Teaching Mathematics in the Visible Learning Classroom, High School

John Almarode, Douglas Fisher, Joseph Assof, John Hattie, Nancy Frey

• Publisher: Corwin
• Year: 2018
• Print ISBN: 9781544333144
• eBook ISBN: 9781544333120
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

Select the right task, at the right time, for the right phase of learning

It could happen in the morning during homework review. Or perhaps it happens when listening to students as they struggle through a challenging problem. Or maybe even after class, when planning a lesson. At some point, the question arises: How do I influence students? learning—what's going to generate that light bulb "aha" moment of understanding?

In this sequel to the megawatt best seller Visible Learning for Mathematics, John Almarode, Douglas Fisher, Joseph Assof, John Hattie, and Nancy Frey help you answer that question by showing how Visible Learning strategies look in action in the mathematics classroom. Walk in the shoes of high school teachers as they engage in the 200 micro-decisions-per-minute needed to balance the strategies, tasks, and assessments seminal to high-impact mathematics instruction.

Using grade-leveled examples and a decision-making matrix, you'll learn to

• Articulate clear learning intentions and success criteria at surface, deep, and transfer levels
• Employ evidence to guide students along the path of becoming metacognitive and self-directed mathematics achievers
• Use formative assessments to track what students understand, what they don't, and why
• Select the right task for the conceptual, procedural, or application emphasis you want, ensuring the task is for the right phase of learning
• Adjust the difficulty and complexity of any task to meet the needs of all learners

It's not only what works, but when. Exemplary lessons, video clips, and online resources help you leverage the most effective teaching practices at the most effective time to meet the surface, deep, and transfer learning needs of every student.

1 Teaching With Clarity in Mathematics

Chapter 1 Success Criteria:

1. I can describe teacher clarity and the process for providing clarity in my classroom.
2. I can describe the components of effective mathematics instruction.
3. I can relate the learning process to my own teaching and learning.
4. I can give examples of how to differentiate mathematics tasks.
5. I can describe the four different approaches to teaching mathematics we use in this book.

In Ms. Norris’s algebra class, students are learning to create equations in one variable and then use those equations to solve problems. On the board, she has clearly provided her learners with a learning intention and success criteria for the lesson:

Learning Intention: I am learning that authentic situations can be modeled or represented with equations.

Success Criteria

1. I can create an equation that models an authentic situation.
2. I can determine which type of function best models the situation (linear, exponential, or quadratic).
3. I can justify my decisions in creating my equation.

A learning intention describes what it is that we want our students to learn.

Success criteria specify the necessary evidence students will produce to show their progress toward the learning intention.

EFFECT SIZE FOR LEARNING INTENTION = 0.68 AND SUCCESS CRITERIA = 1.13

EFFECT SIZE FOR COOPERATIVE LEARNING = 0.40

EFFECT SIZE FOR COOPERATIVE LEARNING COMPARED TO COMPETITIVE LEARNING = 0.53

There are many different approaches for engaging learners in creating equations and inequalities. Today, Ms. Norris provides her learners with a contextual situation and then, after assigning them to cooperative learning teams, asks learners to come up with an equation that models the specific situation.

Four people can sit comfortably at a rectangular table. If two tables are placed together, this arrangement will comfortably seat six people. If three tables are placed together, the arrangement will comfortable seat eight people. How many people can comfortably sit at 10 tables if they are placed together? How many tables are needed for a group of 100 people?

Ms. Norris provides each cooperative learning team with different manipulatives (e.g., tiles, index cards, grid paper, and counters) that they can choose to use in accomplishing this task and deriving an equation. She tells students that they can choose to use some, all, or none of the manipulatives. Furthermore, she encourages them to use any strategy that they believe would be appropriate for completing this task. One cooperative learning team decided to use scissors and paper to construct models of tables. Another cooperative learning team used the manipulatives to model different table configurations and explore if different types of configurations provided more or fewer seating options. One specific student asked, “Does this have to be an equation, or can we develop an inequality?” A third team of learners did not find the manipulatives helpful and began to discuss the information they needed to create an equation. Ms. Norris is pleased that her learners are actively monitoring which strategy works best for them on this particular task.

Teaching Takeaway

As part of learning content, students should have access to and learn to apply a variety of strategies for solving problems.

EFFECT SIZE FOR STRATEGY MONITORING = 0.58

Ms. Norris is implementing the principles of Visible Learning in her algebra classroom. Our intention is to help implement these principles in your own classroom. By providing learners with a challenging task, a clear learning intention, and success criteria, cooperative learning teams are engaging in conceptual understanding, procedural knowledge, and the application of concepts and thinking skills. Ms. Norris holds high expectations for her students in terms of both the difficulty and complexity of the task, as well as her learners’ ability to deepen their mathematics learning by making learning visible to herself and each individual learner. As she monitors the learning progress in each team, holding every student accountable for his or her own learning, she takes opportunities to provide additional instruction when needed. Although her learners are engaged in cooperative learning with their peers, she regularly assesses her students for formative purposes. Ms. Norris is mobilizing principles of Visible Learning through her conscious awareness of her impact on student learning, and her students are consciously aware of their learning through this challenging task. Ms. Norris works to accomplish this through these specific, intentional, and purposeful decisions in her mathematics instruction. She has clarity in her teaching of mathematics, allowing her learners to have clarity and see themselves as their own teachers (i.e., assessment-capable visible mathematics learners). Clarity in learning means that both the teacher and the student know what the learning is for the day, why they are learning it, and what success looks like. This came about from using guiding questions in her planning and preparation for learning:

1. What do I want my students to learn?
2. What evidence shows that the learners have mastered the learning or are moving toward mastery?
3. How will I check learners’ understanding and progress?
4. What tasks will get my students to mastery?
5. How will I differentiate tasks to meet the needs of all learners?
6. What resources do I need?
7. How will I manage the learning?

Clarity in learning means that both the teacher and the student know what the learning is for the day, why they are learning it, and what success looks like.

Effect Size for Teacher Clarity = 0.75

Figure 1.1 How Visible Teaching and Visible Learning Compare

This figure is available for download at resources.corwin.com/vlmathematics-9-12

Ms. Norris exemplifies the relationship between Visible Teaching and Visible Learning (see Figure 1.1).

Video 3 What Does Teacher Clarity Mean in High School Mathematics?

https://resources.corwin.com/vlmathematics-9-12

Now, let’s look at how to achieve clarity in teaching mathematics by first understanding how components of mathematics learning interface with the learning progressions of the students in our classrooms. Then, we will use this understanding to establish learning intentions and success criteria, create challenging mathematical tasks, and monitor or check for understanding.

Components of Effective Mathematics Learning

Mathematics is more than just memorizing formulas and then working problems with those formulas. Rather than a compilation of procedures—isolating the variable, subtracting exponents, plugging numbers into the quadratic formula, FOIL, PEMDAS—mathematics learning involves an interplay of conceptual understanding, procedural knowledge, and the application of mathematical concepts and thinking skills. Together, these compose mathematics learning, which is furthered by the following practice standards:

1. Making sense of problems and persevering in solving them
2. Reasoning abstractly and quantitatively
3. Constructing viable arguments and critiquing the reasoning of others
4. Modeling with mathematics
5. Using appropriate tools strategically
6. Attending to precision
7. Looking for and making use of structure
8. Looking for and expressing regularity in repeated reasoning (© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.)

Teaching mathematics in the Visible Learning classroom fosters student growth through attending to these mathematical practices. As highlighted by Ms. Norris in the opening of this chapter, this comes from linked learning experiences and challenging mathematics tasks that make learning visible to both students and teachers.

Surface, Deep, and Transfer Learning

In their high school years, students grow in mathematics learning through a progression that moves from understanding the surface contours of a concept into how to work with that concept efficiently by leveraging procedural skills, as well as applying concepts and thinking skills to an ever-deepening exploration of what lies beneath mathematical ideas. But understanding these progressions requires that teachers consider the levels of learning they can expect from students. We think of three levels, or phases, of learning: surface, deep, and transfer (see Figure 1.2).

Learning is a process, not an event. With some conceptual understanding, procedural knowledge, and application, students may still only understand at the surface level. We do not define surface-level learning as superficial learning. Rather, we define the surface phase of the learning as the initial development of conceptual understanding and procedural skill, with some application. In other words, this is the students’ initial learning around finding the roots of a quadratic, understanding conceptually what the roots of the quadratic represent, and how to apply this learning to specific problems in algebra. Surface learning is often misrepresented as rote rehearsal or memorization and is therefore not valued, but it is an essential part of the mathematics learning process. You have to know something about the concept of the roots of an equation and process for solving the quadratic formula to be able to do something with that idea in an authentic situation.

Surface learning is the phase in which students build initial conceptual understa...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Illustration List
• Acknowledgments
• About the Authors
• Introduction
• 1 Teaching With Clarity in Mathematics
• 2 Teaching for the Application of Concepts and Thinking Skills
• 3 Teaching for Conceptual Understanding
• 4 Teaching for Procedural Knowledge and Fluency
• 5 Knowing Your Impact: Evaluating for Mastery
• Appendix A Effect Sizes
• Appendix B Teaching for Clarity Planning Guide
• Appendix C Learning Intentions and Success Criteria Template
• Appendix D A Selection of International Mathematical Practice or Process Standards∗
• References
• Index
• Publisher Note