Are Workbooks Necessary for Ray’s Arithmetics?

A visitor to this site wrote to me asking for help with her daughter who is struggling with math. She is in 4th grade.  She is using 3rd grade workbooks designed for use with the Ray’s Arithmetics.  She started using them at the 2nd grade level.  Things have not been going well!

The mother is thinking of quitting using the workbooks and just sticking with the textbooks.  That would be a wise move.  This in spite of the fact that her 6-year-old son simply loves the 2nd grade workbooks!

What’s a mother to do?

I wrote back and told her that she should indeed dump the workbooks.  Here is why I said that.


I actually told her that she needed to do two things: one, ditch the workbooks, and, two, go back and shore up her daughter’s understanding of simple math at the manipulative and “mental image” stages: the first two stages of a child’s (or adult’s) understanding of numbers and arithmetic operations.

What is the manipulative stage, what is the mental image stage and why aren’t the workbooks helping her daughter with her understanding of math?

Here is why, briefly.

The manipulative stage is where physical and visible objects are used to count and add and subtract and perform basic operations and begin to understand numbers and the basic concepts of arithmetic.

The “mental image” stage is where all that counting and numbering and adding, subtracting, multiplying and dividing starts getting done mentally.  Physical and visible objects are no longer necessary. And when the student does see them, he or she can instantly perform whatever quick calculations are needed to solve the problem or answer the question at hand.

Until basic arithmetic is firmly grasped at the manipulative and mental image stages, workbooks — which belong in the abstract stage, the third after the other two — are likely to frustrate the student and become tedious and inhibiting rather than helpful.

With the exception, of course, of this lady’s six-year-old!


I told her that her daughter needed more time in the earlier phases of math to develop her understanding of simple arithmetic operations.  How much time?  As much time as necessary.  Her daughter will let her know when she “gets it.”  Then (and only then) can they and should they move forward.

But that leads to the bigger question, and the topic of this article:

Are workbooks necessary with Ray’s Arithmetics?

I’ve given this a great deal of thought.  Here is my answer:


They are not.

Uh, Okay. . . Then why do so many parents use them and why are they published by the same company that publishes the printed editions of the Ray’s books?

I’ve also given this a great deal of thought. Here is my answer:

1. Because parents think they need them.

2. Because parents are willing to buy them.

The truth is, Ray’s Arithmetics were written to be used without workbooks.  Ray’s textbooks are the workbooks.  The many and varied story problems that are offered in the books are more than adequate for helping students to grasp the fundamental and, as they progress through the books, advanced concepts, principles and operations of math and “higher math.”


These books, as I have said before, are rigorous, despite their apparent simplicity and quaintness.  Thus, (as I have also said before, in different words) they are not for the faint of heart.

But they can and should be supplemented with lots of math games and fun and interesting ways to apply mathematics and arithmetic to situations in real life.

In fact, Ray’s books are written in such a way as to give the student the impression that they are not really “doing math” but are actually solving fun and challenging riddles and puzzles.  Clever!

Some kids like the workbooks.  Most parents probably like the workbooks.  Most parents probably like the workbooks because they keep the students occupied and busy “doing something” while the parent figures what their next move will be in their homeschool curriculum!

Ray’s, especially at the Primary Arithmetic level, are a visual and oral (and only later, mental) approach to teaching and learning math. They are unique in that regard.  The written part comes only after the visual and mental parts are grasped.  Concept by concept.  Brick by brick.  Learn it first.  Then put it down on paper.

Workbooks use a lot of paper.  They also cost a lot of paper.  Parents, especially homeschooling parents, don’t like things that cost a lot of paper.  Not when the return on investment and value received are both lower than expected.


One other thing I suggested to this mother-in-distress:

If she hasn’t already done so, secure a copy of Dr. Ruth Beechick’s excellent and concise little Parent-Teacher’s Guide for RAY’S NEW ARITHMETICS.  That book has more practical advice per page on how to successfully execute/implement an academically rigorous and satisfying math curriculum using Ray’s books than any other that I know of.  Get it!

Don’t drift like a “little rowboat in a vast ocean.”  Trade in your rowboat for a motorized craft that has onboard curricular navigation.

More importantly, don’t throw your student into the deep end to tread through waves and waves of workbook pages.  Make math as fun as a pool party.  But make sure they know how to swim first!

Dr. Ruth Beechick’s book provides a lot of that needful navigation.  Hopefully, in coming months — to tip my hand a little bit — I will be able to as well.


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