Your department head has asked you to prepare a position paper for your department as you plan changes in the algebra curriculum. (a) What are the main goals of the high school algebra curriculum? (b) To what extent do you agree with the NCTM standards regarding algebra topics that should receive increased or decreased attention? What roles will technology play in your revised curriculum? (d) What forms of assessment, if any, should your department be using to assess algebra learning?
According to the National Council of Teachers of Mathematics (NCTM) Standards, the main goals of the high school algebra curriculum should include:
“…the continued study of algebraic concepts and methods so that all students can represent situations that involve variable quantities with expressions, equations, inequalities, and matrices; use tables and graphs as tools to interpret expressions, equations, and inequalities; operate on expressions and matrices, and solve equations and inequalities; appreciate the power of mathematical abstraction and symbolism so that, in addition, college-intending students can use matrices to solve linear systems; and demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations.” (NCTM, 1989)
The goals of our school’s algebra curriculum are very similar to and follow those goals set by the NCTM Standards. There are a few things that we will do to supplement the suggestions of the Standards.
In our curriculum, we want to make sure that our students leave here with the ability to do the following:
· Translate word problems into algebraic forms;
· Use key phrases that are used to become an effective problem solver;
· Learn strategies to solving college entrance exam problems, most notably the SAT and the ACT;
· Use open-ended, real-world situations to bring forth a wider range of thinking skills;
· Feel a sense of confidence and motivation to use algebra in real life.
One of the suggestions that the NCTM Standards have made is that the algebra curriculum should “move away from a tight focus on manipulative facility to include a greater emphasis on a conceptual understanding” (NCTM, 1989). While the teachers of our department agree that greater emphasis should be placed on conceptual understanding, the majority of our mathematics teachers agree emphasis should not be taken away from manipulative facility. We have seen too many instances where some of our more gifted students struggle with manipulations that do not incorporate the use of technology (i.e., the graphing calculator or computer). One of our main goals and primary concerns is that our students realize the value and potential of technology but do not lose sight of the importance of manipulations (e.g. simple arithmetic). The Standards continue by stating that “college intending students who can expect to use their algebraic skills more often “should maintain an appropriate level of proficiency” (NCTM, 1989). We feel that this statement is a valid goal for all of our students. We feel that even though the Standards state that “there will be a trade-off in instructional time,” we should try to maintain a balance without sacrificing a “tight focus on manipulative facility” (NCTM, 1989).
Because some students are led to believe that “mathematics” means the same as “calculations,” we may find ourselves producing “key-pressers” instead of “mathematics learners” (Broman, n.d.). It is easy for us as math teachers to mistake a student who is proficient with a calculator such as a TI-82 for a good mathematics student. It is equally as easy for us to miss a good math student who is not as apt at using the calculator. Although the stance of this department is not the most popular, we feel that the use of technology for items such as arithmetic should be strongly deemphasized, as we will discuss later in the statement.
The Standards also make mention of a change of emphasis “within topics.” For example, “…[spending] less time simplifying radicals and manipulating rational exponents [and devoting] more time to exploring examples of exponential growth and decay…” (NCTM, 1989). We as a department feel that there is great importance in studying real-world problems such as the growth and decay problems to which the Standards refer; however, we do not wish to sacrifice the discipline required from items such as simplifying radicals and the manipulation of rational exponents. We do feel that less time can be spent on working with simplifying radicals, primarily since a calculator can give a very precise solution and a solution that is more representative in terms of the Cartesian coordinate system. Our department shares a feeling that it is in our best interest to develop students to not only be good problem solvers but also disciplined problem solvers. We are deeply concerned that our students will leave this school without the ability to reason through problems and be confident with their solutions. The ability to problem solve is very important, but the ability to recognize possible errors in solutions is vital. To quote from Brownell (1987), “to say in effect that computational skill is of negligible importance…we can hardly justify this position.” He continues by claiming, “When we correct, we tend to over-correct. Just this sort of thing seems to have happened in the teaching of arithmetic.” We, as a department, feel that this type of “over-correction” supersedes the teaching of arithmetic in that we as mathematics teachers tend to over-correct other items in the teaching of mathematics that require the same type of discipline that arithmetic requires. For this reason, we do not choose to de-emphasize, to the point of abandonment, certain topics (e.g., simple arithmetic, the simplification of radicals, the manipulation of rational exponents, etc.).
Although we have previously commented on using calculators less for arithmetic, that statement should not be misunderstood to imply that the use of technology as a whole will be diminished. In fact, nothing could be more to the contrary. As the development of technology continues to increase, our use of technology will also increase. We realize that as our students continue on to post-secondary education and on to the work force, these students will need to be familiar with more and more types of technology. Our school currently uses and maintains a mathematics computer lab with Macintosh computers and a class set of CBLs. Also, each teacher has a class set of TI-83 graphing calculators. As funds continue to grow, we are constantly updating the technology we currently have, and we will add more CBLs to our inventory very soon.
The Professional Standards publication from the NCTM states that the “teacher of mathematics should create a learning environment that fosters the development of each student’s mathematical power by…using the physical space and materials in ways that facilitate students’ learning of mathematics” (NCTM, 1991). We feel that lab activities requiring CBLs and the use of the Macintosh computer lab are two key tools that promote the type of learning environment the NCTM describes. As former students and current learners, the teachers of our department realize that we need to continue to stimulate our students in environments that promote continuous motivation.
The primary application that our school
uses for the Macintosh is the Geometer Sketchpad (GSP). We have been using this software and its
upgrades for the past five school years with greater success each year. We feel that this is one of the most
successful pieces of technology that we use.
Although the GSP’s uses have been heightened
in Algebra, the GSP is primarily used in our Geometry courses. The GSP allows for students to manipulate and
dynamically explore the properties of triangles, quadrilaterals, circles, and
other geometric figures. Although our school
has not completed any statistical studies testing the success of the GSP, the
results of a study done at Athens Academy (GA) show a
significant success level (Hannafin and Hawkins,
· Of 97 students, 55 said that working with the GSP “helped their learning, while only 8 claimed the GSP detracted from their progress;
· 64 out of 99 students said there was a benefit to using the GSP in learning geometry, while 14 said there was no benefit;
· 55 percent of the students said that “teacher explanations to the class were clearer when he/she used the GSP in the discussion.
As for the use of the graphing calculator, as stated above, each mathematics teacher is provided a class set of TI-83 calculators for use in any mathematics class that he/she teaches. Although we have encouraged our teachers to not encourage the TIs use for simple arithmetic, we have encouraged the use of these calculators “to break with long standing curriculum traditions” by paying more attention to iterative solutions of quadratic equations than to exact solutions and dealing with finding maximum and minimum values of functions long before our students encounter calculus (Lowe, Willis, Grace, & Kissane, 1994). Our teachers have all given great testimonies for years as to how much more material they have been able to cover, especially in their algebra classes. The only changes that have occurred in the use of the calculators in the past few years is that the calculator has been used more as the teachers have become more efficient in their use of the graphing calculators.
With the changes that we have made in our algebra curriculum, we need to also focus on how these changes will affect the way we assess our students’ learning in algebra. Although we consider ourselves to be a more “traditional” set of teachers, we are attempting to find new and alternative forms of assessment. No matter what forms of assessment we use, we are still attempting to follow what the NCTM has suggested for assessment. The NCTM states that there are four purposes of assessment (NCTM, 1995):
· to monitor students’ progress toward learning goals;
· to use students’ mathematical understanding to make instructional decisions;
· to evaluate students’ achievement at a particular time; and
· to evaluate an instructional program
We have had several staff development courses describing different forms of alternative assessment, its benefits, and its implementation. We have had several teachers attempt alternative assessment in various forms. Some of the more successful attempts have come in the honors level classes and some of the more limited success has come from the regular, academic-tracked students.
One of the more popular methods of alternative assessment has been the use of journal writing. Journal writing is “a good way to introduce this type of assessment…in order to ask students about their feeling toward mathematics” (Berenson and Carter, 1995). Students tend to feel more comfortable writing in journals and, after a short period of time, the teacher “can use the journal to explore students’ conceptual understandings.”
Another increasingly popular method of assessing our algebra students has been through the use of open-ended problems. This type of questioning “invites multiple solutions and multiple ways to arrive at solutions,” and “assists students in the development of problem solving skills, as well as creative and divergent thinking.” Similar to other teachers’ previous experience with open-ended problems, our teachers who use this form of assessment usually find that students’ grades do not differ dramatically from what they previously achieved, although these students’ conceptual understanding is much stronger (Cooney, Bell, Fisher-Cauble, and Sanchez, 1996).
We will continue to explore these types of assessment and others such as portfolio assessment and interview assessment, while intertwining alternative assessment with the traditional forms of assessment (excluding multiple choice and true/false assessments).
Broman, P. (n.d.). Roles of calculators in the classroom: Possibilities and fears. Retrieved November 14, 2003, from http://ued.uniandes.edu.co/servidor/em/recinf/tg18/Base/ wwwfiles-1.html.
Brownell, W.A. (1987). Meaning and skill: Maintaining the balance. Arithmetic Teacher, 34(8), 18-25.
Hannifan, M.J., & Hawkins, C.H.
(1997). The media across the curriculum project (Evaluation Report).
I, Willis, S., Grace, N. & Kissane, B.
(1994). Access to algebra (Book 3).
Council of Teachers of Mathematics (1989).
Curriculum and evaluation
standards for school mathematics.
Council of Teachers of Mathematics (1991).
Professional standards for
Council of Teachers of Mathematics (1995).
Assessment standards for school