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Focusing on Process, Not Understanding, in Math


In the wake of a recent release of uninspiring test scores and a federal study showing that states lowered their "proficiency" standards, there's been a lot of tough and in some ways surprising analysis being put forward recently about math instruction in this country. Here's a sample:

—In The Baltimore Sun, a college physics professor and parent says schools are rushing students through overly difficult material, rather than ensuring that they are taught rigorous math through "age-appropriate concepts and techniques." Joseph Ganem describes his teenage daughter's struggles with high school trigonometry material that he says is at a level appropriate for upper-level college physics students. Many students, he says, are lost when they get to college-level math because they have been fed math processes but lack a solid understanding of math. "Learning techniques without understanding them," Ganem writes, "does no good in preparing students for college, where emphasis is on understanding, not memorization and computational prowess."

The Des Moines Register looks beyond Iowa's overall state scores to examine how students are faring, by achievement level, when compared to those students' peers in other states. The paper's editorialists are troubled by the fact that Iowa has far fewer students scoring at the "advanced" level than top-performing states, particularly Massachusetts. Students from disadvantaged backgrounds are also lagging behind those from more affluent backgrounds.

—A recent examination of states' tendency to set very divergent, and in many cases very low "proficiency" standards has a lot people asking hard questions of state officials. This story in the Chicago Sun-Times about that state's proficiency standards is one example.

—And on a different note, a new survey reveals just how lost many parents are when it comes to helping their children with math and science homework. Many mothers and fathers, it turns out, find it easier to talk about the perils of illegal drug use than about math and science content. The Orlando Sentinel sums up an Intel survey on parents' math and science knowledge, or lack of it.


National math test scores continue to be disappointing. This poor trend persists in spite of new texts, standardized tests with attached implied threats, or laptops in the class. At some point, maybe we should admit that math, as it is taught currently and in the recent past, seems irrelevant to a large percentage of grade school kids.

Why blame a sixth grade student or teacher trapped by meaningless lessons? Teachers are frustrated. Students check out.

The missing element is reality. Instead of insisting that students learn another sixteen formulae, we need to involve them in tangible life projects. And the task must be interesting.

Project-oriented math engages kids. It is fun. They have a reason to learn the math they may have ignored in the standard lecture format of a class room.

Alan Cook
[email protected]

In lower grades, the relevancy of the math being taught is really not something that is disputed, since the word problems involve situations students have experienced. There has been a prevailing belief that math concepts are separate and apart from procedural fluency and that the latter is "rote learning". Procedural fluency actually leads to understanding, and also results in students being able to solve problems. When students are unable to solve problems or understand the material, they check out. A program like Investigations in Number, Data and Space (developed from a grant from the NSF and used in an alarming number of schools despite protests from parents and mathematicians) proceeds very slowly so that by the time students are in the fifth grade, they are about two grades below their peers who have had a more traditionally taught class. Students must solve problems that ask for three ways to add 23 + 69--many more examples exist. (See http://www.ednews.org/articles/obama-sidwell-friends-and-the-achievement-gap.html for more info on this program).

A middle school math program called Connected Math uses "project-oriented" math. Students are given problems for which they have not had proper instruction, though bits and pieces of what they need to know are fed to them in piecemeal fashion in a "just in time" mode of learning. The result is much confusion and students providing answers to problems that don't have any lasting effect since the experience is tantamount to throwing a kid in the deep end of a swimming pool and telling him that now would be a good time to learn the breast stroke. If the kid happens to make it to the other side, he or she will probably say "I don't know how I did that but I never want to do that again."

Students taught with good instructional material that use distributed practice (repetition of previous problems), mastery and building upon prior knowledge tend to do quite well. Saxon Math is such a program and despite what critics of the program say, students have good conceptual knowledge in addition to procedural fluency. The math program used in Singapore, which is available in the US over the internet and used by many homeschoolers, is also very effective.

In my town, we go from Everyday Math where spiraling instead of mastery and building upon prior knowledge is the basis. In 6th grade, all students take pre-Algebra. Many students have difficulty making the transition to pre-Algebra because they are lacking mastery in basic arithmetic. I live in a town where tutors make up for the lack of knowledge. This is all in preparation to take Algebra I over two years (in 7th and 8th grade). Yes, my town lacks the mastery of basic arithmetic and pushes more difficult concepts to children too young.

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