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Bill, that kind of sounds like fluency with the standard division algorithm in Grade 6 is optional. In practice students will be all over the place in terms of accomplishment, you’re right, but the specific goal for 6.NS.2 is fluency with the standard algorithm, isn’t it?
This comment is a touchy one for me. While the CCLS states the long division algorithm will be an expectation for 6th graders I KNOW there are other efficicient methods for long division. I NEVER understood long division very well as a child and relearned to divide using Marilyn Burns’ Big seven way of dividing as an adult. WOW! Once again I realized that I received horrible math instruction as a student. I see no reason for students to long divide digit-by-digit, removing all place value. Honestly speaking it is more important to me that my students recognize division situations. To sit and practice naked division calculations seems a waste of precious learning time when we will use calculators for that. That being said I still think single digit division fluency is a must to build the Base Ten understandings of division.
“CCLS” means “Common Core Learning Standards” and refers to the New York state standards, right? This blog is about the CCSS. They seem to be identical except for the title and title page.
My sense is that some people associate algorithms with absence of meaning (e.g., association with place value) because that’s the way they are often taught in the US. But, they don’t have to be.
Maybe this excerpt from the Number and Operations in Base Ten Progression draft will help:
Another component of understanding general methods for multidigit division computation is the idea of decomposing the dividend into like base-ten units and finding the quotient unit by unit, starting with the largest unit and continuing on to smaller units. As with multiplication, this relies on the distributive property. This can be viewed as finding the side length of a rectangle (the divisor is the length of the other side) or as allocating objects (the divisor is the number of groups).
On a related note to algorithms, reading through the NBT Progressions, page 3 notes a distinction between “general methods” and “special strategies”. General methods are defined as applicable to all numbers (in base-ten) but not necessarily efficient. They may be efficient but it is not always the case. Special strategies are defined as applicable only to certain cases or applicably to more cases only with “considerable modification”.
The example given on page 3 for a special strategy is 398 + 17, which is rewritten as (398 + 2) + 15. A general strategy example is given as combining like base-ten units, i.e. 300 + (90 + 10) + (8 + 7). Another example of a special strategy is given on page 7 (margin) where you start with one number then count on tens then ones individually, e.g. 46 + 37 –> 46, 56, 66, 76, 77, 78… and so on.
The special strategy given on page 7 doesn’t seem all that difficult to extend to three-digit numbers (i.e. count hundreds, tens, then ones) and beyond, or by adding instead of counting (as noted on page 7). Time-consuming, yes, but not requiring considerable modification. It’s not all that different from counting by ones which was defined on page 3 as a general method. Given its close similarity to counting by ones, and its applicability to all cases, what makes this strategy “special”?
Also, a distinction is made between algorithms and strategies (p.3), with strategies being broken into special and general as discussed above. The top example in the margin of page 7 shows the standard addition algorithm but it is labeled as a general method, i.e. a “strategy”. So I’m confused – is the diagram showing an algorithm or a strategy?
Duane,
Sorry for the long delay in replying to this, but it made me realize I needed to get that revised version of NBT finished. It is now posted. Could you take a look and see if it helps with this confusion? Happy also to answer more questions, now that it is done.Thanks Bill, I finally had a decent chance to read through the new draft. I think the explanations are much clearer than before, especially the discussion surrounding “efficient, accurate, and generalizable methods”. There are a few things I’m interested in clarifying though.
One is that on p.14 there is mention of students adding and subtracting through 1,000,000 using the standard algorithm in Grade 4. I recall reading somewhere else on this blog that the standard algorithms need only be extended as far as necessary to demonstrate that they are generalizable. In light of that comment, is this reach to 1,000,000 viewed as what is necessary to accomplish the light-bulb moment or is it simply a border to stop teachers going any further?
Another is there seems to be a mismatch between an explanation on p.9 and a method in the margin. At the start of the fourth paragraph a description is given of the first (presumably top) method shown in the margin. The statement is given that “The first method can be seen as related to oral counting-on… in which an addend is decomposed…[and] successively added to the other addend.” Further down in that paragraph this is shown as essentially: 278 + 100 = 378 –> 378 + 40 = 418 –> 418 + 7 = 425. However, this is not what the method in the margin shows. Instead the method shown is simply an expanded form of the standard algorithm and relies on splitting both addends into hundreds, tens, and ones. Am I interpreting the paragraph text and margin method correctly? (On a related note, something that may help generally in all final versions of the Progressions documents is labeling any figures “Figure 1”, “Figure 2”, and so on.)
A final query is about a term on p.7, 3rd paragraph. What is a “5-group”?
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