Education has changed over the years. Some change has been for the good and some change has not. I would like to address two changes that took place in the 1960s that have been devastating. One change was made in reading education. The teachers were convinced that the Whole Language approach was better than the Systematic Phonics approach and a wholesale change was made with no research to back it up. We now have seen the results of that change. The other change that was made was in the way math was taught. The systematic way, which had been taught for over 100 years previously, was thrown out in favor of a method called “Spiral Mathematics”. This approach had no research behind it that would suggest that it would be better either. The fall in reading and math scores that followed, was covered up by dumbing-down the domestic standardized tests. The international tests, however, have proven to be a different matter. We routinely score low on those test.
I would like to concentrate on the changes in math, though there are many similarities between the two. Both arise out of the same thought process. For example: both spiral math and whole language reading rely on memorization and guessing. In the case of reading, the whole language approach asks that the child memorize the shapes of some 300, or so, words. If the child runs across a word that he/she has not memorized that child is urged to guess at the word that would be there. In the case of math, the child is expected to memorize the procedure for solving each kind of math problem and is then given a large number of problems, all done the same way, to insure that the process is not forgotten. In math guessing is called estimating and many kinds of problems are solved by guessing at the answer and readjusting the guess until the child comes close or gets the right answer. In spiral math, there is a great deal of review. If the review lessons were removed from most texts, they would be only half as thick. People have come to except that this review is necessary. It wasn’t always that way. Let me pose a question, which no one seems to be asking. If review is necessary to keep from forgetting what has been taught in math, what happens when the student graduates? Does he need to keep reviewing or does he loose everything he learned about math?
In order to explain the difference between systematic mathematics and spiral mathematics, I want to talk about a parallel difference in reading. To do this, I want to examine in depth, exactly how systematic phonics works.
In systematic phonics, we teach the student about the system of our written language. The sounds that each letter makes, the sounds made by certain combinations of letters, the rules that govern the written language. When the child understands, these things, he can use this information to de-code written words. As he gets more practice in de-coding, his reading becomes more natural. After the child is reading, you never have to go back and review to keep him from forgetting what he has learned about phonics. Why is that? It is because he is actually using what he knows about phonics every time he reads. He sees words in terms of phonics and applies phonics to each word in order to read it. You don’t have to have review lessons because review is automatic and continuous as he reads.
In like manner, systematic mathematics is taught by explaining the system of mathematics. We teach how, the base ten number system works, the rules that govern it, the tactics used to solve certain kinds of problems, the proper sequencing of different operations, the different symbols that are used and correct ways to use them. All lessons are put into a context of the system. When the student begins to see math in this light, he begins to apply these rules and concepts to solving problems and so automatically reviews them every time he uses math. Now he is, understanding math, not just memorizing it. Long, tedious, review lessons are no longer necessary. If we give up these review lessons, we can cover much more math in a shorter time period.
At this point I want to give a couple of anecdotal examples of what I am talking about. One of my sixth grade students was shown a linear equation by a fiend. My student had never seen an equation before, but she had learned about ratios and proportions. She looked at the equation and decided that it could be solved, by setting it up as a proportion, which she did, and solved the equation. She applied what she knew about the system to a kind of problem she had never seen. When I was teaching in the public schools, I kept hearing how students forgot so much over the summer. One year I had 7th graders and knew that I was to have 8th graders the following year. At the end of the year I gave my 7th graders the, final test, and kept them all in a file. The following fall, I gave my 8th graders the same test. I had about 30% of my students back for the 8th grade and compared the two final tests of those students and found that most of them had the same grade, with the same problems wrong. A few had one or two more problems wrong in the fall but more than that actually had fewer problems wrong in the fall than the spring. Another case from one of my customers illustrates the same thing. Her son finished a module at the end of the year and was going to take the final test the next day. Something came up and it was postponed and time went by and her son finally took the test at the beginning of the next school year and got only 2 problems wrong. In my opinion, all of this is evidence that review is not necessary if we use the systematic approach to mathematics.
We have come to accept the idea that math is hard and confusing for school children. I reject that notion. It has not always been that way. This is what I believe. If the teacher is doing a good job of teaching and has good material, and the student is trying to learn, then learning should be easy. It is only when the student doesn’t understand, that things get hard. If that is the case; either the material isn’t good, the teacher isn’t teaching it properly, the student isn’t trying to learn, or the student isn’t ready for the material being presented.
You might ask what credentials I have to make such statements. That would be fair.
My name is Paul Ziegler and I have devoted more than 40 years to teaching, researching, and writing math curricula. I am a student of old math textbooks and math history. Much of this came about while I was teaching in a junior high school and seeing a total change in the math textbooks that were available to us. I believe the year was 1965 when we were to get new textbooks. The texts that we had been using were printed in 1955 and we all liked them. When we got the new books all of us in the math department looked at them and then at each other and said; “What is this?” The new books were written on, what was known as the Spiral Approach. This idea had been around since the 1800s and could get no traction among educators. Suddenly; all of the textbooks were written from this point of view. The idea was to just introduce concepts without developing them and then to do the same the next year and the next and, somehow, the students were supposed to figure it all out and understand how all of these bits of information fit together. Continuity and context were thrown out. The material jumped from one concept to another with no apparent sequence or continuity. The first year we used these books, I was very confused and dissatisfied with the effect on my students. I vowed I would never put in another year like that. I started to re-arrange the book in a way that gave it some reasonable sequence. I did this by taking each lesson in a sequence that made sense. One lesson might be on page 68 and the next one might be on page 182 but at least it made some sense. I soon found that there needed to be lessons that made transitions between lessons in the book so I started to write worksheets to fill the gaps. Soon I realized that many of the lessons were poorly arranged and didn’t contain material that needed to be learned and I started re-writing lessons to better fit the situation. Within a few years I had enough worksheets so I didn’t even use the books. Year by year I kept improving the worksheets to make them better. By the time I retired in 1990, I had written more material than all of my colleges in the math department combined and many teachers were using some of my worksheets. I decided to keep a copy of each of them when I retired and they became helpful when I decided to develop my own math curriculum for home school families.
I am not alone in the opinion that spiral is bad. Let me
quote from textbook author, David Eugene Smith. "The extreme
spiral system, in which no topic is ever thoroughly treated at one time, but each is repeated until the pupil wearies of it, is psychologically too unwarranted to be considered seriously." He wrote that in the preface to his 1904 textbook, Advanced Arithmetic.
There is a story that goes something like this: The superintendent of a tall building noticed that the doors started binding and cracks started appearing in the plaster on the 103rd floor of the building. An architect, known for his ability to deal with such things, was called and was to arrive at a certain time. The superintendent of the building waited for him on the 103rd floor, and the time passed for the man to arrive. The superintendent was then told that the expert was in the 8th basement. He took the elevator down to the bottom of the basement and informed the expert that the problem was on the 103rd floor. The architect answered him by saying, "The symptoms may be on the 103rd floor, but the problem is here in the foundation."
What was true about the building is also true about math
education. The seventh grader who is having trouble in math didn't develop that problem in seventh or even sixth grade, and you can't just put a little plaster on it and think it will be OK. That child is functioning the way he/she is, because the foundation is faulty or even missing altogether. You must repair the foundation if that student is to learn properly.
Knowing How vs. Knowing Why
It has been said that the man who knows "how" will always
have a job, but he will work for the one who knows "why." You can always memorize "how," but you must understand "why." I am not alone in the opinion that spiral is bad. Let me quote from textbook author, David Eugene Smith. "The extreme spiral system, in which no topic is ever thoroughly treated at one time, but each is repeated until the pupil wearies of it, is psychologically too unwarranted to be considered seriously." He wrote that in the preface to his 1904 textbook, Advanced Arithmetic.
Paul Ziegler has devoted more than 40 years to teaching, researching, and writing math curricula and has a homeschool math curriculum web site: http://www.systemath.com
