Making Sense of Math
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Making Sense of Math

How to Help Every Student Become a Mathematical Thinker and Problem Solver (ASCD Arias)

Cathy L. Seeley

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eBook - ePub

Making Sense of Math

How to Help Every Student Become a Mathematical Thinker and Problem Solver (ASCD Arias)

Cathy L. Seeley

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About This Book

In Making Sense of Math, Cathy L. Seeley, former president of the National Council of Teachers of Mathematics, shares her insight into how to turn your students into flexible mathematical thinkers and problem solvers. This practical volume concentrates on the following areas: * Making sense of math by fostering habits of mind that help students analyze, understand, and adapt to problems when they encounter them.
* Addressing the mathematical building blocks necessary to include in effective math instruction.
* Turning teaching "upside down" by shifting how we teach, focusing on discussion and analysis as much as we focus on correct answers.
* Garnering support for the changes you want to make from colleagues and administrators. Learn how to make math meaningful for your students and prepare them for a lifetime of mathematical fluency and problem solving.

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Publisher
ASCD
Year
2016
ISBN
9781416622451
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Introduction

When I learned how to teach mathematics many years ago, it seemed like a fairly straightforward task—prepare well and explain clearly. Whenever I could, I tried to elaborate why a particular procedure worked or a particular kind of problem might be solved a certain way. If I wanted the students to really stay with me, I learned to focus on asking good questions and challenging students to go beyond their comfort zone, always with enthusiasm and a smile on my face. I worked hard to find interesting puzzles and games that might have some slim connection to what I was teaching. And I made myself available for students before and after school. I like to think that I got better every year, and I also like to think that most of my students thought I was a pretty good teacher overall. Many of them came to not hate math, and perhaps they even learned most of what they needed to know in order to move on. I tried not to focus on the fact that the math I was teaching might not have been very relevant to their lives or might not make sense to some students, even with my "clear" explanations.
I've learned a lot since those early days in my teaching career. Today we know much more about what it takes to equip students to become mathematical thinkers who can take on any problem they encounter. We also know that students—all students—have more ability and even more intelligence than we might have imagined. As we think about how to nurture and help students develop their abilities and intelligence, I'm convinced that their success in the future depends at least as much on how they think as it does on what they know. Likewise, I'm convinced that if we're going to help them succeed, we need to pay at least as much attention to how we teach as to what we teach. We need to challenge some of our old ideas about struggling and frustration and consider structuring our classrooms differently from how classrooms might have been structured when we were students. We may even need to turn those structures upside down. And we need to recognize that professional learning communities—if used appropriately—can offer a powerful vehicle for teachers to learn how to more effectively help students gain the mathematical knowledge, problem-solving skills, and habits of mind they need for living and working in the 21st century.
In this brief look at mathematics teaching, let's think together about what it takes for every teacher to help every student become a mathematical thinker. To look at broader issues related to prioritizing math at the school level and creating and supporting strong math programs, see my companion volume for leaders, Building a Math-Positive Culture (Seeley, 2016).
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Who's Smart in Math?

Students have difficulties in mathematics for many reasons. They may have learning problems. They may not speak English as their primary language. They may be too shy to ask questions or have behavior or attendance problems. They may have missed or misunderstood an important concept in the past. They may not see the relevance of what they're learning. But the biggest barrier for many students is their belief that some people are naturally good at math and some people simply aren't. Students come to believe this myth largely because the adults around them, including their parents and teachers, may also believe it. Unfortunately, this idea has been reinforced by some long-standing practices and underlying beliefs, particularly structuring classrooms around teacher-centered lectures, using timed tests to assess mathematical fluency, and believing that good students shouldn't make mistakes.
Traditionally, we may have thought of a "good" or "smart" math student as one who is quick to respond to a teacher's question, accurate at computing the answer to a computation problem, and able to apply a newly learned procedure to solve a word problem. But the truth is that there are many ways to be smart in math. Some students may be creative problem solvers. Others may be visual thinkers who can see and analyze quantitative or spatial relationships. Still others may be thoughtful, slow processors of information and generators of multifaceted solutions to complex problems. When we expand our ideas about what it means to be smart in math, and when we help students develop their mathematical talents in a variety of ways, we're likely to see many more smart students. Of greater importance, more students will see themselves as smart. And when they see themselves that way, they're far more likely to be willing to tackle the next mathematical idea or challenging problem they encounter.
We now know that nearly any student can learn mathematics and succeed if we shift our practice a bit. We'll take a look at some of those shifts shortly. For now, let's consider what it means to be smart—smart in general, and specifically, smart in mathematics.

Mindsets About Intelligence

Thanks to advances in cognitive psychology and technological breakthroughs in studying neural connections in the brain, we now know more about the nature of intelligence than ever before. In the groundbreaking book Mindset, Carol Dweck (2006) brought these notions to a broad audience as she discussed the differences between a fixed mindset about intelligence and a growth mindset. In the past, many people—including some experts—believed that intelligence was all about the genes a person is born with. This fixed notion of intelligence would mean that a person is as smart now as he or she ever was and is ever going to be—that even if someone learns new things, his or her basic intelligence would remain the same.
However, researchers are accumulating a growing body of evidence in support of a growth mindset­—the notion that intelligence is far more malleable than some may have thought. We now know that genes are only a starting point in determining a person's intelligence. Researchers studying intelligence and analyzing brain scans of individuals working on different types of problems have found that as people work through an activity that is difficult, they can actually grow new neural connections—in essence, their intelligence increases, and they become smarter. The experiences in a person's life and, more significantly, how that person processes those experiences, can influence that person's intelligence. If someone understands that certain kinds of experiences can increase intelligence, it can influence not only how the person approaches school (especially hard subjects), but even how that person relates to others and functions in everyday life.

Impact of a Student's Mindset

A student's mindset about his or her intelligence plays a huge role in the student's willingness to tackle and persevere through a challenging problem. Imagine how a student with a fixed mindset—who believes he's only as smart as he's ever going to be—reacts when encountering a hard problem that he doesn't know how to solve. He's very likely to believe that the problem is beyond his intelligence and capability. His response may be something like, "You never taught me that. I don't know how to do this. I can't do it."
Ironically, a fixed mindset about intelligence can affect students who see themselves as smart just as negatively as it affects students who see themselves as less smart. The student with a fixed mindset who believe she's smart views any kind of failure as evidence that she's reached her limit of "smartness"—that her innate intelligence has hit its peak. Consequently, she tries at all costs to look smart by providing the right answer and strives to avoid situations in which she might fail, such as tackling a challenging problem.
If, however, a student—either one who sees herself as "smart" or one who doesn't—has come to understand that a person can get smarter by working hard through challenging problems, that student might instead respond by thinking, "Wow. This is a hard problem. I may have to work pretty hard to solve this. It may take me a while." With a growth mindset, both successful and less successful students can become motivated to tackle something they think may be beyond their previous level of success.
Students aren't likely to develop a growth mindset on their own. But we can help them get there. We can teach students about the nature of intelligence, including the findings of recent brain research about a growth mindset. We can help them see the subtle and blatant stereotype threats prevalent in society—for example, "girls can't do math"; "minority students aren't as smart as white students"; "poor children can't learn as well as affluent children"—that may undermine their confidence and the development of their intelligence. The choices a teacher makes about what tasks to assign, how to orchestrate classroom discussion, and how to evaluate and reward learning can have a tremendous influence on a student's mindset about intelligence and on how students view their abilities and potential in mathematics.

Impact of a Teacher's Mindset

A teacher's mindset about intelligence may be at least as important for a student's success as the student's mindset. As much as we may talk about the importance of high expectations for all students, it is all too easy to fall prey to our own unconsciously low expectations for some students. During my Peace Corps experience in Burkina Faso a few years ago, my unconsciously low expectations for students almost kept me from allowing a class to tackle a challenging topic. I was on the verge of deciding to skip the unit on quadratic functions with a class of students who were in the non-math track in my school. A Burkinabè colleague convinced me—actually insisted—that I needed to include it. When I reluctantly did so, I realized how important the experience of working through this challenging content was for my students. When we finished that unit, the class celebrated. They knew they had done something hard. I believed that it would be too hard for them, that they wouldn't get it, that they wouldn't like it, and that they didn't need it. I now know that, in the process of working through something difficult, they may have even grown some new neural connections and become smarter. Even if none of them ever use what they learned outside of that classroom, the experience of working through hard mathematics was rewarding for them, and it was important for their learning, intelligence, and attitudes about mathematics and themselves as math students. It's humbling to this day to realize that I almost withheld that opportunity from them not because of their limitations but because of my own unconscious habits and beliefs. A teacher's underlying beliefs and mindset about intelligence can prevent us from allowing a student or group of students to fully develop their potential. Changing our mindset enables students to engage in and struggle through problems that help them grow smarter.
Mathematics offers a unique proving ground for students to work through challenging problems. As teachers and curriculum developers, we have tremendous power to offer students the kinds of tasks that can help them develop their intelligence.

Implications for Teaching

If we're going to help students see themselves as smart, we need to reconsider the efficacy of some of our long-standing practices. In particular, we should reexamine how often, if ever, teachers should present mathematical procedures via lecture, expecting students to listen and remember what we have presented. Too many students tune out such lectures, seeing them as abstract and unrelated to their lives. Moreover, such a teaching approach reinforces the idea that math is a set of rules and that, in order to succeed, you have to remember which rule to use in which situation. Watching a teacher deliver a step-by-step procedure can make a student believe that the teacher is one of those people for whom mathematics comes easily. If the student doesn't see himself as one of "those" people, the student may disconnect from what the teacher is saying, hoping desperately that he will be able to remember to use that rule at the right time. The double whammy here is that, by teaching in a teacher-centered, listen-and-learn way, we confirm students' wrong ideas about both mathematics and themselves as mathematics students. Watching teachers work through a problem or computation without being engaged with the content themselves can make students believe that math is like magic—and not anything that can make sense. It can reinforce their belief that "I'm probably just not a math person." If a student has adopted a fixed mindset about intelligence, these beliefs can become even more firmly entrenched than otherwise.
The alternative to a teacher-centered classroom is a teacher-structured classroom focused on student engagement and discussion around rich problems and mathematical ideas. We'll take a look at how such a classroom might operate in the section on "upside-down teaching." For now, consider the possibility that there may be ways to structure classrooms that draw out the best of students' thinking and help them become smarter in the process.

Seeing Below the Surface

There are many ways students may inadvertently hide what they already know, what talents they may have, or how smart they might be or become. If students can't speak English well, it's difficult to know what they know or how they think. If students don't pay attention well or if they act out in class, or even if their work looks sloppy or messy, it's easy to think they aren't smart or motivated to learn. And the various acronyms we may assign students—SPED, ED, ADHD, ELL, and so on—all bring with them the weight of low expectations for students.
With respect to language differences, we need to look for ways to help students express their ideas, even if English is not their primary language. The worst thing we can do for limited English-speaking students is to limit their mathematical experience to strictly numerical work, thinking they will succeed better without words. Instead, these students need to be given many opportunities to share their mathematical thinking orally and in writing. For example, when a class activity involves small-group work, we can ensure that there are proficient English readers in every group so that all students can understand the task at hand. We can establish group norms for respect and participation that remind students that everyone in their group is responsible for making sure all other group members understand and can present their work. And we can be sure to call on all students to share the group's work with the class.
Likewise, the worst thing we can do for any student is to believe that because of some label or behavior, a student needs—or can only handle—less mathematics than other students. Too often, telling ourselves we're doing what's best for certain students, we lower the level of the mathematics we give them. We may focus primarily on computation or on one-step word problems strictly related to a rule they just learned. This narrow view of mathematics reinforces to these students that math is boring and irrelevant. Worse, our compassionate tendency to give students only what we think they can handle may just disguise our own low expectations, even to ourselves. Without intending to do so, we may sentence students to fulfill those low expectations and ensure that they will never be able to tackle a relevant, real problem they may eventually face later, simply because they will never have had any experience dealing with such problems.
Regardless of the ways students' talents or potential may be hidden, it's our responsibility as educators to lo...

Table of contents