Linda M. Gojak and Sara Delano Moore, two co-authors of Visible Learning For Mathematics: What Works Best to Optimize Student Learning, agreed to answer a few questions about the book. Their other co-authors are John Hattie, Doug Fisher, Nancy Frey, and William Mellman.

Winner of the Presidential Award for Excellence in Science and Mathematics Teaching, Linda M. Gojak directed the Center for Mathematics and Science Education, Teaching, and Technology (CMSETT) at John Carroll University for 16 years. She has spent 28 years teaching elementary and middle school mathematics, and has served as the president of the National Council of Teachers of Mathematics (NCTM), the National Council of Supervisors of Mathematics (NCSM), and the Ohio Council of Teachers of Mathematics.

Sara Delano Moore is an independent educational consultant at SDM Learning. She has worked as a classroom teacher of mathematics and science in the elementary and middle grades, a mathematics teacher educator, Director of the Center for Middle School Academic Achievement for the Commonwealth of Kentucky, and Director of Mathematics & Science at ETA hand2mind.

LF: How is this book different from previous ones sharing John Hattie’s research?

Linda M. Gojak and Sara Delano Moore:

“Visible Learning for Mathematics” takes Hattie’s important ideas and focuses them in the mathematics classroom. It examines the teaching practices that have high effect sizes and puts them in a mathematics instruction context, using classroom vignettes that illustrate what these strategies and practices look like while teaching a mathematical concept or skill. It also spotlights personal stories from the authors, all of whom are or have been practicing teachers. The book makes the research come alive through a setting that is all about mathematics and mathematics instruction.

“Visible Learning for Mathematics” is a combination of research-based approaches for mathematics instruction and their connection to the surface, deep, and transfer phases of learning. That is what makes this book different.

LF: What guidelines are used to determine if the instructional strategies that work in one subject, like English, will work in another, like Math?

Linda M. Gojak and Sara Delano Moore:

Good instruction is good instruction in any subject. John’s research is built on a broad foundation that identifies what strategies we know to work across the all subjects. In this book we address guidelines for when and in what ways to use the strategies in the mathematics classroom. For example, clear learning intentions and success criteria are as important in mathematics as they are in English, social studies, science, or art. Students need to know the purpose of their work and be able to self-monitor whether and what they are learning.

Sometimes in education, we use the same term to mean different things. Direct instructionis a good example. In the United States, we often interpret direct instruction to mean a lecture followed by worksheets for practice. That’s not the definition in John’s research, and we are very explicit about that in the book. Direct instruction in this work includes seven major steps. Key elements include communicating learning intentions and success criteria to students, building student commitment and engagement, and lesson design in which students are actively involved in meeting the success criteria. This model is consistent with the discourse- and rich task-based instruction considered best practice in the mathematics education community (Editor’s Note: You can learn more about these seven steps of direct instruction here).

Our goal was to build connections and alignment between the best practices the mathematics education community has established and the language used in John’s work so that we can use these two strong bodies of research to support each other, and translate that into practice for teachers.

Although the instructional strategies are the same, the way they are carried out in mathematics (or science) may be different than in English Language Arts (or social studies). For example, the role of student discourse in the classroom is critical in both English Language Arts and mathematics, but the effective questioning techniques teachers use may differ between the two subjects. Vocabulary development is likewise important in both subjects. However, mathematics vocabulary may not always make sense in context and may require different instructional approaches than learning vocabulary in English Language Arts.

“Visible Learning for Mathematics” sets out to clarify what the implementation of Hattie’s work looks like in a mathematics classroom with highly successful students.

LF: The books talks about the role of making errors in learning and in teaching. Can you elaborate on that topic?

Linda M. Gojak and Sara Delano Moore:

Too many people, especially students, think math is all about getting the right answer, and when they have the right answer, all is good! In reality, mathematics is about thinking, reasoning, problem solving, and making sense. Students (and their parents) need to value mistakes and consider them steps toward reaching deep understanding and ultimately transferring learning, which includes using mathematics in contexts outside of the math classroom or environment.

Teachers should use student errors to analyze where misconceptions may be happening and then use those errors to adapt instruction, clarify misunderstandings, and build conceptual understanding. Mathematics that is taught as “show and tell” --in other words the teacher shows and tells and the students copy what the teacher has done--does not build understanding. For example, students who know how to compute to find the answer to a division example, but who do not understand the meaning of division, struggle with when to apply division to a problem situation.

Errors also enable students to think about their thinking (metacognition) and make adjustments so that the mathematics makes sense. Student errors are opportunities for both students and teachers to learn more deeply. When students are just going through the process and getting the right answer with no understanding, they are really not reaching the level of deep learning.

LF: What one-to-three findings/recommendations made in the book do you think mathematics teachers might find most surprising?

Linda M. Gojak and Sara Delano Moore:

Teachers may be surprised by the idea that surface, deep, and transfer learning are three different phases or components of learning, and the fact that students constantly move in and out of these phases throughout their mathematics education may be new to some teachers. “Visible Learning for Mathematics” will give K-12 teachers a lens through which they can evaluate the mathematics content for their grade level and think more deeply about how their instruction enables students to move through these learning phases.

It may also be surprising to some that surface learning does not mean just rote memorization; it involves conceptual understanding as well as procedural skill so that students are able to extend and apply that concept or skill in a variety of situations (deep learning). Too much mathematics instruction focuses on and is limited to building superficial skills such as knowing basic facts or knowing how to calculate with fractions, but misses the importance of conceptual understanding. It is the foundation of conceptual understanding which actually helps students to become proficient with skills and applications. We really tried to emphasize this in the book with many examples of what surface, deep, and transfer learning look like in effective mathematics instruction.

For some teachers, the richness of transfer learning will also come as a surprise. John’s research tells us that we need to think about transfer at both small and large scales. At a small scale, we encourage students to make jumps to the next idea in the progression. At a large scale, we help students see how the mathematics they’re learning applies in more far-reaching ways across the disciplines and even in various careers. There’s a tendency, perhaps, for some teachers to focus on the smaller scale and the book helps them look for opportunities to help their students make greater leaps. At the secondary level, with a focus on college and career readiness, there’s a tendency to focus more on the great leaps and assume that students can make the smaller connections on their own. This book likewise shares strategies to support the small-scale connections to ensure the richest possible mathematical learning for students.

LF: What findings/recommendations made in the book do you think might be most “controversial” and why?

Linda M. Gojak and Sara Delano Moore:

Anyone who keeps up with mathematics education research and professional opportunities from organizations such as the National Council of Teachers of Mathematics (NCTM) should not find these recommendations to be controversial. In fact, they are supported by documents such as “Principles to Actions” (NCTM, 2014), which includes a chapter on research- based effective teaching practices in mathematics. Each of those teaching practices is supported by Hattie’s work, and we include examples of them in “Visible Learning for Mathematics.” We hope that the recommendations in the book will help to continue the discussion, allowing more classroom teachers opportunities to plan and implement effective mathematics instruction so that all students can be successful.

We are excited that the important work of John Hattie has now been expanded to include what visible learning looks like in mathematics education. Since the original Curriculum and Evaluation Standards (NCTM, 1989) and subsequent state standards including the Common Core State Standards for Mathematics (2010) there has been ongoing controversy over the way mathematics is taught, particularly among non-mathematics educators including parents. Questions we hear over and over from parents such as “Should the focus be on skills and procedures? Should the focus be on understanding? Why does my child have to explain her thinking when she can accurately multiply two numbers?” are addressed in Hattie’s findings. Hattie’s meta-analysis of general educational research supports what mathematics research has found to be the most effective instructional practice for helping students to be successful in mathematics. “Visible Learning for Mathematics” provides clear, practical examples for administrators, classroom teachers, and parents to ensure opportunities for all students to have the best mathematics instruction.

LF: Thanks, Linda and Sara!

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