ALN Releases First Chapter of New Book, “Teaching Math for All Learners”
Today, ALN is excited to release the first chapter of its brand-new book on the blog for free! This work is a collaboration of writing by multiple ALN experts focused on the All Learners Network Approach to mathematics.
In coming weeks, subsquent book chapters will be released free of charge here on our website. See the table of contents to see the content that will be covered in this book. Be sure to subscribe by email so you don’t miss out on any new posts! When the chapters are completed, the published book will be available for purchase.
Another book on teaching math? We knew when we set out to write this book that there were already several good resources available. We also felt, though, that too few were offered that focused on supporting students whose race, language, or socioeconomic or IEP status predicted a learning gap in math with their white and/or middle-class gened counterparts. We wanted to explore the ways that math instruction might need to be specially focused when working with underserved populations.
In some ways — fundamental ways — teaching for all students is no different than teaching math well. The difference lies in the clarity, specificity, and “tightness” of instruction. While many children in low-poverty schools seem to achieve reasonably well with less-than-perfect instruction, evidence is pretty clear that children in highpoverty schools need excellent instruction from skillful teachers in the context of supportive administrative frameworks (Tapper, 2010). We will not address the question of administrative frameworks in this book, though that is an interesting question. Our goal is to focus on giving teachers tools they need to be skillful practitioners.
This book offers an approach to instruction for all students that is unique in three ways:
The use of inquiry to guide instruction.
A strong focus on continuous formative assessment to understand student thinking and to inform instructional decisions.
The use of a Main Lesson/Menu approach to lessons to address the need for both inclusion and differentiation.
In Chapter 1, we’ll look at the theoretical framework – the underlying approach – to teaching math for all learners. We’ll look at how learners actually learn math and what teachers can do to facilitate that learning.
Chapter 2 examines an All Learners lesson. This book is focused on the instructional context and pedagogy that form the framework of teaching so that everyone can learn math with conceptual understanding. Through the use of an inclusive Main Lesson and a differentiated math Menu, we create a context for learning that can support success for a broad range of learners. Throughout this book we’ll reference the All Learners lesson model to situate activities and pedagogy. One of the most important elements of creating success with mathematics is focus. A complete vision of math learning (as articulated by documents like the Common Core State Standards) is important for students to understand the breadth of math topics that will lead them to high school and beyond. But sometimes math learning in the U.S. can be “a mile wide and an inch deep” (Schmidt, McKnight & Raizen, 2005). At ALN we’ve done considerable work identifying the math concepts that will be most helpful for student success at each grade level. We’ve done this by parsing the concepts that will lead to success in the next grade. These concepts are called, HLCs or High Leverage Concepts.
Chapter 3 explores what HLCs are and how they might be used to support learning.
In Chapter 4, we explore the important formative assessment practices that support teacher understanding of student learning. Formative assessment is the heart of effective instruction because it asks the critical questions:
“How does a student think about (or understand) a concept?”
“What connections can a teacher make to further (or improve) this understanding?”
Without good formative assessment the creation of appropriate menu activities to support students is impossible.
In Chapter 5, we consider the ways in which student discourse supports, encourages, and facilitates instruction. We’ll unpack a variety of talk moves that teachers can use during the Main Lesson, in small groups, and with individual students. The goal with this pedagogy is to help students find their own meaning with math concepts and be able to articulate that understanding for themselves and others. Some readers of this book may also have read my earlier book on working with struggling math learners, Solving for Why. That book reviewed some basic challenges that students on IEPs face when doing mathematics. A number of teachers have put the principles from Solving for Why to use in their classrooms to support students with specific learning challenges.
In Chapter 6 we will dive into techniques some of these teachers have used. We’ll also explore some general principles teachers use to create classrooms that support students with developmental delays, attention deficit, and specific learning challenges.
Chapter 7 is our take on the way mathematics can lead to greater equity among students. I’m fond of saying “math is the silver bullet.” That’s because there is good evidence that being successful with math leads to greater opportunity both in school and in the greater world. At ALN, we believe that effective math instruction is an act that promotes greater social equity and opportunity for everyone. Chapter 7 will explore this idea.
We’re making this manuscript available as an overview of teaching when the goal is for everyone to learn. We’re making it available with the hope that teachers will find both the perspective and the techniques helpful for realizing the goal that every child can do math.
Who Needs this Book?
Teaching Math for All Learners is aimed at teachers who want to provide effective instruction for all the children in their classes. This book can be valuable for teachers who may feel overwhelmed by the array of learners they must support. The strategies the authors in this book advocate are aimed at a single goal – math for everyone. Most teachers are familiar with the notion that most children can reach most of the standards set for them. While to many this attitude seems “most realistic,” I believe it fails the American vision of what school should be – an education for all.
This book is not comprehensive in identifying all the techniques necessary to ensure that everyone learns, but it’s a start. Teachers will find value information in these pages on how to meet the needs of learners whom they might have thought not capable enough. When we talk about all learners, we really do mean all, and our approach is our best set of tools to meet that goal.
Almost all teachers will have students in their classes with learning challenges of one kind or another. It’s a wonderful fact that students come to school with a wide variety of knowledge and ability. This should be celebrated. The question is whether the schools are prepared to help students make use of what they bring rather than lamenting that they might be different. This book is an attempt to help teachers be prepared to support all the students in their classes to be successful with math – and the opportunities it affords.
Chapter One: What it Means to Teach for All Learners—an Overview
What does it mean to learn mathematics?
For many of us it meant memorizing things. We memorized math facts and procedures for adding and subtracting. We memorized the quadratic formula and the area of a circle. If we could remember all those formulas and facts, we did well on our math grades. Our parents were pleased. We were happy. But were we able to access the kind of thinking that mathematics brings to us? Had we developed the skills to solve complex problems? Did we develop the ability to analyze subtle patterns? Had we become comfortable with the process of discovery?
Certainly, if we have all that information about numbers at hand, we know something about math. Still, at least for Americans, a large percentage of us are functionally unable to use whatever information we were exposed to in high school and college. We see math as information that we don’t really understand. It’s as if we had memorized lots of dates and events but had almost no idea how they formed the story of history. We know about math, but we don’t understand it.
That math has been taught and learned in much the same way for the last 30 years is puzzling. In 1989 the National Council of Teachers of Mathematics (NCTM, 2008) produced a new set of professional standards that should have refocused math instruction away from memorization and toward understanding. For some teachers, in some schools, instruction has changed, and as a result, their students learn math with understanding. For others, often schools that serve poor students and students of color, instruction is still focused on memorization, with less emphasis on understanding mathematics (Wenglinsky, 2000, 2001, 2002, 2004). The result of this poor instruction is that the opportunities that knowing math provides – access to half the majors in college, higher paying jobs, greater economic opportunity – are lost to students from historically marginalized groups. The disparity in math instruction, along with the challenges faced by students living in poverty, students of color, and students who are learning English, has created a math achievement gap that’s lasted more than 50 years (Clotfelter, Ladd, Vigdor, Wheeler & Duke Univ., Durham, NC. Terry Sanford Inst. of Public Policy, 2006).
The goal of this book is to give you the tools to be the kind of math teacher that closes the achievement gap in mathematics, creating opportunity for all the students you teach. What we’re presenting here is an approach to math instruction for all learners. This work has been informed by the design research and collaboration among dozens of teachers and math coaches trying to actualize the ideal that every student can learn math. This started as the All Learners Project and, after expanding to hundreds of classrooms is now called the All Learners Network.
In this book, we’ll first look at a method for teaching math that allows for learning grade-level math while also supporting remediation and scaffolding for learners who have difficulty. We’ll examine how to use inclusion, so students can benefit from each other’s insights, and we’ll explore ways to use differentiation to provide just right instruction for everyone.
To begin a conversation about teaching math, we must first consider what math concepts are and how our minds learn them. If we are going to move from memorization to understanding, we need to know how the brain learns to make sense of and incorporate new mathematical ideas.
Some underlying ideas behind good math instruction:
Teaching math requires us to understand student thinking. Mathematics instruction has been a bit behind reading instruction in the sense that, when teaching reading, teachers are explicitly taught that they must understand learner strategies to inform instruction. Teachers need to know how learners are making sense of the math they do, rather than providing thinking for them to copy. By knowing how a student thinks about the concepts she is working on, we can help her find her way to new understanding.
Students must “do the work” if they are to make sense of the math. There is a large gap between a student’s personal understanding of a concept, however idiosyncratic, and a student’s “borrowed understanding” from the teacher. We want teachers to understand that math instruction is not about students copying their thinking. It’s about offering tasks and problems to develop understanding and then using questions and practice to help students clarify and extend their own in sights. This means that we must abandon the idea that we “model good math procedures” and then have students practice what we’ve done. In our approach, we take the opposite of the gradual release model of instruction: “I do, we do, you do.” In teaching math for all learners, students explore a concept to make personal meaning and to gain conceptual insight. Sharing these insights with others refines and extends the understanding. Finally, the teacher might make connections between student understanding and the greater world of mathematics.
What students have to say to each other is as important as what the teacher says to them. Student discourse is at the heart of learning. Human beings are social animals. Being part of a conversation that leads to understanding is motivating (even contagious!) for many learners. For some students, th e process of interaction will enable a mathematical insight. For others, personal struggle leads to understanding, but that understanding is refined and extended through conversation with others. In either case, the opportunity for frequent student student discourse is a touchstone of instruction aimed at serving all learners. Mathematical conversations also benefit students who are learning English, students who need more time to chew on ideas or students who for any reason might need support from others to make personal connections. Student conversations about mathematics in small groups is an important scaffold to support any learner who might need help with understanding and is an effective support for all learners those who struggle and those we don’t.
Besides the three cornerstones for instruction above, there is a trajectory that we use to describe the process of learning mathematics. While the considerations above ar e about pedagogy, the sequence below is about the learner.
Mathematical thinking is based on conceptual models.
Learners can approach mathematics either as a collection of procedures or as an approach to solving problems. A more traditional approach teaches students to use the same procedures (or algorithms) to solve problems. This can be summed up as “showing them how,” and most of us were taught this way. An algorithm, by definition, is a procedure that requires no thinking. It’s automatic. We program algorithms into computers so they can do the computation for us (more on that later). Taking an algorithmic approach to instruction means that we want students to recognize when to use the correct algorithm and then to apply it accurately. While there is still a place for developing algorithms and knowing when to apply them, real mathematics involves creativity and insight, which are key components of problem solving, logic, and pattern recognition: the real heart of mathematics.
Problem solving, or heuristics , is what math is really about. What good is knowing the quadratic formula, or the area of a circle, or how to multiply and divide fractions, if you never apply it to anything useful? Math is a human endeavor, as interesting and thought provoking as literature or art. Teaching only algorithms is a bit like teaching a student to play scales on an instrument without allowing him to explore what it means to express himself in music.
These days there is a kind of algorithm/heuristic dichotomy, but as math teachers we always want students to work with understanding. How can we help someone to understand a concept without “showing them how”? We give them “tools to think with.” These tools allow students to manipulate mathematical ideas, whether physically or mentally, to fit new situations or to solve new problems. We call these tools conceptual models. or just models.
Models are the cornerstone of learning math. They allow us to think about abstract ideas, like numbers, using physical or represented images. Sometimes numbers and symbols themselves can serve as abstract models. The most basic example of using models as a tool to think with is the way that young children use their fingers to count all, to count back, or to add on.
A conceptual model is an analogue or prototype that behaves the way the math behaves. When I join 3 blocks together with 2 blocks, the total is 5 blocks. The blocks provide an analogy for the abstract act of reasoning additively. Likewise, the use of a double number line to explore proportions allows students to examine the concept of co variation in a way that can be changed to fit a wide variety of situations.
Models come in three flavors: concrete, representational, abstract. Concrete models are those that can be manipulated physically. Representational models are those are drawn or recorded on paper (like number lines and T charts). Abstract models are those that make use of the manipulation of quantities symbolically. When students, for example, decompose numbers for computation (we’ll learn much more about this later), they are manipulating abstract models: the symbols for numbers, in this case, carry the meaning. The numeral “3”, for example stands in for the quantity of three. This makes the numeral a model for the number. Taking the number apart or redistributing it for computation without physical objects is an example of using abstract models.
Some educators believe that a concept can be more fully developed, particularly for learners who have difficulty with mathematical concepts, by following the concrete representational abstract models sequentially. That is, they suggest that there is a benefit to having students explore mathematical concepts in increasingly abstract ways. There is some empirical evidence to support this point of view. In our work on having students demonstrate understanding with all three models (see Tapper, 2012), we have found that students often prefer to use representational models when they are given the choice and are doing their own thinking. In any case, model use is essential for making personal meaning of mathematical concepts.
Students use models to solve problems and, in doing so, develop mathematical strategies for solving new problems.
A model is an effective tool, but a tool is meant to be used for something. In the case of models, we use them to solve mathematical problems. One of the biggest shortcomings in math education over the last 20 years is that teachers have sometimes taught the use of models without bringing them explicitly into problem solving. Learners can use a grouping model to solve problems about groups, usually related to multiplicative thinking. They can use an area model to figure out how to add or subtract rational numbers.
Posing tasks or problems is the heart of math instruction. Once a learner has a model to work with (with luck, several models in the toolbox), they need to apply this learning to a wide variety of situations and problems. When, for example, a learner can use unifix cubes to model a number sentence (5+3=8), they can then use the same model to find all the addends that will make a sum of 8. In this instance, the same model is being used to extend some understanding. The model is the tool, and the teacher’s role is to facilitate stude nt explorations of math concepts using that tool.
When students use a tool repeatedly in similar ways, they develop strategies. Decomposing into tens and ones (a common approach when using place value blocks) can become a strategy with consistent use. “Making 10’s”, “ doubles, doubles plus one ” are common strategies for adding that come from using models in specific contexts. For a concept to develop, then, a model has to be used to solve problems.
Real understanding begins with a moment of conceptual insight.
Learning math with understanding is not a linear process. The process is less like traveling along a straight path and more like wandering in a specific, if not predictable, direction. Problem solving creates productive struggle with conceptual models, which leads to realization. In the moment of realization, a learner is in the “flow” or having “aha” moment (Csikszentmihalyi, 2016). It usually comes from being stuck in a problem and engaging in productive struggle to figure it out. As teachers we are always looking for this moment of insight in our students. When students have a sudden insight, practice and conversation about the concept will cause their understanding to grow. When students attempt to practice before they have that moment of understanding, they can become progressively more tangled and invent less and less logical approaches.
One of the interesting things about flow is that it is connected to a particular state of mind, measurable with brain scans. Insight is far more likely when a person is in a relaxed state than when they are anxious. Students who struggle with understanding math frequently report heightened states of anxiety. Anxiety is likely one of the prime reasons for difficulty understanding, both from its effect on working memory and because it inhibits insight. Classrooms where productive struggle is routine, and where students expect to find problems challenging, produce far less anxiety. The anxiety in classrooms is sometimes a result of adult beliefs that answers must come quickly and easily. In mathematics, though, this is rarely the case.
Opportunities to reflect on and communicate insights deepen understanding. When a student has had some insight into a new mathematical concept, there is benefit to refining and connecting that learning to prior knowledge and to a greater social context. This means that the math a student does has some usefulness, or meaning, in the real world. Connecting math to context is done through classroom discourse during the Main Lesson (see Chapter 2) and through deliberate practice during Menu (also in Chapter 2).
As students discuss and practice new understandings, they make connections to what they already know. “Percents are like a ratio”, “2/4 is the same as ½ because it covers the same amount of space on the same whole”, “Multiplying and dividing are both about making groups with the same number of things” are examples of the kind of connections that students can make through reflection or through conversation with other students. As students make these connections, they broaden their understanding of the newly acquired concept.
Students can also learn to make connections to a mathematical convention or context. In his work on the Zone of Proximal Development, Vygotsky suggested that as we have new experiences, it’s useful to have a teacher or another student help us to name our experience in a way that everyone can understand (Moll, 2014). For example, when I find that I’ve divided 27 things up among 5 groups and there are 2 left over, it’s helpful to have someone tell me that those leftovers are called remainders. Learning this special language allows me to communicate my understanding with the greater body of math knowledge.
Many schools and educators embrace the maxim, “All Children Can Learn.” In practice, not many behave as if this were true. Our approach in this book is to embrace the knowledge that when math instruction is skillful and positive intentions for student success guide our decisions (practice?), “All Children Can Learn” can become a reality.
The Big Ideas of math instruction in this book.
To teach math to all learners, we must allow them to develop their own thinking, rather than copying ours.
We help students by introducing new concepts with models tools to think with. We then have students solve problems using the models.
Student understanding emerges when they solve problems and have mathematical insight. Insight means that they can “see” the concept from a more sophisticated perspective.
After developing a mathematical insight, students benefit from reflecting on it, talking with peers about it, and practicing its application is a wide variety of settings.
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Moll, L. C. (2014). L. S. Vygotsky and education.
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