Math in American High Schools: The Delusion of Rigor, AEI, October 2009
1. The GAP!
Today, more high school kids are tackling advanced math courses and getting higher grades, but this radical change has not improved math skills much since the 1970s, according to data from the NAEP Long Term Trends (LTT) study (1978-2008) for 17-year olds.¹ In short, there is a large gap between what K-12 schools say they are teaching and what students are actually learning. The gap is much smaller in high performing nations like Singapore and Hong Kong.
Student performance on both national and international tests and on math placement tests at colleges and universities indicates the woeful lack of effectiveness of K-12 math instruction. The LTT longitudinal study, which started in the 1970s and given every few years, demonstrates that our students are about the same today as they were in the 1970s, says Mark Schneider¹, a researcher from the American Enterprise Institute for Public Policy Research (AEI). Tests like PISA, TIMSS, the regular NAEP, given every two years, and college placement tests support the LLT study. Attempts to improve math performance (e.g., NCTM math) have not worked as expected. Our students lag behind internationally, not by a little, but by a lot.
Schneider points out a lack of rigor in many high school math courses. He argues, "If policymakers decide that a mark of a successful high school career is the completion of Algebra II, then schools enroll more students into a course called Algebra II. But not all math courses are equal--and it is easier to rebrand courses and still teach low-level math than it is to increase the rigor of math instruction [my emphasis]."¹
Students who want to go to college and who are given weak or substandard algebra courses in high school will end up taking remedial math courses when they apply for postsecondary education. The college math placement test is mostly algebra skills. There are huge numbers of high school graduates in the remedial math trap because of bad instruction and diluted algebra courses. For example, at Pima Community College in Tucson, Arizona, over 380 remedial math classes are offered in its Fall 2009 catalog, ranging from basic math (6th-grade math) to intermediate algebra (Algebra II).
Schneider points out a lack of rigor in many high school math courses. He argues, "If policymakers decide that a mark of a successful high school career is the completion of Algebra II, then schools enroll more students into a course called Algebra II. But not all math courses are equal--and it is easier to rebrand courses and still teach low-level math than it is to increase the rigor of math instruction [my emphasis]."¹
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| Solving Quadratic Equations in Algebra-1 |
Students who want to go to college and who are given weak or substandard algebra courses in high school will end up taking remedial math courses when they apply for postsecondary education. The college math placement test is mostly algebra skills. There are huge numbers of high school graduates in the remedial math trap because of bad instruction and diluted algebra courses. For example, at Pima Community College in Tucson, Arizona, over 380 remedial math classes are offered in its Fall 2009 catalog, ranging from basic math (6th-grade math) to intermediate algebra (Algebra II).
Executive Summary: Focus in High School Mathematics, NCTM, October 2009.
2. NCTM Reforms (Standards-Pedagogy-Focus) Fall Short
The reasons for lackluster improvement can range from bad instruction, ill-trained teachers, misnamed courses, and poorly written curricula to an over-reliance on calculators and ineffective math reforms that do not focus on mastery of content and fluency of skills. The National Council of Teachers of Mathematics (NCTM) reform movement, which has dominated math instruction for over 20 years, falls well short. Its main emphasis on understanding and problem solving, which sounds great, has not produced an upsurge in math achievement because content mastery and skill fluency are not the central nuclei of math programs. The consequence is that our kids are almost standing still, while their peers in other nations are racing ahead.
In NCTM's latest (version) publication ($35), Focus in High School Mathematics: Reasoning and Sense Making (FHSM)², the NCTM philosophy and pedagogy have not changed, except for a new vocabulary. Instead of "understanding and problem-solving, "the NCTM extols "reasoning and sense-making." The Executive Summary³ states NCTM's clear purpose, that "reasoning and sense-making" should dominate math instruction, not mastery of rigorous content and fluency of skills that actually encourage and promote thinking and problem-solving. According to the book's executive summary³, "[FHSM] offers a different perspective, proposing curricular emphasis and instructional approaches that make reasoning and sense-making foundational to the content that is taught and learned in high school." Every few years, NCTM recycles old ideas that didn't work well.
Note. Math-savvy parents are fed up with the way math is taught in schools and cart their kids to private tutors, tutoring centers, or Kumon lessons. And, if parents are lucky, sometimes, they find public, independent, private, and charter schools that actually teach mastery of math content and fluency of key math skills. But education shouldn't be a matter of luck or where one lives. One excellent charter school is BASIS School in Tucson, Arizona.
Stars by Which to Navigate, Fordham, October 2009
3. A Better Way
I am encouraged, somewhat, by The Thomas Fordham Institute interim report called Stars by Which to Navigate (SWN)⁴, in which "subject matter experts reviewed and graded the content, rigor, and clarity of certain tests (TIMSS, NAEP, PISA) and the first public draft of the "Common Core" standards released in September 2009 by the Common Core State Standards Initiation (CCSSI). The interim report also outlines a specific grading metric and a minimal set of content expectations [See Part II Grading in Stars by Which to Navigate]. The math grades are:
TIMSS: A,
NAEP: C,
PISA: D, and
CCSSI: College and Career Readiness Draft: B (A step forward, but far from complete. It is a B for effort.)
Fordham directed the subject matter experts to glean a minimal set of content expectations for 4th grade, 8th grade, and 12th grade (as well as 15-year olds: PISA). Like the National Mathematics Advisory Panel (NMAP), the math experts concluded that much of the DASP content (data analysis, statistics, and probability), often advocated by NCTM, isn't what students must know for college. The report states that "DASP standards are given more prominence than they deserve." Moreover, the shadow instruction of DASP drains valuable time from important topics students must know. According to the report, "Presently many [DASP] standards include deep mathematical content that is far about the level of mathematics accessible to K-12 students. This results in superficial coverage that often lacks mathematical coherence." In addition, the report points out that selecting the right standards frequently relies more on content experts rather than on evidence because the evidence is very limited. Furthermore, the report concludes that international benchmarking should use TIMSS, graded an A because of its high "content, rigor, and clarity" scores, not PISA, which is graded a D.
Like the content experts, I have reservations about CCSSI. The biggest problem I see is that there is no single-minded, independent organization (math board of experts) in charge of the standards, their updates, and assessments. I called for a national math board in a previous post. Also, the public won't know how good (or bad) the K-12 grade-by-grade standards will be until their release in December of 2009. For example, will the K-6 standards be equivalent to what is taught in high performing nations like Singapore and Hong Kong and follow the recommendations of the National Mathematics Advisory Panel? Another concern is that states, with rare exceptions, have done a lousy job writing curriculum. We don't know what states will do with the standards hen they write curriculum for teachers to follow. Also, CCSSI lacks a plan to teach teachers to teach the new standards.
Race to the Top Funds: Strategies for Colorado (and other states)
4. Strategy 6: State-Wide Adoption of an Effective Curriculum
The National Council on Quality Teaching (NCQT) released a report that would help Colorado (and other states) compete for Race to the Top federal grants. The report asserts that only those states that outline strong, large scale reforms in their proposals will be considered for the federal grants.⁵ In the 50-page report, the NCQT recommends that Colorado switch to Singapore Math (Strategy 6). This is the first time NCQT has strongly advocated this.
According to the NCTQ, "The common core standards [CCSSI] are being benchmarked against international standards and Singapore consistently performs first, second or third on international assessments. The desirable characteristics consistently mentioned for the common standards (e.g. fewer topics in each grade) are the elements already present in the Singapore curriculum. If this is a concern, however, a state could choose to wait until the second application period to provide enough time to review the actual standards when they are released and vetted. We would go so far as to say that if the standards were in conflict with the Singapore curriculum, a state ought to consider opting out of the new standards [my empahsis].”
Essentially, NCQT is saying that teachers should focus on the mastery of content and skills that foster thinking and problem solving, which is the essence of Singapore Math. In other words, children should build the right background knowledge by mastering content and skills in long-term memory. This prior knowledge promotes thinking and problem-solving.
Note: The acquisition of background knowledge is key to understanding math and being able to think and do problem-solving in mathematics. If teachers want students to become better at math, then they should make sure students have the adequate background knowledge and lots of practice for mastery because cognition suffers without background knowledge stored in long-term memory. Singapore Math builds background knowledge. The best K-12 teachers have always provided students with increased layers of difficulty in math (i.e., deep practice⁶) to stretch their knowledge and extend their thinking. Moreover, they teach math hierarchically (properly sequenced), linking new ideas and skills to knowledge already learned in long-term memory.
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Draft I
This is a work in progress. Revisions are forthcoming.
This is a work in progress. Revisions are forthcoming.
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Endnotes
1 Reference is made to Math in American High Schools: The Delusion of Rigor, Mark Schneider, American Enterprise Institute for Public Policy Research, October 2009. You should read this report.
2. Focus in High School Mathematics, NCTM, October 2009, $35
3 Executive Summary: Focus in High School Mathematics, October 2009.
4 Stars by Which to Navigate, Thomas Fordham Institute, October 8, 2009. The 60-page pdf file discusses content, rigor, and clarity of the math and English "Common Core" Standards draft for college and career readiness. My comments are about the math part. You should read this interim report.
5. Race to the Top (pdf) from the National Council on Quality Teaching, August 2009. "Read NCTQ's white paper about what Colorado can do to improve its shot at getting a portion of the $4.35 billion in Race to the Top funds."
6. The idea of "deep practice" is from The Talent Code by Daniel Coyle, who states that repetition (to automaticity) is the key to learning.
6. The idea of "deep practice" is from The Talent Code by Daniel Coyle, who states that repetition (to automaticity) is the key to learning.
