Improving Math Instruction: Making the Complex Understandable for Teachers and Students

Posted by John Woodward on May 3, 2018

  • Math
  • Professional Development
John Woodward


Curriculum’s role in the classroom, like so many educational trends, has followed a pendulum swing for decades.

Beginning in the 1960s, there was an attempt to “teacher proof” curriculum so the materials would be consistently implemented from one classroom to the next. Yet, research has shown teachers deviate from even the most detailed, “script-based” curriculum for a range of reasons. 

Some teachers instinctively bristle at the constraints of these materials. Others vary from the prescribed routines because of classroom logistics. Most recently, standards have pushed students to higher levels of communication and problem solving. Detailed scripts cannot program how students are supposed to respond to questions like, “Is there another way you can show me why 3/8 is smaller than 3/5 using manipulatives?” The notion that well-taught students might use number lines, Cuisinaire rods, or pattern blocks in unexpected ways to answer this question goes against a highly prescriptive curriculum. More teacher judgment based on content and teaching or pedagogical knowledge is needed for this kind of instruction.

The standards movement in the 1990s impelled teachers to move away from the curriculum as the mechanism for transforming classroom instruction. Detailed observations and self-reports of expert teachers, as well as interest at the time in Asian mathematics instruction (where curricular materials are outweighed by carefully guided, teacher instruction), led to a major emphasis on teacher content and pedagogical knowledge. Teacher-driven practices, rather than the virtues of a detailed, prescriptive curriculum, came into vogue. And, as a consequence, the need for extensive professional development—as part of teacher preparation programs as well as considerable, ongoing inservice education—became an unavoidable byproduct.

However, we are now at a point where the limits of teacher-driven practices have become apparent. New teachers with sufficient levels of math content preparation are harder to find than before. Elementary teachers still tend to receive more preservice preparation in reading than in math. Those who serve struggling and special education students typically lack the kind of background needed to teach today’s mathematics. For these reasons, and more, the pendulum between curriculum and teacher-driven practices is moving again.

A middle ground, where curriculum is more “user friendly” to novice teachers and their students, has begun to appear in the science education literature. Elizabeth Davis and her colleagues have articulated what they call an, “educative curriculum.1” Their approach, which also draws extensively on work in mathematics education, acknowledges that teachers need to adapt curriculum to their instructional conditions. In this regard, curriculum should be designed to speak to teachers and not through them. Most of all, these materials need to be designed so teachers can and should learn about a topic and its related concepts with their students.

An educative curriculum allows topics to develop in complexity over time. In this way, teachers are not overwhelmed from the beginning with complex information, nor are they asked to push students in overly demanding classroom activities. Instead, teachers and their students move gradually into in high-level tasks. There is increasing support for posing problems and conducting classroom discussions. All of this is structured into the curriculum and the design of text materials.

In mathematics, a judicious use of visual representations and manipulatives can significantly help teachers and students understand key concepts. The best use of these tools is when they are described in the textbook in a way that complements their use in the classroom. This also suggests that textbook explanations are clear and concise. While this recommendation may seem obvious, far too many math textbooks are structured around extremes. The traditional format is two or three procedural examples before a bank of practice problems. A more contemporary or “reform” approach is to present an elaborate description that sets the stage for extended problem solving. A clear presentation of the key concepts behind a topic like fractions—equal shares, part-to-whole relationships, equivalence, and magnitude—is missing. Teacher guides can augment how educators learn the major concepts behind a topic by elaborating on the big ideas that guide a unit or chapter.

There also are emerging, technology-based tools for assisting teachers new to today’s mathematics. One of the more intriguing suggestions that Davis and her colleagues describe in their work is the teacher narrative. Online videos present a teacher describing how he or she implemented a lesson (or unit of instruction). These accounts can include what the teacher learned and how he or she adapted instruction for students. Also, brief technology-based animations of concepts along with clear explanations can be used inside and outside the classroom. These animations initially free teachers from the uncertainty of explaining a concept like regrouping as it appears in the context of long division.

A final component of educative materials is the inclusion of student work, particularly when it involves performance assessments. Far too often, a math unit ends with computational tests. These are fine, but an inherently limited form of assessment. A better measure of communication and problem solving can be found in performance assessments. Including rubrics and samples of student work that show a range of responses to problems can help teachers and students see what an optimal response to a challenging problem should look like.

By starting with a curricular approach that “speaks to and not through” teachers, further professional development is more contained and a logical extension of the curriculum. After all, one of the central problems of the teacher-driven phase of math education was that how much content knowledge and related pedagogical knowledge a teacher needed to know was seemingly unlimited. District budgets simply do not allow for this kind of professional development support. Thus, more elementary teachers, robust professional development in numbers and operations, fractions, and geometry would be a sensible and contained complement to an educative curriculum.

To be sure, the educative curricular approach will have its problems. However, it is a reasonable solution to what is one of the major challenges in math education now and in the near future—how we substantively assist teachers who are new to (or uncomfortable with) the mathematics we must teach in today’s classrooms.

Helping teachers build mathematics connections and problem-solving skills is key to student success. You can listen to Dr. John Woodward's webinar, "Improving Math Instruction: Making the Complex Understandable for Teachers and Students" here.

NUMBERS is an interactive, hands-on mathematics professional development offering for elementary and middle school math teachers. Research-validated findings have been integrated into the program along with numerous strategies and best practices to meet the expectations of rigorous state standards.


1Davis, E., & Krajcik, J. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3-14.

1Davis, E., Palincsar, A., Arias, A., Bismack, A., Marulis, A., & Iwashyna, S. (2014). Designing educative curriculum materials: A theoretically and empirically driven process. Harvard Educational Review, 84 (1), 24-52.

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