Background
Experts agree that algebra is a crucial course in students’ mathematical trajectory, and success in this course has been identified as important to college and career readiness (Stein, Kaufman, Sherman, & Hillen, 2011; Cogan, Schmidt, & Wiley, 2001; Kaput, 2000; Moses & Cobb, 2001). A key learning goal in algebra is the use of symbols to represent and analyze situations and problems, and many textbooks are written with a heavy emphasis on symbolic usage and presentation.
However, working with symbolic representations is challenging for students (Walkington et al., 2012), and evidence suggests that students actually have improved learning when they first learn about a new concept using concrete and familiar formats (Goldstone & Son, 2005) - like verbally presented algebra story problems. By giving students story scenarios first, instead of symbols alone, we can draw on the things they already know and understand. Over time, these verbal supports can be faded, as students begin to understand and work with symbols more.
Textbook Classifications
Current Algebra I textbooks are classified as being traditional or reformed. Traditional texts have the lion's share of the textbook market (Holt, Pearson, Saxon, etc.), and introduce concepts by showing definitions and worked examples, and then presenting problem sets. These texts are typically expected to go along with a teacher-directed approach to instruction. Reformed textbooks are more rare in Algebra I, and often follow NCTM standards for reformed teaching, taking a student-centered approach. They may present students with more complex, open-ended problems or mathematical investigations, and accentuate the use of problem-based learning. Differences have emerged in the way that traditional and reformed curricula introduce the use of symbols in Algebra I, and many reformed textbooks in mathematics have taken less of a symbolic approach and adopted a more verbal presentation style.
Example of problem presented in VERBAL format:
- Maria just got a new cell phone, and on her phone plan each text message she send costs $.10. Write an algebraic expression that relates the number of texts Maria sends to the cost in dollars. How much will it cost to send 7 texts?
- Solve for y when x = 7:
y = 3x + 5
y = 0.25x
y = 2x - 3
In a recent study, researchers found that the presentation format of the examples and homework problem sets in commonly used Algebra I textbooks varied depending on the type of textbook, traditional or reformed (Sherman, Walkington, Howell, 2013). Reformed texts favored a verbal presentation first, and this verbal first approach was faded over time in the text. Thus in reformed texts, when students are first learning about algebra, they get a lot of verbal problems, but as their expertise develops, they get more symbolic problems.
Traditional texts favored symbols first - in each section, symbolic problems were presented to students before verbal problems. Traditional texts also had fewer single format only sections - there were fewer sections that had only verbal problems, or only symbolic problems. Most traditional texts contained a mixture of symbolic and verbally presented problems in the homework, yet the instructional examples provided in the text trended toward symbolic. Other research has also suggested that in traditional texts, the student recommended exercises in the teacher’s edition sometimes excluded the verbal problems from the students’ assignment.
Implications
Because many schools are using traditional textbooks, it is likely that your district adopted text has a heavy prevalence of symbol-first presentation. Be aware of the challenges that this approach may present to your beginning Algebra I students, and use verbal presentations in your initial classroom examples whenever possible. When assigning homework, be aware that the recommended exercises may exclude all of the rich verbal contextual problems, and add a few back in to your homework assignment. Better yet, choose a few to discuss and work on together as a class.
The choice of an Algebra I textbook may not be a decision that you can determine, but how to best use the examples and problems presented in the book is always your choice as the teacher. What type of textbook are you currently using? Take a closer look, and be aware of the presentation style that the book favors. If verbal scaffolding is not prevalent, it is easy to add those supports back in to help your students to succeed. A closer look will help you to see those places in the curriculum where a verbal presentation could be beneficial.
Resources:
Texas State Adopted Textbooks: http://www.tea.state.tx.us/index2.aspx?id=2147499935
References:
Cogan, L.S., Schmidt, W.H., & Wiley, D.E. (2001). Who takes what math and in which track? Using TIMMS to characterize U.S. students’ eighth-grade mathematics learning opportunities. Educational Evaluation and Policy Analysis, 23, 323-341.
Goldstone, R., & Son, J. (2005). The transfer of scientific principles using concrete and idealized simulations. Journal of the Learning Sciences, 14(1), 69-110.
Kaput, J. J. (2000). Teaching and learning a new algebra with understanding. U.S.; Massachusetts: National Center for Improving Student Learning and Achievement.
Moses, R., & Cobb, C. (2001). Radical Equations: Math Literacy and Civil Rights. Boston: Beacon Press.
Sherman, M., Walkington, C., & Howell, E. (April, 2013). A comparison of presentation format in algebra curricula. Presented at National Council of Teachers of Mathematics Research Pre-session, Denver, CO.
Stein, M.K., Kaufman, J. H., Sherman, M., Hillen, A.F. (2011). Algebra: A Challenge at the Crossroads of Policy and Practice. Review of Educational Research, 81(4), 453-492.
Walkington, C., Sherman, M., & Petrosino, A. (2012). ‘Playing the game’ of story problems: Coordinating situation-based reasoning with algebraic representation. Journal of Mathematical Behavior, 31(2), 174-195.
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