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Brian, these are some posts which may help:
http://commoncoretools.me/forums/topic/expanded-notation-and-order-of-operations/
http://commoncoretools.me/2011/05/29/complete-draft-progression-for-cc-and-oa/#comment-92
http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comment-1999
Cathy, I can see the distinction you’ve made between how the students calculate and what they use to illustrate their thinking. The “equal groups” reference I made was in relation to the bottom margin diagram on p.15 of the Progressions – this was a sloppy mistake on my part.
Essentially though, the trouble is with students calculating using the sharing process shown in that diagram, then not having an obvious means to illustrate their calculation from the options of equations, arrays, or area models. I’m guessing the closest one is equations. Is this correct? If so, could you please provide an example of what the matching equations would look like for the bottom margin diagram on p.15?
The language of the progressions is not necessarily the language you would use with students, and I can see that you would want to be careful about the confusion between multiplicative cancellation and additive cancellation. Still, it’s hard to avoid using the word altogether when talking about algebra. In the end the issue is not which words you use but how people understand them, and I agree with your basic point that we want kids to understand the mathematical meaning behind the word.
Exponents are introduced in Grade 5, but not in this standard, rather in 5.NBT.2. “Order of operations” usually refers to what you do if there are no grouping symbols present, but I can see how you might think this standard also has something to do with order of operations, since grouping symbols can be used to enforce an order.
Thanks Kristin, I agree with you about the IM task not fully representing what is required. The example on page 28 of the OA Progressions is too far at the other end of the scale from the IM example to get a feel for what is required. Your idea about having stories match multiple equations to show what is possible is a good one and I agree that you wouldn’t expect students to use symbols all the time, only when necessary.
The tasks of identifying and drawing points, lines, and line segments is what I am querying. Even rays, really. The tasks you mentioned from the Progressions would benefit from being referenced to the Standards. More importantly though, drawing can be completed using the everyday, non-mathematical understanding of “point” as a particular spot on the page (or screen) and “line” as a shorthand for “line segment”. In other words, differentiating between lines, line segments, and rays doesn’t seem necessary for turtle geometry or for drawing parallel lines. Knowing the difference may help at some later stage, but I can’t see it strengthening ability or deepening understanding in Grade 4. Is it all leading to something greater and in the immediate future for these students?
Also, when the students draw lines, line segments, and rays I assume it has to be something like this: http://youtu.be/JcqCf762y9w?t=2m2s
Is that correct?
People have had these problems before, but I’m pretty much at a loss what to do about them, since all these things work for me (Safari, Firefox, Chrome, Preview, Skim) and I am also on a Mac. Is your version of the Mac OS up to date?
I just produced another version of the pdf by printing this one to pdf. I doubt that will help because I think I already did that, but if you get in touch with me privately at william.mccallum@gmail.com I’ll send it to you by email just in case. Otherwise, maybe find someone for whom it works and ask them to print it out for you.By the way, this progression was largely written by Doug Clements, so it’s not surprising that it shows an awareness of the research. Once all the progressions are in final form and put together in a single document, we will probably add an introduction explaining the purpose and the policy on various issues, and acknowledging all those who helped write them.
I’m thinking you mean 7.NS.2a here:
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as $(-1)(-1)=1$ and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
To some extent using the properties will require developing names for them. The standards don’t require that students know the formal names: that’s really up to curriculum developers. The most important one here is the distributive property, and it’s hard to imagine talking about it without naming it. I’m not sure it’s as important to name the commutative and associate properties. Many textbooks give a name to the combination of these two properties, something like the “any order any grouping principle.” I think that’s fine, but others might disagree. Anyway, the standards don’t settle that point.
It’s certainly food for thought and I can see the connections you’re talking about. I’m not concerned that students won’t understand the measurement model of division. It’s more that the “finding group size” model on page 15 of the Progressions seems to fit easily with the standard algorithm that students need to be fluent with in Grade 6, yet it isn’t clearly mentioned in the last sentence of 4.NBT.6:
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Is there a link there that I’m missing? You seem to suggest that if students use blocks and a sharing model then it is symbolically equivalent to any other model, and that this demonstrated via equations. This is true, but arrays and area are probably closer to what teachers think of as models, and equations as ways of representing those models (models of models?). Not including “equal groups” as a model in the last sentence of 4.NBT.6 has probably muddied the waters unintentionally.
Also, speaking more generally, despite the statement that the Standards do not dictate curriculum or teaching methods, to my eyes 4.NBT.6 is a counter-example (among others). When phrases such as “using strategies based on” and “illustrate and explain the calculation by using” are used it directs teachers towards certain methods. To make 4.NBT.6 devoid of direction it would simply say “Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors. Illustrate and explain the calculation.” When lists of strategies and models are presented as well, then the choice of teaching methods becomes skewed towards those strategies and methods. This isn’t necessarily bad, depending on the goals, but the use of “and/or” makes it more difficult.
Just one single method may not be dictated but, in the specific case of 4.NBT.6, three methods are presented instead. To the first part of my original question, is it the intention that all three models be presented over the course of a year? If they use none of them is that okay too? The models in the last sentence of 4.NBT.6 are not given as examples – they read as requirements – so at least one of them must be used. So I’m not asking for which one, but how many?
