The book is intended to serve as a brief companion for mathematical educators of elementary teacher candidates who learn mathematics within a college of education both at the undergraduate and graduate levels. Being informed by mathematics teaching and learning standards of the United States Australia Canada Chile England Japan Korea Singapore and South Africa the book can be used internationally.The teaching methods emphasize the power of visualization the use of physical materials and support of computer technology including spreadsheet Wolfram Alpha and the Geometers Sketchpad.The basic ideas include the development of the concepts of number baseten system problem solving and posing the emergence of fractions in the context of simple reallife activities requiring the extension of whole number arithmetic decimals percent ratio geoboard geometry elements of combinatorics probability and data analysis.The book includes historical aspects of elementary school mathematics. For example readers would be interested to know that twosided counters stem from the binary system with its genesis in the 1st millennium BC China of which Leibnitz 17th century was one of the first notable proponents.