The Effect of Training Grade Eight Students on the Algebraic Problems: Solving Strategy on Their Ability to Solve Other Mathematical Problems and Learning Retaining
Introduction
Students in algebra classes face many difficulties. Among them are an inability to apply algorithmic knowledge to novel problems, poor arithmetic skills, and difficulty with the abstract content associated with higher-level mathematics. With this in mind, I have explored some of the research on the algebra performance of students with disabilities and reasons for their difficulties. In addition, we discuss pedagogical practices that best serve students with learning disabilities in algebra.
Problem representation strategies are needed to process linguistic and numerical information, comprehend and integrate the information, form internal representations in memory, and develop solution plans (Silver, 1985). These strategies facilitate translating and transforming problem information into problem structures or descriptions that are verbal, graphic, symbolic, and/or quantitative in nature (Janvier, 1987; Mayer, 1985). These verbal and visual representations in turn assist in organizing and integrating problem information as the problem solver develops a logical solution plan. Specific problem representation strategies include (a) paraphrasing or restating problems in one's own words; (b) visualizing problems by drawing pictures, constructing diagrams or charts, and making mental images; and (c) hypothesizing or establishing goals and setting up a plan to solve the problem.
Research has also suggested that academic performance may be influenced not only by cognitive factors such as ability to represent problems, but also by noncognitive factors, for example, self-perceptions of ability or academic competence and perceptions of task difficulty (Heath, 1996; Montague, 1997). Students' self-perceptions may directly influence how they approach a task and the amount of effort they put forth.
Partially in response to previous research, developmental educators began to question traditional methods of teaching mathematics (Hackworth, 1994), and to experiment with other instructional techniques (Ahmed, Barks, & Dolega, 1996; Higbee & Thomas, 1999;), including strategies for reading math texts (Campbell, Schlumberger, & Pate, 1997), writing assignments (Williams, 1995), studying word problems (Dooley & Sundeen, 1997), testing (Glover, 1995), and grading (Mollise & Matthews, 1996). Many developmental mathematics educators began to shift from predominantly lecture to a more collaborative learning environment (Higbee & Thomas, 1998).
The difficulty with symbolic reasoning can be heightened when teachers introduce algebra concepts that are more difficult than arithmetic instruction. According to some researchers, educators need to attend to the following instructional techniques to help students make connections between arithmetic to algebra and understand algebraic notation (Brownell, Smith, and Witzel, 2001).
Although it is easy for experts to advocate for the use of explicit, relevant math instruction, it is difficult for teachers to do. One possible way to make instruction relevant and use explicit instruction is to provide students with hands-on experiences. Hands-on experiences allow students to understand how numerical symbols and abstract equations are operating at a concrete level, making the information more accessible to all students (Devlin, 2000; Maccini & Gagnon, 2000).
Problems of mathematics underachievement are greatest for students with mild disabilities and those at risk for mathematics failure (Parmar, Cawley, & Frazita, 1996). Specifically, word-problem solving is difficult for mathematics students who evidence problems in reading, computation, or both (Dunlap, 1982). As a result, some of these students reportedly spend more than one third of their resource room time studying mathematics (Carpenter, 1985).
The importance of providing quality word-problem-solving instruction for students with mild disabilities and at-risk students is clear. Although mathematics instruction in general, and word-problem solving in particular, has not received as much in-depth study and analysis as reading (Bender, 1992), a reasonable number of mathematics word-problem-solving intervention studies with samples of students with learning problems is now available. A recent narrative review of word-problem-solving research (Jitendra & Xin, 1997) provided information of practical importance but was limited by its reliance on published studies only, and by a lack of quantitative techniques for analyzing the magnitude of intervention effectiveness.
In this proposed study, I shall investigate several variables that may influence the mathematical problem solving of eight grade students. These included perceptions of problem difficulty, problem-solving accuracy, persistence as measured by time taken to solve the problems, use of problem-solving strategies, and problem-solving method (silent problem solving versus thinking aloud). I propose that enabling the students to adopt a learning strategy on algebra and how they perceive the mathematical problems will result to lesser time in solving problems, get more correct answers, and use more problem-representation strategies.
Background of the Problem
Mathematics is very sensational for each person. It would be very difficult to live normal life in very many parts of the world in 21st century without making use of mathematics of some kind.
Beginning algebra students are required to move from arithmetic thinking to algebraic thinking. This transition is believed to involve a move from knowledge required to solve arithmetic equation to knowledge required to solve algebraic equation. This transition is referred to as pre-algebraic thinking, the cognitive gab or didactic cut between arithmetic and algebra (1997;Boulton-Lewis, Cooper, Atweh, Pillay and Wilss, 1998;Boulton-Lewis, Cooper, Pillay and Wilss, 1998).
Algebra can be thought of many ways, as pattern, functions and relation, as language, representations and structures based on generalised arithmetic, or as a tool for modelling mathematical ideas and problems” National Council of Teachers of Mathematics, 1998.p221)
Students face difficulty in solving mathematical word problems in general and Algebraic word problem solving in particular (Batler1976). Many capable Australian students experience difficulties with school algebra (Pegg and Hadfield, 1999).
Developing a strategy to problem solving may contribute to enhance the scientific approach among students to solve their problems and to choose the appropriate method to solve any problem (Scandura.1978, Wheatley, 1980).
Teaching students how to solve algebraic word problems is the most difficult task that faces mathematic Teachers (Batler, 1970). One of the most important goals for teaching mathematics is to train students to use some strategy to problem solving and to get feedback how this will reflect the improvement in solving other mathematical problems (NCSM 1988).
Many previous studies have searched for strategies to teach algebraic problem solving and the influence on student’s ability to solve mathematical word problems (Christian, 1985,Perez, 1986). This research will try to develop a strategy to teach algebraic problem solving and apply it on year eight. This study will determine the effects of these strategies on students’ performance and achievement and it can lead to designing better strategies.
Conceptual FrameworkThis proposed study shall utilize the concrete to representational to abstract (CRA) method in testing the student’s ability to solve algebraic problems and in enhancing the algebra education of grade eight students as proposed by Mercer and miller (1994, 1997). A highly touted instructional method for students with disabilities is the concrete to representational to abstract (CRA) sequence of instruction, which incorporates the use of hands-on materials and pictorial representations. The CRA method facilitates abstract reasoning by moving students through three phases of instruction: concrete, representational, and abstract. The concrete phase involves manipulatives, such as toothpicks for counting. The representational phase uses pictures, such as tally marks. Pictorial representations relate directly to the manipulatives and set up the student to solve numeric problems without pictures. Matching pictorial representations to abstract problems, such as using tally marks to solve multiplication problems helps students understand the abstract concept of multiplication.
Students learning basic mathematics facts using the CRA sequence of instruction show improvements in acquisition and retention of mathematics concepts (Miller & Mercer, 1993). This CRA sequence of instruction, although successful for basic facts, can tie directly to algebra when the materials match up specifically with each component to algebra equations. For basic algebra, it is important to include aids to represent arithmetic processes, as well as physical and pictorial materials to represent unknowns (Miller and Mercer, 1997).
HypothesisThis proposed study shall test the following hypothesis:
The purpose of this study is to explore grade eight students' perceptions of problem difficulty, persistence, and knowledge and use of problem-solving strategies in solving mathematical word problems in algebra. This shall be the basis of the proposed strategy that shall be developed and tested in this research using 8th grade students.
Specifically, the following research questions shall be addressed:
1. What are the levels of capability of 8th grade students in terms of their knowledge on equivalent equations and their identification of other known equations?
2. What is the degree of comprehension of 8th grade students in solving simple to complex equations?
3. What is the degree of competence of 8th grade students in solving word problems?
4. How effective is training in their skills in developing strategy problem solving and retention in solving algebraic word problem and geometric word problem?
5. Is there a variation of the result on the students’ gender?
Research Objectives
Given the relevance of problem solving in today's technologically advanced society, Patton, Cronin, Bassett, and Koppel (1997) recommended teaching students with learning problems to be proficient problem solvers in dealing with everyday situations and work settings. This proposed study seeks to contribute in this literature by first outlining the problems of 8th grade students in algebra and propose a strategy of word problem and retention in solving the problem in algebra deficiency.
Specifically, I aim to accomplish the following tasks:
1. to determine the areas of weakness of 8th grade students in algebra specifically in simple and complex equations
2. to illustrate the degree of proficiency of 8th grade students in solving word problems
3. to evaluate the effects of training in the proficiency and retention of knowledge of 8th grade students in terms of solving algebra and geometric word problems
4. to evaluate the effects of training on the strategy formulation and retention of 8th grade students based on the students’ gender
5. to recommend a strategy and approach in improving the algebra and geometric skills of 8th grade students
Significance of the StudyPrevious research indicates that despite a positive attitude toward mathematics, students with are significantly poorer mathematical problem (Montague & Applegate, 1993). Research also suggests that most students have strategy deficiencies that may be related to poor academic performance (e.g., Swanson, 1990). For mathematical problem solving, the most salient deficiency seems to be in problem representation processes and strategies, which are critical to effective problem solving (Hutchinson, 1993; Zawaiza & Gerber, 1993).
This proposed seeks to contribute to the literature by conducting a study involving 8th grade students by assessing their problems in algebra, evaluating their degree of competence in word problems both in algebra and geometry and in their mastery of simple and complex equations and how they relate to each other. Furthermore, by examining the effects of training on the students, I aim to validate the claim by researchers that training can improve the strategy formulation and retention of knowledge of students. Ultimately, I shall be recommending a strategy of training that shall bridge the gap between the gap in students’ knowledge and skills in algebra.
Scope and Limitations of the StudyThis study concerns 8th grade students in a given school. The sample shall be limited to one school and approximately 60 students due to time and financial constraints. Furthermore, only one training program shall be tested. Due to ethical issues, the names of those involved in the study shall be kept confidential. Only when their permission is granted will I divulge personal statements and information. This is in relation to the ethical considerations I apply as a researcher.
Definition of Terms algebra Any formal mathematical system consisting of a set of objects and operations on those objects. Examples are Boolean algebra, numerical algebra, set algebra and matrix algebra. algebraic equation An equation of the form f(x)=0 where f is a polynomial. algebraic number A number that is the root of an algebraic polynomial. For example, sqrt(2) is an algebraic number because it is a solution of the equation x2=2. algebraic expression the variable represents values of a quantity that can vary. We need to define what the variable represents. word problem
Refers to algebra problems expressed in words instead of in numbers
References
Batler and Wern and Bank - The Teaching of Secondary Mathematics.; New York, McGraw-Hill, 5th Edition, 1970.
Bender, W. (1992). Learning disabilities: Characteristics, identification, and teaching strategies. Boston: Allyn & Bacon.
Brownell, M., Smith, S. and Witzel, B. (2001) How can I help students with learning disabilities in algebra? Intervention in School & Clinic, Vol. 37.
Campbell, A. E., Schlumberger, A., & Pate, L. A. (1997). Promoting reading strategies for developmental mathematics textbooks. Selected Conference Papers, National Association for Developmental Association, 3, 4-6
Christian, W.Acomparison of the effectiveness of three strategies for the teaching word proplems at the intermediate algebra level to college students. DAI, vo. 45 No.9, 1985 , p. 2790-A.
Cooney , Davis and Handerson Dynamics of Teaching Secondary School Mathematics; Houghton Mifflin, 1975.
Devlin, K. (2000). Finding your inner mathematician. The Chronicle of Higher Education, 46, B5.
Dooley, J. L., & Sundeen, T. R. (1997). Facilitating word problems effectively and enjoyably: Can it happen? Selected Conference Papers, National Association for Developmental Education, 3, 17-18.
Hackworth, R. (1994). Teaching mathematics effectively. In M Maxwell (Ed.), From access to success (pp.243-251). Clearwater, FL: H & H.
Heath, N. (1996). The emotional domain: Self-concept and depression in children with learning disabilities. In T.E. Scruggs & M.A. Mastropieri (Eds.), Advances in learning and behavioral disabilities (Vol. 10, Part A, pp. 47-76). Greenwich, CT: JAI Press.
Higbee, J. L., & Thomas, P. V. (1998). Daily brainteasers: Promoting collaboration, persistence, and critical and creative thinking. Academic Exchange Quarterly, 2(4), 20-23.
Higbee, J. L., & Thomas, P. V. (1999). Affective and cognitive factors related to mathematics achievement. Journal of Developmental Education, 23(1), 8-10,12, 14, 16, 32.
Hudgins, B. Problem Solving in the Classroom; New York, The Macmillon Company, fourth printing, 1968
Hutchinson, N. (1993). Effects of cognitive strategy instruction on algebra problem solving of adolescents with learning disabilities. Learning Disability Quarterly, 16, 34-63.
Janvier, C. (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum.
Jitendra, A. K., & Xin, Y. (1997). Mathematical word-problem-solving instruction for students with mild disabilities and students at risk for math failure: A research synthesis. The Journal of Special Education, 30, 412-438.
Krulik. S and Rudnick. J Problem Solving Second Edition, Boston, London, Sydney, Toronto, 1987
Mayer, R.E. (1985). Mathematical ability. In R.J. Sternberg (Ed.), Human abilities: Information processing approach (pp. 127-150). San Francisco, CA: Freeman.
Miller, S. P., & Mercer, C. D. (1993). Using data to learn about concrete-semiconcrete-abstract instruction for students with math disabilities. Learning Disabilities Research and Practice, 8(2), 89-96.
Miller, S. P., & Mercer, C. D. (1997). Educational aspects of mathe-matics disabilities. Journal of Learning Disabilities, 30(1), 47-56. National Center on Educational Statistics. (1996). National assessment of educational progress: Graduation requirements for math. Washington, DC: NCES.
Montague, M. (1997). Student perceptions, mathematical problem solving, and learning disabilities. Remedial and Special Education, 18, 21-30.
National Council of Supervisors of Mathematics (NCSM Newsletter,June 1988,Vol. XVII,No 4)
Parmar, R. S., Cawley, J. F., & Frazita, R. R. (1996). Word problem-solving by students with and without mild disabilities. Exceptional Children, 62, 431-450.
Patton, J. R., Cronin, M. E., Bassett, D. S., & Koppel, A. E. (1997). A life skills approach to mathematics instruction: Preparing students with learning disabilities for the real-life math demands of adulthood. Journal of Learning Disabilities, 30, 178-187.
Polya, G. Mathematical Discovery of Understanding Problem Solving. New YOrk, Jowiley, 1965
Scandura. J Problem SolvingAstructural Process Approach with Instructional Implicaation.; Academic press, New York, 1977.
Silver, E.A. (1985). Research on teaching mathematical problem solving: Some underrepresented themes and needed directions. In E.A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 247-266). Hillsdale, NJ: Erlbaum.
Williams, A. C. (1995). Connecting mathematics to the students' world: A writing assignment. Selected Conference Papers, National Association for Developmental Education, 1, 41-42.
Zawaiza, T., & Gerber, M. (1993). Effects of explicit instruction on math word problem solving by community college students with learning disabilities. Learning Disability Quarterly, 16, 64-79.
Other Useful References Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching mathematics with understanding. In D.A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp 65-97). New York: Macmillan. There is a chapter on Algebra and Geometry as well. Lawson, M. J. (1991). Testing for transfer following strategy training. In G. Evans (Ed.), Learning and teaching cognitive skills. Hawthorn, Victoria: The Australian Council for Educational Research. Lesh, R. (1985). Conceptual analyses of problem-solving performance. In E.Silver (Ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspective (pp. 309-329). Hillsdale, NJ: Lawrence Erlbaum. Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Research: Learning, Memory and Cognition, 17, 398-416. Owen, E., & Sweller, J. (1985). What do students learn while solving mathematics problems? Journal of Educational Psychology, 77, 272-284. Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press. Skemp, R.R. (1971). The psychology of learning mathematics. Penguin: Middlesex.
Appendix 1.
TIMETABLE
TASK Month
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
Select topic
Undertake preliminary literature search
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Define research questions
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Write-up aims and objectives
Select appropriate methodology and locate sources of information. Confirm access.
Write-up Dissertation Plan
Undertake and write-up draft critical literature review.
Secondary and Primary Data Detailed
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Sources
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Consulted
Research Findings:
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Analysed
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Evaluated
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Written-up
Discussion:
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Research findings evaluated and discussed in relation to the literature review
Methodology written-up
(including limitations and constraints)
Main body of the report written-up and checked for logical structure
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Conclusions drawn
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Recommendations made
Introduction and Executive Summary written-up
Final format and indexing
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