Build Mathematical Skills by Delaying Arithmetic, Part 2
To my fellow homeschoolers,
Most young children are not developmentally ready to master abstract, pencil-and-paper rules for manipulating numbers. But they are eager to learn about and explore the world of ideas. Numbers, patterns, and shapes are part of life all around us. As parent-teachers, we have many ways to feed our children’s voracious mental appetites without resorting to workbooks.
To delay formal arithmetic does not mean that we avoid mathematical topics — only that we delay math fact drill and the memorization of procedures. Notice the wide variety of mathematics Benezet’s children explored through books and through their own life experiences:
Grade I
There is no formal instruction in arithmetic. In connection with the use of readers, and as the need for it arises, the children are taught to recognize and read numbers up to 100. This instruction is not concentrated into any particular period or time but comes in incidentally in connection with assignments of the reading lesson or with reference to certain pages of the text.
Meanwhile, the children are given a basic idea of comparison and estimate thru [sic] the understanding of such contrasting words as: more, less; many. few; higher, lower; taller, shorter; earlier, later; narrower, wider; smaller, larger; etc.
As soon as it is practicable the children are taught to keep count of the date upon the calendar. Holidays and birthdays, both of members of the class and their friends and relatives, are noted.
— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment
Grade II
There is no formal instruction in arithmetic.
The use of comparatives as taught in the first grade is continued.
The beginning is made in the telling of time. Children are taught to recognize the hours and half hours.
The recognition of page numbers is continued. The children are taught to recognize any numbers that they naturally encounter in the books used in the second grade. If any book used in this grade contains an index, the children are taught what it means and how to find the pages referred to. Children will naturally pick up counting in the course of games which they play. They will also easily and without formal instruction learn the meaning of “half,” “double,” “twice,” or “three times.”
The teacher will not devote any formal instruction to the meaning of these terms if the children do not pick them up naturally and incidentally.
To the knowledge of the day of the month already acquired is added that of the name of the days of the week and of the months of the year.
The teacher learns whether the children come in contact with the use of money at all in their life outside the school. If so, the meaning of “penny,” “nickel,” “dime,” and “dollar” is taught. In similar fashion, and just incidentally, the meaning and relation of “pint” and “quart” may be taught.
Grade III
While there is no formal instruction in arithmetic, as the children come across numbers in the course of their reading, the teacher explains the significance of their value.
Before the year is over the children will be taught that a “dime” is worth 10 cents, and a “dollar” 10 dimes or 100 cents, a “half dollar” 5 dimes or 50 cents, etc. They will learn that 4 quarters, or 2 halves, are worth as much as one dollar.
They add to their knowledge of hours and half hours the ability to tell time at any particular moment. The first instruction omits such forms as 10 minutes to 4; or 25 minutes to 3. They are first taught to say 3:50; 2:35; etc. In this connection they are taught that 60 minutes make one hour.
It is now time, also, for them to know that 7 days make a week and that it takes 24 hours to make a day. They are also taught that there are 12 months in a year and about 30 days in a month.
The instruction in learning to count keeps pace with the increasing size of the textbooks used and the pages to which it is necessary to refer. Games bring in the recognition of numbers. Automobile license numbers are a help in this respect. For example, the teacher gives orally the number of a car [of not over four digits] which most of the children are likely to see, and later asks for the identification of the car. Children are encouraged to bring to class their own house numbers, automobile license numbers, or telephone numbers and invite the class to identify them.
The use of comparisons is continued, especially those involving such relations as “half,” “double,” “three times,” and the like.
— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment
Read all the posts in the Delayed Arithmetic Series.
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I haven’t heard of this method, so I find it interesting. I disagree though. Each child should be allowed to go at their own pace and to say that all children aren’t ready at an early age is wrong. There are some children who love arithmetic and understand it at a very young age. The problem would be the pushing or the pressure, not the exposure. I look forward to the remainder of the article to see how it further develops.
@ Michael: I find myself in a strange position regarding the “Math Wars.” On many things, I agree with Mathematically Correct, but on other things I find them much too reactionary. But then, I don’t agree with Benezet on everything, either. I guess I belong on the Island of Misfit Toys.
Maybe there should be an Island of Misfit Math Teachers?
@ homeschoolmom: I don’t think Benezet would disagree with you. He wanted to delay the formal teaching of arithmetic rules and recipes in a classroom situation. Similarly, I believe most homeschoolers will benefit from postponing such formal rules and recipes.
But a child who loves numbers should definitely be encouraged to play with them and learn whatever he is interested in — in fact, all children should play with and explore numbers, patterns, and shapes in many different situations.
To delay formal arithmetic does not mean we avoid mathematics.
Love this blog!
I think young children, especially today, especially BOYS, need discipline. And I think that an hour of workbook math is a great vehicle for that plus it lays a tremendous intellectual foundation. I’ve had ridiculous success by pushing math on my young children. My son is 7.6 and studying calculus. But beyond that, he has a photographic memory, reads 200 pages a day, and just picked up the piano at an astounding rate. I have to assume that his overall mental development benefited from the very hard push I gave him early on in math. I don’t know anyone that has tried what I did – no less anyone who has tried my style and destroyed a child…..so I try to convince others to give it a go.
From age 3.625 to 5.125…he did 2200 workbook pages starting with counting and got through 6th grade computation. Then he began algebra. It seems like a lot, but that’s only 4 workbook pages per day.
A 4 year old is awake 14 hours per day. I don’t think it’s too much to ask for one solid hour to be devoted to math – not when the upside is so high. (You can break it up in halves or thirds too.)
Denise, I love what you said: “To delay formal arithmetic does not mean we avoid mathematics.” This is certainly true with how things are going with my own daughter and our math explorations.
I think I get what Benezet was trying to do, something I subscribe to after almost fifteen years as a teaching artist, and that is to have children learn in context first, experiment with the medium of language and math as a living, breathing, useful tool for life. Not to get too self referential, but the symbols (or map) are only a representation of the math ideas, not the ideas (territory) itself. I think the collective “We” often ignores or forgets this.
It’s too bad he laid it out in grade levels steps — my view is that when kids are ready (on no one’s time table but their own) for more formal mathematics, then by all means bring it on. But, I suspect most kids need time to explore and find connections and make sense of math and reading and language in context before it moves to the page.
Thanks for sharing Benezet’s work. I love this series!
The grade-level steps were an artifact of working in traditional, classroom-oriented public schools. Those of us who have more freedom to follow our child’s individual development should rejoice in that freedom.
Cyberscholar, really?
“A 4 year old is awake 14 hours per day. I don’t think it’s too much to ask for one solid hour to be devoted to math – not when the upside is so high.”
Really? You don’t think it’s too much to ask? What else is worth a “good solid hour” of the bountiful 14 hours of a four year old? Heck, if you plan it correctly you could likely spend “only” 3 solid hours a day on academics and you could skip childhood altogether and go directly to college.
Pity.