Resources
Annotated Bibliography
Ciobanu, Mirela, “In the Middle-Misconceptions About the Equal Sign in Middle School,” OAME/AOEM Gazette (2014), 14-16.
This article defines the “operational” meaning of the “=” sign (calculation equals answer), and describes how this interpretation causes errors, especially when solving algebraic equations.
Fuchs, Lynn S. et al., “The Effects of Schema-Broadening Instruction on Second Graders’ Word-Problem Performance and Their Ability to Represent Word Problems with Algebraic Equations: A Randomized Control Study,” Elem Sch J. (2010), 446-463.
This is a study on teaching students to representing the structural, defining features of word problems with the schema of “overarching equations.” Essentially, this unit is doing something analogous, but using the “problem types” and their corresponding bar models rather than “overarching equations.”
Howe, Roger, “From Arithmetic to Algebra,” Mathematics Bulletin—A Journal for Educators, 49(2010), 13–21.
This is a paper on teaching algebra through analyzing and discussing both arithmetic and algebraic word problems. In particular, I found his discussion on units of variables, and of terms in an addition equation, to be helpful in my unit.
James, Jonathan “The Surprising Impact of High School Math on Job Market Outcomes,” Economic Commentary, (2013).
This is a study on the effect of high school math courses on unemployment rates and income, both for high school graduates who attend college, and those who go straight into the workforce. I was surprised at the effect for those who don’t attend college!
Kinzel, Margaret Tatem, “Understanding Algebraic Notation from the Students’ Perspective”, The Mathematics Teacher, Vol. 92, No. 5 (1999).
This paper examines students’ various struggles with representing and interpreting word problems using algebraic symbols (variables).
MacGregor, Mollie and Stacey, Kaye “Cognitive Models Underlying Students’ Formulation of Simple Linear Equations”, Journal for Research in Mathematics Education, Vol. 24, No.3 (1993), 217-232.
This paper shows students’ struggles in representing compared unequal quantities. Most notably, reversal errors can occur even if the student is not translating word-for-word (syntactically).
National Governors Association Center for Best Practices, Council of Chief State School Officers, Common Core State Standards for Mathematics, (Washington D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010), 88-89.
This is the list of the one-step problem taxonomies. In my unit, I changed the labels used in the table to ones we used in seminar, because I found them to be more descriptive than the ones in the CCSS guide. For example, the unknowns in the “Compare (Multiplication/Division)” problems changed from “number of groups” to “factor”, “group size” to “smaller”, and “product” to “bigger.” I also took some of my example problems in Appendices B-F from this document.
Ng, Swee Fong and Lee, Kerry, “The Model Method: Singapore Children’s Tool for Representing and Solving Algebraic Word Problems,” Journal for Research in Mathematics Education, Vol 40, No. 3 (2009), 282-313.
This paper studies the effect of using models to represent key information in a problem, in order to solve both arithmetic and algebraic problems. What I found most helpful were both the general, and applied examples of the Part-Whole model, the Comparison model, and the Multiplication and Division model.
Polya, G. How to Solve It: A New Aspect of Mathematical Method (Princeton and Oxford: Princeton University Press, 2004).
This is a book on methods of problem solving, broken up into four phases. In my curricular unit, I focus on the first phase, which is “understanding the problem.” The rest of Polya’s phases are devising a plan, carrying out the plan, and reviewing/extending the plan.
Powell, Sarah, "The Influence of Symbols and Equations on Understanding Mathematical Equivalence," Interventions in School and Clinic (ISC), Vol. 50(5) (2015), 266-272.
This article focuses on the “relational” meaning of the “=” sign, where one side of the equation is “the same as” the other side of the equation. The article goes on to explain how instructing on this relational meaning of equality improved student outcomes in setting up and solving word problems.
Teacher Resources
http://www.mathplayground.com/ThinkingBlocks/thinking_blocks_modeling%20_tool.html
This is a tool for creating bar models. Requires Adobe Flash. My diagrams in the “Key Ideas” section for this unit plan were screen-grabs from this tool.
Forsten, Char, Step-by-Step Model Drawing: Solving Word Problems the Singapore Way (Peterborough, NH: Crystal Springs Books, 2010), 30-32. I only used this book to see how she treated an unknown number of groups in the multiplication/division model. However, she is very step-by-step in explaining the process for beginners of the Singapore Bar Method. While this may be helpful as an entry, it is a bit prescriptive and too procedural.
The following are places where I got most of my word problems featured in the appendices.
http://www.math-aids.com/Algebra/Algebra_1/Word_Problems
http://cdn.kutasoftware.com/Worksheets/PreAlg/One-Step%20Word%20Problems.pdf
http://www.onlinemathlearning.com/comparison-word-problems.html
https://learnzillion.com/resources/11807?card_id=68694
https://www.ixl.com/math/algebra-1/solve-linear-equations-word-problems
https://www.khanacademy.org/math/algebra/one-variable-linear-equations/alg1-linear-eq-word-probs/e/linear-equation-world-problems-2
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_OneVariableWritingEquations.xml
https://www.cliffsnotes.com/study-guides/algebra/algebra-i/word-problems/age-problems
https://www.algebra.com/algebra/homework/word/age/Solving-Age-Problems.lesson
http://www.purplemath.com/modules/ageprobs.htm
Here is one version of group role cards you can give to students, including sentence starters for each group member.
http://www.readwritethink.org/files/resources/lesson_images/lesson277/cooperative.pdf
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