Delayed Formal Math

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Dear Mr. & Mrs. Bluedorn,

First of all, thank you for speaking at the CHEACT conference in Austin. My wife and I both benefited from all of your talks. We especially enjoyed the Seven Undeniable Truths of
and The Practical Trivium. After your talk on Teaching Classical Languages we decided to add Greek to our list of languages to learn, along with Latin and Spanish — we’re still praying about Hebrew.

Anyway, I have a question regarding the study of math. We have read the Research on the Teaching of Math article and one of the Raymond Moore books. We also read your book. We, too, initially rejected the Moore’s findings but are reconsidering. We have used Saxon successfully for 1st and 2nd grade level work — apparent mastery with our 8 and 7-year-old daughters with no tears. My initial concern is that we are somehow implanting math time bombs in their heads that will explode at some future time, causing them unnecessary trouble. However, when we review the Saxon materials we have covered so far, we find that most of the course work appears to be of the informal variety, e.g. filling out calendars, telling time, counting money, adding/subtracting every day objects, rudimentary graphing to learn proportions, reading thermometers, etc. Even though worksheets are involved, the problems all appear to be of the informal variety and are not too different from the list of examples in your book (p. 373). From my understanding, this type of math is not problematic. Is it possible that Saxon has changed to be more developmentally appropriate since you last reviewed it?

How do you differentiate between informal and formal math? In other words, what constitutes the formal math that we should avoid?

Thank you for your time.
Craig, Austin, TX

If you feel that using this math curriculum:

1) is a good use of your time — or, are there more important things to fill that time;
2) allows the students to understand the concepts (as opposed to simply memorizing them — and I’m not talking about memorizing the math facts here — memorizing the math facts is important);
3) is an enjoyable experience for the students — or at least isn’t dreaded by them,

then perhaps you should continue.

Here is a quote from an article on our web site:

A few years ago, we were invited to answer some specific questions regarding early FORMAL workbook math — the idea of teaching math informally until about age ten seems most uncustomary to many. We gave some of the documented history and research on the question, and told them that if anyone knew of any contrary historical evidence or research, we very much wanted to learn of it. Everything which we encounter on the question continues to confirm our common sense view on the matter. We are satisfied that the time spent studying math — which the young child is not yet developmentally equipped for — could better be spent developing verbal skills — which the child is a sponge for at these early ages. The ideal would be to learn to speak and write several languages and become familiar with a wide scope of literature before age ten, which lays a wide and solid foundation for formal math and grammar beginning around age ten. Everything seems to point to this as the best course to take. But we have never said don’t ever teach math before age ten. The whole idea is as ridiculous as it sounds. You cannot avoid exposing your child to arithmetic concepts. They will discover it on their own at a very early age. Teach them what they are ready to learn. But teach them in a concrete way, not in an abstract way. That’s what informal math is. Also, we have never said, don’t ever teach formal math before age ten. We have always said that that was a judgement call to be made by the parent, and if you should have a precocious little tyke who loves math and wants to learn it, then you would probably be mistaken if you were to hold him back. But if you force him beyond his developmental capabilities, then you are more prone to cause developmental abnormalities. In other words:

Ephesians 6:4 And O ye fathers, do not aggravate your children, rather, nurture them to full maturity in the correction and counsel of the Lord.

Colossians 3:21 O ye fathers, do not over stimulate your children, in order that they should not be broken in spirit.

The Greek words for aggravate [parorizete] and stimulate [erethisete] convey the idea of pushing the child too far, too fast, beyond his capacity, to the point of justifiable anger and broken desperation. An exaggerated example of this in English literature would be the Charles Dickens’ character Paul Dombey. Fathers are to nurture their children to full maturity, not drive them there. The process is different with each and every child, requiring plenty of micro management. We cannot just run our children down an academic conveyor belt and expect them to emerge at the end like well manufactured machines. There is a lot more which can be said for this analogy.

The Apostle Paul gives this warning (in Ephesians and Colossians) precisely because it is necessary and ought to be heeded. We discourage the formal abstract workbook type of rigor as an aggravation on the child developmental level. Just because something can be characterized as disciplined does not mean it is the most profitable in the end. We encourage other avenues and other time schedules which reach precisely the same goals with as much or more discipline, but with much less time commitment and more satisfactory results. In other words, don’t bring the classroom school home, but tutor your child instead. This frees up time for other good and important things, such as reading aloud. Nevertheless, as we have always said, the parents are the best judges of what is best for their own family, and what may be too much for one may not be enough for another. The one-size-fits-all academics is incompatible with homeschooling.

Formal vs. Informal Math Example

Teaching fractions formally — using a workbook/textbook which uses symbols/numbers and pictures to teach the concept of 1/2 of something

Teaching fractions informally — using everyday objects to teach the concept of 1/2 of something


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