Do Algorithms Sometimes Get In The Way Of Mathematical Thinking?

As a professor of mathematics education at Fresno Pacific University, I often have the opportunity to work with K-6 teachers in their schools. One such visit sticks out in my mind. A lesson study team comprised of myself, another university math educator, a district math specialist, and four fourth grade teachers had planned a lesson together. The objective of the lesson was to have students solve various word problems. However, we chose the numbers in the word problems carefully so that the numbers could be decomposed and composed to facilitate mental math. Our hope was that students would “see” the compatible numbers and discover that they could solve the problems without resorting to the standard algorithms.

One of the team members volunteered her class, which was comprised of students who were proficient or advanced on the state tests in math. She began the lesson by putting the following problem on the board.

Juan had 97 Jolly Ranchers, Tenisha had 62 Tootsie Rolls, and Kaleb had 113 peppermints. How many pieces of candy were there altogether?


The fourth graders were not the least bit intimidated by the extra adults in the room, and most quickly came up with the correct answer of 272 candies. The students had written the three addends vertically and used the column addition algorithm. (This was anticipated by the team because students had been taught this algorithm earlier in the year.)

Because none of the students had seen the easier way to add the numbers, the teacher asked her class to try and come up with a different way—not using column addition—to find the answer. After a long pause a few students added the first two addends for a subtotal and then added the third addend to this subtotal. After sharing this approach most of the class saw that you could add two of the numbers first and then add the remaining number as an alternative to column addition.

To our amazement, none of the students had yet seen that you could decompose 113 into 3 and 110 and then put the 3 with the 97 to make 100. Then you could add 100+110+62 to get a total of 272.

Once more the teacher challenged her students to see if they could find the answer in a different way. After another long pause most of the students began working again. In their minds a different way must have meant using different operations, because they were trying various combinations of adding, subtracting, multiplying and dividing. One of the students I was watching multiplied 97 by 62 and then began dividing this product by 113 before he gave up.

After seeing that her students were probably not going to come up with the idea of decomposing and recomposing the addends, the teacher began asking leading questions. How far away from 100 is 97? Do you see that 3 anywhere? Finally the students were able to see what we had expected them to see before the algorithms got in the way.

Have you had a similar experience where the algorithm (or procedural knowledge) got in the way of the mathematics (or conceptual knowledge). If so I’d love to hear from you.


2 Responses to Do Algorithms Sometimes Get In The Way Of Mathematical Thinking?

  1. Thanks for sharing this experience. It seems to support the philosophy of Phil Daro in the blog entry by Beverly Ford, “Moving Beyond the Answer”. I also believe that a teacher’s understanding of the importance of teaching mathematical thinking as opposed to “teaching to the algorithm” will be the biggest challenge to the successful implementation of Common Core.

  2. If I remember correctly, it was the author of the classic, How to Solve It, Dr. G. Polya, that said it was better to solve the same problem five different ways than to solve five of the same kind of problem! The algorithms are designed to get to the answer, one way. Learning just the algorithm does keep students from developing number sense and ‘getting to the answer’ makes it hard for the teacher to go back and work on developing a broader sense of number relations in students.

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