Chapter 1 - NSF-funded exemplary comprehensive mathematics curricula

Figure 7

NSF-funded exemplary comprehensive mathematics curricula

Elementary school (K-5):

• Everyday Mathematics

(http: //ars-www.uchicago.edu/ucsmp-el/)*

• Investigations in Number, Data and Space

(http://www.terc.edu/investigations)*

• Math Trailblazers

(http://www.math.uic.edu/IMSE/MTB/mtb.html)*

Middle school (5-8):

• Connected Mathematics Project (CMP)

(http: //www.math.msu.edu/cmp)*

• Mathematics in Context (MiC)

(http://www.ebmic.com)*

• MathScape

(http: //www.edc.org/mcc/cscape.htm)*

• Middle Grades Math Thematics

(http: //www.math.umt.edu/~stem/)*

• Middle School Mathematics through Applications Project (MMAP)

(http://mmap.wested.org)*

High school (9-12):

• Contemporary Mathematics in Context (CORE-Plus)

(http://www.wmich.edu/cpmp)*

• Interactive Mathematics Program (IMP)

(http://www.mathimp.org)*

• Math Connections

(http: //www.mathconnections.org)*

• Mathematics: Modeling our World (ARISE)

(http://www.comap.com)*

• SIMMS Integrated Mathematics

(http: //www.montana.edu/~wwwsimms)*

* web addresses are current at time of publication

In order to be considered “exemplary,” a unit or comprehensive curriculum must be consistent with the NCTM Standards, designed by groups of specialists in mathematical content and pedagogy, and revised based on field tests in various instructional settings.

Exemplary instructional materials are much more than a textbook for students. They usually include a rich collection of documents to support learning experiences. The documents may include suggestions for planning lessons and orchestrating class discussions, examples of student work, tools and rubrics for assessment, and opportunities for teachers to learn more about the mathematical concepts to be taught.

While there is certainly a value for teachers to create their own innovative lessons and units, the results of the multitude of Teacher Enhancement and Local Systemic Change projects supported by the NSF in the last two decades suggest that the use of exemplary comprehensive mathematics curricula is critical to the success of systemic reform. That is, if the goal is to reform the entire mathematics program within a given school or district, not just to improve the practices of a few committed teachers, it is very difficult to achieve significant success unless the system adopts a coherent curriculum that ensures that students engage in a well-constructed sequence of worthwhile mathematics experiences, and frees teachers to focus their energy on improving their instructional practices and evaluating their students’ learning.

While exemplary instructional materials can revolutionize the way we approach school mathematics reform (Ball & Cohen, 1996; Russell, 1997), they also require considerable time (and, in some cases, special expertise) to be used efficiently. Therefore, professional development programs should include opportunities for teachers to become familiar with at least some exemplary instructional materials, selected so as to maximize the participants’ opportunities to implement reform in their classes.

Teachers also need to learn about high quality software and other technological tools if they are to implement mathematical learning experiences consistent with the most recent calls for reform. Indeed, new technologies such as graphing calculators, spreadsheets, and programs like the “Geometer’s Sketchpad” and statistical packages like “Fathom,” have radically changed the way certain mathematical topics can be taught in school (e.g., Dunham & Dick, 1994; Rojano, 1996). Teachers need to become proficient users of these technologies and to learn to consider how using these tools could affect not only their teaching practices but also their instructional goals.

Understanding equity issues and their implications for the classroom

At the forefront of the current call for school mathematics reform is the directive that all students should have opportunities to learn mathematics (NCTM, 1989, 2000; Secada, Fennema & Adajian, 1995). The underachievement of some ethnic minorities and women has been the cause of serious concern and one of the reasons that led to the recent critical scrutiny of curricula and teaching practices (Chipman & Thomas, 1987; National Science Foundation, 1986; Oakes, 1990; Secada, 1992). Students with disabilities may also perform much better in mathematics if they have appropriate learning opportunities and support (Silver, Smith & Nelson, 1995; Thornton & Langrall, 1997).

Because the new instructional goals and teaching practices articulated in the NCTM (2000) Standards are meant to recognize and respond to student diversity, researchers and policy makers are confident they will help bridge the achievement gap. Our vignette is evidence of how mathematical tasks can be designed to provide access to students with diverse learning styles, strengths and background experiences. An open-ended task, such as finding the area of a “fish,” offers many more opportunities for success for all students than traditional tasks that recognize only one correct solution and one way to achieve it. Multiple forms of assessment, as exemplified in our vignette by the combination of a group performance assessment and more traditional paper-and-pencil tests, may also help students with different strengths and learning styles to show more easily what they know.

However, taking on new instructional goals and teaching practices will not be enough for teachers to fully address equity issues in school mathematics. Each teacher must first gain a good understanding of the many issues related to equity and diversity and their implications for mathematics instruction (Darling-Hammond, 1998). Teachers must also become aware of their own biases and privileges and learn how these may affect their relationship with students who are different with respect to race, class, gender, primary language, sexual orientation, etc. (Weissglass, 1996). Teachers must also believe that all students can learn mathematics when they are provided with ample opportunities, conditions conducive to learning and high teacher expectations.

Teachers also need to know how to identify their students’ unique needs and how to differentiate instruction to address those needs. For example, it was important for the teacher in our vignette to recognize the different strengths and abilities of her students in order to place them with an appropriate partner for the final project; the same knowledge enabled her to offer additional scaffolding for some students who needed it. To respond to students with specific learning disabilities, teachers may need knowledge that is even more specialized.

Coping with the emotional aspects of engaging in reform

Several reform projects have noted that emotions, both positive and negative, inevitably accompany efforts to change one’s teaching practices (Clarke, 1994; Ferrini-Mundy, 1997). A participant in one of our professional development projects aptly described her initial experiences in instructional innovation as an “emotional roller-coaster”; at times she felt elated by her students’ success and the depth of their mathematical thinking, but she could also sink into dejection from an unsuccessful instructional experience she had spent hours putting together or from the opposition presented by a parent or administrator. Some teachers may suddenly feel inadequate after years of perceiving themselves as successful teachers and may even blame themselves for “doing it wrong.”

Studies of learning and problem solving show that behavioral changes often engender strong feelings of anxiety, frustration and elation (McLeod, 1992). Teachers need to know that conflicting feelings will inevitably arise and they need to find ways to cope with these feelings. If emotional needs are not directly addressed, teachers may even drop out of professional development programs and reform efforts. Weissglass (1993) has suggested that “any reform that does not provide methods for people to systematically and profoundly address their feelings, emotions and values related to reform will be inadequate.” (p. 3)

For teachers to recognize and deal constructively with feelings, they need, among other things, to break the isolation that so often characterizes teachers’ work. The need for teachers to share ideas and feelings with other teachers involved in research and reform has been long recognized in the teacher education literature (e.g., Clark, 1994). Quality professional development programs should strive to meet this need by creating opportunities for teacher collaboration.

Continued




		Figure 7 NSF-funded exemplary comprehensive mathematics curricula Elementary school (K-5): • Everyday Mathematics (http: //ars-www.uchicago.edu/ucsmp-el/)* • Investigations in Number, Data and Space (http://www.terc.edu/investigations)* • Math Trailblazers (http://www.math.uic.edu/IMSE/MTB/mtb.html)* Middle school (5-8): • Connected Mathematics Project (CMP) (http: //www.math.msu.edu/cmp)* • Mathematics in Context (MiC) (http://www.ebmic.com)* • MathScape (http: //www.edc.org/mcc/cscape.htm)* • Middle Grades Math Thematics (http: //www.math.umt.edu/~stem/)* • Middle School Mathematics through Applications Project (MMAP) (http://mmap.wested.org)* High school (9-12): • Contemporary Mathematics in Context (CORE-Plus) (http://www.wmich.edu/cpmp)* • Interactive Mathematics Program (IMP) (http://www.mathimp.org)* • Math Connections (http: //www.mathconnections.org)* • Mathematics: Modeling our World (ARISE) (http://www.comap.com)* • SIMMS Integrated Mathematics (http: //www.montana.edu/~wwwsimms)* * web addresses are current at time of publication In order to be considered “exemplary,” a unit or comprehensive curriculum must be consistent with the NCTM Standards, designed by groups of specialists in mathematical content and pedagogy, and revised based on field tests in various instructional settings. Exemplary instructional materials are much more than a textbook for students. They usually include a rich collection of documents to support learning experiences. The documents may include suggestions for planning lessons and orchestrating class discussions, examples of student work, tools and rubrics for assessment, and opportunities for teachers to learn more about the mathematical concepts to be taught. While there is certainly a value for teachers to create their own innovative lessons and units, the results of the multitude of Teacher Enhancement and Local Systemic Change projects supported by the NSF in the last two decades suggest that the use of exemplary comprehensive mathematics curricula is critical to the success of systemic reform. That is, if the goal is to reform the entire mathematics program within a given school or district, not just to improve the practices of a few committed teachers, it is very difficult to achieve significant success unless the system adopts a coherent curriculum that ensures that students engage in a well-constructed sequence of worthwhile mathematics experiences, and frees teachers to focus their energy on improving their instructional practices and evaluating their students’ learning. While exemplary instructional materials can revolutionize the way we approach school mathematics reform (Ball & Cohen, 1996; Russell, 1997), they also require considerable time (and, in some cases, special expertise) to be used efficiently. Therefore, professional development programs should include opportunities for teachers to become familiar with at least some exemplary instructional materials, selected so as to maximize the participants’ opportunities to implement reform in their classes. Teachers also need to learn about high quality software and other technological tools if they are to implement mathematical learning experiences consistent with the most recent calls for reform. Indeed, new technologies such as graphing calculators, spreadsheets, and programs like the “Geometer’s Sketchpad” and statistical packages like “Fathom,” have radically changed the way certain mathematical topics can be taught in school (e.g., Dunham & Dick, 1994; Rojano, 1996). Teachers need to become proficient users of these technologies and to learn to consider how using these tools could affect not only their teaching practices but also their instructional goals. Understanding equity issues and their implications for the classroom At the forefront of the current call for school mathematics reform is the directive that all students should have opportunities to learn mathematics (NCTM, 1989, 2000; Secada, Fennema & Adajian, 1995). The underachievement of some ethnic minorities and women has been the cause of serious concern and one of the reasons that led to the recent critical scrutiny of curricula and teaching practices (Chipman & Thomas, 1987; National Science Foundation, 1986; Oakes, 1990; Secada, 1992). Students with disabilities may also perform much better in mathematics if they have appropriate learning opportunities and support (Silver, Smith & Nelson, 1995; Thornton & Langrall, 1997). Because the new instructional goals and teaching practices articulated in the NCTM (2000) Standards are meant to recognize and respond to student diversity, researchers and policy makers are confident they will help bridge the achievement gap. Our vignette is evidence of how mathematical tasks can be designed to provide access to students with diverse learning styles, strengths and background experiences. An open-ended task, such as finding the area of a “fish,” offers many more opportunities for success for all students than traditional tasks that recognize only one correct solution and one way to achieve it. Multiple forms of assessment, as exemplified in our vignette by the combination of a group performance assessment and more traditional paper-and-pencil tests, may also help students with different strengths and learning styles to show more easily what they know. However, taking on new instructional goals and teaching practices will not be enough for teachers to fully address equity issues in school mathematics. Each teacher must first gain a good understanding of the many issues related to equity and diversity and their implications for mathematics instruction (Darling-Hammond, 1998). Teachers must also become aware of their own biases and privileges and learn how these may affect their relationship with students who are different with respect to race, class, gender, primary language, sexual orientation, etc. (Weissglass, 1996). Teachers must also believe that all students can learn mathematics when they are provided with ample opportunities, conditions conducive to learning and high teacher expectations. Teachers also need to know how to identify their students’ unique needs and how to differentiate instruction to address those needs. For example, it was important for the teacher in our vignette to recognize the different strengths and abilities of her students in order to place them with an appropriate partner for the final project; the same knowledge enabled her to offer additional scaffolding for some students who needed it. To respond to students with specific learning disabilities, teachers may need knowledge that is even more specialized. Coping with the emotional aspects of engaging in reform Several reform projects have noted that emotions, both positive and negative, inevitably accompany efforts to change one’s teaching practices (Clarke, 1994; Ferrini-Mundy, 1997). A participant in one of our professional development projects aptly described her initial experiences in instructional innovation as an “emotional roller-coaster”; at times she felt elated by her students’ success and the depth of their mathematical thinking, but she could also sink into dejection from an unsuccessful instructional experience she had spent hours putting together or from the opposition presented by a parent or administrator. Some teachers may suddenly feel inadequate after years of perceiving themselves as successful teachers and may even blame themselves for “doing it wrong.” Studies of learning and problem solving show that behavioral changes often engender strong feelings of anxiety, frustration and elation (McLeod, 1992). Teachers need to know that conflicting feelings will inevitably arise and they need to find ways to cope with these feelings. If emotional needs are not directly addressed, teachers may even drop out of professional development programs and reform efforts. Weissglass (1993) has suggested that “any reform that does not provide methods for people to systematically and profoundly address their feelings, emotions and values related to reform will be inadequate.” (p. 3) For teachers to recognize and deal constructively with feelings, they need, among other things, to break the isolation that so often characterizes teachers’ work. The need for teachers to share ideas and feelings with other teachers involved in research and reform has been long recognized in the teacher education literature (e.g., Clark, 1994). Quality professional development programs should strive to meet this need by creating opportunities for teacher collaboration. Continued