While the K-8 common-core mathematics standards have garnered praise from many mathematicians and math educators, even some of the most ardent supporters of the Common Core State Standards agree that the high school math standards have weaknesses and should be revisited.
There鈥檚 less agreement, though, on exactly where the high school standards fall short: Some say they鈥檙e too dense, while others argue they don鈥檛 adequately prepare students for college. Still others point to specific skills the standards fail to address. One expert even claims that a standard is completely missing from the published document鈥攐ne that was there in the drafts.
The high school standards were tougher to reach a consensus on, experts involved in writing them said. 鈥淓verybody had their pet topic. But all those topics, they鈥檙e all good things for kids to learn,鈥� said William G. McCallum, a math professor at the University of Arizona in Tucson and one of the lead writers of the common standards. 鈥淗igh school was just hard. The whole exercise was trying to bring people together to agree on things.鈥�
As some experts see it, the high school standards were shortchanged because reviewers were rushed looking over them. 鈥淭he amount of time given to the high school standards was definitely inadequate,鈥� said Hung-Hsi Wu, a professor emeritus of mathematics at the University of California, Berkeley, who served on the standards鈥� development team. 鈥淲e were so busy with K-8.鈥�
The criticisms come as the first round of tests aligned to the math standards roll out this spring in most states.
Forty-two states and the District of Columbia have adopted the common-core math standards, and many have been implementing them in classrooms for several years now.
Evaluating Depth
One of the ways the common core aims to change mathematics instruction is by emphasizing 鈥渇ocus,鈥� or by teaching fewer topics more deeply. The standards, as the common-core document states, 鈥渁sk math teachers to significantly narrow and deepen the way time and energy are spent in the classroom.鈥�
But some educators say there isn鈥檛 less content at the high school level鈥攊ndeed, there may be more than there was previously.
The standards are 鈥渧ery dense,鈥� said Bill Barnes, a secondary math coordinator for the 52,000-student Howard County public school system in Maryland. 鈥淚 don鈥檛 know that the authors of the common core were able to achieve the kind of focus [in high school] they were able to achieve in the early grades.鈥�
While students in Mr. Barnes鈥� district are technically being introduced to fewer individual standards now鈥攊n Algebra I, for instance, they went from around 70 standards to 30鈥攅ach of the new standards contains about 鈥渢hree or four objectives from the old curriculum,鈥� he said. 鈥淚 don鈥檛 think we鈥檙e realizing a reduction in standards.鈥� Districts can choose which standards are taught in which high school courses, though the number of standards students are responsible for by the end of high school remains the same.
鈥淓ach standard has so many ideas built into it, you really have to sit down and think through all the implications of that,鈥� said Bobson Wong, a math teacher at Bayside High School in Bayside, N.Y. 鈥淚 could easily make each of these courses a two-year course.鈥�
Mr. Wong, who has twice received master teacher fellowships from the New York City-based teacher-training group Math for America, said the standards are 鈥渧ague, embedding many difficult concepts in one sentence.鈥� For example, he pointed to the following geometry standard:
鈥淧rove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180掳; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.鈥�
鈥淚n one sentence, this standard summarizes almost half a year鈥檚 worth of proofs,鈥� said Mr. Wong.
Thin on Content?
On the other hand, though, several mathematicians have argued the high school standards don鈥檛 contain enough content, in terms of both breadth and depth.
Unlike the K-8 standards, the high school standards are grouped by topic, and not by grade level or course. Mr. Wu, an adamant supporter of the standards, highlighted 鈥渟ome obvious curricular gaps鈥� he saw in the standards in 2011 when he wrote out two proposals for how the standards could be grouped into courses. Among the gaps listed were: 鈥渢he explicit definition of congruence and similarity transformation, the explicit definition of an inverse function, an explicit explanation of why the graph of a linear equation of two variables is a line, etc.鈥�
One of the most vocal opponents of the math standards, R. James Milgram, a professor emeritus in mathematics at Stanford University, has long contended that the high school standards don鈥檛 contain enough content to prepare students for college.
鈥淭hey鈥檙e very low level,鈥� said Mr. Milgram, who served on the standards鈥� validation committee but refused to sign off on their approval. He points to a study for the U.S. Department of 91制片厂视频, called 鈥淭he Toolbox Revisited: Paths to Degree Completion From High School Through College,鈥� indicating that, of the students from the high school class of 1992 whose highest level of math was Algebra II, just 39 percent obtained bachelor鈥檚 degrees.
Meanwhile, about 75 percent of students who went through precalculus eventually graduated with a bachelor鈥檚. Mr. Milgram fears that students will not take courses beyond Algebra II, which the data show are correlated with an increased likelihood of college graduation, because the common core does not explicitly require them. 鈥淚n low socioeconomic areas, [those courses] simply won鈥檛 be offered,鈥� he said. 鈥淭hey鈥檙e not offered now, and there鈥檚 no reason to assume that鈥檚 going to change.鈥�
The common core does include additional standards, known as 鈥減lus standards,鈥� which go beyond Algebra II but will not be tested. And while the standards generally assume students will not take Algebra I until 9th grade, the appendix shows an accelerated pathway, in which students combine three courses into two years. Districts also have leeway in how they organize courses.
Several experts have said the standards suffered because the review process was rushed. 鈥淲e didn鈥檛 have enough time to really polish the high school standards,鈥� said Mr. Wu.
Mr. McCallum explained that the drafts went out to reviewers in succession鈥攆irst K-5, then K-8, and finally K-8 plus high school. 鈥淪o high school didn鈥檛 get as many read-throughs as other standards,鈥� he said. 鈥淵ou could always use more time. But it鈥檚 not like it would鈥檝e been a good idea to do it the other way around. There鈥檚 no point in getting high school right if you get K-8 wrong.鈥�
鈥淚 think people were enough concerned about elementary and middle school that they focused more on that in the review,鈥� said Richard A. Askey, a professor emeritus at the University of Wisconsin-Madison, who was a member of the math standards鈥� feedback group.
Concerns About Haste
In fact, he said, the process toward the end was so hurried that an entire high school standard was left out of the final draft. Mr. Askey noticed the omission of a geometry standard鈥攐ne that explains that, under similarity transformations, area scales by squares and volume by cubes鈥攁fter the document was published. 鈥淚t鈥檚 mentioned in Appendix A but isn鈥檛 in the standards,鈥� he said. 鈥淎nd it was there in one of the drafts.鈥�
Mr. McCallum chose not to offer a response. He did, however, point out that states can add standards, and that California added a standard to address the geometry skill Mr. Askey said was missing.
According to Chris Minnich, the executive director of the Council of Chief State School Officers, the Washington-based group that led the standards development, 鈥渢he entire process took about 14 months.鈥� However, as the common-core website notes, independent reviews of the K-12 grade by grade college- and career-readiness standards鈥攖he document that would eventually become the common core鈥攄id not begin until January 2010.
The common standards were officially released five months later, after being pushed back several times so the writers could incorporate more feedback.
There鈥檚 no formal mechanism in place for a wholesale review of the common core, but it鈥檚 likely that states will鈥攁s they always have鈥攔eview their standards at times and decide whether they need to be altered.
Francis M. 鈥淪kip鈥� Fennell, a professor of education at McDaniel College in Westminster, Md., and a former president of the National Council of Teachers of Mathematics, says the standards should be reviewed once there are data on how much students have learned under them. 鈥淚n my mind, standards should be a work in progress,鈥� said Mr. Fennell, who also worked on the standards development team. 鈥淚 think you could say that about any set of standards in any subject. Wouldn鈥檛 you want to revisit them periodically?鈥�
Until then, even some of those with criticisms say they鈥檙e pleased the common core is in high school classrooms.
鈥淭hese [standards] are the best that have been done in the U.S.,鈥� said Mr. Askey. 鈥淚 want them to succeed.鈥�
