Mathematics, in general, and algebra courses, in particular, have been categorized as "gatekeepers" for higher education, better jobs, and even citizenship. For many low-income and working adults, community college is the institution where they choose to develop their mathematics understanding so they can pursue their dreams. Unfortunately many fail in their attempts. In an effort to better understand their plight so that the community colleges can better meet their needs, I studied community college students' foundational fraction understanding. Specifically, I examined students' procedural skills and problem-solving strategies to determine evidence of fragmented knowledge and fragile learning. I investigated a sample of 373 adult students in four tiers of community college developmental education mathematics courses: Computational Arithmetic, Pre-Algebra, Beginning Algebra, and Intermediate Algebra. In Phase 1, I quantitatively examined students' performance on a written assessment of foundational fraction problems. I compared groups of students to determine if differences might be due to factors of course level, age, and number of years out of school. In Phase 2, I interviewed 33 of the lowest performing students and examined their explanations and categorized students' problem-solving strategies and levels of procedures and explanations while using the strategies. My analysis revealed five major findings. 1. Students' average score on an 11-item foundational fraction assessment was 74%, below what I considered mastery level on the assessment. 2. The assessment scores differed based on course level rather than other demographic factors. 3. On specific NAEP items, Algebra and Intermediate Algebra students scored similarly to United States eighth-graders, whereas Arithmetic and Pre-Algebra students scored higher than 4th graders yet lower than eighth-graders. 4. The foundational fraction items related to magnitude tended to be the most difficult for the students. 5. The major characteristics of students' conceptual understanding were fragmented, fragile, non-fluent and only rarely, sophisticated. While community college developmental education students know something about fractions, my research indicated that their knowledge was held as multiple unconnected knowledge chunks, bits and pieces of prior knowledge mixed with inaccurate, imprecise and partial notions and procedures making students' resulting "fraction sense" tenuous. Although they sometimes successfully solved problems, occasionally with sophisticated self-generated strategies, students were not fluent in their fraction knowledge. The dissertation ends with some recommendations for instructors to address students' limited fraction understanding along with some suggestions for the system as a whole to make fraction instruction a greater priority in developmental courses so that more students can achieve their goals.
|Advisor:||Ambrose, Rebecca C.|
|Commitee:||Martin, Lee, White, Tobin|
|School:||University of California, Davis|
|School Location:||United States -- California|
|Source:||DAI-A 75/07(E), Dissertation Abstracts International|
|Subjects:||Mathematics education, Mathematics, Education|
|Keywords:||Algebra, Community college, Conceptual understanding, Developmental education, Formative assessment, Fractions|
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