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The first couple of pages of the geometry progression might help give some context here. In particular, this paragraph:
Thus, learning geometry cannot progress in the same way as learning number, where the size of the numbers is gradually increased and new kinds of numbers are considered later. In learning about shapes, it is important to vary the examples in many ways so that students do not learn limited concepts that they must later unlearn. From Kindergarten on, students experience all of the properties of shapes that they will study in Grades K–7, recognizing and working with these properties in increasingly sophisticated ways.
In particular, the uptick from Kindergarten to Grade 2 in the standards you refer to is more in the way that students perceive these shapes than in the shapes being presented. In Grade 2 the emphasis is on recognizing shapes in terms of their attributes rather than simply being able to name them. Thus a second grader is expected to be able to say “this is a triangle because it has three angles,” whereas a kindergartener is expected simply to name the triangle.
From my fading memory, your analysis of the rationale behind the recommendation in Appendix A is correct. I also remember there was an active debate about this, and your point of view also had supporters. My personal view is that enrichment should be considered before acceleration. That is, students who are eating up the material should be given harder problems on grade level material rather than an accelerated and possibly superficial treatment of material from later grades. Acceleration should be an option for students who are truly prepared for it, and hopefully that fraction of the population will increase over time. This doesn’t, of course, answer your question about whether to start acceleration in Grade 6 or 7 … sorry!
Jim, all good points, and ones that will inform the revision, thanks. (It looks like a rectangle to me, too.)
It’s hard to answer this question at this level of generality, it would be better if we have a test case to discuss. There are two opposing tendencies in implementing standards, both undesirable if carried to an extreme. One is the tendency to want to cover everything with equal intensity, so that the curriculum becomes choked with undergrowth; the other is the tendency to ignore things that you find inconvenient or that are not in the textbook you are using, so that the curriculum becomes parched and arid. So what I want to say in answer to this is “use your own judgement!”; but I also want the person using the judgement to have a thorough knowledge and understanding of the standards.
Trying to be more helpful, I would say that some high school standards are clearly more important than others. This is not a matter of subjective judgement, but a matter of seeing how the standards fit together and detecting which ones are really consequential because they have lots of connection to other standards. So, for example,
N-RN.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
is an important piece of knowledge, but the student’s world isn’t going to fall apart without it. Whereas
A-SSE.1. Interpret expressions that represent a quantity in terms of its context.
is foundational to all the standards on algebra and functions.
In other words, there are peaks and valleys in the standards, and it isn’t wrong to point them out. This might be a good point to revisit Jason Zimba’s essay on examples of structure in the standards.
The standard is limited by what else students know at this grade level. This includes the cases you mention, and it might make sense for a curricular implementation of the standards to leave it at that. But there are other transformations that are possible, it seems to me. You could rotate by a multiple of $\pi/2$ around any point on the coordinate grid, for example.
The first interpretation is the correct one: the rules for the patterns are given to the student. But then students might see a relationship between the two patterns that is not given explicitly, but that follows from the two rules.
I suppose there are many standards that, when taken to their fullest extend, become quite involved. As a matter of common sense, we can’t take all standards to their fullest extent all the time. That said, this strikes me as a nice activity aligned to 7.G.3 if the teacher has the time in the curriculum. The standard does not ask students to describe all possible cross-sections, and I think it would be absurd to make that a curricular requirement. But the activity of trying to find all possible cross-sections is another matter. Not all students will be able to find them all, but the process of trying is good mathematical exercise, bearing not only the specific content standards but also on the mathematical practice of perseverance (MP1).
This is up to curriculum and assessment writers to decide, and the answer is probably not fixed across all implementations.
So the question is whether “when there are no parentheses to specify a particular order” as an adverbial clause modifying the verb “perform” or an adjectival clause modifying the noun “order”? Since that noun occurs in the adverbial phrase “in the conventional order,” the second interpretation is quite awkward. If that had been the intended meaning it would have been better to write “in the order that is conventional when there are no parentheses …”. And, as you point out, this interpretation suggests that there is a special convention for problems with no parentheses, which you quite correctly debunk. So all in all I think the first interpretation is the most natural. There does remain some ambiguity, however. Are the parentheses to be absent from the entire expression in order for the standard to apply, or merely absent from the part of the expression where the matter of order is at stake? I think the latter interpretation makes more sense. That is, the standard expects students to interpret 5 + 2(8+7) correctly.
[I’ve rewritten this answer a few times since I posted it a couple of hours ago.]
The mention of mode on page 9 of the publisher’s criteria comes under the heading: “Materials do not assess any of the following topics before the grade level indicated” (emphasis added). That is, a criterion for alignment is that materials avoid assessing topics not mentioned in the standards. The table mentions mode in Grade 6 because measures of center are introduced in Grade 6, and curriculum writers might include mode as a measure of center. But it should not be construed as suggesting that mode must be introduced in Grade 6. Rather it is saying “assess mode in Grade 6 if you must but certainly not earlier.”
In short, my answer is no, you do not need to introduce mode in Grade 6. Of course, the assessment consortia might not agree with me.
The middle school curriculum before the Common Core was full of trivial, meaningless questions about small, artificial, and context-free data sets. You know the sort of question I mean: “What is the mode of 1,2,6,6,6,6,6,23?” Who on earth cares what the mode of this set is? Or the median or the range? Summary statistics serve to describe large data sets. There is no need for such statistics for the set 1,2,6,6,6,6,6,23. Everything you need to know about that set you can see just by looking at it (if indeed you need to know anything at all).
A curriculum could meet the Grade 6 standards on Statistics and Probability by working with large data sets arising from real contexts, using technology to plot them and compute their summary statistics. Students should be able to answer statistical questions, display data graphically, choose appropriate summary statistics and interpret them in terms of the context. That’s what the Grade 6 standards say.
Vocabulary should serve understanding, not replace it. A test question that fails a student who meets the Grade 6 standards as described above but forgets which is the mode and which is the median is a bad question. We can’t stop bad test questions, but we can avoid letting them drive the curriculum.
I’d also be interested to know of efforts in this area. CCSS is already more substantial than many previous middle school curricula, so accelerating it has to be done carefully and should be a matter of choice, not mandate. An alternative to acceleration is enrichment; instead of rushing on to the next topic, do harder and more complex problems on the current topic. For many students, having solid grasp of ratios and proportional relationships by the time they reach high school will serve them better than having been exposed to quadratic equations.
