Making irrational numbers real

Abstract One of the greatest conceptual difficulties faced by middle level mathematics students is developing a rich understanding of irrational numbers that includes recognizing that irrational numbers are truly real numbers with an exact value and an exact place on the number line. Developing a deep conceptual understanding of irrational numbers is essential for extending students’ mathematical knowledge of number sets from the set of rational numbers to the set of all real numbers. In this article, we identify some pedagogical missteps that lead to misunderstandings and then explore pedagogically sound methods for combating student misconceptions. Here, we introduce, by means of two hands-on activities, the use of multiple representations for helping middle level students develop a more robust, less fragile understanding of irrational numbers.