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Duane, I’ll answer both posts here. You are certainly on the right track with all of this; let me add a few comments. In addition to written methods and algorithms, the standards refer to strategies in a number of places. The distinction between a strategy and an algorithm is that an algorithm is general, it works in all possible cases, whereas a strategy might be specialized (e.g. shifting a 1 when one of the addends ends in 9). There are a number of places where the standards ask students to use strategies and then relate them to a written method (in fact I think all the occurrences of the phrase “written method” are side by side with strategies). So the idea here is really that students should be able to formulate the strategy in writing (“you break a 1 of the second number and add it to the first to make a number in the tens, and then you add the two numbers”). So I would modify your progression a bit and say it is strategies/written methods -> algorithms -> the standard algorithm.
As for your question about adding and subtracting decimals, the standards don’t require use of the standard algorithm for that until Grade 6; students can use other algorithms in Grade 5.
And, as for assessment, I don’t think you can assess standards before the grade level in which they occur. But, as you noted, a curriculum would probably be moving towards the standard algorithm before the grade in which it is mentioned in a standard. Actually, I’m not sure how you can tell in assessment which algorithm a student used, so the standards which require fluency with the standard algorithm might be classroom standards, not assessment standards.
Yes, number lines and graphs are tools; for that matter, equations are tools. Or, more precisely, they can be tools if students make conscious use of them to solve problems. A concrete model is any physical setup that models a piece of mathematics; that includes manipulatives in elementary school, but could also include things like models of sliced cones used to visualize conics in high school, e.g. Dandelin spheres.
I’m hoping a California reader with more knowledge of the current situation will chime in here. I did hear a California official say at a meeting a few months back that some commission was going to recommend moving to the Common Core which, as you say, has a fair amount of Algebra I in Grade 8, and has a carefully designed progression for getting students there.
Fred, well, this really is just a matter of opinion, so I’m speaking here as a private citizen, not as a writer for the Common Core. But, for what it’s worth, my opinion is that acceleration is greatly over-used, and that for most bright students it would be better to deepen within grade level rather than accelerate beyond grade level. That is, study harder and more interesting problems on the material in the current curriculum, rather than race through it and go on to the next grade. And I would add that the Common Core provides a structure of coherence and connections that make this easier to do than it might have been previously. All that said, I know that many parents demand acceleration, and school districts have to deal with that.
Duane, thanks again for these careful questions. The standards regard mixed numbers and decimals as fractions; or, more precisely, a mixed number such as 3 1/2 is a sum of fractions, namely 3/1 + 1/2, and a decimal such as 0.3 is just a different way of writing the fraction 3/10, not a different sort of number from a fraction. So the Grade 4 example is an example of multiplying a fraction by a whole number.
The same philosophy pervades the second example you mention, which illustrates both 4.NF.5 and 4.NF.6:
4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.
4.NF.6. Use decimal notation for fractions with denominators 10 or 100.
The text at the top of the blog has a link to them. I am working on getting them incorporated into the new forum.
Brian, the work with story problems starts in Grade 7 with 7.EE.4: “Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.” Part (a) of this restricts the equations to those that can be solved in one-step. 8.EE.7 then lifts this restriction, but it should be assumed that work on using equations to solve real-world and mathematical problems continues, i.e, they don’t become suddenly decontextualized in Grade 8.
She is quite right that the divisibility rules are based on the place value system. But the point of focus is not to include everything you might want to include just because it’s a nice topic. That’s how we ended up with the mile-wide-inch-deep curriculum in the first place. I think middle school is already pretty full, and would advise against this. In general, LCM and GCF are quite circumscribed topics in the Common Core, not intended to take up a lot of time in the curriculum. (I’m actually working on a draft of the NS progression as we speak.)
Joshua, the standards do indeed quite consciously avoid the word “simplify”, the point being that different forms of expressions are useful for different purposes, and there is often no mathematical reason to call one of those forms the simplest. This is in accord with MP7, Look for and make use of structure. Students are expected to be able to make strategic choices about what manipulation they perform for the purpose at hand, rather than respond mechanically to commands like “simplify”.
Sorry, that problem should be fixed now. I am still learning how to run this blog. It’s getting more complicated!
Send them to illustrativemathematics@gmail.com, the email address for Illustrative Mathematics. I know they are eager for ideas on this.
Great question, Eric. I think the standards have a lot to offer for adult education in Algebra. At least some of the students in adult education are there because they either didn’t get (or are not getting) something out of their high school experience. The standards emphasize a coherent viewpoint of Algebra which might help them where manipulating symbols without meaning did not.
