There seems to be an ever-escalating concern about the adoption of the Common Core State Standards in Mathematics (CCSSM). Every decade or so, there are suggested innovations as to how to best teach mathematics. Naturally, as times change so might the emphasis we place on topics in this field. Political events, such as Sputnik, have spurred on what in the early 1960s was known as “new math.” Technological advances and international test results in mathematics achievement also have had an influence on what and how we teach mathematics. The confluence of these factors and a lack of uniform state standards for the development and implementation of a purposeful, national mathematics curriculum have led to the development of the CCSSM.
These standards have become a political football. Parents are upset when their children come home with homework assignments that the parents do not understand. By the same token, teachers are being highly challenged to learn new mathematics content and strategies, for which they often receive no additional training or support.
What are the standards and why should we care? In particular, the CCSSM is simply a set of skills that each student should be able to perform upon graduation from high school. According to the CCSSM, these are skills necessary for success in “college, career and life after high school graduation.” New York State has always had standards. We were members of the commission that produced the most recent New York State Mathematics Standards in 2005, and were not drastically different from those currently published in the CCSSM. In fact, they were lauded across the country as among the best produced.
What is equally important to understand is what the standards do not say. According to the CCSSM, “These standards do not dictate the curriculum or teaching methods.” For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A. This statement is critically important to geometry teachers, to counter the prevailing belief that the CCSSM has “changed” the geometry curriculum.
Because of its heavy emphasis on the topic of transformational geometry, the CCSSM appears to have created a major shift in the approach to teaching geometry in New York State. This is simply not the case. For example, the intuitive notion of congruence (same shape and same size) remains the same. The actual definition of congruence will depend on the choice of an underlying set of axioms. The CCSSM does not specifically identify which set of axioms is to be used – a choice of axioms is part of the geometry curriculum. However, the state adaptation of the CCSSM requirements include using transformations to introduce congruence and similarity – an approach that is not only unnecessary, but also causes confusion among teachers and can distract from some of the classical aspects of the subject.
The choice to develop the notions of congruence and similarity from a transformational perspective was grounded in the CCSSM authors’ assumption that it is somehow more intuitive than a traditional approach. The jury is still out on that assumption, and until the schools and the teachers have had sufficient time to develop high-quality curricular materials for a purely transformational approach, a better strategy would be to use a traditional approach and address the CCSSM transformational standards as an extension. Using transformations to inspect geometric relationships is sometimes more rewarding when you look back over familiar geometric properties, than during the initial introduction of geometric concepts.
The required use of the transformational approach has angered most mathematics teachers in the state for a number of reasons. First, there is a general insecurity about the topic of transformations with which most teachers may not have strong familiarity. As a result, they struggle to understand what is being expected of them as they teach high school geometry. Once again, we are faced with the situation where students will not be properly motivated toward a study of mathematics.
Some have felt over the past decades that the many beautiful geometric relationships – some even counterintuitive – have been overshadowed by the strong emphasis of proof. This beauty could be further camouflaged by the introduction of transformations. We need to highlight the power and beauty of mathematics as we progress in this ever-increasing technological society.
Controversies about how mathematics should be taught are not new. Each time new ideas are presented, some parts remain and some do not. As we look at the CCSSM, let us follow the path the standards suggest, without dictating to teachers how they should meet them. What makes education in Finland so superior – as seen through international comparisons – is its willingness to give teachers the freedom to teach as they feel most effective, yet holding them responsible to meeting standards.
Alfred S. Posamentier is dean of the School of Education and a professor of mathematics education at Mercy College in Dobbs Ferry.