by Carolyn Forte
Many years ago – it was in the spring of 1973- I found myself in the agonizing position of having to ignore the pleas of my first grade students for better instructions as I administered the state mandated standardized test for math. My students were frustrated by the confusing directions that I was reading to them. I was not allowed to re-word or amplify the directions, only to repeat them over and over. Most of the students in my class knew and understood the material that was supposedly being tested, but due to the unfamiliar wording of the directions, they were confused and a couple of my brightest students became so upset that they cried.
That was my first really negative experience with the “new math.” In the following years, I made it a point to pay close attention to the constant changes California was making to its math curriculum. I began to notice some disturbing trends. California students have been on this “new math” path for over 40 years. That means that more than two generations of students have been confused and frustrated by the math they had in school. The situation is now so bad that two mothers with PhDs in math have complained to me that they couldn’t figure out their child’s elementary school math homework!
For years, I have advised parents to avoid math books printed after 1970 because they all confuse arithmetic with algebra. A very frustrated algebra teacher pointed out to me the result of this madness: his students had developed such illogical and jumbled thought patterns that they couldn’t learn and apply the most basic principles of algebra. Dr. Jane Healy explains in her book, Endangered Minds, how children, when confronted with conceptual tasks beyond their maturity, develop inappropriate brain pathways to solve the task they have been given. However, since these pathways are not correct, they lead eventually to dead ends and the children are blocked from further learning in that area. Remediation is difficult and time consuming. This is happening every day in our math classrooms and alas, since nearly all math curriculums follow the national standards, our homeschool parents are falling into the same trap.
The “new math” was an abject failure from the start, but instead of trashing it, the math textbook planners just went to work to make it even worse. After 40+ years of revision and “reform,” we now have an unintelligible jumble of arithmetic, algebra, geometry, calculus (yes – calculus!), statistics and probability thrown at our children from kindergarten on up. Is it any wonder that American students need tutors to learn math?
This, however, is only the most obvious problem with our math books. I noticed that what educators call the “scope and sequence” was being accelerated little by little. Math concepts I taught in the first grade forty years ago are now being thrust on four-year-olds. Children in the 1950’s did not learn to “borrow and carry” or “regroup” until the third grade, but today’s texts push this difficult concept on first graders. While I began to learn multiplication in the forth grade, children are now being introduced to multiplication in the second grade and sometimes even earlier!
This vastly accelerated and developmentally inappropriate math curriculum would be bad enough by itself, but there is more. It has to do with the nature of learning: how our brains receive, store and retrieve information. Much research has gone into this arena and much has been learned, but most of it is studiously ignored by the curriculum developers.
Our brains are marvelous and logical instruments of learning. Every second, a vast amount of information is received through our senses. If our brains did not have a filtering mechanism, we wouldn’t be able to focus on anything much less make sense of it and choose what to remember. Most of the new things we encounter are discarded automatically, so we can focus on a few things that we decide are important. Some things, like language, we learn almost automatically through repeated exposure, but others require more purposeful focus. The more meaningful a new idea or task is, the more easily and quickly it will be lodged in long -term memory.
Numbers in isolation are not very meaningful, especially to young children. Numerals (the marks we call 1,2,3…) are especially abstract. Some children catch on to the symbols easily, but others take longer to learn them. Most current math books past the kindergarten level present math mainly as isolated numerals to manipulate. This was not always the case. The earliest American math textbooks for young children presented math in the context of life. Problems using numerals in isolation were relatively rare. The problems given were meaningful and thus easier to understand and absorb. Still, children had to move from introduction of a new concept through four levels of memory and understanding before they had that concept in long-term (essentially permanent) memory.
When we first encounter a new bit of information – like a new word – we have had exposure only. Suppose you hear the word anathema in a conversation. You have been introduced to an unfamiliar word. Perhaps you look it up and discover what it means. You are now at the first level of learning. If you don’t encounter anathema again for many months, you will probably forget what it means. But if you hear it again or see it in print soon after that first encounter, you will recognize it and perhaps even remember what it means. Now you are at the second level of learning. In school, a test in which you match words to their meanings or where you have a multiple choice of answers, measures this second level of learning. This level is not very advanced and again, if you do not encounter the word anathema for a long time, you may forget it or confuse it with another word.
In order to advance to the next level of learning, you must be able to recall the word and have some sense of its appropriate usage. Fill-in-the-blank tests are at the third level of learning. At this point, you can do more than just recognize the word when you hear it, you can recall it and maybe even use it cautiously. You are not at the forth and final level of learning until you understand the word so well that you can easily explain it to someone else and are comfortable using it in conversation. At this point, you are not likely to forget it. You are at the fourth or fluency level and more or less own the word in your long-term memory.
Whenever you learn something new, you go through the four stages. The speed with which you move to fluency will vary with the difficulty of the task, your prior experiences and (if you are a child) your developmental level. Some tasks, such as learning to turn on a faucet, will move very quickly to level four, but others, such as playing a musical instrument, take a very long time.
Different children learn at varying rates but our school curriculums handle these differences very clumsily. Often, a worried parent will tell me that her child can’t remember something he learned the day before. Once I explain the four levels of learning, she is somewhat relieved but still needs help to shepherd her child through the four levels in each math concept presented in the text. Often, the solution involves setting the text aside and working on one or two concepts for a while, hopefully making them more meaningful to the child. This approach makes many parents nervous. Understandably, they worry that their children will fall even farther behind.
This worry has its roots in a basic misconception about learning math. Instead of worrying about how far or how fast your child is moving through math textbooks, worry about how well your child understands the relatively few concepts presented in them. If you subtract the superfluous advanced math concepts (anything that looks like algebra, calculus, probability or statistics), you will have a sane and workable list of arithmetic concepts, which can be conquered bit by bit, always working toward that forth level of learning in each concept. Once a child has mastered arithmetic (addition, subtraction, multiplication, division, fractions, decimals, percent, measurement, area and volume), moving into algebra and geometry is easy.
Unfortunately, today’s math texts begin presenting algebra-type math in kindergarten. This is where the confusion starts. Algebra has its own set of rules, terminology and signs that are quite different from those of arithmetic, the only exceptions being the signs for plus (+) and minus (-). Bowing to the requirements of National Standards, the writers have tried to marry the two, which is something like trying to mix oil and water. The inevitable result is confusion, accelerating with each passing year. Faced with a largely unintelligible hodge-podge of facts and algorithms (education jargon for “procedures”), children resort to memorization in place of understanding.
Math is basically logic applied to numbers. It was invented to make life easier, not to give us headaches, but in order to be helpful, math must be understood. For understanding to develop, math must be presented in a logical sequence using practical, familiar examples. Each concept must be understood before another is tackled. Forcing children to memorize the procedure for long division or multiplying fractions without understanding why it works will lead to a dead end. Yet, this is how nearly everyone learns math today: little understanding and a whole lot of memorizing.
Memorization without understanding is ultimately useless. Think about it. When you memorize an algorithm you are limited to exactly the same type of problem as the one you memorized. If your next problem is varied just a little bit, you will be lost because you only memorized a procedure and have no understanding of the principles needed to solve the problem. This method ultimately leads to frustration and failure. Whereas it takes a little longer at first to develop an understanding of the principles used in arithmetic (and later algebra, geometry, etc.) and then to build up speed and proficiency, in the end much time and effort will be saved.
On December 28, 1928, a physics teacher named O.A. Nelson was invited to take part in a meeting of high-level educational planners, including Drs. John Dewey and Edward Thorndike. The meeting was chaired by a Dr. Ziegler. Later, Mr. Nelson reported:
“We spent one hour and forty-five minutes discussing the so-called “Modern Math.” At one point I objected because there was too much memory work, and math is reasoning; not memory. Dr. Ziegler turned to me and said, “Nelson, wake up! That is what we want . . . a math that the pupils cannot apply to life situations when they get out of school!” (quoted from The Deliberate Dumbing Down of America by Charlotte Iserbyt, page 14)
Mr. Nelson went on to report that the radical changes then proposed were finally introduced in 1952. It took another generation for the changes to really take hold, but by 1970 they were almost universal. That year the “New Math” textbooks were adopted by California and introduced in the school where I was teaching. Since I taught Kindergarten that year, I didn’t have any math book to contend with but I heard all the other teachers complaining bitterly that the children weren’t getting the new math. Our Principal told them to just plow through the books anyway. They were told to do two pages a day regardless!
Today, most people can’t even remember when math was taught in a logical manner and the “New-Modern Math” seems normal. Everyone is frantically pushing memorization of both facts and algorithms at younger and younger ages. This is understandable since most math books are more confusing than not. Memorization is the only way to get through it. Algebra, Euclidian and Cartesian geometry, probability, calculus and more are mixed in a crazy jumble with old- fashioned arithmetic.
To make matters immeasurably worse, this Franken-math is being pushed on younger and younger children, creating a whole generation of math-phobic children. Tortured by developmentally inappropriate concepts they have no hope of understanding, many either work themselves to a frazzle or give up on math altogether. In order to hide the dismal failure of the new math, new courses were invented: Pre-Algebra and Pre-Calculus, two year Algebra and Geometry. What we used to learn in four years, now takes at least six. Planners have pretended to make math instruction more “rigorous” by forcing everyone into a so-called college prep math track. Many fall by the way side as thousands of students are thrown into limbo each year when they can’t pass the required math test to get a high school diploma.
Homeschool families spend huge sums on math programs and tutors trying to force their children through the absurdities of our continually deteriorating textbooks. They think that their children just aren’t good at math, or don’t work hard enough, or have memory problems. They will go from program to program and system to system trying to find something that works for their children. The gold ring often eludes them, because, in many cases, they are trying to accomplish an impossible task.
I’m sure you know children who have no problems with math. There will always be those whom I call “math-naturals.” They have a natural affinity for numbers and enjoy figuring out why things work. They will do well in math no matter what you put in front of them. Because their brains are wired for math, they go along with the program and gradually figure out how things work on their own, often without realizing what they are doing. I’m worried, however, about the majority, who aren’t getting it. Besides, why should the “math-naturals” have to work extra hard to learn what should be relatively easy if presented properly?
The promoters of Common Core insist that they are addressing some of the above-mentioned problems. Don’t be deceived. Common Core just muddies the water further. Common Core math is sold to the public with a promise of deeper “understanding.” This is the same old rhetoric we have been hearing for half a century and it is just as false as the old “New Math.”
Finding a usable math series, however, is problematic. Most people are so addicted to workbooks that the very thought of teaching arithmetic without them is terrifying. This need not be so. As stated before, there are just a few arithmetic concepts that must be learned and understood. Taken a step at a time, the task is not daunting and there are guides to help you along. Here is a list of some materials that are useful if you want to teach your child (and yourself) real arithmetic before moving on to algebra and higher math.
Ray’s Arithmetic Series (a complete set of four textbooks — approx.. K-8 although the problems in the last book would stretch most college graduates- even with a calculator). This is by far the best math program I have ever seen, but it requires parental involvement. You can’t just hand it to your child. The first two books (approx.. K-4) are for the teacher. There are workbooks available for these two volumes. There is also a parent guide by Ruth Beechick.
The Three R’s and You Can Teach Your Child Successfully Grades 4-8 both by Ruth Beechick. These are parent guidebooks which will give you an idea of what to teach and when. You will need to find your own practice sheets or workbooks.
How To Tutor by Samuel Blumenfeld. This single volume outlines how to teach both reading and math through approximately 4th grade. There are two workbooks that accompany it.
Math-It and Advanced Math-It from the Weimar Institute. This kit includes an excellent guide for learning math facts in a logical manner, several solitaire games for increasing speed, and a parent guide, which lists all the math concepts from kindergarten to grade 8. It also teaches math tricks our great-grandparents knew that make mental math much easier and faster. You need to provide your own practice sheets from online or purchased workbooks. You will need to study a little to understand the great math tricks that we weren’t allowed to learn but you will find the time well spent!
I realize that using the materials listed above is too radical a move for most, so I have this advice if you want to use a current set of math workbooks:
Ignore and skip anything that looks like algebra or higher math.
- Consider carefully your child’s learning style and maturity when choosing your math book. Remember that most concepts have been accelerated a full two years. This is not a race! The same concepts will be learned 10 times faster if you wait a few years.
- Make sure your child is learning and understanding each concept even if you have to stop and work on it with outside materials before moving on in your text.
- Trash the book if your child is not able to understand and/or retain the information. Maybe your child needs more hands-on practice with real things, cooking, money, games, etc.
Don’t worry if it takes you two or even three years to get through a math book. If all the concepts presented are learned to memory level 3 or 4, your child will move more quickly through the next book. As each volume only presents 30-50% new material, there isn’t really much to learn each year. You needn’t worry about the algebra concepts you skip either. They can be learned very quickly when they are needed for real algebra. Remember, since most students don’t ever really master many mathematical concepts, the books review all of them ad nauseam year after year.
Most children can learn all of basic arithmetic in two or three years once they are developmentally ready. The problem we have today is that by the time they are ready to take it all in easily, they are usually too burned out and discouraged. If you go at a reasonable pace and make sure your child understands each concept, he will learn it well without undue stress.
If you want an algebra program you can go to without slogging through “pre-algebra” first, I highly recommend the courses offered atwww.mathrelief.com. Once you have a good grasp of basic arithmetic (addition, subtraction, multiplication, division, fractions, decimals, percents, measurements), you can go directly to this algebra course. With brief and clear explanations, it has been used successfully by both advanced and struggling students.
©Excellence In Education 2015