Publication Date



Open access

Embargo Period


Degree Type


Degree Name

Doctor of Philosophy (PHD)


Teaching and Learning (Education)

Date of Defense


First Committee Member

Walter G. Secada

Second Committee Member

Luciana de Oliveira

Third Committee Member

Mary A. Avalos

Fourth Committee Member

Ji Shen

Fifth Committee Member

Cengiz Zopluoglu

Sixth Committee Member

Robert Schoen


Cognitively Guided Instruction (CGI) is a professional development framework that informs participating teachers regarding research-based knowledge of elementary arithmetic-problem types and children’s developmental strategies for solving them. A classroom taught led by CGI participants includes one where heterogeneous learners (including English Language Learners, ELLs) are using and discussing representational systems and modeling tools (such as base-10 blocks, snap cubes, fingers, and verbal narratives) as part of the collaborative whole-class and small group problem solving process. This exploratory study investigated the effects of classroom instruction by CGI teachers on ELLs’ use of representational systems as they independently solved arithmetic word problems. A subset from a large-scale randomized control trial, totaling fifty-two first graders and fifty-three second grade ELLs (including nineteen and twenty-five CGI ELLs, respectively), were observed and videotaped during an individualized, statistically-validated arithmetic assessment comprised of several question formats. Students’ use of different, total, and simultaneous representational modeling tools were coded and tallied using video-coding software as up to five different word problem types were solved. The resulting data were analyzed using analyses of variance and effect sizes, comparing between subjects (Treatment versus Control) and within subjects (by Problem Type). This study found that CGI ELLs were more likely to use a greater number of different and total representational modeling tools in more complex ways when solving a variety of word problem types, than their non-CGI peers. The first grade ELLs exhibited these findings more often and more consistently than their second grade peers. The significant differences in representational tool use by first grade CGI ELLs (when compared to their non-CGI peers) as they solve several word problem types emphasized in the curriculum (such as Addition/Subtraction with a missing addend”; and “Multiplication with grouping”) appear to be reflecting CGI effects during assessments. Similar (though less pronounced and frequent) effects were found as second graders solved "Multiplication with Grouping” problems, an emergent 2nd grade curricular emphasis. Both grades showed CGI effects on representational modeling when solving “Subtraction with an unknown result” problem types. This study’s findings are consistent with prior research in CGI that notes the mediating power of the word problem type. It contributes to that work by adding a representational modeling perspective as it suggests a transfer of the representational modeling practices present in CGI classrooms by ELLs when they are independently assessed on similar word problems and allowed to use similar representational modeling tools.


CGI; ELL; Representations; Modeling; Assessment; Word Problem Types