We start with a two-page introduction, with lots of quotes from people who helped write these standards, and then a paragraph on mathematical understanding.

Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from.Me like...the hard part now will be writing valid and reliable assessment questions.

A page describing how to read the standards follows, and explains that grade level standards will be broken up into three hierarchies: standards, clusters, and domains.

"Standards define what students should understand and be able to do. Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Domains are larger groups of related standards. Standards from different domains may sometimes be closely related."

"grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians"

This concerns me a tad. We're making decisions based on what's been done in the past (isn't the whole point that what we've done in the past hasn't worked particularly well...I really hope this is more than just trying to get all 50 states to do the same mediocre thing) and what's being done in places like Singapore (where reportedly everyone is good at math). Now while Singapore Math and I are not friends on facebook, there are aspects of this curriculum that I think are fantastic. Leinwand and Ginsburg wrote a good article about what they believe make it successful. I wonder if the core standards goes far enough to link problem solving with concepts and skills, especially since the authorsexplicitly claim that these standards "do not dictate curriculum or teaching methods." I feel that this was a political decision, and question whether this was a really bad idea. If we are trusting "math experts" to determine the content that is taught in schools to students who will be entering the workforce in 2030, why not entrust experts to dictate best teaching practices? I've heard there's lots of research on the subject.

Three pages are then devoted to defining and describing "standards for mathematical practice."

They are:

problem solving, reasoning and proof, communication, representation, connections... adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).These practices are organized into eight categories that the authors elaborate on.

1 Make sense of problems and persevere in solving them (very Polya-esque).

2 Reason abstractly and quantitatively (ability to decontextualize and contextualize).

3 Construct viable arguments and critique the reasoning of others (conjecture, proof, and critique).

4 Model with mathematics (connect math to "the real world"...not sure how this is different from #2 and I definitely worry about how this will be done in lame, inauthentic ways).

5 Use appropriate tools strategically (can you in good faith make this 1 of the 8 principals of doing mathematics and not allow students to use these tools on high stakes tests?).

6 Attend to precision (accuracy, on the other hand, is overrated...I jest, precision is a practice that will be beneficial to students in every realm of their life).

7 Look for and make use of structure (understanding symbols/structures and characteristics of these symbols/structures).

8 Look for and express regularity in repeated reasoning (looking for shortcuts and generalizations).

It's unclear whether the authors believe that these practices will lead to a better understanding of the content (which will be assessed) or whether they believe these practices to be an integral part of doing mathematics (which would then imply that these practices should be assessed independently of content). Boy, I could probably be convinced to do some unsavory things to see the latter. The authors leave this decision largely to textbook writers: "Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction." which, in my experience, are beholden to test writers. Unfortunately, my guess is that these will be treated much like the current "mathematical reasoning" standards that are"inherently embedded in each of the other strands", in my humble opinion a specious claim confirmed by the low cognitive levels involved in some of the standards and most of the test questions.

Also...where's pattern sniffing? Did I miss it? That was a surprise.

What about problem posing? Ok...I'm less surprised that this wasn't in there. At least I still have a purpose in life.

And I still can't help think that this isn't as good as Cuoco's Mathematical Habits of Mind, but I digress.

Anyway, that ends the first section. The remainder of the document is devoted to grade level standards. This post is already way too long, so I'm going to stop here and do a separate write-up on the grade specific standards.

Thank you! What a great analysis! I've started reading Cuoco's article, and love it. I'm guessing you referenced it earlier ... thank you for that too.

ReplyDeleteI just finished the Cuoco article. Some of it is above my head. I'd love to turn their ideas into suggestions for standards.

ReplyDeleteThey say they're writing some curriculum, but standards based on these habits of mind would help us focus our lessons in good directions.

They are writing a curriculum...or I should saw were...it's done now. It's called the CME Project.

ReplyDeleteI haven't finished reading this long and thoughtful post yet, but in case it takes me a while, I wanted to share one snarky point: If Singaporeans are so good at math, they shouldn't need to be barred or discouraged from going to their own casinos, right? (http://www.nytimes.com/2010/06/07/world/asia/07singapore.html)

ReplyDeleteYou might be interested in reading the "summary of public feedback". From that document: "Elementary teachers are most alarmed that 'patterning' is missing. Many of the respondents asked that it be added back in the standards at every grade level, even as they suggest removing the Properties of Arithmetic. (Rather than 'Algebra as patterns,' the CCSS emphasizes 'Algebra as generalized arithmetic,' and so the Properties of Arithmetic are the appropriate precursor to Algebra.)"

ReplyDeleteIt's at http://corestandards.org/assets/k-12-feedback-summary.pdf (linked, among other interesting documents, from http://corestandards.org/about-the-standards).