The most common virtual manipulatives used were geoboards, pattern blocks, tangrams, and base-10 blocks. We were not surprised to find these specific virtual manipulatives used most frequently across the grade levels, as these are commonly used physical manipulatives with which many teachers are familiar.

As noted above, the number line stands in contrast to other manipulatives and mathematical models used within the number realm. Some of the reasons for developing the number line as a foundational tool are illustrated below as key ideas for this textbook.

➢ Concrete manipulatives appear to be more effective for building pre-service teachers' conceptual understanding, with virtual manipulatives used to reinforce those concepts. ➢ The majority of students found concrete manipulatives to be easier to use but not necessarily more readily available than virtual ones.

Virtual manipulatives effectively provide a visual and interactive approach to understanding math concepts that can easily adjust to meet individual learning needs. While one student may prefer to build with fraction bars, another may see fractions as shaded parts of a clock or maybe even jumps on a number line.