How Does It Add Up? Views on Math Education
Once again it seems that mathematics has garnered front stage on the education agenda. Many people have had less than euphoric experiences with mathematics instruction in their formative years. Consequently there is anxiety about children’s learning of mathematics, especially with parents who have had less than favorable experiences themselves. Currently there is a battle between two widely divergent philosophies in teaching mathematics. On the one side there are the mathematics educators who believe passionately in the “constructivist” philosophy and on the other side there is a group of conservative mathematicians who would like to see mathematics taught as it has been for the last many decades.
I have tried to look at the situation as objectively as possible. This is somewhat difficult for someone, as myself, who has grown to love mathematics early on and has been successful with the way it has been taught. It is a natural tendency to like what you are good at and then to praise the process that made that happen.
Our current dilemma is facing off those who successfully learned mathematics with those who feel that there must be a better way to learn mathematics, since so few learn it successfully and then develop a love for the subject. We are obviously not doing this task as well as we should, or else there wouldn’t be so many people ready to admit (and be proud of it) that they were never good in mathematics. Would we have this math teacher shortage today if we had taught mathematics better at the lower grades?
In one of the best performing school districts in NYC a group of parents became unhappy with their perception that their children were coming home from schools incapable of doing arithmetic. Or, perhaps, in a way that they learned it. We should not yet abandon the notion that everyone should have a proper facility with arithmetic computations. These algorithms, properly presented and having passed the test of time, provide an important foundation for future study as well as useful insights into number theory. By the same token, youngsters should be educated to view arithmetic challenges in sophisticated ways that allow a deeper understanding of number relationships. For example, a youngster can use an algorithm to multiply 13 times 7, or be encouraged to approach this problem by considering this as the sum of the easy (mental) calculations: 10x7 and 3x7 and then (still without writing) come up with 70 + 21 = 91. Understanding the relationships between fractions with different denominators provides greater facility in understanding quantitative comparisons. These understandings bode well when using the calculator. When the elementary instruction focused exclusively on rote memorization of algorithms, mathematical understanding was usually minimized or lost.
Since the calculator is so ubiquitous, the average person might then question why bother learning these algorithms when they will be rarely used. A mathematician might argue that a good knowledge of the division algorithm greatly facilitates algebraic manipulations later on. Is this a sufficient reason to drill our youngster with this skill? In the past our models of psychology and teaching were about behaviors. We thought that all we had to do as teachers was to explain concepts and procedures clearly, have learners practice, and then give them reinforcement and feedback. Now with technology, we have been studying the brain and how meaning develops. Neurobiologists have shown that algorithms are performed on a different side of the brain than the side used for mathematical thinking and that focusing on the practice of algorithms in the early years actually can impede the development of mathematical reasoning. Cognitive developmental researchers have also proven repeatedly over the last fifty years that meaning develops progressively. Mathematics cannot just be explained or transmitted. The brain chunks information in order to organize it and make sense of it, and thus new ideas are connected to prior conceptions.
We must keep some basic and indisputable tenets above the fray. Surely both sides of the argument will agree that it is of primary importance that a chief goal of mathematics instruction is to imbue our youngsters with the problem-solving skills necessary for the future study of mathematics and beyond. Done well, this should result in many more individuals appreciating the beauty of mathematics—something we thus far just haven’t been able to achieve!#
Dr. Alfred S. Posamentier is Dean, School of Education, and Professor of Mathematics Education at City College of NewYork.