Logical statements are prevalent in mathematics, the sciences, law, and many areas of everyday life. The most common logical statements are conditionals, which have the form “If H..., then C...,” where “H” is a hypothesis (or condition) to be satisfied and “C” is a conclusion to follow. Reasoning about conditionals is a skill that is only superficially understood by most individuals and depends on four main conditional contexts (e.g., intuitive, abstract, symbolic, or counterintuitive). The purpose of this study was to test a theory about the effects of context ordering on reasoning about conditionals. To test the theory, the researcher developed, tested, and revised a virtual manipulative educational mathematics application, called the Learning Logic App. This study employed a convergent parallel mixed methods design to answer an overarching research question and two subquestions. The overarching research question was “How does the order of teaching four conditional contexts influence reasoning about conditionals?” The two subquestions examined this influence on reasoning in terms of performance and perceptions. This study involved two phases. During Phase I, 10 participants interacted with the Learning Logic App in a clinical setting. The researcher used information gathered in Phase I to revise the Learning Logic App for Phase II. During Phase II, 154 participants interacted with the Learning Logic App in a randomly assigned context ordering in an online setting. In both phases, the researcher collected quantitative and qualitative data. After independent analyses, the researcher made meta- inferences from the two data strands. The results of this study suggest that context ordering does influence learners’ reasoning. The most beneficial context ordering for learners’ performance was symbolic-intuitive-abstract-counterintuitive. The most beneficial context ordering for learners’ perceptions was intuitive-abstract-counterintuitive-symbolic. Based on these results, the researcher proposed a new context ordering: symbolic-intuitive-abstract-counterintuitive-symbolic. This progression incorporates a catalyst at the beginning (symbolic context) which aids the learner in reassessing their prior knowledge. Then, the difficulty of the contexts progresses from easiest to hardest (intuitive-abstract-counterintuitive-symbolic). These findings are important because they provide an instructional sequence for teaching and learning to reason about conditionals that is beneficial to both learners’ performance and their perceptions.

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This exploratory qualitative study investigated the presence of and relationships among constructs that contribute to children's interactions with educational technology, leading to the development of the modification of attributes, affordances, abilities, and distance (MAAAD) for Learning framework. For this study, each of 10 fifth-grade children participated in one individual video-recorded semistructured interview session, during which they interacted with two mathematics virtual manipulative iPad apps and responded to follow-up questions. Video recordings and observation field notes were analyzed for evidence of attributes, affordance-ability relationships, distance, and relationships among these constructs. Constant comparative data analysis using memoing and eclectic coding provided evidence of the presence of each focus construct. Further analysis and interpretation, including quantization of qualitative data for visualization using novel rhombus plots, also led to the identification of emergent themes related to each construct and revealed relationships among the constructs. Emergent themes included categorization, alignment, and modification of attributes, variations and interrelationships among affordance-ability relationships, and the identification of and interactions among mathematical and technological distance. Furthermore, each construct related to each other construct. The evidence and interpretations led to the development of the MAAAD for Learning framework. The results of the study suggest that the MAAAD for Learning framework models relationships among attributes, affordance-ability relationships, and distance in the context of user-app interactions. the framework could serve as a tool for app developers designing apps, educators using apps to support children's learning, and researchers characterizing user-app interactions and the outcomes of those interactions. The constructs, relationships, and framework identified in this study advance the literature on children's interactions with educational technology tools, in particular literature concerning children's interactions with mathematics virtual manipulative iPad apps.

In recent years, the Internet has increasingly been used to provide significant resources for student to learn mathematics and to learn about mathematics, as well as significant resources for teachers to support these. Effective access to and use of these has been hampered in practice by limited facilities in schools and the limited experience of many mathematics teachers with the Internet for mathematical purposes. This paper offers models for understanding the effective use of Internet resources, based on typologies of resources for learning and teaching mathematics. Six categories of Internet resources for mathematics student use are identified: (i) Interactive resources; (ii) Reading interesting materials; (iii) Reference information; (iv) Communication; (v) Problem solving; and (vi) Webquests. Similarly, five categories of Internet resources for mathematics teacher use are identified: (i) Lesson preparation; (ii) Official advice and support; (iii) Professional engagement; (iv) Commercial activity and support; and (v) Local school web sites. The paper recognises that web resources can be used in a range of ways, including supporting both teaching and learning. The prospects for sound use of the Internet are briefly described in terms of these models of use.

In recent years, the Internet has increasingly been used to provide significant resources for student to learn mathematics and to learn about mathematics, as well as significant resources for teachers to support these. Effective access to and use of these has been hampered in practice by limited facilities in schools and the limited experience of many mathematics teachers with the Internet for mathematical purposes. This paper offers models for understanding the effective use of Internet resources, based on typologies of resources for learning and teaching mathematics. Six categories of Internet resources for mathematics student use are identified: (i) Interactive resources; (ii) Reading interesting materials; (iii) Reference information; (iv) Communication; (v) Problem solving; and (vi) Webquests. Similarly, five categories of Internet resources for mathematics teacher use are identified: (i) Lesson preparation; (ii) Official advice and support; (iii) Professional engagement; (iv) Commercial activity and support; and (v) Local school web sites. The paper recognises that web resources can be used in a range of ways, including supporting both teaching and learning. The prospects for sound use of the Internet are briefly described in terms of these models of use.