Learning mathematics involves abstract thinking and the use of symbols. To be successful in a mathematics classroom, students must master what symbols mean and how they work. Much prior work has focused on the nature of the external representations used in mathematics instruction, including manipulatives and symbolic representations. In this work, I shift the focus to consider, not only external representations, but also the nature of the "actions" in which children engage when learning mathematics with manipulatives. Working within an embodied cognition perspective, I have highlighted the importance of action for thinking and learning. I investigated whether an "action fading" progression--from producing action, to viewing action, to imagining action--is an effective instructional approach for children learning about mathematical equivalence. First and second grade children participated in a pretest, lesson, posttest design in which they received either an action fading lesson, an action-only lesson or a symbolic-only (control) lesson. I addressed two research questions. The first research question concerned the effectiveness of action: do lessons that include action lead to better learning than a lesson that does not include action? There was a significant interaction of this contrast with grade level, such that first graders learned more from lessons that included action than from the control lesson, but this was not the case for second graders. The second research question concerned the relative effectiveness of an action fading sequence: does a lesson that includes action fading have benefits for learning that are greater than the benefits of a lesson that includes action without fading? This contrast was significant for one subcomponent of the posttest; participants in the action fading condition performed better than those in the action only condition on items that assessed understanding of the meaning of equality. Further research is warranted to explore action fading as an instructional approach. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]