Teaching Mathematics in Grades 3-5 Classrooms

by Thomas L. Good - April 07, 2008

Background: The National Mathematics Advisory Panel was appointed by President Bush in April of 2006. The panel’s charge was to make recommendations for improving mathematics learning and capacity in the United States. The National Math Advisory Group was divided into four major task groups; Conceptual Knowledge and Skills; Learning Processes; Instructional Practices; and Teachers and Teacher Education. Besides these major task groups there were three sub groups: Standards of Evidence; Instructional Materials; and A National Survey of Algebra teachers.

Purpose/Objective: I first describe the historical failure of reform efforts in American Education. I then briefly discuss reasons for this failure and note that calls for reform generally ask for too much change too quickly. I suggest that one major problem in recent math reform is that reformers have called for expanded math content (e.g., estimation, problem solving, statistics, infusion of technology, and so forth) to be added to the traditional elementary school curriculum without removing anything. Thus, reformers have created a crowded curriculum without sufficient time to teach all content well, and when teaching the cluttered curriculum, teachers place too little attention on students as social beings.

Research Design: This paper elaborates on an invited address to the National Mathematics Advisory Panel (NMAP), in October, 2006 at Stanford University. My invitation was to speak on mathematics instruction. I narrowed my scope to focus on mathematics instruction in grades 3-5. The paper reviews and interprets past research and raises new issues.

Conclusions/Recommendations: After a critique of the NMAP report Good offers strategies for dealing with the so-called “math problem,” including eliminating some of the math taught in 3-5 grade classrooms, making mathematics more meaningful, and an increased use of active teaching. Finally, I suggest the need for field experiments that alter the normative or typical curriculum and instructional practices in measured ways.

In October of 2006 I delivered an invited address to the National Mathematics Advisory Panel (NMAP), which was meeting at Stanford University. My invitation was to speak on improving mathematics instruction. I narrowed my scope to focus on mathematics instruction in grades 3-5. Subsequently, I was invited by the editors of Teachers College Record to submit an article based on this address so that my views might be distributed more widely for critical comment. Since my presentation, I have reviewed an embargoed copy of one section of the NMAPs report on instruction. More recently I read two draft committee reports (instruction and curriculum) which became a part of the final report entitled, Foundations for success: Report of the National Mathematics Advisory Panel,  released on March 13, 2008.  The major goal of this paper is to present my ideas for improving students learning of math, especially in grades 3-5. However, at the conclusion of this paper I provide a few comments on the final NMAP report.

National Mathematics Advisory Panel

The NMAP was appointed by President Bush in April of 2006. The panels charge was to make recommendations for improving mathematics learning and capacity in the United States. The specific dimensions of the panel's charge, its membership, and the extensive processes used by the panel to obtain information can be found on the web at http://www.ed.gov/news/pressreleases/2008/03/03132008.html. The final report can also be accessed through this site. The National Math Advisory Group was divided into four major task groups; Conceptual Knowledge and Skills; Learning Processes; Instructional Practices; and Teachers and Teacher Education. Besides these major task groups there were three sub groups: Standards of Evidence; Instructional Materials; and A National Survey of Algebra teachers.

Comments to the Panel

In my opening comments to the NMAP, I noted that I had firsthand experience in educational reform as I had provided invited public testimony on two occasions as the National Commission on Education developed its historic report, A Nation at Risk. I noted that A Nation at Risk and many other reform efforts had not had the expected transformative effects on normative instructional practice in American schools. I wished NMAP members luck on their task, but noted that the history of the many failed reform efforts did not leave me with high expectations for their success. I also said that I was encouraged by their broad approach to reform (including, curriculum, instruction, assessment and so forth), which was not typical of past reform efforts.

I located myself in this reform issue noting that I am an educational psychologist, and obviously bring that professional disposition to my point of view.  In my presentation to the NMAP, I explained that beyond my academic training I have also spent many hours observing teaching and learning in math classes. And, that I have come to believe that good mathematics instruction reflects a variety of curriculum goals, as well as pedagogical skills, beliefs, dispositions, and mathematical knowledge of teachers; and students. Different instructional formats (individualized, small group, large group, technology-infused, and so forth) can provide effective learning environments. Students can learn from other students as well as their teachers. There are no panaceas or preferred formats per se that transcend allor even mostlearning contexts, and the quality of the format for teaching and learning is vastly more important than any given format itself (e.g. the quality of small group instruction is considerably more important than its presence) (Good, 1996).

I indicated that my purpose in addressing NMAP was to suggest why past reforms had failed and to provide my conception of the current math problem in grades 3-5, and what could be done about it. I stressed that my comments were focused primarily on instruction and that other important issues such as curriculum, teachers mathematical knowledge, teacher education, and professional development could be better addressed by experts in those areas, and noted that much important work was taking place in these areas (e.g., Hill, Rowan, & Ball, 2005; Reys, 2006; Kersting, in press).

The Intent of this Paper

In this adaptation of my NMAP address, I first describe the historical failure of reform efforts in American Education. I then briefly discuss reasons for this failure and note that calls for reform generally ask for too much change too quickly. I suggest that one major problem in recent math reform is that reformers have called for expanded math content (e.g., estimation, problem solving, statistics, infusion of technology, and so forth) to be added to the traditional elementary school curriculum without removing anything. Thus, reformers have created a crowded curriculum without sufficient time to teach all content well, and that when teaching the cluttered curriculum, teachers place too little attention on students as social beings. I offer strategies for dealing with the so-called math problem, including eliminating some of the math taught in 3-5 grade classrooms, making mathematics more meaningful, and an increased use of active teaching. Finally, I suggest the need for field experiments that alter the normative or typical curriculum and instructional practices in measured ways (elaborations on this program of field experiments with teachers has been suggested elsewhere; McCaslin et al., 2006; McCaslin & Good, in press; Good & McCaslin, in press). I end my paper with a brief critique of the NMAPs report.

REFORM DE JOUR: A PLETHORA OF REFORM EFFORTS

My conclusion that quality of instruction is vastly more important than its form is not revolutionary but it stands in direct contradiction to the many reform movements that emphasize a particular mode of instruction or a philosophy of learning, and is supported by considerable research evidence. Whole-class teaching or small group instruction can be dreadful or wonderful, yet reformers often insist upon the superiority of one single format. In education, fads come and go only to return again.  We have curriculum reform, then instructional reform, and so forth. Despite my argument that good math teaching takes many forms, the history of reform suggests that at different points in time only certain approaches to curriculum or teaching have been defined as good teaching practice by policy makers, teacher educators, or even foundations. Narrow conceptions of what defines good teaching continue today. At present, I believe the field suffers, in part, by its lack of respect for the role of active teaching in mathematics instruction teaching that involves helping students to understand the math they do through providing illustrations and clear explanations and helping students to apply this knowledge.

Past reforms in American education have had but little impact on classroom practice and this conclusion seems to be fairly common knowledge. Thus, a quick history lesson should suffice. I briefly review four important reform movements in American education: the Sputnik crisis, the Individualized Instruction/Open Classroom movements, the Nation at Risk report, and the current era of the No Child Left Behind legislation.

Specific Reforms: A Brief Review

The Soviet Unions launching of Sputnik in 1957 was widely perceived to demonstrate that science and mathematics instruction in American classrooms was so outdated and weak that it left us at military peril. The policy response to this threat was to radically reform the mathematics curriculum and to introduce abstract set theory (New Math) to whole classes of students as a solution to our scientific problems. As most of us know, New Math appeared only to suddenly quickly disappear. And it is arguable that we won the space war largely with scientists and mathematicians who were trained in the 1940'sand who had not studied new math. Some critics felt strongly that New Math was arguably more self-serving and political than driven by research evidence showing the need for a fundamental change for curriculum reform.

The purposes, effects, and failure of the New Math reform have been analyzed extensively (e.g., Moise, et al., 1965). The three quotes that follow capture a sense of the types of criticism directed at New Math. I especially like the third quote that suggests that the New Math was as much about advancing personal status and power as it was about advancing scientific knowledge.

Students are asked to learn operations with sets and the notion of subset, finite and infinite sets, the null set (which is not empty because it contains the empty set), and lots of other notations which are abstract and in fact rather remote from the heart and essence of arithmetic. Yet on this abstract basis, students are required to learn arithmetic. The whole theory of sets should be eliminated. On the elementary and high school levels it is a waste of time. (p. 30)

There are other differences between the modern mathematics curricula and what I and others would recommend. The modern curricula insists on precise definitions of practically every word they use and, by actual count, the first two years of the school mathematics study group curriculum asks students to learn about 700 precise definitions. This is pure pedantry. The common understandings which students have acquired through experience are good enough and formal definitions are usually not needed. After reading the formal definition of a triangle, I had to think hard to be sure that it really expressed what I knew a triangle to be. Another point of issue is symbolism. Because symbolism has made mathematics more effective many naïve mathematicians now seem to think that the more symbols they introduce the better the mathematics. What they have done is to make a vice out of a virtue. (p. 30)

In many ways the new math movement has the character of the childrens crusade of the middle ages. It is recognized as such by many responsible educators, but it is difficult to stop because of the very large and tightly knit web of bested interest preying on the mathematical unsophistication of the press, the public, and the foundations themselves. Under these circumstances, I urge administrators to forget the prestige of the sponsors and view with restraint and enthusiasm anything which does not really make sense to them, rather than be Sputnik-panicked into the hysterical adoption of new programs. (p.31)

In the 1960s educators became interested in creating more individualized instruction. Whole class instruction was seen as outdated and many contended that a better educational future would be created with use of emerging computer technologies that could individualize instruction and enhance learning.  However, disappointment with individualized instruction set in quickly and educators then advocated for a humanistic approach to education that would reduce student isolation and dependency on teachers and curriculum materials (see Good & Braden [2000] for extended comment). The more humanistic, open classroom movement asserted the need for choice and opportunity for self-direction that putatively would allow students to become more committed and, in time, more thoughtful and creative learners (Barth, 1970; Silberman, 1970).  As with the New Math, the open school movement also came and left relatively quickly, in part, because it was soon evident that given a smorgasbord curriculum (take what you want), many (high school) students made poor curriculum choices (Powell, Farrar, & Cohen, 1985). The shopping mall approach was soon replaced with reform that called for more adult control of structure and more specified content.

In 1983 the report by the National Commission of Excellence in Education entitled A Nation at Risk sounded the alarm that America was in economic peril because our students education was inferior to that found in Germany, Japan and elsewhere. This educational reform called for more instruction in core academic subjects, longer school days, longer school years, more home work, and so forth. The emphasis on A Nation at Risk was largely to spend more time on the same curriculum. Doing more of the same was soon seen as an inadequate reform. Yet, the economic war that spawned it was soon won by economists, business managers, psychologists and so forth who had graduated from American schools long before 1983 and, hence, had not received the educational value of the more curriculum.  

Currently, we are in the midst of another educational reform as defined by federal law No Child Left Behind (NCLB). NCLB was fueled by various claims including low student test scores, especially in comparison to those in other countries, large ethnic group differences in achievement, low quality teachers who are inadequately trained, outdated conceptions of high schools that allow students to be lazy, and learning standards that expect too little from students. Since we are in the middle of this reform, I will leave it to history to judge NCLBs effectiveness; however, my tentative judgment is not high and I share many of the concerns that others have expressed about NCLB and its possible effects (Nichols & Berliner, 2007; Glass, 2008; Linn, 2005; McCaslin, 2006; Stipek, 2006).

Why Have Reforms Failed?

Why have these reforms failed to meet their intended goals? First, these reforms have largely focused on discrete concerns: curriculum or teaching format, the quality of teachers' characteristics or their practices, student motivation or volition, school level reform or teacher level reform, teacher centered instruction or student-centered learning, enriched technology or not and so forth. Second, such reform efforts failed because single variables (and even single themes) do not have an independent effect on student learning. Since the late seventies I and others have argued that teacher characteristics are mediated by teaching practices, which are mediated by student characteristics, which in turn are mediated by those opportunities that students have to apply content concepts and so forth. The usefulness of a variable depends upon both its quality and how it fits into a learning system (Good, 1996).

In discussing the futility of studying single variables, I offer two examples. First, I discuss homework because it has been studied so widely and because it is recommended and condemned so frequently. Consider the role of homework in stimulating student learningis it needed or irrelevant? Homework might be useful if it is a format for practice that is informative to students and teachers about the degree of student learning (or the lack there of), and if the obtained information is then used to plan subsequent instruction in math. If ungraded or poorly graded and not used in subsequent teaching or testing, homework is at best irrelevant and can be harmful.

Research demonstrates that the effects of homework on student learning, and their attitudes toward learning and schooling, are extremely diverse (Cooper, Robinson, & Patall, 2006). How would you answer this multiple choice question?

Research shows that homework has what effect on learners?

A. It lowers students attitudes
B. It improves students achievement
C. It lowers students achievement

D. It improves students attitudes
E. All of the above

Given my previous comments, you likely know that the answer is E. This is because some research does exist to support the "truth" of A, B, C, and D in one or more conditions. Further, meta-analysis (Cooper, et al., 2006) and careful logical analysis (Corno 2000; 1996) allow for some understanding as to when and why homework does or doesnt improve student achievement and attitudes. Clearly, the usefulness of homework depends on several factors.

A second example is the believed need for math to be interesting or fun.  This reform variable has recently garnered much media attention, including some members of the NMAP. Should math be fun? The effect of fun on learning was recently considered in an international study (see Brown Policy Center, 2006). Why should math be fun? Is there theory or research to suggest that enjoyment and math proficiency are highly correlated? In my comments to the NMAP, I noted that I enjoy singing and listening to music but I do not sing well. I asked the panel members, Would you want me to sing to you now simply because I find singing fun? I continued, and noted that I am a reasonable writer, but do I enjoy all the proactive reading and rewriting that precedes a finished product? Did I enjoy preparing this paper? I did not. Apparently, some scholars have not studied the open classroom reform movement where fun or, at least personal choice was arguedonly to be rejected lateras a critical component of a major reform to improve American education. My point is not to poke fun at fun. I raise this issue to question why this variable was included in an international study. The literature is replete with examples to suggest that fun per se or positive affect, by itself, is not a self-sufficient conduit to achievement. For example, Oakes (2005) reported that lower-track students in junior high school and high school settings liked school and reported classes to be satisfying at the same level as did high- track students. Although highly desirable, student satisfaction with instruction does not by itself yield gains in achievement. Clearly, any single variable has meaning only as part of an instructional system. I submit that until we give up simple conceptions of the problem and its solution classroom practice will continue to be what it is and what it has been for sometime.

WHAT IS THE MATH PROBLEM IN GRADES 3-5?

First, I describe my conception of the current math problem in grades 3-5. Then I comment on a few instructional issues/opportunities in grade 3-5 mathematics classes that might enhance students learning of math (coupled with better teacher knowledge of math, better curriculum, better use of technology, better assessment, and so forth).

Time Allocation for Mathematics

Arguably among the most important predictors of learning are the opportunity to learn and the time needed to learn (Carrol, 1963). Given this important principle, it is critical to ask if we allocate enough time for mathematics instruction in grades 3-5 for the range of content recommended to be taught. I believe the answer is a clear No. Although many feared that the effects of NCLB would be to reduce the elementary school curriculum in the 21^st century to only the study of reading and math, these predictions were only partly correct. There is strong evidence to suggest that the elementary school curriculum has become a literacy curriculum. In one national study (NICHD, 2004), and in one large study in a single state (McCaslin et al., 2006) it was found that time spent on mathematics instruction in grades 3, 4, and 5 was about the same amount as spent in transition between subjects.

Robert Pianta and his colleagues (NICHD, 2004) describe what took place in a single day in 780 third-grade classrooms sampled from about 250 school districts. They found that over half the time available in the day was spent on literacy instruction. The ratio of time committed to other activities was found to be: .29 in mathematics, and in transitions .24, science .06, technology .03, and free time (where students pursued tasks of their own choosing) .008. In a study of grade 3-5 teaching in one state, Mary McCaslin and her colleagues visited 145 teachers on 447 occasions (McCaslin et al., 2006). Observations were organized into 10-minute intervals, and overall 2,736 ten-minute intervals of observational data were collected. Of these observation intervals, 587 were devoted to instruction in math and 1642 intervals were devoted to reading and literacy. Clearly students received much more instruction in literacy than in math.

The amount of time allocated for math instruction is further reduced by the fact that time during the math period is not always spent on instruction. Teachers vary enormously in their use of time and in some cases as much as 50% of available instructional time is spent on various things other than mathematics such as announcements, "house keeping" activities, and so forth (Berliner, 1979; Fisher et al., 1980; Freeman & Porter, 1989). There is no evidence to suggest that math teachers as a group have become more efficient in their use of instructional time. Thus, the amount of time spent on mathematics instruction/learning may be considerably less than that reported by Robert Pianta and Mary McCaslin and their colleagues.

Current descriptive studies of time allocated for math suggest that the allocation of time has not increased. Research results continue to illustrate that teachers vary notablyeven teachers at the same school and grade levelin how they allocate and use time (Rowan, Correnti, & Camburn, 2008). This issue of limited time is made more acute by the sharp increase in the number of topics that have been recommended to be added to the elementary school curriculum.

Expansion of the Math Curriculum

Time issues have intensified for grade 3-5 teachers because in the last twenty years more ambitious math content has been recommended for inclusion in grade 3-5 instruction. For example, topics and activities like estimation, measurement, problem solving, statistics, calculator usage, and computer usage have been added to the curriculum. Specifically, in 1989 NCTM made the following recommendations that new content be added to the K-4 and 5-8 grades:

The K-4 curriculum should include a broad range of content. To become mathematically literate, students must know more than arithmetic. They must possess a knowledge of such important branches of mathematics as measurement, geometry, statistics, probability, and algebra. These increasingly important and useful branches of mathematics have significant and growing applications in many disciplines and occupations.

An ideal 5-8 mathematics curriculum would expand students' knowledge of numbers, computation, estimation, measurement, geometry, statistics, probability, patterns and functions, and the fundamental concepts of algebra. (http//nctm.org)

The logic of adding more math to the K-8 curriculum has never been fully articulated. Seemingly these topics were added because they could be and because a powerful group like the National Council of Teachers of Mathematics (NCTM)¹ recommended that they be added to the curriculum (NCTM, 1989; 2000). Most importantly, the addition of these topics was not supported by evidence that the basic curriculum could be taught just as effectively with less time. Further, there was no evidence to support that introducing new topics like estimation would enhance later learning in a more powerful fashion. These topics were added to a full range of arithmetic functions and operations that were already in the curriculum and reformers did too little to specify what content was to be removed from the curriculum. One hypothetical example of the problem is depicted in Table 1.

Table 1 Time allocation for mathematics in a hypothetical grade 4 classroom
Activity	Percent of time allocated for math function	Percent of time spent on facts, skills, & concepts
Fractions*	15	12
Multiplication	10	8
Division	15	12
Computation	15	14
Review** (daily/weekly)	20	15
Testing	5	4
Transitions***	10	-
Geometry	5	4
Measurement	5	2
*Fraction work includes percents and related ideas. **The percent for review may be underestimated. Our naturalistic research found that some teachers spend an inordinate amount of time discussing homework.
*** Small group work may necessitate more transition time.

Table 1 illustrates how a hypothetical teacher in the early 1990s might have allocated time for math instruction during a typical year. These hypothetical estimations would vary from class to class depending upon variations in learners knowledge of math. For example, some third and fourth grade teachers would spend time reviewing subtraction work while others would not. Currently, teachers are asked to teach more content (e.g., statistics, estimation, pre-algebra skills, place more emphasis on geometry and problem solving, as well as using more technology), and to place more meaning on higher conceptual levels at which math is studied. As suggested by the second column in Table 1, most instruction in the 1990s would have been directed at computational issues. How would teachers then or now to find time to add content and to increase the cognitive level at which math is taught?

Given an assumption that time allocations for math at the early grade levels are fixed, adding new topics to the curriculum means that traditional topics like fractions (including ratios and proportions) will receive a smaller fraction of allocated time. So, in making the recommendation for more ambitious content, do NCTM, publishers, policy makers, and other educators believe that teachers were sand bagging and that teaching fractions and other content was so easy that teachers should be able to teach math materialsuch as fractionsmuch more quickly than they had been prior to the Nineties? Although the breadth of the recommended curriculum has expanded in recent years, time for mathematics instruction has not followed suit. Reformers suggest that too much time was used for practice and review and hence reducing time spent on these activities would allow time to teach new math content. However, a reduction in the time for practice and review may have made it likely that some, if not many, teachers would then spend less time on computational instruction and practice than teachers did 20 years ago. If so, it is not surprising that elementary school students computational proficiency has decreased (Brown Policy Center, 2006). This reduction in practice and drill was made on the advice of the NCTM but not on the basis of evidence.

In the past 20 years NCTM (along with many policy makers, and educators) has argued that several new strands of mathematical content need to be added and simultaneously (as opposed to first introducing statistics, and then turning to estimation content and so forth). Further, NCTM has advocated for more problem solving and higher order thinking, both as a new mathematical content strand and as an everyday lesson component that encourages students to spend much more time thinking about the math they do (e.g., why? how does it apply?). I will address this recommendation later in the paper. I believe that asking teachers to add more new content and to notably change the cognitive level at which it is taught has had less impact on typical practice than if a more measured approach to reform had been taken. The complexity of this reform recommendation, coupled with the absence of any data to support its value and effects on students seem unlikely to have motivated teachers to respond with their full support.

I can understand the frustration that math educators and other teacher educators have about certain enduring aspects of extant practice that could benefit from well-reasoned and well-measured change. But it is important for reformers to understand that much of what teachers do is reasonable and to recognize that some perceived problematic aspects of practice endure for logical reasons (Kennedy 2005; Lortie, 1977; Waller, 1932/1961). Reformers would be well served by seeking small incremental change (McCaslin et al., 2006; Weick, 1984).

This is not an argument against teaching more ambitious content; it is an argument for considering how best to allocate time for math instruction in grades 3-5. However, if certain topics like fractions and ratios have not been fully mastered by students, it seems counter- productive to infuse much new math content because this necessarily takes time away from core topics. Further, as Brophy (in press) has noted, much of what is in the American curriculum seems poorly justified and some pruning of the current math curriculum might be beneficial. But math reformers have not identified ways to reduce the curriculum to make room for new content.

If time cannot be increased or used more effectively, then the curriculum must be reduced. Spreading the same amount of instructional time over more and more content (even if time on practice is reduced) guarantees that teachers cannot reach, let alone teach, all the content included in the math curriculum. Increasing the time on math instruction by 10 minutes a day is a straightforward and inexpensive policy action that might have powerful impact. An equally powerful and considerably less expensive action is to delay some math topics to grades taught after 3-5 or to reduce emphasis upon some content markedly.

CONCEPTUAL LEVEL AT WHICH MATH IS TAUGHT

As noted earlier, many math educators wanted to change the conceptual level at which mathematics is discussed with students, as well as adding new content; and the need to increase the conceptual level at which math is taught has received at least as much attention as the need for adding new math content (and the evidence to show the benefits of shifting the focus is also scant). Porter and colleagues (1989a; 1989b) documented that the study of math in elementary schools was largely the study of bask skills and facts. Recent evidence suggests that efforts over the past 20 years to change the conceptual level at which math is taught were not successful.  Teachers appear not to have been persuaded to increase the conceptual level of their math lessons.

Data from the NICHD study (2005) of third grade classes (and an earlier national sample of first grade classrooms, NICHD 2003) revealed that the focus of instruction in most classes was basic skill instruction. For example, the ratio of basic skill instruction to analysis/inference opportunities was roughly 11:1. McCaslin et al. (2006) also found that the learning focus in grades 3-5 was on basic skill instruction. Wiley et al. (2006) differentiated these general findings to compare math and reading instruction. In mathematics, students were virtually never asked to engage in tasks that involved higher order thinking and reasoning; rather, students were 3 times more likely to engage basic facts and skills tasks than tasks that integrated basic facts and related thinking. So the normative (typical) math curriculum today in grades 3-5 focuses upon the study of math facts, skills, and concepts. An alternative intervention plan for teaching math at a higher conceptual level would have been to accept the content of the traditional math curriculum and to have gradually infused problem solving, problem finding, and thinking in dealing with the traditional curriculum.

IMPROVING MATH INSTRUCTION IN GRADES 3-5

To improve mathematics instruction, I argue that we need to make instruction as meaningful as possible. My suggestion is not novel, as math educators have long supported and continue to note the need to focus on meaning (Brownell, 1947; NCTM, 2007; Grouws & McNaught, in press; Hiebert & Grouws, 2007). Such an endeavor may mean that less content can be covered. Further, I argue that teachers need to place more emphasis in responding to students as social beings. Finally, I argue that to improve the mathematic curriculum, we need to recognize that much of the traditional curriculum endures; we need to understand why this is the case; and we need to develop ways to modify it gradually.

Make Math Instruction Meaningful

Given that considerable time is spent on facts and skills, it seems reasonable to argue that mathematics in grades 3-5 could be taught somewhat differently and focused on the meaning of the mathematics studied. Helping students to make sense of the math they do is potentially a powerful strategy for increasing students' understanding and ability to apply mathematics. The idea of meaningful instruction has deep historical roots in education as seen generally in the theoretical writings of John Dewey and in mathematics specifically (Brownell, 1947; NCTM 1989, 2007). And, there is evidence to believe that some time spent on understanding ideas and concepts, as opposed to only or primarily practice and review, can lead to better understanding and retention (Greathouse, 1996; Fraiving, Murphy, & Fusion, 1999).

Doug Grouws and I addressed how to make mathematics more meaningful and to increase students' learning in the mid 1970s. This research, supported by the National Institute of Education, became known in time as the Missouri Math Project (MMP). The conception of the research, methods, and findings are available elsewhere (Ebmeier & Good 1979; Good & Grouws, 1975; Good & Grouws, 1977; Good, Grouws, & Ebmeier, 1983). I make but a few comments about MMP here.

Doug and I addressed two goals in this project. First, we wanted to assess the degree of teacher effects on student learning. We began MMP in an era when scholars and policy makers alike assumed that individual differences in teachers were unimportant determinants of student learning. When Doug and I began our research we had no preconceived idea about the desirability of any teaching/learning format. In our naturalistic study there were teachers who used individualized instruction, who grouped students for math instruction, and who taught in a whole class format. We found that teachers varied both in their effectiveness from year to year in terms of student achievement and students perceptions of classroom climates. In selecting teachers to study, we wanted to find teachers who produced achievement effects that were relatively stable over consecutive years in their effects on students. As it turned out, those teachers who had relatively stable and relatively high achievement (and relatively low) effects primarily used whole class rather than individualized or group teaching methods.  Clearly, whole class teaching did not predict teacher effectiveness in this correlational study, as effective and less effective teachers used whole class teaching. We looked within the format of whole class teaching to establish a strong correlational link between teaching practices and student achievement. We then pursued the question of causalitycan these practices and beliefs be taught to other teachers in ways that will improve their students achievement in comparison to students in matched control groups?  In our experimental studies, we found that teachers who used the practices we taught them did have an important positive effect on student achievement. In todays language, the effect size was large.

In building our teacher training program we drew upon our correlational work that described how teachers who obtained high student achievement scores taught differently than teachers who obtained lower achievement scores from similar students under similar conditions. We also drew upon a small but powerful set of basic studies in mathematics that showed in several experimentals that student achievement was higher when the ratio of time spent on the meaning of the content was greater than time spent on practice (Dubriel, 1977; Shipp & Deer, 1960; Shuster & Pigge, 1965; Zahn, 1966). These data suggested that spending time initially to provide a rich conceptual base allows for meaningful practice and increased skill proficiency. Practicing mathematics teachers at that time, however, tended to spend most of their instructional time on practice. Our goal was to see if we could increase the time that teachers and students spent discussing the meaning of the math they studied, and then work with students to apply and practice the math they learned during seat work and in small groups. Our study demonstrated that teachers in general could implement most aspects of the treatment, except for the development of a rich conceptual base prior to application and practice. Clearly more work was needed on how teachers can best develop mathematical meaning in students, but we had made at least a small dent in the problem. Others have implemented MMP and have consistently reported positive impacts on student achievement in other experimental studies, and some have adjusted the treatment for successful application with older students (e.g., Sigurdson & Olson, 1992).  

However, Missouri Math is not an answer to enhancing students mathematical achievement. This was made clear in work by Howard Ebmeier and I that illustrated that teachers and students beliefs and needs mediated the impact of the treatment (Ebmeier & Good, 1979), and that classroom composition factors also mediated findings (Beckerman & Good, 1981). Although teachers using MMP were able to produce greater student achievement compared with teachers in control classes, our treatments worked better for some teachers and students than for others. These data were consistent with Cronbachs (1975) critical analysis of the lack of attention in educational research to context and student aptitude x treatment interactions (see e.g., Domino, 1968; Domino, 1971).  Different students often benefit differentially from the same treatment.

The publication of our findings was met with enthusiasm in many quarters, and indeed
too much enthusiasm in some cases as some concluded that our approach, which we called active teaching, was the way to teach math. Others rejected our findings out of hand and criticized our conception of practice as too narrow. Some of these concerns were legitimate but many were political; the notion of active teaching was not on the preferred how to teach menu that teacher educators served.  Teacher educators at that time and professional groups like NCTM (1979; 1980) were beginning to recommended more constructivist concepts of teaching and learning. Although teachers and policy makers were markedly favorable to our findings, teacher educators generally were dismissive. Teacher educators views of what desirable practice should look like differed from our findings, and hence many of them rejected (or at least ignored) these findings rather than clarified them through further research.

Although there were no compelling data by 1989, NCTM was firm in its advocacy for a constructivist curriculum:

The K-4 curriculum should actively involve children in doing mathematics. Young children are active individuals who construct, modify, and integrate ideas by interacting with the physical world, materials, and other children. Given these facts, it is clear that the learning of mathematics must be an active process. Throughout the Standards, such verbs as explore, justify, represent, solve, construct, discuss, use, investigate, describe, develop, and predict are used to convey this active physical and mental involvement of children in learning the content of the curriculum.

The importance of active learning by children has many implications for mathematics education. Teachers need to create an environment that encourages children to explore, develop, test, discuss, and apply ideas. They need to listen carefully to children and to guide the development of their ideas. They need to make extensive and thoughtful use of physical materials to foster the learning of abstract ideas. (http://nctm.org)

Whether intended or not, the NCTMs recommendations for active learning were erroneously interpreted as rejecting the role of active teaching in stimulating active learning. There is no reason to relive the 1980s and 1990s, but I do want to say that our basic claim was that the MMP project was a good way (though not the only way) to teach math concepts to fourth grade students. Further, by using two different performance means for classifying students (coupling students previous achievement along with self-report measures provided by students describing their beliefs and preferences about math learning to create student typologies) and one analytical assessment of teacher characteristics (a cluster analysis to provide teacher types based on teachers reports of beliefs and preferences about the teaching of mathematics), we found that differences in teachers and students preferences mediated treatment effects. Although largely ignored by critics, these published data showed that the MMP treatment was mediated by teacher and student beliefs and those findings invited basic research on how and why the MMP treatment could be modified to benefit more students (and other outcome measures, for that matter). I mention this because one criticism of MMP was that it was insensitive to teachers and students beliefs. My goal here is not to pull MMP off the shelf (not that in some contexts that would be a bad thing!); rather, it is to argue that the role of active teaching has been woefully underconsidered by many teacher educators, especially given the bloated math curriculum. However, as important as it is, active teaching is not enough. Like any format, it can be implemented well or poorly.

Active teaching is not enough. Active teaching or, as some now call it, explicit teaching, is dependent upon teachers mathematical knowledge and ways of teaching math that connect with students as learners and social beings. Active teaching also is dependent upon teachers ability to listen to students, respond to students questions appropriately, and co-regulate and scaffold student learning, as well as a myriad of social communication skills (e.g. conveying and developing a sense of community, pace, clarity, and so forth). These various dimensions that must be in place if active teaching is to be effective are discussed in detail elsewhere (Good & Brophy, 2008).

I discuss but one of these needed components herethe role of appropriate teacher expectations for student learning. Teacher expectations are relevant to the current reform movement, in part, through the widely acknowledged need to reduce ethnic differences in student test scores (Ferguson, 1998; Ladson-Billings, 2006), and for dealing with students who live in poverty and attend schools with few resources (Darling-Hammond, 2007; Good & McCaslin, in press; McCaslin & Good, in press).

Teachers' expectations for student performance as a function of students' achievement level, gender, ethnicity, social class and other student characteristics have often been found to
influence teachers' actions toward students. Teachers expectations can fuel differentially inappropriate opportunities in ways that can lower student achievement (Brophy & Good, 1970; Rubie-Davies, Hattie, & Hamilton, 2006; Raudenbush, 1984; Weinstein, Gregory, & Strambler, 2004). For example, students who are believed by their teacher to have less ability sometimes receive fewer and less substantial academic opportunities than do their peers who are perceived as having more talent. Students perceived as having more talent benefit from teacher confidence and expanded opportunities and perform better than they would have if they had not received their teachers higher performance expectations. Hence, teachers might create problems by having overly low expectations for certain groups of students. However, many differences between students are valid and reliable, and some students need more practice and structure than do others. Still, teachers need to be sure that all students understand the math that they do. As noted earlier in this paper, single variables have various effects. The issue here is that teachers need to develop appropriate expectations.

Math teachers need to ensure that low achieving students engage challenging work like problem solving and not just an exclusive series of drill and practice, or work that is separated from a meaningful understanding of what is being practiced. Finding the balance between time spent on practice and conceptualization varies with a multitude of factors including the age of students, the topic being studied, and the quality of instruction.

The belief that teachers make a difference has been argued on logical grounds (Gage, 1985; Good, Biddle & Brophy, 1975) and over time various researchers have shown empirically that teachers have a major impact on students' learning of subject matter (Good & Grouws, 1979; Good, Grouws, & Ebmeier, 1983; Hanushek, Rivkin, & Kaim, 2005; Nye, Konstantopoulos, & Hedges, 2004; Rowan, Correnti, & Miller, 2002; Sanders & Horn, 1994; Sanders & Rivers, 1996). However, most of the new evidence on teacher effects are black box studies that do not include measures of classroom process so there is no way to describe what happened differently in more and less effective teachers classrooms. Ironically, as the evidence that teachers make a difference on student achievement continues to mount, our ability to explain why and how teachers impact student achievement differentially has not improved.

We now know that teachers make a difference in student learning and, of all the school- related variables, the quality of teaching by far exerts the most powerful influence on student learning. Teachers who know math well and who can teach math well significantly enhance student math achievement. Although the claim that teachers make a difference now is accepted widely, if not universally, among researchers, citizens and policy makers, this has not always been the case. As recently as the early 1970s, researchers and policy makers believed that the impact of schooling (including teachers) on student learning was very small and that hereditary and home factors were the major determinants of student performance (Coleman et al., 1966). And unfortunately even today, as Rhona Weinstein has noted, some policy makers and educators still cling to a belief that intelligence is largely fixed and not malleable through life experiences (Weinstein, Gregory, & Strambler, 2004).

Our data on variation in teacher effects on student mathematics achievement suggests otherwise (Ebmeier & Good, 1979; Good, Grouws, & Ebmeier, 1983). Yet this experimental treatment is not a prescription for practice. Active teaching can be done well or poorly. However, it does seem that high quality active teaching is something that math teachers should be able to do while also using other high quality instructional formats in appropriate contexts. Also, active teaching² can be used to scaffold students ability to assume more responsibility for their own learning as shown in Table 2.

Table 2  

A Scaffolding Model of Teaching

1. Pre-engagement.  Guided by instructional goals and anticipating possible student difficulties, needs, and strategies, the teacher selects an appropriate learning activity. 2. Establishing a shared goal.  The teacher introduces the activity in ways that help the students see its value and thus adopt the goal of accomplishing its intended outcomes. 3. Actively diagnosing learners understandings and needs.  This requires teachers to be familiar with the learning domain involved, including knowledge of students typical trajectories of progress as they move from novice to more expert status. 4. Providing tailored assistance.  This might include modeling ideal performance, providing explanations, questioning or leading discussion, or coaching (cuing or prompting) performance. 5. Maintaining pursuit of the goal.  The teacher may need to support students to help them stay with a difficult or complicated task by requesting clarifications, asking questions, or offering praise and encouragement. 6. Giving feedback.  This includes both summarizing the progress made so far and indicating what still needs to be accomplished (including making students aware of mistakes and helping them to remedy them). 7. Controlling for frustration and risk.  This is done at a macro level by establishing a learning community in which mistakes are appreciated as part of the learning process, and at a micro level by giving feedback that helps students to cope with frustration and overcome mistakes as they carry out the activity. 8. Assisting internalization, independence, and generalization to other contexts.  This is accomplished by fading assistance as students gain expertise, by engaging them in activities that allow them to generalize what they are learning, and by providing cues, feedback, or other coaching that helps them to develop strategies for addressing problems or to recognize the implications and applications of what they are learning.

Source:  This table appears in Good, T., and Brophy, J., Looking in Classrooms (10th Ed.) (2008). Allyn & Bacon: Boston, MA. Adapted from Hogan, K., & Pressley, M. (1997).  Scaffolding scientific competencies within classroom communities of inquiry.  In K. Hogan & M. Pressley (Eds.), Scaffolding student learning:  Instructional approaches and issues (pp.74-107).  Cambridge, MA:  Brookline

What constitutes quality teaching remains under debate, in part, because of the ecological complexity of teaching and learning (Good, 1996; Good & Brophy, 2008; Hiebert and Grouws, 2007; Grouws & McNaught, in press). Yet, I want to stress that in the past 25 years I have seen mathematics educators increasingly argue for small group learning, learning communities, discourse communities and so forth, but the role of math teacher as presenter of knowledge has typically been painted as a negative practiceeven though there are ample data to show that sometimes telling students is sufficient, if not preferred (Schwartz and Bransford 1998; McCarthey in press; and Rosenshine in press). Apparently it has been acceptable for college math professors to present and explain math ideas, but not elementary school math teachers.

I acknowledge that encouraging students to be active learners is important and can be a very effective learning format under many circumstances (e.g., Fraiving, Murphy, & Fuson, 1999). However, when students become collaborators with and teachers of other students they have to be able to fulfill those roles. There is evidence to suggest that students can communicate low expectations to peers quickly and powerfully (Mulryan, 1992; 1995).  Also, Nuthall (2002; 2004) in his extensive field work noted that sometimes more confident but mistaken students cause their classmates to shift from accurae to inaccurate ideas. Many students' knowledge of math and teaching are inadequate to scaffold or to co-regulate the learning of their peers. Abdicating the teaching role to students is not a panacea and at times may be exceedingly poor practice.

The press for a uniform practice of constructivist learning and teaching has become prevalent in the last 20 years. Yet, advocates seldom stop to consider that this theory of teaching can be poorly implemented or that it can have differential effects on students of different ability levels. This was one of the major findings of Cronbach and Snow (1977) in their summary of extant research on aptitude-instructional method interactions.  Teacher educators have largely not heeded the advice of some scholars who have suggested that constructivist approaches are but one form of useful instruction, but rather confuse the form with the quality of the instruction. Richardson (2003 ) put it this way, ...In our enthusiasm for constructivist pedagogy and our advocacy of this particular vision as represented in national and state standards, in our teacher education classes, professional development, and call for reform, we may be imposing a dominant model of pedagogy on those who wish-and may have very good reasons-to operate differently (p. 1635).

In the past couple of decades, whole class or large group teaching has often been characterized as ineffective practice by many teacher educators. If teachers are told not to use active teaching, it is akin to telling a softball pitcher that she cannot use her fast ball and must only use a change up, a curve, and a rise ball. Why should teachers or softball players not use their entire repertoire? Active teaching is not a panacea, and it is important to understand its limitations and to recognize the useful role of the principles of social construction of knowledge for certain types of learning (Good & Brophy, 2008). Still, my goal here, in part, is to rekindle interest in active teaching as an important aspect of a teacher's repertoire. If teachers graduate from teacher education programs without extensive practice and informed feedback about their use of active teaching models, they have not been fully trained in the repertoire. Of course teachers need to know how to use other instructional formats like small group instruction (Webb, in press) and how to adapt instruction to address student differences within the classroom (Corno, in press). Others have much more expertise than I in how such formats and strategies can best be used and better knowledge about such formats can be found elsewhere.

Students as Social Beings

Teachers need more than a good grasp of math content and instructional procedural knowledge if they are to teach students effectively. Many other types of knowledge, dispositions, and beliefs are needed. For example, Tom Carpenter et al. (1989) showed how research on childrens thinking could be used by teachers to improve their students learning. Elsewhere, Lee Shulman has logically pointed out that content knowledge can be separated from knowledge of how to teach the content. Shulman (1986) called this knowledge pedagogical content knowledge &in a word, the ways of representing and formulating the subject that makes it comprehensible to others. He pointed out the need for teachers to possess knowledge about the types of examples, analogies, and activities that make content accessible and understandable to particular learners.

Math teachers also need to see that students are social beings as well as academic learners and to understand how to use that knowledge to enhance instruction (McCaslin, in press; McCaslin, 2006; McCaslin, 1996; McCaslin & Good, 1996). When Mary McCaslin, Barbara Reys, and I conducted a project on small group mathematics learning supported by the National Science Foundation in the 1990s, we conducted an intake interview with participating teachers (Good, McCaslin, & Reys, 1992). Grade 3-5 teachers had volunteered to teach content (statistics, geometry, estimation and problem solving) using small group formats. The interview assessed both teachers' knowledge of mathematics and their social knowledge of students (what are fourth grade students like?). We found that some teachers knew a lot about math and others a lot about students. Very few teachers knew enough about both. Although there are many reasons for this differential knowledge base (some teachers want to teach kids while others want to teach math) I think teachers can be effective only if they understand both mathematics and students.

Social knowledge concerns knowledge of students life experiences such as the media they consume and technology they use (Nichols & Good, 2004) and issues such as students affiliative needs within the social structure of the class or the group (McCaslin, et al., 1994). Mary McCaslin (2006) has noted math content and students' interests may become fused or confused; dividing the pizza correctly for some students is more about fairness than fractions. Awareness of social knowledge may help teachers to make decisions about what type of data topics to graph, what story problems to solve; and, for a group learning project in measurement, whether to develop a food court, a bank, or a ball park economy.

Researchers also have reason to be concerned about students as social beings as they recruit students to their research and then attempt to learn from them. For example, offering students a sweat shirt as an incentive is not as unimportant a decision as it may seem at first glance.  Many variables will mediate the attractiveness of the shirt. Does it have a hood? Is it the right color? Does it have a desired logo? Is that logo readily available elsewhere? In our recent research in Comprehensive School Reform schools, we found that the right pencil can be attractive to grade 3-5 students, the wrong pencil an insult (descriptions of this research can be found elsewhere (McCaslin and Good in press; Good and McCaslin in press). Researchers also need knowledge of students as social beings if they are to engage students in meaningful conversations and interviews to understand students mathematical knowledge, their likely interest in math or science as a career, and so forth. Similarly, if they want to assess student problem solving abilities, knowledge of students as social beings might aid in designing tasks that have appropriate levels of task difficulty and authenticity.

Normative Practice Endures

I believe that some aspects of normative (again, I use normative to mean typical) practice require change (excessive focus on drill, facts, low levels of cognitive activity, and too little chance for student exploration, choice, or contingent rewards and so forth). How should we change practice to improve student proficiency in math? I hypothesize that students are not spending enough time thinking about the meaning of the mathematics they study. They are not finding, solving, or creating problems. For the most part students deal with basic facts and skills. Is this bad? Not necessarily so. Facts, skills and concepts can be learned and applied in ways that enhance students understanding of math. Can what students learn be improved? I think so. If this normative picture described all instruction throughout K-16, the situation would be dire. But it doesnt, and it isnt. We know that in later grades students spend much time solving problems in algebra classes. In trigonometry classes students attend to relational aspects of math, and so forth. Thus, all math topics need not be addressed (let alone be resolved) in grades 3-5. Yet, some reform in grade 3-5 math instruction may be useful. Eliminating some of the math concepts taught would allow the cognitive level at which math is taught to be increased in some contexts. It might also generate some time to work on problem solving using basic skills in some contexts with an emphasis on thinking as well as computational proficiency.

We know that certain teaching practices have endured for over a century in various school subjects (Cuban, 1984). Why might math teachers have resisted contemporary reform recommendations? I suspect, in large measure, they find that teaching traditional arithmetic consumes most of the available math time. With limited time, teachers may believe that large group instruction is more effective than formats that require the grouping of students. This may be particularly the case when teachers engage students in problem solving and dealing with more ambitiousand ambiguousmath, such as general problem solving, estimation, and probability. As noted, recent data on elementary school math instruction suggests that highly touted mathematical reforms have not impacted normative practice (McCaslin et al., 2006; NICHD, 2005), and we need to consider why this is the case.

Although teachers may find aspects of a reform attractive, they may also believe required changes and the speed with which they are enacted will not be more effective than their current practices. So why spend time developing new lessons, and activities, and modes of evaluation? Typically, a new curriculum or instructional method is first touted in teacher education programs or some professional organization, and then some high-profile research occurs in a limited number of schools. But, by and large, as Larry Cuban (1984) has long noted, the typical teacher continues practice that is largely unaltered by the newest prevailing fad. Each reform is costly. New textbooks and materials are produced and teacher education students spend much time learning an approach to math instruction that (if enacted) is then abandoned in a few years. Part of teacher rejection of the new reform is that it arrives on the basis of assertion not evidence. More complex discussions are available as to further describe why teachers resist reform (Coburn, 2001; Spillane, 2004). I conclude that reform recommendations can involve too much as well as too little; however, historically, reformers have erred by attempting to do too much too soon. I suspect that changing normative practice can best be seen as a series of small steps that allows for major progress over time. And, these changes must occur with teacher support and the reform process must encourage teacher thought and innovationnot simple compliance (Randi & Corno, 1997).

Changing Normative Practice

Teachers can only teach so much content in the allocated time; decisions need to be made about what math to teach. Further, if our goal is to change the math curriculum in ways that increase student learning and understanding of math, then we need to ensure that somewhat more time is spent considering the meaning of math (and, when appropriately designed, practice and review can contribute much to this goal); we need to allow and enhance the use of active teaching when appropriate, and we need to encourage teachers to acquire and use knowledge of students as social beings.

Based on our recent research program Mary McCaslin and I (Good & McCaslin, in press; McCaslin & Good, in press; McCaslin, et al., 2006) we have hypothesized that many Comprehensive School Reform (CSR) classrooms could be enhanced by the provision of more opportunities for students to apply knowledge and to elaborate on facts, concepts, and skills intended to sharpen their understanding of the content they study. We suspect that these same issues apply in grade 3-5 instruction in many non-CSR classes. More opportunities for student choice also seem important so that students can learn how to make choices, develop their interests, and meet the responsibilities that come with choice. Expectations for student understanding and responsibility can be too high as well as too low, and achieving the appropriate balance between breadth and depth, and other and self-direction is no easy task. Instructional balance requires more research and development. Teachers involved in these research and development efforts should know that they are involved in the search for balance.  

As argued elsewhere even a relatively small shift in instructional practice that attempts to balance discrete and elaborated knowledge could yield a notable gain in student learning and understanding (McCaslin, et al., 2006). What constitutes an enabling balance, however, is likely to vary as a function of such factors as teacher knowledge of mathematics and beliefs they hold about teaching math, student prior knowledge and beliefs and dispositions that students hold about learning, subject matter, grade level, and the type of learning outcome sought (Connor, Morrison, & Katch, 2004; Connor, Morrison, & Petrella, 2004; Good & Brophy, 2008; Hiebert & Grouws, 2007; Ebmeir and Good 1979; McCaslin & Burross, in press). Further, it is useful to recognize that Van Horn and Ramey (2003) found that the most successful programs for younger students were generally more structured.

THE NATIONAL PANEL REPORT: REFORM DE JOUR 2008

On March 13, 2008 NMAP released its report, Foundations for Success: Report of the National Mathematics Advisory Panel³. I comment briefly on two group task reportsthe Instructional (Task Group on Instructional Practices) and Curriculum (Task Group on Conceptual Knowledge and Skills)and on the report itself. It is beyond the scope of this paper to summarize the NMAP's report except in the briefest terms. As mentioned earlier, the report itself is readily available (http://www.ed.gov/about/bdscomm/list/mathpanel/index.html) and brief summaries are also available (e.g., Cavanagh 2008). However, it is useful to note that, in my words, the report concluded that the curriculum was cluttered and needed to be refocused. In broad terms, the call was for the current curriculum to be narrowed and to become more focused by moving from a major focus on conceptual skills to a more focused effort on learning, understanding and using basic arithmetic. The previously discussed preference as argued earlier by the NCTM for infusing new and different math into the elementary school curriculum was sharply reduced (if not rejected). Rather, the report called for more attention to fractions (including negative fractions, proportions and ratios). The rationale for expending more time on the teaching of fractions, decimals, and percents was based on the belief that student mastery of these topics was not adequate at present. NMAP presented the idea that such knowledge is required for success in algebra, and especially algebra II, which is seen as the primary gateway for more advanced work in mathematics. Hence, NMAP declared that the cluttered and unfocused elementary math curriculum needed adjustment if students were to be more successful in algebra and beyond.

The NMAPs report will receive wide distribution and undoubtedly much commentboth positive and negativefrom a wide array of audiences. Time will tell if this reform is to have more impact and value than have previous ones; however, I have already expressed my low expectations for the amount of positive, direct impact that this report will have on enhancing mathematical literacyeven though, on balance, I think it is a good report.

I make three comments about the instructional report. First, I acknowledge the reports clear and strong recognition that teacher-centered instruction can enhance students understanding of math. However, I lament that the instructional panel continues to confuse the format of teaching/learning with quality. I devote most of my discussion to this point. Secondly, the report inadequately addresses the social aspects of the student being taught, and third, it brings little theoretical or research coherence to the issue of improving mathematical capacity in the United States. Finally, I briefly comment favorably upon the curriculum panels report on the need to narrow and refocus the math curriculum and then note the need for more integration across the individual task groups.

Instruction Report

The committee reviewed much research and analyzed many instructional issues. Importantly, the committee concluded that no research supports an exclusive teacher-centered or student-centered approach to instruction. This is a laudable and important conclusion.

However, one disappointing aspect of the instructional report is its continued reliance on the teacher-centered instruction or student-centered instruction dichotomy as an analytical lens. I have visited many mathematics classrooms and this distinction is more illusionary than real; often, lessons are a blend of instructional approaches. For example, the teacher chats with students about division with remainders while reviewing the previous two weeks of work. The conversation with the class takes 30 minutes with the teacher talking 90% of the time. A couple of times the teacher asks the fourth graders to individually work out a couple of problems and occasionally she asks, When do you use division? Why? Are you sure? Then for the last 15 minutes the teacher says, Now I want you to write two sentences to describe when you might use division yourself and then write a problem for someone else to solve. Start to work now and finish as much as you can. Today's homework is to finish what you start now. Tomorrow we will trade problems! Is this teacher-centered instruction or is it student-centered instruction? If a teacher does whole class skill work one day, say, using mental math, Whats 11+6-2? Jane thats right! Now Dick, add 20 and divide by 2. And, then, after this mental arithmetic exercise the teacher uses the remainder of the period for individual drill work, then the next day students work cooperatively in four person groups for the entire period solving word puzzles where the solution method is not immediately obvious, is this a teacher-centered teacher or not? Do we make that distinction within lessons? Between lessons? Or on the basis of an instructional week? Is it teacher- or student-centered instruction based on use of concrete activity manipulatives, so that if students manipulate something its student-centered? If a teacher question stimulates student thinking, then is that student-centered? If all students manipulate the same object in the same way, is that student-centered (Bozack, Vega, McCaslin, & Good, in press).

Many years ago I visited a science educator in Australia who was developing a program to move instruction from whole-class to an individualized format. Videotapes were available showing instructors before and after the treatment. In one tape I saw an instructor standing in front of the class with a test tube and various chemical containers. He said, Now I am pouring&. One year later the same instructor taught the same lesson to a new class. Now all students have their own test tubes and chemicals in front of them. The teacher said, Now I want you to pour.... Have we now moved from teacher-centered to student-centered learning? I think not. But would it have been teacher- or student-centered instruction in year 1 if the teacher had largely demonstrated procedures but asked questions along the way, such as: Now what chemical do we add next? Each of you jot down what comes next and why. The teacher waits an appropriate amount of time and then says, Caitlin, what should we do and why? After Caitlin responds, the science teacher asks, "Randy, do you agree with Caitlin? Why? Assume in this 45-minute period the teacher asks four such questions, talking 38 minutes out of the 45. What type of teaching is this? What if the teacher talked 15 minutes and asked 15 questions?

The failure of the task groups report to discuss different types of teacher-centered and student-centered instruction and to then suggest when one format might be most useful is disappointing. Even more disappointing is the lack of work on instructional theory and model building in ways integrating teacher-centered and student-centered approaches.

A second criticism is that the report places little emphasis on the need to understand students better as social beings. I have discussed this previously, and will not elaborate this point further here. I simply note that the need for enhancing our conception of what it means to be a student was not part of the report and this seems tragic given both the increasing and changing diversity of students in American classrooms. The panel does not consider that different types of students can and often do benefit differently from the same type of instruction. Although there are sections on teaching gifted students and on teaching students who have experienced difficulty in learning mathematics, and general issues of student ethnicity, there is no discussion of students as active social beings who live in particular communities, who have identifiable interests, and so forth.

Finally, my third criticism is that the report does not bring to light new theoretical conceptions or even present a clear description of what is the instructional math problem. Even if clinging to the teacher-/student-centered dichotomy, it seems potentially important to relate particular methods to the potential for helping students understand particular mathematical content. Are there times when teaching fractions from a specific instructional approach might yield unique advantages? If fractions are a gateway to more advanced math, what do we know about how fractions are taught? or how they should be taught?

To reiterate, I do appreciate that the instruction task group made it clear that teacher-centered instruction is at least a potentially useful instructional method, and thus addressed directly the long-standing trend among teacher educators who have largely rejected teacher-centered instruction. Further, any reader of this report will find much helpful descriptive information that will be useful to them. My criticisms are addressed at the analytical level of the report and its failure to articulate an integrative directionespecially after reviewing many studies examining the effects of mathematics instruction on student learning. But, in fairness, perhaps my expectation is inappropriate as I may be asking the committee to do more than it reasonably could within the time and resources it possessed.

Curriculum Panel

I was very pleased with the conceptual knowledge and skills task group report. It offers a sharp call that the currently crowded (I have often referred to it as cluttered) math curriculum must become smaller and more focused. The call for teaching less content is a substantial response to the current math problem. Further I like the panels call for the elementary curriculum to focus on three content areas: fluency with whole numbers, fluency with fractions, and particular aspects of geometry and measurement. Given a more focused and narrower set of content to address, teachers and students will theoretically have the needed time for conceptual development, appropriate practice, reflective review, and application. I am glad this sub-group recognized that little real problem solving will occur if students are not mathematically fluent in whole numbers and fractions. My major disappointment with this report is its lack of any extended discussion of how and why the math curriculum became so cluttered. The failure to address the reasons that caused the cluttered curriculum in the first placeand the reduced role of fractions in itmay leave many teacher educators still recommending the inclusion of more ambitious math content. It would have been important to have addressed questions such as what led to the misconception that new content should be added to the elementary school math curriculum, and why that view has now changed.

NMPA

In terms of the entire report, I would have liked some effort to combine and to make sense of what the individual reports might collectively have provided. The final report simply summarizes each sub-panels report and does no integrative work. The separate reports would have benefited from a synthesis linking parts together. In some ways the separate sections are akin to possessing separate reports on horse care, protecting the needs of cowboys, preventing cattle disease, and finding dependable trails with no integrating piece on how the trail drive assumes meaning and direction. As a case in point, the instructional panel included an extended discussion of problem solving instruction (and much of this is very informative). In contrast, the curriculum report discusses the need to refocus on the traditional math curriculumespecially fractionsand does not discuss problem solving. Thus, nowhere in the report is there a discussion of how to teach fractions effectivelywhich is a key component of the report. I understand that the instructional panel could not have foreshadowed the curriculum's emphasis far in advance, or vice versa, but some attempt at integration across the panel reports would have been useful. In the absence of any integrative framework, consumers of the report are more likely to pick and choose among the separate reports rather than to attempt a more comprehensive approach to reform.

Further, the report might have placed itself in relation to other broad issues. Discussion of the disparities in the math curriculum from state to state (Reys, 2006) and a discussion of the advantages and disadvantages of a national curriculum (Porter & Polikoff, in press) would have been important issues to mention. Can the NMAPs advice be implemented without a national curriculum or some reduction in state to state variation? Finally, the report is not grounded or informative about the coersive effects of poverty on student learning (Good, & McCaslin, in press; McCaslin & Good, in press).

Having critiqued aspects of NMAP (its instructional and curriculum task group reports), I do want to recognize the energy and countless hours that the panel members, and the multitude of individuals who provided input to them expended in preparing their reports. Their active involvement  represent an important symbol of the high value of mathematical knowledge that is held today by policy makers, citizens, and educators. Teachers, students, the media, other constituents, and readers of this report can take comfort in this public embracement of the immense value of enhancing mathematical capacity today and tomorrow. As noted, I do not believe that this report will transform mathematics instruction in America. However it provides momentum, language, and knowledge to add to a continuing evolution of understanding, valuing, and improving mathematical literacy. It also promotes a continued discussion about how to improve mathematics instruction in ways that enhance the social and economical aspects of our lives. The call for a sharper and more limited definition of the elementary school math curriculum is laudable. However, it is not clear that the voice of Foundations for Success will trump the voice of the many teacher educators who have called for a more ambitious curriculum with more advanced content.

CONCLUSION

I have placed the current reform document entitled Foundations for success: Report of the National Mathematics Advisory Panel, in the context of past reform movements. I reviewed selective past reform efforts that had little impact on instructional practice. I suggested why reform efforts have failed and noted that among other factors reform efforts failed because they asked teachers to change too much too quickly and because they failed to recognize and build upon adaptive aspects of the normative (typical) curriculum. I defined the current problem of the math curriculum and suggested ways to address that problem.

I believe that normative practice can be enhanced in small ways over time, and can have powerful effects on studentsand eventuallythe nations mathematical capacity. These small but continuous steps need and in accordance with NMAPs call for a reduced curriculum to be made collaboratively with teachers in field experiments in which core design features are implemented generally but with enough room for teachers to adapt the general treatment to their context. Teachers must understand the social aspects of their students as well as their mathematical knowledge. Application of both types of knowledge is needed if students are to develop more mathematical knowledge and skills and to develop the dispositions to value learning mathematical concepts and to actively apply that knowledge in intelligent, creative and appropriate ways.

Finally, I have reviewed the NMAP⁴ report and I have noted certain limitations, especially its lack of attention to the need for teachers to have rich knowledge of students as social beings as well as math learners. Further, the continued use of tired and useless dichotomies such as teacher-centered vs. student-centered instruction is not helpful. As I noted the panel's biggest contribution may be the symbol it represents in terms of the value that the nation places on mathematical learning. And, this report will provoke much debate. Clearly I hope some of these deliberations will focus on a more measured approach to enhancing mathematical reform (including NMAPs call for reduced math content in elementary schools) in ways that retain the best of normative practice, addresses its weakness, and to integrate missing parts into normative practice.

Acknowledgements

This is an edited and somewhat expanded version of an invited paper presented at the National Mathematics Panel (NMAP), at Stanford University in 2006. I appreciate the extensive editorial assistance of Alyson Lavigne Dolan, Amanda Rabidue Bozack, and Toni Sollars in preparing the paper as well as the editorial advice of Lyn Corno and three anonymous reviewers who examined the paper for the Teachers College Record. Thanks to Jere Brophy who provided helpful feedback and special thanks to Mary McCaslin for her helpful criticisms of several previous drafts of this paper.

Notes

1. Given space limitations, I have focused mainly on NCTMs role in advocacy for sweeping changes in what math should be taught and how. However, in fairness,  NCTM actions often followed conferences sponsored by the National Science Foundation to study the math curriculum, and/or after parent/public protest. For more details on the political, professional, & business (e.g., publishers) influence on the math curriculum, see Grouws and McNaught (in press).

2. There are many terms that have been used to suggest a role for teachers active involvement in presenting information (e.g., explicit teaching, active teaching, direct instruction, etc.). Direct instruction typically has been seen as teachers telling students what to do. I believe that direct instruction is sometimes sufficient (see Rosenshine, in press). Active teaching, in my mind, is a more focused attempt to impact students understanding of math as well as just learning it.

3. The final report is 90 pages. Also available are the final draft reports of each sub-committee task group. Some of the individual reports are longer than the final document. Readers may want to read these reports for greater detail and to see what parts were dropped in the editing room.

4. Please note that my comments about the report refer only to the two sections that I have examined. These are the Report of the Task Group on Instructional Practices and to a considerably less extent the Report of the Task Group on Conceptual knowledge and Skills.

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