Chapter 1: Introduction. The use of writing in mathematics classes is now widespread and generally

Transcription

1 Chapter 1: Introduction The use of writing in mathematics classes is now widespread and generally acknowledged as a useful instructional choice. For more than fifteen years now, the Writing Across the Curriculum (WAC) and Writing to Learn movements, the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (1989, 2000), and numerous other sources have promoted the use of writing in the classroom. They suggest that writing helps students learn mathematical concepts more deeply as well as how to communicate them, and gives teachers insight into what students understand. More recently, legislators and standardized test designers have joined the movement, using writing on standardized assessments to give deeper insight into students understanding. The use of writing in mathematics instruction has expanded over the same time period to match this emphasis on communication, conceptual development and assessment, especially at the middle school level. This can be seen in data from the National Assessment of Education Progress (NAEP). When asked about how often they wrote a few sentences about how they solved a problem, 62% of 8 th graders reported in 1992 that they never did this, while only 37% reported that they never or hardly ever did this in Likewise, the number of students who wrote about how they solved a problem almost every day went up from 7% to 16% (NAEP Questions, May 12, 2005). Similar changes are found at the 12 th grade level, although they are not as pronounced. Likewise, teachers at the 8 th grade level reported more use of writing in mathematics classes in 2000 than they did in 1992: the number of teachers who reported using short or long written responses once or twice a week rose from 9% to 24%, while the number who reported never or hardly ever using written responses fell from 33% to 14% over the same time period (NAEP Questions, May 12, 2005). 1

2 2 The emphasis on writing and communication has also impacted curriculum, with many recent textbooks including more opportunities for students to use writing in many ways. Writing Movements The Writing across the Curriculum movement (WAC) began in the late 1970s and early 1980s (Sully, June 17, 2005) in an effort to promote writing in all subject areas. This movement began mostly at the college level. It now contains two basic subdivisions: Writing in the Disciplines (WID) and Writing to Learn. The former is more concerned with helping students learn to communicate according to the standards of a discipline while the latter focuses on using writing to help students think about and learn the content of any discipline. Part of the development of WAC grew out of a similar movement in Britain, and was motivated by fears of a literacy crisis in the United States. The Writing in the Disciplines (WID) movement is closer to the original intentions of WAC than Writing to Learn, and grew out of the realization that students are increasingly being asked to communicate within fields that require specialized communication skills. Each discipline has particular genres and expectations for writing in the field, and one of the tenets of the WID movement is that they should be taught how to write within specific disciplines so that they can learn how to write in that specific field. There is also a desire that students should encounter writing in every discipline as a necessary communication skill. Sully writes practising writing results in improved student writing (June 17, 2005), emphasizing that better communication through writing results from using writing. This tendency to treat writing in mathematics as a discipline specific skill, with certain vocabulary and standards of justification is closely related to the emphases of the WAC movement.

3 3 William Zinsser s book Writing to Learn was the beginning of the movement by the same name. It took the idea of WID a step further, and thought of writing as a means to learn the material and not just to communicate it. It contends that learning and understanding the content better was an outcome of using writing, and that writing was not just a means of communicating about the discipline. Inherent in this movement is the belief that writing brings thought onto paper, making it external and therefore easier to act on (Applebee, 1987; Zinsser, 1988). Revision of written work is therefore akin to rethinking an idea: reorganizing and modifying thoughts. This leads to deeper understanding and learning. With this assumption of the connection between writing and thinking, writing also becomes a means to promote metacognition. The NCTM Standards The NCTM Principles and Standards for School Mathematics (1989, 2000) have been a catalyst in mathematics education reform. The first Standards were published in 1989 and a revised edition was published in These guidelines provide direction for reform, and ideas for teacher education and practice. In the 2000 revision, there are five process standards and five content standards. One of the process standards is Communication and emphasizes both spoken and written communication. This standard is mainly concerned with helping students deepen their understanding of mathematical concepts, improve their communication of mathematics to others clearly and precisely, using mathematical language and work to analyze others thinking and strategies. Inherent in this standard is the assumption that communication can help students understand the mathematics more deeply. One of the main goals of the standard is that students organize and consolidate their mathematical thinking

4 4 through communication (NCTM, 2000, p. 59). The theoretical foundation for this is found in the belief that ideas become objects of reflection, refinement, discussion, and amendment (p. 59) when they are communicated. Communication also supports a sense of meaning and permanence (p. 59) for concepts. This assumption that communication helps students learn is a strong connection with the Writing to Learn movement. The Communication Standard supports the introduction and expansion of students mathematical writing throughout their school career. In particular, the middle grades are a time for students to develop a sense of audience and purpose (NCTM, 2000, p. 61) in their writing, as well as learn to use some formal language. At the end of high school, students are expected to be able to write well-constructed mathematical arguments using formal vocabulary (p. 61). This is consistent with the WID movement s emphasis on learning to write within specific disciplines. As with WID, the Standards also emphasize the necessity of practice within mathematics: The process of learning to write mathematically is similar to that of learning to write in any genre. Practice, with guidance, is important (p. 61). Students are also expected to think about discipline specific writing principles: possible structures for argument, standards for justification, and precise use of mathematical language. They highlight attention to the specifics of mathematical argument, including the use and special meanings of mathematical language and the representations and standards of explanation and proof (p. 61). The Standards also recognize that [c]ommunication can be used in many ways as a vehicle for assessment (p.348), as well as learning. Writing can communicate to teachers what students know, either at the beginning or end of instruction. Writing at the start of instruction can help teachers know what they need to emphasize in their instruction, while writing after instruction can help the teacher

5 5 evaluate the success of their teaching methods. It is assumed that spoken or written explanations are more likely to highlight student misunderstanding than symbolic manipulations because they give in-depth information about students ideas and make the connections in students minds more explicit. Standardized Testing Standardized testing has always been a concern because these tests are used to compare students and schools and can be determining factors in student placement within the school system and entry into college. Testing has become an increasing concern, though, as recent legislation has increased the extent and importance of mandated assessments. Both the Elementary and Secondary Education Act (ESEA) and No Child Left Behind (NCLB) have called for increased state assessments. Most recently, NCLB has mandated annual tests in Grades 3-8 in mathematics and reading/language arts by the school year, and continued the mandate for annual tests given once during grades (U.S. Dept. of Education, 2003). These tests are intended to keep schools accountable for student learning, and help identify schools, sub-groups and students who are in need of extra attention. States have the right to choose or make their own assessments, and may allow some variation in assessments in different localities as long as these tests are shown to be equivalent. These assessments are required to be closely aligned to the state standards, however. Norm-referenced tests may be used, but must be augmented with questions that make sure the standards are adequately assessed (U.S. Dept. of Education, 2003, p.13-14). Preference seems to be given to criterion referenced tests, which are specifically designed to test the state standards. The state assessments are required to assess the full depth of the cognitive demand of the challenging state standards (U.S. Dept. of Education, 2003, p.12). As a result of these standards and changes in expectations for

6 6 large-scale assessment, most of these state assessments include short answer questions that require students to share their own thinking. For example, this is a sample question for an 8 th grade test in New York: after students have drawn the image of a triangle PQR after reflection through the x-axis, they are asked to explain how you decided where to locate P and given three lines to write their answer (New York State Education Department, March 18, 2005). Standardized assessments have been changing in nature even before these regulations, including more short and long answer questions that require students to write their own answer rather than just pick a one of a list of possible answers. The NAEP tests exemplify these changes in standardized tests. These tests include multiple-choice questions, short constructed-response questions and extended constructed-response questions. Students are expected to spend at least half their time working on the constructed response questions (NAEP, 2003c, p. 12). One of the questions used at the 12 th grade level is found in Figure 1. This question would require a significant amount of writing, which is expected to communicate clearly. NAEP includes constructed-response questions to allow students to communicate their ideas and demonstrate the reasoning they used to solve problems (NAEP, 2003c, p. 12). This demonstrates that standardized test makers no longer expect to assess student understanding with just multiple-choice questions. Teachers are responding to this use of writing in standardized tests by giving students more opportunity to practice these skills. This is especially a concern in middle school since many schools have high-stakes tests during 8 th grade. This concern is echoed in the NCTM Standards: Since written assessments of students' mathematical knowledge are becoming increasingly prevalent, students will need practice responding to typical assessment prompts (p. 61).

7 7 This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work. One plan for a state income tax requires those persons with income of $10,000 or less to pay no tax and those persons with income greater than $10,000 to pay a tax of 6 percent only on the part of their income that exceeds $10,000. A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusions. (NAEP Questions, Block M12 No. 9, May 12, 2005) Figure 1: 12 th grade NAEP Extended Constructed Response Question Classifying Writing Assignments It is clear that there are many motivations for including writing in the mathematics classroom. The WAC and Writing to Learn movements advocate using writing to help students learn to communicate in the discipline of mathematics and to use writing as a means of learning the subject more deeply. The increased use of written responses in standardized assessments and the growing number of assessments provides a more immediate motivation for teachers to include writing in their instruction. Taking these elements into account, NCTM has encouraged a number of forms of communication, including writing. Teachers are responding and including more writing, as seen in the NAEP data.

8 8 However, even as more teachers are using writing in more and more ways, we are still at the beginning of the journey to determine if writing meets the goal of increased student learning in mathematics. The work to study how writing affects mathematical learning has been progressing for fifteen years, but slowly. This is a complex task since many factors may play a role: students beliefs about mathematics, student characteristics, and the assignment used are some of the main pieces that affect the outcome. Student characteristics can be classified by variables such as age, location, and socio-economic status and Schoenfeld (1992) has identified how some beliefs may affect how students do mathematics. Writing tasks, on the other hand, are not well-classified and frequently sketchily described in the literature. There is a need to study many different types of writing tasks because they are likely result in different types of learning. Teachers ask questions about which assignments will be most effective in meeting their instructional goals. Which types of writing assignments are effective in deepening students understanding of algebra? What ways of assessing student work are most helpful in teaching them disciplinespecific writing habits? Is practice on similar problems or use of a variety of writing tasks better preparation for open-ended questions on standardized tests? These are but a few questions that could be addressed in the research. However, we lack a comprehensive classification of types of mathematical writing tasks, which makes it difficult to describe writing tasks and correlate them with particular outcomes. Therefore, descriptions of tasks in the literature are difficult to compare and even more difficult for teachers to implement effective tasks accurately. A focus on writing tasks, rather than student factors, can also empower teachers. The type of writing task is a variable over which they have significant control; they may not be able to choose their students, but they can choose instructional activities.

9 9 Detailed classification of writing tasks can enable research that focuses on certain areas of learning achieved by certain types of writing tasks. Different types of writing assignments promote different types of learning (Applebee, 1987); therefore, writing tasks need to be closely examined before we can easily consider which writing assignments are most effective for different instructional objectives in mathematics. A classification of writing assignments would help researchers in this area to better examine and write about the differences between writing assignments. Research that takes into account more detail of the assignment may be more productive since more factors are recognized as affecting the outcome. Therefore, it is less likely that some factor that might influence the outcomes, thereby making the results hard to interpet, will be overlooked. Some factors may also be shown to make negligible difference in a given context. Also, a classification of writing tasks can motivate an organized research program. These factors could help to overcome the current lack of clear research results on instructional uses of writing in mathematics. If research is based on a common classification, communication about writing tasks will also be easier, making research more usable to practitioners who try to replicate successful uses of writing. The elements that made the tasks successful in promoting learning can be more clearly identified if other factors are ruled out, so teachers are better informed about which elements of the task are important and should not be changed as they adapt the task to their classroom. The classification also attempts to make the details of the task clear and understandable to teachers so that they know how to implement it. This is currently difficult, since the literature frequently does not describe writing tasks in depth or includes vocabulary that is used differently in different studies. For example, it is difficult to know what is meant by a journal, since it is used to describe a variety of written work. This can cause

10 10 confusion, and more detailed descriptions of writing tasks are needed for teachers to implement research results. Communication about writing tasks can also be more difficult for mathematics teachers because, traditionally, mathematics has been viewed as separate from writing. Therefore mathematics teachers tend to have less training in and exposure to various types of writing assignments, and fewer ways to communicate about them. In order to use writing most effectively, teachers need to know the breadth of writing tasks available for their use. They also need a vocabulary to help them communicate about these variations. Then teachers will have more choices about tasks that they could use in their classroom so they can make informed decisions about the instructional activities they choose. Therefore, there is a need to find a way to describe writing tasks in detail to promote better communication among researchers, among teachers, and between researchers and teachers. Since different writing tasks are likely to lead to different types of learning, this would allow research to be more targeted and perhaps fit within a larger program. It would also make any findings about which types of tasks are effective for certain learning goals more practical for teachers to implement since they will have more detail about the tasks and a common vocabulary for both teachers and researchers to communicate this detail. A classification of writing tasks can help make teachers aware of the wide variations possible in writing tasks and provide a vocabulary to describe these variations. Research Goals For these reasons, this study undertakes the task of classifying mathematics writing assignments, focusing on factors that teachers can control in making the assignment. The goal of this research is to promote communication among and

11 11 between researchers and teachers so that research on the use of writing can be more coordinated, effective, and applicable. This study focuses particularly on secondary mathematics writing tasks and situates them within the classrooms of the teachers who use them to carefully describe variations in writing tasks that currently exist. This classification is intended to provide a foundation upon which more research may be done, to start the process of determining how we can use writing most effectively to meet instructional goals by suggesting variations in writing tasks that should be considered and providing a context in which to locate all research on mathematics instructional uses of writing. The classification can also provide a common basis for communication, so that teachers will find this research more accessible and easier to implement in their classrooms. I also hope that this classification will be useful to teachers, making them aware of the wealth of possibilities for using writing in the classroom, so that they can be more intentional in designing assignments to meet their instructional goals.