Diagnosing and Remediating Common Student Errors in Math
Recently I have been reading Robert Ashlock's book, Error Patterns in Computation, which discusses common errors made by students in elementary and middle school mathematics. Most of these mistakes seem to arise from misunderstandings students have accumulated based on their experience. For example, many students believe that multiplication always results in a larger number. Their confusion is understandable. When people refer to objects multiplying outside of math class, they almost invariably mean increasing in number. Therefore, when students encounter a problem in which a whole number is multiplied by a fraction or a decimal less than one, many will automatically assume that the whole number will grow or multiply. (It doesn't, but the fraction does.)
In his book, Ashlock presents a series of computation errors made by students and asks the reader to find the patterns in order to uncover the students' thought processes. The best instruction should target each student's specific misunderstanding, whether it be computational or conceptual. Only when students have been redirected towards a fuller understanding of the material can they effectively move forward in math.
After diagnosing student errors, Ashlock recommends using a variety of tools and strategies to enrich instruction in basic math. By using manipulatives, such as Base 10 blocks, students develop a richer understanding of place value. Diagrams of mathematical vocabulary help eliminate confusion caused by arithmetic syntax. Ashlock focuses on experiential learning because it is the most meaningful to students. Experience is, after all, how they have accumulated most of their knowledge in the first place. Furthermore, when students can understand and make connections, they will be more motivated to learn.
How can teachers correct student misconceptions while still encouraging them to learn from experience? And in increasingly diverse classrooms, how can teachers find and address these "error patterns"?
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