Sunday, May 25, 2014

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

I teach mathematics content courses for prospective elementary school teachers. Some of my students are parents of elementary school children. Sometimes I hear them complain about the "Common Core math." One complaint I often hear is that their children are made to practice different strategies - strategies like those discussed in 1.OA.6. However, I believe such teaching practice misses the point of 1.OA.6. The standard expects students to add and subtract within 20 (fluency within 10) using strategies. The standard does not say that each students needs to master or be fluent with each strategy.

Let's think about 8 + 9. There are a variety of strategies students can use. Here are some that I can think of:

* count all
* count on (from 8)
* count on (from 9)
* "I knew 8 + 8 is 16, so 1 more is 17."
* "I knew 9 + 9 is 18, so 1 less is 17."
* "I knew 10 + 10 is 20. Since 10 is 2 more than 8 and 1 more than 9, I took away 3 from 20, and the answer is 17."
* "I took 2 from 9 and 8 + 2 = 10. Then added 7 more to get 17."
* "I took 1 from 8 and 9 + 1 = 10. Then added 7 more to get 17."
* "I knew 5 + 5 = 10, and 3 + 4 = 7. 10 + 7 = 17."

I'm sure there are others. When students figure out 8 + 9 using their own reasoning, what is the point we want to emphasize? Are these strategies equally good? If not, how do we decide which strategy is "better than" others? Better in what sense?

If the focus is getting the correct answer to 8 + 9, perhaps all of these strategies are equally good. However, if our goal is more about helping students to think about ways to calculate, then perhaps these strategies have "good" in different ways. For example, those who recognize counting on from the larger added has begun to focus on the more efficient way to find the sum. Those who use doubles, either 8 + 8 or 9 + 9, realize that the known sums can be used to figure out an unknown sum. They have also figured out that if an added increases (or decreases) by 1, the sum also increases (or decreases) by 1. Figuring out such a relationship is an important mathematical practice - perhaps noticing and making use of mathematical structure. Using 10 + 10 also requires the realization that known facts may be useful to figure out unknown sums. In addition, these students are beginning to focus on 10 as a useful benchmark in our number system. Perhaps we can say the similar thing with those who use 5 + 5. Finally, the other two strategies, 8 + 9 = 8 + 2 + 7 or 8 + 9 = 7 + 1 + 9, are not only focusing on 10 as an important benchmark, they are also taking advantage of the way numbers between 10 and 20 can be thought of as "10 and some more."

So, these strategies have different strengths, and it makes no sense to force all students to be fluent with all of these strategies. Moreover, if the strength of a double strategy is the fact that students are beginning to develop the disposition to seek what they already know to figure out something they don't know yet, giving students several problems to solve using this particular strategy seems to be totally inappropriate.

Of these strategies, only the make-10 strategies are the strategies we want to make sure that all students understand and be able to use. So, perhaps assigning some homework problems where students practice these strategies are appropriate. However, we should remember that whether or not students themselves will realize the usefulness of these strategies really depend on how easily they can compose/decompose numbers to 10 and their understanding that numbers between 10 and 20 can be thought of as 10 and some more. If they don't have that prerequisite understanding, they may not see the point of these strategies. On the other hand, if they have that prerequisite understanding, they may not need much "practice" to master these strategies. It is perfectly appropriate to assign some practice problems, but we should be careful about what we want students to practice.


Claire Hallinan said...

Your idea sounds like relating with the mathematical practices beyond the content. Students should be able to decide which strategy makes more sense than others.

Barry Garelick said...

Not sure this is what you had in mind; in fact, I'm sure it isn't. But you might find it interesting.

Tad said...

Thank you for an interesting article. I agree with some of your ideas and disagree with others.

I think both "traditional" and "reformed" teaching are often mischaracterized. I also think there are teachers who teach well using the "traditional" approach (of course, there may be a question whether or not if there is THE traditional method) and there are those who teach well with the "reform" approach. Unfortunately, we probably have more who teach poorly with both approaches.

I do think it is appropriate to include the discussion of the standard algorithms (addition and subtraction) in Grade 2. When students understand we can/must add ones with ones and tens with tens, it gives the reason for writing numbers vertically - I think it is rather illogical to write two single digit addition problem vertically as the vertical notation does nothing in those situations. But, how you approach needs to be carefully thought.

Then, there is a question about what is THE standard algorithm. The authors of the CCSS have said that "the standard" algorithms are locally defined. Here is an interesting article about subtraction algorithm in the US.

Barry Garelick said...

There really is no question about what THE standard algorithms are unless you believe what Beckmann and Fuson wrote in their paper that posits that all are equivalent. (

That paper seems to seal the fate of the standard algorithms so that either one can ignore THE standard algorithm for multi-digit addition and subtraction until 4th grade, OR you can interpret it that it can be taught earlier. Unfortunately, many go with the former--delay until 4th grade.

Mastery of the standard algorithm can actually help students incorporate the alternative strategies much easier; some even discover them on their own.

Barry Garelick said...

Meant to add: Please read Wayne Bishop's reaction to Elizabeth Green's article--in particular the information about Japanese methods of teaching:

Tad said...

First, just to abide by the principle of full disclosure. I collaborate with Dr. Takahashi frequently. I have observed his teaching of mathematics both in Japan and in the US, and I truly believe he is a great mathematics teacher. I also have worked with Dr. Beckmann.

As for THE standard algorithm, I think the more appropriate label should be the CONVENTIONAL algorithm - the most commonly seen/used algorithm. What is the conventional algorithm varies across the world - and across the history. The equal addition subtraction algorithm was often seen in the US while back - and till much more recently in European countries and Australia. The article I linked in my previous comment suggest that what is conventional today seems to have become the conventional method almost by accident.

Every teacher who taught the conventional algorithm knows that "borrowing (regrouping) across 0" is often difficult for young students. However, with the equal addition algorithm, there is no such thing as "borrowing across 0."

As for the characterization of Japanese mathematics teaching, I would tend to believe someone who spend many years in Japanese classrooms and recognized as an exemplary mathematics teachers by their colleagues (i.e., Dr. Takahashi) than someone who probably has never been to a Japanese school.

Here is a link to videos of 4 consecutive 6th Grade mathematics lesson from Japan. I encourage you to watch the video and make your own conclusions about Japanese mathematics teaching. By the way, this teacher's name is also Mr. Takahashi, but it is not the same person as Mr. Takahashi referenced in Green's paper.

Barry Garelick said...

Thank you; I will look at the video. You may be interested in this post written by a Japanese woman in response to Elizabeth Green's article in the NY Times:

Ze'ev Wurman said...

Tad is arguing that "the standard algorithm" should be better called "the conventional algorithm." This is a distinction without a meaning.

But where he really goes astray is when he argues that "the conventional algorithm varies across the world." Wrong! The conventional/standard algorithm does not vary across the world. Its so-called "variations" are minor purely cosmetic differences. Calling those "different algorithms" is either ignorant or intended to confuse.

Further, while it is true that the standard algorithms for the four arithmetic operations evolved over time (as did much of mathematics), they have mostly stabilized since Viete's days, and barely changed over the last century. What's Tad's point in bringing this up? That the Egyptian way was better? Such arguments should be better left to anthropology class rather than to math.

The point of offering a single standard way of doing grade school arithmetic is that students will develop fluency and automaticity with them, so they will not get in the way of learning higher math. The push for teaching multiple algorithms for arithmetic emanates from the belief that, somehow, by teaching multiple ways children will acquire some "deeper understanding" of arithmetic (whatever THAT may be). Wrong. What students acquire by being taught multiple and non-standard ways is confusion. What they don't acquire is automaticity or fluency.

But the Fuson-Beckmann wrong headed paper essentially saying "anything base 10 is a "standard algorithm" is not rooted in ignorance -- it is an intentional effort to confuse and muddle the waters offered with a patina of "scientific rigor."

Tad said...

Are the variations in algorithms merely cosmetic? I encourage people to read the article by Ross and Pratt-Cotter I referenced above and make their own judgement. Here is the link one more time:
Also, some might be interested in looking at this video on the equal addition subtraction algorithm:

It seems like "offering a single standard way of doing grade school arithmetic is that students will develop fluency and automaticity with them, so they will not get in the way of learning higher math" is something most US schools have been doing over the last several decades. It has been successful with some, and it failed miserably others. Of course, the success/failure of that approach might have something to do with the quality of instruction, too. I am not saying that you cannot teach the conventional algorithm meaningfully so that students can make sense of them.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.