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This sounds awful. No, there is no goal that students be able to name the practices. The standards are not intended to be read by children, but rather by their teachers and those who design curriculum for children.
I think highly of both thinkers, and have read both their writings on this, but don’t have a detailed comparison of their different interpretations of the geometry standards to throw out right now.
I think highly of both thinkers, and have read both their writings on this, but don’t have a detailed comparison of their different interpretations of the geometry standards to throw out right now.
I don’t think this standard gets down to this level of detail (distinguishing hexagons from octagons) and I think it is certainly beyond the standards to be assessing Kindergartners on this distinction. The list in parentheses is neither inclusive nor exclusive. Like most other lists in the standards it is there for guidance, and such guidance must always be wedded to common sense. There are many overly focused assessment schemes that are simply trying to read more fine grained resolution from the standards than is there; the standards were not designed to be friendly to such schemes, rather preferring schemes that focus on larger coherent units of mathematical knowledge.
This is a good discussion, and I don’t have much to add, except to confirm that students are not required to calculate standard deviation by hand. Students should be seeing real data sets where this would be absurd.
Yes, you are right, this isn’t called out explicitly. And, in fact, the main point for students to understand eventually is nesting of grouping symbols, rather than the hierarchy itself. That is to say, if you can parse a complex expression with parentheses nested 2 or 3 deep, then that is the main point; it doesn’t particularly matter if you go parentheis, bracket, brace, as I was taught. It was really this idea of nesting that I wanted to get across, and mainly the idea that it doesn’t have to happen in Grade 5.
This whole subject is an area where the standards are pretty agnostic. Reading the conventions of mathematical notation is important, but conventions themselves are not mathematical concepts. So, the ability to read nested grouping symbols is implicit in A-SSE.1–3, for example, but not explicitly mentioned.
I guess in Kindergarten students might just be saying “bigger” and “smaller,” but I don’t think they need to wait until Grade 6 to see “greater than” and “less than.” In fact, you want bigger and smaller to go away sooner than that, because of the confusion this could cause with comparisons of negative numbers.
On “wide part,” I would say that’s a little different from alligators eating something, because it relates quantities visible in the symbol (the width of each end of the symbol) to the quantities in the comparison; there is no eating action here, which I agree is extraneous!
This follows from the fact that all circles are similar. It’s quite fun to figure out why using the definition of a circle and the definition of a similarity transformation. (I can supply the answer later if you want.)
Now, a similarity transformation with scale factor $k$ transforms any length by a scale factor of $k$, including the arc length. So, just as the radius gets multiplied by $k$, so does the arc length. This means that the ratio between the arc length and the radius stays equivalent no matter what the radius; in other words, the arc length is proportional to the radius.
So this tell us that
$$
\mbox{arc length} = \mbox{constant} \times \mbox{radius}.
$$
Setting the radius equal to 1 tells us what the constant is: it’s just the arc length for a circle of radius 1, which is exactly the radian measure of the angle.Sorry, the link didn’t paste correctly: http://www.tutorvista.com/content/math/hexagon/
Aside from anything else you’ve asked, regarding hexagons it may be helpful to emphasize the number of sides/corners as being the defining feature. If students are getting confused between a hexagon and an octagon it may be that they are reasoning purely visually, rather than counting the sides. To check if this is the case try using a variety of hexagons for identification. This page has some examples:
You are very welcome! Sorry if I sounded a little bit crabby in my last answer. The term “research-based” sometimes does that to me.
We looked at many sources, including national reports, well-regarded state standards, and standards of high achieving countries. There is no agreement on where to put operations with integers. Singapore puts them in Grade 7. Massachusetts started with addition and subtraction in Grade 6, excluding subtraction of negative numbers. Curriculum Focal Points puts them in Grade 7. And so on.
As for research, the way you ask the question makes it sound as if you think that the placement of operations with integers in Grade 6 is itself research-based. But is it? How would you design a research experiment to determine the correct grade placement of operations with integers, or any other topic? Doesn’t it depend on what else you are doing in Grade 6, and what you have been doing in Grades 1–5, and what you plan to do in Grades 7–12? The research would have to look at the entire sets of standards. There has been some such research, for example the research of William Schmidt and Richard Houang, but they don’t have conclusions about specific grade level placement of specific topics. Rather they address large scale properties of the standards, such as coherence. My guess is that you could have a coherent set of standards which places operations with integers in Grade 6 and one which doesn’t; and you could have an incoherent curricula which do the same two things. The important thing is not the exact grade level placement but the coherence.
The standards do not specify how the high school standards should be arranged into courses. Maybe you are looking at Appendix A, which is one example of how it might be done. But it’s just that, an example. Notice that Appendix A also gives a sample integrated sequence. There may be all sorts of practical reasons to start with Algebra I (e.g., because that’s what everyone else is doing and it is useful to coordinate with them) but there is no specification of that in the standards.
