Teaching Math

by | Math | 3 comments

The following is a summary of our article Research on the Teaching of Math: Formal Arithmetic at Age Ten, Hurried or Delayed? by Harvey Bluedorn

Historically, the age for instruction in arithmetic and mathematics seems to have slowly shifted from age fifteen or later down to age ten. Then, about a century ago, this was shifted again to about age seven, or six. In very recent times it has shifted again to age five or four. But recorded history may not be the place to go in order to find substantive support for the practice of beginning formal instruction in arithmetic at any age – five, ten, or fifteen.

There is more material in arithmetic and mathematics to learn and to use today than the ancients studied. Some may argue that starting earlier allows more time to learn more material. It seems obvious to them that if a child learns to do multiplication and division at age ten, then he is five years ahead of the child who learns to do multiplication and division at age fifteen. Right? Perhaps. So if we teach him to multiply and divide at age five he would be ten years ahead. At birth, fifteen years ahead. Get the point? This is more than merely an issue of enough time. This is an issue of development.

How much math there is to learn, and how early children may have been forced to “learn” some math – these considerations do not give us data to define the time when it is most effective and most efficient to begin teaching arithmetic and mathematics. Most obviously, there is a time when it is too early. Those who advocate formal arithmetic at age five appear to have ignored this developmental issue, and when the results are not like they want, they patch them up with experimental classroom methods which try to emulate informal experiences in arithmetic – a tacit witness to informal instruction before age ten.

In our culture, we erroneously perceive that the only way anyone anywhere at any time can learn arithmetic is from early formal instruction – usually in a classroom school. But young children have learned the basic concepts of number in every culture without any formal instruction. Games, measurements, and commercial activity have been the primary childhood instructors. They are still the best instructors of young children. Withholding formal instruction until age ten will by no means guarantee failure. Depending on what arithmetic activities are done, it may actually guarantee the child’s success.

What we suggest is:

1. Formal textbook or workbook instruction in arithmetic may begin at age ten. It is about age ten that the developmental light bulb goes on, and the child becomes capable of a great deal more mental and physical skill. (Of course that’s not an absolute rule. With a few children, it is as early as eight. We call them “bright” children because the developmental light bulb goes on early.) Waiting until the child is developmentally prepared to handle the concepts makes instruction in arithmetic very easy, because the child learns very quickly.
2. There is no necessity for formal teaching in arithmetic before age ten. Once all of the developmental parts are there, most children can learn – in a few weeks – everything which they might have spent six years learning (kindergarten through fifth grade), that is, if they haven’t already learned it through questions and experiences and working things out on their own — which is generally the case.
3. Depending upon the child, upon the method, and upon the subject matter covered, there exists the potential for developmental harm from the formal teaching of arithmetic before age ten. Small children cannot understand many arithmetic concepts at an early age. We can teach them to perform the process, but we cannot make them understand the concepts. The child “learns” to hate “learning.” The child’s understanding develops along the wrong lines. He may actually develop mental “blocks” to arithmetic – actual physiological blocks in the brain. (This may give new meaning for the term “blockhead.”)
4. Not formally teaching arithmetic before age ten frees up a lot of time for other activities which will build the vocabulary of the child. Vocabulary is the number one index of intelligence. Developing vocabulary was one of the deliberate foci of ancient education. We waste valuable time for developing vocabulary and verbal language skills if we instead spend those hours teaching a five year old to count by fives. (He’ll know it intuitively by age ten anyway, without ever being taught.) Instead, we ought to spend those hours reading to him. We only have so much time in the day. Do we want to spend it trying to force math skills into a child who developmentally is not optimally prepared, or spend it doing what is developmentally natural to a young child – learning new words and associating them with new ideas and experiences. Stretch the child’s vocabulary during the formative years, and when he’s developmentally ready to do some deeper thinking, he’ll have a mind prepared to take on the task, and he’ll take off like a rocket.

Please note: We are not saying that no child should ever utter the name of a number before age ten. Not at all. About age four, most children discover money, and there is no hiding numbers from them after that. They encounter numbers all of the time. If we encourage learning, then they’ll be asking lots of questions, and we’ll be full of opportunities to teach numbers and measurement. But we would not encourage using a formal workbook before age ten, unless the child has a genuine desire to do so, he shows that he is competent to handle the work, and it does not take away time from other valuable activities. We are not going to ruin the child if we wait until age ten before beginning formal teaching of arithmetic.


  1. Amy

    Thank you for the article about teaching math. I have two questions. First, did any of your children request to do formal math before age 10? My husband and I both love math, but I’m not sure if love for and ability in math are traits that are passed down. Second, we live in a state with mandatory standardized tests yearly once the child is seven. We don’t have to report the scores to anyone, oddly enough, but we’re concerned that if we don’t teach our children math formally before age ten, they will perform poorly on the exams and feel like they are being cheated on their education. I have a friend who used a certain well-known homeschool curriculum for science for her sons, and they were clueless on a few topics on the science section of the standardized test they took (because they were not taught in that curriculum.)

  2. Barbara

    Our son “hated” math his last year in a traditional school (grade 2), even hiding his math workbook in an attempt not to have to do the work.

    When we began homeschooling him, we forgot to order a math curriculum at first. I felt bad about this till I asked a friend, a doctor who homeschooled her four children till college, what math curriculum she would recommend for a 7-year-old, and she said she never started formal math instruction till her kids reached the age of 10. So we didn’t “do” math that year or the next–though our son began (or continued) playing around with numbers, such as figuring out in his head how many days it was between Christmas and Valentine’s Day and other simple computations like that.

    He was 9, going on 10, the summer he suddenly asked me if he could “do math” that fall. So we got some workbooks and he did just fine. What had been like pulling teeth when he was 6 and 7 was almost easy when he was 9 and 10. He just hadn’t been ready before.

  3. Bec

    I agree that young children arrive at school with a wealth of informal math knowledge. Sadly, too often teachers do not take advantage of this and it is lost. Games are a great way to develop number sense and the web provides some great free resources. One of my favorites is
    which provides a range of free math teaching resources, math games, and hands-on math activities for K through 5th grade that you can instantly print and all activities are correlated with the Common Core State Standards.



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