Place Value and Standard Algorithms

For some years now I have been thinking about the role of standard algorithms (carrying, borrowing, long division, invert and multiply, etc.) in mathematical development. Constance Kamii has research evidence to show that students "unlearn" place value as they work with the standard algorithms. I well remember the first time I read some of her research findings. I actually thought I must have misread, so I had to read it a second time! Since that time, I have spent a lot of time thinking about the role of standard algorithms in math education. Should they be taught at all? If so, when? I don't think these questions have easy answers, and I believe that we will be better teachers if we grapple with these questions. We need to seriously consider what criteria we would use to make such a decision. In this blog entry I will share my experience in an interview with Anna during the summer after she had completed third grade. I had seen Anna during the school year. She had been with a teacher who was using Cognitively Guided Instruction principles to inform math instruction. My observations of Anna led me to believe that she was a very capable third grade mathematician. Her teacher told me, however, that Anna's parents had been trying to get her to use the standard algorithms. The teacher felt that her meaning-making processes had suffered as a result. During my interview with Anna, I noticed that she typically used the standard algorithm for addition, but relied on an open number line for subtraction. Interestingly, she seemed to illustrate better number sense for subtraction than she did for addition. This makes sense, since she her approach to subtraction required her to think and make decisions about the numbers, whereas she was following a procedure for addition. The following three anecdotes from the interview illustrate this point. The original problems were in context, which she conceptualized well, creating an appropriate numerical expression.

Subtraction: 301 - 199
***Anna drew an open number line and showed a jump of 1 to get to 200, a jump of 100 to get to 300, and a jump of 1 to get to 301. Then she said, "It's 102, because 100 + 1 + 1 = 102." In this situation, Anna used what she knew about the numbers to quickly generate a solution. Although she drew out the number line, this method was more efficient than the standard algorithm, and has the advantage of easily translating to a mental math strategy in the future.

Addition: 2015 + 1587
***Anna solved this problem using the carrying algorithm.
A: 5 + 7 is 12 (she wrote down "2")
K: Oh, are you splitting up the 12?
A: Yes, into a 1 and a 2
K: Doesn't 1 and 2 make 3, not 12? How did you split the 12?
A: (uncertainly) 6 and 6?
K: I don't see any 6.
Long Pause
K: I see you took off 2. Then what did you have?
A: Oh, 10...pause...yes, this 1 up here is the 10.

Anna was not in tune with her place value understanding when she was using the standard algorithm. She was able to talk about it with meaning only after probing and her response was far from automatic. It is important to remember that Anna was a girl who had been generating her own strategies and using place value understanding to guide that process. I was dismayed at how quickly she gave that up when introduced to the standard algorithm. The next example was from later in the same interview.

Addition: 1999 + 1999
***Anna solved the problem with carrying, but struggled a great deal and there were many crossed out marks. At times she had to stop what she was doing and model parts of the problem with base 10 blocks in order to figure out what she was doing well enough that she could explain it to me (I was asking her to explain her thinking to me, so blindly following steps wasn't really an option). Finally, she came up with her answer: 3998. Then I rewrote the numerical expression, 1999 + 1999, in a clean space on her paper.

K: What do you know about the number 1999?
A: It's close to 2000.
K: Could you use that information to solve the problem?
A: Yes, I could make the 1999s into 2000s and then add those and then at the end just take off 2. That would have been a lot faster than what I did actually.

Anna's place value understanding was so much more in evidence when she was using her own thinking. The place value is not very transparent in the standard algorithm, and clearly she wasn't keeping her own place value understanding paramount when she was using the standard algorithm. The discouraging aspect here is that Anna has good place value understanding. With probing, she was able to finally figure out what was happening in the algorithm. This would be much less likely to be the case if she had been introduced straight away to a procedure without having to use her own place value understanding to make sense of personal algorithms.

To be honest, I still struggle with the question of whether standard algorithms are ever necessary, and (if so) when. Based on this experience (and others) as well as the research findings, I suspect that students need more time than we might anticipate to solidify their understanding before being introduced to the standard algorithms. However, I am also willing to admit that the way that the standard algorithm was introduced to Anna may have contributed negatively to her sense-making. Her parents probably introduced it as a procedure to follow, not as an algorithm to explore and find out why it makes sense. Her parents probably referred to all numbers by their digit name rather than their actual value. They probably merely expected her to follow steps rather than make sense of them. Anna had built up some outstanding personal strategies for addition by the end of third grade. Perhaps if her parents had shown her how they used to do it and challenged her to make sense of why it worked, maybe some of her difficulties may have been minimized. Likewise, if they had referred to all numbers by their actual value, she may have been able to see how to use her own understanding of place value to make sense.

My daughter enters 5th grade this fall. So far she has never seen any of the standard algorithms. I cannot see at this point that it has done any damage at all. She is mathematically very competent. Still, this is the year when typically standard algorithms get really crazy, because the procedures and rules vary according to each of the operations and whether you have whole numbers, mixed numbers, fractions, decimals, percents... I will continue to ponder the issue of when and how for introducing standard algorithms and hope that you will also.