Recommendation 4 – Solving Word Problems Based on Common Underlying Structures
Connected to the CCSSO Standards and Instructional Recommendations
This is the second in a series of five postings that connect the Institute of Education Sciences (IES) practice guide recommendations, Assisting Students Struggling with Mathematics: Response to Intervention (RTI) for Elementary and Middle School Students with corresponding CCSSO Standards indicators, and an instructional look alike for that recommendation.
The IES guide identifies eight recommendations that are designed to help teachers, principals, and administrators use Response to Intervention for early detection, prevention, and support of students struggling with mathematics. This guide is a synthesis of research that provides instructional recommendations in support of engaging those struggling mathematics students. See IES Practice Guides.
RTI Recommendation 4: Interventions should include instruction on solving word problems that is based on common underlying structures.
CCSSO Standards:
- Solve word problems by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
- Use properties of whole numbers; look for and make use of structures.
- Represent and solve problems involving addition, subtraction, multiplication, and division.
- Work with addition, subtraction, multiplication, and division equations.
- Build number sense through activities.
What would solving word problems based on common underlying structures look like instructionally?
I found the following from the IES RTI Guide interesting and helpful; it also reminded me of the instructional approaches advocated by the Singapore mathematics series. “The two problems here are addition and subtraction problems that students may be tempted to solve using an incorrect operation. In each case, students can draw a simple diagram like the one shown below, record the known quantities (two of three of A, B, and C) and then use the diagram to decide whether addition or subtraction is the correct operation to use to determine the unknown quantity.” The model provides a structure or schema that assist students in formulating a strategy for solving the problem. Another problem from the suggestions is quoted.
Version 1:
Brad has a bottlecap collection. After Madhavi gave Brad 28 more bottlecaps, Brad had 111 bottlecaps. How many bottlecaps did Brad have before Madhavi gave him more?
C = 111 total caps after receiving bottle caps from Madhavi, B = 28 that Madhavi gave Brad and A = amount that Brad had started with…so that C – B = A
Version 2:
Brad has a bottlecap collection. After Brad gave 28 of his bottlecaps to Madhavi, he had 83 bottlecaps left. How many bottlecaps did Brad have before he gave Madhavi some?
C = Brads total caps, A = 28 that Brad gave to Madhavi, and B = 83 caps that Brad had left over …so that C = A + B
Version 3:
There are 21 hamsters and 32 kittens at the pet store. How many more kittens are at the pet store than hamsters?
C = total number of kittens or 32 kittens, A = 21 hamsters and B = number of more kittens than hamsters, C – A = B
All three problems use the same model that provide a visual for what a student knows and what is unknown and thus enables the student to determine the appropriate operation for solving the problem. This modeling also reminds me of strategies used and reported in Algebra for Everyone from National Council of Teachers of Mathematics (1990), Edwards, Edgar L. Jr..
What other examples could we include for solving word problems based on common underlying structures?