When seeing isn't believing: Influences of prior conceptions and misconceptions

When seeing isn’t believing: Influences of prior conceptions and misconceptions Eliane Stampfer (stampfer@cs.cmu.edu) Kenneth R. Koedinger (krk@cs.cmu.edu) Human Computer Interaction Institute, 5000 Forbes Ave Pittsburgh, PA 15213 USA Abstract Instruction often employs visual representations to support deep understanding. However, students‟ prior misconceptions may override the meaning in these scaffolds. We investigate fraction bars, a common representation intended to promote sense-making. Our prior work found that students often did not use the fraction bars effectively. This difficulty factors assessment compares four scaffold types: pictures only, two forms of pictures with numbers, and numbers only, to assess which interpretation steps were difficult. On equivalence items, students performed equally well with all scaffolds that included pictures, but worse with the numbers-only scaffold, indicating that fraction bars improved scores for equivalence. However, including numbers with the pictures decreased performance for fraction addition. Although students demonstrated competence with fraction bars in fraction equivalence, they did not transfer this knowledge to addition. These results suggest caution in designing and teaching representations for sense-making. Keywords: graphical representation; fraction addition; symbolic fractions. Many researchers strive to identify ways to support deep understanding, as it is thought to promote robust and adaptable learning. One strategy has been to use multiple representations, particularly ones that connect to students‟ prior knowledge and aid sense-making. However, there is little data on what representations will make sense to the students. Singapore textbooks and the NCTM standards, for example, advocate using concrete visual representations in mathematics as a bridge to more formal, abstract thinking (NCTM, 2013; Leinwand & Ginsburg 2007). But, perhaps we should question the benefits of these representations (Rittle-Johnson & Koedginer, 2001; Booth & Koedinger, 2012): Are they actually easy entry points for students? Our tutors for 5 th graders aim to support sense- making by providing conceptual representations as feedback, a strategy that appears effective with adults (Mathan & Koedinger, 2005; Nathan, 1998). In our fraction-addition tutor interfaces, equally-divided rectangles, or fraction bars, provide immediate feedback by dynamically showing the fractions that students enter numerically. We hypothesized that fraction bars would be a more intuitive representation than symbolic fractions, and having students input symbolic fractions and get feedback from fraction bars would prompt thinking on how the two representations were related. Also, we thought it would show students that the common mistake of adding both numerators and denominators was incorrect. We termed this feedback grounded feedback because it was grounded in student‟s prior knowledge, and grounded an unfamiliar representation (fraction symbols) in a more intuitive one (fraction bars). An initial think aloud study showed promise. The 5 th grade participants seemed to understand what the fraction bars meant, and used them to find and correct fraction- addition errors (Stampfer, Long, Aleven, & Koedinger, 2011). An experimental study found learning benefits with a fraction bar tutor (interface in Figure 1) (Stampfer & Koedinger 2012). This tutor does not indicate explicitly if an individual step is right, but students cannot advance to the next problem until all steps have been solved correctly. Although students learned from the tutor (improved from pre-test to post-test), process measures show incorrect interpretations of the fraction bars. Students often indicated they were done solving the problems even though the fraction bars did not line up. They clicked the “done” button on the tutor screen an average of about 2.5 times per problem (rather than the one necessary click). This finding revealed that one of our key assumptions about this form of grounded feedback for these students was not fully satisfied. It appeared that the fraction bar representations of addition were not as meaningful to all students as the think-aloud results suggested. Thus, we were led to investigate more deeply the cognitive