NCTM Standards and 7th Grade Algebra Lesson Plan Analysis

Introduction: NCTM Framework and Algebra in the Curriculum

The National Council of Teachers of Mathematics (NCTM) provides a comprehensive set of principles and standards for developing curriculum for grades K through 12. Chapter two of their text Principles and Standards for School Mathematics specifies the six principles considered vital for the development of a coherent mathematics plan. The principles are general enough to apply across a wide variety of disciplines, as they are "not unique to school mathematics" (p. 16). However, chapter three, which deals with the ten standards themselves, makes clear — and rightly so — that mathematics, unlike other disciplines, can benefit from a truly integrated approach: "Because mathematics as a discipline is highly interconnected, the areas described by the Standards overlap and are integrated" (p. 30). In other words, the standards cannot be easily divided into particular grade levels (e.g., numbers and operations in K–2, geometry in 3–5, algebra in 6–8). All ten standards are relevant to all grade levels from K through 12. The only thing that changes at each particular level is the emphasis placed on any given standard relative to the others.

Consider algebra — the main topic of this paper. Teaching in this area of mathematics should not be saved for middle school or early high school. Instead, it should be seen "as a strand in the curriculum from pre-kindergarten on, [and] teachers can help students build a solid foundation ... as a preparation for more-sophisticated work in algebra in the middle grades" (p. 36). Later in the text, chapter six, which details the 6–8 grade standards, notes that spending an entire middle-grade year dealing only with algebra, geometry, or any other single subject type is inadvisable: "Instruction that segregates the content of algebra or geometry from that of other areas is educationally unwise and mathematically counterproductive" (p. 212).

With this in mind, the purpose of this paper is to analyze three specific lesson plans, each of which concentrates on a separate aspect of algebra covered in a typical 7th grade classroom: Solving Equations, Positive Exponents, and Translating Words into Equations. The following topics will be addressed as they relate to the plans:

1. The NCTM Principles and the Lesson Plans
2. The NCTM Standards and the Lesson Plans
3. Addressing Algebra Misconceptions in the Lesson Plans
4. Addressing the Transition from Arithmetic to Algebra in the Lesson Plans

The NCTM Principles and the Lesson Plans

The six principles are woven into the three lesson plans in both subtle and direct ways. To begin with, equity is addressed through the emphasis on technology. PowerPoint presentations with animation make specific topics come alive for students. More importantly, each student can progress at a rate appropriate to his or her individual ability level. Because "technology can be effective in attracting students who disengage from non-technological approaches to mathematics," this approach helps bridge the equity gap (p. 13).

The curriculum principle deals with "important mathematics — mathematics that will prepare students for solving problems in a variety of school, home, and work settings" (p. 14). In other words, mathematics is to be seen as relevant to life outside the classroom. The three lessons that form the subject of this paper deal with topics that have broad application to everyday life.

Regarding the teaching principle, the NCTM text is explicit. Teachers need "to understand and be committed to their students as learners of mathematics and as human beings and be skillful in choosing from and using a variety of pedagogical and assessment strategies" (p. 16). This need for diversity in approach is vital. Too often students see mathematics as a boring array of numbers and formulas. Because "well-chosen tasks can pique students' curiosity and draw them into mathematics" (p. 18), introducing a novel PowerPoint and computer-based approach can add variety to otherwise dry material.

This connects directly to the learning principle. The hands-on approach of these three lessons contrasts sharply with rote memorization. One of the worst things a mathematics student can do is "memorize facts or procedures without understanding, [because they] often are not sure when or how to use what they know, and such learning is often quite fragile" (p. 19). The ultimate "goal of school mathematics programs is to create autonomous learners" (p. 20). Emphasizing self-paced computer learning, as these lessons do, is a meaningful step toward that goal.

Regarding the assessment principle, this is addressed by the final worksheets due upon completion of each PowerPoint lesson. However, "[f]ormal assessments provide only one viewpoint on what students can do in a very particular situation — often working individually on paper-and-pencil tasks, with limited time to complete the tasks. Over-reliance on such assessments may give an incomplete and perhaps distorted picture of students' performance" (p. 22). With these lessons, the final worksheets serve as only one concrete measure of student progress. The instructor is also expected to circulate during the lesson to observe, answer questions, and offer hands-on assistance. This active involvement serves two purposes: providing assistance and offering an additional, interactive dimension to the evaluation of student performance.

The final principle is technology, already referenced above. Technology provides an enriching experience in any subject area, and mathematics is no exception. "Technology offers teachers options for adapting instruction to special student needs. Students who are easily distracted may focus more intently on computer tasks, and those who have organizational difficulties may benefit from the constraints imposed by a computer environment" (p. 24).

The NCTM Standards and the Lesson Plans

There are ten standards to be considered when formulating mathematics lesson plans. These standards can be discussed from a general perspective — applying across all grade levels from K through 12 — and from a more specific middle school perspective (grades 6–8). Not all ten standards are directly relevant here. The standards addressed in this discussion are: number and operations, algebra, problem solving, and representation.

Under number and operations, one specific student expectation is to "develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation" (p. 213). This expectation is directly addressed in the lesson plan on Positive Exponents. Also listed under that standard's expectations is that students "understand and use the inverse relationships of addition and subtraction" — a technique applied directly in the Solving Equations lesson plan.

Naturally, the algebra standard bears most directly on the lesson plans under consideration. It sets expectations that students will "develop an initial conceptual understanding of different uses of variables," which is addressed in the Solving Equations lesson (p. 221). Specifically, the standard calls for increased use of algebraic symbols and the development of "an understanding of several different meanings and uses of variables through representing quantities in a variety of problem situations" (p. 222). These are all items that appear in that first lesson.

The problem-solving standard emphasizes that mathematical problem solving must "allow applications of mathematics to other contexts. Many interesting problems can be suggested by everyday experiences" (p. 255). All three lesson plans acknowledge this by consistently referencing real-life examples and situations to which students can relate. Related to this is the connections standard, which calls on students to "recognize and apply mathematics in contexts outside of mathematics" (p. 273).

The final standard considered here is representation. Mathematical representations "can help students organize their thinking. Students' use of representations can help make mathematical ideas more concrete and available for reflection." As applied to middle-grade instruction, representations are used "more to solve problems or to portray, clarify, or extend a mathematical idea" (p. 67). This standard is evident in all three lesson plans: variables in equations, scientific and exponential notation, and mathematical language are all representational forms.

Conclusion

Solving Equations. Retrieved March 29, 2003, from [redacted] web site:

Tall, David. (1992). The transition from arithmetic to algebra: Number patterns, or proceptual programming? Retrieved April 3, 2003, from University of Warwick, Mathematics Education Research Centre web site:

Translating Words into Equations. Retrieved March 29, 2003, from [redacted] web site:

Unit 16 Algebra: Linear Equations Teaching Notes. (n.d.). Retrieved April 1, 2003, from http://ex.ac.uk/cimt/mepres/book7/y7s16tn.pdf.