about the puzzles, by the way. Jigsaw puzzles, logic puzzles, word puzzles -- even those that seem to have no direct connection with math seem to stimulate the mathematical part of the brain.
I'm sure I would agree more with your position (as I understand it; I may still be wrong) if I thought this were and either/or choice. I see no reason why we can't have it all. :)
By the way, I highly recommend visiting Perla's Classical Mommy site. Even though we don't agree on everything -- as I've said before, I don't even agree with myself on everything! -- she's a remarkable woman and has assembled an enormous collection of resources.
Hello Linda:
Well, as the Bluedorns said in the second article of my post “On Early Academics”, the research until now shows that delay formal arithmetic instruction give better result in the long term.
They wrote about the delaying of formal math:
…We very much want to learn if there is any contrary research or historical evidence. Everything which we encounter on the question continues to confirm this common sense view on the matter…
We can take decision based in our personal believes or in individual and personal experiences, or we can take decision based in research which are based in historical investigations and studies that use the scientific method with hypothesis and analysis of data, etc.
Until now, while we do not find another study that invalided the previous research, we do not have a reason to believe the research false.
As you know, all is allowed to us, but not all is convenient. I think the workbooks and flash cards styles of learning for yound children are of those kinds of things that are not convenient. At least not have prove to be convenient.
Bye, bye
Ps: Sorry my English, as child I only study English in textbooks.And thank you for your words. I visit your blog sometimes, but my english is not in the level of understand all what you wrote, I am working on that :)
Perla, your English is wonderful. My Spanish is nonexistent, or I'd appreciate still more of your blog. I do think you mean "profitable" or maybe "expedient" instead of "convenient."
Ruth Beechick's booklet (An Easy Start in Arithmetic) is indeed a very good one; thank you for bringing it to my attention. I love most of her ideas for exploring and practicing math concepts, especially because she includes a favorite game of our family's called Bang. She offers a simpler version and calls it Zap, but the idea is the same.
I do quarrel with her reliance on Piaget; the division of learning into Manipulative, Mental Image, and Abstract Modes seems useful, but I find these categorization too restrictive as well as inaccurate in my experience. Perhaps my prejudice comes from having had a child who could easily handle all three modes before she even entered school, and from our frustration at what the school had -- or more accurately, didn't have -- to offer her. That was another reason we chose to begin homeschooling. If she had had to wait until Piaget's recommended age 12 for abstract thinking, we would have both gone mad. And she probably would have given up on math altogether.
One of homeschooling's great advantages is that we don't need to rely on experts and their research. We can learn from them, but in the end it's the "individual and personal experiences" with our own children that matter most.
She hasn't spoken up here yet, but you might be amused to know that the same mathematical child I'm speaking of would probably agree with you more than I do, since she is very much an unschooler and favors natural learning whenever possible. Yet as a child those math workbooks and worksheets were just another type of delightful puzzle to her.
Heather came through: Here's the post with Jonathan reciting Psalm 1.
