My research interests lie in the broader field of Cognitive Development, with particular focus on the development of math skills--how students learn math. My research program has several facets, all linked in their aim to discover how prior knowledge affects performance and learning in math and how instruction can be designed to help fill gaps in student knowledge.    
       
  Development of a Linear Representation of Numerical Magnitudes  
       
 

Research in the domain of number line estimation has revealed an interesting developmental trend: when estimating magnitudes of numbers on a 0-100 scale, kindergartners produce logarithmic patterns of estimates while second grade students produce a pattern of estimates more consistent with a formal, linear representation of numbers (Siegler & Booth, 2004); a similar trend was found between second and sixth grade on the 0-1000 scale (Siegler & Opfer, 2003). Students' use of a linear representation of number is associated with higher mathematical achievement (Siegler & Booth, 2004) and better performance on a variety of estimation tasks (Booth & Siegler, 2006). It has also been shown to influence learning of arithmetic facts with sums up to 100; providing students with accurate external representations of magnitude was found to be useful for facilitating learning of those facts, as well as improving students' internal representations of magnitude (Booth & Siegler, 2008).

 
       
  Presentation Format in Pre-Algebraic Lessons    
       
 

For students who are in the process of transitioning to algebra (e.g., those taking Pre-Algebra courses), one of the main challenges is development of abstract, algebraic reasoning skills (Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005). Previous research has shown that presenting simple algebraic problems in story format rather than as equations can make the problems more concrete, which facilitates students’ thinking and can potentially help them to develop these important skills (Koedinger & Nathan, 2004); providing corresponding diagrams may also help students to parse the problems conceptually. However, our findings suggest that while diagrams are often included in instructional materials with the intent to aid learning, those students who have the most to gain (young, low-ability students) are unable to utilize the diagrams to their advantage; in fact, the inclusion of diagrams actually seems to hinder their learning compared with that of older and high-ability peers (Booth & Koedinger, 2007; in preparation). Providing low-ability students with training in construction of vertically arranged diagrams improves their use of both more complicated vertical and horizontal diagrams when solving algebraic story problems (Booth & Koedinger, 2009; in preparation)

 
       
  Prior Conceptual Knowledge in Algebraic Problem-Solving    
       
 

Previous research has shown that students often use ineffective procedures when learning to solve algebraic equations (Lerch, 2004; Sebrechts, Enright, Bennett, & Martin, 1996), and our findings confirm that use of these incorrect strategies may be attributed to misunderstandings or gaps in students’ conceptual knowledge of Algebra; students’ prior conceptual knowledge (or lack thereof) has also been shown to influence their learning of correct procedures (Booth & Koedinger, 2008).  Further, encoding of conceptual features of equations is associated with both conceptual knowledge of those features and equation solving performance; poor conceptual knowledge may lead to difficulty with both encoding and equation-solving procedures (Booth & Olsen, 2009).

 
       
 

Worked Examples and Self-Explanation in Algebra Learning

   
       
 

One possible way to help low-achieving students gain the prerequisite conceptual knowledge while promoting learning of correct procedures is to provide them with exercises in which they are asked to self-explain incorrect worked examples, forcing them to explicitly describe why the procedures are incorrect. This form of corrective self-explanation has been shown to aid learning in other domains (Siegler, 2002) and is hypothesized to help students to reject the incorrect procedures as well as to facilitate construction of correct procedures by drawing students’ attention to the important conceptual features of the problem. In our work, we provide students with this type of experience—self-explaining both correct and incorrect examples—while they are learning to solve equations. Students who receive this experience show greater gains in conceptual knowledge, and gain as much procedural knowledge as control students who receive more guided practice solving equations (Booth, Koedinger, & Siegler, 2008). We anticipate that individual differences in the degree of effectiveness of corrective self-explanation will depend on the amount and quality of students' background knowledge.