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Hi Bill,
I have a question about how the recording is done for the division algorithm in your progressions document, “K-5, Number and Operations in Base Ten.” There is one example (the picture on page 16, underneath 5.NBT.5) that shows 1655/27. It shows 50 x 27 = 1350 as part of the calculation, but does not show how students record that multiplication—many students will employ the multiplication algorithm for such a calculation.
My question: Where do you want them to put the 3 hundreds (from 7 x 50 = 3 hundreds 5 tens)? (Obviously they can use mental math for this particular problem, but I’m thinking of general problems where employing the multiplication algorithm really is the smart strategy.)
I think the way the multiplication algorithm shown on page 14 is very clever and I would like to use it as part of the Story of Units curriculum. However, I also have to think ahead to how students might be able to use that form of the multiplication algorithm as part of performing a division algorithm.
In the case I just mentioned, it does not seem prudent to put the 3 hundreds above the 0 in the next calculation (i.e. the 0 in 20 x 50 = 10 hundreds) using the same technique as the multiplication algorithm shown on page 14, because, ultimately, then you have to add the 0+3 hundreds and then subtract the result from 6 hundreds in the dividend. This “adding and subtracting” in the same column would probably lead to student confusion and error–what am I adding, what am I subtracting?
A natural place to record the “3 hundreds” would be above the 2 tens in 27, as in the old division algorithm because that technique separates the addition and subtraction into two different regions of the algorithm. But if this is desired then probably the best way to teach the multiplication algorithm is the old way as well, for then the recording of the multiplication algorithm in the old division algorithm would then match very closely with the way of recording the “carry’s” in the old multiplication algorithm.
I’m truly interested in your ideas for keeping the “addition subtraction” issue separate on the long division algorithm in such a way that allows me to use the multiplication algorithm as depicted on page 14—I would like to use that algorithm if I can!
Best,
ScottScott, just to summarize your point to make sure I have it clear, you are worried about the conflict between the method of recording multiplication on p. 14 of the progression (bottom right) being out of sync with the traditional way of recording long division. I can certainly see this as a worry. But I guess the method on p. 14 is trying to get away from the other possible error students might make with the traditional way of recording the algorithm, namely that they add the carried 3 to the 2 above which it sits before multiplying by the 5. I don’t really have a definitive answer here; you are deep in the problems of curriculum design, for which the progressions are intended to provide ideas and support, but not all the answers. However, I do agree that it is worth noting this point in the progression, and will make sure it is included in the final draft. Thanks very much for this detailed reading.
Bill,
I’d like to repeat a question I posted to another discussion that may fit better here:
Do the “general methods” shown in the margin on pages 13 and 14 of the 7 April 2011 “K–5, Number and Operations in Base Ten” progression qualify as “the standard algorithm” for the purposes of 5.NBT.5?
I understand and appreciate your reluctance to hold forth on what you see as details of curriculum design, but on such a basic question as the above, I think it’s fair to expect you to respond. When CCSS demands “the” standard algorithm, it’s hard to comply unless we know exactly what that’s supposed to be.
Andy
Andy, I just replied in the other forum, here. I’m not reluctant at all, but I work through replies to comments in the order in which I receive them. Since our semester started in late August I have found it more difficult to keep up, and unfortunately I cannot guarantee any specific turn-around time.
Hi Bill,
I discussed the multiplication algorithm on page 14 of the NBT Progressions document with a number of mathematicians. Many pointed out how completely baffled they were with the “44” and “720” appearing on separate lines. (How do you get 44 from any partial product of 94 and 36?) I explained that there was a mini-version of the lattice method built into the algorithm (one needs to read the numbers down and to the right to give 54 tens and 24 ones), but agreed that changing the font size of the 5 and 2 was bad form—at a minimum they should be the same size as the 44 so that it is easier to recognize the “54” and “24.” Changing the font size does not alleviate the problem I discussed in my first post, however.
Finally, while talking with a Chinese mathematician about this problem, we realized that this algorithm was actually an incorrect modification of the standard Chinese multiplication algorithm.
In the standard Chinese algorithm, the first line under the 36 would contain a small “2” and the second line would read “564.” The 564 would then accurately represent the sum of the partial products. The process would go like this: 4 x 6 = 24 = 2 tens 4, put the 4 in ones column and put a small 2 in the tens column (where it is now). Next, 9 tens x 6 = 54 tens. Add the 2 tens to get 56 tens and write 56 in the hundreds and tens column to get 564. In this algorithm, the “2” is only a reminder and isn’t used after 564 is established (which is why it written smaller).
The standard Chinese multiplication algorithm is as fast, efficient, and as error free as the standard U.S. multiplication algorithm (with extra benefit of placing “carries” in the correct place value locations). Also, and this is the main point of this post, it embeds very nicely into the division algorithm without the big concern I had in the first post above. (Students still have to be careful with recording small numbers versus big numbers, but now at least, “small” and “large” mean something.)
I understand why you might be loathed to make a major change to the NBT Progression document, but in this particular case I really think it is warranted. The algorithm as presented doesn’t make sense (again, the 44 means?) and cannot be nicely embedded into the long division algorithm.
As of now, the multiplication and division algorithms in the NBT Progressions document are not strong enough to risk switching away from the standard U.S. algorithms in my curriculum (or even to suggest it as an alternative to the standard U.S. algorithm). Frustrating!
Best,
ScottScott, just to let you know, I’m not ignoring this, but I asked Sybilla Beckmann or Karen Fuson to respond, and they haven’t gotten to it yet. If they don’t answer soon I’ll give it a shot.
Here is a response from Sybilla Beckmann to Scott’s question:
Hi, Below is a response from Karen Fuson and me. We will also respond to another posting about the multiplication algorithm (which is somewhere, and I saw it before, although I don’t know where it is now), but that will require referring to a page of figures, which we will post at the Mathematics Teaching Community
https://mathematicsteachingcommunity.math.uga.edu/
so stay tuned for that. (It may take us a few days.)
Here is from Karen Fuson and me:
Great question. We address division first and then relate that answer to multiplication. The best option for division is for students to multiply the tens place in the divisor and mentally add in the trades from the product of the ones place because then there will be no extra confusing numbers interfering with the subtraction of the partial product, which as you pointed out could be problematic. For students for whom this is difficult, a second option is to write that number to be added (3 hundreds in the 27 into 1655 problem) just below the 6 hundreds where it waits to be added. Or it could be written below on the subtracting line. The key thing here is to cross out that number after it has been added in so nothing further is done with it. A third option is to do the multiplication of 50 x 27 out to the side of the division problem, record the 3 wherever in that case (see below), and then put the product 1350 under the 1655. For weaker students that may be a good way to start to show what is really happening, at least initially.
It does not seem to be a good idea to put the 3 hundreds above the 2 tens in the divisor 27 because that is putting hundreds in the tens place. Also, if students are writing some estimated divisor such as 30 in this case, that estimate 30 might be written above the divisor.Here is a reply to Scott’s September 30, 2012 posting from Sybilla Beckmann and Karen Fuson:
We strongly disagree that the multiplication method Scott refers to is an “incorrect modification.” First, it is a systematic and practical way of recording the steps of multiplication. Second, it has some advantages for students, which we discuss at the Mathematics Teaching Community https://mathematicsteachingcommunity.math.uga.edu in a posting titled “What is the standard algorithm?” at https://mathematicsteachingcommunity.math.uga.edu/index.php/230/what-is-the-standard-multiplication-algorithm where we also discuss the Chinese algorithm that Scott mentions.
Here is my reply to Sybilla and Karen’s post on October 18, 2012:
Dear Sybilla and Karen,
Maybe you are misreading my misgivings: I’m actually asking for your help.First, it is nice to see that you have already thought through the different methods/algorithms displayed in the figure and discussed at:
(For other readers of this forum, Method 4 is the one I discussed in my previous post above.)
In the curriculum I’m helping to create, we will use Methods 1 and 2 (and a combination of both of them) to help explain multiplication by a 2-digit number in the curriculum. (We will probably avoid Method 5 for the same reasons you state in the discussion.) The plan is to help students understand multiplication in several ways (area, partial product, distributive property, etc.) and, as a capstone to that experience, to learn a standard algorithm that is “accurate and reasonably fast” as required by the K-8 Publishers’ Criteria for CCSS-M.
The statement “accurate and reasonably fast” seems to imply Methods 3, 4 or 6 (with Method 2 as a slightly slower backup). Students also need to learn an “accurate and reasonably fast” division algorithm. Thinking of a particular set of addition, subtraction, multiplication, and division algorithms as grouped together into a system of algorithms, Methods 4 and 6 stand out for their ability to embed into their corresponding division algorithms without confusion (at least without confusion in regards to the embedding–Method 6 has other problems as you know).
In my previous post I was basically begging that a figure similar to the one at the link above be included in the NBT Progressions together with a corresponding figure for division algorithms. The discussion below the figure is insightful and I hope can be incorporated as well (together with similar descriptions for the division algorithms).
Having Method 4 clearly articulated in the NBT document together with how it embeds into its corresponding division algorithm will help tremendously any attempt to move to a system of algorithms that makes sense with regards to place value.
With warmest regards,
ScottIn relation to the article Sybilla posted here ( https://mathematicsteachingcommunity.math.uga.edu/index.php/230/what-is-the-standard-multiplication-algorithm ), I’m interested in the longer term implications of seeing all written recordings in Sybilla’s article as demonstrations of the standard algorithm (as suggested in para. 2 on p.1). Presumably this means that any of the methods could be used from Grade 5 onward if they are equivalent in content, if not in presentation. This thought is softly echoed on p.13 of the NBT Progressions: “…minor variations in methods of recording standard algorithms are acceptable.”
When I look at 5.NBT.5 and 6.NS.3 the extent of the number range is only given as “multi-digit”. For some of the methods in Sybilla’s article the number of digits is irrelevant – if you had the time and inclination, Methods 3, 4, and 6 could see you through whatever range you wanted. Method 2, however, rapidly becomes cumbersome once you start multiplying 2-digit by 3-digit numbers.
To assist teachers in making decisions about when a method outlives its usefulness and plan for that eventuality, it would be useful to know what “multi-digit” means for:
– 5.NBT.5
– 6.NS.3
– and for good measure, 6.NS.2Multi-digit for all these standards means “enough digits to reveal the algorithm in all its generality, but not so many as to constitute pointless torture of children.” Exactly where that point is depends a lot on the classroom context, and is a matter of opinion anyway. I would guess that 3 x 3 and 2 x 4 are enough for 5.NBT.5.
I agree! Past the limit you mentioned it becomes rather pointless. I think giving some guidelines in the Progressions would help both under- and over-enthusiastic teachers of algorithms.
We’re wondering what the benefit is to the requirement that “the” standard algorithm for multiplication “relies on the fact that the order of computing the partial products allows you to keep track of the addition of the partial products while you are computing them, by storing the higher value digit of each product until the next product is calculated” (Bill McCallum, September 13, 2012 at 12:08 pm #939).
This requirement seems to rule out the “all-partials” algorithm as “the” standard algorithm. For example, as we read the requirement above, it seems that Beckman and Fuson’s Method D on page 24 (or Method A or Method B on page 23) of their NCSM Journal article (http://www.mathedleadership.org/docs/resources/journals/NCSMJournal_ST_Algorithms_Fuson_Beckmann.pdf) would not qualify as “the” standard algorithm. Given that Fuson and Beckman contend that Method D is conceptually clear and fast enough for fluency we’re wondering what the benefit is to the requirement above.
Requirement?
Here is a link to the comment: http://commoncoretools.me/forums/topic/algorithms-grades-2-5/#post-939. In it, I said
Some think it is the algorithm exactly as notated by our forebears, some think it includes the expanded algorithm, where you write down all the partial products of the base ten components and then add them up. Ultimately this is a question that has to be settled by discussion, not fiat.
I then went on to state my opinion:
My opinion is that the standard algorithm has two key features; like the expanded algorithm it relies on the distributive law applied to the decomposition of the number into base ten components, but in addition it relies on the fact that the order of computing the partial products allows you to keep track of the addition of the partial products while you are computing them, by storing the higher value digit of each product until the next product is calculated.
I don’t see how I could have made it clearer that I was not stating a requirement, just giving an opinion. And I didn’t say anything about benefit, either. That would take a much longer discussion, since benefit or harm would depend on the context: the students, the classroom culture, the curriculum, the time constraints, and so on.
I agree that students can be fluent with methods such as those described by Beckmann and Fuson. They would satisfy
4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. …
However, as I’ve said elsewhere, I don’t think that the partial products algorithm is included in what people meant when the used the term “standard algorithm” circa 2009, whether they were for it or against it. So I don’t think 5.NBT.5 includes the partial products algorithm. That said, I don’t think kids should be beaten to death with this; the standards form a pathway along which some students will be ahead, some behind. And some of those behind will need to take shortcuts to catch up. They are not a catechism, they are a shared agreement about what we want students to learn.
Bill,
I appreciate this response. Thanks. My apologies for taking your opinion
as a “requirement.” It’s sometimes hard to know when you all are speaking
as private citizens and when you are speaking ex cathedra.So, the bottom line seems to be that (i) since “the” standard
multiplication algorithm is not defined either in the standards or in the
progressions, our operational definition for it should be what most people
meant by the term in 2009; so (ii) you think the partial products
algorithm is OK in G4 but is not sufficient for G5; but (iii) kids should
not be “beaten to death with this” in seeking to meet 5.NBT.5 (or,
presumably, anywhere); and (iv) we should not think of CCSS as a catechism
(though maybe as a hymnal?). Also, (v) questions about the advantages or
disadvantages of specific features of “the” standard multiplication
algorithm would involve lengthy discussion, for which probably nobody has
the stomach.One more question. Is there an example of “the” standard multiplication
algorithm anywhere in the progressions document athttp://commoncoretools.me/wp-content/uploads/2011/04/ccss_progression_nbt_2
011_04_073_corrected2.pdfor in some other place you could point to? If not, then given the
confusion over what qualifies, maybe you all should consider providing an
example or two.Andy
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