by Barry Garelick
Mathematics education in the United States is at a pivotal moment. At this time, forty-five states and the District of Columbia have adopted the Common Core standards, a set of uniform benchmarks for math and reading. Thirty-two states and the district have been granted waivers from important parts of the Bush-era No Child Left Behind law. As part of the agreement in being granted a waiver, those states have agreed to implement Common Core. States have been led to believe that adoption of such standards will improve mathematics and English-language education in our public schools.
My fear (as well as that of many of my colleagues) is that implementation of the Common Core math standards may actually make things worse. The final math standards released in June, 2010 appear to some as if they are thorough and rigorous. Although they have the “look and feel” of math standards, their adoption in my opinion will not only continue the status quo in this country, but will be a mandate for reform math — a method of teaching math that eschews memorization, favors group work and student-centered learning, puts the teacher in the role of “guide” rather than “teacher” and insists on students being able to explain the reasons why procedures and methods work for procedures and methods that they may not be able to perform.
I base my opinion on what I see being discussed at seminars on how to implement the Common Core. The emphasis in such forums is not on the content standards, but on the 8 Standards of Mathematical Practice (SMP). The SMP are a slight repackaging of the National Council of Teachers of Mathematics (NCTM’s) process standards. And while some maintain that “process” doesn’t mean the same as “practice”, from what I see, process is still trumping content. The popular interpretation of SMP is a pedagogical agenda that features student-centered and inquiry-based approaches. The practice of “making sense of mathematics” sounds great on paper. But what it means to those of the thoughtworld of the education establishment is what is also called “habits of mind” in which students are taught habits of analyzing problems long before they have learned the procedural knowledge and content that allows such habits to develop naturally. They are called upon to think critically before acquiring the analytic tools with which to do so. More precisely, they supposedly are acquiring the analytic tools by being given problems to solve and learning via their groups and exploration (with teachers “facilitating”) and being forced to learn the techniques in a “just in time” basis. Such a process while eliminating what the edu-establishment views as tedious “drill and kill” exercises, results in poor learning and lack of mastery.
In addition, both the SMP as well as the content standards themselves are predicated on a belief that conceptual understanding MUST precede procedure. Evidence that this belief will have widespread implementation is seen in a recently published document that provides guidance to publishers on criteria for aligning textbooks to the standards. ( See http://www.corestandards.org/assets/Math_Publishers_Criteria_K-8_Summer%202012_FINAL.pdf ) Two of the writers of this guidance document –Phil Daro and William MacCallum–are the lead authors of the math standards. The document states that “conceptual understanding needs to underpin fluency work,” or that “[sufficient] fluency can be practiced in the context of applications.” (Found on page 11 of the referenced document.) It is untrue that conceptual understanding “needs” (implying it always does) to underpin fluency. Often it does, often it does not..Understanding and procedure work hand in hand; sometimes students learn procedure before understanding the concept.
While the math standards may be an improvement over existing standards in some states, they are still largely deficient. Members of the U.S. Coalition for World Class Math have addressed the content standards in comments submitted to CCSSO and NGA. These comments are on the web site of the U.S. Coalition for World Class Math: (http://usworldclassmath.webs.com/U.S.%20Coalition%20for%20World%20Class%20Math%20Comments%20on%20June%202010%20CCSSI%20Math%20Standards.pdf)
Ze’ev Wurman has also written extensively about these standards in a report published by the Pioneer Institute. (See http://www.pioneerinstitute.org/pdf/common_core_standards.pdf ) He is an executive in the high tech industry in Silicon Valley and was a member of the 2010 California Academic Content Standards Commission that evaluated the suitability of Common Core’s standards for California. He served as a Senior Policy Adviser with the Office of Planning, Evaluation, and Policy Development at the U.S. Department of Education from 2007 to 2009. I echo his concerns with the content standards as summarized below:
– Common Core replaces the traditional foundations of Euclidean geometry with an experimental approach. This approach has never been successfully used in any sizable system; in fact, it failed even in the school for gifted and talented students in Moscow, where it was originally invented. Yet Common Core effectively imposes this experimental approach on the entire country, without any piloting.
– Common Core excludes certain Algebra II and Geometry content that is currently a prerequisite at almost every four-year state college. This effectively redefines “college-readiness” to mean readiness for a nonselective community college, as a member of the Common Core writing team acknowledged in his testimony before the Massachusetts Board of Elementary and Secondary Education.
– Common Core fails to teach prime factorization and consequently does not include teaching about least common denominators or greatest common factors.
– Common Core fails to include conversions among fractions, decimals, and percents, identified as a key skill by the National Research Council, the National Council of Teachers of Mathematics, and the presidential National Advisory Mathematics Panel.
– Common Core de-emphasizes algebraic manipulation, which is a prerequisite for advanced mathematics, and instead effectively redefines algebra as “functional algebra”, which does not prepare students for STEM careers.
More specifically, at the K-8 grade span:
– Common Core does not require proficiency with addition and subtraction until grade 4, a grade behind the expectations of the high-performing states and our international competitors.
– Common Core does not require proficiency with multiplication using the standard algorithm (step-by-step procedure for calculations) until grade 5, a grade behind the expectations of the high-performing states and our international competitors.
– Common Core does not require proficiency with division using the standard algorithm until grade 6, a grade behind the expectations of the high-performing states and our international competitors.
– Common Core starts teaching decimals only in grade 4, about two years behind the more rigorous state standards, and fails to use money as a natural introduction to this concept.
– Common Core fails to teach in K-8 about key geometrical concepts such as the area of a triangle, sum of angles in a triangle, isosceles and equilateral triangles, or constructions with a straightedge and compass that good state standards include.
There is already evidence that book publishers’ revisions to texts that align with the standards are highly likely to be “inquiry-based”. (See http://www.virtual-strategy.com/2012/08/22/houghton-mifflin-harcourt-offers-math-expressions-common-core-2013-support-national-scien Of note is this statement : “With special emphasis on Mathematical Practices and Mathematical Progressions, teachers create an inquiry-based environment and encourage constructive discussion.”) Discovery and group learning approaches to math have had poor results when they have been used in classrooms across the country. The Common Core math standards will in effect be a national mandate for reform math. I do not believe they will be good for this country.