Chapter 4: Assessment and Opportunity to Learn
In this chapter, the focus shifts away from task development and issues that impinge on students' opportunity to perform and toward assessment tasks as seen in the social milieu of the classroom because that is where students are (or are not) provided opportunity to learn. Providing such opportunity means creating access for students to the procedural, conceptual, and strategic knowledge to support a deep and robust understanding of mathematics and the know how necessary to demonstrate this multifaceted knowledge. A number of key research reports have focused attention on the importance of collecting opportunitytolearn data to inform the interpretation of assessment data (NCTM & NRC, 1997; NRC, 1989, 1997, 1998; Porter, Kirst, Osthoff, Smithson, & Schneider, 1993; Schmidt, McKnight, Valverde, Houang, & Wiley, 1997; Stigler & Hiebert, 1997). In particular, opportunitytolearn issues emerge as an essential strategy in any effort to improve education and address equity issues (Massell, Kirst, & Hoppe, 1997; Black & Wiliam, 1998).
The concept of opportunity to learn is linked to the concept of opportunity to perform as described in Chapter 3. If students lack opportunities to perform, they will not be able to show what they know and can do. If students lack opportunities to learn, they win not be able to avail themselves of opportunities to perform. In Chapter 3, issues concerning opportunity to perform led to a focus on the development of assessment tasks and the dimensions of a balanced assessment. Here, issues concerning opportunity to learn lead to a focus on teaching and learning.
The task development experience that underpins both the model for balanced assessment in Chapter 2 and the discussion of opportunitytoperform issues in Chapter 3 was not conducted behind the closed doors of assessment designers' offices. Instead, much of this experience has been obtained in mathematics classrooms across the country, because each assessment task is put through several rounds of systematic classroom trials, including initial task trials implemented in two or three mathematics classrooms, and also largescale field tests where the tasks are put through trials with a large stratified sample of students.
Each classroom trial is observed by at least one of the following:

a fulltime assessment developer,

a fulltime mathematics teacher who is participating as a codeveloper in the assessment development process,

a classroom teacher who is responsible for providing written evaluations of the task in action, or

a teacher who is participating in a professional development program focusing on assessment.
This chapter draws upon this extensive body of classroomgenerated experience to present a series of recommendations and conclusions based on observations of assessment tasks in the social context of the classroom. Many of the sections of this chapter highlight barriers to opportunity to learn, such as tight sequencing of teaching and testing, inappropriate emphasis on skills acquisition activities, inappropriate task modification, the need to cover the curriculum, preconceptions of teachers, and gaps in the curriculum. Each of these discussions is framed by locating it within the context of relevant and recent research. Where appropriate, larger implications for the teaching and learning of mathematics are identified. Other sections discuss how complex tasks may be used to enhance learning opportunities in the classroom. While working through such a task, for example, misconceptions and mistakes may be viewed as opportunities to learn rather than as complications to be avoided. Furthermore, class work on complex tasks is necessary to develop problemsolving tenacity and the ability to communicate about mathematics. The purpose of this chapter is to identify how efforts to improve assessment might be used to improve mathematics instruction and learning. Therefore, the primary concern in this chapter is not just finding better ways to assess students but finding ways to enable students to perform better on worthwhile assessments. The chapter closes with a list of recommendations based upon these issues and informed by Black and Wiliam's (1998) contention that learning is driven by what teachers do in classrooms.
Tight sequencing of teaching and testing
A common obstacle to success on nonroutine tasks is the tendency of students to attempt to apply the specific mathematics that they are currently studying to the task at hand, whatever that task might be. For example, when students were presented with either Shopping Carts or Paper Cups immediately after they had studied area or volume, many began their work by trying to find the area or the volume of the cart or cup. Similarly, when a large number of students tried to set up and solve a system of linear equations in response to one of these two tasks, it turned out that the class had just been studying systems of linear equations. When a disproportionate number of students in a class provided solutions involving y = mx + b to Broken Plate, a task investigating the relationship between percent decrease and percent increase, their teacher confirmed that his students were working on the slopeintercept form of the equation of a line. When considering what might have inspired such seemingly incongruous responses, it seemed that students were using whatever tool was most readily at hand rather than grappling with and making sense of the task. In fact, the premise on which most classroom assessment rests is to assess what has just been taught.
The problem with this what we just studied phenomenon is that it usually does not work well in creating access to the task, and often makes the task less accessible for the student. In the words of one teacher commenting on her students' work: ''Most students made the task harder than it was, when they tried to make it fit with what we were currently studying."
When teachers discussed this at professional development workshops on assessment or at assessment codevelopers meetings, they realized that they shared a common problem. Teachers are frequently quite taken aback by this realization and hypothesize that it is their common practice of teaching a topic then testing it, teaching another topic and then testing, that might cultivate this behavior in their students. These teachers acknowledge that they rarely test the mathematics that their students have learned twelve, six, or even just two months previously. Therefore, it is also rare that their students are required to attempt challenging nonroutine tasks.
This what we just studied phenomenon is not reserved for those students who are underprepared in mathematics or for those who find learning mathematics difficult, but can be observed even in honors classes where student participation is usually marked by a high level of success. Many highly successful students, well on their way to successful completion of Algebra II and Trigonometry courses, failed to solve a problem such as that posed by the unscaffolded version of Shopping Carts because, rather than trying to make
sense of the task, they attempted to bring only their most recently learned but inappropriate mathematics to bear. It was as if these students had in their heads a directory of template problems that they had learned how to solve. Instead of thinking about the task at hand and making decisions about the mathematics that might be needed to solve it, these students simply forced aspects of one template problem after another onto Shopping Carts. When students of this caliber work on nonroutine assessment tasks, it is evident that they know a lot of mathematics—far more than is actually needed to solve the task. But it also is evident that their mathematical understanding is fragile and inflexible (Lesh, Lamon, Lester, & Behr, 1992). Further, it is evident that these students have had little practice either making sense of mathematics or using mathematics in a practical fashion. These observations also reinforce an earlier conclusion stated by Schoenfeld: "'Knowing' a lot of mathematics may not do some students much good if their beliefs keep them from using it" (1987, p. 198).
The most serious aspect of this what we just studied phenomenon is that when students attempt to make only their most recently learned mathematics relevant to the task at hand, they are providing evidence of a routine that may characterize all of their learning of mathematics. The students' habit is to do mathematics without having to think about the task. Unfortunately, these students are not only doing what they usually do, but also are doing what usually works for them.
The coupling of teaching and testing in this way has consequences both for the mathematics that is learned and for students' perception of the learning of mathematics. On the one hand, it leaves students illequipped to tackle nonroutine tasks where an important hurdle is the selection of relevant mathematics. On the other hand, it instills in students the notion that mathematics can be learned by applying recently learned mathematics without a great deal of thought. It also runs the risk of teaching students that making mathematical sense or using common sense are not appropriate behaviors for the mathematics classroom.
Coupling teaching and testing in this way also has consequences for what a teacher can say about her students' learning of mathematics. How does a teacher know whether her students have really learned the mathematics? How does the teacher know that her students will retain what they have been taught over the longer term? How does the teacher know what her students can do with the mathematics they have learned?
The view of teaching and learning that is evidenced by such tight coupling of teaching and testing also has been criticized by Schoenfeld:
All too often we focus on a narrow collection of welldefined tasks and train students to execute those tasks in a routine, if not algorithmic fashion. Then we test the students on tasks that are very close to the ones that they have been taught. If they succeed on those problems, we and they congratulate each other on the fact that they have learned some powerful mathematical techniques. In fact, they may be able to use such techniques mechanically while lacking some rudimentary thinking skills. To allow them, and ourselves, to believe that they understand the mathematics is deceptive and fraudulent. (Schoenfeld, 1988, p. 30)
One of the more farreaching effects of this practice, as shown by our own experience and addressed in the discussion by Schoenfeld, emerges most acutely when narrowly defined tests are used for stateor districtwide accountability purposes. In such cases, teachers report that they find themselves under increasing administrative pressure to spend greater and greater amounts of time preparing for the test (Romberg, Zarinnia, & Williams, 1990). This can breed an everexpanding culture of test preparation and, at its most extreme, runs the risk that test preparation could completely replace instruction. Mathematics classes might then become characterized by students working repetitively on set after set of questions that mimic those that are on the test.
When narrowly defined tests are used in this way to address accountability needs, the consequences for learning are inevitable. Students' opportunity to learn is replaced by the opportunity only to practice a narrow range of test questions. There is a well grounded fear that this approach will fail to prepare students for higher level mathematics courses. Such an approach will do little to inculcate a mathematical disposition or to encourage students to invest in further study of mathematics. Finally, the costs of this kind of testing can become hidden—large amounts of teacher time, and classroom resources are diverted away from teaching and learning and are used instead to prepare students for narrow tests that are at best loosely connected to a balanced curriculum.
Inappropriate emphasis on skills acquisition activities
Teachers often say that although some nonroutine tasks are interesting, rich, and target worthwhile mathematics, they are not appropriate for their students.
When we explore this perception further, we find that many of their students are considered by these teachers to be underprepared in mathematics. In the view of their teachers, these students lack basic skills. Teachers described how, in an effort to rectify this situation, they restricted their students to sets of short, closed, procedural exercises. They have the perception that their students must acquire some basic level of achievement in rudimentary
mathematics before they can be permitted to attempt challenging nonroutine tasks. In these teachers' views, the full range of tasks illustrated here would be far too challenging for their students, and so they believe it necessary to restrict students to skillsbased tasks.
One serious problem with this common approach in teaching mathematics to students who are underprepared is that there is very little evidence that it works to do anything more than teach simple calculation procedures, terms, and definitions (Hiebert, 1999). Hiebert draws on the most recent National Assessment of Educational Progress (NAEP) to answer the question, "What are students learning from traditional instruction? " He reports:
In most classrooms, students have more opportunities to learn simple calculation procedures, terms, and definitions than to learn more complex procedures and why they work or to engage in mathematical processes other than calculation and memorization. (Hiebert, 1999, p. 12)
Another serious problem is that it is simply inequitable for large numbers of students to emerge from high school without ever having had the opportunity to engage in mathematics work that has been designed to develop conceptual and strategic capabilities. Clearly, the intention is not to deny students this opportunity. Teachers usually intend to shift to a more interesting gear after their students provide evidence that they have acquired the basic skills. Unfortunately, this frequently does not happen, and many students leave school without ever having been given the opportunity to learn mathematics in a broad and balanced way.
The problem can be approached somewhat differently. There is increasing evidence that the memorization of decontextualized fragments of mathematics does not work well in helping students learn mathematics (Hiebert, 1999). But there also is increasing evidence that students can learn when instruction regularly emphasizes engagement with challenging tasks (Stein & Lane, 1996; Schoen & Ziebarth, 1998), or when teachers regularly use technological tools to develop mathematical ideas (Heid, 1988; Hiebert & Wearne, 1996). The Carnegie Learning Program makes extensive use of technology and is currently demonstrating great success in motivating reluctant learners in large urban districts (Hadley, personal communication, February, 1999).
Inappropriate task modification
As noted above, teachers will often argue that tasks of the type presented in Chapters 2 and 3 are more appropriate for students who are better prepared mathematically. Perhaps as a consequence, when teachers administer these tasks to their students, many of them massage the challenge of each task, in the hope that
their students will become neither too frustrated nor too confused by its demands. This practice of massaging the challenge of particular tasks to close the gap between the teachers' perception of what their students know and the perceived demands of the task has also been reported by others (Doyle, 1988; Henningsen & Stein, 1996).
Many teachers are particularly adept at deploying gapclosing strategies. They will often provide directive hints, construct pertinent demonstrations, introduce task scaffolding, and when all else fails they will sometimes try to walk their students through the task. The evidence presented in Chapter 3 shows how scaffolding and other wellintended challenge reduction techniques can radically alter the assessment target of the task. This evidence suggests that teachers' gapclosing processes can have farreaching implications for students' opportunity to learn through challenging nonroutine tasks, and that these strategies can restrict the actual range of tasks that their students will truly have the opportunity to tackle.
Covering the curriculum
Another factor that can inhibit the use of worthwhile assessment tasks in classrooms is teachers' perception of the length of time that it will take their students to do the tasks. Teachers sometimes fear that, if they were to invest the time necessary for administering rich assessment tasks, they might be unable to cover large portions of the material they are expected to cover. Choices about the allocation of precious classroom time are difficult. However, many involved in the reform of mathematics teaching and learning urge teachers to cover less but spend more time going deeper, thus creating a broader and more balanced system of instruction (NCTM, 1989, 1995; NRC 1993b; Schmidt, McKnight, & Raizen, 1997; Schmidt, McKnight, Valverde, Houang, & Wiley, 1997; Stigler & Hiebert, 1997).
The implementation of worthwhile assessment tasks in classrooms is not the only innovation that is labeled as timeconsuming. Indeed, most effective teaching strategies are timeconsuming and therefore regarded as untenable by teachers who are faced with a large amount of material to cover. There is little doubt that if teachers are to be freed to provide opportunity to learn for all, they must be freed from the burden of covering large amounts of material.
Preconceptions of teachers
It is interesting to observe teachers as they consider assessment tasks with a view toward possibly embedding them in their instruction. Frequently, teachers will work through a task and
then draw extensively on this experience in their appraisal of the task's appropriateness. As a consequence, this process leads some teachers to reject certain tasks outright. One teacher said,
This task would not be appropriate for my students. If it took me this long to complete the task, my students would never be able to stay at it.
Another stated,
This task is too abstract, I had to really think about this task. My students would never be able to start it.
Clearly, teachers are very concerned about overwhelming their students and about selecting appropriately demanding tasks for them. This is not surprising given the large number of students who give up all too quickly when they are presented with an assignment that does not immediately resemble one that they have been taught how to do. These findings about teachers' perceptions of the appropriateness of such assessment tasks in their classrooms corroborates research that addresses teachers' perceptions of the appropriateness of instructional materials. For example, teachers' perceptions have been found to be affected by both their perceptions about their students' backgrounds and abilities and the mathematical knowledge of the teachers themselves (Floden, 1996).
Some teachers do recognize that even tasks that challenge the teachers themselves sometimes can be appropriate for their students. It is difficult, however, to persuade other teachers that almost all of their students can learn to do challenging mathematics tasks and that students can learn mathematical skills at the same time that they are working on challenging tasks. This is in contrast to what seems to be a deepseated belief that students' ability (or inability) to do mathematics is immutable and not something that can be improved upon by creating new or enhanced opportunities to learn
Through classroom observations and interviews about assessment tasks, some revealing aspects of student beliefs about learning mathematics have also been identified. Many students have clearly defined and somewhat narrow views of what counts as appropriate behavior for the mathematics classroom. For example, many students have great difficulty in formulating a workable approach to a nonroutine task. When students evaluate such tasks and describe how the tasks might be improved, they almost invariably judge the tasks as not giving them a clear enough indication of what they are supposed to do. They gave responses such as,
"Be more specific about what you want us to do on paper."
"Tell us more information on what we are actually supposed to figure out."
"You do not make it clear what you want us to do. It is better if you saydo this, then do this."
In these responses, the students have revealed that they do not expect to have to formulate an approach to challenging tasks. They expect that their assignments will make clear not only what they are supposed to do but also the steps that they should take to do it. Many students simply do not perceive doing challenging mathematics as appropriate work for mathematics classrooms. Many students just want to be told what to do by their teachers. By the same token many teachers believe that, with so much content to cover, there is little time to do anything but tell their students as much as possible. Many students also will express a lack of confidence in teachers who wish to delve deeply into mathematics rather than rush through larger amounts of material at great speed (Borasi, 1996). Far from relishing the opportunity to dwell on fundamental mathematics with new eyes, students are often concerned that they will never be able to cover the given curriculum, or that focusing on specific aspects of mathematics in greater depth will adversely affect their final grade.
Gaps in the curriculum
Developing assessment tasks sometimes highlights limitations in the ways in which curriculum content is determined. For example, the study of solids and their volume is a content area that is often deemphasized in the current high school curriculum, and students invariably find our tasks involving solids and their volume difficult. As an illustration, Table 2 shows the distribution of scores for responses to Snark Soda (Figure 5, p. 19).
To earn a score of 4, a student must fully accomplish the task. To do so, the student must model the entire bottle using two or more solids, consider the curvature of the top and bottom of the bottle, address accuracy, and communicate each step of the work. This level of success requires significant integration of mathematical skill, conceptual understanding, and problem solving, but it is reasonable to expect that students in the eleventh grade will have fully absorbed these specific skills and concepts. Therefore it is disappointing that so few students are able to make use of these skills and concepts to fully accomplish the task.
Table 2. Scoring of 11^{th}grade responses to Snark Soda
Score 
Off task 
Score 1 
Score 2 
Score 3 
Score 4 
N = 877 
2 
431 
291 
129 
24 
% 
0.3 
49.1 
33.2 
14.7 
2.7 
To earn a score of 3, a student must prepare a response that, while not fully complete, can be characterized as ready for revision. It should be reasonable to infer that the student has the mathematical knowledge and ability to solve the task. The student can show this by modeling the entire bottle using two or more solids and addressing the curvature of either the top or the bottom of the bottle. The student might or might not address the accuracy of the volume and might not fully communicate each step of the work. Even so, just one student in six was able to reach or exceed this level of achievement on the task.
To earn a score of 2, a student must show partial success by modeling the bottle using more than one geometric solid (e.g., two cylinders). The student might not address either curvature and may use a combination of area and volume formulas. For most of the students who were able to achieve any significant success with this task, this level of achievement was as far as they got.
To earn a score of 1, a student must engage with the task but will have done so with little or no success. For example, the response might use only one cylinder to model the entire bottle. When the response contains only words or drawings that are unrelated to the task, the response is scored as "off task." Notice that these two categories account for almost onehalf of all of the eleventhgrade student responses.
What can be said about such disappointing performance? In this version of the task, students were advised to use a ruler, so the problem was not that students did not think to use a ruler to measure the bottle. One hypothesis is that the issue has less to do with specific task characteristics and more to do with students' experience with solids in their classroom.
Many teachers readily confide that they often have only a few days left at the end of the tenth grade to devote to volume. Others indicate that they feel that the large body of knowledge they are obliged to cover during the tenth grade sometimes makes it impossible to cover volume at all. Why would a teacher choose to leave out volume rather than any other topic? It appears that this decision often reflects teachers' perceptions of what is or is not necessary for the next mathematics course their students will take. For many students studying geometry, Algebra II is the next course in the sequence, and there seems to be a belief that a study of solids and their volume is not critical for success in Algebra II. As a consequence, the topic is often neglected. Unfortunately, even though the study of solids and their volume may not be a prerequisite for Algebra II as it is traditionally defined, a sound conceptual understanding of this subject area truly is a prerequisite for calculus. Clearly, if high school curriculum is determined solely by a perception
of what is required for the next course, then the longitudinal coherence of school mathematics is jeopardized.
Another place where a study of solids and their volume is deemphasized is in largescale assessments. In New York, for example, the Spring 1997 Pilot Questions that are used by many teachers to prepare their students for the Mathematics A Examination (which will soon replace Course I in the New York Regents sequence) suggest that a study of solids and their volume will be confined to finding the volume of a rectangular prism. Undoubtedly, this deemphasis at the assessment level will bring about a deemphasis on all other solids in the curriculum.
Study of solids and their volume should not be added to the curriculum in a cursory way because the problems described here cannot be addressed without placing such study at the firmly in the mathematics curriculum. The marginalization of solids within the curriculum has unfortunate consequences that extend beyond student preparation for the study of calculus. A study of solids and their volume provides an abundance of useful material for those seeking to enhance the learning of mathematics through an emphasis on connections, both within mathematics and with worthwhile and relevant contexts outside of mathematics. The Principles and Standards for School Mathematics: Discussion Draft (NCTM, 1998) goes a long way toward placing the study of solids and their volume firmly in the curriculum, and does so in a way that provides a coherent sequence across the PreK12 curriculum.
Unfortunately, the problems identified by this assessment work are not confined to the study of solids and the tenthgrade curriculum. Schmidt and Cogan write:
Review of the TIMSS U.S. mathematics achievement and curriculum analysis results forms a rather compelling notion that the fundamental problem with our mathematics education system lies not with students or teachers but primarily with the way in which we think about and develop our mathematics curriculum. (Schmidt & Cogan, 1999, p. 7)
The TIMSS study, which characterized the U.S. curriculum as repetitive and lacking depth, indicates that the learning of mathematics can be tackled effectively only after something is done to reconceptualize the mathematics curriculum. Assessment development experience also has demonstrated that a fragmented and cluttered curriculum puts teachers under enormous pressure and restricts their opportunities to deepen the focus of their instruction.
Misconceptions and mistakes as opportunities to learn
Attempts to develop tasks designed to assess the robustness of students' understanding of mathematics have been met by an interesting mix of teacher reaction. For some teachers, this approach has validated their own classroom practice, characterized by a constructive focus on the robustness of conceptual development. In part, such an approach requires putting a constructive focus on misconceptions that are brought into the open by presenting students with thoughtprovoking and sensitive tasks. According to these teachers, assessments that made misconceptions visible were an invaluable aid to longterm learning and retention.
For many other teachers, however, conceptually oriented tasks that have the power to realize student misconceptions are to be avoided, lest these confuse students who already find learning mathematics difficult. In recent development work, New Standards staff prepared a formative assessment package to be used in a conceptual approach to the teaching and learning of slope (NCEE, 1998). Students use this package to investigate slopes of ramps, slopes of stairs, and slopes of lines, and to work through a range of challenging and conceptually oriented assignments on slope. A final assignment invites students to imagine a world where slope is defined not as rise over run but as run over rise. Students are asked to discuss the implications of this redefinition of slope. This final task is designed to assess the robustness of students' conceptual understanding of slope. Many teachers have reacted vehemently, arguing that this will run the risk of confusing students, or even leave them with the erroneous view that slope is defined as run over rise.
To steer away from conceptually oriented tasks is to adopt a view of student learning that is characterized by memorization of isolated fragments of knowledge, inherently fragile and unlikely to be retained. Indeed, teachers should use assessments that do operationalize student misconceptions, not to confuse students, but as part of the process of developing mathematical understandings that are robust and can withstand both the test of time and of counterargument. Tasks that ferret out student misconceptions will provide insight into how the student has internalized the body of knowledge that the teacher is attempting to teach. It is only from a clear sense of what the student understands that subsequent instruction can be tailored to benefit the student. Seen in this way, tasks that illuminate student misconceptions will be crucial to the process of benchmarking growth in student understanding (Borasi, 1996).
Assessment practice makes perfectdeveloping tenacity
When teachers regularly administer quality nonroutine tasks to their students and provide feedback to students about their progress, it becomes possible to distinguish those aspects of student performance that are more resistant to change from those that are less so.
One aspect of student performance where there are real opportunities to foster improved learning behaviors is that of tenacitystudent readiness to stay with nonroutine problems. If students can be led to recognize and accept that it often takes time and effort to know what to do when they look at a task, they are less likely to give up prematurely. Teachers can foster this recognition and acceptance by giving the following directions each time their students are asked to work on a nonroutine assessment task:
This task is designed to assess how well you can solve nonroutine problems.
You will not have learned how to solve this problem in class. But you will have learned the mathematics that you will need to solve this problem.
Remember that when you look at this task for the first time you will probably not know what to do. This task is designed to see what you do when you don't know immediately what to do. So don't give up immediatelyread the question again and again. Try to say in your own words what you are being asked to do.
Teachers report that some variation on this theme was important in focusing student attention on the task, reducing student frustration, and increasing student tenacity.
Assessment practice makes perfectdeveloping communication
Communication is one aspect of student performance that is frequently difficult to develop. It is common to see a group of students making substantial inroads in solving a challenging task, where classroom discourse is characterized by focused mathematical discussion and quality thought. It is disappointing to read student responses later, only to find that little of their engaging work has been committed to paper. Even when the work is recorded, it is often difficult to see complete chains of thought.
To encourage students to communicate more effectively and so earn the credit their work suggests they deserve, we recommend providing students might be provided with the following opportunities:

to score other student responses to tasks, trying to follow the line of reasoning, and to provide feedback to the student;

to be given true mathematical statements and asked to explain why they are true;

to represent ideas using more than one form of mathematical representation;

to represent mathematical ideas using their own words and to practice writing down the main tenets of these ideas.
When these strategies are used with students, the students gain a better understanding of how to communicate their efforts. For example, when students scored other students' work, they were provided with a model of different levels of communication and were able to use this model to evaluate the effectiveness of responses. When students were asked to say why a given statement was true, they were relieved of the manipulative challenge of the task and could concentrate entirely on communicating their understanding.
What can be done—some recommendations
This section presents recommendations for those who are interested in using assessment to enhance instruction, including several recommendations advanced by Black and Wiliam (1998), who make the centrally important point that learning is driven by what teachers and students do in classrooms.
Black and Wiliam draw upon a great number of research studies to argue persuasively that to enhance learning, specific attention must be paid to formative assessment. This is an aspect of teaching that Black and Wiliam posit as indivisible from effective teaching, and indeed as the heart of effective teaching:
We use the general term assessment to refer to all those activities undertaken by teacher—and by their students in assessing themselves—that provide information to be used as feedback to modify the teaching and learning activities. Such assessment becomes formative assessment when the evidence is actually used to adapt the teaching to meet student needs [italics in original]. (Black & Wiliam, p. 140)
Nonetheless, for formative assessment to be effective, it cannot be simply bolted on to existing practice. Instead, there is a need for a radical reconceptualization of what teachers and students do in the classroom. Teachers will need a great deal of support as they attempt to rethink and restructure their classroom practice. Here are some key recommendations for enhancing instruction and learning:

Provide teachers with a rich and varied supply of worthwhile assessment tasks to be used as classroomembedded instruction and that will provide students with opportunities to perform.

Encourage schools to move toward the use of highquality, endofcourse assessments that are standardized across an entire school, district, or state. This will help teachers appreciate the importance of teaching to a set of publicly agreed upon and challenging standards.

Encourage teachers, parents, and students to deemphasize grades and emphasize feedback to students.

Provide professional development for teachers that will help them provide their students with useful and constructive feedback that can improve learning rather than compare or rank students.

Structure professional development to enable teachers to recognize and appreciate student growth. All students can learn to complete challenging mathematics tasks. Student work that demonstrates growth in tenacity, communication, and procedural, conceptual, and strategic knowledge should be generated and shared with teachers in the school, district, or state.

Work with teachers to develop a view of the student as an active rather than as a passive learner.

Provide professional development that will enable teachers to incorporate student selfassessment as a useful tool for learning.

Provide teachers with tasks for class and homework that are aligned with standards or learning expectations.

Demonstrate that all students can learn mathematics (either by using videos or student work that shows growth). This is important in encouraging teachers to regard students as having potential to be tapped rather than having innate inability.

Encourage students and teachers to become willing participants in a diagnostic approach to learning, where errors and misconceptions are exposed and resolved, rather than left as unacknowledged and invisible obstacles to learning.

Create approaches to learning where students are given time to communicate, to explore, to receive feedback on, and to reorient their evolving understanding. In such classrooms, students can work for mathematical understanding rather than only for coverage.

Develop approaches to curriculum adoption that are coherent within and across grades.

Enable teachers to use a curriculum that encourages a more integrated and connected approach to learning mathematics. Many of the curricula developed with funding from the National Science Foundation are excellent resources.

Encourage teachers to avoid textbooks that take a superficial approach to mathematical connections.

Consider organizing teaching in a way that enables teachers to teach across an entire grade span. For example, a teacher might continue to teach the same cohort of sixthgrade students through grades seven and eight. This would provide continuity for students and enable teachers to develop a longitudinal view of the larger curriculum.