• Exploring everyday examples to explain basis: insights into student understanding from students in Germany

      Zandieh, Michelle; Adiredja, Aditya; Knapp, Jessica; Univ Arizona, Dept Math (SPRINGER HEIDELBERG, 2019-12)
      There is relatively little research specifically about student understanding of basis. Our ongoing work addresses student understanding of basis from an anti-deficit perspective, which focuses on the resources that students have to make sense of basis using everyday ideas. Using data from a group of women of color in the United States, we previously developed an analytical framework to describe student understanding about basis, including codes related to characteristics of basis vectors and roles of basis vectors in the vector space. In this paper, we utilize the methods of the previous study to further enrich our findings about student understanding of basis. By analyzing interview data from students in Germany, we found that this group of students most often used ideas that we describe by the roles generating, structuring, and traveling, and the characteristics different and essential. Some of the themes that emerged from the data illustrate common pairings of these ideas, students' flexibility in interpreting multiple roles within one everyday example, and the ways that the roles and characteristics motivate students to create additional examples. We also discuss two ways that differences between the German and English languages were pointed out by students in the interviews.
    • A metacognitive intervention for teaching fractions to students with or at-risk for learning disabilities in mathematics

      Hacker, Douglas J.; Kiuhara, Sharlene A.; Levin, Joel R.; Univ Arizona, Educ Psychol (SPRINGER HEIDELBERG, 2019-08)
      Assessment data from the United States and international reports of student achievement indicate that upper elementary students are failing to meet basic levels of proficiency in fractions and writing, and that this is particularly prevalent with students with or at-risk for learning disabilities in mathematics. Proficiency with fractions has been identified as foundational for learning higher-level mathematics but remains one of the most difficult skills for students to learn. In addition, students' difficulty with fractions is exacerbated because of increased chances of comorbidity with language learning problems, particularly difficulties constructing arguments and communicating using writing. We describe FACT+(RC2)-C-2, a language-based, metacognitive instructional intervention that was designed using the Self-Regulated Strategy Development model (SRSD) for teaching foundational concepts of fractions. The results from two studies in which the intervention was administered to upper elementary students who exhibit mathematics difficulties indicated selected increases in students' computational accuracy, quality of mathematical reasoning, number of rhetorical elements, and total words. With evidence of improved performance in these areas, FACT+(RC2)-C-2 holds promise for helping these students become proficient self-regulated learners.