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Properties of arithmetic operations are important in mathematics and play a significant role in elementary school students' learning of mathematics. At the same time, it is quite challenging to help students learn arithmetic operational properties beyond memorization, so that they know when and how to use a specific arithmetic property. This paper reports a study containing two parts to explore students' learning of the Commutative Property of Addition (CPA) as related to the content presentation in textbooks. Part one is to assess selected students' performance on CPA at the end of Grade two, and part two is to explore possible contributions of textbooks to students' performance in Shanghai. Firstly, we evaluated the performance of 41 second-graders on CPA by collecting data through a paper-and-pencil test, which was followed up with individual interviews. Through analyzing how textbooks present and organize this topic, we sought to better understand from a curriculum standpoint how students' performance on CPA is impacted by textbooks. Results revealed that Shanghai textbooks make a positive contribution to students' learning performance. The findings from this study help to shed light on how those textbooks aid student learning, and provide an example for further reflection on student learning of arithmetic operational properties in different curricula.

This study focused on an examination of how type, quality and children's awareness of design features in digital math games, along with an awareness of the mathematics goals of the game, were related to learning outcomes. We conducted a parallel conversion mixed methods study with 45 students in Grades 3 and 4 (ages 9-10). Students participated in clinical interviews using three digital math games. The results suggest a moderated mediation relationship in which the quality of the design features moderated the mediating impact of children's awareness of the game's design features, specifically when the child was aware of the mathematics content learning goal (MCLG) of the game. These findings show how important it is that design features are of high quality in a digital math game, and how this is intertwined with children's awareness of the features and the MCLG. When these variables intertwined in just the right way, the interactions between the children and the digital math game afforded mathematical learning growth.

This paper introduces the Affording Understanding Conceptual Framework as a way for mathematics teacher educators to analyze student 's perceptions of a digital math tool 's helpfulness (e.g., GeoGebra) versus the instructor's perceptions when seeking insights into how a lesson design could be improved. An example of this action research process is provided using GeoGebra to help pre-service middle grades mathematics teachers (MGMTs) develop their conceptual understandings of definite integrals. Findings showed that the pre-service MGMTs noticed four categories of what they considered to be primarily helpful features: "Ease of Use," "Pictorial Representation," "Colored," and "Slider/Dynamic Movement" types. While the pre-service MGMTs all described these as helpful, the content of their comments showed that there was a material difference in how they viewed the "Ease of Use" type from the other three suggesting ways in which the lesson could be revised to be more effective.