SAMPLE RESPONSE TO A COMPREHENSIVE EXAM QUESTION
Question:
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?
Response:
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,
1997). The
·
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).
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References 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. Cooney,
T.J, Hannifan, M.J., & Hawkins, C.H.
(1997). The media across the curriculum project (Evaluation Report). Lowe,
I, Willis, S., Grace, N. & Kissane, B.
(1994). Access to algebra (Book 3).
National
Council of Teachers of Mathematics (1989).
Curriculum and evaluation
standards for school mathematics. National
Council of Teachers of Mathematics (1991).
Professional standards for
teaching mathematics. National
Council of Teachers of Mathematics (1995).
Assessment standards for school
mathematics. |