Sunday, April 06, 2014

Math Standards

At our last department meeting we looked a little more deeply into the Common Core Algebra 2 standards, and reading them reminded me of the 1989 National Council of Teachers of Mathematics standards.  From the Wikipedia:
The controversial 1989 NCTM Standards called for more emphasis on conceptual understanding and problem solving informed by a constructivist understanding of how children learn. The increased emphasis on concepts required decreased emphasis on direct instruction of facts and algorithms. This decrease of traditional rote learning was sometimes understood by both critics and proponents of the standards to mean elimination of basic skills and precise answers, but NCTM has refuted this interpretation.[4]

In reform mathematics, students are exposed to algebraic concepts such as patterns and the commutative property as early as first grade. Standard arithmetic methods are not taught until children have had an opportunity to explore and understand how mathematical principles work, usually by first inventing their own methods for solving problems and sometimes ending with children's guided discovery of traditional methods. The Standards called for a de-emphasis of complex calculation drills.

The standards set forth a democratic vision that for the first time set out to promote equity and mathematical power as a goal for all students, including women and underrepresented minorities. The use of calculators and manipulatives was encouraged and rote memorization were de-emphasized. The 1989 standards encouraged writing in order to learn expression of mathematical ideas. All students were expected to master enough mathematics to succeed in college, and rather than defining success by rank order, uniform, high standards were set for all students. Explicit goals of standards based education reform were to require all students to pass high standards of performance, to improve international competitiveness, eliminate the achievement gap and produce a productive labor force. Such beliefs were considered congruent with the democratic vision of outcome-based education and standards based education reform that all students will meet standards. The U.S. Department of Education named several standards-based curricula as "exemplary", though a group of academics responded in protest with an ad taken out in the Washington Post, noting selection was made largely on which curricula implemented the standards most extensively rather than on demonstrated improvements in test scores.[citation needed]

The standards soon became the basis for many new federally funded curricula such as the Core-Plus Mathematics Project and became the foundation of many local and state curriculum frameworks. Although the standards were the consensus of those teaching mathematics in the context of real life, they also became a lightning rod of criticism as "math wars" erupted in some communities that were opposed to some of the more radical changes to mathematics instruction such as Mathland's Fantasy Lunch and what some dubbed "rainforest algebra". Some students complained that their new math courses placed them into remedial math in college, though later research found students from traditional curricula were going into remedial math in even greater numbers. (See Andover debate.)

In the United States, curricula are set at the state or local level. The California State Board of Education [1] was one of the first to embrace the 1989 standards, and also among the first to move back towards traditional standards.[5]
That description certainly fits Common Core, doesn't it?

Finding California's version of the Common Core standards online is easy, getting the 1989 NCTM standards is a bit harder.  Anyone have a link?

One thing I definitely don't like about the Common Core standards is their imprecision.  California's 1997 math standards were easy to understand--any math teacher reading them would know exactly what students were to learn.  I certainly am not clear on what this standard, for example, means:
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 
How about these?
Prove simple laws of logarithms.
Prove the Pythagorean identity...
Prove polynomial identities and use them to describe numerical relationships.
What constitutes a "proof" of these at the Algebra 2 level? What does the third one even mean?

I believe in clarity.  I'm sure that if someone were to provide me with more detail, I would absolutely know what and how to teach.  However, if I cannot read the standards and know for sure what to teach, what good are the standards?  If there is room for (mis)interpretation, then the standards are not standard at all.  California's Common Core Algebra 2 standards start on page 82/164 at the link above, how many of them would you question?

This is a goatscrew.


maxutils said...

I think I see what they are trying to do ... and it's evidence of a lack of precision in terminology, not necessarily a bad idea. Each of the standards seems like something in ALg 2 that you, as a teacher should be guiding the students throughout: not something that they should be able to do themselves to pass the class. But ... that's not what the standard implies. I certainly don't want my kids to just have to memorixe the laws of logarithms and use them because they work; you can make them memorize them and show them WHY they work, easily. I don't need them to be able to do it on their own; but, I always learned math better if I knew why it worked. People who can do it by rote ... more power to them, but they get theirs, too. The only place I differ on that? Simple computation. But the times tables are patterns, aren't they? So even then... memorize, but explain. The standards you listed I think are perfectly acceptable (the first IS vague ...but probably because it was over broad) for teaching ... way too tough for student performance.

Mr. W said...

I feel like any common core post you make I will comment on now. You sound like you have been in on our meetings again.

We feel the same way, they are very vague. How deep do they go? How much are they expected to know?

For example, geometry standards talk about conic sections. So we assumed it meant parabolas, circles, hyperbolas, and ellipses. It took awhile to find out it was just circles and parabolas.

Also what happened to logic in geometry? No mentionof converse, inverse, contrapositive, or biconditional statements. How does geometry work without logic?

maxutils said...

Mr. W ... I haven't seen them. But based on your comment? Having been the guy who always taught geometry classes, (which is weird ... my experience is that the 'real' math teachers often want no part of geometry, which I found to be beautiful... (and, no disrespect implied. I just know that when I was Dept. Chair, Geometry was not a highly requested course.)) But ... conics in geometry? WHY? On the other hand LOGIC is vital. And I swear ... I was an English major, so I'm biased. But, if you do logic correctly ...all those kids who hated math the previous year will start to buy in. It's a completely different skill, and yet , it isn't.

Anonymous said...

This site appears to have the 1989 NCTM standards: