A Reformer's 'Retrogression': Speaking Out for Kumon Mathematics
l am not a person to stop lightly into controversy. Some people feed on conflict. Debate brightens their day. I am not one of them. So my decision to speak out for Kumon Mathematics came only after considerable thought and commitment to a program I have seen in operation for over a year.
Were we still in the back-to-basics swing of the pendulum, I would feel little trepidation. But the current "educationally correct" procedures for teaching math involve math manipulatives, calculators, and higher-level thinking activities and problem solving.
In contrast, Kumon is an extremely sequential set of over 4,000 pencil-and-paper worksheets designed to develop students' speed and accuracy in math. This individualized math program, developed by the Japanese, relies on old-fashioned &ill-and-practice. Many educators believe students should forgo the drudgery of learning math facts and simply use calculators. They rightly point out that most of us use calculators anyway. Many of the current recommendations from such professional groups as the National Council of Teachers of Mathematics place heavy emphasis on higher-level thinking skills and problem solving. Educators of the gifted often make similar recommendations.
I know how negatively &ill-and-practice is viewed by my colleagues. I can see a scarlet R stamped on my forehead-not standing for reformer, as I like to think of myself, but for retrogressive. What I hope to explain is how different math is from all other school subjects and how Kumon uniquely addresses itself to that difference.
For the nine years of its existence, the University of Tulsa School for Gifted Children, where I teach, has used math manipulatives, calculators, and higher-level thinking and problem solving to teach math. And we plan to continue to do so. Students here understand math concepts, enjoy math, and do well on standardized tests. Yet, until about a year ago, we had one small problem. Students had not committed math facts to memory, and no amount of jingles, jumping to numbers, or classroom games had brought them to the point of owning those basic math facts. I am not saying they could not figure out answers and do well on tests. They could and did. But my concern was for their futures.
Too many students who love math in elementary school drop by the wayside in middle school or high school, particularly girls. This was certainly true of me. Though I enjoyed elementary and junior- high math and made good grades, by the time I took high-school algebra I was struggling. In college, I took the minimum amount of math required for an undergraduate degree. It was not until taking statistics courses for my doctorate that I wished I had a stronger math background.
I believe girls are particularly vulnerable to dropping out of math during the adolescent years because their physical development, which occurs earlier than boys', causes them to be more sensitive to criticism and aware of the possibility of failure. They have a heightened sensitivity to anything that might bring ridicule.
Math is quite unforgiving. Usually in math, the problem is either right or wrong. With this type of subject matter, with so many opportunities to be wrong and face ridicule, many girls choose not to continue.
My concern has to do with this premature severing of options. I believe that only an exceptionally strong background in math, like the one Kumon develops, will overcome this problem. Let me explain: Math is the most sequential of all academic subjects. And because of this, as the authors David Paul Ausubel and Floyd G. Robinson point out in their classic text, School Learning (1969), a failure to learn each link in the sequence jeopardizes subsequent understanding. The authors put it this way: "Mathematics learning exhibits such a high order of sequential dependence that unless the student masters each step in the development of the subject, further progress is impossible." In math, failure to learn (or indeed overlearn) any task will, they say, put the learner in the position of being "unable to comprehend what follows in the sequence."
This to me is the answer to educators who find math drill-and-practice an anathema. Unlike in other subjects, skills in math must be overlearned in order to be successful in subsequent math.
Kumon is so sequential that, for the most part, it is totally self-teaching. A classroom teacher can have 20 students working on 20 different stages of Kumon. At our school, in fact, we had 95 students working on Kumon at 95 different stages. Students time themselves as they work on five or 10 small worksheets for approximately 20 or 30 minutes. They record their work and correct prior work. They may confer with a teacher to see if their speed and accuracy are meeting the required standards. If they are not, the student may repeat some prior worksheets to gain speed and accuracy, much as an athlete might repeat certain exercises in training.
The comparison to physical training may take some attitude adjustment. We understand the value of repetition in sports and music but we have failed to see how similar the situation is for math. At some juncture in a child's education we, as parents and teachers, should be saying, "Yes, math is hard but you can do it with practice. Math is like learning to play the piano or getting better at basketball. It takes practice, and that means doing some things over and over again until numbers become a part of you." Instead we say, "You understand the concepts and can do the problems; I know you are bored by having to repeat work." We thus unwittingly provide the perfect excuse for future failure.
I know of no job that does not require at least some repetitive work. In math, the need for overlearning is the justification for this repetition. Likewise, many occupations require a level of competence and understanding that must be built up steadily, link by link. In order to be--and feel--competent in math, all the links must be strong. A certain amount of overlearning must take place at each stage.
Again, Ausubel and Robinson explain it well: "In the learning of number combinations, the child is not only required to comprehend a proposition such as '3 + 2 = 5,' but overlearn it to the extent that he can instantaneously provide the sum ('5') when the numbers to be added are given. Consequently, a very large amount of practice is required before the child will have acquired the number facility necessary if these arithmetical propositions are to be used for computational purposes. Such 'drill' should not be regarded as rote learning, but rather as the overlearning of propositions which are already meaningful to the learner. In undertaking such practice the learner is in a somewhat similar position to the actor who is memorizing lines, the meaning of which he has already apprehended."
I believe we are not requiring enough of our students if we are not willing to have them work hard for 30 or 40 minutes a day to ensure that these links of math learning are strong. When I was a child, much was written about the small portion of our brains we actually use. I always wanted to be told a way to use more of my brain. Kumon makes the brain work harder, makes numbers "stick in your head," as one of our students put it. Another student, age 10, wrote that Kumon had helped his math skills by getting him to constantly repeat the facts until they became automatic.
Based on these and other responses our students gave to a questionnaire on the subject, here are what I perceive to be the chief benefits of using Kumon:
- Improved concentration. One of the quickest and most obvious positive changes noted by the University School staff was in concentration. Students were able to focus on their work more than they had previously. On the questionnaire, 68 percent thought Kumon had helped them concentrate better. A 9-year-old wrote, 'I think Kumon has helped me by making me do the work instead of talking to my friends and being weird and awful at math."
- Improved math skills. Ninety-one percent of the students said their skills had improved since doing Kumon. One 10-year-old wrote that Kumon had made him "quicker and still accurate," and that he would recommend it for other students. An 11-year-old said Kumon had helped her math skills by getting her to work hard and stay on-task. Some students confessed that they liked Kumon but did not want others to know.
- Greater self-esteem. Many of the students mentioned that they had gained the confidence to, as one put it, "try harder stuff." Their answers could best be summarized by one 9-year-old girl who, at the end of school, was working at Kumon Level G, which is approximately equal to 7th-grade math. Wrote Jennifer: "Kumon helped me by letting me know that I'm good in math."
Still, Kumon is not meant to be a total math program. The reforms that the NCTM and other groups are proposing are needed changes. But the need for them does not provide the basis for an either/or dichotomy. A program like Kumon that is highly sequential, self-teaching, and provides for the necessary "overlearning" in mathematics is an essential adjunct to any program that aims at making the most of all students' math potential.
Kumon has helped our students understand what they are capable of doing, if they are willing to put forth the effort. And they have learned that the effort itself can be satisfying. They know with Kumon that they are in control of their learning. These are reasons enough to make me glad to enter the stream of controversy.
Vol. 11, Issue 13, Pages 23, 25Published in Print: November 27, 1991, as A Reformer's 'Retrogression': Speaking Out for Kumon Mathematics