Foundations
of Math
Trailblazers
Math Trailblazers is based on the ideas that mathematics is best learned in real-world contexts that make sense to children; that all students deserve a richer and more challenging curriculum; and that a balanced and practical approach to mathematics learning is what students need and what teachers want.
Features of Math TrailblazersAlignment with Reform Recommendations
NCTM’s Principles and Standards provides the most current and comprehensive set of recommendations for improving the teaching and learning of mathematics. The Principles and Standards is an update of three, groundbreaking volumes published a decade earlier that collectively became known as the "NCTM Standards." The Principles and Standards, which were released in 2000, largely reflect the same consensus about how to reform mathematics education as found in the earlier NCTM recommendations.
The NCTM documents outline a vision for school mathematics that includes: a curriculum that is mathematically challenging, coherent, and covers a broad range of mathematical content; a strong focus on engaging students in mathematical problem solving; instruction that is conceptually oriented and stresses thinking, reasoning, and applying; appropriate uses of calculators and computers; and a strong commitment to promoting success in mathematics among all students—not merely those who traditionally have done well. This vision for mathematics teaching and learning was incorporated into the mathematics standards of most states.
The first edition of Math Trailblazers was the result of more than six years work, that included extensive pilot and field testing of the materials. After publication of the first edition, continued NSF support enabled the TIMS Project to develop implementation models and professional development materials that assisted teachers and schools as they adopted the program. TIMS researchers also studied how the program impacted student achievement. The second edition of Math Trailblazers therefore represents over twelve years of NSF-supported work related to the curriculum.
In developing the second edition of Math Trailblazers, we looked carefully at the NCTM Principles and Standards. The reorganization of the Math Trailblazers math facts program in the second edition reflects new recommendations in the Principles and Standards and keeps Math Trailblazers closely aligned with NCTM recommendations for mathematics curricula.
The importance of a coherent, well-developed curriculum is clearly articulated in the Principles and Standards which state:
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A curriculum is more than a collection of activities: it must
be coherent, focused on important mathematics, and well articulated
across the grades. A school mathematics curriculum is a strong determinant of what
students have an opportunity to learn and what they do learn.
In a coherent curriculum, mathematical ideas are linked to and
build on one another so that students' understanding and knowledge
deepens and their ability to apply mathematics expands. An effective
mathematics curriculum focuses on important mathematics—mathematics
that will prepare students for continued study and for solving
problems in a variety of school, home, and work settings. A well-articulated
curriculum challenges students to learn increasingly more sophisticated
mathematical ideas as they continue their studies. (NCTM, 2000; p. 14)
Math Trailblazers was developed with that goal in mind. It is a Standards-based curriculum in the truest sense— developed explicitly to reflect national standards for K–5 mathematics, and now revised to maintain close alignment with those standards as they have evolved.
More specific connections between Math Trailblazers and the NCTM Principles and Standards are discussed in Section 5 of the Teacher Implementation Guide Scope and Sequence & the NCTM Principles and Standards.
High Expectations and Equity
The first principle in the Principles and Standards is the Equity Principle. It challenges a myth prevalent in the United States that only a few students are capable of rigorous mathematics. The Equity Principles states:
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Excellence in mathematics education requires equity-high expectations
and strong support for all students. (NCTM, 2000, p. 12)
Accordingly, we introduce challenging content in every grade: computation, measurement, data collection, statistics, geometry, ratio, probability, graphing, algebraic concepts, estimation, mental arithmetic, and patterns and relationships.
Contexts for this demanding content begin with students’ lives. Lessons are grounded in everyday situations, so abstractions build on experience. By presenting mathematics in rich contexts, the curriculum helps students make connections among real situations, words, pictures, data, graphs, and symbols. The curriculum also validates students’ current understandings while new understandings develop. Students can solve problems in ways they understand while being encouraged to connect those ways to more abstract and powerful methods. Within the same lesson, some students may work directly with the manipulatives to solve the problems while other students may solve the same problems using graphs or symbols. The use of varied contexts and diverse representations of concepts allows children of varying abilities to access the mathematics.
Problem Solving
A fundamental principle of Math Trailblazers is that mathematics is best learned through active involvement in solving real problems. Questions a student can answer immediately may be worthwhile exercises, but problems, by definition, are difficult.
The importance of problem solving is echoed in the NCTM Principles and Standards, which state:
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Problem solving is the cornerstone of school mathematics….The
goal of school mathematics should be for all students to become
increasingly able and willing to engage with and solve problems.
Problem solving is also important because it can serve as a vehicle for learning new mathematical ideas and skills. A problem-centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion. (NCTM, 2000, p. 182)
As recommended by NCTM, problem solving in Math Trailblazers is not a distinct topic but permeates the entire program, providing a context for learning concepts and skills. Throughout the curriculum, students apply the mathematics they know and construct new mathematics as needed. Students’ skills, procedures, and concepts emerge and develop as they solve complex problems.
Connections to Science and Language Arts
Real-world problems are naturally interdisciplinary, so that any problem-solving curriculum should integrate topics that are traditionally separated. Accordingly, we have integrated mathematics with many disciplines, especially science and language arts.
Connections to Science
Math Trailblazers is a full mathematics program that incorporates many important scientific ideas. Traditionally, school science has focused on the results of science. Students learn about plate tectonics, the atomic theory of matter, the solar system, the environment, and so on. Knowing basic facts of science is seen as part of being educated, today more than ever. However, the facts of science, important and interesting as they are, do not alone comprise a comprehensive and balanced science curriculum.
Science has two aspects: results and method. The results of scientific inquiry have enriched human life the world over. More marvelous than the results of science, however, is the method that has established those results. Without the method, the results would never have been achieved. Math Trailblazers aims to teach students the method of science through scientific investigation of everyday phenomena. During these investigations, students learn both mathematics and science.
The method scientists use is powerful, flexible, and quantitative. The TIMS Project has organized this method in a way that is simple enough for elementary school children to use. Students use the TIMS Laboratory Method in a rich variety of mathematical investigations. In these investigations, students develop and apply important mathematical skills and concepts in meaningful situations. Their understanding of fundamental scientific concepts is also enhanced through the use of quantitative tools.
Investigations begin with discussions of experimental situations, variables, and procedures. Then, students draw pictures in which they indicate the experimental procedures and identify key variables. Next, students gather data and organize it in data tables. Then, they graph their data. By analyzing their data or studying their graphs, students are able to see patterns in the data. These patterns show that there is a relationship between the variables. The relationship can be used to make predictions about future data. The last phase of the experiment is an in-depth analysis of the experimental results, structured as a series of exploratory questions. Some questions ask students to make predictions and then to verify them. Other questions probe students’ understanding of underlying concepts and explore the role of controlled variables. As students advance through the curriculum, the questions progress from simple to complex, building eventually to problems that require proportional reasoning, multiple-step logic, and algebra.
The TIMS Laboratory Method initiates children into the authentic practice of science. Identifying variables, drawing pictures, measuring, organizing data in tables, graphing data, and looking for and using patterns are a major part of many scientists’ work. This is a major goal of science: to discover and use relationships between variables—usually expressed in some mathematical form—in order to understand and make predictions about the world.
The science content in Math Trailblazers focuses on a small set of simple variables that are fundamental to both math and science: length, area, volume, mass, and time. Understanding these basic variables is an essential step to achieving scientific understanding of more complex concepts. Measurement is presented in meaningful, experimental situations.
Emphasizing the scientific method and fundamental science concepts helps students develop an understanding of how scientists and mathematicians think. These habits of mind will be important in all aspects of life in the 21st century. See the TIMS Tutor: The TIMS Laboratory Method in the Teacher Implementation Guide for more discussion of these ideas.
Connections to Language Arts
Reading, writing, and talking belong in mathematics class, not only because real mathematicians and scientists read, write, and talk mathematics and science constantly, but also because these activities help students learn. The NCTM Principles and Standards emphasizes the importance of communication and discourse for students to achieve at higher levels.
In school mathematics, results should be accepted not merely because someone in authority says so, but rather because persuasive arguments can be made for them. A result and a reason are often easier to remember than the result alone. By discussing the mathematics they are doing, students increase their understanding of that mathematics. They also extend their ability to discuss mathematics and so to participate in a community of mathematicians. Talking and writing about mathematics, accordingly, are part of every lesson. Journal and discussion prompts are standard features in the teacher’s guide for each lesson.
Reading is also built into this curriculum. Many lessons, especially in the primary grades, use trade books to launch or extend mathematical investigations. The curriculum also includes many original stories, called Adventure Books, that show applications of concepts being studied or sketch episodes from the history of mathematics and science. Literature is used to portray mathematics as a human endeavor, so that students come to think of mathematicians and scientists as people like themselves. Mathematics embedded in a narrative structure is also easier to understand, remember, and discuss. And, of course, everyone loves a good story. In addition, students regularly write about their mathematical investigations, even in the early grades.
Collaborative Work
Scientists, mathematicians, and most others who solve complex problems in business and industry have always worked in groups. The reasons for this are not hard to understand: Most interesting problems are too difficult for one person working alone. Explaining one’s work to another person can help clarify one’s own thinking. Another person’s perspective can suggest a new approach to an unsolved problem. Ideas that have been tested through public scrutiny are more trustworthy than private notions.
All these are reasons for collaborative work in schools as well. But there are other reasons, too. Students can learn both by receiving and by giving explanations. The communication that goes with group work provides practice in verbal and symbolic communication skills. In group discussion, a basic assumption is that mathematics and science ought to make sense—something, unfortunately, that many students cease to believe after only a few years of schooling. Social skills, especially cooperation and tolerance, increase, and the classroom community becomes more oriented towards learning and academic achievement.
Assessment
There are three major purposes for assessment. First, assessment helps teachers learn about students’ thinking and knowledge; this information can then be used to guide instruction. Secondly, it communicates the goals of instruction to students and parents. Finally, it informs students and parents about progress toward these goals and suggesting directions for further efforts.
Assessment in Math Trailblazers reflects the breadth and balance of the curriculum. Numerous opportunities for both formal and informal assessment of student learning are integrated into the program. Many of the assessment activities are incorporated into the daily lessons; others are included in formal assessment units. Assessment activities include a mix of short, medium-length, and extended activities. Some are hands-on investigations; others are paper-and-pencil tasks. In all cases, assessment activities further students' learning.
For a detailed discussion of assesment in Math Trailblazers, refer to Assessment (Section 8) and the TIMS Tutor: Portfolios in the Teacher Implementation Guide.
More Time Studying Math
One cannot reasonably expect to cover all the concepts in a traditional program, add many new topics, and utilize an approach that emphasizes problem solving, communication, reasoning, and connections in the same amount of time that is used to teach a traditional math curriculum. In developing Math Trailblazers, we have attempted to achieve some efficiencies, such as building review into new concepts and making use of effective strategies to ease the learning of math facts and procedures. However, we make no appeal to magic with Math Trailblazers. Implementing a comprehensive, reform mathematics curriculum will require a significant amount of time. We assume that in Grade 1–5 one hour every day will be devoted to teaching mathematics. In Kindergarte, the make-up of the class and the activity will determine the length of the class.
In some schools, finding an hour per day for math instruction may require a restructuring of the daily school schedule. Please note, however, that because Math Trailblazers includes extensive connections with science and language arts, some time spent with Math Trailblazers can be incorporated within science or language arts time. Thus, it may be possible to allot the one hour per day of mathematics instruction by simply scheduling math and science instruction back to back or including literature connections and journal writing in language arts.
Hard work on the part of students and more time spent engaged in mathematical problem solving are important ingredients for success in mathematics—no matter what program you are using. Because students using Math Trailblazers will be actively involved in applying mathematics in meaningful contexts, our experience is that students will be highly motivated to spend this extra time studying mathematics.
Staff Development and Broad School Support: Essential Ingredients
Our experience developing the curriculum and working with schools over the last decade has taught us that implementing a Standards-based mathematics program will go much more smoothly if it is accompanied by a solid support system for teachers. Ideally, this includes a professional development program that includes workshops on content and pedagogy, leadership developmet, in-school support, and the means to address a variety of concerns as they arise.
Overall implementation of Math Trailblazers will also be much easier in a school that organizes support for the new program. Necessary tools such as manipulatives, overhead projectors, and transparency masters need to be provided in adequate quantities. Institutional considerations, such as classroom schedules that allot necessary time for instruction and even provide time for teachers to meet and plan together, need to be implemented. In-classroom support from resource teachers is extremely helpful. Storage and check-out systems for shared manipulatives need to be in place. Ways to engender parental support for the program, including possible ways to involve some parents in providing classroom assistance to interested teachers during math time, should be developed.
Administrators need to maintain a long-term perspective on program implementation, recognizing that it will take some time before teachers and students are in “full gear” with the program—and they need to communicate this clearly to teachers. In short, schools need to examine their current situations and make necessary modifications to develop a school environment that supports the kind of teaching and learning that characterizes Math Trailblazers.
A Balanced Approach
A reform mathematics program should take a balanced and moderate approach. Math Trailblazers is balanced in many different ways. Whole-class instruction, small-group activities, and individual work each have a place. New mathematical content is included, but traditional topics are not neglected. Children construct their own knowledge in rich problem-solving situations, but they are not expected to reinvent 5000 years of mathematics on their own. Concepts, procedures, and facts are important, but these are introduced thoughtfully to engender the positive attitudes, beliefs, and self-image that are also important in the long run. Hands-on activities of varied length and depth, as well as a variety of paper-and-pencil tasks, all have their place. The program’s rich variety of assessment activities has been designed to reflect this broad balance.
Either-or rhetoric has too often short-circuited real progress in education: problem solving vs. back-to-basics, conceptual understanding vs. procedural skill, paper-and-pencil computation vs. calculators. Math Trailblazers is based on the view that these are false dichotomies. Students must solve problems, but of course they need basic skills to do so. Both concepts and procedures are important, and neglecting one will undermine the other. There is a place in the curriculum both for paper-and-pencil algorithms and for calculators, and for mental arithmetic and estimation. The careful balance in Math Trailblazers allows teachers and schools to move forward with a reform mathematics curriculum while maintaining the strengths of their current teaching practices.
Research Foundations of Math Trailblazers
The first edition of Math Trailblazers was completed in 1997.
The roots of the curriculum, however, date back to the late 1970s
and the work of Howard Goldberg, a particle physicist at the University
of Illinois at Chicago (UIC). Building on the work of Robert Karplus
and others, Goldberg developed a framework for adapting the scientific
method for elementary school children and applied that framework
within a series of elementary laboratory investigations (Goldberg
& Boulanger, 1981). Goldberg was joined in 1985 by UIC mathematician
Philip Wagreich and, together with others, formed the Teaching
Integrated Mathematics and Science Project (TIMS).
Thus, the motivation behind TIMS came from two sources, one with
roots in science and one with roots in mathematics. The scientific
impetus stemmed from the desire to teach science to children in
a manner that reflects the practice of scientists. Modern science
is fundamentally quantitative; therefore, quantitative investigations
should have an important role in children’s science education.
The mathematical impetus was to find a way to make mathematics
meaningful to children. Engaging children in quantitative investigations
of common phenomena harnesses their curiosity in a way that builds
upon their understanding of the natural world, which, in turn,
helps promote study of rigorous mathematics (Isaacs, Wagreich,
& Gartzman, 1997).
Initially, TIMS focused primarily on the development and implementation
of a series of quantitative, hands-on activities, The TIMS Laboratory Investigations (Goldberg, 1997).
The laboratory investigations continue the tradition of Dewey (1910) and others in focusing on the method
of science and on exploring a relatively small set of fundamental
scientific concepts in depth. Several studies provided clear evidence
that the TIMS Laboratory Investigations are effective in promoting students’ development of mathematics
and science concepts (Goldberg & Wagreich, 1990; Goldberg, 1993).
Based largely upon the success with the supplemental TIMS Laboratory Investigations, the National Science
Foundation (NSF) supported the TIMS Project
to develop a comprehensive, elementary mathematics curriculum
that would align with the National Council of Teachers of Mathematics
(NCTM) Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). The first edition of
Math Trailblazers is the result of this work. As part of the revision process for
the second edition, the authors made changes so that the new edition
aligns with the current NCTM recommendations as outlined in the
Principles and Standards for School Mathematics (NCTM, 2000).
In developing Math Trailblazers, the authors drew upon research findings from a wide variety
of sources, as well as from the broad experiences of the authoring
group. In this section we highlight a portion of the research
that helped guide the curriculum’s development and other current research that validates the approaches used.
A Problem-Solving Curriculum
A distinguishing element of any Standards-based mathematics curriculum is that problem solving is used as a context for students to learn new concepts and skills. Throughout Math Trailblazers, students apply the mathematics they know and construct new mathematics as needed. Students’ skills, procedures, and concepts emerge and develop as they solve problems. Students also practice skills and procedures as they apply them in diverse and increasingly challenging contexts. Problem-solving contexts can emerge from real-life investigations or from more mathematical situations. The problems serve as the motivation for purposeful use of mathematics.
Using problem solving as a cornerstone of the mathematics curriculum has been promoted by educators dating back to Dewey and has received considerable support from current mathematics education researchers. (See Hiebert et al., 1996; Schoenfeld, 1985 and 1994; National Research Council, 2001; and NCTM 1989 and 2000 for a discussion of problem solving in the curriculum.)
A curriculum that emphasizes solving problems can use many different contexts for problem situations. In developing Math Trailblazers, the authors first identified the key mathematical concepts and skills to be developed and then selected problem-solving contexts that would support student learning in these areas. Our earlier work and that of others (e.g., Cognition and Technology Group at Vanderbilt,1997) had confirmed the value of using quantitative laboratory investigations for this purpose. Drawing from this work, eight to ten laboratory investigations were adapted and integrated into Math Trailblazers in each grade, beginning with first grade. These investigations are supplemented by many other contexts for problem solving.
Challenging Mathematics in All Grades for All Students
An assumption in the development of Math Trailblazers was that students can learn more challenging mathematics than is covered in traditional mathematics curricula. This has been underscored in international comparison studies (Heibert, 1999; McKnight et al, 1987; Stigler & Perry, 1988; Schmidt, McKnight, & Raizen, 1996; Stigler & Hiebert, 1997; Stevenson & Stigler, 1992) and in analyses of U.S. mathematics instruction (Flanders, 1987; Lindquist, 1989). Maintaining high expectations and providing access to rigorous mathematics for all students has proven effective regardless of the socioeconomic status and ethnicity of the students (National Research Council, 2001; Newmann, Bryk, & Nagaoka, 2001; Smith, Lee, and Newmann, 2001). Therefore, mathematical expectations in Math Trailblazers are high and grow steadily over time. Review of concepts and skills is carefully built into new and increasingly challenging problems as the program builds upon itself both within and across grades.
Balancing Concept and Skill Development
The importance of developing a strong conceptual foundation is emphasized in research that guided the development of all content strands in Math Trailblazers. Many educators have long recognized the importance of balancing conceptual and skill development. New conceptual understandings are built upon existing skills and concepts; these new understandings in turn support the further development of skills and concepts.
A wide body of research affirms that instructional programs that emphasize conceptual development can facilitate significant mathematics learning without sacrificing skill proficiency (see Hiebert, 1999). Research affirms that well-designed and implemented instructional programs can facilitate both conceptual understanding and procedural skill.
This research had a profound effect on development of Math Trailblazers. For example, such research helped define the curriculum’s program for developing number sense and estimation skills (Sowder, 1992) and for learning math facts and whole-number operations (Carpenter, Carey, & Kouba, 1990; Carpenter, et al., 1999; Fuson, 1987, 1992; Fuson & Briars, 1990; Fuson et al., 1997; Isaacs & Carroll, 1999; Lampert, 1986a and 1986b; Thornton, 1978, 1990a, 1990b). Work with fractions and decimals was informed by research that stressed the need to develop conceptual understandings of fractions and decimals prior to introducing procedures with the four arithmetic operations (see Behr & Post, 1992; Ball, 1993; Cramer, Post, & del Mas, 2002; Lesh, Post, & Behr, 1987; Mack, 1990, Hoffer & Hoffer, 1992). Similar research findings on student understanding of geometric concepts shaped the development of lessons in geometry (see Burger & Shaughnessy, 1986; Crowley, 1987; Fuys, Geddes, & Tishler, 1988).
In direct response to research such as that cited here, every content strand within Math Trailblazers interweaves the promotion of conceptual understanding with distributed practice of skills and procedures.
Field Testing and Research on the Curriculum in Classrooms
NSF funding allowed sequential development of the program to be coupled with extensive field testing in schools. As a result, curriculum development became an iterative process involving considerable interaction between the developers and field-test teachers. Comments and suggestions from teachers were incorporated into revisions that were often retested in classrooms one or more times before final revisions were made. Field-test teachers met regularly with the program’s authors, and classrooms were often visited as part of the development process. Content, language, format, practical considerations, and other issues related to each lesson were addressed by teachers who were using the early versions in their classrooms.
In addition to assessing feedback from field-test teachers, independent researchers and TIMS Project staff conducted preliminary studies that helped affirm the efficacy of the content placement and approaches in the early versions of the materials (Burghardt, 1994; Perry, Whiteaker, & Waddoups, 1996; Whiteaker, Waddoups, & Perry, 1994). These preliminary studies provided an additional means for the authors to monitor the effectiveness of the developing program.
Early studies of student achievement in Math Trailblazers classrooms in both urban and suburban schools have shown that students using the curriculum performed as well, and often better, on mandated standardized tests than students in those schools prior to implementation of Math Trailblazers. (Carter, et al., 2003; Putnam, 2003) "[The results] indicate that the balanced problem-solving approach found in Math Trailblazers has been successful in improving student learning and achievement in mathematics." (Carter, et al., 2003)
In developing the second edition, we benefited from the suggestions and comments of teachers throughout the country who were using the published Math Trailblazers materials. Input from teachers, for example, resulted in the major changes to the second edition’s teacher guides. Teachers’ feedback also directed us to modify or eliminate some lessons.
Extensive field testing during the program’s development, current research and evaluation in schools, and our ongoing conversations with Math Trailblazers teachers have helped us develop a challenging, yet grade-level-appropriate program that reflects the needs and realities of teachers and students.
References
The following references are cited in the above document. Additional references to research literature are included in many Unit Resource Guides following the Background and in many sections of the Teacher Implementation Guide including the Assessment Section and the TIMS Tutors.
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NOTE: Above text taken from Math Trailblazers Teacher Implementation
Guide (TIG)
Copyright © 1998 by Kendall/Hunt Publishing Company. Used with
permission.
Copyright © 1999 Institute for Mathematics and Science Education.
All rights reserved.
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