The development of the Common Core State Standards for Mathematics was guided by research-based studies about how students’ mathematical knowledge, understandings, and skills develop over time. These standards define a balanced combination of mathematical conceptual understanding and procedural knowledge. The balance between these two areas is important because students who do not develop conceptual understanding are more likely to rely on procedures and may be less likely to engage in robust problem solving and reasoning about mathematical ideas.
Research studies in mathematics conducted over the last decade have shown the importance of and the need for focused, coherent curricula in order to improve math achievement in this country. The importance of a focused coherent mathematics curriculum is also clearly embodied in the NCTM Principles and Standards, which states:
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 or solving problems in a variety of school, home, and work setting. A well-articulated curriculum challenges students to learn increasingly more sophisticated mathematical ideas as they continue their studies. (NCTM, 2000 p. 14)
The Common Core Standards demand a focused study such as that embodied in the Math Trailblazers curriculum. Instead of a “mile wide and an inch deep” curriculum, the reorganized units in Math Trailblazers 4th Edition allows students to concentrate deeply on a narrow set of main topics by including the scaffolding students need while carefully considering which topics need more time and attention. Math Trailblazers is not designed to “cover” a topic in one unit or one month because that is not how students learn. Learners need opportunities to explore, develop, build upon, revise, solidify, and make their own meaning of a topic. This requires time. The topics are revisited throughout the year, with meaning and understanding growing deeper as the year progresses.
Within each unit, students are very focused on a narrow definition of expectations. Each unit is organized around what is assessable and what relates to each other. Consider third grade as an example. In Unit 3, students explore multiplication with models and stories and build their conceptual understanding because understanding a problem requires models, not just an algorithm. Daily Practice and Problems between units of study ensure that exploration, invention, and retention are continued. In Unit 8, students study patterns, develop multiplication strategies, and formalize what they have been inventing. Later, in Unit 10, students apply those strategies to a new setting using models to solve multiplication and division problems and reason quantitatively. They apply multiplication as scale strategies, not just repeated addition. These concepts serve as a foundation for development of whole number computation with multiplication and division. In Unit 13, students use the context of volume to solve multiplication and division problems with larger numbers. In Math Trailblazers, contexts such as volume do not detract from the focus. Rather these secondary stories support the main topic while giving students an interesting, meaningful context for real-world problems. With this design, students have the opportunity to access material such as volume or area through natural contexts that are developmentally appropriate without it distracting from the main topics of the grade level. In addition, math fact fluency is spread across the entire year at each grade level.
Math Trailblazers was developed as a coherent curriculum that provides students with robust opportunities to develop both the necessary skills and conceptual understandings needed to become mathematically proficient. The importance of developing a strong conceptual foundation is emphasized in research that guided the development of Math Trailblazers. There is a careful, deliberate, and progressive development of ideas within each grade. Major topics are linked between grades, and major content is developed over time.
The Common Core State Standards for Mathematics was written to address the challenges of coherence and focus by “stressing conceptual understanding of key ideas [in mathematics] but also by continually returning to organizing principles such as place value or the properties of operations to structure those ideas.” (CCSSM, p. 4)
Figure 1: Key Ideas from Math Trailblazers
The Math Trailblazers curriculum is organized around a set of Key Ideas that embody both mathematical content and mathematical processes. There is a set of Key Ideas defined for each content strand: Number, Algebra, Geometry, Measurement, and Data. These Key ideas are based on the “big ideas” in mathematics and describe what students will be able to do within each strand. See Figure 1.
James Kaput (2008) defines algebra in the early grades has having three focuses, the first focus is building generalizations from arithmetic and quantitative reasoning, the second involves identifying and using patterns, and the third focus involves modeling mathematics. Each area of focus is used to help elementary students make sense of the world quantitatively. In Math Trailblazers, Algebra is not viewed as separate but rather works within the other strands, offering opportunities to provide a more connected mathematical experience for students.