Defining a High-Standards Math Curriculum for Struggling Students, Part 2 of 2
I made the case in my previous blog that adjusting the pace of instruction for struggling students in a high-standards curriculum is imperative. We all have different aptitudes for a given endeavor—from music to mathematics—and it is unrealistic to expect that all students can learn the same set of complex ideas in the same, fixed period of time.
Adjusting the pace of instruction does not mean that we should teach topics like fractions or integers in “twice the time” as much as it means that we need to sequence carefully the flow of concepts within these topics. It also means trade-offs such as not teaching every standard. There is little reason to believe that every standard is equally weighted in its importance, particularly if time is an issue.
There are also a number of occasions where a curriculum sequence might dictate that standards from previous grade levels be taught as a lead-up to a grade-level standard. For example, struggling students in fifth grade may need to review third and fourth grade standards such as equivalence, magnitude, and part-to-whole relationships prior to learning about operation on fractions. This kind of structure takes less for granted, and it helps solidify the necessary background knowledge that is crucial for successful understanding.
A further elaboration on how we adjust instruction is to control for cognitive load. Mathematics research on struggling students indicates that instructional tasks that place too much cognitive demand or “load” on struggling students can overwhelm their learning.1 Excessive cognitive load can take the form of complex, high-level tasks that last for an entire period. Problems may be simply too challenging, or the pace of instruction too fast for these students.
Another place where cognitive load issues arise is when a significant portion of an instructional period is devoted to highly routine tasks like computation procedures. This pattern is all too common in special education settings, and struggling students are either overwhelmed by the repetitive task of accurately performing calculations or, more likely, they simply lose interest in the task because they are bored. It is fair to say that 50 minutes of adding, subtracting, multiplying, and dividing fractions is likely to generate less than optimal results in most students, if not adults.
A novel structure for adapting curriculum for struggling students is to create dual topics for each unit of instruction. For example, consider a unit on simple algebraic equations and geometry. Let the primary topical strand be algebra, where students work equations such as 2(x + 3) = 26. The focus at this stage of instruction may be on either distributing the coefficient or showing how distribution is unnecessary because the first step in solving this equation is to multiply both sides by one-half. This is a complex task for struggling students, and it requires careful sequencing and opportunities to compare both strategies for solving this equation. Students will need time to work with coefficients, but it can be counterproductive to work on something like this all period, day after day.
This is the role of the second topical strand. Following our example, this could be geometry, and it is useful for several reasons:
- First, it moves students away from the kind of intense symbol manipulation that is involved in solving algebraic equations.
- Second, there are fruitful connections between the two topics (e.g., finding the degrees of a missing angle of a quadrilateral based on the measure of other angles).
- Third, the shift in topics helps students remain engaged in instruction for the entire period.
It should be understood that a dual topic structure does not imply an equal weighing of each topic in a lesson. Depending on the sequencing and distribution of practice, a lesson one day might place major emphasis on the primary topic. For example, current research in problem solving2 recommends that students compare strategies for solving equations. This would be a significant component of one day’s lesson, with more routine work on a recently taught geometry concept. As we move to the next day, students would continue solving equations with coefficients, but a major focus might be on geometric problem solving (e.g., given the measure of one interior angle in a parallelogram, what are the measures of all of the other interior angles?).
Another advantage of the dual topic structure is that it helps promote variation in instructional tasks within a lesson. Mary Kay Stein and her colleagues3 underscore the importance of varied instructional tasks as a way of thinking about a unit of instruction. Too often, classrooms can contain little more than low-level tasks such as memorization and “procedures without connections.” It is critical that students experience higher-level, more challenging tasks on a regular basis. The ebb and flow of dual topics across an instructional unit help ensure that students receive this kind of variation on tasks.
In my next blog, I will discuss the research support for the dual topic approach for struggling students. Stay tuned!
References
1Baxter, J., Woodward, J., & Olson, D. (2001). Effects of reform-based mathematics instruction in five third grade classrooms. Elementary School Journal, 101(5), 529-548.
2Star, J., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565-579.
3Stein, M., Smith, M., Henningsen, M., & Silver, E. (2009). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.
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