A Fourth Grade Teaching Experiment on Fraction Magnitude: Investigating Student Reasoning Through Mathematical Discourse and Design Research
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Abstract Students in the United States continue to struggle with fraction mastery despite the fact that understanding fractions is essential for further progress in mathematics (NAEP, 2008). This research project investigated the potential of using high-press mathematical discourse (Kazemi & Stipek, 2008) to investigate fourth graders’ emerging understanding of fraction magnitude. Fraction magnitude refers to the measurement quality of the fraction, or how much of a given unit the fraction represents. Students who understand the properties of fraction magnitude have demonstrated a more promising trajectory for fraction mastery and for success in algebra (Fuchs et al., 2014; Schneider, Grabner, & Paetsch, 2009; Siegler, Fazio, Bailey, & Zhou, 2012). This study investigated fourth graders’ reasoning about the measurement construct of fractions that emerged during a whole-class teaching experiment (Lamberg & Middleton, 2009; Lamberg, 2007; McKenney & Reeves, 2013). Students participated in solving tasks which involved fraction magnitude, measurement, and density. Design research methodology was used to carefully analyze the learning ecology (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003). The domain specific understanding of fraction magnitude was investigated using flexible design revisions based on student work and discourse which emerged during the unit. This was used to produce a fraction magnitude unit reflecting an actual learning trajectory of fourth grade students (Lamberg & Middleton, 2009; McKenney & Reeves, 2013). An instructional unit based on developing fraction magnitude and measurement understanding was created and used in this study. The developed unit incorporated research-based findings on using the measurement construct for instruction (e.g., Fuchs et Al., 2014: Schneider et al., 2009; Siegler et al., 2012) as well as research based curriculum, such as the Rational Number Project (Behr, Cramer, Harel, Lesh, & Post, 1979). Data was collected from each teaching episode. The lessons were video recorded, student work was documented and analyzed, and teacher/researcher interviews, as well as student interviews, were recorded, transcribed, and evaluated. An iterative design was implemented in which the data collected from one lesson informed changes made in the following lesson. The purpose of the iterative design was to test emerging theories and hypotheses of what was being understood and observed (Gravemeijer & Cobb, 2006; Lamberg & Middleton, 2009) The teaching experiment provided insights into students’ reasoning during fraction magnitude tasks and how episodes of whole class mathematics discourse supported both teacher insight and student understanding. The realized learning trajectory along with mathematical mindset norms that emerged are documented. Implications for curriculum development and teaching fraction magnitude are discussed.