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Posted by John Woodward on Jun 7, 2017
Editor's Note: This is part 3 of a 3-part series on Math PD
American educators have a well-honed way of thinking about curriculum. Typically, district committees compare and then adopt a curriculum to meet specific goals or guidelines. More recently, curriculum adoption in math has been driven by state or national standards.
A subsequent step in many districts is curriculum training, where teachers spend one or two days learning about the scope and sequence of topics for each grade level, the structure of the lessons, the types of assessments, and the availability of print or online supplementary materials. All of this renders a strong impression that publishers have thought out all of the details. For teachers, then, it’s a matter of following individual lessons or sections of a chapter from one day to the next.
This process is far different in countries deemed highly successful in math instruction, even if they are not the top performers on international math assessments. Put simply, there is far more to a day’s math lesson than what is in the text. Teachers have to work beyond the printed page, because their textbooks are radically more streamlined than those found in the United States. This compels teachers in these successful countries to use central or “big ideas” to guide their instruction as they move across lessons or a chapter.
Big ideas add coherence to what is taught. They also enable teachers in the Netherlands, Japan, or Singapore to grasp better what students do and do not understand on a daily basis—well before any kind of formal assessment. This ongoing, informal way of diagnosing student understanding is what has come to be known as formative assessment.
In fairness, it should be said that any number of math educators in the United States have long agreed with this approach to curriculum. Stein and Kaufman made an elegant argument for this in a recent study of elementary grade teachers who were using one of two, standards-based curricula. Teachers who participated in the research were interviewed, surveyed, and observed during the course of two years. Stein and Kaufman focused on how: a) understanding big ideas helped teachers add coherence to their lessons (particularly in cases where some teachers were pushed beyond their background knowledge and experience by the curriculum), and b) teachers were able to engage students more consistently in challenging or “high-level tasks” throughout the lesson. This latter concern was particularly important in light of what has been a central feature of national math standards for decades.
Stein and Kaufman found teachers who were guided by big ideas—regardless of curriculum—implemented their lessons better and more coherently. Specifically, teachers were less likely to be distracted by peripheral content in the textbook. An understanding of the big ideas behind the lesson (or chapter) also enabled them to focus on sustained, high-level tasks. It was no surprise, then, to find students were much more engaged in math day after day.
As the senior author of NUMBERS, a professional development program for K-8 mathematics, I find these results fully congruent with our thinking about how to help teachers in the classroom. My fellow authors and I have created and piloted five PD modules (Number Sense, Fractions and Decimals, Measurement and Geometry, Ratios and Proportions, Algebraic Thinking) that unpack the core math domains in today’s state and national standards. One key purpose of each of these two-day modules is to translate the standards into big ideas that teachers can understand and use in their classrooms. Furthermore, our goal is to show what high-level tasks look like in each domain.
For example, our Fractions and Decimals module stresses a number of big ideas before we ever begin presenting the concepts behind operations on fractions. We begin this module by discussing the importance of equal shares, part to whole relationships, equivalence, and magnitude. We also have activities where teachers use tools like Cuisinaire® Rods and pattern blocks to explore part to whole relationships. Area models and number lines become critically important as ways to develop number sense around the magnitude of fractions. These tools also provide a powerful foundation for helping teachers assess student understanding on a daily basis as well as create a well-articulated framework for classroom discussions. They also are readily transportable to the classroom irrespective of what curriculum is being used.
The emphasis on big ideas in NUMBERS PD is important for many other reasons. Having a better sense of the big ideas found in the standards or a textbook can dramatically help teachers differentiate their instruction. As we all know, most American math curricula are chock full of optional activities and peripheral tasks. By focusing on big ideas, teachers can help more of their struggling students succeed by reinforcing day after day what an operation like division of whole numbers means or the dynamic relationship between area and perimeter.
A final and compelling reason for why we link big ideas to high-level tasks in our NUMBERS modules is that it yields a deeper, more flexible understanding of math. This is the central takeaway from the Stein and Kaufman research. It also is the essence of today’s standards, mathematical practices, and high-stakes tests.