Posted by: degarcia | November 24, 2008

Multi-digit Addition and Subtraction

The most important thing to understand about multi digit addition and subtraction is that it should be a concept rather than a procedure. Most teachers can perform the standard algorithms for both multi digit addition and subtraction, but that does not necessarily mean they have a conceptual understanding of the problem. The same is true with their students. So, it is essential for the teachers to understand the concept behind the problem so that whatever algorithm the student decides to use, she will understand the student’s thinking. In the article, Educational Studies in Mathematics, the results of a study showed that it is beneficial for the teacher to have some background knowledge in the concept that she is teaching. It says, “Because the teacher has a deep understanding of both the concept and possible student responses, she is able to anticipate the discussion and develop the necessary underpinning concepts. The teacher’s strong knowledge base is fundamental to her ability to work with student generated solutions” (p. 104).


So, what are some of these algorithms that students come up with, and how are we to understand them? Carpenter includes some of the algorithms the students might use: counting single units, direct modeling with tens, and other invented algorithms. When a student counts using single digits, there is no base 10 concept involved. The invented algorithms a student might use would be incrementing, combining tens and ones, and compensating. Carpenter (1999) says that in order to solve a problem with incrementing, “the tens and ones are operated on separately and results subsequently combined” (p. 70). So if one were to use this strategy, they would have an understanding that tens and ones can be combined, just not the ability to combine them within the problem. Another invented algorithm is compensating, which is adjusting a number to compensate for changes in another number. In the article Educational Strategies in mathematics, it discusses the difference between the way American teachers solved problems compared to how Chinese teachers solved the problems. It said that the Chinese teachers spoke about decomposing 53 into 40+10+3, which are easier numbers to add and subtract (p. 104). A teacher must understand what knowledge these different algorithms represent in order to know how the student performed the problem.


            In conclusion, in order for a teacher to be successful in her classroom, she must have a conceptual understanding of the problems she gives to her students. She should give the students credit for coming up with their own way of solving the problems, and she should try to relate the algorithm to the concept to give other students a more concrete understanding. The article Educational Strategies in mathematics quotes a Chinese teacher, “Sometimes they [students] don’t agree that the standard way I am teaching is the easiest” (p. 104). If a teacher wants her students to develop a concrete understanding, students will do more than just figure out the standard algorithm, they will use their own algorithms.




Carpenter, T. P. (1999). Children’s mathematics: Cognitively guided instruction. New Hampshire: Heinemann. 


Lawson, Alex. (2000). Knowing and teaching elementary mathematics [Review of the book Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States]. Educational Studies in Mathematics, 42,(1), 104.




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