Commentary on excerpts from the Mathematics Core Common Standards

There is an intiative in the United States to form a common set of standards across the entire country; 48 states (excluding Texas and Alaska) have signed up.

Link to the Common Core Standards website

The plan is for these standards will eventually override any state standards being taught with right now.

The draft of the first phase of standards-making is up; comments are only being accepted from the public until October 20th. Hence there is something of an urgency in deciphering what’s going on.

In Language Arts, according to those in the know there’s enough worry to start sending a flood of concern.

I have read through the Mathematics standards and I’m not as worried, but there are some worthy excerpts to discuss.

The College and Career Readiness Standards for Mathematics will anchor the next phase of the Common Core State Standards Initiative: development of K–12 Mathematics Standards.

First note these are “general goal” standards, although there are clues (which I’ll get into in a moment) for what exact topics are of interest. Don’t expect to read the draft and get an exact curriculum yet.

The evidence tells us that in high performing countries like Singapore, the gap between what is taught and what is learned is relatively smaller than in Malaysia or the U.S. states. Malaysia’s standards are higher than Singapore’s, but their performance is much lower. One could interpret the narrower gap in Singapore as evidence that they actually use their standards to manage instruction; that is, Singapore’s standards were set within the reach of hard work for their system and their population. Singapore’s Ministry of Education flags its webpage with the motto, “Teach Less, Learn More.” We accepted the challenge of writing standards that could work that way for U.S. teachers and students: By providing focus and coherence, we could enable more learning to take place at all levels.

Those writing the standards appear to be updated on current research: there are far too many standards in the US.

An extra common point is textbooks in higher performings countries are the fraction of a size of the United States; however, I’m unsure how indicative this is of the actual material, since cutting pictures and paring down to the problems in US texts will also cut material significantly. However this point works, it suggests US textbooks may need some serious revision.

Overview of the Mathematical Content Standards

Number … Quantity … Expressions … Equations … Functions … Modeling … Shape … Coordinates … Probability … Statistics.

(I have cut the descriptions for the categories.)

An interesting approach: the categories in the document cut down to the objects of mathematics, rather than, say, grouping expressions and equations together as Algebra. One gets a strong sense the authors of the draft wanted to ensure the concepts were separate conceptually, even when manipulated in roughly the same way.

Part of the implication here is that each area will likely get some emphasis all the way down to K level, so to an extent even 3rd graders will have exposure to functions. (Likely as discussing input and output; I recall an educational game I played at elementary level where the “number machine” metaphor became quite clear to me.)

This also likely explains why “coordinates” are an entirely separate topic: the goal is to teach them at an early grade so we don’t have fumbling in later grades over if (0,5) is on the x-axis or the y-axis.

Extra-quirky: listing geometry as “shape”. Shape can be described by lines, and curves and so forth, but the lines can be studied in themselves without any shapes involved. I realize the intent here was to avoid the clumping word “geometry”, but I’m not sure any way around it. Here’s the full description:

Shape. From only a few axioms, the deductive method of Euclid generates a rich body of theorems about geometric objects, their attributes and relationships.

Technically I think they mean “geometric objects” as “shape”, but they’re trying to stick to single-word descriptions. While I understand the intent I worry teachers along the line may get confused, so a retitle of the category may be worthwhile.

Note also how the description immediately mentions Euclid. The draft gets a strong sense that curriculum needs a stronger proof basis, rather like how the NY Regents have swapped back to formal proofs very recently. This could cause issue with teachers who were never taught geometrical proofs at any level.

For example, systems of linear equations are covered by all states, yet students perform surprisingly poorly on this topic when assessed by ACT. We determined that systems of linear equations have high coherence value, mathematically; that this topic is included by all high performing nations; and that it has moderately high value to college faculty. Result: We included it in our standards.

One deep question of concern: if the authors are serious about cutting standards, which ones get cut?

This excerpt gives three hints to the overall strategy:

1. Weight is given to how highly colleges rate the skills in importance.
2. Topics are judged in comparison with high performing countries.
3. Standardized tests like the SAT and the ACT are additionally used as indicators.

Why were exponential functions selected for intensive focus in the Functions standard instead of, say, quadratic functions? What tipped the balance was the high coherence value of exponential functions in supporting modeling and their wide utility in work and life. Quadratic functions were also judged to be well supported by expectations defined under Expressions and Equations.

Two more hints:

4. Redundancies can be removed; quadratics are already covered in equations, so their coverage in functions is less important.
5. Topics that support applications are given higher priority.

Mathematically proficient students consider the available tools when solving a mathematical problem, whether pencil and paper, ruler, protractor, graphing calculator, spreadsheet, computer algebra system, statistical package, or dynamic geometry software. They are familiar enough with all of these tools to make sound decisions about when each might be helpful. They use mathematical understanding and estimation strategically, attending to levels of precision, to ensure appropriate levels of approximation and to detect possible errors. They are able to use these tools to explore and deepen their understanding of concepts.

(Emphasis mine.)

The writers of the standards draft seem to be of the technological bent, for this portion explicitly mentions the use of the technology. The practical ramification may be that when the common national test gets written, technology use will be included in a portion of the test.


3 Responses

  1. […] Interesting and thorough commentary on the proposed Mathematics Core Common Standards just posted over at The Number Warrior. […]

  2. Thanks for sharing your thoughts here. I’ve been reading a lot of the commentary on the language arts standards and I’m glad to know more about the math standards. As an elementary school teacher they all impact me.

    You wrote, “Part of the implication here is that each area will likely get some emphasis all the way down to K level”. As I understand it (and I am not an expert, by any means) there is some concern, at least at the elementary level, that we are trying to do too many different things each year. There is some thought that it would be better for kids if we dug deeply into a couple of areas each year rather than shallowly explore lots of areas. Do you think these standards will push us to continue working on many areas rather than being able to really focus on one or two?

    • Do you think these standards will push us to continue working on many areas rather than being able to really focus on one or two?

      Given the second excerpt I quoted, I believe the goal is to focus.

      The proper thing to do would be to sort the vertical integration so that teachers know what they should NOT teach, and that they can feel comfortable doing so because the standard is handled explicitly at a different grade level.

      In any case, your concern is a highly relevant one and would be a great one to send the authors while the public comment period is still open.

      The feedback survey is here.

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