William McCallum's Common Core testimony in Wisconsin
Who I am and why I decided to work on the Common Core State Standards
I am a university-distinguished professor of mathematics at the University of Arizona. My doctorate in mathematics is from Harvard University, and I have been a fellow at the Mathematical Sciences Research Institute at Berkeley and the Institute for Advanced Study in Princeton. In addition to mathematics research and university teaching, I have been involved in K–12 education for 20 years. For my work in this area, I was honored to receive the National Science Foundation Award for Distinguished Teaching Scholars in 2005 and the American Mathematical Society’s Award for Award for Distinguished Public Service in 2012. I have come to be known in the mathematics and mathematics education communities as someone who can be trusted to care both about the rigor of the mathematics curriculum and about how children learn.
When I was asked to work on the standards, I decided to use that trust, knowledge, and experience to the utmost, to help build a world where all people know, use, and enjoy mathematics. I saw a once-in-a-lifetime opportunity to improve our children’s prospects for college and career, to give them the sort of mathematics education they deserve and need in order to prosper. Our children are no less capable than the children of other countries; they can meet high standards and they deserve the opportunity to do so.
How the standards were written
The Common Core originated in 2007 with a meeting of the Council of Chief State School Officers (CCSSO). For many years the states had been hearing that our mathematics curriculum was covering too many topics too superficially. They recognized the power of an agreement to share standards that were focused, coherent, and rigorous. In Spring 2009 CCSSO was joined by the National Governors Association (NGA). Forty-eight states signed a memorandum of understanding to develop common standards in Mathematics and English Language Arts. NGA and CCSSO put together a team of about 80 mathematicians, teachers, educators, policy makers, and state department of education staff, divided into a working group and a feedback group. University of Wisconsin mathematician Richard Askey was an active member of the feedback group.
Three of us—myself, Phil Daro, and Jason Zimba—were designated as lead writers. The states were our bosses. We started from raw material produced by the working group and produced periodic drafts for them to review. Many states put together teams of teachers at each grade level to provide detailed feedback. We also received reviews from the feedback group, from national organizations such as the National Council of Teachers of Mathematics and the American Federation of Teachers, and from prominent individuals and researchers. I remember in particular one exhausting and exhilarating weekend with my fellow writer Jason Zimba listening to teams of teachers put together by the AFT, who had meticulously read the standards and shared detailed comments with us. In March of 2010 the standards were released for public review, and received over 10,000 public comments. Three months later, after significant changes in reponse to these comments, the standards were released on 2 June 2010.
Throughout, we focused not on our opinions but on the evidence. Our job was to listen carefully and make considered decisions in response to the evidence, and to the amazing quantity of feedback we received from many sources, including from the state of Wisconsin, feedback that we found incredibly helpful.
Evidence and support for the standards
These standards are built for American students, based on the evidence of the best standards in this country and around the world. For years, major national reports have called for us to abandon our mile-wide, inch-deep approach and embrace focus and coherence in school mathematics. The standards finally act on those reports. Research on high performing countries shows that their teachers tend to focus on fewer topics in each grade, teach them to greater mastery, and build on them the next year in a coherent sequence of topics.
Research by William Schmidt, a leading expert on international mathematics performance and a previous director of the U.S. TIMSS study, has compared the Common Core State Standards to high-performing countries up through grade 8. The agreement was found to be high. Moreover, no state's previous standards were as close a match to the high performing countries as the Common Core State Standards.
This agreement is no accident. Evidence from international comparisons strongly informed the development of the standards. The bibliography of the standards on pp. 91–93 lists some of the numerous studies, major reports, and international and state standards that were used during the development process.
These standards have been widely praised not just by the presidents of every major mathematical society in the country, but by classroom teachers. They know the standards won’t be easy but they know they are the right thing for our students. To quote a teacher from a mixed rural/suburban school district in Missouri:
My dear colleagues teaching in my high school are no longer asking, “We never understood this stuff so why should the students be expected to?” ... We are recognizing the difference between students trained as robots vs. students who can think. ... Elementary school teachers are welcoming professional development so that fractions make sense to them.
The three principles on which the standards are based: focus, coherence, and rigor
The first two evidence-based shifts embodied in the standards are focus and coherence.
The strong focus of the standards in early grades is arithmetic. That includes the concepts underlying arithmetic, the skills of fluent arithmetic computation, and the ability to apply arithmetic to solve problems. Arithmetic in the K–5 standards is an important life skill, as well as a thinking subject and a rehearsal for algebra in the middle grades.
Focus remains important through the middle and high school grades in order to prepare students for college and careers. National surveys have repeatedly concluded that postsecondary mathematics instructors value greater mastery of a smaller set of prerequisites over shallow exposure to a wide array of topics, so that students can build on what they know and apply what they know to solve substantial problems.
Coherence is about making mathematics make sense. Mathematics should not seem like a sequence of disconnected tricks, but like a story in which ideas grow naturally on a trellis of sound basic principles, such as place value and the properties of operations.
Maintaining focus and coherence means not trying to fit everybody’s pet topic in Grades K–5, where the focus is on arithmetic. Tools of data analysis, such as mean, median, and range, can wait until Grades 6–8, where students have the solid number sense to work with the more complicated data sets for which these tools are appropriate.
Focus and coherence also imply teaching students to draw on what they know, and make connections, instead of turning every single thing into its own separate topic. The standards require least common multiples, so that students can find least common denominators if helpful. The standards require factoring and recognizing prime numbers, so that students can find prime factorizations if helpful. But least common denominators and prime factorizations themselves are not turned into separate requirements.
Some important topics in arithmetic were moved earlier than was previously the case in many state standards (e.g. fluency with two-digit addition from Grade 3 to Grade 2), while others were moved later (division of fractions from Grade 5 to Grade 6). Taken as a whole, the reorganization of topics replaces the plate-piled-high smorgasboard approach of previous standards with a carefully thought out sequence of courses. This represents a smartening up of the curriculum.
The Common Core received full marks for content and rigor in a 2010 review by the Fordham Institute. The standards call for a rigorous balance in what we seek to instill in students of mathematics during the K–12 years. Conceptual understanding, procedural skill and fluency, and applications are all required by the standards.
Let me emphasize some important specifics about the Common Core.
- The Common Core requires students to demonstrate fluency with the standard algorithm for each of the four basic operations with whole numbers and decimals, as you will see on pages 29, 35, and 42 of the standards.
- The Common Core requires students to know addition facts and multiplication facts from memory, as you will see on pages 19 and 23.
- There are no standards in the Common Core that call for students to invent, construct, or discover algorithms.
How do the standards prepare students for college?
The definition of college readiness in the standards is readiness for entry-level, credit-bearing courses in mathematics at four-year colleges as well as courses at two-year colleges that transfer for credit at four-year colleges.
The high school standards consist of easily three years of mathematics at the level of Algebra II. This certainly fits the definition of college readiness. But college readiness and STEM readiness are two different things. The mathematical demands that students face in college will vary dramatically depending on whether they are pursuing a STEM major or not. Students who intend to pursue STEM majors in college should know what is required. That was true before the Common Core, and it remains true today. States still can and still should provide a pathway to calculus for all students who are prepared to succeed on that pathway—not only because it is at the heart of many STEM fields, but also because the calculus is one of the greatest intellectual developments in human history.
The Common Core has every promise of increasing the number of students in our country who actually attain advanced levels of performance. Just because the Common Core standards end with Algebra II, that doesn’t mean the high school curriculum is supposed to end there. California, for example, had calculus standards before adopting the Common Core, and the state still has them now. The difference in California today is that better standards can help more of California’s students gain the strong foundations they need to succeed in calculus.
The standards are an historic agreement between the states and they are also a long overdue promise to our children. But without action the agreement is just empty words, and the promise is broken. We should be standing forward today to deliver on that promise. The road to faithful implementation of the standards is not easy. Tough standards don’t implement themselves; that’s up to states and local districts. There are many challenges ahead: improving curriculum, preparing teachers, and thoughtfully improving assessments. Shared standards help us meet those challenges. Let us take advantage of tough shared standards to give our nation’s children a chance to learn the skills they need in order to prosper.
William G. McCallum is a university-distinguished professor of mathematics and head of the Department of Mathematics at the University of Arizona.
Certain portions of this testimony were exerpted from the testimony of Jason Zimba before the Indiana legislature, with permission.