A
DRIVE FOR ALTERNATIVE LESSONS, ACTIVITIES,
AND
METHODS FOR TEACHING ALGEBRA
by
Mark
Karadimos
March
10^{th}, 2004
karadimosmd@mathguide.com
In an attempt to deal with a student body that suffers
from a high failure rate in algebra at a certain school in Illinois,
institutional changes that support alternative lessons, activities, and
teaching methods is warranted.
The discoveries presented within this body are intended
to help not only this school, but also other educational institutions with
similar demographics, issues, and challenges.
Table of Contents 

Background Information

1 
Potential
Solutions 
2 
Implementing
Solutions 
7 
Creating
an Atmosphere for Continuous Growth 
9 
Conclusion 
10 
Resources 
12 
Background Information
This certain school in Illinois has a mathematics
department that ails from numerous problems.
Incoming students are 93.2% Hispanic, from 69.2% low socioeconomic
families, and have low ability levels in language skills and in mathematical
reasoning and mathematical computation (Kurth, 2003). The school has attempted enrichment programs
for these learners with little sustained success and has two levels of algebra
courses, one that carries beginning algebra over a twoyear period.
The students with the lowest mathematics ability who take
beginning algebra over the longer time frame suffer from poor comprehension
skills and high failure rates. The
students who do pass beginning algebra and eventually go on to a secondyear
algebra course (advanced algebra, also known as algebra 2) find themselves
struggling there in part due to weak basic algebra skills.
When students fail classes, it breaks their potential for
success. It interferes with the natural,
seamless progression of mathematics courses because students who fail
mathematics courses must often wait a semester before they can take a repeat
course in the summer. Failing mathematics
courses also undermines student confidence where it is already low due to poor
ability and comprehension.
Teachers and administrators alike are troubled by student
performance in algebra. They have
expressed a strong desire to change this slow, growing, negative
development. It is apparent that a break
from the exclusive use of traditional methodologies and a change of pedagogy is
in order.
Potential Solutions
Many schools across the country are trying to overcome
educational problems with MexicanAmerican students (Henderson & Landesman,
1992). The similar conditions are:
incoming students with low ability and substandard confidence that result in a
high number of failures and dropouts, poor achievement on standardized tests,
and possible behavior problems in and out of school.
Steps can be taken to break this impasse. Educational pedagogy must be broad enough to
encompass the many learning styles of students.
Teaching methods under a holistic approach can include incorporating
visual tools and models, utilizing handson lessons, allowing cultural
connections, acknowledging the multipleintelligences of every student, and
advocating gaming both inside and outside of school.
There are plenty of ways to approach visualization within
algebra even though it may seem to be a strictly numerical field of
mathematics. Algebra lends itself well
to the visual learner. This suits
MexicanAmerican students because they possess a field dependant cognitive
style (Henderson & Landesman, 1992).
Field dependence is a type of learning preference. Students who are field dependent learn best
with group situations and when presented with a high degree of
organization. They also do well when
their environment is structured and material is visually presented (Didkovskaya,
n.d.).
Calculators that perform visual representations of
equations exist and can help provide simple methods for teaching what many
students find to be troublesome in a format that can help Hispanic students,
especially when the population is field dependent.
For instance, a lecturebased approach to teaching
factoring would have students finding patterns with numbers. When students first learn to factor they must
find a pair of numbers that multiply to a certain product and add to a certain
sum. It is a relatively simple game at
first that many students can successfully accomplish if they possess a great
deal of number sense, which has been gained by memorizing multiplication tables
and playing like games in the past.
However, for students who do not possess such skills or
have not had the benefit of playing number games, factoring is laborious. It frustrates these students because it
exists as a number game that they cannot begin to appreciate, in part do to its
lack of relevance to their lives. To
side step this dilemma, teachers may use graphing calculators and/or algebra
tiles.
Graphing calculators allow students to see connections
from graphing semicomplex polynomials to transforming them into the binomial
factors they need to find. Through the
use of graphing calculators, students can find special points on the graphs of
these polynomials to factor trinomials, be successful at it, and even find the
process to be meaningful. As is often
the case with meaningful lessons, the endeavor also lends itself to further
work with factoring much more complicated polynomials and solving equations
that involve polynomials.
There are numerous handson ‘discovery’ lessons that can
be used to demonstrate mathematical properties and expedite learning not otherwise
gained from noncalculator use (Gage, 1999).
Besides visual models that can demonstrate proportionality possibly
through the use of similar figures or lengths of shadows and the triangles they
form, there is a technique used by biologists to count fish that involves
proportions. By actually mimicking the
process that biologists use to tag, release and capture fish using a physical
model to create a mathematical model, students can appreciate and thereby understand
exactly what they are doing and why they are doing it. It provides a complete picture of the full
process that might otherwise escape the learner who is taught without the
context. Furthermore, it promotes a healthy
classroom environment.
Graphing calculators help students visualize problems,
discover mathematical theorems on their own, instantly check the validity of
their answers, test out their own hypotheses, and explore different ways of
solving problems. Graphing calculators allow topics to be discovered by
students on their own, even before the teacher formally introduces them. They
facilitate an active approach to learning, converting a classroom from a place
where students sit back passively listening to the instructor, to one where
students work with their classmates and produce their own ideas and solutions.
Graphing calculators improve communication among students, and they allow
students a faster, better way to produce graphs. (Pomerantz, 1997)
In fact, it is believed that no learner truly learns
unless it is done from within one’s culture (Nieto, 1999). Therefore, teachers may include lessons that
use elements of student culture. Since the
school is predominantly Hispanic, math lessons can include Mexican art, dance,
cuisine, and music.
Although it may be extremely difficult to conduct
complete algebra lessons using art, dance, cuisine and music, it would be
extremely simple to introduce topics that way.
For instance, to begin discussion on rational expressions, students must
understand how to use fractions. One
could easily produce a Mexican recipe to demonstrate fractions and proportions
that would have students ready to accept a higher order of algebra for work on
rational expressions.
There is a school of thought that says challenges with
language affects mathematics comprehension.
Hispanic students arrive at common misconceptions in mathematics similar
to Anglos but with a greater frequency due to language deficiencies (Mestre,
1999). To address the problem, teachers
must alter pedagogy and break free from a teachercentered classroom, which
will be handled more closely within the section Implementing Solutions below.
Teachers who maximize their potential as educators,
act as facilitators. They vary the
delivery methods and make use of techniques that are receptive to all
learners. According to
The Multiple Intelligence model has suggests vehicles teachers can use that students can
also enjoy. The vehicles are games,
which can be found with many software products and physical games. Games cross many dimensions with
One game specifically worth mentioning that is useful for
a broad range of ages is the game of Chess.
The game hits across
Implementing Solutions
To promote continued growth at the school and any other
educational institution, there are three strategies that may be invoked:
keeping institutional portfolios, training teachers according to the student
population being served, and performing ongoing site research.
Teachers at the school are very collegial and the
mathematics department is no exception.
Like similar schools that have encountered problems dealing with its
population, the change to occur in order to reach beneficial results in student
performance is not about teacher motivation nor is it about teacher
ability. The problem is specific to
technique and knowing the student population.
The teacher toolbox may never be completely full of
strategies and ideas, but a method for encouraging the sharing of productive
work, cleaver ideas, and success lesson plans is a necessity. Teachers have adopted portfolios as a means
for students to mark progress, invite dialogue, identify personal traits or
goals, and offer a tool for reflection (NCREL, n.d.). Since this activity can be beneficial for
students, it can be equally beneficial for teachers and the educational leaders
who adopt such activities.
The record keeping of work is not a new concept, but as a
departmental or possibly interdepartmental activity, it can be used as a tool
for constructive criticism between professionals. Such a tool would help determine what is best
for the population being served, the teachers who serve the population, and the
leadership that guides the institution.
A portfolio could be extremely extensive and hit a complete index of
topics or generic and be an open, less detailed venue, as the situation may
require.
To facilitate the use and growth of a departmental and/or
interdepartmental portfolio system, one could invite an incentive program. Teachers who become engaged in the
development of the portfolio system could be offered 1) a positive mention
within formal observation reports, 2) recognition at school meetings, or 3)
curriculum compensation to promote the portfolio, 4) credits (CPDUs) toward
state recertification, or 5) a stipend toward certification costs.
Another avenue to promote institutional growth is teacher
training. The leadership in education
speaks of studentcentered classrooms, teachers as facilitators and using
cognitive techniques within instruction.
This model is akin to the practice of Socrates (Nenney, 2001). The teacher evokes higher order thinking and
understanding from the students and does not necessarily deliver it as a neat,
complete package. It is a strategy used
to promote independence, personal confidence and propagate dialogue conducive
for a democracy (Reich, 1998).
Studentcentered classrooms, where teachers evoke
knowledge from students instead of handdelivering it, invites
Creating an Atmosphere for Continuous Growth
Site research brings a level of professionalism and
legitimacy to education that it deserves.
It indicates to administrators, teachers, students and community members
alike that continual growth comes from learning and serves as a model for
education. The act of basing
decisionmaking and practice on research informs participants within education
that learning is important and our understanding of it is limited, which
facilitates the need for research as a cyclical, selfperpetuating event.
The Hawthorne Effect that arose from the study of a
Western Electric plant in
The method of perpetuating educational research can be
accomplished through the use of institutional portfolios. Institutional portfolios would serve as a
means for producing anecdotal artifacts for the process of continued
research. They would direct serious
research toward target problems generated by the specific artifacts and the
reflections made as a result of them.
Conclusion
Education in the 21^{st} Century is beginning to
reflect the changes made in the late 20^{th} Century. Due to the nature in which information and
society changes, education has abandoned hardfast rules in academia. It is leaning toward process, communication,
critical thinking, and other generalized skills that closely reflect
Within mathematics education, there are tools to remain
current with these changes. Graphing
calculators allow what Gage describes as learning that moves faster due to the
removal of cumbersome and timeconsuming tasks.
This would appear to serve Hispanic populations that suffer from lower
abilities and consequently need to catch up to their peers in other
communities.
As schools target
Such an institutional model will serve our ultimate
societal goal of maintaining a healthy democracy. Hispanic students will assimilate into
American society if they can continue achievement. Achievement depends, in part, on fostering a
firm grasp of mathematics  specifically algebra  as it pertains to
exercising critical thinking, communication, properties of the world around
them, and reasoning. By providing
Hispanics and populations with similar hurdles a meaningful, understandable
framework for learning algebra, the school can overcome its problems.
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