To my fellow homeschoolers,

How can our children learn mathematics if we delay teaching formal arithmetic rules? Ask your librarian to help you find some of the wonderful living books about math. Math picture books are great for elementary students. Check your library for the Time-Life “I Love Math” books or the “Young Math Book” series. You’ll be amazed at the advanced topics your children can understand!

Benezet’s students explored their world through measurement, estimation, and mental math. Check out my PUFM Series for mental math thinking strategies that build your child’s understanding of number patterns and relationships.

Still there is no formal instruction in arithmetic.

By means of foot rules and yard sticks, the children are taught the meaning of inch, foot, and yard. They are given much practise in estimating the lengths of various objects in inches, feet, or yards. Each member of the class, for example, is asked to set down on paper his estimate of the height of a certain child, or the width of a window, or the length of the room, and then these estimates are checked by actual measurement.

The children are taught to read the thermometer and are given the significance of 32 degrees, 98.6 degrees, and 212 degrees.

They are introduced to the terms “square inch,” “square foot,” and “square yard” as units of surface measure.

With toy money [or real coins, if available] they are given some practise in making change, in denominations of 5′s only.

All of this work is done mentally. Any problem in making change which cannot be solved without putting figures on paper or on the blackboard is too difficult and is deferred until the children are older.

Toward the end of the year the children will have done a great deal of work in estimating areas, distances, etc., and in checking their estimates by subsequent measuring. The terms “half mile,” “quarter mile,” and “mile” are taught and the children are given an idea of how far these different distances are by actual comparisons or distances measured by automobile speedometer.

The table of time, involving seconds, minutes, and days, is taught before the end of the year. Relation of pounds and ounces is also taught.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

[fall semester]

There still is no formal instruction in arithmetic except that the children are asked to count by 5′s, 10′s, 2′s, 4′s, and 3′s. This work is done mentally at first with no written figures before them, either on paper or on the blackboard. This leads naturally to the multiplication tables of 5′s, 10′s, 2′s, 4′s, and 3′s which, in this order, are given to the children before the end of the semester.

With toy money, or with real coins if available, the children practise making change in amounts up to a dollar, involving, this time, the use of pennies.

The informal work of previous grades in the estimating of distance, area, time, weights, measure of capacity, and the like, is continued. The ability to guess and estimate by games is developed. Each child in the class writes his estimate before these are checked up by actual measurement.

The children compare the value of fractions and discover for themselves that 1/3 is smaller than 1/2 and greater than 1/4; i.e., that the larger the denominator the smaller the fraction. This is illustrated concretely or by pictures.

Toward the end of the semester the children are given the book, Practical Problems in Mental Arithmetic, grade IV. The solution of these problems involves a knowledge of denominations which the children have not had and the use of tables and combinations which have not yet been taught to them. Nevertheless, children with a natural sense of numbers will be able to give the correct answers.

The teacher will not take time to explain by formula or tables the solution of any problem to those who do not grasp it quickly and naturally. The purpose of the mental arithmetic book is to stimulate quick thinking and to get children away from the old-time method of using the fingers to do the work of the head. If some of the children do not grasp the problems easily and quickly, the teacher simply passes on, knowing that the power to reason will probably develop in them a year or two subsequently.

The one thing which is avoided is that children shall get the idea that a fixed method or formula can be used as a substitute for thinking.

Notice that the multiplication facts were not taught as abstract information to be memorized, but as natural extensions of the patterns and relationships among numbers which the children had been exploring. This is much like the approach I take in my Times Table Series of blog posts.

I could not find the textbook Benezet mentions, but from his description of how the teachers used it, this one might be a good substitute:

Benezet’s teachers would skip the abstract number exercises and go directly to the story problems, which were a natural extension of their students’ work with measurement and estimation.

[spring semester]

The children are asked to count by 6′s, 7′s, 8′s, and 9′s. This work is done mentally without written tables before them, either upon paper or on the blackboard. After a time this leads naturally to the multiplication tables of 6′s, 7′s, 8′s, and 9′s.

The attention of the children is called to the fact that in the table of 9′s the second digit is always diminished by one [18, 27, 36, etc.] and the reason is explained that adding 9 is the same as adding 10 and taking away 1. In similar fashion it is shown that adding 8 is the same as adding 10 and taking away 2, so that in the table of 8′s the second digit of each successive product is 2 less than the second digit of the product above it [48, 56, 64]. In similar fashion it is shown that adding 7 is the same as adding 10 and taking away 3.

After the tables have been learned the teacher makes sure that the children know the products in any order; i.e., that it is not necessary for the child to start at the beginning of the table and run through until he reaches the product which he is asked to give. They learn that 2 times 3 is always equal to 3 times 2.

Children are given a little idea about the relative value of the fractions 1/2, 1/4, 1/5, and 1/10. Concrete examples assist in this; e.g., when the children remember that 2 quarters are worth one half dollar, it is easy to show them that twice 1/4 equals 1/2, or that twice 1/10 equals 1/5.

The problems listed under December to June, inclusive, in the book Practical Problems in Mental Arithmetic, grade IV, are covered in the course of the semester.

If the children do not grasp the problem quickly and easily, the teacher does not stop to explain the method or prescribe any formula for solution. Of course as new terms occur in the problems [pecks, gallons, etc.] the teacher explains, incidentally, what they mean.

— 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!

Don’t miss our new math tips and games:  Subscribe in a reader, or get updates by Email.