Virginia Mathematics Teacher Spring 2017

the whole on both models is 7. Jamal makes the first bar model for 5+2 quickly. After thinking a bit and checking with his rekenrek, Jamal creates a second bar model for 7+0. Jamal then writes a new number sentence 5 + 2 = 7 + 0. To bring closure to the lesson the students share their equations with partners. Then Mr. Dominguez engages them in whole class discussion to share and compare equations. Jamal is able to share his mistake, his strategy to fix his mistake, and what he learned from his mistake. Discussion and Implications Time is a critical factor in teachers’ instructional decisions about both what and how to teach mathematics. Effective teaching practices improve student achievement as well as student retention and application of concepts and skills over time. Choosing to implement evidence-based, effective teaching practices maximizes instructional time. “Effective mathematics teaching focuses on the development of both conceptual understanding and procedural fluency” (NCTM, 2014; p. 42). Procedural fluency and conceptual understanding are both important components of developing mathematical proficiency. One should not be sacrificed for the other; in fact, Mr. Dominguez’s lesson is an example of developing procedural fluency built on conceptual understanding. As the students problem-solved to become more efficient and flexible with basic addition facts, they relied on their understanding of part-whole relationships. Their fluency worked in tandem with their number sense for composing and decomposing numbers. The tasks, materials, and discussion within Mr. Dominguez’s lesson built upon their conceptual understanding and continued to deepen this understanding by allowing students to extend their understanding to a new context, new representations, and further mathematical discourse. Additionally, students began to explore the new concept of equality with an emphasis on meaning-making. CRA (also known as CSA or CPA) is an effective teaching strategy for all populations of students (including students with learning disabilities and English Language Learners). Rekenreks and part-whole bar models are effective tools for developing procedural fluency from

conceptual understanding. Mr. Dominguez’s lesson is an example of making the CRA connection across all levels in one lesson. It is not necessary that these connections are made within one lesson; in fact, this not often the case. Rather, connections are made across a unit or series of lessons. As teachers plan instruction, they should engage students in using meaningful concrete, representational, and abstract representations and explicitly making connections among them. CRA is also an effective teaching practice for differentiation. Students may be ready for problem solving and explaining their reasoning with different levels of representation (concrete, representational, abstract). Students can choose to use the level of representation for which they are ready or teachers can assign students to use one or more levels of representation based on their readiness. Teachers need to develop procedural fluency that is built upon and reinforced by conceptual understanding using concrete, representational, and abstract representations for all students. Two powerful tools are rekenreks and part-whole bar models. These effective teaching practices will maximize instructional time and grow mathematically proficient learners who are deeply engaged in the complex, rigorous problems of mathematics. References Baroody, A. J. (2006). Mastering the Basic Number Combinations. Teaching Children Mathematics 13 (1), 23–31. Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics , 5 (7), 400-405. Crawford, L. & Ketterlin-Geller, L.R. (2008). Improving Math Programming for Students at Risk: Introduction to the Special Topic Issue. Remedial and Special Education, 29 (1), 5-8. Fosnot, C. T., & Dolk, M. (2001). Addition and subtraction facts on the horizon. In V.Merecki& L. Peake (Eds.), Young mathematicians at work: Vol. 1. Constructing number sense, addition and subtraction (pp. 97-113). Portsmouth, NH: Heinemann.

Virginia Mathematics Teacher vol. 43, no. 2

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