Focusing on Essential Knowledge and Skills

A coherent system recruits, prepares, develops, and retains strong teachers and leaders, leading to an effective teacher for every student and principal for every school. MORE

 

Read the papers that informed The Opportunity Equation report recommendations.

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Common standards, linked with rigorous assessments, set the bar for all students—from struggling to advanced—to master academically rigorous content and succeed in the global economy. MORE

 

Michele Cahill responds to probing questions about why stronger math and science education is crucial for all American students. MORE

 

Connecting to Your Work

How can you mobilize to help focus teaching and learning on essential math and science knowledge and skills? Read recommended actions from The Opportunity Equation report. MORE
 

Math and Science Standards That Are Fewer, Clearer, Higher to Raise Achievement at All Levels

David Coleman, Student Achievement Partners
Jason Zimba, Bennington College
2008

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

Introduction

In early 2000, we started an organization called The Grow Network. Our mission was to provide large school systems with better tools for using assessment to inform instruction. To design these tools, we worked closely with district superintendents and state school officers across the country. But we spent even more time working with hundreds of teachers, principals and professional developers, visiting schools in New York, Chicago, Los Angeles and other cities. Our initial goal was humble: just get the existing score data out into the field more effectively. Achieving this proved far more difficult than we imagined. But when our first printed reports went out into the field, more than a few teachers told us that it was the first time they’d ever seen all of their students’ scores on one page.

We also wanted to help teachers use the test scores to inform their standards-based instruction. Now we encountered some truly thorny problems, and we made some sobering discoveries as well. Instructional leaders told us that their teachers didn’t always understand the standards well enough to teach them. Principals in our focus groups often proved incapable of drawing appropriate conclusions from hypothetical score numbers we showed them. And when we began to develop our own content keyed to the standards, we were floored by the magnitude of the task schools were facing. Teachers wanted all the information they could get, but the value was limited, because behind every number was a forest of detailed content areas. With so many standards in play, standardized tests would have to be prohibitively long in order to assess them well.

Now we believe that a consensus is emerging around the need to revise math standards, as evidenced by recent and current efforts...

What we saw convinced us that the math standards at the heart of the system were far too vast to effectively guide instruction and assessment across large school systems. Now we believe that a consensus is emerging around the need to revise math standards, as evidenced by recent and current efforts by NCTM, Florida, Pennsylvania and Washington to clarify and distill their standards. Science standards have a shorter history, but the National Academy’s recent work suggests that the time to get those right is also now. In what follows, we discuss these trends and attempt to sketch a vision for how to proceed. We offer these suggestions not as representatives of the organizations to which we belong, but rather in our private capacities as concerned citizens and observers of American education.

We invite the Commission to consider taking the following actions:

  1. Issue a call for more pragmatic analyses of what readiness for work actually requires, and what this implies for the teaching of math and science.
  2. Outline a program for dramatically raising the number and diversity of students performing at the highest levels.
  3. Evaluate deliberate pracice as a means of achieving a significant increase in the number and diversity of high performers and also as a guide for the reform of secondary math and science instruction altogether

to ensure standards that are fewer, clearer, and higher.

Although these actions are interrelated, they are not intended to form a comprehensive proposal for reform. Nor is any one of these actions being put forth as a magic bullet. The hope, rather, is that elements of any or all of these ideas may prove valuable as part of the Commission’s overall thinking.

In the following we offer some rationale and commentary on each of the four actions. In some cases, we attempt to stake out a position in a way that invites counterarguments; in other cases, we do not argue a position, but only attempt to raise issues for debate and consideration.

Action 1. Call for states to distill content standards in math and science so that they are fewer, clearer and higher.

The rationale for Action 1 begins with some fairly detailed remarks about math standards. For reasons of space, the remarks about science will not be as extensive. However, to the greatest extent possible, we intend for all of the suggestions described here to be understood as recommendations about both the math standards and the science standards.

A consensus seems to be emerging around the need to distill math standards. The National Council of Teachers of Mathematics has recently published Curriculum Focal Points for Pre-Kindergarten through Grade 8 Mathematics: A Quest for Coherence (NCTM, 2006), which aims to distill math standards in grades K-8. At first glance, NCTM’s Focal Points appear to make math standards fewer and clearer. However, the Focal Points actually leave the existing NCTM Standards intact, and are meant to extend the existing standards documentation. Thus, while the Focal Points do represent “an important, initial step in advancing collaborative discussions about what mathematics students should know and be able to do” (NCTM, 2006, p. 11), the Focal Points do not admit that the best way to solve the problem of “long lists” (NCTM, 2006, p. 1) is to shorten them.

The selections made in the Focal Points are principled, thoughtful and helpful. But while the Focal Points may show the way, they will not automatically lead the way to shorter math standards for the states. A case in point is Florida and its “Big Ideas” initiative. Florida has aligned this distillation effort to the Focal Points (FL, 2007, p. 13). Yet Florida’s Big Ideas have not made the Florida math standards any fewer, any clearer or any higher. Indeed, in addition to the Big Ideas, the Florida standards still contain plenty of “Supporting Ideas.” These, we are told, are “not less important than the Big Ideas” (FL, 2007, p. 13). But if everything is still required, does introducing a hierarchy clarify the teacher’s task, or obscure it?

The standards must be made significantly fewer in number, significantly clearer in their meaning and relevance for college and work, and significantly higher in terms of the expectations for mastery of what is covered.

Revising the standards in ways like these will only make existing problems worse. The standards must be made significantly fewer in number, significantly clearer in their meaning and relevance for college and work, and significantly higher in terms of the expectations for mastery of what is covered.

As educators focus overwhelmingly on bringing students to proficiency, there is little incentive to increase the number of our students doing advanced math and science, especially in K-8. Of equal concern is the way the current system prevents most students from mastering anything at all. Even if they reach the proficient level in every standard, they will be a long way from mastery of any of them—with no realistic way to get there, because of the insistence on covering everything.

What’s more, our current approach is not positioning us to respond to international competitive pressure. As Schmidt et al. report (Schmidt et al., p. 2):

This preoccupation with breadth rather than depth, with quantity rather than quality, probably affects how well U.S. students perform in relation to their counterparts in other countries. It thus determines who are our international ‘peers’ and raises the question of whether these are the peers that we want to have. In today’s technologically oriented global society, where knowledge of mathematics and science is important for workers, citizens, and individuals alike, an important question is: What can be done to bring about a more coherent vision and thereby improve mathematics and science education?

A fewer/clearer/higher approach should help our nation’s students compete better once they enter the workforce. Instead of a weak recall of a vaster terrain, perhaps it is more effective to have true mastery of the essential parts of math and scientific thinking, so that our citizens will readily apply them to jobs we cannot even yet envision today.

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A Closer Look at Focal Points and Big Ideas: When Less is Still More

We believe an innovation is required in the development process for math and science standards. The process should attend to relevance in more pragmatic ways, and there should be procedural safeguards against the “pork barrel” effect that occurs when multiple stakeholders all advocate for pet topics.

Without a new way to think about the process, attempted revisions will fail to make a real break with the current standards. Evidence for this can be seen in the NCTM Focal Points and Florida’s “Big Ideas.”

Procedurally speaking, the Focal Points used a three-part “filter” to identify focal points in mathematics (NCTM, 2006, p. 5):

  1. Is it mathematically important, both for further study in mathematics and for use in applications in and outside of school?
  2. Does it “fit” with what is known about learning mathematics?
  3. Does it connect logically with the mathematics in earlier and later grade levels?

Identifying standards that pass through this filter is a good start, but taking the next step requires that we examine, and generally discard, those standards which fail to pass the test. Along these lines, one could envision adding even more filters, such as:

4. Can it wait until later grades? And should it?

For example, can much of formal geometry be put off until middle school? And should it? Figure 1 shows the Focal Points for Grade 1 math (NCTM, 2006, p. 13).

fn35_Coleman-ZimbaGraphic-1

Figure 1. NCTM Focal Points for Grade 1 math (from NCTM, 2006, p. 13).

The Florida math standards also list three “Big Ideas” in Grade 1. By design (FL, 2007, p. 13), they resemble the Focal Points a good deal (FL, 2007, pp. 21-23):

BIG IDEA 1: Develop understandings of addition and subtraction strategies for basic addition facts and related subtraction facts.

BIG IDEA 2: Develop an understanding of whole number relationships, including grouping by tens and ones.

BIG IDEA 3: Compose and decompose two-dimensional and three-dimensional geometric shapes.

A child who leaves first grade without mastering the first two ideas is in serious trouble. The stakes are much lower for the third idea. Addressing the geometry strand in a later grade would give first-graders more time to solidify the must-have concepts at the beginning of their trajectory in mathematics. Doing so would also significantly clarify the mission of first-grade teachers. And doing so would allow teachers to invest more in refining their key curriculum materials over time, and developing their craft in teaching the subtle concepts of numeration.

To be sure, engaging students in geometry early gives them a useful and powerful form for expressing mathematical ideas, and multiple forms of knowing are often the path to deeper understanding. But high-quality teaching of early numeracy will always involve multiple representations, including spatial ones, and this would not disappear if Big Idea 3 were to move.

Arguments about geometry aside, an observation made earlier bears repeating: the Big Ideas as a whole have not, in fact, shortened the Florida standards at all. Making fewer, clearer and higher standards a reality in the states will not happen without some new thinking about the standards development process itself.

The next filter is the converse of #4:

5. Can it be started earlier, and is there a benefit in starting earlier?

For example, if a significant amount of the formal geometry moves out of early grades, then can a few things profitably move in—such as the strategy of using boxes or other symbols to represent unknown numbers? Another scenario: If most of the formal polynomials move out of Grades 7 and 8, then can certain useful elements of “calculus” reasoning move in? For example, Florida secondary benchmark MA.912.C.3.3 (FL, 2007, p. 94) is to “Decide where functions are decreasing and increasing….” This broadly applicable idea could be introduced well before high school. And yet, it is not even a “sunburst” benchmark in Florida. (Sunbursts indicate benchmarks that are required for all students.)

Applying filter #4 would likely result in large quantities of material moving later in time, with some of the material moving all the way into the high school years, where it might no longer even be required for all. Meanwhile, a small number of powerful ideas would move earlier under filter #5.

The next filter asks us to take a hard look at claims that any particular piece of content is truly necessary for work and college.

6. Is it truly necessary for college and work and thus should be provided for all, or an element of advanced math for only some students to pursue?

The viewpoint of this filter is that for each separate item in the standards, it should be possible to articulate the reasons, and the evidence, for why that item specifically is important for all. (This theme is pursued further in the next section below.)

Filter #6 would put substantial pressure on standards like Florida Benchmark MA.912.A.4.4, “Divide polynomials by monomials and polynomials with various techniques, including synthetic division.” (See Figure 2.)

fn35_Coleman-ZimbaGraphic-2

Figure 2. Benchmark MA.912.A.4.4 from the Florida math standards (FL, 2007, p. 84). The sunburst symbols at left “denote benchmarks that include content that all students should know and be able to do. These benchmarks are considered to be appropriate for statewide assessment.” (p. 76)

One has to wonder how much time, money and effort will be spent to ensure that every student in Florida can divide polynomials by polynomials using synthetic division. And yet, the standards offer no specific justification for this benchmark. Our burden of proof needs to shift so that each piece of content earns its way in.

One final filter is

7. Is this content most effectively learned years in advance of its use, or closer to the actual moment of application in work or college?

Identifying what students need to know is one thing, but there is also the question of when they need to know it—and when it is most appropriately learned. Action 2 below invites the commission to explore this distinction in the context of some examples.

Identifying what students need to know is one thing, but there is also the question of when they need to know it—and when it is most appropriately learned.

After the “filtering” process is carried out, there is still another crucial step to take. The standards that pass through the filters should be written in a way that makes transparent the depth of mastery envisioned. The idea is to write the standards so that proficiency essentially implies command. For example, instead of saying,

“Students use exponential notation” the standard could say “students use exponential notation, without prompting, when it is valuable to do so.

Tasks and assessments would have to be rich enough to allow students to display these capacities. Tasks such as “Write 0.0000352 in scientific notation” would not suffice, nor would cues such as “Express your answer in scientific notation.”

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The Situation in Science

In common with mathematics, science standards and science curriculum suffer from a “kitchen sink” problem. The leading textbooks in introductory college physics courses run to 1,000 or even 1,200 pages. Physics educators and researchers in physics education now argue that fewer topics will lead to better outcomes; but just as in K-12, no process seems to exist for bringing the stakeholders to agreement on a new approach.

Just as the Focal Points may help show the way towards distilling math standards, the National Academy’s report Taking Science to School (Duschl et al., 2007) is a key starting point for scientific subjects. Here we draw attention to the report’s second recommendation:

Recommendation 2: The next generation of standards and curricula at both the national and state levels should be structured to identify a few core ideas in a discipline and elaborate how those ideas can be cumulatively developed over grades K-8. (Duschl et al., from the Executive Summary)

However, as the case of the Focal Points and the Big Ideas illustrate, the next generation of standards may end up resembling the previous generation, absent some hard-headed thinking about the standards development process.

Getting Advanced Content into the Math and Science Standards

Currently, the standards framework takes the form “All students must learn…”. We would like to suggest bringing some nuance to this dictum. Imagine, for example, a content framework with two categories:

Category 1. Everyone learns…

Category 2. Everyone can learn…

Category 1 asks us to identify those ideas, understandings, skills and powers from math and science that are so clearly crucial to an educated and productive life that everyone must know them. Here one thinks of ideas like proportional reasoning, drawing valid inferences from data, algorithmic thinking, and formulating and testing hypotheses.

Keeping Category 1 manageable allows Category 2 to come to life. Everyone can learn…complex numbers, matrix multiplication, organic chemistry.

What hope is there of inserting topics like these into the standards-based system we have today? And even if we did, what good would it do? Distilling the standards in Category 1 is what makes it possible to envision raising achievement for a large group of people.

The suggestion here is not that every student in the United States would actually learn advanced topics. Many, perhaps most, would prefer not to take on the sustained discipline that is required to succeed with this material. Rather, when we say that “Everyone can learn,” what we mean is that structures are in place to ensure that everyone who wants to, and who remains committed to doing what it takes, can.

The important questions for Category 2 now become questions about what those structures will have to be like. How can we ensure that everyone willing and able to do what it takes can learn highly sophisticated material in every grade? Who will teach it, and what structures or institutions must exist for this to happen? How should we think about accountability for these subjects?

The Category 1/Category 2 distinction is emphatically not intended to evoke a two-track system of mathematical haves and have-nots. For one thing, a hallmark of the distinction we are making is that while the material in Category 2 is technically advanced, the material in Category 1 is profound. There is no hierarchy of value in this distinction. Nor is there even a hierarchy of difficulty: Students who soak up advanced techniques will often not have an easy time with the infinitely rich material in Category 1.

Closing thoughts on benefits that could be derived from fewer, clearer and higher standards are:

  • Fewer topics will allow more time for in-depth treatments, more time for practice, and more time for students of different learning styles to grasp the material.
  • Fewer topics will also make standards-based tests more reliable. Currently there are so many standards that a reliable evaluation usually cannot be given for each one, due to the small number of test items in each standard. With fewer standards, tests would provide much more useful and reliable information on students’ strengths and weaknesses to support instruction.
  • As long as state standards in reading and math remain vast, aligning instruction is doomed from the start; textbooks become sprawling and assessments remain invalid.
  • Briefer standards will allow more energy to be devoted to refining shared lesson plans and assignments, refining content-specific approaches to professional development, and refining resources for evaluating student work in specific standards.
  • Briefer standards will likely bring different states’ standards closer together, leading to greater ease of transfer for best practices.
  • Life and work present unexpected, unpredictable challenges. Non-standards-based exams like the SAT reward a kind of flexibility and power that we should be seeking to instill in everyone. Raising standards should mean giving assessments that not only assure topic proficiency but also put students off-balance, asking them to demonstrate a forward stance towards unfamiliar problems. Emphasis on topic coverage has prevented us from focusing on teaching the expert mentality.

With fewer standards, tests would provide much more useful and reliable information on students’ strengths and weaknesses to support instruction.

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The Commission can call for states to revise standards so that they are fewer, clearer and higher. There is already movement in this direction across the country, but the Commission can accelerate it, make it practical, and give it shape. In our efforts to get the standards right—a crucial step, as they drive assessments, text and teacher training—we may continue to get them wrong unless we apply new criteria. We need our leading mathematicians, scientists and educators to do the really hard work of being ruthless about fewer and clearer, at the same time that we need them to be visionary about higher, unlocking the power of math and science to transform personal lives and national possibilities. The Commission can make a powerful case that we need a plan ensuring that everyone reaches much deeper comfort with math and science, even as we allow many more students to do advanced work than is currently the case.

Action 2. Issue a call for more pragmatic analyses of what readiness for work actually requires, and what this implies for the teaching of math and science.

This is a thread from section (1) that deserves further discussion. The American Diploma Project Network (ADP Network), a joint effort by Achieve, The Education Trust, and The Thomas B. Fordham Foundation, recently undertook a groundbreaking effort to correlate math standards to readiness for college and work. ADP has found that existing standards and high school exit exams do not ensure readiness for college and work (ADP, 2004). We applaud ADP for taking up this work, but we believe that more pragmatic work still needs to be done to ascertain what math and science is really crucial for the world of work.

We need our leading mathematicians, scientists, and educators to do the really hard work of being ruthless about fewer and clearer, at the same time that we need them to be visionary about higher, unlocking the power of math and science to trans-form personal lives and national possibilities.

Consider the ADP Sample Tasks. The sample tasks were developed in partnership with businesses in order to show how ADP benchmarks figure in the workplace:

The workplace tasks vividly illustrate the practical application of the ‘must-have’ competencies described in the benchmarks themselves, helping states answer questions such as ‘Why do I have to learn this stuff?’ (http://www.achieve.org/node/197)

The task at http://www.achieve.org/node/610 is for a Machine Operator in a chemical company to “Determine the percent concentration by weight of 5g Peters fertilizer and 50g distilled water.” This task is offered as an example of algebra benchmark J1.5. Yet the content given under J1.5 is actually to express 1/x + 1/y in the form (x + y)/xy and to simplify ((a2 − b2)/2b)(6ab/(a + b)) in the form 3a(a − b). The content quoted to illustrate the benchmark significantly overshoots the Sample Task in question.

Is ADP’s Benchmark J1.5 really necessary for Machine Operators? Did the employer who asked for concentrations really mean to sign on for the simplification of rational expressions as well? There seems to be a mismatch here.

We agree with ADP that increasing the number of students who are proficient, without ensuring that they can also meet real-world demands, “will only mislead high school students about their chances for success as adults” (ADP 2004, pp. 5, 6). But we would like to see better evidence than we have now about how manipulating ((a2 − b2)/2b)(6ab/(a + b)) prepares students for real-world demands.

There is a methodological issue here as well. Inviting employers to present their wish lists to the schools will inevitably lead to a bloated set of expectations. It is convenient for employers to ask that their employees know X when they walk in the door; but people may only be ready to learn some Xs when they themselves perceive the need for them. Consider what college students are currently asked to do in order to become doctors. All premedical students have to take a year of physics in college. Medical schools would probably cite content-related reasons for this: “Students have to understand X in order to deal with Y, and they learn about X in physics class.” But as research shows, even students who get high grades in premedical physics classes don’t know the material very well soon after the term ends. This may be a source of alarm to physics instructors, but it hasn’t yet crippled the medical profession. The reason is that the doctor will later pick up whatever he or she needs to know about X as the need arises; and at that time the doctor will learn about X at the level of detail and depth appropriate for his or her particular needs. Having a generalized learning experience several years in advance of this encounter will almost always overshoot the actual content coverage required, and will also lack the intensity of self-motivated, “just in time” learning.

To judge from the full set of Sample Tasks in the ADP report, most high-growth, highly-skilled jobs involve only a small body of technical mathematics content. It appears that a few powerful topics of wide applicability are crucial, such as proportional reasoning, drawing inferences from data, thinking algorithmically, and estimating orders of magnitude as well as orders of importance. The most valuable capacities should be identified and their relevance made clear to everyone.

We recognize how counterintuitive it may sound for a Commission of this caliber to question frequent claims about where math and science are needed in life; to demand that the burden of proof be shifted. But we believe this will be necessary in order to achieve the promise of standards-based education. How many people today can sketch a graph of a relationship, not only when specifically asked to do so in school, but also unprompted and with confidence as a way of attacking a problem on the job? What we teach in math and science needs to become a tool in someone’s hand—one frequently used in times of need. We urge the Commission to chart a course towards a workforce and a society in which virtually everyone is vastly more comfortable using mathematics and thinking scientifically than they are now. We still need to achieve what John Dewey long ago characterized as a “widening spread and a deepening hold of the scientific habit of mind.”

The Commission can stimulate new research on readiness, with some or all of the following features:

  • Readiness should be defined in ways that are more pragmatic than prior efforts with respect to topic coverage, yet more ambitious than prior efforts with respect to depth of mastery of powerful ideas at the core of math and science.
  • Case studies can be made of the most successful individuals working in areas with high numeracy demands, such as finance, insurance, real estate, management consulting and entrepreneurship. Our hypothesis is that the most successful individuals use a few ideas powerfully over and over again to deal with problems of unpredictable variety that arise in their work.
  • Studies should specifically identify which of the existing standards are unimportant for most foreseeable jobs.

How many people to-day can sketch a graph of a relation-ship, not only when specifically asked to do so in school, but also unprompted and with confidence as a way of attacking a problem on the job? What we teach in math and science needs to become a tool in someone’s hand—one frequently used in times of need.

Action 3. Outline a program for dramatically raising the number and diversity of students performing at the highest levels.

Our country’s educational effort overwhelmingly focuses on bringing all students to proficiency. Much less systematic thinking, work and funding has been devoted to bringing more students to exceptional levels of performance. States and districts typically do not even report widely the statistics of how many students proceed beyond proficiency, and almost none disaggregate these data to reveal how many minority students achieve the highest levels of performance. When examined, this data reveals a persistent and stark racial, economic and often gender gap in the number of exceptional performers.

A recent research effort by the Jack Kent Cooke Foundation (Wyner et al., 2007) has begun to document the way low-income students in the top quartile of academic performance in early grades gradually slip to mediocrity or failure (Wyner et al., pp. 5, 6). They are not on our radar; once proficiency is achieved they are off the charts. Bill Sanders (personal communication) has found that the most neglected subgroup in our schools is high-performing minority students in poor schools, based on a comparison of their later performance to the promise of their early performance. Reversing these currents will require moving beyond a strategy that focuses solely on failing students, and introducing a new emphasis on multiplying the number and composition of exceptional performers.

Taking action on this problem is both a smart thing to do and the right thing to do. The systematic frustration of human potential that is evidently now occurring on a national scale cries out for action on ethical grounds—especially when the dynamics are strongly biased racially and economically. The health of society as a whole suffers when poor and minority communities lose potential leaders. And when we look to the schools to address questions of national competitiveness, keeping our best students on a rising trajectory amounts to wisely conserving a precious natural resource. The United States may not be able to double its population to compete with India and China, but we believe it is possible to double the pool of exceptional performers who will lead and innovate in the decades to come.

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Reasons for Believing that this is Possible

Any call to increase radically the number and diversity of our highest performers will have to confront some unspoken, yet deeply-held beliefs about inborn talent. How can schools possibly raise the number of high performers by a multiplicative factor, given a fixed distribution of inborn mathematical and scientific talent? This question will be returned to in the next section on deliberate practice, a body of research that questions common perceptions about how high expertise is gained.

But one immediate answer to this concern is that the country already has a wealth of talent on hand; it’s just that our current system is neglecting it. Modifying the accountability system so that it measures more than mere proficiency will be necessary if we want to tap into the wellspring of talent we already have.

Modifying the accountability system so that it measures more than mere proficiency will be necessary if we want to tap into the wellspring of talent we already have.

The standards themselves also currently stand in the way. Action 1—making standards fewer, clearer and higher—is an important element of the drive to dramatically increase the number and diversity of students achieving at the highest levels. A more focused and coherent set of standards makes it more feasible for school systems to bring everyone to meaningful proficiency while still devoting resources to the highest performers. And when the standards become higher—when they use more phrases such as “unprompted, when valuable to do so”—when powerful ideas move into earlier grades—and when advanced content enters the system, as in “Everyone can learn…”—we give high-performing students a more reliable supply of challenge and enrichment.

Another basis for believing that the number of high performers can be raised by a multiplicative factor comes from research on expert performance (described in Action 4), which shows that the acquisition of high expertise in many fields, including academic ones, is better explained through special patterns of work and practice than by general ability or overall intelligence. We urge the Commission to consider whether this research has implications for teaching and the structure of the school day.

The Commission has an opportunity to highlight the problem of high-achieving, low-income students falling by the wayside. Additional research is needed to better document the scope of the phenomenon and evaluate the effectiveness of various interventions. The Commission can also make recommendations for changes to standards, curriculum, assessment, teaching and institutions that will increase the number and diversity of high performers by a multiplicative factor.

Action 4. Evaluate deliberate practice as a means of achieving a significant increase in the number and diversity of high performers and also as a guide for the reform of secondary math and science instruction altogether.

How is High Expertise Attained?

Research in the psychology of expert performance indicates that the attainment of high expertise in diverse fields is better explained by special patterns of work and practice than by overall intelligence or general ability. These special patterns of work and practice—termed “deliberate practice” by Ericsson and others (Ericsson, 2006)—have been found to be strikingly similar among the highest achievers in diverse endeavors, including mathematics, chess playing, sports, games of skill such as darts, typing and musical performance (Ericsson, 1996). See Figure 3.

We believe that the research on expert performance has implications for K-12 and college education. For example, expertise is gained over a period of 5-10 years; but what institutions exist that can take responsibility for growth on timescales this long? Graduate schools are the only example that comes to mind. Can institutions be built in K-12 that monitor and support patterns of student work over long timescales?

Some other possible implications are explored below.

Research on exceptional performers in diverse areas has found they share a set of behaviors described as deliberate practice. Deliberate practice has several characteristics, including:

Solitary. Deliberate practice is something you do alone. It is not group work, so much so that world class musicians describe practice sessions with a chamber group as “leisure” in comparison to their practice alone (Ericsson, 1996, p.26). This is not to say that group work is not important but that sustained solitary labor is an essential condition of excellence.

Highly-focused concentration: Not only no cell phones, e-mails, but so intense that a person will take a nap if they can not sustain the concentration (Ericsson, 2006, p. 699). In sum, “they found that concentration is the most essential aspect of deliberate practice. … Master teachers argue that full concentration is essential and that when it wanes a musician should rest, because practice without full concentration may actually impair rather than improve performance.” (Ericsson, 1996, p.24)

Feedback and revision. Deliberate practice is not just repetition, which can reinforce errors, or the accumulation of experience (Ericsson, 2006, p. 698). The “deliberate” aspect of deliberate practice means that the person doing it has a personal understanding of how the tasks fit into their growth. Exceptional performers design their practice to “stretch” them in ways that break new ground. A coach or master teacher often plays an essential role in providing feedback and guidance. There is a mechanism for feedback that enables the individual to revise continually based on observations/monitoring of performance outcomes.

4-5 hour daily limit. There is a limit, common across various fields of endeavor, of the amount people can productively practice each day. The concentration level is so intense that people build up to that four hours gradually; e.g., at first one might only be able to do an hour a day and build endurance (Ericsson, 2006, pp. 699, 700).

Sustained: World class talent in various fields typically develops over roughly 10 years (Ericsson, 2006, p. 31). Deliberate practice means a daily set of habits. Hence, sustaining the concentration both within each day, and then across the weeks, months and years, distinguishes world-class performers.

Figure 3. Five aspects of deliberate practice.

Intensity and Motivation

The research we have referred to verifies what many of us already know about our own lives, as experts in one field or another: that expertise is not attained casually or reluctantly. The sustained, intense, and solitary work that typifies the highest achievers is not a moderate form of behavior. Our schools are a long way from fostering anything like the productive intensity we need in order to increase the ranks of high performers. We need new ways of motivating young people and their families to bring their work to a higher level. Vague gestures to future success in life or college do not motivate most students, to say nothing of policymakers’ worries about economic competitiveness. Tests motivate students, but do not animate them. Traditionally, public events like science fairs, spelling bees, and essay contests have brought out the best in many students. Exhibiting individual work beyond the classroom involves a healthy blend of the pressure of expectations with the excitement and pride of performance. Such events could be made more prevalent, and evaluation and feedback on the work made much more systematic.

In the research on deliberate practice, one thing that distinguishes productive practice from mere repetition is its “deliberate” aspect. Deliberate means that the person doing it has a personal understanding of how the tasks fit into their growth. Making standards clearer and more transparently relevant therefore becomes an essential part of the strategy for generating performance gains.

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Shifting the Rhythms of Teaching to Allow the Fostering of High Achievement

The theory of deliberate practice may also have significant implications for teaching. Studies of world-class performers in various fields demonstrate that the core of their training is several hours spent each day practicing alone. School systems should build-in the practices, systems and support that scaffold independent work and thereby build student resilience. Deliberate practice depends on a cycle of feedback between student and teacher. In tandem with a focus on student patterns of work, we need a focus on the quality and timeliness of teachers’ feedback on student work.

This focus need not be seen as multiplying a teacher’s workload. If students spend more time during the school day independently creating work, teachers will have more time during the day to evaluate it. Teachers in other countries spend less time in front of the classroom, leaving more time for evaluating student work and collaborating with other teachers on lesson development and planning.8

Homework will not go away, but in recognition of the central role that working independently plays in fostering high achievement, we should reorient the school day to promote independent work, instead of trusting as we do now that this work is going on in the home. Differing parental expectations among different American cultures make homework an ineffective strategy for closing achievement gaps, particularly at high levels where the material being learned may render family members unable to provide assistance.

Group work will not go away either, but we need to encourage the productive patterns of group work that lead to high performance. In Treisman’s early calculus studies, for example, successful Chinese-American students worked both alone (8-10 hours per week) and in groups (4-6 hours per week). Students worked individually first, then came together to compare answers, not letting go of a problem until discrepancies were resolved (Treisman, 1992, p. 366).

Treisman also highlights the importance of the particular math problems being worked on by the group: “Most visitors to the program thought that the heart of our project was group learning. They were impressed by the enthusiasm of the students and the intensity of their interactions as they collectively attacked challenging problems. But the real core was the problem sets which drove the group interaction” (Treisman, 1992, p. 368). Curriculum development should concern itself with crafting and refining excellent problems over time—and finding ways to disseminate the best of them beyond individual classrooms. We cannot promote high quality student work on a large scale without developing common assignments, common tools for evaluating those assignments, and common tools for monitoring student work. Secondary schools in particular have been slow to adopt common approaches. Foundations and the government can help forge a much more productive, practical link between research and practice by focusing on the specific tools that can be built.

Additionally, technology offers new possibilities for directly engaging students in their work, tracking and sustaining their growth. Foundations and businesses should be investing in ways to support common approaches, such as delivering common assignments, aiding feedback, engaging students in their work, and following their growth.

...technology offers new possibilities for directly engaging students in their work, tracking and sustaining their growth. Foundations and businesses should be in-vesting in ways to support common approaches, such as delivering common assignments, aiding feedback, engaging students in their work, and following their growth.

Refocusing secondary schooling to make successful transitions to college and work

Secondary schooling can focus on enabling students to work independently and productively so they can make successful transitions to college and work. So many high school graduates, particularly low-income students, fail the demands of college. A factor leading to success in college is the student’s capacity to work independently and productively, even in the face of enormous emotional and social pressure. Secondary schools today focus very little on developing students’ habits of working when they are alone: their capacity to sustain their concentration, to work well in advance of the need to perform, to complete their work even during a personal crisis. This may explain why the results of even strong programs like KIPP evaporate as students enter high school and college; these kids learn to work in school and classroom settings with a lot of guidance and structure – this is very different from the world of college and work where these structures fade. A recent study by Duckworth and Seligman (2005) emphasizes the role of self-discipline in academic achievement among adolescents, and lends support to this analysis.

The Commission should consider whether deliberate practice has a role to play in achieving the goal of substantially increasing the number and diversity of high performers. The Commission should also address related questions of motivation and intensity, and their relationship to success. Finally, the Commission should consider the prospect that restructuring patterns of work in high school might smooth the transition to college.

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References

ADP (2004), Ready or Not: Creating a High School Diploma that Counts, The American Diploma Project Network, report and executive summary available at http://www.achieve.org.

Duckworth, A.L. and Seligman, M.E.P. (2005), “Self-Discipline Outdoes IQ in Predicting Academic Performance of Adolescents,” Psychological Science 16(12), pp. 939-944.

Duschl, R.A., Schweingruber, H.A., and Shouse, A.W. (2007), Taking Science to School: Learning and Teaching Science in Grades K-8, National Academies Press, 2007.

Ericsson, K.A. (1996), The Road To Excellence: the Acquisition of Expert Performance in the Arts and Sciences, Sports, and Games, Erlbaum Publishers.

Ericsson, K.A., Charness, N., Hoffman, R.R., and Feltovich, P.J. (2006), The Cambridge Handbook of Expertise and Expert Performance, Cambridge University Press.

FL (2007), Florida Department of Education mathematics content standards, 2007_FL_Mathematics_Standards_9_13_07.pdf, document accessed on September 29, 2007

NCTM (2006), Curriculum Focal Points for Pre-Kindergarten through Grade 8 Mathematics: A Quest for Coherence, National Council of Teachers of Mathematics, ISBN 0-87353-595-2

NECTL (1994), Prisoners of Time, Report of the National Education Commission on Time and Learning, http://www.ed.gov/pubs/PrisonersOfTime/index.html

Plattner, L. (2007), Washington State Mathematics Standards: Review and Recommendations; may be accessed at http://tinyurl.com/yuq3gt

Schmidt, W.H., McKnight, C.C., Raizen, S.A., and others, A Splintered Vision: An Investigation of U.S. Science and Mathematics Education, Executive Summary, U.S. National Research Center for the Third International Mathematics and Science Study, Michigan State University.

Stevenson, H.W., Lee, S.-Y., Chen, C., Stigler, J.W., Hsu, C.-C., Kitamura, S., and Hatano, G. (1990), Contexts of Achievement: A Study of American, Chinese, and Japanese Children, Monographs of the Society for Research in Child Development, Vol. 55, No. 1/2.

Treisman, U. (1992), “Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College,” Mary P. Dolciani Lecture, in College Mathematics Journal 23(5), November 1992.

Wyner, J.S., Bridgeland, J.M., DiIulio, J.J. (2007), Achievement Trap: How America is Failing Millions of High-Achieving Students from Lower-Income Families, Report by the Jack Kent Cooke Foundation and Civic Enterprises, with original research by Westat.

David Coleman is the founder of the Student Achievement Partners. The organization assembles leading thinkers and researchers to design actions that will substantially improve student achievement. The group integrates rigorous policy analysis, research, and design to focus entirely on the most significant outcomes for students. David also serves as Senior Vice President of Education Policy for McGraw-Hill Education. In this role, David cultivates a thought leadership role as a liaison with key leaders in nonprofit organizations and government.

David founded The Grow Network in 2000, acquired by McGraw-Hill in 2004. The Grow Network’s mission is to transform assessment into an opportunity for meaningful instruction. Grow quickly became the leading provider of customized solutions that make assessment results truly useful for students, families and educators. Prior to founding Grow, David was Senior Engagement Manager with McKinsey & Company, where he was a leader of McKinsey’s Business-to-Business Electronic Commerce Practice and its pro bono education work. David, a Rhodes Scholar, holds a bachelor’s degree from Yale University, and master’s degrees from both Oxford University and Cambridge University.

Jason Zimba earned his B.A. in mathematics and astrophysics from Williams College, where he graduated summa cum laude and earned highest honors for research. He attended Oxford University as a Rhodes Scholar, where he earned an M.Sc. by research in mathematics as a student of Roger Penrose. As a graduate student at the University of California at Berkeley, he was chosen by Bruce Birkett to be the lead curriculum developer for an NSF-sponsored project to reform introductory physics courses; this project received the University’s Innovations in Education Award in 1999. Jason earned his doctorate in mathematical physics from Berkeley in 2001. Meanwhile, in early 2000, he helped to co-found The Grow Network, a K-12 education technology company (now a McGraw-Hill Education company). As Grow’s Vice President for Education and Product Development, he was responsible for standards alignment, curriculum design, product development, educational strategy and psychometrics, working closely with nonprofit organizations, businesses and school leaders at the local and state levels. In 2003 he returned to teaching, taking a position as Assistant Professor of Physics at Grinnell College, and in 2004 he moved to Bennington College, where he is now a Faculty Member in Physics and Mathematics.

Cited in this section


1 Deliberate practice is a term used in the field of experimental psychology to describe patterns of work common to expert performers in diverse fields. (Described further within.)

2 Grades 9-12 are under development.

3 From the Curriculum Focal Points Questions and Answers, on www.nctm.org.

4 Admittedly in high schools there has been a successful push to increase the number of students taking AP courses—but ExxonMobil’s recent $125M investment seems to recognize that there is a problematic lack of state incentives even in high school.

6 Interestingly, manipulatives and diagrams as strategies for numeration are relegated to Supporting Ideas in Florida. (See MA.1.A.6.2, FL 2007, p. 25.)

7 From Stevenson et al. (1990): “Whereas children’s academic achievement did not appear to be a central concern of American mothers, Chinese and Japanese mothers viewed this as their child’s most important pursuit. … Chinese and Japanese mothers held higher standards for their children’s achievement than American mothers and gave more realistic evaluations of their child’s academic, cognitive, and personality characteristics. American mothers overestimated their child’s abilities and expressed greater satisfaction with their child’s accomplishments than the Chinese and Japanese mothers. In describing bases of children’s academic achievement, Chinese and Japanese mothers stressed the importance of hard work to a greater degree than American mothers, and American mothers gave greater emphasis to innate ability than did Chinese and Japanese mothers.” (Italics added.)

8 “Japanese teachers are typically in ‘front of the class’ for only four hours a day. Time spent outside the classroom is not consideed wasted, but an essential aspect of professional work. The same phenomenon can be seen in Germany—teachers are in front of a class for 21 to 24 hours a week, but their work week is 38 hours long. Non-classroom time is spent on preparation, grading, in-service education, and consulting with colleagues” (NECTL, 1994, “Lessons from Abroad”).

9 The Chinese-American students in Treisman’s study did solitary work for many hours each week, and to do their group work, they organized themselves into an “academic fraternity” (Treisman, 1992, p. 366).