Both the Catholic School Blogger and ReformK12 recount the "horror story" of changes to the Algebra I curriculum, as described by a student at Montgomery Blair high school in Maryland.
Bottom line: The Maryland State Dept. of Education wrote a new set of standards that "take the algebra out of algebra," according to one critic. The statewide test (the High School Assessment, or HSA) was then re-written to match the new standards. Schools such as Montgomery Blair re-wrote the curriculum to match the tests. And the response of teacher at the Blair school (which, if the online newspaper is any indication, is a top-notch school) is overwhelmingly negative:
Concerned about her students’ performance, Pre-Calculus teacher Julie Greenberg asked her colleagues via email about the competency of their current students. Forty teachers responded, 29 of whom indicated that their students were "less competent" than those of their earliest teaching experience. The majority of those 29 cited basic algebra skills as the root of their students’ deficiencies...
One of the main complaints about the new MCPS algebra curriculum is the emphasis on data analysis, a topic that was not included in Algebra I until the introduction of the HSA. Costello considers data analysis to be displacing some of the algebra topics that used to be covered. "The new Algebra I curriculum consists of 45 percent data analysis," said Costello. Fourteen of the teachers who responded to Greenberg "explicitly criticized" the algebra curriculum’s emphasis on data analysis.
Data analysis represents one of the seven Algebra I units, and that unit takes less time than most of the other units, said MCPS Mathematics Instructional Specialist Lauren Duff. However, MCPS documentation for the Algebra I curriculum recommends spending six weeks on the "Data Analysis and Probability" unit, more time than on any of the remaining six units in the curriculum.
The battle lines in mathematics education appear to be drawn, with the National Counsel of Teachers of Mathematics (who guided the development of the new math standards) on one side, and those who believe in teaching fundamental skills on the other side.
Many believe that this problem is not only limited to Algebra I and other courses with HSA requirements, but that it is a symptom of a larger movement in math instruction that spans K-12.
According to [Montgomery County Gifted and Talented Association president]Hoven and [University of Maryland Associate Professor of Mathematics] Dancis, math curriculum and instruction have been factionalized into two sides of a "math war." One side is represented by the National Counsel of Teachers of Mathematics (NCTM), who, according to Hoven, "wants to emphasize geometry vocabulary and fake data analysis." The other side, according to Dancis, is represented by people like Hoven and Magnet calculus teacher Eric Walstein, who advocate for teaching methods that promote a deep understanding of material. According to Hoven, MCPS has "enthusiastically" embraced NCTM’s goals.
Walstein believes that this new instruction is preventing students from learning fundamental math concepts which are needed to understand higher level material. "The kids are not learning the foundations of the material. They’re just sitting and memorizing formulas, and they don’t have any idea what it means," said Walstein.
"It all relates to one word…intuition. The kids just memorize, and they can’t intuit anything," said Walstein. "The things that should come out of their heads automatically are just not there," agreed Costello.
Walstein cited the increased use of calculators in MCPS curriculum as an example of this problem. "If students can just punch things into a calculator, and it spits out the answer, that’s not math. They’re not learning anything," said Walstein. While Walstein believes that calculators should be used in certain circumstances, they are not a substitute for understanding the material.
Mathematically Correct has a good summary of a set of sound mathematical standards that go from kindergarten through Geometry. The website also provides a link to a scathing critique of the NCTM standards. Key comments (with respect to revisions to the NCTM standards in 2000):
The NCTM has toned down the constructivist language, but they still stress content-independent "process skills" and student-centered "discovery learning". Similar to the NCTM Standards, PSSM emphasizes manipulatives, calculator skills, student-invented methods, and simple-case methods.
Although PSSM contains five "Connections" sections, there continues to be no acknowledgement of the vertically-structured nature of mathematics. Mastery of math requires a step-by-step build up (in the brain) of specific content knowledge. PSSM omits this aspect of the "connections" within mathematics...
Each of the following skills serves as a preskill for acquiring all higher skills. To move up to the next skill level, the student must remember all preskills.
The ability to instantly recall basic multiplication facts
The ability to factor integers
The ability to reduce a fraction to lowest terms.
The NCTM says they want to maximize "understanding", but they still fail to recognize that specific math content must first be stored in the brain as a necessary precondition for understanding to occur. Although rarely the preferred method, intentional memorization is sometimes the most efficient approach. The first objective is to get it into the brain! Then newly remembered math knowledge can be connected to previously remembered math knowledge and understanding becomes possible. You have to "know math" before you can "understand math", "do math", or "solve math problems."
Also mentioned is an article by Berkeley Professor of Mathematics H. Wu about the "false dichotomy" in mathematics education:
Education seems to be plagued by false dichotomies. Until recently, when research and common sense gained the upper hand, the debate over how to teach beginning reading was characterized by many as "phonics vs. meaning." It turns out that, rather than a dichotomy, there is an inseparable connection between decoding—what one might call the skills part of reading—and comprehension. Fluent decoding, which for most children is best ensured by the direct and systematic teaching of phonics and lots of practice reading, is an indispensable condition of comprehension. -Wu, Page 1
"Facts vs. higher order thinking" is another example of a false choice that we often encounter these days, as if thinking of any sort—high or low—could exist outside of content knowledge. In mathematics education, this debate takes the form of “basic skills or conceptual understanding.” This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics runs counter to the acquisition of conceptual understanding. The truth is that in mathematics, skills and understanding are completely intertwined. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding. There is not 'conceptual understanding' and 'problem-solving skill' on the one hand and 'basic skills' on the other. Nor can one acquire the former without the latter. - Wu, Page 1
And speaking of "false dichotomies," there are, unsurprisingly, comments following the original SilverChips article that make the following assumption: On one side is genuine learning and a set of meaningful standards; on the other side are bad standards and all high-stakes tests.
Although the tests in this Maryland case are compounding the problem, the tests aren't at the heart of the problem. Yes, a test based on poor standards will not help education; in fact, it will probably make matters worse. But this is not evidence that high-stakes testing is necessarily flawed. The problem here is the set of standards, not the test.
Posted by kswygert at January 12, 2004 02:08 PM