Algebra Lesson Plans and Curriculum for the 7th Grade Classroom
The National Council of Teachers of Mathematics (NCTM) provides a comprehensive set of principles and standards for developing curriculum for grades K. through 12th. Chapter two of their text Principles and Standards for School Mathematics specifies the six principles considered vital for the development of a coherent math 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 dealing with the ten standards, themselves, makes quite clear (and rightly so) that math, 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 (i.e. numbers/operations in K-2, geometry in 3-5, algebra in 6-8, etc.). All ten standards are relevant to all grade levels from K. through 12. The only thing that changes at each particular level is the actual emphasis placed on any given standard over another.
Consider algebra - the main topic of this paper. Teaching in this area of mathematics should not be saved until middle 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 on in the text (chapter six which details the 6-8 grade standards) mention is made of the fact that spending an entire middle grade year dealing only with algebra or geometry or any other single-type subject is a bad idea. "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 will be 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, 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 referred to above are insinuated into the three lesson plans in both subtle and direct ways. To start, Equity is obtained through the emphasis on technology. The Power Point presentations with animation make the specific topics come alive for the students. More importantly, each student can progress at a rate appropriate to his respective ability level. Because "technology can be effective in attracting students who disengage from non-technological approaches to mathematics" this unique approach will help 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, math is to be seen as important to life outside the classroom. The three lessons that form the subject of this report deal with three topics which clearly have broad application to everyday life.
Regarding the teaching principle the text is very clear. 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 to ordinary math problems is vital. Too often students see math as a boring array of numbers and formulas. Because "well-chosen tasks can pique student's curiosity and draw them into mathematics." (p. 18), introducing a novel Power Point and computer-based approach can add variety and spice to otherwise bland numbers.
And this last ties in well with the learning principle. The hands-on approach of these three lessons contrasts sharply with a mere memorization style/approach to learning math. One of the worse things for a student of math to do would be to simply "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). What better way than to attain this but to emphasize self-paced computer learning as these lessons provide.
Regarding the assessment principle, this is being handled by the final worksheets that are due upon completion of each of the Power Point lessons. 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 student's performance." (p. 22). And with these lessons, the final worksheets serve as only the one concrete measurement of student progress and ability. The instructor would also be expected to walk around the classroom to observe, answer questions and give hands-on assistance to any student who would need it. This active involvement of the instructor would then serve two important purposes: 1) providing assistance and 2) providing an additional dimension to the evaluation of students' performance as the students could be watched on an interactive basis.
The final principle is technology that has been referenced above. Technology will provide an enriching experience in any subject area - and math is certainly 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 constrains imposed by a computer environment." (p. 24).
The NCTM Standards and the Lesson Plans
There are ten standards to be considered when formulating lesson plans in mathematics. These standards can be discussed from a general perspective (standards which apply across all grade levels from K. through 12) and from a more specific middle school perspective (6-8). Not all of these ten standards are relevant in the current discussion however. Considered here will be the standards dealing with: number and operations, algebra, problem solving, and representation.
Under the category of number and operations one of the specific student expectations is to "develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation." (p. 213). This expectation is specifically addressed in the lesson plan dealing with Positive Exponents. Also listed under expectations on that same page is that students "understand and use the inverse relationships of addition and subtraction..." A technique which is applied directly in the lesson plan entitled Solving Equations.
Naturally the standard entitled algebra deals more directly with the lesson plans under consideration here. The standard sets expectations that the student will be able to "develop an initial conceptual understanding of different uses of variables" which is addressed in the first lesson on Solving Equations (p. 221). Specifically the standard suggests an increase in the use of algebraic symbols as well as the development of an "understanding of several different meanings and uses of variables through representing quantities in a variety of problem situations." (p. 222). Again, these are all items that appear in the first lesson.
The issue of the problem-solving standard revolves around the fact 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 fact by consistent reference to real-life examples and use of real-life situations to which students can relate. And related to this standard is that of connections which should "recognize and apply mathematics in contexts outside of mathematics." (p. 273).
The final standard is representation. The issue here is that "[r]epresentations can help students organize their thinking. Student's use of representations can help make mathematical ideas more concrete and available for reflection." As applied to the middle grades that the lessons under discussion deal, representations are used "more to solve problems or to portray, clarify, or extend a mathematical idea." (p. 67). The use of this standard can be seen in all three of the lesson plans. Variables in equations, scientific/exponential notation and even math language are all examples of representational forms.
Addressing Algebra Misconceptions in the Lesson Plans
Many students appear to have difficulty with the concept of a variable. Younger students see variables as mere placeholders. It is not until much later on that they learn that situations are not always that simple. Because of this fact "[a] thorough understanding of variable develops over a long time, and it needs to be grounded in extensive experience." (p. 38). The method in which these lesson plans introduce the variable clearly aids in this development process. Furthermore, these lessons help develop a…