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:

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

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.

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.

For more ideas about teaching math informally, check out my book Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It!

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July 26, 2012 10:16 am

Great of this to share Benezet with readers who might not know of him. This is great stuff. You may draw flack from the Mathematically Correct crowd: they hate Benezet.

2. July 26, 2012 10:59 am

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.

3. July 26, 2012 11:34 am

@ 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.

4. July 26, 2012 8:12 pm

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.)

July 26, 2012 8:56 pm

Uh, oh. I am biting my tongue here and will not be promoting my famous math textbooks for Intrauterine Education (“Calculus In the Womb” was a huge best seller before it went out of print).

Getting back to Denise and the MC/HOLD folks: I’d like to say I found some wisdom in what they have to say, but even if I did, the WAY they say it is awfully offensive to me and people I really respect. But let’s not stir up the Math Wars here. Instead, let’s talk about child-rearing.

I’ve always been of the opinion that only a fool holds back a child from learning something s/he is truly self-motivated to learn. I love all the really fun stuff that many parents do with their pre-school kids. Not all of them are planning to be home schooling or unschooling parents, and I’m fine with that, too. I couldn’t home school my son without having full custody (he moved in with me full-time last September, just as he began his junior year in h.s., a bit late in the game) or his mom’s consent which I never received. I’m of mixed feelings about the (mis-)education he got in public school, but most of that is a combination of the sort of community his mom moved to and the usual values of far too much of US education that I find objectionable, particularly the ways in which school teaches kids not to trust themselves and to lose their natural love of learning. It needn’t be that way, not even for large schools, but I see little likelihood that it will change any time soon beyond little pockets of resistance in and out of the system.

Some intriguing blogs I read by people who write a lot about their own kids’ math thinking and learning are OVERTHINKING MY TEACHING by Christopher Danielson and THE MAP IS NOT THE TERRITORY by Malke Rosenberg. There are many others, including this one, but I only read a few. If all K-5 teachers thought as well as a few of these bloggers, we’d be 1000 times further ahead in providing solid, intriguing math for US kids. But unfortunately, great elementary math teachers are the exception, not the rule. And to my mind, it only gets worse in secondary school, though not for all the same reasons. Sometimes it’s math ignorance, but often it’d pedagogical insanity, on my view.

I will respect your blog and you and not spout off on the implications of pushing math or any other subject onto kids, even one’s own. It’s hard enough to argue about education or parenting without trying to do both at once. My views on pushing kids and believing you’re doing them a service, as well as on the questions of how we psychologically push kids in ways that make it look to all appearances that the motivation is purely intrinsic would result in the spilling of blood here and I know I’d have zero impact. So I’ll just end by asking, “What’s the rush?”

6. July 27, 2012 11:35 am

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!

7. July 27, 2012 11:45 am

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.