About

We created this workbook for our senior project at Carleton College with the goal of addressing student challenges with developing algebraic understanding. Throughout the project we worked closely with Mr. Dan Meyers, a teacher at Northfield Middle School, who instructs remedial math courses for 6th-8th grade students. He proposed the idea of creating new supplementary algebra curriculum for his students. He has introduced bar modeling to his students in the past using one of his favorite iPad apps for the classroom, Thinking Blocks. However, since Thinking Blocks has little algebra material, he asked us to develop a curriculum to extend the structure of Thinking Blocks to algebra. We observed his 7th and 8th grade classes for 10 weeks and then worked with his 6th grade remedial math class and taught our curriculum for 20 minutes 2 days a week for 6 weeks. We want to extend a huge thank you to Dan and his students for welcoming us into their classroom! This workbook has been 
dramatically shaped by student feedback and our observations from teaching the curriculum.

The goal of this curriculum is to use explicit instruction, paired with scaffolded, procedural practice, of a schematic modeling technique in order to develop students’ conceptual knowledge of the relationship between variables and numbers in equations. This method should also help students gain insights and understanding into the standard procedural methods they have probably learned in school.

Bar modeling can be beneficial for students in two ways: by providing a second strategy for solving problems, and by offering a spatial strategy for students. Providing students with multiple strategies for solving problems has been shown to improve problem-solving abilities, especially when students compare strategies (Hattikudur 2016). Additionally, students with better spatial reasoning skills tend to perform better in high school mathematics, so spatial strategies can help not only students that prefer spatial explanation, but also students whose spatial reasoning could be further developed (Chrysostomou et. al 2012).

We also believe bar modeling can address several misconceptions that students may have about algebra, including misconceptions of the equal sign, the negative sign, and variables (Busha & Karp 2013). See more about the misconceptions and how they are addressed on our misconceptions page.

Sources
  • Busha, Sarah, and Karen S. Karp. “Prerequisite algebra skills and associated misconceptions of middle grade students: A review.” Journal of Mathematical Behavior 32, no. 3 (2013): 613–632. http://dx.doi.org/10.1016/j.jmathb.2013.07.002.
  • Chrysostomou, Marilena, Demetra Pitta-Pantazi, Chara Tsingi, Eleni Cleanthous, and Constantinos Christou. “Examining number sense and algebraic reasoning through cognitive styles.” Springer Science+Business Media Dordrecht 83 (2012): 205-223. http://dx.doi.org/10.1007/s10649-012-9448-0.
  • Hattikudur, Shanta, Pooja G. Sidney, and Martha W. Alibali. “Does Comparing Informal and Formal Procedures Promote Mathematics Learning? The Benefits of Bridging Depend on Attitudes Toward Mathematics.” The Journal of Problem Solving 9, no. 1 (2016). doi:10.7771/1932-6246.1180.
  • Lucariello, Joan, Michele T. Tine, and Colleen M. Ganley. “A formative assessment of students’ algebraic variable misconceptions.” The Journal of Mathematical Behavior 33 (March 2014): 30-41. doi:10.1016/j.jmathb.2013.09.001.