Visible Learning for Mathematics, Grades K-12

What Works Best to Optimize Student Learning

John Hattie, Douglas Fisher, Nancy Frey, Linda M. Gojak, Sara Delano Moore, William Mellman

• Publisher: Corwin
• Year: 2016
• ISBN:
  9781506362953
• Edition: 1
• Topic: Pedagogía
• Subtopic: Enseñanza de matemáticas

Rich tasks, collaborative work, number talks, problem-based learning, direct instruction…with so many possible approaches, how do we know which ones work the best? In Visible Learning for Mathematics, six acclaimed educators assert it’s not about which one—it’s about when—and show you how to design high-impact instruction so all students demonstrate more than a year’s worth of mathematics learning for a year spent in school. That’s a high bar, but with the amazing K-12 framework here, you choose the right approach at the right time, depending upon where learners are within three phases of learning: surface, deep, and transfer. This results in “visible” learning because the
effect is tangible. The framework is forged out of current research in mathematics combined with John Hattie’s synthesis of more than 15 years of education researchinvolving 300 million students. Chapter by chapter, and equipped with video clips, planning tools, rubrics, and templates, you get the inside track on which instructional strategies to use at each phase of the learning cycle: Surface learning phase: When—through carefully constructed experiences—students explore new concepts and make connections to procedural skills and vocabulary that give shape to developing conceptual understandings. Deep learning phase: When—through the solving of rich high-cognitive tasks and rigorous discussion—students make connections among conceptual ideas, form mathematical generalizations, and apply and practice procedural skills with fluency. Transfer phase: When students can independently think through more complex mathematics, and can plan, investigate, and elaborate as they apply what they know to new mathematical situations. To equip students for higher-level mathematics learning, we have to be clear about where students are, where they need to go, and what it looks like when they get there. Visible Learning for Math brings about powerful, precision teaching for K-12 through intentionally designed guided, collaborative, and independent learning.

1 Make Learning Visible in Mathematics

Copyright © Erin Null

2 + 2 = 4.

It just adds up, right? But think about how you know that two plus two equals four. Did you memorize the answer from a flashcard? Did someone tell you that and then expect that you accept it as truth? Did you discover the answer while engaged in a relevant task? Were you asked to explore a concept, and when you grasped the concept, someone provided you with labels for the ideas? In all likelihood, it was a combination of these things that led you to come to understand the concept of the number two, the possibility of combining like items, and the idea that the sum is a result of these combinations. Over time, you were able to consider an unknown term such as x in the equation 2 + x = 4 and master increasingly complex ideas that are based on algebraic thinking. Your learning became visible to you, your teachers, and your family.

And that’s what this book is about—making learning visible. By visible learning, we mean several things. First and foremost, students and teachers should be able to see and document learning. Teachers should understand the impact that they, and their actions, have on students. Students should also see evidence of their own progress toward their learning goals. Visible learning helps teachers identify attributes and influences that work. Visible learning also helps teachers better understand their impact on student learning, and helps students become their own teachers. In this way, both teachers and students become lifelong learners and develop a love for learning. Importantly, this is not a book about visible teaching. We do, of course, provide evidence for various teacher moves, but our goal is not to make teaching visible but rather the learning visible. Before we explore the research behind visible learning, let’s consider the ways in which you may have been taught mathematics. We need to accept and understand that high-quality learning may require that we discard ineffective pedagogy that we may have experienced as learners of mathematics.

Forgetting the Past

Do you remember the Men in Black movies? The agents who are protecting the universe have neuralyzers, which erase memories. They use them to erase encounters with intergalactic aliens so that people on planet Earth are kept in the dark about threats to their world. We wish we had that little flashy thing. If we did, we’d erase teachers’ memories of some of the ways they were taught mathematics when they were younger. And we’d replace those memories with intentional instruction, punctuated with collaborative learning opportunities, rich discussions about mathematical concepts, excitement over persisting through complex problem solving, and the application of ideas to situations and problems that matter. We don’t mean to offend anyone, but we have all suffered through some pretty bad mathematics instruction in our lives. Nancy remembers piles of worksheets. Her third-grade teacher had math packets that she distributed the first of each month. Students had specific calculation-driven problems that they had to do every night, page after page of practicing computation with little or no context. A significant amount of class time was spent reviewing the homework, irrespective of whether or not students got the problem wrong or right. In fact, when she asked if they could skip the problems everyone completed correctly, she was invited to have a meeting with the teacher and the principal.

In algebra, Doug’s teacher required that specifically assigned students write out one of their completed homework problems on the chalkboard while the teacher publicly commended or criticized people. Doug wasn’t academically prepared for entry-level algebra, so he hid outside the classroom until the teacher ran out of problems each day. (He took the tardies rather than show everyone he didn’t understand the homework.) When this ritual was completed, the teacher explained the next section of the textbook while students took notes. The teacher wrote on an overhead projector with rollers on each side, winding away, page after page. Doug learned to copy quickly into his Cornell notes since the teacher often accidentally erased much of what he wrote because of his left-hand hook writing style. When finished with this, students were directed to complete the assigned odd-numbered problems from the back of the book in a silent classroom. Any problems not completed during class time automatically became homework. Doug copied from his friend Rob on the bus ride home each day but failed every test. This spectator sport version of algebra did not work for students who did not already know the content. Doug’s learning wasn’t visible to himself, or to his teacher.

If you’re worrying about Doug, after failing algebra in ninth grade, he then had a teacher who was passionate about her students’ learning. She modeled her thinking every day. She structured collaborative group tasks and assigned problems that were relevant and interesting. Doug eventually went on to earn a master’s degree in bio-statistics.

John did okay in mathematics and enjoyed the routines, but if offered, he would have dropped mathematics at the first chance given. But his school made all students enroll in mathematics right to the last year of high school. It was in this last year that he met Mr. Tomlinson—rather strict, a little forbidding, but dedicated to the notion that every one of his students should share his passion for mathematics. He gave his students the end-of-the-year high-stakes exam at the start of the year to show them where they needed to learn. Though the whole class failed, Mr. Tomlinson was able to say, “This is the standard required, and I am going to get you all to this bar.” Throughout the year, Mr. Tomlinson persistently engaged his students in how to think in mathematics, working on spotting similarities and differences in mathematical problems so they did not automatically make the same mistakes every time. This teacher certainly saw something in John that John did not see in himself. John ended up with a minor in statistics and major in psychometrics as part of his doctoral program.

Learning isn’t linear; it’s recursive.

These memories of unfortunate mathematics instruction need to be erased by Men in Black Agent K using his neuralyzer, as we know that one of the significant impacts on the way teachers teach is how they were taught. We want to focus on the good examples—the teachers we remember who guided our understanding and love of mathematics.

We’ve already asked you to forget the less-than-effective learning experiences you’ve had, so we feel comfortable asking you one more thing. Forget about prescriptive curricula, scripted lesson plans, and worksheets. Learning isn’t linear; it’s recursive. Prescriptive curriculum isn’t matched to students’ instructional needs. Sometimes students know more than the curriculum allows for, and other times they need a lot of scaffolding and support to develop deep understanding and skills. As we will discuss later in this book, it’s really about determining the impact that teachers have on students and making adjustments to ensure that the impact is as significant as possible.

A major flaw of highly scripted lessons is that they don’t allow teachers to respond with joy to the errors students make. Yes, joy. Errors help teachers understand students’ thinking and address it. Errors should be celebrated because they provide an opportunity for instruction, and thus learning. As Michael Jordan noted in his Nike ad, “I’ve missed more than 9,000 shots in my career. I’ve lost almost 300 games. 26 times, I’ve been trusted to take the game winning shot and missed. I’ve failed over and over and over again in my life. And that is why I succeed.”

Linda remembers playing a logic game using attribute blocks with her students. The beginning of the game required that students listen carefully to the ideas of others and draw some conclusions as to whether those ideas were correct or accurate. At one point, she commented to an incorrect response, “That’s a really important mistake. I hope you all heard it!” The reaction of almost every student was a look of surprise. It was as if the students were thinking, “Have you lost your mind? The goal in math is to get it right!” That response made a real impact on Linda’s teaching moves in terms of recognizing how important it is for students to understand they learn and develop understanding from making mistakes (and, in fact, she still says that to this day!). The very best mathematicians wallow in the enjoyment of struggling with mathematical ideas, and this should be among the aims of math teachers—to help students enjoy the struggle of mathematics.

When students don’t make errors, it’s probably because they already know the content and didn’t really need the lesson. We didn’t say throw away textbooks. They are a resource that can be useful. Use them wisely, and make adjustments as you deem necessary to respond to the needs of your students. Remember, it is your students, not the curriculum writers, who direct the learning in your classroom.

What Makes for Good Instruction?

When we talk about high-quality instruction, we’re always asked the chicken-and-egg question: “Which comes first?” Should a mathematics lesson start with teacher-led instruction or with students attempting to solve problems on their own? Our answer: it depends. It depends on the learning intention. It depends on the expectations. It depends on students’ background knowledge. It depends on students’ cognitive, social, and emotional development and readiness. It depends where you are going next (and there needs to be a next). And it depends on the day. Some days, lessons start with collaborative tasks. Other days, lessons are more effective when students have an opportunity to talk about their thinking with the entire class or see worked examples. And still other days, it’s more effective to ask students to work individually. Much of teaching is dependent on responding to student data in real time, and each teacher has his or her own strengths and personality that shine through in the best lessons. Great teachers are much like jazz musicians, both deliberately setting the stage and then improvising. Great teachers have plans yet respond to student learning and needs in real time.

But even the most recognized performers had to learn techniques before applying them. Jazz musicians have to understand standards of music, even if they choose to break the rules. Similarly, great teachers need to know the tools of their craft before they can create the most effective lessons. Enter Visible Learning.

Errors help teachers understand students’ thinking and address it. . . . They provide an opportunity for instruction, and thus learning.

The Evidence Base

The starting point for our exploration of learning mathematics is John’s books, Visible Learning (2009) and Visible Learning for Teachers (2012). At the time these books were published, his work was based on more than 800 meta-analyses conducted by researchers all over the world, which included more than 50,000 individual studies that included more than 250 million students. It has been claimed to be the most comprehensive review of educational research ever conducted. And the thing is, it’s still going on. At the time of this writing, the database included 1200 meta-analyses, with more than 70,000 studies and 300 million students. A lot of data, right? But the story underlying the data is the critical matter; and it has not changed since the first book in 2009.

Meta-Analyses

Before we explore the findings, we should discuss the idea of a meta-analysis because it is the basic building block for the recommendations in this book. At its root, a meta-analysis is a statistical tool for combining findings from ...