What Students Learn From The Standard Algorithm

What Students Learn from Instruction on
The Standard Subtraction Algorithm
Linda Levi, ©2006

Two third grade boys, Andy and Terry, were playing one of those popular trading card games. These boys were in a non-academic setting. Although I watched them play through the lens of a math teacher, the boys were clearly focused on playing the game. They were very interested in the cards they had collected and the points and powers of the monsters in their decks

At the start of the game each boy had 6000 points. On the first move, Andy’s monster attacked with 347 points which needed to be subtracted from Terry’s initial 6000 points. Each boy figured out how many points Terry would have; not surprisingly, they got different answers. I knew Andy; he was in a CGI classroom. I didn’t know what type of math instruction Terry had, but I knew that he and Andy attended different schools. After the boys argued a bit about what 6000 – 347 was, Andy said, “why don’t we just show each other how we got our answers?” Terry showed Andy his work; he had done the standard subtraction algorithm. He got the right answer and the marks on his page indicated that he had performed all the standard steps. They then discussed Terry’s work.

Andy: What is that?

Terry: This is the real way to do subtraction, I learned it at school. And see, I got 5,653.

Andy: You have to show me what you did.

Terry: Ok, this is the real way to do subtraction. You have to write it like this. (Writes 6000 with 347 lined up below it.) Now you have to start with the ones. When do subtraction you always have to start with the ones.


Andy: What do you mean you have to start with the ones? You can start with the hundreds, the thousands – you can start anywhere.

Terry: No, you can’t, when you do subtraction the real way, you have to start with the ones. That is what my teacher told me.

I had never thought carefully about exactly what we tell children when teaching the standard algorithm until I heard this discussion. When we teach the standard subtraction algorithm, we tell children to start with the ones. Of course, you don’t have to start with the ones when you subtract. For example, in a problem such as 5,000 – 3,002, it makes much more sense to first subtract 5,000 – 3,000 (the thousands) and then subtract 2 (the ones). We want children to know that 5,000 – 3,000 – 2 is the same as 5,000 – 2 – 3,000. Children with a strong understanding of subtraction know that they don’t need to start with the ones. Understanding how subtraction works will help students solve algebraic equations such as 3x – 48 – x = 49 or 3x – 48 = 49 – x.

Terry and Andy’s discussion then went on,

Andy: Ok, with that way you have to start with the ones. Then what did you do next?

Terry: Well, you can’t subtract seven from zero so I had to borrow.

Andy: What do you mean you can’t subtract seven from zero! Don’t they have negative numbers at your school?

Terry: Well, yeah, we have negative numbers. Like with the temperature or the number line, but you can’t subtract seven from zero.

Andy: Of course you can, zero minute seven is negative seven.

Terry: Well, I guess.

I again thought about the directions we give when teaching the standard algorithm. Perhaps Terry’s teacher didn’t tell Terry, “You can’t subtract seven from zero,” and said something like, “We have more ones in the number we are subtracting from than the number we are starting with so we are going to need to borrow.” However, what Terry learned was you can’t subtract seven from zero. It is not atypical for children to say something like, you can’t subtract a bigger number from a smaller one when explaining how they compute with the standard subtraction algorithm. This misconception may interfer with students' learning about negative numbers.

Andy: Ok, so now what did you do?

Terry: Well, I needed to borrow, and since there were no tens in six thousand I needed to go the hundreds, and there are no hundreds in six thousands so I needed to go to the thousands.

Andy: What do you mean there are no tens in six thousand? I don’t know how many tens are in six thousand, but there are a lot of tens in six thousands. There are ten tens in one hundred, so there has to be lots of tens in six thousand.

Again, even though Terry’s teacher might have stated the borrowing rule without stating misconceptions what Terry took from this instruction was that there were no tens in six thousand.

At this point Andy seemed to give up and listened as Terry went on through his explanation of the standard algorithm. When Terry was finished, Andy explained his strategy.

Andy: Well first I did 6,000 minus 300 and that was 5,800. Oh no, I did that wrong, it should be 5,700. And then it would be 5,700 minus 40 that would be 5,660 and then minus 7 is 5,653. Oh we got the same thing.

Terry: I was right, 5,653.

I later asked Andy if he had ever seen the way that Terry solved that problem. He said he thought he had, that some kids in his class showed that way during math times. I asked him if he had ever been taught that way and he said, “No. My teacher wants us to solve problems in our own ways, she doesn’t teach us how to solve problems.”

Knowing Andy’s teacher, I wouldn’t say that she doesn’t teach her students how to solve problems. She is careful in choosing problems for her students that will help them progress in their understanding of number and operation. She also carefully scaffolds her conversations with students and facilitates conversations among students to lead them to more sophisticated understandings of number and operation. As their understanding grows, their solution strategies become more efficient. Andy’s teacher does not teach standard algorithms.

It is very difficult to consider the possibility of not teaching the standard algorithms. These algorithms are part of our culture. Most elementary school teachers spend many hours teaching these algorithms to their students. Many elementary school teachers who are committed to teaching math for understanding throw in some instruction on the standard algorithms as another way of solving problems.

In deciding whether or not to teach the standard algorithm, we need to think about what our students learn from our instruction on standard algorithms. We want all students to leave elementary school with a strong understanding of addition, subtraction, multiplication and division. We also want students to leave elementary school with efficient strategies for computation. One of my goals is to help teachers learn to increase students’ computational proficiency in conjunction with their understanding of number and operation. Terry was quite proficient in performing the standard algorithm; he was efficient and accurate. However his proficiency came at the expense of his understanding of number and operation. We can teach children to be proficient with computation at the same time as we help them develop their understanding of number and operation. I will be writing more about this in future postings.