A DRIVE FOR ALTERNATIVE LESSONS, ACTIVITIES,
AND METHODS FOR TEACHING ALGEBRA
March 10th, 2004
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
Creating an Atmosphere for Continuous Growth
This certain school in Illinois has a mathematics department that ails from numerous problems.† Incoming students are 93.2% Hispanic, from 69.2% low socio-economic 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 two-year 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 second-year 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.
Many schools across the country are trying to overcome educational problems with Mexican-American 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 hands-on lessons, allowing cultural connections, acknowledging the multiple-intelligences 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 Mexican-American 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 lecture-based 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 semi-complex 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 hands-on Ďdiscoveryí lessons that can be used to demonstrate mathematical properties and expedite learning not otherwise gained from non-calculator 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 oneís 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 teacher-centered 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
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 student-centered 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).
Student-centered classrooms, where teachers evoke
knowledge from students instead of hand-delivering 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 decision-making 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, self-perpetuating 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.
Education in the 21st Century is beginning to
reflect the changes made in the late 20th Century.† Due to the nature in which information and
society changes, education has abandoned hard-fast 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 time-consuming 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.
Ballantyne, P. (2000)
Didkovskaya, O. (n.d.) Cognitive
Styles and Problems of Psychological Education,
Fogarty, R. & Stoehr, J. (1995) Integrating curricula with multiple intelligences. Teams, themes, and threads. K-college. IRI Skylight Publishing Inc.
Gage, J. (1999) Teaching Algebra Using Graphing Calculators, Online: http://www.tech.plym.ac.uk/maths/CTMHOME/ictmt4/P53_Gage.pdf (Retrieved February 9th, 2004)
Henderson & Landesman (1992) Mathematics and Middle School Students of Mexican Descent: The Effects of Thematically Integrated Instruction, Online: http://www.ncela.gwu.edu/miscpubs/ncrcdsll/rr5.htm (Retrieved January 20th, 2004)
Kurth, J (2003) Illinois School Report Card: Online: http://188.8.131.52/publicsite/reports/2003/school/english/1401620100001_2003_e.pdf (Retrieved February 13th, 2004)
MacFarlane, Sparrowhawk & Heald (2002) Report on the Educational Use of Games, Online: http://www.teem.org.uk/publications/teem_gamesined_full.pdf (Retrieved February 3rd, 2004)
Mestre, J. (1999) Hispanic and Anglo Students' Misconceptions in Mathematics, Online: http://www.penpages.psu.edu/penpages_reference/28507/285073220.HTML (Retrieved February 3rd, 2004)
Nanney, B. (2001) Student-Centered Learning, Online: http://www.gsu.edu/~mstswh/courses/it7000/papers/student-.htm (Retrieved February 10th, 2004)
Nieto, S. (1999) The Light in Their Eyes: Creating Multicultural Learning Communities. Teachers College Press.
North Central Regional Educational Laboratory [NCREL] (n.d.) Portfolios, Online: http://www.ncrel.org/sdrs/areas/issues/students/earlycld/ea5l143.htm (Retrieved February 10th, 2004)
Pomerantz, H. (1997) The Role of Calculators
in Math Education Research, Online: http://www.educalc.net/135569.page
(Retrieved February 9th, 2004)
Reich, B. (1998) Confusion about the Socratic Method: Socratic Paradoxes and Contemporary Invocations of Socrates, In Philosophy of Education Society Yearbook