Build Mathematical Skills by Delaying Arithmetic, Part 4

To my fellow homeschoolers,

While Benezet originally sought to build his students’ reasoning powers by delaying formal arithmetic until seventh grade, pressure from “the deeply rooted prejudices of the educated portion of our citizens” forced a compromise. Students began to learn the traditional methods of arithmetic in sixth grade, but still the teachers focused as much as possible on mental math and the development of thinking strategies.

Notice how waiting until the children were developmentally ready made the work more efficient. Benezet’s students studied arithmetic for only 20-30 minutes per day. In a similar modern-day experiment, Daniel Greenberg of Sudbury School discovered the same thing: Students who are ready to learn can master arithmetic quickly!

Grade VI

[20 to 25 minutes a day]

At this grade formal work in arithmetic begins. Strayer-Upton Arithmetic, book III, is used as a basis.

[Note: Essentials of Arithmetic by George Wentworth and David Eugene Smith is available free and would probably work as a substitute.]

The processes of addition, subtraction, multiplication, and division are taught.

Care is taken to avoid purely mechanical drill. Children are made to understand the reason for the processes which they use. This is especially true in the case of subtraction.

Problems involving long numbers which would confuse them are avoided. Accuracy is insisted upon from the outset at the expense of speed or the covering of ground, and where possible the processes are mental rather than written.

Before starting on a problem in any one of these four fundamental processes, the children are asked to estimate or guess about what the answer will be and they check their final result by this preliminary figure. The teacher is careful not to let the teaching of arithmetic degenerate into mechanical manipulation without thought.

Fractions and mixed numbers are taught in this grade. Again care is taken not to confuse the thought of the children by giving them problems which are too involved and complicated.

Multiplication tables and tables of denominate numbers, hitherto learned, are reviewed.

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

Grade VII

[25-30 minutes a day]

The assignment in the text is Strayer-Upton, books IV and V.

[Note: Continue working through Essentials of Arithmetic by George Wentworth and David Eugene Smith.]

Tables of denominate numbers, including United States money, found in the rear of book IV are reviewed. In addition to the table of linear measure, as given, it is taught that there are 1760 yards in a mile, 880 yards in a half mile, 440 yards in a quarter mile, etc.

The teacher will omit any problems in the book which, because of the length of numbers involved, cause the child in using the four fundamental processes to lose sight of the reasoning process which, after all, is the main purpose of the problem.

There is a great deal of work in mental arithmetic, involving the solution of problems without reference to paper or blackboard. This is far more important than accuracy in the four fundamental processes. Wherever possible the work is done mentally.

The practise of estimating the probable answer and checking the result with this preconceived estimate is constantly followed.

Again the teachers remember that ability to reason the problem correctly is far more important than errorless manipulation of the four fundamental processes.

Grade VIII

[30 minutes a day]

The assignment is the latter part of Strayer-Upton, books V and VI.

[Note: Complete Arithmetic, Part II by George Wentworth and David Eugene Smith is available free and might work as a substitute.]

The practise of making a preliminary estimate or an approximation at the answer before attacking the problem is continued. The ability to guess closely and promptly what the answer will be is one of the most important objectives to be gained from the study of arithmetic.

Tables of denominate numbers are kept fresh in the minds of the children. The practise of estimating lengths, heights, and areas of familiar objects and the checking up by actual measurement is constantly kept up.

The work of this grade must necessarily be a summary of everything that has been learned in arithmetic, but, above all, the ability to approximate and estimate in advance the probable answer is kept as the important objective.

The children are shown reasons for the various processes employed; why it is that a correct answer is obtained in the division of fractions by inverting the divisor and multiplying, etc.

The ability to read problems intelligently and explain how they should be attacked is far more important than the ability to add large columns of figures without an error.

The teacher will bear in mind that a great deal of work in mensuration will be difficult for some pupils to understand. Of course this work is really using geometrical formulas without giving the geometrical reasons why they work, and some children will be unable to grasp the meaning of it all. It will be found worthwhile to have models in class and to perform experiments like filling a cylinder with water from 3 times the contents of a cone of equal base and altitude, etc.

Again as much of the work as possible is done mentally. Problems are chosen to illustrate principles and give practise in reasoning rather than practise in the manipulation of large figures or complicated fractions.

— 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 Families Can Learn Math Together — and Enjoy It.