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I did a little searching and found a description from Ravitch of “field testing.” From: http://dianeravitch.net/2012/07/09/my-view-of-the-common-core-standards/
“I have worked on state standards in various states. When the standards are written, no one knows how they will work until teachers take them and teach them. When you get feedback from teachers, you find out what works and what doesn’t work. You find out that some content or expectations are in the wrong grade level; some are too hard for that grade, and some are too easy. And some stuff just doesn’t work at all, and you take it out.”
The comment above doesn’t mention education research at all, but it does help me understand the quote about reaching consensus. My impression after reading three Ravitch books (from 1995, 2010, and 2013) is that she’s not a user of subject-specific education research, so probably doesn’t see the relevance of work on learning trajectories for early grades and their implications for later grades. She does cite studies that use test scores as measures, noting sometimes that reading scores rose and math scores didn’t for certain studies. That’s about as subject-specific as she gets when reporting research—at least in those three books.
This Ravitch comment about development may also be helpful. From http://gothamschools.org/2010/10/29/city-official-and-biggest-critic-find-slivers-of-common-ground/#more-48933:
“I’m very supportive of the idea of developing new assessments, and I think it’s a very important thing. But it will take years.
Just as these common core standards were written in a little over a year — it took me three years working on the California history standards. I worked on history standards in other states, and it was never done in only a year. So I would like to think that it’s going to take a lot of time to do this well because anything that’s done hurriedly is not going to survive….
I’m very happy that there’s money out there to develop new tests, but don’t think that they’re going to be available next year or the year after. If they’re good tests, it could be three to five years. And then they have to be tried out….So this is not going to be in time for the next election.”
I can remember also being skeptical about the short turnaround for standards development. (I worked on PSSM in its third and last year as an additional writer.) I think that one difference is that people working on CCSS worked very intensely via email. A second difference is that PSSM had longer illustrative examples. PSSM is 402 pages. CCSS is 93 pages.
Ravitch worked on the CA history framework for 1997 and its 2005 update. The 2005 version is 249 pages long and has suggested courses and appendixes. I found no references to education research in it. That doesn’t rule out its use but does reinforce the impression that I gained that subject-specific education research isn’t one of Ravitch’s considerations.
Here are a few more comments on “multistage” and “compound.”
I did a google search on “multistage event”. I don’t get a lot of hits related to probability and statistics for that but I do for “multistage experiment”, which gives the term a slightly different emphasis. An experiment is something like tossing a coin, tossing two coins, rolling a die, etc. It’s what you do to generate an outcome.
“Compound event” appears to have two definitions which are not equivalent. Sometimes the idea that there are two different definitions of a term comes as a shock, but it does happen. Trapezoid is one example.
The NCTM book Navigating Through Probability, 6–8 says on p. 11: “An event is the outcome of a trial. A simple event (usually called an event) is a single outcome. A compound event is an event that consists of more than one outcome.” For the experiment “roll a die,” it gives the example of “six on top face” for simple event, “prime number on top face” for compound event. Neither of these is the result of a multistage experiment (“roll a die” has only one stage).
That’s the definition of compound event that I grew up with. Using that definition and using the sample space described above (i.e., there are four 7th graders on the list), the event “picking a 7th grader” is a compound event and “picking an 8th grader” is a simple event.
In the CCSS, “compound event” is more akin to “an outcome of a multistage experiment,” e.g., an outcome of rolling two dice, as Bill has already discussed. Under that definition, “picking a 7th grader” as described above is not a compound event. The experiment is “picking a student” which has only one stage.
I can see that we need a note about this in the S&P Progression.
This might be answered by the Publishers Criteria here: http://www.achievethecore.org/files/1413/6545/4893/Math_Publishers_Criteria_HS_Spring_2013_FINAL.pdf
Some comments though . . . one thing that I notice in discussions of curriculum and instruction is that a given topic can be taught in different ways. Just saying that a given topic is included isn’t necessarily evidence that something (e.g., curriculum materials) is standards-aligned or not. Obviously, you’re thinking of a particular approach rather than just a topic. The question might become “How does this approach fit with the standards?” I’d suggest thinking of “standards” (plural) rather than just the Pythagorean theorem or just standards that involve the Pythagorean theorem.
Have you looked at p. 13 of the Functions Progression? It can be downloaded here: http://ime.math.arizona.edu/progressions/.
Does it help to look at p. 17 of the Number System & Number Progression posted here: http://ime.math.arizona.edu/progressions/. I see that the document label on that page is misleading, you need to click on the link that says “Draft 6–8 Progression on The Number System”.
I’m not sure how the standard points to physical situations, could you explain?
It may help to look at the discussion under HS Number and Quantity here: http://commoncoretools.me/forums/topic/use-of-significant-figures/
I just downloaded the Algebra Progression. This seems to be fine on my mac, both with Adobe Reader and Preview. I just checked out Functions which also seems fine. I wonder if your OS needs updating.
I tend to wait a long time to update, so am familiar with the problems caused by not updating. With one of my old systems, I couldn’t open some progressions with Adobe but they were fine with Preview. You may want to download another copy and stay with Preview as your progressions reader.
Not surprisingly, some of the examples that lhwalker mentions bear a strong resemblance to examples in in the NBT Progression.
kimbergunn, it sounds as you might be thinking that an area model must be carved up into units. When students begin using area models, it seems that initially they should maintain the connection between understanding the connection between units of area and units of numbers, but that certainly becomes unwieldy to show explicitly by drawing unit squares when numbers get large (and we hope the connection has been built in the context of smaller numbers).
The area model on p. 15 of the NBT Progression does not show individual units of area. (It shows a 3-digit dividend and 1-digit divisor, I hope it’s obvious how an area model might be drawn for a 4-digit dividend and 2-digit divisor.) Also, the standard allows an equation as an illustration.
The discussion of introducing the commutative property for addition here (http://lipingma.net/math/One-place-number-addition-and-subtraction-Ma-Draft-2011.pdf) might be helpful in thinking about how to introduce it for multiplication.
Note that there are various different definitions and notations for ratios and fractions. This got discussed a while back (November 2011) on the blog here: http://commoncoretools.me/2011/09/12/progression-on-ratios-and-proportional-reasoning/.
CCSS treats a ratio of two numbers as a pair of numbers rather than a fraction. So, one answer is that a/b is one number (assuming that b isn’t zero) and doesn’t determine coordinates of a point in the plane. (I’m assuming that we’re not dealing with complex numbers.)
Maybe this helps to make an answer more obvious because the only choice is how to plot the pair of numbers a and b. That depends on what the coordinate axes are supposed to be representing. If you’ve got a ratio of 5 cups of grape juice to 2 cups of peach juice, and cups of grape juice corresponds to the horizontal axis and cups of peach by the vertical axis (as in RP Progression, p. 4), then the ratio corresponds to the point (5, 2). If cups of peach were represented by the horizontal axis, then the corresponding point would be (2, 5).
Thanks for mentioning this. It was also mentioned in February (I think somewhere on the forum). Anyway, the revised figure is now up on my blog: http://mathedck.wordpress.com/?attachment_id=431.
I started a forum thread with some curriculum ideas here http://commoncoretools.me/forums/topic/prove-that-all-circles-are-similar-2/#post-1966.
Putting equations and expressions in particular forms comes up under a variety of headings in the Algebra forum: http://commoncoretools.me/forums/forum/public/hs-algebra/.
A general principle of the Standards is described in the Algebra Progression, p. 4, http://ime.math.arizona.edu/progressions/:
“The Standards emphasize purposeful transformation of expressions into equivalent forms that are suitable for the purpose at hand. . . . Each is useful in different ways. The traditional emphasis on simplification as an automatic procedure might lead students to automatically convert the second two forms to the first, before considering which form is most useful in a given context.”
In the quote above, for “simplification” one could substitute “putting the equation of a line into standard form” or “putting the equation of a line into slope–intercept form.”
Re systems of equations: there is the beginning of discussion here: http://commoncoretools.me/forums/topic/a-rei-5-what-does-it-meanlook-like/. I’ll try to contribute more to that thread.
Thanks for this comment. Yes, the composing shapes are adjacent not superimposed.
I don’t know if you’ve looked at the Progressions, but you might check out the K–6 Geometry Progression, http://ime.math.arizona.edu/progressions/
There’s a discussion of composing shapes to build pictures and designs on p. 7, but I see that the discussion might say more about other shapes before it gets into pictures and designs.
There’s an illustration of shapes composing another shape on p. 10.
The grades 6–8 Statistics and Probability Progression discusses box plots in grade 6. It can be downloaded here: http://ime.math.arizona.edu/progressions
Two important building blocks for understanding relationships between fraction and decimal notation occur in Grades 4 and 5. In Grade 4, students’ understanding of decimal notation for fractions includes using decimal notation for fractions with denominators 10 and 100 (4.NF.5; 4.NF.6). In Grade 5, students’ understanding of fraction notation for decimals includes using fraction notation for decimals to thousandths (5.NBT.3a).
Students identify correspondences between different approaches to the same problem (MP.1). In Grade 4, when solving word problems that involve computations with simple fractions or decimals (e.g., 4.MD.2), one student might compute 1/5 + 12/10 as .2 + 1.2 = 1.4, another as 1/5 + 6/5 = 7/5; and yet another as 2/10 + 12/10 = 14/10. Explanations of correspondences between 1/5 + 12/10, .2 + 1.2, 1/5 + 6/5, and 2/10 + 12/10 draw on understanding of equivalent fractions (3.NF.3 is one building block) and conversion from fractions to decimals (4.NF.5; 4.NF.6). This is revisited and augmented in Grade 7 when students use numerical and algebraic expressions to solve problems posed with rational numbers expressed in different forms, converting between forms as appropriate (7.EE.3).
