On+Strategic+Additive+Reasoning+(SAR)


 * = On Strategic Additive Reasoning =

**Strategic additive reasoning** (SAR) is reasoning that involves solving addition and subtraction problems through invented strategies. Usually these strategies involve reasoning with bases and single units. For example, the first grade student Joel, who we saw on the first day of class (8/30), used strategic additive reasoning to figure out what was left when 27 cupcakes were eaten and he started out with 45 cupcakes. (Do you recall his solution?)

Elementary school students invent many strategies for solving addition and subtraction problems! Often they were not explicitly taught (told how to do) these strategies. They develop these strategies because they make sense to the students, based on their evolving understanding of the base ten number system. **These strategies are very powerful—often a lot more powerful than more standard strategies that are part of traditional curricular materials.** So, as future teachers we need to understand how to think about what kids do to reason strategically, and how we can do it too.

Two strategies that children use early on are not considered SAR, but these are very important for children to use! They are:


 * **Counting all by ones:** The first grade student Matt, who we saw on the first day of class (8/30), used this strategy to figure out how much he had if he started with 18 pennies and got 25 more for his birthday.
 * **Counting //on// by ones:** On Monday 9/6 we saw Vanessa, a first grader, solve 5 clay animals plus 9 clay animals. She started at nine and then added five onto that rather than doing what Matt did and counting all (Melissa Kilpatrick).

Eventually we want children (and ourselves!) to be able to use strategies other than counting all or counting on by ones, because these two strategies are not efficient with large numbers. Strategies that involve reasoning with bases and single are not only more efficient, they help us organize and extend our understanding of number, addition, and subtraction.

Addition Strategies, as used by elementary school students
Here is a list of the strategies (other than counting by ones) that we will observe children do to solve 1- and 2-digit addition problems. **Can you use all of them to solve problems in base ten? In base five?** We will focus on using base five to develop these strategies, because doing so puts you in a position similar to children who are building up their understanding of base ten as they develop their additive strategies.


 * **Adding bases and then adding ones:** On Wednesday 9/8 we saw Antonio, a first grader, solve 28 + 39 by adding the "complete" tens to get 50, and then adding the "leftover ones" to get 17. Then he added 50 and 17 to get 67.
 * ** Counting on by bases and then by ones (or by ones and then by bases): ** On Monday 9/6 we saw Lauren, a second grader, solve 46 + 37 by counting on starting at 46. However, she did not count on by 37 ones! She counted on by tens, and then ones. Can anyone describe how she did it? One way to notate it is the following: 46, 56, 66, 76, 77, 78, 79, 80, 81, 82, 83. **In our own work, we need to be careful of writing run-on inequalities when we use this strategy. (What is a run-on inequality, a.k.a., a ROI?)**
 * ** Rounding: ** On Monday 9/6 we saw Robert, a second grader, solve 8 + 5 by rounding. He rounded 8 to 10, added 10 + 5, and then adjusted. His method was to say "if it was 10 + 5, it would be 15; if it was 9 + 5, it would be 14; since it's 8 + 5, it's 13." We would probably more likely adjust by subtracting 2, since we added 2 to 8 to make the 10 in the first place.
 * ** Compensation: ** On Monday 9/6, we saw Sara, a second grader, solve 8 + 5 by compensation. She took 2 from the 5 to give to the 8, making it 10, and making the 5 now 3. Then she added 10 + 3. [Maeghan noted that you could do it the other way too: Take 5 from the 8 to make the 5 in "8 + 5" into a 10, leaving 3.]
 * So, what is the difference between **Rounding** and **Compensation**? What is the difference between **Adding bases and then adding ones** versus **Counting on by bases and then by ones**?

The SCA for multi-digit addition
A strategy that we did not actually see a child use is the **standard computational algorithm (SCA) for multi-digit addition**. On 9/13, we discussed this strategy. **[posting opportunity: Answer one of the questions below.]**
 * Can you describe the thinking involved in this strategy?
 * Can you describe the notation that is usually used for this strategy?
 * We have discussed that the word "carry" focuses solely on moving notation on the page. What are better words to describe the underlying concept involved in "carrying"?
 * Sometimes, teachers focus on the notation of the strategy at the expense of the thinking. Why can that be harmful?
 * On 9/20, we discussed an expanded quantitative notation for the SCA, in which we always label any numeral we write with the units associated with it. Can you show an example of this (from class, or you can use another example)?

Subtraction Strategies
Then we complied and developed strategies for subtracting 2- and 3-digit numbers in base ten and base twelve. We have discussed: **[posting opportunity: Describe one of the strategies listed below, with an example in base TWELVE.]** //﻿// //You can use this method to easily solve a problem in any base. I am going to show an example of one that is using the method of of counting back by bases and ones using a base 12 problem.// //(73)-(25)= ?// //To count back by bases then ones the sequence would look like this// //(63)---> (53)---> (43)--->(33)---> Since we dont want to go down to (23) because then we would have passed (25), we will start counting back by ones from (33):// //(33)---> (32)---> (31)---> (30)---> (2E)---> (2T)---> (29)---> (28)---> (27)---> (26)---> (25)...// //We have just counted all of the ones. Now lets count up all the bases we counted back....// //We counted back four bases...// //Now lets count up the ones we counted back...// //Eleven, which in base 12 is represented by E.// //Therefore (73)-(25) is 4E.// //(Lindsey Heller)// One method that we frequently see children use to do subtraction is a method called the "Chunking" Method. In this method, children break up numbers in order to make them easier to deal with, similar to rounding. An example of this is when students have the problem (in base 12) of (82) - (26). This problem is hard for students to do because the "loose" ones in the number you are trying to take away is greater than the "loose" ones in the number you are subtracting from. In order for children to make this an easier problem to deal with, they could make the (26) = (22) + 4. The problem is now (82) - (22) = (60) and now (60) - 4 = (58). This is an easier way for the students to subtract because they are finding a way to create a base, which is easier to work with (Ashley Gilchrist). //(PARTIAL) You can use this strategy in order to break down a number into loose ones and bases. For example if you have (23)-(15) you have two whole bases and three loose ones- one full base and 5 loose ones. If you subtract (15) from that (23), since the ones place in the first number is bigger than the ones place in the second number you have to take decompose the first number by taking apart one of the bases and giving at least two more of the ones in the base to the 3 loose ones. This makes the subtraction a little bit easier. Therefore, you now would have a problem that looks like this : 1 dozen + T ones+ 5 ones- 1 dozen five. That way the fives in both problems drop out, leaving me with 1 dozen T-1 dozen. Therefore, the answer is just T. By using partial decomposition, the problem was made to be much simpler.// //(Lindsey Heller)// //(FULL) You can use this strategy in order to break down a number into loose ones and bases in order to make subtraction easier. Using the problem (T3)-(34), you first have to decompose (T3). This will then turn into 9 bases and one dozen three ones. The problem will then look like (90)-(34). Then To make subtraction easier, I broke up the (30) and the (4). Then I put all of the full bases on one side of the problem and all of the loose ones on one side of the problem to make the problem look like this : (90)- (30) + (13) - (4). I know that (90)- (30) is 6 dozen, however I used the counting on by bases strategy in order to solve the problem (13)-(4). So I went from (13) (12) (11) (10) (E). This gives me my answer of 6 bases and E ones, which transfers to (6E) as my final answer.// //(Suzie Drazner)//
 * **Counting on or back by bases and then by ones (or by ones and then by bases):**
 * **Chunking:**
 * **Rounding:** (If you describe this one, be sure to tell how you adjust when you round the number you are "taking away," sometimes called the subtrahend.)
 * **Sliding:**
 * **Decomposition**

Elementary school students' subtraction strategies On 9/22 we saw some elementary school students using the first two strategies above. **[posting opportunity: Complete one of the items below.]** joel used subtracting bases and subtracting ones, he took 20 off first on base ten, then he took 5 off. later he separeted ten in ten ones, and took off two from this chart. finally he got 18. also we can say he used chuncking which is 45-27=45-25-2=18 (VanessaNiu)
 * Describe the solution that Kim, a 3rd grader, made of 400 - 294, and identify the strategy/ies she used:
 * Describe the solution that Rita, a 3rd grader, made of 65 + ___ = 94, and identify the strategy/ies she used:
 * Describe the solution that Megan, a 2nd grader, made of 74 - 29, and identify the strategy/ies she used:
 * Describe the solution that Joel, a 1st grader, made of 45 - 27, and identify the strategy/ies he used: (As noted above, we also saw Joel on the first day of class! Now we can say more about his reasoning!)

The SCA for multi-digit subtraction
On 9/27 we talked about decomposition and the **standard computational algorithm (SCA) for multi-digit subtraction**.

[posting opportunity: Answer one of the questions below.] > Can you describe the thinking involved in this strategy? //Some teachers believe that using SCA strategies can unteach place value concepts because students are so focused on where to put each number, especially when it comes to borrowing that they lose the understanding of the place values. For example, ones tens or in base twelve land "loose ones" and "dozens".// //(Suzie Drazner)//
 * Why is "borrow" a poor word to describe what is going on mathematically in this SCA
 * Some teachers focus only on this strategy as the main way to do subtraction. Why could that be harmful for students?
 * Where/when is this SCA useful?
 * Why do some mathematics educators believe that a focus on this SCA can "unteach" place value concepts?