Multicultural Mathematics and Alternative Algorithms
Philipp, Randolph A., Teaching Children Mathematics
Up until recently I wasn't even aware that other people in the world did things [arithmetic algorithms] differently. I thought God sent these. That's the way of the world. The first day you [to another teacher] were talking about some way you did things differently in Ireland. It never occurred to me. I thought there was a world standard.
- A sixth-grade teacher reflecting on alternative-mathematical algorithms
A teacher's beliefs about mathematics significantly affect the manner in which he or she teaches (Thompson 1992). Teachers, from school experience, often believe that there is one right way to solve a particular mathematics problem or to apply a computational algorithm for adding, subtracting, multiplying, and dividing. In turn, these beliefs become the beliefs of their students. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) has called for decreasing the attention paid to isolated treatment of paper-and-pencil computations and the memorization of rules and algorithms and suggests instead that we increase the attention paid to students' creating algorithms and procedures. Implicit in this suggestion is that the algorithms we have come to learn and to use are not the only way, and may not even be the best way, to compute.
Although teachers are usually aware that various cultures have historically used algorithms that are different from those currently taught in United States schools, these teachers may not be aware that, various algorithms are being used currently in the United States. Many of these algorithms are culturally based and are used by people with common ethnic and cultural backgrounds. This article describes how preservice elementary school teachers developed an awareness that the algorithms we teach in school are not the only algorithms for operating on numbers and that if they look, they may find alternative algorithms in their community and school.
An Invented Algorithm
Dictionaries define an algorithm as a rule. or procedure for solving a problem. Computational algorithms are invented by people to streamline the process by which we compute. The fact that algorithms are a convention is often lost on our students, who come to think of a particular algorithm as the way, instead of as a way, to compute. The following example illustrates the role that algorithms play in school mathematics.
A colleague recently told me a story about his third-grade daughter, who came home from school crying because of long division. The girl, whom I shall call Michelle, could not understand why she needed to learn a procedure for 63 [divided by] 7 or 88 [divided by] 8. After all, she said, "Can't everyone see what the answers to those are?" Michelle was struggling with the procedure for long division taught in class and was getting confused about when to multiply, when to subtract, and when to "bring down the next number." That afternoon her father sat with her and took a fresh approach. He first asked her whether she could explain a way of thinking about 126 divided by 3. Michelle said, "If you share 126 with 3 people, how much would each person get?" Her dad then asked if she could think of another approach, and she said, "How many 3s are in 126?" He told her to think about division that way. He asked her to imagine having a large number of ones, to take out groups of three, and to keep track of how many groups she "moved aside." Michelle thought that the explanation made sense, and without any other prompting, she solved 579 [divided by] 3 [ILLUSTRATION FOR FIGURE 1 OMITTED].
Although Michelle's dad suggested that she write down only what she needed, Michelle said that it helped her to write "How many 3s are in 579?" so she could remember what she was doing. Notice the unconventional approach that Michelle invented for this problem. This "algorithm," although nonroutine, was based on Michelle's understanding of the meaning of division and her sense for numeration. …