Effective teaching is the non-negotiable core of any mathematics program. As mathematics educators, we continually strive to improve our teaching so that every child develops the mathematical proficiency needed to be prepared for his or her future. By mathematical proficiency, we mean the five interrelated strands of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (National Research Council, 2001).
There is a plethora of “research-based” recommendations about instructional practices that we should employ to build students’ proficiency, such as peer tutoring, using worthwhile tasks, building meta-cognitive capabilities, using manipulatives, project-based learning, direct instruction . . . the list goes on and on. But which practices have a strong research foundation? And which are likely to produce the most significant pay-off in terms of students’ learning?
Several recent reports indicate considerable consensus about the essential elements of effective mathematics teaching based on mathematics education and cognitive science research over the past two decades. The National Council of Teachers of Mathematics’ (NCTM) publication Principles to Actions: Ensuring Mathematical Success for All describes effective teaching as “teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically” (NCTM, 2014, p. 5). It also identifies the following eight high-leverage teaching practices that support meaningful learning:
The 2012 National Research Council report Education for Life and Work identifies the following essential features of instruction that promotes students’ acquisition of the 21st century competencies of “transferable knowledge, including content knowledge in a domain and knowledge of how, why, and when to apply this knowledge to answer questions and solve problems” (p. 6) in mathematics, science, and English/language arts:
These features are strikingly similar to the NCTM effective teaching practices described above.
Consensus on these effective practices, while critically important, leaves open the questions of their relative effectiveness, the conditions in which they are most effective, and details of their implementation in the classroom. Visible Learning for Mathematics addresses these questions and more, which makes it an invaluable resource for mathematics educators at all levels.
First, Visible Learning for Mathematics extends John Hattie’s original groundbreaking meta-analysis of educational practices in Visible Learning (2009) to specific mathematics teaching practices. The book goes beyond identifying research-based practices to providing the relative effect a teaching practice has on student learning—the effect size. For example, the second effective teaching practice calls for implementing tasks that promote reasoning and problem solving. In Visible Learning for Mathematics, the authors describe how engaging students in problem-based learning by using tasks that require them to apply their prior knowledge and skills in new situations has a strong effect on student learning (effect size of 0.61). The authors emphasize the importance of selecting tasks that are appropriate, given students’ prior knowledge and the learning goals for the lesson, and describe criteria for doing so.
High-quality instruction involves both implementing effective practices and eliminating ineffective practices. A second extremely valuable feature of Visible Learning for Mathematics is that it identifies ineffective practices, including ones that have face validity and are widely used, such as ability grouping in elementary grades (effect size of only 0.16) and stopping instruction prior to high-stakes testing to teach test-taking (test prep) (effect size of only 0.27). The authors then provide alternatives to these ineffective practices such as effective grouping strategies and distributed practice (effect size of 0.71) in place of test prep.
Further, the book situates highly effective teaching practices in three phases of learning—surface, deep, and transfer learning. It productively redefines “surface learning” as the phase in which students build initial conceptual understanding of a mathematical idea and learn related vocabulary and procedural skills. Unfortunately, many teachers stop here, which doesn’t give students the complete picture. It is through the subsequent phases of deep and transfer learning that students begin connecting ideas, making generalizations, and applying their knowledge to new and novel situations.
This framework offers a precise way to consider when particular teaching practices most benefit students’ learning, considering where students are in the learning process. For example, it recommends the kinds of mathematical tasks and talk that are likely to be most beneficial in each phase. It clarifies when and why practices like direct instruction or problem-based learning are most useful and effective and gives specific tools for implementing them. In short, it helps us know more specifically what to do, when, and why to achieve maximum impact in our classrooms. The authors illustrate this through the use of vignettes and concrete tools that show readers how to incorporate particular practices into one’s teaching. The supplemental videos offer classroom-based models of what these practices look and sound like for each phase of learning across the K–12 spectrum, along with teachers’ personal reflections on how they incorporate these practices into day-to-day instruction.
Finally, the book is designed to support individual and collaborative professional learning. We know teachers are more effective when they’re working together. The reflection and discussion questions at the end of each chapter give teachers an opportunity to digest the book a chapter at a time, considering and discussing how what they’re learning can be applied in their own situations. It is an extremely valuable extension of the ideas in Principles to Actions in that it supports taking action. And while the book is written for teachers, it will surely be an equally valuable resource for all mathematics educators, including leaders, administrators, and teacher educators.
In short, with its focus on true student-centered teaching, this book brings all the research together into a coherent and precise structure that can guide our practice, making the learning visible both to our students and to us. It’s a must-read. I highly recommend it.
—Diane J. Briars
Past President (2014–2016) National Council of Teachers of Mathematics