Defining a High-Standards Math Curriculum for Struggling Students, Part 1 of 2
It takes time for research to be translated into practice, particularly when it comes to textbooks. For example, it was nearly 20 years ago when U.S. math educators examined the textbooks and instructional practices of highly successful countries around the world, only to determine what we already knew. American math textbooks were “a mile wide and an inch deep.” In contrast, international curricula typically contained fewer topics that were addressed in greater depth.1
The traditional structure of math textbooks as you move across the grade levels has been unfortunately predictable. James Flanders’ analysis of elementary and middle school texts in the late 1980s characterized the typical text as bloated with all kinds of review and extra content.2 Almost 30 years later, we still have the same problem in many of our math textbooks.3 This problem remains in spite of the fact that efforts like the National Council of Teachers of Mathematics (NCTM) standards to infuse more conceptual understanding and problem solving in textbooks occurred in the intervening years.
For all of the controversy today over the Common Core State Standards, one thing can be said. They are a serious effort once again to address the fewer topics and greater depth features of standards and curriculum found in high-achieving countries. Allowing teachers the time in a year’s curriculum to “go deep” on a topic like the magnitude of fractions or part-to-whole relationships produces far better results than the superficial treatment of such topics in most American math curricula. These characteristics need to be reflected in curricula for all students, particularly for those who struggle to make progress from one grade to the next.
Yet controlling the number of topics and adding depth is only the beginning. The field of psychology attests to the fact that asking all students to learn challenging content in the same, fixed amount of time is futile. Students vary in auditory and visual processing, working memory, and quantitative reasoning.4 Students also end up in the middle grades with a quality of instruction that is highly varied—not to mention those students who are affected by profound factors that lie outside of school. Interestingly, some of the high-performing countries try to account for this variation by using instructional resources flexibly to close achievement gaps. Finland, for example, dynamically groups students in grades K–8 in a way that approximately 50 percent of eighth graders receive some kind of specialized instruction. In contrast, Singapore maintains high expectations for all students but moves students through secondary education in eight different tracks of instruction.
Put simply, the rate at which struggling students are taught concepts and given opportunities to solve problems needs to vary from other, more successful students. This statement should not be translated as, “struggling students need the same context at half the speed.” The issue is more intricate, and one implication is that not all topics (or standards) in a grade level are equally important. For example, it isn’t clear that struggling students need to learn all of the middle school statistics and probability standards in the Common Core. That might be nice, but it may not be possible in light of more important topics in ratios and proportions or expressions and equations.
It goes without saying that curriculum also needs to weave together conceptual and procedural understanding. While this statement may be obvious to those in the math education community, it is an idea that has been slow to develop in special education. Far too much time in remedial and special education settings is spent on worksheet drills and the practice of procedures.
Furthermore, it is imperative that visual representations be used to help students understand concepts and solve problems. This is a major recommendation that can be found in a recent, major synthesis of educational literature on the needs of at-risk and special education students.5 Visual representations such as tape diagrams and number lines (pictured here as a digital manipulative in TransMath Third Edition) are critical tools that scaffold learning for these students. They provide another window into the symbols that are so often difficult to comprehend and manipulate.
Finally, problem solving needs to occur on a regular basis, not just once in a while. Again, a careful examination of research across grades 4 through 8 recommends that ongoing, challenging problem solving should be a regular feature of instruction for all students.6 It should be at the center of classroom instruction, not just “end of section” homework assignments. Not only is problem solving a central feature of the Common Core and so many previous state standards of the 1990s; it is essential to the way we think about mathematics today.
In Part 2 of this blog post, I will talk about a further refinement of these features of a high-standards curriculum: the way mathematical ideas can be structured in the form of “dual topics” as a technique for maintaining student engagement, structuring appropriate review, and controlling the cognitive load of a math lesson.
References
1 Schmidt, W. et al. (1997). A splintered vision: An investigation of US science and mathematics education. New York: Kluwer Academic Publishers.
2Flanders, J. (1987). How much of the content of mathematics textbooks is new? Arithmetic Teacher, 35(1), 18-23.
3Polikoff, M. (2012). The redundancy of mathematics instruction in US elementary and middle schools. The Elementary School Journal, 113(2), 230-251.
4Lynch, S., & Warner, L. (2012). A new theoretical perspective of cognitive abilities. Childhood Education, 88(6), 347-353.
5Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009a). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides.
6Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides.
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