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The thesis processes data from the research of the company H-edu, which developed a course to help students understand the issue of fractions in the difficult times of Covid-19. Using a large number of materials, an interactive environment, videos and professional guidance, the company helped those interested in understanding the basic principles for perceiving and counting fractions. The thesis compares the results before entering the course and after its completion, which presents the details associated with the course and allows for an evaluation of the course design. The data also helps to identify where students make the most mistakes. Furthermore the thesis contains information on how to identify more demanding tasks or why students might need more time during calculations. At the same time, it explains the importance of gradation of tasks associated with higher success of respondents. The thesis also deals with the structure of questions that affect the representation of incorrect answers. For the most frequently recorded incorrect answers the thesis explains its cause and the procedure by which the respondents came to the wrong result. Recurring errors highlight the need for thorough explanation and practice of the substance. Moreover, the thesis contains a detailed analysis of the...

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This qualitative study examined how my practice of using manipulatives to teach multiplying and dividing fractions to 8th grade students facilitated their conceptual understanding of those operations. The students who participated in the study were enrolled in one of my intensive mathematics classes. Before the lessons began, I interviewed the students and gave them a pre-assessment to determine their content knowledge and comfort level with manipulatives. The students engaged in activities that included solving problems using various manipulatives. During the activities, I made observations of their problem solving techniques and how they used the manipulatives. At the conclusion of the unit I gave them a post assessment and conducted post interviews to determine any change in their content knowledge and comfort level with using manipulatives. I concluded through my research that by giving the students a hands-on, minds-on approach to learning they were able to develop an understanding of the concepts and apply that knowledge to multiplying and dividing fractions.

Social norms are patterns of behavior expected within a particular society in a given situation. Social norms can be shared belief of what is normal and acceptable shapes and enforces the actions of people in a society. In the educational classroom, they are characteristics that constitute the classroom participation structure. Sociomathematical norms are fine-grained aspects of general social norms specifically related to mathematical practices. These can include, but are not limited to, a student-centered classroom that includes the expectation that the students should present their solution methods by describing actions on mathematical objects rather than simply accounting for calculational manipulations. For this action research study, my goal was to determine if the role of the teacher would influence the social and sociomathematical norms in a mathematics classroom and in what ways are sociomathematical norms reflected in students' written work. I focused specifically on students' mathematics journal writing and taped conversations. I discovered that students tended to not justify their work. Also, I discovered that my idea of justification was not really justification. I learned from this and was able to change my idea of justification. By encouraging the students to socialize in mathematics class, I found that the quality of their dialogue improved. Students readily discussed mathematical concepts within small groups and whole class discussions.

In an attempt to examine my practice of providing conceptually-based professional development on fractions to fifth grade teachers, I conducted a series of four one hour professional development workshops. I focused on the conceptual understanding of ordering, adding, and subtracting fractions. I examined the solution process that teachers used to solve fraction problems and their abilities to explain and justify their solutions in an attempt to interpret their understanding. My data showed the effects of this workshop series. The study helped determine the effects of conceptually-based professional development on fractions as demonstrated in the teachers' discussions, participation, and written explanations.

Many U.S. students possess only a weak knowledge of fraction arithmetic. I hypothesize that textbooks are a critical reason for students’ poor performance on fraction arithmetic. This is not because of what textbooks contain but rather because of what they lack. Distributions of fraction arithmetic problems in textbooks are imbalanced, with certain types of problems almost never presented (Braithwaite et al., 2017). As a result, students often err when they attempt to solve those rare types of problems (Siegler & Pyke, 2013). Two experiments, a pilot study (n = 40 students in grades 5 through 7) and a larger study (n = 127 students in grades 6 and 7), were conducted. These experiments utilized a pretest-intervention-posttest design to empirically test the benefits of providing either a complement of the typical textbook distribution of problems (hyperbalanced practice) or an equal distribution of problems (balanced practice) compared to the benefits of providing a practice set that followed the typical distribution found in math textbooks. A MANCOVA and follow-up ANCOVAs revealed significant differences between students in the textbook condition and students in the balanced and hyperbalanced conditions.For items involving adding fractions with unequal denominators, students who received typical textbook practice showed greater improvement and made fewer strategy errors than students who received hyperbalanced practice. For items involving multiplying two fractions with equal denominators, the opposite was true. Students who received hyperbalanced practice showed greater improvement and made fewer strategy errors than students who received typical textbook practice on items involving multiplying two fractions with equal denominators. Finally, students who received fully balanced practice showed greater improvement and made fewer strategy errors than students who received typical textbook practice on problems involving multiplying one whole number and one fraction. This last finding was of particular interest since none of the practice conditions included practice with that item type. The results of this study demonstrate that even a brief intervention in which students received extra practice with rare item types could improve performance. It also showed that gains in one type of item often resulted in decrements in others, which must be considered when making recommendations to textbook publishers and educators. In sum, this dissertation seeks to make a scholarly contribution to the field by discussing the role that textbooks play in student performance and by analyzing the benefits of supplementing typical textbook instruction with differently balanced fraction arithmetic practice.

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