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Math in the Grammar Stage

by A. Becky Rathbun


Math instruction at the grammar level ought to concentrate upon the installation of foundational factual material into the minds of our children. The best installation of math facts into young heads requires that teachers present those facts in a connected and sequential manner as well as utilize constant repetition and speed drills.

The foundational facts of math include the basic addition facts, the multiplication table, the procedure for performing multi-digit multiplication and other similar procedures, and concepts such as the commutative property of addition. Furthermore, certain symbols and vocabulary terms are math facts that grammar age children must learn (e.g., the numerals and the four basic operators; the names of the operations and the names of the results of these operations).

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Mathematical Language

The use of terminology such as “symbols” often causes confusion at this juncture in our own understanding as instructors. As classical educators, we often find ourselves boxed into difficult corners because we impose strict definitions upon ourselves. The trivium paradigm might cause us to insist that symbols and other abstractions be left to higher levels of learning such as the dialectic stage. Aren’t symbols inherently difficult for young children to comprehend?

Indeed, from our own experiences, we know that it is a bad idea to push children into the use of symbols too quickly. However, in order to teach children the most rudimentary facts of math beyond counting, teachers must use abstractions. When we write the numeral “5” on the chalk board, we actually expect our kids to understand what that symbol means. When we write a “+”, we expect our kids to recognize that this symbol means they should perform the addition operation.

It is legitimate to expect children to easily learn to recognize these symbols, just as they learned to recognize the letters of the alphabet in order to read. The symbols of math are simply part of the language of math. When children have learned how to read well (at the 2nd or 3rd grade level, which is what many 1st graders can do), you can be sure that they are ready to handle the demands of the language of math.

When teaching math with symbols, the teacher is really using a modeling tool to convey meaning to this particular language. This type of modeling may be termed abstract modeling. We must not confuse the abstract modeling tool with higher realms of cognition simply because the term “abstract” describes the model. So, the issue should not be whether or not to use abstracts, or any other modeling tool, but to what degree.

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Modeling Tools

There are actually several modeling tools which a good math instructor will use in order to teach math. These models are: the real world (fingers, pencils, dogs); concrete manipulatives (Cuisennaire rods); musical (skip count songs); verbal (numerals spoken aloud); written (numerals written in long hand, e.g. “five”); the pictorial (diagrams, pictures of blocks or other objects); and the abstract (symbols such as “+”, “x”, numerals, and the symbol for pi). For more information on these models and their importance in mathematics instruction, I refer you to the articles in Practical Home Schooling magazine by Bob Hazen (particularly PHS #16 p.28-29).

Each one of these models may be used to represent grammar material during the grammar stage. As representations, these models are language tools. Languages can be memorized and translated. These are grammar level skills, and thus, all the models are appropriate conveyors of grammar material.

Good teaching requires the use of several models together in order to represent the math facts. This is useful because it simply presents several, complimentary ways of learning the facts and organizing them. Children retain facts and become more adept in using them at upper levels if the facts are presented with complimentary tools and in a sequential manner.

This type of presentation is one of the primary jobs of the math teacher and in order to accomplish this, we must teach our kids how to translate between mathematical models. We achieve connectivity between the models by teaching our students how to translate properly. Connectivity makes for better retention of the facts. Repetition and speed drills are the other means we must incorporate into our instruction at the grammar level to insure that our children will have a firm foundation in math.

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Choosing a Curriculum

The curriculum we choose will either hamper or enhance our efforts to properly instruct our children. I have had the opportunity to use Miquon and Cuisennaire materials, Saxon materials, Calculadders and Math-U-See. Of these, I have found that Math-U-See combined with Calculadders does the best job of utilizing all the models of mathematics in a complimentary and sequential fashion. Since this combination provides for the necessary connectivity of facts, repetition and speed drills, I have found success as an instructor. Most importantly, my children have been able to efficiently and effectively learn the foundational math facts.

Any curriculum you use must accomplish several objectives:

  1. If you aren’t already a master math teacher, your curriculum must give you the tools to be a master teacher. In The Seven Laws of Teaching, John Milton Gregory lists the seven laws, along with rules for the teacher, and the common mistakes and violations of each law. The rules are a little more specific. The first rule is to know thoroughly and familiarly the lesson you wish to teach--teach from a full mind and a clear understanding.

    If we agree with Mr. Gregory, as many classical educators do, then we agree that teachers must become masters at teaching the subject. A strong math background doesn’t guarantee you will be able to communicate on a child’s level. Neither does a weak background preclude teaching success. Math U See does an excellent job of giving the teacher a clear understanding as well as a full mind. Other curricula can be modified or used in a way to accomplish the same ends.

  2. Your children should “see” the concept and know how to apply it rather than only learning a technique in a mechanical manner. Miquon and Cuisenaire do not do as good a job of grounding the children in a concept and extending them into the use of it. Saxon uses repetition as the major means of pounding facts into the heads of little scholars. However, consistent translation between complimentary models is not a highlight of the Saxon program. The manipulatives are dropped after Saxon 3. Furthermore, the repetition present in the Saxon series is not necessarily sequential. Thus, new material doesn’t always build on already well-imbedded concepts. Math-U-See avoids these problems while retaining the good features of constant review and complimentary modeling all the way through the grammar years and beyond.

  3. Your children should become speedy at basic arithmetic operations. Both Saxon and Math-U-See include speed drill material. I believe that Calculadders does a better job and requires more speed. I am not fond of math drill software. Keyboarding and nice graphics may actually retard your childrens’ speed.

Lest anyone think that I am a Math-U-See distributor, I will tell you that even Math-U-See has a few other shortcomings. There is no absolutely perfect math curriculum.

Having the perfect curriculum isn’t as important as being committed to the right approach. Math instruction at the grammar level and beyond requires committing yourself to the installation of the facts in a connected and sequential manner. After installation of the facts, repetition allows your child to maintain those facts in a clear and ordered mental matrix. Speed drills require a student to recall the material efficiently and correctly. The combination of connectivity, sequence, repetition and speed will produce students who are firmly grounded in mathematics.

Copyright 1998 by A. Becky Rathbun. All rights reserved. No part of this article may be reproduced, in any manner whatsoever, without written permission of the author, except as provided by USA copyright law.


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