The Massachusetts Board of Elementary and Secondary Education

How good are the Common Core Math Standards?

H. Wu

Are the Common Core Math Standards (CCMS) worthy of being adopted by California? Yes, because, as of 2010, CCMS offers the best hope of leading our students to a higher level of achievement.

The first requirements of a set of math standards are that they cover all the essential topics in K-12 and that each standard makes mathematical sense. Since both CCMS and the California Mathematics Standards are above this level, we can concentrate on the higher order learning issues. For example, is there sufficient mathematical guidance to ensure that students learn the major topics? Is the progression of mathematical ideas from grade to grade sufficiently coherent so that students who have mastered the materials of one grade are in a position to learn the materials of the next? Let us take a close look at these two issues in the context of the three critical areas of the K--12 curriculum: fractions, introductory algebra, and high school geometry.

Most readers are likely to be surprised that the first issue is even a concern. After all, if mathematics is objective and clear-cut, isn't there only one way to present each topic, and wouldn't it be always the correct one? To see why such a simplistic vision is far from reality, think of how most of us were taught how to add fractions, for example. To add 7/8 to 5/12, we were told that we first get the least common denominator (LCD) of the denominators 8 and 12, which is 24. Then because 24 = 3 8 and 24 = 2 12, we are told that 7/8 + 5/12 = (3 7 + 2 5)/24 = 31/24. This is well and good, but what ever happened to our intuitive idea that adding is "combining things together"? Many students ask this question and, not getting any good answers, balk at going any further with mathematics. Such an unsatisfactory state of affairs notwithstanding, almost all state standards, including our own, would simply say that in grade 5, students learn to perform calculations and solve problems involving the addition and subtraction of fractions and decimals, and that in grade 6, they learn to solve simple problems, including ones arising in concrete situations involving the four arithmetic operations of fractions. There is absolutely nothing wrong with such standards per se, except that, at this stage in the evolution of our nation's mathematics education, they do nothing but legitimize this kind of incomprehensible teaching of adding fractions using LCD. The same scenario replays itself in all aspects related to the teaching of fractions.

CCMS tries to improve mathematics education from K to 12 by outlining clearly the progression of mathematical ideas behind the major concepts and skills, especially those involving fractions. This is the difference between a set of standards that is mathematically transparent and one in which detailed guidance is lacking. Given the generally abysmal state of math textbooks and our inability to consistently produce math teachers with adequate content knowledge, a set of standards has the obligation to give the needed guidance if learning is to take place. What sets CCMS apart from any of the existing standards is the fact that it provides this guidance.

This is but one example of the above-mentioned coherence of the curriculum outlined by CCMS that facilitates the learning of mathematics in K--12.

The learning of fractions has a direct bearing on the learning of algebra. According to the National Mathematics Panel report (2008), the critical topics directly supporting the learning of algebra are fractions and the geometry of similar triangles. Beyond doing a much better job than other sets of standards in guiding students through fractions, CCMS distinguishes itself by delivering the necessary background information to students so that they can learn about the graphs of linear equations in two variables without resorting to brute force memorization. As is well known, the study of linear equations is a dominant topic in beginning algebra. If we adopt CCMS, California's students can finally learn how to correctly define the slope of a line. Whereas students of generations past had to scramble to memorize the four forms of the equation of a line, students can now use reasoning to write down the equations of a straight line when certain conditions of the line are given. There is hope that our students will finally get the preparation they need for the learning of algebra.

A burning question of the moment is whether CCMS prescribes the teaching of Algebra I (or equivalent) in grade 8. The answer is that CCMS asks that roughly half of the topics of Algebra I be taught in grade 8, but devotes the other half of grade 8 to the teaching of the needed geometric materials which will prepare students for not only similar triangles but also high school geometry. In addition, it must be emphatically pointed out that what CCMS does regarding algebra is completely aligned with the curricula of other advanced countries as revealed by the data of TIMSS.

The mathematical coherence of CCMS also lies at the heart of the discussion of high school geometry. Briefly, the better standards, such as California's, insist on teaching proofs. This is a good thing, but it does place an unreasonable burden on a high school course on geometry as the only place where any kind of proof can be found in school mathematics. As a result, some of these courses begin with formal proofs based on axioms from the beginning, with no motivation. There is another kind of reaction, however. Giving up entirely on proofs as unlearnable, some courses treat plane geometry as a sequence of hand-on activities that do not mention proofs. In addition, both kinds of courses are disconnected from the teaching of rigid motions (translations, rotations, and reflections) in middle school. What CCMS does is to add the teaching of dilations to rigid motions in grade 8 using hands-on activities, and on this foundation, develops high school geometry by proving all the traditional theorems. For the first time, the school geometry curriculum provides a framework in which all the apparently unrelated pieces of information now begin to form a coherent whole. It holds the promise that learning geometry in K--12 can finally become a reality.

It is perhaps not common knowledge that our school mathematics curriculum is broken. We would like to believe that all the missing pieces for success are already in place and that all that is required to fix it is to apply some clever tinkering. The truth is that we must do some serious hard work before we are out of the woods. What CCMS has done is to make a tentative first step in the right direction. It is not perfect, but it is well thought out and deserves our firm support. When we get good textbooks and mathematically knowledgeable teachers behind it, then our nation will be in a position to speak about good school mathematics education.