TEACHING AND LEARNING

TEACHERS’ ATTITUDES AND PERCEPTIONS ABOUT MATHEMATICS TEACHING AND LEARNING

Teachers’ own beliefs about mathematics, how children learn mathematics, and what constitutes good teaching affect the way teachers choose to teach mathematics and what they choose to teach. Research has shown us that many graduates of teacher education programs still end up teaching the way they were taught as students (usually in a traditional manner) despite the quality of their teacher education program. This must change in light of our changing society and the current research on effective mathematics teaching and learning. It is a major challenge for school administrators if a teacher does not believe that change is necessary in the mathematics program.

Teacher beliefs, and the choices teachers make, can have a major impact on how students view mathematics and their learning of it. According to research, it is suggested that teachers’ beliefs about mathematics are often limited and may be dualistic, in the sense of having a traditional right/wrong orientation and using mostly single procedures to arrive at the correct answer. A consistent theme found in Cooney, Wilson, Albright, and Chauvots (1998) RADIATE study was that teachers equated good teaching with good telling. In other words, students should understand mathematics step by step and should not be confused. A second  theme that was found was that of “caring.” Because teachers cared about their students’ success in mathematics class, they felt that caring meant enabling students tomaster basic skills, often putting aside challenging tasks on assessments for those that mimicked the traditional skill-based lessons done in class. This is a reductionist

orientation that is counter to reform efforts in mathematics.

Baroody (1998) provides a summary of research on three different views of mathematics that have been identified among teachers:

1. Mathematics as a collection of unrelated basic skills

2. Mathematics as a coherent network of skills and concepts (mathematics as a static body of knowledge)

3. Mathematics as away of thinking (inquiry process,mathematics as a dynamic field)

Knowledge and beliefs are inextricably intertwined. Our beliefs are like a filter through which new phenomena are interpreted. A teacher’s sense of purpose as a mathematics teacher, philosophy of learning and teaching, and sense of responsibility in terms of the community in which he or she teaches are all fused with what the teacher “knows.” As well, it is important for teachers to be reflective practitioners. In the case of mathematics, teachers need to see mathematics as a creation of knowledge rooted in rationality. Mathematics knowledge is not static; it is fluid. Context and reflection play an important role in allowing the knowledge required by reform to be fluid and flexible. Both “what the teacher knows” and the way the knowledge is acquired are important issues.

Administrators can push teachers to change their classroom activities, but we also need to change their fundamental beliefs and attitudes about teaching and learning, the roles of teachers and students, and how teaching and learning should be carried out. For change to be successful, teachers’ beliefs, attitudes, and practices need to be aligned. It seems logical that influencing teachers’ beliefs may be essential to changing teachers’ classroom practices. At one end of the beliefs continuum are traditional beliefs. Stipek, Givvin, Salmon, and MacGyvers (2001) found that teachers who scored high on these more traditional beliefs were less self-confident about teaching mathematics and enjoyed it less. In their data analysis, five dimensions of beliefs (more traditional beliefs linked to teachers’ being less confident about teaching mathematics) were strongly associated with each other:

1. Mathematics is a set of operations to be learned.

2. Students’ goal is to get correct solutions.

3. The teacher needs to exercise complete control over mathematics activities.

4. Mathematics ability is fixed and stable.

5. Extrinsic rewards and grades are effective strategies for motivating students to engage in mathematics.

If one looks at the opposite end of the dimensions (reform based beliefs linked to teachers’ being more confident about teaching mathematics), there was consistency in the following beliefs:

1. Mathematics is a tool for thought.

2. Students’ goal is to understand.

3. Students should have some autonomy.

4. Mathematics ability is amenable to change.

5. Students will want to engage in mathematics tasks if the tasks are interesting and challenging (not for extrinsic rewards).

The authors speculate that building teachers’ self-confidence in mathematics (which requires building their mathematical understanding) could be an important and perhaps necessary criterion in moving teachers toward more inquiry-oriented beliefs and practices. If this suggestion is valid, the school administrator’s challenge in this area is to be able to provide the intensive, sustainable professional development required to improve teacher understanding of mathematics, to improve confidence in the subject area, and to change beliefs and attitudes about teaching and learning mathematics.

TEACHER CHANGE IN MATHEMATICS

As a school administrator supporting your school community in its mathematics instruction and assessment initiatives, it is important for you to be informed of current research in mathematics education. The school administrators as well as the teachers must have some understanding of the change process in education. Research has shown us that the one-shot, workshop-based professional development opportunities that teachers have usually been offered may result in some improved mathematics in individual classrooms. However, necessary long-lasting, school wide change calls for substantive, ongoing, school wide support with continuous formal and informal job-embedded learning strategies being used with your school faculty. For real change to happen, schools need to plan carefully. According to Michael Fullan and Andy Hargreaves (1992), there are three parts to the change process:

1. Providing new materials. This is the tangible part of a change innovation and relatively easy to accomplish.

2. Introducing new behaviors and practices, or the “doing” part of the change process. This is introducing and supporting the different pedagogical style, skills, and practices in which a person will be involved. This would involve changing a teacher’s mathematics instruction from a traditional skill-based program to a more balanced, reform-based program based on rich learning tasks, problem solving, and deep conceptual understanding.

3. Embedding new beliefs and understandings. This is what makes innovation happen—where one internalizes and understands the rationale for the change. This understanding is very important when making the decision about whether or not to implement the change and how to use it. This third part is the crux of the change process in mathematics education. It is absolutely imperative that school administrators, teachers, and the school community have an understanding of the reform movement in mathematics and the change process itself, as both are essential for any meaningful change in mathematics teaching and assessment to happen. The decades of the 1990s and the first decade of the new millennium have had the ongoing theme of “large-scale reform.”We have seen those testing initiatives change in a variety of ways over the past decade. The data sometimes show improvement, but they never seem to show enough improvement. The question is, how do we do better at closing the achievement gap?What qualities do our educational leaders and faculty need to best drive the improvement agenda for a sustainable period of time? “Sustainability is the capacity of a system to engage in the complexities of continuous improvement consistent with deep values of human purpose” (Fullan, 2005). Fullan goes on to explain that sustainability isn’t just about ensuring that an initiative lasts a long time; it also addresses the fact that new initiatives may need to be developed without compromising existing initiatives. As a school administrator, you face the challenge of deciding which initiatives are worthwhile and necessary for a school community and then moving the school in that direction. Once you have decided, together with your staff and school community, that mathematics is a focus for your change initiative, it has been shown that the success or failure of an education program is determined more by whether it upholds or challenges the school community’s beliefs than by the number or breadth of the changes involved in the initiative.

As stated above, teachers’ values, beliefs, and reflections are important in the change process. Several elements have been observed in studies of successful educational reform: shared leadership, awareness of need, adult interaction with each other, ongoing commitment in spite of conflict and tension, desire to learn, and parental support. Elizabeth Smith Senger (1999), in her study of three elementary teachers who were struggling with issues of reform and traditional mathematics teaching in relation to their personal values and beliefs, found some observed paths of teacher change:

The first, newly gained awareness, was initially held without commitment in a tentative questioning mode and was pooled from various sources. Essentially, the teacher has simply gained an awareness of the need for change in his or her teaching; however, no change or commitment to change has been made at this point. If the teacher did not reject this new information and awareness, with time and reflection, produced mental images of new forms of teaching practice (pre-images) are formed. The teacher is thinking about new or different ways to teach and is beginning to visualize them. The pre-images inspired a double-faceted experimentation which involved both verbal and classroom practice trials. The teacher willingly begins to experiment with teaching methods as a result of the visualizations. Verbal experimentation involved the teachers using expressions and descriptions of their pre-images as a means of expressing and assessing several aspects which included:

(a) Their own comfort level

(b) Their confidence that the ideas would work in their classrooms

(c) The ways in which the ideas “fit” with the teacher’s own past history

(d) The teacher’s reputation with her colleagues as “traditional” or “reform-oriented” in terms of initiative and

mathematics teaching. (Senger, pp. 210–211) Senger’s data revealed that the integration of a new belief occurred as a thoughtful and complex process over time, which included imaging, experimentation, and reflection on those values.

One complication for teachers when attempting to implement change in the area of mathematics is that,unlike teaching other subjects such as literacy or history, most teachers face the extra challenge of not having a solid understanding of the mathematics content itself. Borko,Mayfield,Marion, Flexer, and Cumbo (1997), in a study of a group of third-grade teachers who participated in a University of Colorado assessment project on designing and implementing mathematics performance assessment tasks, found the following supports and impediments to teacher change:

Theme 1: Situating the change process in the actual teaching and learning contexts where the new ideas will be implemented is an effective strategy in helping teachers change their practice. This is job-embedded learning in the teachers’ own classroom and school with their own colleagues.

Theme 2: Group discussion of instructional and assessment issues can be an effective tool for the social construction of new ideas and practices. Just as students need to have math conferences or a math congress, so do teachers. This discussion and sharing of practice helps teachers solidify their understanding of the mathematics being taught and their knowledge and understanding of what and how their students are learning. It also helps build confidence so that teachers are willing to continue with the change process.

Theme 3: Staff development personnel and other persons with specific expertise can facilitate change by introducing new ideas based on teachers’ current levels of interest, understanding, and skill.

Theme 4: When teachers’ beliefs are incompatible with the intentions of the staff development team and are not challenged, the teachers are likely either to ignore new ideas or inappropriately assimilate them into their existing practices.

Theme 5:Time is a major obstacle to changing classroom practice. Competition among priorities for limited classroom time is particularly troublesome, as well as for time for necessary job-embedded staff development that occurs during the school day. (pp.14–26) Researchers also acknowledged that teachers’ knowledge and beliefs about learning, teaching, and mathematics subject matter are critical in determining how and whether teachers implement new educational ideas. If teachers believe in traditional teaching (put simply, the teaching of isolated rote skills out of context and not connected to real-world problems) and ignore the research on mathematics reform and on how students construct learning in mathematics, then it is highly unlikely that teachers will change their practice.

Another major factor in the change process is trust. For trust to be developed in a school, both teachers and leaders need to have discovered that it is safe to take risks and chances and that it is safe to make mistakes. The staff need to feel that they are working in a protected learning environment where mistakes will be made but that they will be learned from and that the small successes will be celebrated along the way.

McLaughlin and Talbert (1993) suggested that, for teachers to adapt their teaching practices to meet the new reform agenda, they must participate in a professional learning community that supports risk taking and that discusses new teaching strategies and materials. Borko et al. (1997) also discuss the issue of time in educational reform. They suggest that staff development programs be at least one year in length and that they provide release time for teachers. Change efforts must take into account the feeling that classroom time is insufficient for teachers to accomplish what they feel they need to with the students.

Vision of Learning Mathematics - A Perspective

The mathematics that one needs to know has shifted. Technology has been a major factor in changing our homes, workplaces, and daily lives. As new technological applications emerge, new mathematics is being created. Has the mathematics teaching and learning in your school changed, or has it remained relatively the same as when you went to school? Do the mathematics programs in your schools still rely heavily on traditional procedures, such as “Yours is not to reason why, just invert and multiply?” Or are teachers and students engaged in mathematics for understanding? Mathematical power can, and must, be at the command of all students in a technological society. Mathematics is something one participates in and does, sees, hears, and touches in meaningful ways. It has broad content encompassing many fields. It is imperative that mathematics teaching and learning in schools is changed to reflect the current research and the changes in society in general.

The vision of mathematics today includes beliefs that students should learn to value mathematics. This new vision espouses that the teaching of mathematics well calls for increasing our understanding of the mathematics we teach, seeking greater insight into how children learn mathematics, and refining lessons to best promote children’s learning. Students should be able to reason and communicate mathematically. Problem solving is a life skill that students will need to be successful in their daily lives and in their chosen careers. In today’s mathematics classrooms, students and teachers should be part of a mathematical community working together to solve problems rather than only being involved in independent work. Students need to use logic and mathematical evidence to provide verification for correct answers, rather than the teacher being the sole authority in mathematics. Mathematical reasoning needs to become more important than the memorization of procedures. The overall objective in teaching mathematics is to develop, in each and every student, an understanding and love of mathematics that lasts a lifetime and evolves to meet changing demands.An effective mathematics program should focus on conjecturing, inventing, and problem solving rather than merely finding correct answers. Frequent discussions in the form of individual and small- and large-group conferences; debriefings; or mathematical congresses about problem solving strategies, mathematical processes, and solutions are important. It is imperative that we move away from treating mathematics as a body of isolated skills and that we present mathematics by connecting its ideas and applications, particularly focusing on the connections of mathematics and the students’ real lives.

A clear vision for learning mathematics is one where students engage in meaningful mathematics experiences through the use of concrete materials and manipulatives, visuals, technology, and other resources (see Chapter 4: “Tools for Success”). It is important for students to build on their prior learning and knowledge of keymath concepts and to make connections to their own world. Inquiry, problem solving, discussion, and question posing are all important parts of mathematics learning. Teachers need to use a variety of learning and assessment strategies to accommodate the different learning styles of their students.Mathematics investigations are important for students to engage in, as they can provide multiple opportunities for students to learn and apply mathematics in creative and purposeful ways. It is important for teachers to provide meaningful feedback (formative assessment or assessment for learning) to their students throughout the mathematics learning process. This is important for student improvement and for teacher reflection on the effectiveness of the mathematics opportunities they are providing for their students. It is important for teachers and administrators to engage in ongoing professional growth opportunities and for them to reflect continuously on the mathematics teaching and learning that is happening in their classrooms and schools. This can be done most effectively through job-embedded learning techniques, such as professional learning communities or teams (PLCs or PLTs); peer coaching; group lesson study; and collaborative planning, scoring, and marking. All members of the school community, including educators, students, and parents, should be actively involved in meaningful mathematics .

MATHEMATICS REFORM

A clear vision for learning mathematics is one where students engage in meaningful mathematics experiences through the use of concretematerials andmanipulatives, visuals, technology, and other resources . It is important for students to build on their prior learning and knowledge of keymath concepts and tomake connections to their ownworld. Inquiry, problem solving, discussion, and question posing are all important parts of mathematics learning. Teachers need to use a variety of learning and assessment strategies to accommodate the different learning styles of their students.Mathematics investigations are important for students to engage in, as they can providemultiple opportunities for students to learn and apply mathematics in creative and purposeful ways. It is important for teachers to provide meaningful feedback (formative assessment or assessment for learning) to their students throughout the mathematics learning process. This is important for student improvement and for teacher reflection on the effectiveness of the mathematics opportunities they are providing for their students. It is important for teachers and administrators to engage in

ongoing professional growth opportunities and for them to reflect continuously on the mathematics teaching and learning that is happening in their classrooms and schools. This can be done most effectively through job-embedded learning techniques, such as professional learning communities or teams (PLCs or PLTs); peer coaching; group lesson study; and collaborative planning, scoring, and marking. All members of the school community, including educators, students, and parents, should be actively involved in meaningful mathematics (see Resource 1: Ontario Association for  Mathematics EducationVisionStatement).

In the early 1980s, educators were faced with a cry for a “back to basics”mathematics curriculum, which was a reaction to the “new math” of the 1960s and 1970s. At the same time, there was increasing interest in problem solving as a focus of mathematics education. As a result of the research of Piaget and other developmental psychologists, mathematics educators were shifting the focus from content to how children can best learn mathematics. Mathematics curriculum, pedagogy, and epistemology have undergone intense rethinking in the past decade and a half. A great responsibility for the success of reformhas been placed on the classroom teacher, according to recent mathematics education documents (NCTM, 1991;NRC, 1989;Romberg&Carpenter, 1986).

The role of the teacher has shifted from expert “information/ answer giver” to guide, facilitator, listener, and observer. The emphasis in the classroom has moved from traditional, skill-based procedural tasks to problem solving and reasoning. There is now cross-strand, cross-category, and cross-subject integration of mathematics tasks. There is communication and discourse about mathematics topics and increased use of technology, manipulatives, and group work. Students are involved with contextual, real-life problems that focus on developing problem-solving strategies rather than finding one single correct answer. Teaching for meaning and understanding is the goal, with the use of a variety of strategies to help students visualize abstract ideas (e.g., pictures, graphs, models, technology, language). The mathematics program also includes authentic, complex, multidimensional assessments. To some classroom teachers, many of these ideas may be new and, therefore, may require changes in practice to make them a reality.

Implementing educational reforms for teaching and learning places profound demands on teachers. If teachers are to move toward these reform visions, all teachers (novices and experts)will need to make major changes in their knowledge and beliefs about mathematics learning and teaching, as well as in their teaching practices. The changes teachers are expected to make require large amounts of time and professional development support. Many teachers’ images and beliefs about mathematics and what mathematics learning involves may still be incompatible with current research and reform efforts in the field. Several factors have influenced mathematics reform. The National Council of Teachers of Mathematics has been the main driving force in the current mathematics reform movement in North America. Other factors include national and international assessments that compare student performance, whose results appear regularly in the news media. For example, in many states, the state test results are published in local newspapers with rankings from lowest to highest. This creates a huge stigma for schools, particularly if the contextual data about mobility and socioeconomic and demographic information are not published along with the test scores. As well, curriculum documents and commercial textbooks are major factors in the reform movement.

Unfortunately, many teachers have not been given the appropriate professional development to understand the philosophy and pedagogy behind the reform-based textbooks. As a result, many teachers may be using these textbooks in a more traditional manner (for example, open the text book to the next page, teach the lesson, assign the questions, assign homework, take up the homework at the beginning of the next day’smath class),which is a less effective way of teaching. This approach also renders the new textbooks a poor investment, as they are not being used for their intended purpose (for example, the new textbooks aremeant

to enhance problem-solving skills and promote deep conceptual understanding through the three-part lesson model: explore the problem in a group or with a partner using a student-generated strategy, connect via a teacher lesson related to the problem and what was observed while students were solving the problem, and a reflection or math debrief/congress where the group discusses effective strategies and understanding of the concepts).

HOW WERE THE MATHEMATICS ASSESSMENT PROBES DEVELOPED?

Developing an assessment probe is different from creating appropriate questions for summative quizzes, tests, or state and national exams. The probes in this book were developed using the process described in Mathematics Curriculum Topic Study: Bridging the Gap Between Standards and Practice (Keeley & Rose, 2006).

The process is summarized as follows:

• Identify the topic you plan to teach, and use national standards to examine concepts and specific ideas related to the topic. The national standards used to develop the probes for this book were NCTM’s (2000) Principles and Standards for School Mathematics and the American Association for the Advancement of Science’s (AAAS, 1993) Benchmarks for Science Literacy.

• Select the specific concepts or ideas you plan to address, and identify the relevant research findings. The source for research findings include NCTM’s (2003) Research Companion to Principles and Standards for School Mathematics, Chapter 15 of AAAS’s (1993) Benchmarks for Science Literacy, and additional supplemental articles related to the topics of the probes.

• Focus on a concept or a specific idea you plan to address with the probe, and identify the related research findings. Choose the type of probe structure that lends itself to the situation (seemore information on probe structure following the Gumballs in a Jar example on page 9). Develop the stem (the prompt), key (correct response), and distracters (incorrect responses derived from research findings) that match the developmental level of your students.

• Share your assessment probes with colleagues for constructive feedback, pilot with students, and modify as needed. Concepts and specific ideas related to the probability of simple events. The  information was used as the focus in developing the probe Gumballs in a Jar .

A probe is a cognitively diagnostic paper-and-pencil assessment developed to elicit research-based misunderstandings related to a specific mathematics topic. The individual probes are designed to be (1) easy to use and copy ready for use with students; (2) targeted to one mathematics topic for short-cycle intervention purposes; and (3) practical, with administration time targeted to  approximately 5 to 15 minutes.

Each one-page probe consists of selected response items (called Tier 1) and explanation prompts (called Tier 2), which together elicit common understandings and misunderstandings. Each of the tiers is described in

more detail below.

Tier 1: Elicitation

As the elicitation tier is designed to uncover common understandings and misunderstandings, a structured format using a question or series of questions followed by correct answers and incorrect answers (often called distracters) is used to narrow ideas to those found in the related cognitive research. The formats typically fall into one of seven categories.

Tier 2: Elaboration

The second tier of each of the probes is designed for individual elaboration of the reasoning used to respond to the question asked in the first tier. Mathematics teachers gain a wealth of information by delving into the thinking behind students’ answers not just when answers are wrong but also when they are correct (Burns, 2005). Although the Tier 1 answers and distracters are designed around common understandings and misunderstandings, the elaboration tier allows educators to look more deeply at student thinking as sometimes a student chooses a specific response, correct or incorrect, for an atypical reason. Also, there are many different ways to approach a problem correctly; therefore, the elaboration tier allows educators to look for trends in thinking and in methods used. Also important to consider is the idea that in order to address misconceptions, students must be confronted with their own incorrect ideas by participating in instruction that causes cognitive dissonance between existing ideas and new ideas. By having students complete both tiers of a probe and then planning instruction that addresses the identified areas of difficulty, teachers can then use students’ original responses as part of a reflection on what was learned. Without this preassessment commitment of selecting an answer and explaining the choice, new understanding and corrected ideas are not always evident to the student.

In Designing Professional Development for Teachers of Science and Mathematics, Loucks-Horsley, Love, Stiles, Mundry, and Hewson (2003) describe action research as an effective professional development strategy. To use the probes in this manner, it is important to consider the complete implementation process. We refer to an action research quest as working through the full cycle of

• questioning student understanding of a particular concept;

• uncovering understandings and misunderstandings using a probe;

• examining student work;

• seeking links to cognitive research to drive next steps in instruction; and

• teaching implications based on findings and determining impact on learning by asking an additional question.

Grade-span bars are provided to indicate the developmentally appropriate level of mathematics as aligned to the NCTM Standards and cognitive research. The dark band represents the grade levels where the mathematics required of the probe is aligned to the standards, and the lighter band shows grade levels where field testing of the probe has indicated students still have difficulties. The grade spans, although aligned to the standards, should be considered benchmarks as some students at higher grades may have misunderstandings based in understandings from lower grades, while others may be further along the learning progression and need probes designed for older students.

Student answers may reveal misunderstandings regarding methods of addition, including a lack of conceptual

understanding of number properties. Responses also may reveal a common misconception that there is only one correct algorithm for each operation or that, once comfortable with a method, there is no need to understand other methods.

What Assessment Encouraged Mathematics Learning?

While addressing curricula objectives, the teachers made high priority of planning relevant activities that connect mathematics with the real world and creating a rich learning environment. The teachers tried innovative approaches and teaching strategies to address the mathematical content in a hands-on, minds-on manner. Teachers used a variety of assessment approaches in a traditional and non-traditional manner Assessment Insights for student assessment. Different learning styles were more easily addressed by alternative assessment.

These teachers developed and implemented some effective approaches to alternative assessment that fostered student learning and helped to address motivation concerns. Some strategies that encouraged students to learn math were: doing extra credit assignments, using peer tutoring, valuing classroom discourse, and finding ways to justify their answer. Teachers used manipulatives and exploration through questioning to assist students in developing as independent thinkers. Showing the application and necessity of mathematics while bringing in real-world scenarios was also an effective and valuable strategy. Teachers reassured students that they can do the work and encouraged risk taking. Students developed self-confidence as they were asked to provide their opinion on problems in classroom discourse and in writing. This created a safe learning environment more conducive to learning. Motivation appeared to be the ultimate goal for ensuring student encouragement and interests. Challenging students with competitions and games was a good motivator for the middle school student. Teachers also reported that cooperative grouping encouraged students in problem solving and logic while they learned to help each other. Teachers encouraged students to justify why they did what they did, focusing on the thinking processes rather than just the answer.

The teachers’ learning experiences focused on developing and promoting better classroom assessment. Initially, the teachers explored the recent trends in changes from behavioral to cognitive views of learning and assessment, as well as changes to authentic, multi-dimensional, and collaborative assessment. Teachers learned about the constructivist perspective of teaching and learning school mathematics that is predominant in the NCTM Standards documents (NCTM, 1989, 1991, 1995, 2000). Teachers confronted their own perspective of the nature of mathematics by participating in learning activities that encouraged deep reflection and discourse. Davis, Maher, & Noddings (1990) believe that this perspective has a direct bearing on the ways reform can be approached. Unveiling or developing one's own conception of the nature of mathematics was an enlightening experience that promoted a deeper understanding of reflective teaching and learning mathematics. Teachers developed a better conceptual understanding as they explored mathematics topics as

learners and teachers to better inform instruction and assessment. Teachers examined and explored reasons for evaluating and assessing student achievement. Being aware that teachers evaluate and assess in order to enable decision-making about mathematics instruction and classroom climate was a critical aspect of these teachers' learning. The protocols presented above communicate important tensions for the middle school mathematics teachers among testing expectations, assessment of student understanding, and  the need to assign grades.

Appreciating the need for reform was another area of study for the teachers. For the teachers, this meant acknowledging that current testing procedures are inadequate and realizing the need for further research. Through the workshop experiences and the teachers' own personal classroom action research, teachers discovered why there is a need for reform in assessment. It was apparent that using multiple assessment strategies was a significant step toward creating a more complete picture of the student's mathematical understanding and achievement. New evaluation models and technologies that utilize assessment procedures that reflect the changes in school mathematics are needed. Ultimately, the middle school teachers demonstrated a belief that classrooms should be active learning environments where instruction is interactive and multiple forms of assessment are interwoven with teaching.

Multiple forms of assessment are being advocated as we come to understand that traditional means of assessment have not addressed the needs of all learners. Richard Stiggins estimates that educators spend about a third of their time involved in assessment-related activities that guide the instructional and classroom decisions which directly affect learning (1993). A time investment such as this demands that teachers examine their current assessment practices. Simply testing student achievement with traditional instruments and protocols is insufficient. Empowering all students with mathematical literacy demands methods of assessment

that reflect and enhance the present state of knowledge about learning, teaching, mathematics, and assessment. Implementing improved assessment in the mathematics classroom begins with combining instruction with assessment to better meet the needs of the learner. In order to plan and implement new strategies for assessment, mathematics teachers should have opportunities for professional development, as did these middle school teachers. It is crucial that a support system in the mathematics learning community be

developed along with any efforts to change, alter, and improve assessment in the classroom. Mathematics teachers must personally explore alternative assessment strategies. They should be involved in creating and implementing tasks that are exemplars of mathematics instruction as envisioned by the NCTM. As part of this effort to develop tasks, teachers should have opportunities to observe students doing mathematics and to examine the their products. A solid basis for mathematics teaching, learning, and assessment is created when teachers value and comprehend recent trends, perspectives towards mathematics teaching and learning, evaluation and assessment, and the need for reform. The informed mathematics teacher has the ability and the tools to offer the best learning environment for improving student achievement and understanding. In this article, I have attempted to present a multi-perspective approach toward understanding and implementing assessment reform. The middle school mathematics teachers encountered many problems on this journey from traditional classroom assessment to implementing alternative assessment strategies. Some problems were unique, but many were common among all teachers. Some problems were collectively resolved, while others, such as student motivation, remain as ongoing obstacles to address. These teachers learned about assessment and implementing innovative strategies in a collaborative environment. As a result, the need for a strong support system to implement change was revealed and valued. The experiences and insights of these teachers may promote and encourage other middle school mathematics teachers to move outside the comfort zone of traditional assessment protocols and begin implementing innovative and alternative approaches to assessment.

Assessment Insights from the Classroom

Reform efforts in mathematics education challenge teachers to assess traditional forms of assessment and to explore and implement alternative forms of assessment. Empowering all students with mathematical literacy demands methods of assessment that reflect and enhance the present state of knowledge about learning, about teaching, about mathematics, and about assessment. This discussion highlights insightful perspectives on assessment strategies and techniques currently being addressed and implemented. A cohort of middle school mathematics teachers reveal their experiences and reflections in addressing current assessment practices and ventures in innovative and alternative approaches to assessment.

Assessment is the central aspect of classroom practice that links curriculum, teaching, and learning. (NCTM, 1995). In the Principles and Standards for School Mathematics (NCTM, 2000) assessment is designated as one of the six underlying principles of mathematics education. The Assessment Principle states: "Assessment should support the learning of important mathematics and furnish useful information to both teachers and students" (NCTM, 2000, p. 22). The emerging theme in assessment reform is to do more assessment than evaluation; to become assessors rather than evaluators. The aim is better assessment, not more. Standards were created to provide guidelines to improve mathematics education and to value the importance of alternative, as well as authentic assessment procedures and protocols. Traditional forms of assessment have been utilized in mathematics classrooms for many years. However, reform efforts in mathematics education challenge teachers to reconsider traditional forms of assessment and to explore and implement alternative approaches. Assessment of school mathematics is addressed in some manner in all of the NCTM documents (1989, 1991, 1995, 2000). It is essential that mathematics teachers be informed and proactive in addressing issues of assessment in mathematics classrooms. In response to the call for changes, a cohort of middle school mathematics teachers in a large metropolitan area reflected on their current assessment practices and ventures in alternative forms of assessment in the classroom. These teachers were participants in a grant focusing on the strengthening of mathematical content knowledge, the improvement of instructional strategies, and the implementation of new curricula fulfilling national standards and state-mandated guidelines. In the light of education reform along with the looming accountability of state-mandated guidelines, these teachers began to realize the vision of achieving mathematical power for all students. This discussion highlights middle school mathematics teacher's new perspectives as they implemented alternative assessment strategies and techniques.

What Assessment Tasks were Explored and Implemented by the Middle School Teachers?

Teachers used a variety of traditional and nontraditional approaches to student assessment. The teacher-developed  assessment strategies explored and currently in practice in the classrooms of these middle school

teachers. The third column indicates the teacher’s reflections on the uses of these assessment strategies. Teachers found that short answer tests, journal writing, manipulatives, projects, concept mapping, and performance assessments revealed a broad range of capability, understanding, and communication of mathematical concepts. Many different tasks were used to create a complete picture of the students' mathematical knowledge. Strategies for evaluating performance on assessment activities also varied. Teachers used rubrics quite extensively, as they became comfortable with this system through the workshops. In addition, concept maps, journal entries, textbook assignments, and worksheets were very informative. Teachers identified sources of feedback such as group grades, participation grades, praise, peer

evaluation, and self-evaluation.

What Mathematics was Assessed? How Did State and National Guidelines and Accountability Affect Assessment Strategies?

The mathematical skills and concepts assessed by the middle school teachers in the workshop were typical for grades 6-8. As with many other state-mandated curricula objectives, as do many districts.  For each grade level and each subject area, there are specific learning objectives and goals for Pre-K through 12th grade.  The final an exit test that must be passed as a prerequisite for high school graduation. Within this framework of curricula are thirteen objectives that are assessed in mathematics. These have been determined by the state, but are also related to the national standards identified by the NCTM (cf. 2000). These objectives were created to help ensure quality and consistency. Learning accountability, in some school districts, is even more defined by specific objectives and goals for the grade levels.

Direct test preparation is widespread. Many teachers used the item analysis from the previous year's test to determine the areas of strengths and weaknesses to improve on the objectives that were deficient. Practice tests, six-week tests, quizzes, and a section of the student's daily homework are formatted so students can practice on how the questions are structured as well as practicing and applying the objectives. Teachers and students review and practice test-taking strategies to develop more confidence. Many of the workshop teachers felt that too much focus was placed on the standardized test, thus limiting the time available for alternative assessments. Even so, teachers valued the need for change and explored the potential of other forms of assessment. Tutoring, motivation techniques, and parental involvement were common efforts.

How Can Information from Alternative Assessment be Integrated into Grading and Reporting Progress?

The teachers shared strategies to integrate information from alternative assessment into grading policy. Alternative assessments were sometimes counted as a test grade and sometimes as a daily grade, depending on how much time was required. For example, some teachers used notebooks as test grades. It was common for teachers to offer extra credit opportunities when implementing new forms of assessment. Extra points were given for creativity and originality, hoping to build student confidence. Most of these teachers used homework to determine the depth of student understanding and which concepts needed re-teaching. Projects and journals offered students opportunities to express their ideas, understanding, and concerns. Some students worked better with manipulatives; others with pen and paper. The teachers reported a creative variety of alternative forms of assessment implemented into traditional protocol. Each type of assessment determined a certain percentage of the grade.

Discussion of the variety of assessment practices and grade recording encouraged all teachers to try more alternative forms of assessment as well as developed increase confidence in this endeavor. The teachers communicated the types and importance of assessment strategies and approaches to students and parents through many venues. Some of the ways used by the teachers include weekly reports to parents, scheduled progress reports, promoting ways parents can help students at home, tutorials after school, and Saturday school. As a result, parents and teachers participated in workshops, conferences, and conversations to encourage and support student learning. Positive reinforcements included special privileges at school and at home, award certificates, and other classroom and school acknowledgements.

Teachers reported that sometimes students do not see any correlation between what they are learning in class and what is tested. Teachers tried to address these issues by using real-world problems and scenarios.

Typical problems encountered involved students that do not study or complete homework, or that do not ask questions. These were addressed in parent conferences and student-teacher conferences. Through these many approaches, students were able to ask questions about concepts they had not mastered.

What Results Did the Teachers See as they Used Assessment to Improve Curriculum And Instructional Practices?

The teachers studied and shared strategies to improve mathematics curriculum and instructional practices. They found that different assessment instruments helped to take the focus off the "computation and accuracy" aspect of mathematics, and helped to encourage mathematical thinking. New sorts of tasks in classrooms created a more complete picture of the students' mathematical knowledge. The workshop teachers reported that assessment informed re-teaching, addressed students with math anxiety, and identified students' need for more instruction and/or reinforcement. Students were able to see the objectives mastered and not mastered, as well as their own strengths and weaknesses. Alternative assessment took the emphasis away from right/wrong answers and concentrated students and teachers on thought processes.

THE IMPORTANCE OF PROBLEM SOLVING

“Problem solving is not only a goal of learning mathematics but also a major means of doing so.”

                                                        (NCTM, 2000, p. 52)

An information- and technology-based society requires individuals who are able to think critically about complex issues, people who can “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998, p. 2-1). To prepare students to function in such a society, teachers have a responsibility to promote in their classrooms the experience of problem-solving processes and the acquisition of problem-solving strategies, and to foster in students positive dispositions towards problem solving.

In promoting problem solving, teachers encourage students to reason their way to a solution or to new learning. During the course of this problem solving, teachers further encourage students to make conjectures and justify solutions. The communication that occurs during and after the process of problem solving helps all students to see the problem from different perspectives and opens the door to a multitude of strategies for getting at a solution. By seeing how others solve a problem, students can begin to think about their own thinking (metacognition) and the thinking of others and can consciously adjust their own strategies to make them as efficient and accurate as possible.

In their everyday experiences, students are intuitively and naturally solving problems. They seek solutions to sharing toys with friends or building elaborate structures with construction materials. Teachers who use problem solving as the focus of their mathematics class help their students to develop and extend these intuitive strategies. Through relevant and meaningful experiences, students develop a repertoire of strategies and processes (e.g., steps for solving problems) that they can apply when solving problems. Students develop this repertoire over time, as they become more mature in their problem-solving skills. The problem-solving processes that Kindergarten students use will look very different from those that Grade 6 students use. Initially, students will rely on intuition. With exposure, experience, and shared learning, they will formalize an effective approach to solving problems by developing a repertoire of problem-solving strategies that they can use flexibly when faced with new problem solving situations.

“We want children to take risks, to tackle unfamiliar tasks, and to stick with them – in short, to try and persevere. We want children to be flexible in their thinking and to know that many problems can be modeled, represented, and solved in more than one way.” (Payne, 1990, p. 41)

In fostering positive dispositions in their students towards problem solving, teachers deal with the affective factors that have an impact on student behaviour in both positive and negative ways (Schoenfeld, 1992). Students who believe that they are good problem solvers are not apt to give up after a few minutes when faced with a challenging problem. Because beliefs influence behaviour, effective mathematics programs always consider students’ beliefs and attitudes, and teachers work to nurture in students confident attitudes about their abilities as mathematical problem solvers and their beliefs that everyone can make sense of and do mathematics. As students engage in problem solving, they participate in a wide variety of cognitive experiences that help them to prepare for the many problem solving situations they will encounter throughout their lives. They:

• learn mathematical concepts with understanding and practise skills in context;

• reason mathematically by exploring mathematical ideas, making conjectures, and justifying results;

• reflect on the nature of inquiry in the world of mathematics;

• reflect on and monitor their own thought processes;

• select appropriate tools (e.g., manipulatives, calculators, computers, communication technology) and computational strategies;

• make connections between mathematical concepts;

• connect the mathematics they learn at school with its application in their everyday lives;

• develop strategies that can be applied to new situations;

• represent mathematical ideas and model situations, using concrete materials, pictures, diagrams, graphs, tables, numbers, words, and symbols;

• go from one representation to another, and recognize the connections between

representations;

• persevere in tackling new challenges;

• formulate and test their own explanations;

• communicate their explanations and listen to the explanations of others;

• participate in open-ended experiences that have a clear goal but a variety of solution paths;

• collaborate with others to develop new strategies.

“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.” (Burns, 2000, p. 29)

Problem solving is central to learning mathematics. Problem solving is an integral part of the mathematics curriculum and is the main process for helping students achieve the expectations for mathematics outlined in the curriculum documents because it:

• is the primary focus and goal of mathematics in the real world;

• helps students become more confident mathematicians;

• allows students to use the knowledge they bring to school and helps them connect mathematics with situations outside the classroom;

• helps students develop mathematical knowledge and gives meaning to skills and concepts in all strands;

• allows students to reason, communicate ideas, make connections, and apply knowledge and skills;

• offers excellent opportunities for assessing students’ understanding of concepts, ability to solve problems, ability to apply concepts and procedures, and ability to communicate ideas;

• promotes the collaborative sharing of ideas and strategies, and promotes talking about mathematics;

• helps students find enjoyment in mathematics;

• increases opportunities for the use of critical-thinking skills

(estimating, evaluating, classifying, assuming, noting relationships, hypothesizing, offering opinions with reasons, and making judgements).

Problem solving needs to permeate the mathematical program rather than be relegated to a once-a-week phenomenon – the “problem of the week”. In this guide it is not considered to be one approach among many; rather, it is seen as the main strategy for teaching mathematics. Problem solving should be the mainstay of mathematical teaching and should be used daily.

Not all mathematics instruction, however, can take place in a problem-solving context. Certain conventions of mathematics must be explicitly taught to students. Such conventions should be introduced to students as needed, to assist them in using the symbolic language of mathematics. Examples of mathematical conventions include operation signs, terms such as numerator and denominator, the decimal point, the numerals themselves, the counting sequence, the order of the digits, and the is less than (<) and is more than (>) signs.

“Developing mathematical power involves more than simply giving students harder problems. It means asking them to focus on

understanding and explaining what they are doing, digging deeper for reasons, and developing the ability to know whether they can do a better job on working on a task.” (Stenmark & Bush, 2001, p. 4)

Although students are natural problem solvers, they benefit from guidance in organizing their thinking and approaching new problem solving situations. Teaching about problem solving focuses on having students explore and develop problem-solving strategies and processes. Teaching about problem solving allows students and teachers to create strategies collaboratively and, at all stages of the problem-solving process, to discuss, informally and formally, the thinking and reasoning that they use in determining a solution. When teaching about problem solving, teachers provide students with opportunities to solve interesting and challenging problems.

In many cases, teaching about problem solving occurs simultaneously with teaching through problem solving. As students are engaged in a problem that focuses on a mathematical concept (as described earlier in the chapter), the skilled teacher integrates the discussion of problem-solving strategies and processes into the discussion of the mathematical concept. For some students, this approach may not be sufficient; such students may require additional and more focused opportunities to learn about problem solving. Whether the approach to learning about problem solving is an integrated or a more isolated one, teachers should ensure that, in their instruction about problem solving, they encourage students to develop their own ways of solving problems. To find their own ways, students must be aware of the variety of possible ways. They learn about new strategies by hearing and seeing the strategies developed by their peers, and by discussing the merits of those strategies.

Teaching Mathematics- General Problem

Teaching General Problem- Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics

Problem solving is central to mathematics. Yet problem-solving skill is not what it seems. Indeed, the field of problem solving has recently undergone a surge in research interest and insight, but many of the results of this research are both counterintuitive and contrary to many widely held views. For example, many educators assume that general problem-solving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge. The best known exposition of this view was provided by Pólya (1957). He discussed a range of general problem-solving strategies, such as encouraging mathematics students to think of a related problem and then solve the current problem by analogy or to think of a simpler problem and then extrapolate to the current problem. The examples Pólya used to demonstrate his problem-solving strategies are fascinating, and his influence probably can be sourced, at least in part, to those examples. Nevertheless, in over a half century, no systematic body of evidence demonstrating the effectiveness of any general problem-solving strategies has emerged. It is possible to teach learners to use general strategies such as those suggested by Pólya (Schoenfeld, 1985), but that is insufficient. There is no body of research based on randomized, controlled experiments indicating that such teaching leads to better problem solving.

Recent “reform” curricula both ignore the absence of supporting data and completely misunderstand the role of problem solving in cognition. If, the argument goes, we are not really teaching people mathematics but rather are teaching them some form of general problem solving, then mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general, and that will make them good mathematicians able to discover novel solutions irrespective of the content. We believe this argument ignores all the empirical evidence about mathematics learning. Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach was never the most effective or efficient way to learn mathematics. The alternative route to acquiring problem solving skill in mathematics derives from the work of a Dutch psychologist, De Groot (1946–1965), investigating the source of skill in chess. Researching why chess masters always defeated weekend players, De Groot managed to find only one difference. He showed masters and weekend players a board configuration from a real game, removed it after five seconds, and asked them to reproduce the board. Masters could do so with an accuracy rate of about 70% compared with 30% for weekend players. Chase and Simon (1973) replicated these results and additionally demonstrated that when the experiment was repeated with random configurations rather than real-game configurations, masters and weekend players had equal accuracy (±30%). Masters were superior only for configurations taken from real games.

Chess is a problem-solving game whose rules can be learned in about thirty minutes. Yet it takes at least ten years to become a chess master. What occurs during this period? When studying previous games, chess masters learn to recognize tens of thousands of board configurations and the best moves associated with each configuration (Simon & Gilmartin, 1973). The superiority of chess masters comes not from having acquired clever, sophisticated, general problem-solving strategies but rather from having stored innumerable configurations and the best moves associated with each in long-term memory.

De Groot’s results have been replicated in a variety of educationally relevant fields, including mathematics (Sweller & Cooper, 1985). They tell us that long-term memory, a critical component of human cognitive architecture, is not used to store random, isolated facts but rather to store huge complexes of closely integrated information that results in problem-solving skill. That skill is knowledge domain-specific, not domain-general. An experienced problem solver in any domain has constructed and stored huge numbers of schemas in long-term memory that allow problems in that domain to be categorized according to their solution moves. In short, the research suggests that we can teach aspiring mathematicians to be effective problem solvers only by providing them with a large store of domain-specific schemas. Mathematical problem-solving skill is acquired through a large number of specific mathematical problem-solving strategies relevant to particular problems. There are no separate, general problemsolving strategies that can be learned.

How do people solve problems that they have not previously encountered? Most employ a version of means-ends analysis in which differences between a current problem-state and goal-state are identified and problem-solving operators are found to reduce those differences. There is no evidence that this strategy is teachable or learnable because we use it automatically. But domain-specific mathematical problem solving skills can be taught. How? One simple answer is by emphasizing worked examples of problem-solution strategies. There is now a large body of evidence showing that studying worked examples is a more effective and efficient way of learning to solve problems than simply practicing problem solving without reference to worked examples (Paas & van Gog, 2006). Studying worked examples interleaved with practice solving the type of problem described in the example reduces unnecessary working memory load that prevents the transfer of knowledge to long-term memory. The improvement in subsequent problem-solving performance after studying worked examples rather than solving problems is known as the worked example effect (Paas & van Gog).

Whereas a lack of empirical evidence supporting the teaching of general problem-solving strategies in mathematics is telling, there is ample empirical evidence of the validity of the worked-example effect. A large number of randomized controlled experiments demonstrate this effect (e.g., Schwonke et al., 2009; Sweller & Cooper, 1985). For novice mathematics learners, the evidence is overwhelming that studying worked examples rather than solving the equivalent problems facilitates learning. Studying worked examples is a form of direct, explicit instruction that is vital in all curriculum areas, especially areas that many students find difficult and that are critical to modern societies. Mathematics is such a discipline. Minimal instructional guidance in mathematics leads to minimal learning (Kirschner, Sweller, & Clark, 2006).

Students view themselves as passive consumers of others' mathematics.

One of the most vivid memories of my education comes from an upper division probability class, when my instructor was about to introduce the binomial theorem. She stopped writing the statement of theorem at the point where she needed to write the formula. "I never remember this formula," she said, " but it's so easy to derive that you don't need it anyway." Then she showed us how to derive the formula. What she showed us made sense. To this day I can't remember the formula, but I can derive it, either when I need it (which is rare) or because the thought of it brings back pleasant memories. The idea that was brought home in that class -- that mathematics really makes sense, and that you can figure something out if you need to -- was exhilarating. It is (or should be) part of the pleasure of learning mathematics.

Such moments were rare in my experience as a student, and they were almost completely absent from the classes we observed. The mathematics instruction that we observed consisted almost exclusively of training in skill acquisition. For each of the years K-12 (and beyond; calculus instruction in college is pretty much the same), there was an agreed-upon body of knowledge, consisting of facts and procedures, that comprised the

curriculum. In each course, the task of the teacher was to get students to master the curriculum. That meant that subject matter was presented, explained, and rehearsed; students practiced it until they got it (if they were lucky). There was little sense of exploration, or of the possibility that the students could make sense of the mathematics for themselves. Instead, the students were presented the material in bite-sized pieces so that it would be easy for them to master. As an example, recall the step-by-step procedure for constructions, described above, that was used by the teacher of the target class. Constructions were introduced that way, and students were given practice that way. When, for example, a student had difficulty with a particular problem, the teacher reminded him that the problem called for a construction with which the student was familiar. He then asked: "In your construction, what is step number one?" The student replied correctly. The teacher continued. "Good. In your construction, what is step number two?" And so on. In this way, students got the clear impression that someone else's mathematics was theirs to memorize and spit back. Nor was step-by-step memorization limited to constructions. Recall that the Regents exam had required proofs as well; students were told to commit them to memory. This was standard practice, and was promoted as being both efficient and desirable. For example, an advertisement for a best-selling series of review books for the Regents exams proudly announced: "Students like these books because they offer step-by-step solutions."

The point I wish to stress here is that students develop their understanding of the mathematics from their classroom experience with it. If the "bottom line" is error-free and mechanical performance, students come to believe that that is what mathematics is all about. In the target class, for example, the teacher talked about how important it was for students to think about the mathematics, and to understand it. He pointed out the fact that they should not memorize blindly, because if they did "and forgot a step" they would be in trouble. In truth, however, this rhetoric -- in which the teacher honestly believed -- was contradicted by what took place in his classroom. The classroom structure provided reinforcement for memorization, and the reward structure promoted it. One of the items on our questionnaire, for example, asked students to agree or disagree with the statement "the math that I learn in school is mostly facts and procedures that have to be memorized." With a score of 1 indicating "very true" and a score of 4 indicating "not at all true," this item received an average score of 1.75 -- the third strongest "agree rating" of seventy questions. Yet the statement "When I do a geometry proof I get a better understanding of mathematical thinking" received an average score of 1.99 -- again very strong agreement. These data parallel the secondary school data, where students claimed that mathematics is mostly memorization but that mathematics helps a person to think logically. Our classroom observations supported Carpenter et al.'s (1983, p. 657) conjecture that the "latter attitudes may reflect the beliefs of their teachers or a more general view rather than emerge from their own experience with mathematics." More importantly, the latter attitudes did not influence behavior: when working mathematics problems, the students behaved in accord with the three mathematics beliefs discussed

above.

Good Teaching, Bad Results

The past decade has seen a radical shift in theories of learning, brought about in large part by progress in the cognitive sciences. Through perhaps the mid-1970's, learning theories were for the most part domain-independent. Such theories attempted to characterize general principles of learning, the specifics of which were hypothetically applicable in different domains such as reading, social studies, and mathematics. The details of the subject matter were not important in such theories, for the most part playing the role of "context variables" that (as the theory had it) could be taken into account in experimental design. The corresponding paradigms for investigating research on teaching are described by Corno (this volume); see also Doyle (1978), Dunkin and Biddle (1974), and Shulman (1985). The first major paradigm described by Corno, process-product research, largely used correlational methods to explore relationships between teacher classroom behavior and student learning. The classroom behaviors explored were for the most part straightforward and easily quantifiable: e.g. time spent in questioning, "active learning time," amount of praise, amount of feedback. Other classroom variables included type of ability grouping, whether students worked in small or large groups, and so on. Learning was operationally defined as performance on achievement tests -- tests which, as we shall see below, may fail in significant ways to measure subject matter understanding.

Mediating process research (see, e.g., Corno, this volume) provides a means of overcoming some of the significant limitations of the process-product paradigm. Such work signals the beginning of a rapprochement with cognitive science research on learning, specifically with its focus on the child as active interpreter of its experience. Doyle's study in this volume provides some compelling examples of the importance of this perspective. Doyle suggests that the presentation of subject matter as familiar work  routinized exercises that can be worked out of context, and without significant understanding of the subject matter -- can trivialize that subject matter and deprive students of the opportunity to understand and use what they have studied. That suggestion is explored at length in this paper.

Recent cognitive research on learning diverges from the domain-independent work described above in that it lays a much greater emphasis on the particulars of the subject matter being studied. In elementary arithmetic, for example, Brown and Burton (1978) developed a diagnostic test that could predict, about 50% of the time, the incorrect answers that a particular student would obtain to a subtraction problem -- before the student worked the problem! The literature indicates that misconceptions in arithmetic, in algebra, in physics, and other domains, are quite common and consistent (see, e.g., Helms & Novak, 1985.). From this and related work follow two main consequences.

The first consequence is that one of the treasured pedagogical principles on which much current instruction is based is, if not plain wrong, certainly inadequate. The predominant model of current instruction is based on what Romberg and Carpenter (1985) call the absorbtion theory of learning. "The traditional classroom focuses on competition, management, and group aptitudes; the mathematics taught is assumed to be a fixed body of knowledge, and it is taught under the assumption that learners absorb what has been covered" (p. 26). According to this view, the good teacher is the one who has ten different ways to say the same thing; the student is sure to "get it" sooner or later. However, the misconceptions literature indicates that the students may well have "gotten" something else -- and that what the student has gotten may be resistant to change. Dealing with this reality calls for a significantly different perspective on the part of the teacher. It also calls for different perspectives regarding the appropriate domain of study of research on teaching, and different measures of competence. The second consequence is that one must look at the subject matter in detail. Arithmetic mistakes differ from misconceptions in algebra and physics, and from misapprehensions about reading; we will understand each of these only by studying it on its own terms. Thus studies of learning and teaching in particular subject areas must be grounded in analyses of what it means to understand the subject matter being taught. It is to that kind of analysis, in mathematics, that we now turn. Some relevant research on mathematical cognition and teaching may be found in Romberg and Carpenter (1985), Leinhardt & Smith (1984), Resnick (1983), Schoenfeld (1985), and Silver (1985).

The issue of classroom practice and its relation to students' understanding of mathematical structure was one of the main themes of Wertheimer's (1959) Productive Thinking, which provides our first two examples. In the first, Wertheimer asked elementary school students to solve problems.  Many of the students, who were fluent in all four of the basic arithmetic operations, solved such problems by laboriously adding the terms in the numerator and then performing the indicated division. By virtue of obtaining the correct answer, the students indicated that they had mastered the procedures of the discipline. However, they had clearly not mastered the underlying substance; if you see repeated addition as equivalent to multiplication and you see division as the inverse of multiplication (i.e., the multiplication and division by the same number cancel each other out), there is no need to calculate at all. This example illustrates that being able to perform the appropriate algorithmic procedures, while important, does not necessarily indicate any depth of understanding. (We note here that virtually all standardized testing for arithmetic competency -- and, de facto, much standard instruction in arithmetic -- focuses primarily if not exclusively on procedural mastery.)

Wertheimer's more famous example comes from his observations of classroom sessions devoted the "the parallelogram problem," the problem of determining the area of a parallelogram of base B and altitude H. The students had been taught the standard procedure, where cutting off and moving a specific triangle converts the parallelogram to a rectangle whose area is easy to calculate. They did quite well at the lesson, and they were able to reproduce the argument in mathematically correct form. But when Wertheimer asked the students to find the area of a parallelogram in non-standard position, or to find the area of a parallelogram-like figure to which the same argument applied, the students were stymied. Wertheimer argues that although they had memorized the proof, they had failed to understand the reason that it worked; although

they had memorized the formula, they used it without deep understanding. With that understanding, he argues, the students would have been able to answer his questions without difficulty; without it they could solve certain well specified exercises but in reality had acquired only the superficial appearance of competence. (We note again that typical achievement tests, which examine students' ability to reproduce the standard arguments, are unlikely to examine the kinds of understandings Wertheimer considers fundamental.)

There are numerous contemporary parallels to these examples. For example, word problems of the following type are a major focus of the elementary mathematics curriculum: "John has 8 apples. He gives 5 apples to Mary. How many apples does John have left?" Perhaps the most commonly used instructional procedure to help students solve such problems is the "key word procedure," which is used as follows. The student is told that certain words in problem statements provide the "key" to selecting which arithmetic operation to employ. For example, the key word in the problem just quoted is left, which indicates subtraction. One can "solve" the problem by identifying the two numbers in the problem statement, and then -- since the key word is "left" -- subtracting one from the other. Note that one can do so without even reading the whole problem, and without understanding the situation it describes. Research indicates that many students work the problems in precisely that fashion. In interviews some students revealed that they circled the numbers in the problem statement and then read the problem statement from the last word backwards, because the key word usually appears near the end of the problem! Thus the key word procedure, initially introduced to help students make sense of word problems, had (at least in these cases) precisely the opposite effects. It allowed students to obtain the right answers without understanding -- and gave them the option of not seeking understanding at all. Worse, it may have suggested to them that understanding is not necessary when solving mathematics problems; one simply follows the procedure, whether it makes sense or not.

The most extensive documentation of students' performance on word problems, without understanding, comes from the third National Assessment of Educational Progress (Carpenter, Lindquist, Matthews, and Silver, 1983). On the NAEP mathematics exam, which used a stratified national sample of 45,000 students, 13-year-olds were given the following problem: "An army bus holds 36 soldiers. If 1128 soldier are being bussed to their training site, how many buses are needed?" Seventy percent of the students who worked the problem performed the long division algorithm correctly. However, 29% of the students wrote that the number of buses needed is "31 remainder 12" and another 18% wrote that the number of buses needed is 31. Only 23% gave the correct answer. Thus fewer than one-third of the students who selected and carried out the appropriate algorithm produced the right answer -- a step that required a trivial analysis of the meaning of the problem statement. There are a number of plausible explanations for this behavior, one of which will be suggested in the case study below (See also Silver, in press, for a discussion of related problems.). But data of this type document an almost universal phenomenon: Students who are capable of performing symbolic operations in a classroom context, demonstrating "mastery" of certain subject matter, often fail to map the results of the symbolic operations they have performed to the systems that have been described symbolically. That they fail to connect their formal symbol manipulation procedures with the "real world" objects represented by the symbols constitutes a dramatic failure of instruction.

A set of similar phenomena motivated the present study. I have conducted a series of studies exploring students' understandings of geometry. Those studies have focused, in particular, on the relationship between geometric proofs and geometric constructions. To sum things up briefly, I had found that high school and college students who had taken a full year of high school geometry -- which focuses on proving theorems about geometric objects -- uniformly approached geometric construction problems as empiricists. They engaged in empirical guess-and-test loops, completely ignoring their proof-related knowledge. In one series of interviews, for example, college students were asked to work two related problems. The first was a proof problem. Solving this problem directly provided the answer to the second, a construction problem (The second problem asked how to construct a circle whose properties had been completely determined in the first.). Yet, after solving the first problem, nearly a third of the students began the second problem by making conjectures that flatly violated the results they had just proved!

Such behavior indicated that these students saw little or no connection between their "proof knowledge," abstract mathematical knowledge about geometric figures obtained by formal deductive means, and their "construction knowledge," procedures and information they had mastered in the very same class for working straightedge and compass construction problems. I make this statement more provocatively as Belief 1, below; some other typical beliefs are also given. I conjecture students may develop these beliefs as a result of their experiences with mathematics. (Extended discussions of the students' beliefs may be found in chapters 5 and 10 of my (1985) Mathematical Problem Solving. A discussion of the "ideal" relationship between geometric empiricism and deduction may be found in Schoenfeld, in press.)

Belief 1: The processes of formal mathematics (e.g. "proof") have little or nothing to do with discovery or invention. Corollary: Students fail to use information from formal mathematics when they are in "problem solving mode."

Belief 2: Students who understand the subject matter can solve assigned mathematics problems in five minutes or less. Corollary: Students stop working on a problem after just a few minutes since, if they haven't solved it, they didn't understand the material (and therefore will not solve it.)

Belief 3: Only geniuses are capable of discovering, creating, or really understanding mathematics. Corollary: Mathematics is studied passively, with students accepting what is passed down "from above" without the expectation that they can make sense of it for themselves. In listing these beliefs we note the parallel to research Doyle describes in this volume. Doyle described a student who took a teacher's instructions for an assignment as a recipe for completing the task, rather than a way of learning the material. In terms more provocative than Doyle might like, one can characterize that student's perspective as follows:

Belief 4: One succeeds in school by performing the tasks, to the letter, as described by the teacher. Corollary: learning is an incidental by-product to "getting the work done."

[S]tudents may not understand some of the problems they do solve. Most of the routine problems can be mechanically solved by applying a routine computational algorithm. In such problems the students may have no need to understand the problem situation, why the particular computation is appropriate, or whether the answer is reasonable... The errors made on several of the problems indicate that students generally try to use all of the numbers given in a problem statement in their calculation, without regard for the relationship of either the given numbers or the resulting answers to the problem situation. (Carpenter et al., 1983, p. 656)

In sum, research on the teaching and learning needs to be expanded both in scope and in breadth. "Learning outcomes" must be broadly defined if we are to provide adequate characterizations of behaviors such as those described in the previous paragraph. But explorations of learning also need to become more focused and detailed as we begin to elaborate on what it means to think mathematically. It is also essential -- both for research purposes and because measurement is the "bottom line" for much real world instruction -- for our efforts to include the development of measures that will adequately characterize this expanded notion of mathematics learning. And if we really intend to affect practice, we will need to become deeply involved in the development and testing of instructional materials. This list of tasks may seem daunting, but it is not beyond our reach. There is, as noted above, an increasing rapprochement between researchers on teaching and cognitive scientists. Similarly, there are closer ties between psychologists of learning and subject matter experts as the result of perceived need for collaborative efforts. As our sense of the task grows, so does our capacity to deal with it.