Musings on Multiplication Tables and Associated Mathematics and Teaching Practices

This paper is based on my reflections on a deceptively simple tabular representation of a combined 12x12 multiplication table showing multiplier and multiplicand, starting at a time when I taught mathematics full time at a primary (elementary) school through to my present teaching of mathematics education modules to prospective teachers. A historically-motivated framework on the importance of tables as expressions of complex information in two-dimensional form is used to gain insight into and understand multiplication tables. Through this framework it is shown that the modal practice of "knowing one's tables" in the primary grade levels is really about knowing sequenced and separated lists of whole number multiplications. In contrast, tabular multiplication sequences in a combined multiplication table can, through appropriate teaching practices, enable the discovery of multiple relationships beyond multiplication or arithmetic, resulting in significant mathematics that spans the grade levels. Implications for mathematics teacher education practice with its current focus mathematical knowledge for teaching, are considered.