Improving Mathematics Learning: Where Are We and Where Do We Need to Head?

Deborah Lowenberg Ball
University of Michigan, School of Education
2008

Prepared for the Carnegie-IAS Commission on Mathematics and Science Education

Are we at a pivotal moment for mathematics education in the United States? It is true that many agree that substantial mathematical proficiency will be needed to participate fully in society and the economy of this century. Mathematical competence is no longer needed only by some; knowing and being able to use mathematics is increasingly seen as an essential form of literacy. Additionally, some occupations will continue to require even higher levels of mathematical skill.

But we also have a problem. We have been doing a lot of talking, for quite a few years. And we have a tendency to gravitate repeatedly to the same strategies. For all the rhetoric about the centrality of mathematics to the life and progress of the twenty-first century, the United States did not manage to achieve high levels of mathematical proficiency among most adults by the end of the last century. Despite continuing decades of critique and attempts at reform, U.S. mathematics has changed relatively little over the last 50 years. Moreover, we manage generally to educate well only a small fraction of our citizenry in mathematics.

We have an increasing pile of reports calling for major efforts to improve mathematics education and proposing the need to make such improvement a national priority². The soon-to-be released National Math Panel report will be one more such document. To take the next steps and make real improvements in mathematics education in the U.S. requires sizing up in plain terms what the problem is and what we know about solving it, and organizing strategically beyond myths and hopes. This will require specific attention to the core––to instruction and to professional education.

I begin with a brief examination of “the problem” and consider what is known about it, and about what it would take to solve it.

What Is The Problem?

When we say that U.S. mathematics education is failing, what exactly do we mean?

Consider first how you would answer the following questions. Is a pizza with a 14” diameter for $10.99 a better or worse buy than a 16” pie for $12.99? Is 1 an even number? What about 0? Why is 1 not a prime number while 2 is? What is a “rectangle”? Is either of the figures below a rectangle ––and how could you decide:

What is an “irrational number” and could you explain this to your great-aunt or your neighbor’s 17-year old? What is the probability that in a group of 25 people, two will share a birthday? As you respond to these questions, consider, too, the nature of your own mathematical understanding, and how it developed as it did.

Not many well-educated adults in the U.S. can answer the above questions comfortably, nor make judgments about orders of magnitude, estimate the likelihood of particular events, or reason effectively about quantitative relationships. Many cannot remember what they learned in their mathematics classes, and when they do, all too often they cannot make appropriate use of what they learned.

Mathematical competence is no longer needed only by some; knowing and being able to use mathematics is increasingly seen as an essential form of literacy and, additionally, some occupations will continue to require even higher levels of mathematical skill.

When adults are asked how they use mathematics in everyday life, the most frequent answers are either that they use mathematics to balance their checkbooks, or that they do not use mathematics at all. Distaste for mathematics is both widespread and socially acceptable. Well-educated people feel little compunction in announcing that they were never good at math. Imagine if those same people said similar things about reading or writing, such as, “I never could read!” In an age where everyone agrees that the workplace and life demands for mathematics will be greater than ever before, our educational system does not reliably prepare people to be mathematically proficient: skillful, able to reason about and solve problems using mathematical tools and ideas, confident in their own abilities and interested in mathematical questions for their aesthetic appeal.

That is not the worst of it. Significant inequities exist in educational resources and opportunities with resulting disparities in mathematical proficiency. Mathematical failure is disproportionately associated with race and family income. The proportion of African American and Latino eighth graders scoring above the basic level on the National Assessment of Educational Progress is less than two-thirds that of their white counterparts. The mathematical underachievement of minority students and students living in poverty remains a pervasive problem in the United States. The changing demographics of the population suggest that this problem could worsen in the future without a serious effort to understand and address the sources of educational inequalities and what part schools can do in redressing them. The role played by mathematics as a critical tool for civic engagement and opportunity makes these inequities a matter of urgency.

The problem, then, is that so many Americans overall leave their formal experience of learning mathematics uninterested in and unskillful with the subject. How can we best mobilize the resources to do something about it in the coming years?

How can we best mobilize the resources to do something about it in the coming years?

What Do We Know?

When confronted with the disappointing outcomes of U.S. mathematics education, many compare this country’s education to that of other industrialized countries and ask about their practices and policies that might inform our efforts. Most often noticed are other systems’ curricula. Less often noticed is the support afforded by a common national curriculum and how it supports more effective and practice-based professional training. When others look to the business sector for ideas, they also often miss things. Most commonly cited is the effectiveness of market competition and salary incentives. Less frequently observed is how business attends to the continuous learning of employees, and the connection of training to real problems of practice through cases and action-based learning. The lesson here is that the tendency to see curriculum, standards, and accountability as the main levers of school improvement makes it difficult to see the emphasis that other sectors place on professional learning.

Of course, critics do blame the poor quality of teachers. Others blame students, as less able or less well-prepared, or criticize homes and communities. Many place the blame on curriculum materials. But critics do not appreciate the centrality of instruction––that is, what teachers and students do together with mathematics, in the real environments of schools––and how the professional support we provide for teachers to develop skilled instructional practice matters.

Fortunately, some significant resources do exist for the problem’s solution: Much effort has already been invested in developing approaches, materials, and knowledge that can support the improvement of mathematics teaching and learning. Oriented in promising directions, more could be learned as well.

Humility demands that we bear in mind that these problems that we are facing are not new. The past forty years have seen several waves of mathematics reform, each entailing serious efforts to improve mathematics learning. Each has attempted to upgrade what counts as “mathematics” in school, to alter students’ mathematical experience, and to improve their grasp of fundamental ideas and skills. And yet change has been difficult, and much has remained the same as it was in 1950, or even 1900. Students still practice pages of sums and products and are still asked to solve improbable story problems. Students are still told to “invert and multiply” to divide fractions and to use “My dear Aunt Sally” to remember to multiply and divide before adding and subtracting in a expression. Teachers still explain how to do procedures, offer rules of thumb, give tests on definitions and procedures, and provide applications. These practices in and of themselves are not necessarily unhelpful. However, the prevalence of instruction that consists only of such practices helps to explain why the number of students who leave school as proficient with mathematics as they are literate with English remains small.

Despite pervasive concerns about mathematics education over the past fifty years, what do we know now from research on school reform, teacher quality, and teaching that––if we used this knowledge to design solutions––could make a difference in these patterns? I identify five that top the list.

1. Curriculum materials alone cannot determine instruction. Teaching is what matters. The development, or adoption, of particular curriculum materials has been the main strategy in attempts to improve mathematics learning. It is not difficult to understand why. It seems, from the outside, to be the central ingredient of classroom lessons; thus, improving curriculum seems like a direct input to improving learning. This is of course, in part, true. Curriculum materials are the major professional tool of instruction. What this strategy overlooks, however, is that teachers exercise substantial discretion in their use of curriculum, making decisions about what to emphasize, augment, and omit. They make decisions about the order of topic presentation, and adapt the book’s treatment of a topic in order to meet their students’ particular needs. Moreover, teachers’ own knowledge of mathematics influences their interpretation of the textbook authors’ intentions and thus shapes their use of the material. Despite common assumptions, there is no such thing as completely “following the text.” Students also interpret textbooks, and their interpretations shape their teachers’ use of the curriculum. They may struggle with particular lessons and require re-teaching, or additional practice. They may already know the material and require extension. They may not understand the book’s examples, and need alternative models. Effective instruction demands such adaptation between teachers and their students. Hence, even in districts where “curriculum pacing” is the instructional policy, how teachers and students interpret and use the textbook lessons results in variability from class to class.

Although “teacher-proof” curriculum is anathema to some and the goal of others, this debate misses the point. Helpfully-detailed materials can be useful for skillful instruction, much as guides for practice are useful in other professions. The question is not whether or not to “script” teaching; this argument distracts attention from the important one: What aspects of instruction can be detailed enough to provide reliable guidance for practice with their students? What sorts of detail are useful to teachers and usable in their work? How much of good practice must be precisely tailored to particulars and what is actually quite regular and predictable? There are mathematical topics––subtraction with regrouping, the meaning of the equals sign in equations, ordering of fractions, for example––about which we know a great deal about students’ typical difficulties. It is possible to design sequences of tasks and to provide specific guidance to teachers about the questions to pose and what to look for in students’ responses and work, and how to follow with subsequent tasks and questions. Japanese teachers produce lesson plans with these features through their work in lesson study, based on close observation of students’ learning of specific topics and careful polishing of particular tasks and detailed plans for their use. Because these instructional materials combine detailed unpacking of the mathematics coordinated with students’ understanding and performance, such guidance can help teachers attend to both students and the progression of the content³.

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2. In order to teach mathematics well, teachers must know and be able to use mathematical knowledge flexibly to help students learn. Teachers’ mathematical knowledge is often a focus of concern. Many argue that we should improve recruitment by attracting mathematically-trained people to the profession by some combination of reduced professional requirements and increased salaries. Others argue for increasing the mathematical preparation of teachers by requiring more math courses. Neither solution directly addresses the need for teachers who have and can use the mathematical knowledge needed for effective teaching. Advanced study in college mathematics does not provide opportunities for developing this sort of mathematical knowledge for it is not currently designed for that purpose. Knowing mathematics for teaching is different from knowing it for yourself. For example, being able to to divide fractions is not sufficient to figure out which of these diagrams represents the problem 2 ÷ 2/3:

⁴Likewise, being able to multiply does not provide the mathematical perspective needed to see what this student has done:

⁵To develop mathematical proficiency in students, teachers need flexible knowledge of mathematics that permits them to “unpack” ideas and procedures to make their reasons available to students. Teachers need to use mathematics in ways that others do not. They need to be able to see the mathematics from someone else’s perspective, not simply their own. This means, for example, being able to select an appropriate model for a particular mathematical idea a model that illuminates the meaning of the idea, and that does not distort or obscure its essence. Teaching involves modifying problems to make them simpler or more difficult. Teaching requires asking the right mathematical question at precisely the crucial moment. And it involves being able to develop and deliver explanations that are comprehensible to learners.

Teaching is not a generic skill. Each of the tasks mentioned above is fundamentally mathematical work. Simply knowing how to ask a good question in general does not equip a teacher to ask a good question about a particular algebraic expression, or about a specific solution.

Knowledge of mathematics needed in teaching is not easily developed in conventional mathematics courses that are designed for people intending to enter other kinds of careers – engineers, professionals in the mathematical sciences, physicists – who must use mathematics in ways significantly different from the uses to which teachers must deploy mathematical insight and knowledge.

3. In order to teach mathematics well, teachers must be able to understand and work from where their students are. Teachers cannot learn for students. No matter the format of instruction – whether lecture, small group activity, or individualized assignment – students make their own sense of what they are taught. Ideas do not flow directly from teachers’ or peers’ minds, or from well-designed worksheets or activities into learners’ minds. Of course, instruction can be more or less well-designed to make it more likely that students will learn what teachers intend and that they will not be misled. Still, effective teaching requires teachers to be able to investigate skillfully what their students are taking from instruction, and to make adjustments along the way to ensure student learning. This requires teachers to mobilize their knowledge of mathematics as discussed above – to ask a good question to probe what students know, or to listen attentively to a stumbling or unexpected student explanation, or to design an additional assignment to provide appropriate help. Teachers must be able to do this with an ever-increasing variety of students, and to listen and work across wide gulfs of experience, language, and culture, to help each of their students learn.

4. Good teaching is something to learn, not an inheritance. Implied in all of this is that teachers are made, not born. Learning to teach effectively requires that teachers have sustained opportunities to learn mathematics, about students, and about ways to help students learn particular mathematical ideas and procedures, both before they begin teaching and as practicing professionals.

These opportunities to learn are most effective when they are connected to the work of teaching. Just as other professionals learn through clinical work, work in field settings, and professional practice, so, too must teachers. They need opportunities to learn mathematics not like those appropriate for engineers, but in the ways that they must use it in teaching. They should learn about the primary resources of their practice – curriculum materials, for example – and about how to examine, modify, and use them effectively and with integrity. They must learn about common student difficulties with specific mathematical ideas and about ways to prevent and remediate those.

This sort of professional education is central to the educational systems in high-performing countries, where teachers’ opportunities to develop professional knowledge and skill are part of their work. Not only do teachers in Japan and China, for example, have time to learn and to improve their practice as part of the regular work week, but what they work on is practice ––curriculum, mathematical content, students’ learning––together with other professionals. Their learning is ongoing, systemic, and systematically connected to the professional career. Just as the Japanese do not believe that mathematical accomplishment is the result of innate ability, neither do they leave good teaching to inheritance: They believe that teaching is a complex practice that can be learned and continually improved.

5. Most improvement efforts do not focus sufficiently on instruction, and are not designed adequately for what it takes to make them work in real contexts. Improvement efforts are most likely to work if they focus on teaching and learning. This is where content and students come together, and without attention to that dynamic, improvement efforts take big changes. Indirect efforts are chancy, for they may or may not––and often do not––impact what teachers do with students, around particular mathematical content, in classrooms. For example, a new curriculum can be used in a wide variety of ways. But a new curriculum accompanied by ongoing opportunities for teachers to learn, to work with other teachers on the use of that curriculum, and to make wise accommodations of that material in their environments, with their particular students, would be much more likely to make a difference in the quality of instruction, and the effectiveness of learning.

The improvement efforts that fail do so because they are too vague, too far from the core of instruction to make a difference, and because they overlook the learning needed to enact the improvement. Incentives alone cannot produce professional learning. Neither can sanctions. If effective instruction is something to learn, then reform efforts must be designed for learning.

What Would It Take to Solve Our Problems in Mathematics Education?

Across the various things we know, one big point stands out: Instruction must be the central focus of efforts to improve students’ learning. Hence, investments in knowledge about effective practice, coupled with substantial opportunities to develop the resources for such practice, must be our main investment. What we know about our past efforts at improvement shows that if we are serious about developing a mathematically-proficient American public for the twenty-first century, we will require a new orientation to our efforts and investments, an orientation that centers on teaching and learning, on teachers’ learning, and on the continued development of knowledge about effective teaching.

Across the various things we know, one big point stands out: Instruction must be the central focus of efforts to improve students’ learning.

We must build on what we know and use the resources that we have. We will need to change our assumptions, and the policies that flow from those. We will need to make the development of high-quality mathematics teaching and learning the central thrust of our efforts. This means:

1. Providing sustained, systemic opportunities for teachers to learn and develop the effectiveness of their practice in ways that are connected to the complex tasks of their work. Professional work and education must afford opportunities for teachers to develop useful and usable knowledge of mathematics, of students’ learning of mathematics, of curriculum and its effective uses, and of teaching itself.

Imagine a group of teachers working on developing an assessment to ascertain whether or not their students had learned to order decimals. They consider alternative lists of decimal numbers and discuss which would be most strategic:

Professional work and education must afford opportunities for teachers to develop useful and usable knowledge of mathematics, of students’ learning of mathematics, of curriculum and its effective uses, and of teaching itself.

One teacher claims that it would make no difference. All require students to put decimals in order. But a second teacher points out that a child could get the numbers in the first two strings in the correct order without even attending to the decimal point, whereas the third string requires students to read the numbers as decimals––.565 is the least, for example, although children might see it as the largest number. The teachers discuss this for a few more minutes, and eventually agree that the third list is the most strategic assessment question. For several of the teachers, this work has illuminated an important part of what there is to understand about learning decimals that was not transparent when they considered ordering a set of decimals themselves. Learning to unpack mathematics from the perspective of what there is to understand is different from knowing the content oneself.

Another group of teachers examines student work on subtraction. One paper attracts their attention. They see that the student has gotten several questions right, but is getting many wrong as well:

They puzzle over his work, trying to unlock the mysterious pattern of errors on his paper. Several teachers declare that he “has the basic idea” but needs more practice. One insists that he does not understand how to “borrow.” As they scrutinize his paper, however, they are able to decipher the pattern: When there is a 0 in the tens place, he regroups from the hundreds, skips over the tens, and adds ten to the ones place. He subtracts, and then when subtracting from 0, reverses the order (5 – 0, 8 – 0):

The teachers check among the papers, and discover that several students are making the same pattern of errors. They begin a discussion of what might have led to this, and how it might be corrected. They examine the textbook to see if there are clues in the way the material is presented that might have reinforced this pattern.

In examples such as these, teachers work together on artifacts or problems taken directly from practice. Their work affords them opportunities to work on the mathematical ideas from the perspective of what it takes to learn them, to see the ideas in a finer grain, and to consider alternatives for helping students develop proficiency with the content.

Although time to meet is important, time is not enough. Teachers need robust examples with which to work, either from their own classrooms, or collected systematically from others’ classrooms. They can work profitably with well-selected video segments, with student work such as the examples above, with curriculum materials, and with cases. As “curriculum” for teachers’ learning, such records and artifacts need careful collection and development. This is an area in great need of support if teachers’ professional learning – as that of other professionals—is to be grounded in problems and cases of professional practice.

2. Building a knowledge base for excellent practice. Research is needed to build shared and usable knowledge about effective instruction. Such research must be focused on problems of practice and its inquiries and products made widely available in usable form. Such studies might examine successful teachers and schools to understand and articulate the elements that contribute to their effectiveness. Other studies might probe the most frequent challenges faced by teachers as they work to develop mathematical proficiency in particular environments. Studies might also design interventions aimed at improvement and then study closely whether and how they work, including successes achieved and obstacles met. Studies should track students over time, through different instructional approaches, in different environments, so that we understand the long-term trajectories of students’ mathematical development. And we need, perhaps most of all, reliable and careful studies of what helps teachers learn – the substance, materials, approaches, and organizational structures that enable teachers to develop the sort of professional knowledge and skill fundamental to effective mathematics instruction and its continuous improvement.

3. Mobilize interdisciplinary expertise to work on the improvement of mathematics instruction. Teaching and learning, and improving instruction in the complex environments of American schools, present policymakers and practitioners with compound challenges. These challenges require multiple kinds of expertise of many different kinds, including that of experienced expert practitioners, scholars of teaching and learning, and of school improvement, research mathematicians, and psychologists and sociologists, along with many others. At the core, the problems of mathematics education are problems of practice, and to be useful, these different sorts of expertise must be mobilized and integrated to focus on practice.

4. Attend to and coordinate conflicting signals and initiatives in the environments in which instruction takes place. If teachers and others in schools are to focus intensively on the improvement of mathematics teaching and learning, these efforts depend on better coordination of the multiple messages and policies. For example, high stakes testing often works against the need for teachers to attend closely to what students actually understand and whether they can use what they are learning. Such high stakes testing creates incentives for rapid coverage, and favors teaching that will help students remember for the test. Such testing also takes time away from instruction as teachers spend increasing time equipping students with the skills needed to succeed on tests. Assessment of learning is unquestionably fundamental to good instruction, but tests and policies for their use must be designed to provide information in ways connected intimately to students’ progress.

Another area in need of coordination can be seen in the pervasive commitment to local educational decision making which results in students who move frequently––most often students from the poorest environments––needlessly re-starting their mathematics learning over and over as they shift from one curriculum and assessment environment to another. And there are many other needs for coordination in the very fragmented environments in which mathematics teaching and learning, and schools, exist.

The challenge is clearly before us. Simply recruiting different people to teaching will not improve students’ learning. Good teaching requires specialized skill and knowledge and is more than the exercise of one’s own knowledge of mathematics plus some common sense. The biggest payoff will come from improvement of professional practice about which we have learned a great deal in the past decades. It is time to take the investments and learning of the past century to make this the century when mathematical proficiency in the United States is as common as competent reading and writing.