Algebra in Middle School

Page Contents

A      B      C      D      E      F      G      H      K      L      M      N      O      P      R      S      T      W      Y      Z

A

Anderman, L. H., & Midgley, C. (1998). Motivation and Middle School Students. ERIC Digest. Research has shown a decline in motivation and performance for many children as they move from elementary school into middle school; however, research has also shown that the nature of motivational change on entry to middle school depends on characteristics of the learning environment in which students find themselves. This Digest outlines some suggestions for middle school teachers and administrators for enhancing student motivation and discusses three theories that are currently prominent and that have particular relevance for young adolescent students and their teachers. Attribution theory emphasizes that students' perceptions of their educational experiences generally influence their motivation more than the objective reality of those experiences. Through instructional practices, teachers can unknowingly communicate a range of attitudes about whether ability is fixed or modifiable and convey their expectations for individual students. Goal theory focuses on the reasons students perceive for achieving: a task goal orientation represents the belief that the purpose of achieving is personal improvement and understanding; an ability goal orientation represents the belief that the purpose of achieving is the demonstration of ability. Studies find that the adoption of task goals is associated with more adaptive patterns of learning than is the adoption of ability goals. A third motivational theory of importance for middle school educators is self-determination theory. This theory describes students as having three categories of needs: needing a sense of competence, of relatedness to others, and of autonomy. Most of the research focuses on the last of these three needs. Within the classroom, autonomy needs could be addressed through allowing student choice and input on classroom decision making. It is important to recognize that supporting student autonomy does not require major upheaval in the classroom or that teachers relinquish the management of students' behavior. Even small opportunities for choice can increase students' sense of self-determination. Contains 13 references. (LPP) ED421281

Anderson, V., & Others, A. (1995). Changing Middle School Students' Models of Literacy through Cognitive Strategy Instruction. Technical Report No. 610. 17p. A study investigated the effects of strategy instruction on the reading and writing abilities of sixth-grade middle-school students with delayed literacy. An experimental group of 10 sixth-grade, English-speaking inner-city students in the midwestern United States was matched on race, gender, and reading comprehension test scores with 10 sixth-grade control students. All students were at least two years behind in their literacy skills. Students in the experimental group engaged in strategic reading and writing instruction for 2 hours daily for 14 weeks. Results of structured performance-based student interviews in reading and writing showed that the short-term instructional intervention in strategy instruction not only improved students literacy performance but also extended and enhanced their models of literacy. (Contains 10 references, and one table and two figures of data.) (RS) ED380789

Applied Algebra Curriculum Modules.(1995). 230p. This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students in both vocational-technical and academic programs. Topics are as follows: (1) real number properties and operations; (2) problem solvinggeometric figures; (3) graphing skills; (4) exponents and roots; (5) estimation skills; (6) word problems; (7) problem solvingrates; (8) linear equations and inequalities; (9) quadratic equations and inequalities; (10) functions; and (11) use of statistics. Modules 1, 2, 8, and 9 consist of these components: objectives; equipment list; handouts/activity or exercise sheets; and informative material for the teacher. Modules 3, 5, and 10 have this format: performance objective, investigations/demonstrations each followed by an activity, evaluation instrument, and list of required materials. Module 4 follows this format: performance objective, background information, demonstrations followed by activities, handouts, workplace/technical problems, posttest, and equipment/materials list. Modules 6 and 7 have these components: performance objective, statement of connection, activity, list of evaluation instruments, and supply list. Module 11 follows this format: introduction, materials list, lesson plan, handouts, list of course objectives, skill check with answer key, and glossary. (YLB) ED395185

author., N. s. (1998). The Nature and Role of Algebra in the K-14 Curriculum: Proceedings of a National Symposium (Washington, DC, 27-28, 1997). Methods of effectively teaching algebraic thinking in elementary schools as well as secondary schools is the topic of the following 19 papers. Papers include: (1) "Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by 'Algebrafying' the K-12 Curriculum" (J. Kaput); (2) "Developing a Coherent and Focused K-12 Algebra Curriculum" (E. Phillips); (3) "Enhancing Algebraic Reasoning with Technology" (G. Akst); (4) "Algebra for Everyone? With or Without Technology?" (M. Norman); (5) "How Might Technology Enhance Algebraic Reasoning?" (R. Zbiek); (6) "What Do We Know about K-14 Students' Learning of Algebra?" (J. Confrey); (7) "Algebra: What All Students Can Learn" (S. Williams and D. Molina); (8) "Improving K-14 Algebra Instruction: A Discussion of Teachers' Responsibilities and Students' Opportunities" (B. Moore-Harris); (9) "Capturing Patterns and Functions: Variables and Joint Variation" (G. Lappan); (10) "Functions and Relations: A Unifying Theme for School Algebra in Grades 9-12" (C. Hirsch); (11) "Middle School Algebra from a Modeling Perspective" (G. Kleiman); (12) "Why Modeling Matters" (L. Godbold); (13) "Modeling: Changing the Mathematics Experience in Postsecondary Classrooms" (R. Dance); (14) "Algebraic Structure in the Mathematics of Elementary-School Children" (C. Tierney); (15) "Structure in School Algebra (Middle School)" (M. van Reeuwijk); (16) "The Role of Algebraic Structure in the Mathematics Curriculum of Grades 11-14" (G. Foley); (17) "Language and Representation in Algebra: A View from the Middle" (R. Billstein); (18) "Teaching Algebra: Lessons Learned by a Curriculum Developer" (D. Resek); and (19) "The Nature and Role of Algebra: Language and Representation" (D. Hughes Hallett). (ASK) ED429801

B

Bankhead, M. P. L. (1997). Reducing "Math Anxiety" in College Algebra Courses Including Comparisons with Elementary Statistics Courses. The high levels of anxiety, apprehension, and apathy of students in college algebra courses caused the instructor to create and test a variety of math teaching techniques designed to boost student confidence and enthusiasm in the subject. Overall, this proposal covers several different techniques, which have been evaluated by both students and the instructor. The paper proposes a series of study techniques, which are covered on the first day of class and throughout the course. Armed with practical advice about approaching the algebra course, the students learn by example. The instructor uses examples, such as rules of cricket matches as a way to make the material come to life. Other suggested techniques are making chapter notes available to students, and providing example tests before final exams and additional study sessions. Sometimes students are allowed to work in groups and complete group projects. Within this paper are several examples of the handouts and examples of worksheets given to students. The paper concludes that in addition to succeeding in reducing students' anxiety level so that the instructor could engage in more teaching, the instructor also developed a greater interest and enthusiasm for the topic. (AF) ED440672

Beaman, C. J. (1994). Solving the Homework Problem in Algebra through the Use of Grade Control Charting. 88pp. Action Research Final Report, Saint Xavier University-IRI. This report describes a program for improving homework on-time completion with high school Fundamental Algebra students in an urban gifted and arts magnet high school. The Grade Control Chart was selected as a strategy for presenting students with a visual reminder of the value of timely completion of homework. The skills needed to produce this chart are in keeping with the National Council of Teachers of Mathematics (NCTM) Standards that call for the incorporation of statistical representations, self-analysis, goal setting, problem solving, and journal writing throughout the mathematics curriculum. The experiment produced negative effect sizes, indicating no practical significance to the intervention, but some slowing of the decline in homework was noted toward the end of the experiment period. Student response to the intervention was mixed, though generally positive. Appendices include: a homework record/seating chart, baseline data on rate of homework completion, teacher homework survey form, student and parent homework survey forms, and the Grade Control Chart. (Contains 76 references.) (Author/MKR) ED383527

Beaton, A. E., & Others, A. (1996). Mathematics Achievement in the Middle School Years. IEA's Third International Mathematics and Science Study (TIMSS). 243pp. Funding for the international coordination of the Third International Mathematics Study is provided by the U.S. National Center for Education Statistics, the U.S. National Science Foundation, the International Association for the Evaluation of Educational Achievement, and the Canadian government. The Third International Mathematics and Science Study (TIMSS) is the largest and most ambitious study undertaken by the International Association for the Evaluation of Educational Achievement. Forty-five countries collected data in more than 30 languages. Five grade levels were tested in the two subject areas, so that more than half a million students were tested around the world. This report addresses middle-school mathematics achievement (grades seven and eight) in six content areas: (1) fractions and number sense; (2) measurement; (3) proportionality; (4) data representation, analysis, and probability; (5) geometry; and (6) algebra. Results cover 41 countries with complete data collection. Singapore was the top-performing country at both grade levels, with Korea, Japan, and Hong Kong also performing very well. There were large differences in average achievement between top performers and bottom performing nations. Gender differences in mathematics achievement were small or nearly nonexistent in most countries, but when they did appear, they favored boys. In nearly every country there was a strong positive relationship between student enjoyment of mathematics and higher achievement. Home factors were strongly related to mathematics achievement in every participating country, but relationships between instructional variables and achievement were less clear. In every country, the pattern was for the eighth grade student whose parents had more education to also have higher achievement in mathematics. The amount of television viewing was negatively associated with mathematics achievement. The document's introduction provides information on each country's characteristics including demographics, public expenditures on education, organization of educational system. Chapters address: (1) International Student Achievement in Mathematics; (2) Average Achievement; (3) Performance on Items within Each Mathematics Content Area; (4) Students Backgrounds and Attitudes towards Mathematics; and (5) Teachers and Instruction. Appendixes include: Overview of TIMSS Procedures; Test-Curriculum Matching Analysis; Selected Mathematics Achievement Eighth-Grade Results for the Philippines, Denmark, Sweden, and German- Speaking Switzerland; and Percentiles and Standard Deviations of Mathematics Achievement. (SLD) ED406419

Bennett, R. E., & Sebrechts, M. M. (1994). The Accuracy of Automatic Qualitative Analyses of Constructed-Response Solutions to Algebra Word Problems. GRE Board Professional Report No. 91-03P. 111p. This study evaluated expert system diagnoses of examinees' solutions to complex constructed-response algebra word problems. Problems were presented to three samples (30 college students each), each of which had taken the Graduate Record Examinations General Test. One sample took the problems in paper-and-pencil form and the other two on computer. Responses were then diagnostically analyzed by an expert system, GIDE, and by four Educational Testing Service mathematics test developers. Results were highly consistent across the samples. Human judges generally agreed in describing responses as right or wrong, but concurred at lower levels in categorizing the specific bugs they detected in incorrect solutions. The expert system agreed highly with the judges' right/wrong decisions, but less closely with bug categorizations that judges agreed on. Causes of machine-rater disagreement were identified, and suggested remedies were proposed. These results suggest that highly accurate diagnostic analysis through knowledge-based understanding of complex responses be difficult to achieve at the fine-grained level used by GIDE. Increasing accuracy is discussed. Appendixes A, B, and C present probabilities and canonical solutions for each of the samples; and Appendixes D, E, and F contain Sample 2 judges' instructions, and Sample 2 and Sample 3 Bug Classification Scheme and Detailed Error Descriptions with Examples. Twenty-one tables present study data. (Contains 13 references.) (Author/SLD) ED385550

Berenson, S., Ed., Dawkins, K., Ed., Blanton, M., Ed., Coulombe, W., Ed., Kolb, J., Ed., Norwood, K., Ed., & Stiff, L., Ed. (1999). Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 1 (20th, Raleigh, NC, October 31-November 3, 1998). For Volume 2, see SE 061 960. This conference proceedings contains three plenary session reports, 12 working group and 79 research reports, 35 short oral reports, 60 poster session reports, and two discussion group reports. The titles of all papers (excluding "short orals", "posters", and brief discussion group reports) are: (1) "On Relationships between Psychological and Sociocultural Perspectives" (Pat Thompson and Paul Cobb); (2) "On Theory and Models: The Case of Teaching-In-Context" (Alan H. Schoenfeld); (3) "Building Mathematical Structure within a Conjecture Driven Teaching Experiment on Splitting" (Jere Confrey); (4) "The Role of Advanced Mathematical Thinking in Mathematics Education Reform" (M. Kathleen Heid, Joan Ferrini-Mundy, Karen Graham, and Guershon Harel); (5) "Visions of Algebra in Diverse Instruction" (David Kirshner, Carolyn Kieran, Tom Kieren, and Analucia D. Schliemann); (6) "Representing Algebraic Relations Before Algebraic Instruction" (Analucia D. Schliemann); (7) "As It Happens: Algebra Knowing in Action: The Polynomial Engineering Project" (Tom Kieren); (8) "Theories and Experiments in Collegiate Mathematics Education Research" (Ed Dubinsky, Francisco Cordero, Joel Hillel, and Rina Zazkis); (9) "Gender and Mathematics: Integrating Research Strands" (Suzanne Damarin, Diana Erchick, Jere Confrey, Dorothy Buerk, Linda Condron, Ruth Cossey, Laurie Hart, Peter Appelbaum, and Patricia Brosnan); (10) "Geometry and Technology" (Doug McDougall, Dan Chazan, Chronis Kynigos, and Rich Lehrer); (11) "Learning to Reason Probabilistically" (Carolyn Maher, Susan Friel, Cliff Konold, and Robert Speiser); (12) "Research on Rational Number, Ratio and Proportionality" (Tom Post, Kathleen Cramer, Guershon Harel, Thomas Kieren, and Richard Lesh); (13) "Representations and Mathematics Visualization" (Fernando Hitt, James Kaput, Norma Presmeg, Luis Radford, and Manuel Santos-Trigo); (14) "Using Socio-Cultural Theories in Mathematics Education Research" (Judit Moschkovich, Karen Fuson, Yolanda de la Cruz, Joanna Masingila, and Marta Civil); (15) "Developing a Conceptual Framework for Mathematics Teaching" (Martin Simon, Deborah Ball, Rijkje Dekker, and Susan Jo Russell); (16) "Teacher Change in a Reform Calculus Curriculum: Concepts Related to the Derivative" (Thomas Fox); (17) "A Logo-Based Microworld as a Window on the Infinite" (Luis Moreno-Armella and Ana Isabel Sacristan R.); (18) "Sociomathematical Norms and Student Autonomy in Calculus II Honors Students" (Chris L. Rasmussen and Karen D. King); (19) "The Role of a Formal Definition in Nine Students' Concept Image of Derivative" (Michelle J. Zandieh); (20) "What Kind of Notation Do Children Use to Express Algebraic Thinking?" (Carol W. Bellisio and Carolyn A. Maher); (21) "Effects of Algebra Instruction on the Recognition of the Mathematical Structures of Word Problems" (Emily D.F. McFadden); (22) "Middle School Students' Algebraic Reasoning: New Skills and Understandings from a Reform Curriculum" (John P. Smith III, Elizabeth A. Phillips, and Beth Herbel-Eisenmann); (23) "Developing an Imagistic Basis for Understanding Binomial Multiplication" (Erna Yackel and Diana Underwood); (24) "The Effects of Staff Development Focused on Assessment Practices" (Daniel J. Brahier); (25) "Case Studies of Preservice Elementary Teachers' Development in Mathematics Assessment" (Anne Raymond); (26) "Assessing and Documenting Student Knowledge and Progress in Early Mathematics" (Bob Wright, Rita Stewart, Ann Stafford, and Richard Cain); (27) "The Dualistic Nature of School Mathematics Discourse" (Eric J. Knuth and Dominic Peressini); (28) "Examining Teacher and Classroom Supports for Student Invention" (Elizabeth S. Senger); (29) "Tracing Children's Construction of Fractional Equivalence" (Elena Steencken and Carolyn Maher); (30) "Graphing of Discrete Functions versus Continuous Functions: A Case Study" (Fernando Hitt and Orlando Planchart); (31) "Perception of Graphs and Equations of Functions and Their Relationships in a Technology-Enhanced College Algebra Course" (Armando M. Martinez-Cruz); (32) "Understanding Mathematical Experience" (Tracy Noble, Ricardo Nemirovsky, Tracey Wright, and Cornelia Tierney); (33) "Unspecified Things, Signs, and 'Natural Objects': Towards a Phenomenological Hermeneutic of Graphing" (Wolff-Michael Roth); (34) "Re-Thinking Covariation from a Quantitative Perspective: Simultaneous Continuous Variation" (Luis Saldanha and Patrick W. Thompson); (35) "Analyzing Children's Length Strategies with Two-Dimensional Tasks: What Counts for Length?" (Jeffrey Barrett and Douglas Clements); (36) "How Has Measurement Been Taught in Mexico?" (Mariana Saiz); (37) "Understanding Angles from the Perspective of a High School Cerebral Palsy Student" (Kenneth L. Shaw and Paul Durden); (38) "Analyzing Students' Learning in Social Context: A Student Learns to Measure" (Michelle Stepha); (39) "The Learning of Concepts of Stochastic Implied in the Binomial Distribution by Means of the Use of Different Representations and Contexts" (Jesus Colin Miranda); (40) "Comparing Data Sets: How Do Students Interpret Information Displayed Using Box Plots?" (Susan N. Friel); (41) "Students' Statistical Thinking" (Graham Jones, Carol Thornton, Cynthia Langrall, Edward Mooney, Bob Perry, and Ian Putt); (42) "Tracing the Origins and Extensions of Mathematical Ideas" (Regina Kiczek and Carolyn Maher); (43) "Justifications to 5-8 Year-old Students' Responses to Decision Problems" (M.en C. Araceli Limon Segovia); (44) "Supporting Students' Reasoning about Data" (Kay McClain and Paul Cobb); (45) "An Analysis of Students' Statistical Understandings" (Maggie McGatha, Paul Cobb, and Kay McClain); (46) "Middle School Students' Misuse of the Phrase '50-50 Chance' in Probability Instruction" (James E. Tarr). (ASK) ED430775

Bishop, J. W. (1997). Middle School Students' Understanding of Mathematical Patterns and Their Symbolic Representations. 11pp. Paper presented at the Annual Meeting of the American Educational Research Association (Chicago, IL, March 24-28, 1997). This study explores seventh- and eighth-grade students' thinking about mathematical patterns. Interviews were conducted in which students solved problems about sequential perimeter and area problems modeled with pattern blocks and tiles, generalized the relationships related to the patterns and represented the relationships symbolically, identified other valid symbolic expressions of the pattern, and encountered equation-evoking situations. Research questions pertained to the strategies middle school students use to reason when solving pattern problems, symbolic representations the students develop, the students' interpretations of symbolic representations, and the students' strategies for solving equation-evoking situations. The results of this study support the use of mathematical patterns to promote algebraic reasoning and provide descriptions of middle school students' reasoning as they engage in solving a specific type of pattern problem. Findings also suggest that experience exploring the relationships in sequential perimeter and area patterns help students develop an appreciation for the meaning of expression. Contains 16 references. (DDR) ED410107

Bosworth, K., & Hammer, R. (1995). Urban Middle School Students Responses to Anger Situations. 8pp. Paper presented at the Annual Meeting of the American Educational Research Association (San Francisco, CA, April 18-22, 1995). The situations in which young adolescents identify anger and the strategies they use in response to anger were studied with students from a midwestern urban middle school health class. The sample included 53 sixth graders, 41 seventh graders, and 41 eighth graders. Responses to a one-page survey indicated that students reported more anger situations in the family than in any other setting. White students were somewhat more likely to report family conflict, especially with siblings, than were teens of other racial backgrounds. Anger with parents remained relatively stable over the three years, but anger with siblings decreased. Unhealthy responses to anger, such as hitting, swearing, revenge or even the silent treatment, are the more common responses from middle schoolers, but about 15 percent reported responding in a nonviolent way. Students reported the most violence with siblings and the least with adults. With friends they are most likely to try to work it out using a number of strategies. Some students reported the strategies they used to prevent angry reaction, such as taking a time out or reacting with humor. Two tables summarize some responses. (Contains four references.) (SLD) ED386504

BouJaoude, S., & Khalick, F. A. E. (1995). Lebanese Middle School Students' Definitions of Science and Perceptions of Its Purpose and Usage. 24pp. Paper presented at the Annual Meeting of the National Association for Research in Science Teaching (San Francisco, CA, April 22-25, 1995). Students' beliefs about science and their perceptions of its purpose and usage in everyday life affect their science world view which forms the framework they use to interpret their experiences and influence their learning of science and their choice of science related careers. The purpose of the study reported in this paper was to investigate the following questions: (1) How do Lebanese middle school students (N=80) define science? (2) What is the purpose of science according to Lebanese middle school students? (3) Where and how do middle school students see themselves using science? (4) Where do middle school students see others using science? and (5) What are the perceptions of middle school students of how others use science in everyday situations? Students participated in semi- structured interviews and filled out a questionnaire while teachers were interviewed. Results show that most students defined science as an academic subject and perceived its purpose as preparation for higher grades, higher studies, and careers; and saw themselves and others using science in academic settings. Moreover, most of the teachers of the students participating in the study defined science as an academic subject whose purpose was to give students information about the world and perceived themselves and others using science in school related settings. Contains 38 references. (JRH) ED387328

Brahier, D. J. (1995). Mathematical Dispositions of Students Enrolled in First-Year Algebra. 8pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. Dispositions of eighth graders accelerated into first-year algebra were described in this study. Data were collected through surveys, observations, interviews, and cumulative academic files. The most frequently reported reasons for enrolling in algebra were for acceleration of course-taking and preparation for high school. Males demonstrated a higher level of self-efficacy to perform in algebra and secondary mathematics. Students showed a high level of perseverance in terms of sacrifices made to take the course, but classroom performances indicated negative dispositions toward mathematics. Students were driven by a desire to please the teacher and earn grades rather than out of natural curiosity and interest. Neither students nor their parents recognized the real-world applications of algebra. Certain teaching methodologies appeared to evoke positive dispositions. Contains 17 references. (Author) ED389590

Brenner, M. E., & Others, A. (1995). The Role of Multiple Representations in Learning Algebra. 55pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 20, 1995). Middle school prealgebra students (n=157) learned about functions in a 20-day unit that emphasized: (1) representing problems in multiple formats, (2) anchoring learning in a meaningful thematic context, and (3) discussing problem- solving processes in cooperative groups. They produced smaller pretest-to- posttest gains on symbol manipulation tasks, such as solving equations, and larger gains in problem representation tasks, such as translating word problems into equations, tables, and graphs, than did a comparison group taught in the standard way. Although the groups did not differ in their pretest-to-posttest gains in calculating correct answers for word problems, the treatment group produced a larger gain in using mathematical representations while solving word problems than did the comparison group. The same pattern of results was obtained for lower-achieving students and language-minority students. Implications for cognitive theory and educational practice are discussed. Contains 72 references. (Author) ED391659

Brumbaugh, D. (1994). Scratch Your Brain Where It Itches: Math Games, Tricks and Quick Activities, Book D-1 Algebra. 81pp. For other books in this series, see ED 401 158-160 and SE 059 308. This resource book for algebra contains games, tricks, and quick activities for the classroom. Categories of activities include puzzlers, patterns, manipulatives, measurement, graphing, and a section that contains reproducible statement and value cards. Twenty one puzzle problems, four pattern activities, and 11 quick activities that engage students in graphing are provided. Nine manipulative activities that use simple supplies and four measurement problems are also included. Solutions to all the activities are provided at the end of the book. (DDR) ED402170

C

Cai, J., & Moyer, J. C. (1995). Middle School Students' Understanding of Average: A Problem-Solving Approach. 8pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. This study used an open-ended problem-solving approach to teaching and assessing middle school students' understanding of the concept of arithmetic average. Three main results of this study show evidence of positive instructional impact on students' understanding of the concept of average: (1) the number of students who gave correct answers increased from pretest to posttest; (2) on the posttest, more students used appropriate strategies to solve the average problems than on the pretest; and (3) more students used multiple representations on the posttest to explain their solutions than on the pretest. The findings of this study indicate that learning the concept of average is cognitively more complex than the computational algorithm suggests. However, with appropriate instruction, students can have an understanding of the concept beyond the computational algorithm. (Author) ED389574

Caison, B., & Others, A. (1997). Resources for Algebra. 365p. This document presents ideas and activities for teaching algebra. The section on "Week by Week Essentials" provides seven resources in a weekly format. It includes writing ideas that provide an algebra prompt and requires students to organize their thoughts and present them in a coherent fashion, and connections to the real world that identify situations or problems where algebra is an important tool in their investigations and explanations. Also included are specific mathematics vocabulary; ideas from teachers about organization, management, assessment, curriculum, standards, projects, and grading; calculator tips that identify the calculator routines that are most likely to be used; problems that review a recently covered concept or skill; and problems for students to do outside of class that further extend concepts or skills students have used in the past. The "Activities" section contains detailed layouts for classroom exercises designed to last for an hour or more. Supporting materials are provided in the "Blackline Masters." (JRH) ED406205

Capani, A., & De Dominicis, G. (1996 Length: 9 Page(s); 1 Microfiche). Web Algebra. In: WebNet 96 Conference Proceedings (San Francisco, CA, October 15-19, 1996); see IR 019 168. Pages contain light type. This paper proposes a model for a general interface between people and Computer Algebra Systems (CAS). The main features in the CAS interface are data navigation and the possibility of accessing powerful remote machines. This model is based on the idea of session management, in which the main engine of the tool enables interactions with the Session Database. The model was implemented using the client-server capabilities of the World Wide Web. Possible applications of this model include educational purposes, computing, data navigation, communication of structured data, and simulation of operations on distributed systems. Topics discussed include the goal of the project; interface requirements; the session manager; sessions; applications, processes, and agents; queries; reports; scheduling of queries; crash recovery; and educational issues. (DLS) ED427654

Carifio, J., & Nasser, R. (1994). Algebra Word Problems: A Review of the Theoretical Models and Related Research Literature. 55pp. Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 5-8, 1994). Research indicates that students have great difficulty solving algebra word problems and that few high school seniors have mastered the fundamentals of algebra, let alone algebra problem-solving skills. Improving students' algebra problem solving skills is considered to be critically important by those who have worked on reforming mathematics education over the past 10 years, as algrebra has become almost an entry level skills for most scientific, business, and technical jobs in western economies. This paper reviews the current models and theories of algebra word problems and algebra word problem solving and integrates these models into a more comprehensive view and model of algebra word problems and problem solving behavior. The empirical research literature is then reviewed in terms of the models presented and summarized, and the implications of each line of inquiry is discussed as well as the types of studies that needs to be done to advance knowledge in this area. Contains 97 references. (Author) ED373965

Carson, C. L., & Day, J. (1995). Annual Report on Promising Practices: How the Algebra Project Eliminates the "Game of Signs" with Negative Numbers. 29p. This paper argues that operations with negative numbers should be taught using a curriculum that is grounded in algebraic geometry. This position is supported by the results from a study that compared the conceptual understanding of grade 9 students who received the Algebra Project transition curriculum to a control group of grade 6 gifted students who received a traditional introductory algebra course. The overall scores on an open-ended examination showed that the Algebra Project students, who performed lower than the traditional students at the beginning of the year, had surpassed the traditional group by the end of the year. Further examination of the students' problem-solving strategies revealed that the Algebra Project students had developed an understanding of integer addition and subtraction, based on vector operations, while the traditional group of students still exhibited confusion from the use of the conventional sign rules. The study results show how all operations with integers can be made more intuitive to students by providing them with physical experiences that correspond to vector operations in space/time coordinates. These results not only reinforce the view that all students should have the opportunity to learn the important ideas of mathematics, but that all students need to learn the traditionally "higher-order mathematics" that provide geometrical grounding for abstract algebraic concepts. (Contains 19 references.) (Author) ED394828

Chazan, D. (1994). Algebra for All Students? Craft Paper 94-2. 21p. This paper brings a teacher's perspective to bear on recent policy initiatives requiring all high school graduates to complete an Algebra One course. The paper is based on the teacher's experience in teaching high school students who have not been successful in mathematics, at Holt High School in Michigan. The teacher is caught between a desire to provide students in a lower track class with access to mathematics, which plays an important role in providing opportunities for a college education, and students' questions about why study algebra. This leads the teacher to rethink his own understanding of algebra and his approach to teaching the subject. The teacher becomes convinced of students' sense-making abilities in mathematics, but is dismayed by the complexity of the task of working with the class as a social unit. The paper finds that the current algebra curriculum is deeply flawed and makes it impossible for many students to study successfully. An alternative view of the central objects of study in algebra is explored, one that suggests connections between algebra and calculations with quantities that are performed repeatedly in everyday work situations. (Contains 35 references. (JDD) ED377167

Clay, D. W. (1998 Length: 28 Page(s); 1 Microfiche). A Study To Determine the Effects of a Non-Traditional Approach to Algebra Instruction on Student Achievement. Masters Thesis, Salem-Teikyo University. The purpose of this study was to determine if there would be a significant difference in the achievement levels of two groups of eighth-grade Algebra I students when one group receives instruction from a non-traditional (Saxon) method of instruction, and the other receives instruction from a traditional method (Fair and Bragg text). The study was conducted at Oceana Middle School in Oceana, West Virginia, and Glen Rogers Grade School in Glen Rogers, West Virginia during the first 9-week period of the 1996-97 school year. Thirty-three students participated in the study. Nineteen Algebra I students at Oceana Middle School served as the control group. Fourteen students at Glen Rogers Grade School were the experimental group. The classes were held at the same time and had the same amount of instructional time per day. A pretest was given on the first day of class. After 45 days of instruction, a post test was given. Both tests consisted of concepts covered in both textbooks. A two-sample t-test was performed on both sets of data as well as the difference between the scores from the pretest to the post test in both groups. The results indicated that there was statistical difference in the two groups at the beginning of the study. The control group scored much higher than the experimental group but on the post test, results indicated no statistical difference between the two groups. The experimental group improved approximately 65% more than the control group. (Author/ASK) ED428963

Clay, D. W. (1998). A Study To Determine the Effects of a Non-Traditional Approach to Algebra Instruction on Student Achievement. Masters Thesis, Salem-Teikyo University. The purpose of this study was to determine if there would be a significant difference in the achievement levels of two groups of eighth-grade Algebra I students when one group receives instruction from a non-traditional (Saxon) method of instruction, and the other receives instruction from a traditional method (Fair and Bragg text). The study was conducted at Oceana Middle School in Oceana, West Virginia, and Glen Rogers Grade School in Glen Rogers, West Virginia during the first 9-week period of the 1996-97 school year. Thirty-three students participated in the study. Nineteen Algebra I students at Oceana Middle School served as the control group. Fourteen students at Glen Rogers Grade School were the experimental group. The classes were held at the same time and had the same amount of instructional time per day. A pretest was given on the first day of class. After 45 days of instruction, a post test was given. Both tests consisted of concepts covered in both textbooks. A two-sample t-test was performed on both sets of data as well as the difference between the scores from the pretest to the post test in both groups. The results indicated that there was statistical difference in the two groups at the beginning of the study. The control group scored much higher than the experimental group but on the post test, results indicated no statistical difference between the two groups. The experimental group improved approximately 65% more than the control group. (Author/ASK) ED428963

Connell, M. L. (1995). A Constructivist Use of Technology in Pre-Algebra. 7pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. This paper presents two examples in which technology, in this case a fairly sophisticated authoring systemToolBook, was used as a tool to construct student understandings in mathematics. In doing so, students were able: (1) to successfully identify the variables (unknowns) and the information given (data) in the problem; and (2) to create meaningful links between the data and givens which enable successful problem solution. Examples are from work in a seventh grade pre-algebra class of below-average-ability students in a middle class urban setting and are from a single classroom. (Author/MKR) ED389552

Critchfield, S., Hall, N., & Pittman, D. (1998). K-8 Building Blocks for Algebra: Patterns, Functions, Relationships. The ability to use algebra in describing and analyzing real-world situations is becoming a basic skill for all students. The main focus of this book is on patterns, functions, and relationships. Information and ideas on how algebraic thinking can be developed in the years leading to a formal algebra course are included. K-8 teachers are offered a series of appropriate activities that will excite students about mathematics in general and algebra in particular. Activities, grouped by grades K-2, 3-5, and 6-8, are organized around the instructional process and include a list of necessary materials and background information for students and instructors. (ASK) ED429839

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Davis, B. M. (1998). Middle School Algebra: Variables for Success. Middle school algebra classes were observed to examine the multiple variables interacting in learning of algebra. Middle school algebra classes were especially informative because of the significant impact of the students' cognitive development transitions from concrete to formal operations. The National Council of Teachers of Mathematics (1989) informed much of the instructional practices in those classes observed. In contrast to much of the literature (i.e., Dickey, 1997), these teachers were effectively implementing developmentally appropriate and challenging algebra instruction. However, teachers acknowledged some challenges in meeting some of the NCTM standards such as use of real-life examples. Some suggestions were offered in this area as well as some recommendations to enhance metacognition in the middle school classes. (Contains 13 references.) (Author) ED436363

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Education's Weak Link: Student Performance in the Middle Grades.(1998). 20pp. Southern Regional Education Board, 592 Tenth St., N.W., Atlanta, GA 30318. Ways in which middle schools can better prepare their students are presented in this report. It discusses the national pattern of underachievement in basic mathematics and science and details how schools served by the Southern Regional Education Board (SREB) report that almost 50 percent of their eighth-graders are below the basic level in math. Of particular concern are students in low-income districts, from rural areas, and girls. The report explores why these gaps in performance exist and what can be done to ease them. One method is to revaluate the standards and expectations of middle schools and redirect the current emphasis from arithmetic to higher mathematics, such as geometry and algebra. Deciding what students are expected to know is vital to the process, in particular, schools must determine what constitutes high standards and expectations. Parents, also, must be brought into any reform efforts. More information about classroom practices is needed. Suggestions as to how states can evaluate student performance and an example that compares a high-performing to a low-performing school are offered. (RJM) ED419278

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Ford, B., & Klicka, M. A. (1998 Length: 29 Page(s); 1 Microfiche). The Effectiveness of Individualized Computer Assisted Instruction in Basic Algebra and Fundamentals of Mathematics Courses. An individualized Computer Assisted Instruction (CAI) mastery learning format was offered to sections of Fundamentals of Mathematics and Basic Algebra courses over four semesters (two academic years). The effectiveness of this method compared to a traditional lecture approach was examined in the areas of passing the course, passing the final examination, course retention, and passing the next math course. For the Fundamentals course, no significant differences were found among methods in all of the above areas except course retention; course retention was significantly higher in traditional sections. In the Basic Algebra course, traditional sections had significantly higher pass rates and course retention rates. CAI sections had significantly higher exam pass rates. Discussion and recommendations are also included. Contains 15 references and 9 tables. (Author/WRM) ED428962

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Gallardo, A. (1995). Negative Numbers in the Teaching of Arithmetic. Repercussions in Elementary Algebra. 8pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. This article reports the results of a questionnaire applied to secondary school students (n=35) to explore efficiency in the resolution of equations in the domain of whole numbers and the spontaneous responses to problems leading to negative solutions. The most significant results obtained with the questionnaire are the lack of knowledge of the double use of brackets in arithmetic expressions, the partial comprehension of the operation of subtraction, and the difficulty in the operativity of expressions with double signs. The conclusions of this research suggest recommendations for the teaching of whole numbers. Contains 13 references. (Author/MKR) ED389549

Getting Ready for College Early: A Handbook for Parents of Students in the Middle and Junior High School Years.(1997). 19p. Steps that parents and children can take to ensure that students properly prepare for college are covered in this guidebook. The guidebook is divided into four steps. In step one, reasons why it is important to go to college are covered. Some of these reasons include better job opportunities, more earning potential, and the increased variety of jobs one can get with more education. In step two, the types of courses that middle school students should take to prepare for college are covered; subjects such as algebra, geometry, a foreign language, English, science, and history are noted. A chart provides a breakdown of the variety of courses children should take and for how many years. Step three looks at college costs and what students and parents can do to prepare for this significant expense. This theme is continued in step four where ideas for paying for college are presented. Some of these payment suggestions entail applying for federal aid, scholarships, loans, and military enrollment. (RJM) ED412460

Goodson-Espy, T. (1995). Understanding Students' Transitions from Arithmetic to Algebra: A Constructivist Explanation. 40pp. Paper presented at the Annual Meeting of the American Educational Research Association (San Francisco, CA, April, 1995). This study sought to provide an explanation of (n=13) university students' understanding of the concept of linear inequality as presented in a problem- solving setting. In the course of the analysis of the data, the transition from arithmetic to algebra emerged as a critical issue. Therefore, the study examines the differences in the problem-solving activities of solvers who were able to make a transition from arithmetical methods to algebraic methods in contrast to those solvers who were unable to make such a transition. Results suggest that if solvers were to make a successful transition to algebra, they needed to attain post-representational levels of reflective abstraction. In addition, it is indicated that imagery is an inherent part of the development from one level of reflective abstraction to the next. The paper includes the learning tasks. Contains 30 references. (MKR) ED393663

Goodson-Espy, T. J. (1995). A Constructivist Explanation of the Transition from Arithmetic to Algebra: The Role of Reflective Abstraction. 26p. This paper reports a study that sought to explain student understanding of the concept of linear inequality as presented in a problem-solving setting. In the course of data analysis, the transition from arithmetic to algebra emerged as a critical issue. Therefore, the study examined the differences in the problem- solving activities of solvers who were able to make a transition from arithmetical methods to algebraic methods as opposed to those who could not. Thirteen college students participated in unstructured interviews in which they solved problems, but were not told that these problems involved linear inequality. Detailed analyses are given for two of the problem solvers. Results showed that, to make a successful transition to algebra, a solver needed to attain post- representational levels of reflective abstraction and that imagery is an inherent part of the transition from one level of reflective abstraction to the next. (MKR) ED394796

Gregoire, M. (April 1999). Paradoxes and Paradigms in an Eighth Grade Pre-Algebra Class: A Case Study of a "Good" Math Teacher. Paper presented at the Annual Meeting of the American Educational Research Association (Montreal, Quebec, Canada, April 19-23, 1999). Evidence is presented from an ethnographic study of an eighth grade pre- algebra teacher's classroom in support of the idea that teachers' beliefs about the ontology and epistemology of math profoundly influence how they teach and thus what students learn. Specifically, beliefs based on a traditional view of mathematics as procedurally-based are explored. Classroom observations and teacher interviews provide evidence that expert, rule-based instruction be beneficial in providing strong foundations for teacher and student efficacy, a clear belief system about the nature of mathematics, unambiguous guidelines for instruction, including areas such as scaffolding and assessment, and specific methods for classroom management. On the other hand, procedural instruction in this classroom was related to student misconceptions, unremarkable mathematics achievement, excessive student dependence on the instructor, and extrinsic student motivation. The implications of this analysis for implementing mathematics reforms based on the differing metaphors of learning are explored. Contains 42 references. (Author/WRM) ED431600

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Ham, S., & Walker, E. (1999). Getting to the Right Algebra: The Equity 2000 Initiative in Milwaukee Public Schools. MDRC Working Papers. Funding also provided by the Ambrose Monell Foundation. Page Length: 78. This paper describes the Milwaukee Public Schools' involvement in Equity 2000, a standards-based reform initiative to enhance mathematics education and achievement among students of color, thereby increasing their likelihood of college enrollment and completion. The study highlights efforts to support and sustain a key component of Equity 2000: districtwide policy change to end tracking and raise academic standards for all students, beginning with the requirement that all students complete algebra by 9th grade and geometry by 10th grade, and including curriculum reform to reflect National Council of Teachers of Mathematics standards. Section 1, "Introduction," presents key themes. Section 2 discusses "Equity 2000: Policy Rationale and Implementation in Milwaukee." Section 3 discusses "Increasing Student Achievement in Mathematics." Section 4, "Barriers to Student Achievement in Mathematics," discusses student attendance and preparation in mathematics, large class size and meeting diverse learning needs, and limitations on building teacher capacity. Section 5, "Getting to the Right Algebra," discusses algebra curriculum content, when algebra should be taught, and how algebra should be taught. Section 6 discusses "The Legacy of Equity 2000 in Milwaukee." Two appendixes offer the context of reform in Milwaukee's schools and other professional development math initiatives in Milwaukee Public Schools. (Contains 58 references.) (SM) ED441896

Hamilton, J. (1998). Relationship of Remedial Reading Needs to First-Attempt Grade Distributions in Introductory College Algebra at a Two-Year College: Fall 1992 to 19p. Georgia's Gainesville College conducted a study to determine the relationship between students' remedial needs and their grades in two required algebra courses: Math 104, introductory algebra for non-science majors, and Math 151, introductory algebra for science and math majors. Grades for the first attempt in these courses were collected for 6,130 students between ED416914

Harman, P., & Others, A. (1994). The Northwest Guilford High School Heterogeneous Grouping, Algebra IA and IB, and Guided Studies Programs. Evaluation Report. 24pp. Paper presented at the Annual Meeting of the North Carolina Association for Research in Education (Greensboro, NC, March 1994). Northwest Guilford High School, Guilford County (North Carolina), is an essentially rural, largely white school that serves about 1,200 students from all socioeconomic levels. An evaluation was conducted of a heterogeneous grouping project involving students in a 2-year sequence of algebra for those who scored below the 40th percentile on a standardized mathematics test and the Guided Studies program, which is for students having difficulty in English, science, and social studies. Heterogeneous grouping was begun in 1990-91 in response to the high percentage of students planning postsecondary education and the apparent polarity between college preparatory and vocational students. A survey completed by 18 teachers, interviews with a 4-member Parent Advisory Group, and a survey of the junior class provided information about the program and responses to it. Although teachers reported initial misgivings, they agreed that the change has resulted in better learning and better student grades. Parents were supportive, and student attitudes toward the program were good. Students generally felt that teachers expected more and worked to make sure students had learned the material. Program descriptions and the student survey are attached. (Contains 6 references.) (SLD) ED368792

Heid, M. K. (1995). The Interplay of Mathematical Understandings, Facility with a Computer Algebra Program, and the Learning of Mathematics in a Technologically Rich Mathematics Classroom. 7pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. As teachers begin to implement mathematics curricula that capitalize fully on computing technology and that are focused on concepts and applications instead of on execution of by-hand symbolic manipulation routines, their well-established routines of thinking about mathematics and its teaching no longer apply in seamless fashion. This case study, a part of which is reported here, examines the ways that an experienced teacher who participated in Computer-Intensive Mathematics Education (CIME), a 4-week program on the teaching and learning of mathematics in technology-intensive environments, confronted some of the mathematical issues inherent in technology-intensive mathematics. This report gives some insight into one teacher's understanding of functions, independent variables, and parameters, and the ways that this understanding interacts with her use of the new computing tools. (Author) ED389611

Higginbotham, S. (1999). Reading Interests of Middle School Students and Reading Preferences by Gender of Middle School Students in a Southeastern State. Master's Thesis, Mercer University. Page Length: 137. A survey was conducted to examine the reading interests of middle school students. Subjects were sixth, seventh, and eighth grade students in a metropolitan, public middle school located in a southeastern state. It was hypothesized that the data would reveal statistically significant categories of reading interest, and would reveal significant differences between categorical interests by gender. A one-dimensional chi square analysis was used to analyze the reading interests for the sample as a whole. A two-dimensional chi square analysis was used to compare the reading interests by gender. The students clearly indicated a strong preference for the categories of Humor and Horror. The students also reported an interest in Mystery, Historical Fiction, Adventure, Science Fiction, and the non-fiction category of Animals. The results of this study showed differences in interest by gender, which are congruent with many of society's stereotypes. The females reported a stronger interest in Romance, Friendship, Animal Stories, Adventure, and Historical Fiction, while the males reported stronger preferences for the categories of Sports and Science. In addition, the results indicated that the male respondents had a stronger preference for non-fiction than did the female respondents. (Author/RS) ED429279

Hinzman, K. P. (1997). Use of Manipulatives in Mathematics at the Middle School Level and Their Effects on Students' Grades and Attitudes. 65pp. Master's Thesis, Salem-Teikyo University. This paper reports on a study that examines mathematics scores when hands on manipulatives and group activities are used in the classroom at the middle school level. The study also explores changes in students' opinions of mathematics, assesses the intentions of females to take higher mathematics classes in high school, and surveys the amount of parental assistance students receive on homework. Students (N=27) in two pre-algebra classes participated in the study by responding to a pre-test, then participating in various activities related to the curriculum. Their scores and information collected via a survey complete the data. Survey data provides insight into student attitudes and perceptions. Results indicate that: (1) student performance is enhanced by the use of manipulative materials; (2) students' attitudes toward mathematics are significantly more positive than those in previous years; and (3) homework did not seem to positively affect student performance. The analyses suggest that the benefits of using manipulative materials in teaching algebraic concepts not be evidenced by student grades. Contains 22 references. (DDR) ED411150

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Kang, D.-H. (1997). Assessing Korean Middle School Students' Language Learning Strategies in Input- Poor Environments. 40p. This study investigated the strategies used by middle school students in Korea for learning English as a Second Language (ESL) in an environment that provides little English language input. Subjects were 60 students in 3 rural public middle schools. Students were selected randomly from advanced and low-level ESL groups in grades 7, 8, and 9. Using a retrospective interview and think-aloud protocol, strategies for vocabulary learning, listening comprehension, reading comprehension, and writing were elicited. Results indicate that over grade levels, student use of both cognitive and metacognitive strategies increased somewhat. In ninth grade in particular, good students used more effective strategies than did poor students. No difference between male and female students was found. Specific strategy use depended on task type, with students consistently using memory strategies during vocabulary learning, compensation strategies during listening or reading comprehension tasks, and metacognitive strategies in listening and writing tasks. Most students used noncommunicative strategies such as repeating, translation, or rote memorization. In higher grades, advanced students relied less on the mechanical process. Poor students used traditional cognitive strategies across grade levels. Overall, every student was actively involved in the learning process. Appended materials include notes on learning strategies and study data. Contains 68 references. (MSE) ED413778

Kaput, J. J. (1995). A Research Base Supporting Long Term Algebra Reform? 26pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. For reaction papers, see SE 057 183-184. This paper discusses three dimensions of algebra reform: breadth, integration, and pedagogy. Breadth of algebra includes algebra as: generalizing and formalizing patterns and constraints; syntactically-guided manipulation of formalisms; study of structures abstracted from computations and relations; study of functions, relations, and joint variation; and cluster of modeling languages and phenomena-controlling languages. Also discussed are research supporting algebra reform and how research can lead the practice of teaching algebra in new directions. Three phases of reform (short, intermediate, and long term) are discussed using examples of current research projects. Short term reform is seen as first attempts that leave course structures in place, but which contain significant enrichments, such as technology. The second or intermediate phase of reform centers on the integration of algebra into the middle school and offered to all students. The third, long term phase of algebra reform involves full integration of the development of the many forms of algebraic reasoning across all grades with the learning of important mathematics. Contains 108 references. (MKR) ED389539

Kaput, J. J. P. L. (2000). Teaching and Learning a New Algebra with Understanding. This paper suggests a route to deep, long-term algebra reform that begins not with more new approaches but with elementary school teachers and the reform efforts that currently exist. This route involves generalization and expression of that generality using increasingly formal languages, beginning with arithmetic, modeling situations, geometry, and virtually all mathematics that can or should appear in the elementary grades. This route involves infusing algebra throughout the mathematics curriculum from the very beginning. A model of this approach, suggestions, and examples are provided. (Contains 10 references.) (CCM) ED441662

Kaput, J. J. P. L. (2000). Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by "Algebrafying" the K-12 Curriculum. This paper asserts that the key to algebra reform is to integrate algebraic reasoning across all grades and all topics, to "algebrafy" school mathematics. The distinction is made between algebra "the institution" and algebra "the web of knowledge and skill," which is also clarified. Finally, suggestions are made as to how the educational community might work towards a genuine algebra for all. The appendix provides concrete, classroom-based illustrations of the different aspects of algebra at the elementary grade level. (Contains 11 references.) (CCM) ED441664

Karneboge, L., Smith, S. B., VandeSchraaf, C., Wiegardt, C. G., & Wormer, G. (1999). Improving Elementary and Middle School Students' Abilities To Manage Conflict. Master's Action Research Project, Saint Xavier University and IRI/SkyLight. Some appended pages not reproduce well. This action research project evaluated the effectiveness of a program to enhance students' social skills with peers. The targeted population was comprised of elementary and junior high school students in an economically diverse, predominantly blue collar community in central Illinois. The problem of inability to problem solve, listen actively, resolve conflict, and deal with anger was documented by means of teacher observational checklists of student behavior, office disciplinary referrals, teacher surveys, and teacher journal entries. The 5-month intervention was comprised of cooperative learning activities, conflict resolution and anger management techniques, and a modified school-based student management program. Program effectiveness was assessed by comparing pre- and post- intervention measures in the number of disciplinary referrals, behavior checklists, student interviews, and teacher observations recorded in journal entries. Post-intervention data indicated that students showed increased interpersonal relationship skills and improved abilities to manage conflict. (Fourteen appendices include data collection instruments and sample classroom materials. Contains 20 references.) (KB) ED434752

Keating, J., & Others, A. (1996). Alternative Delivery Systems for Introductory Algebra. 8pp. Paper presented at the Annual Conference of the American Mathematical Association of Two-Year Colleges (Long Beach, CA, November 14, 1996). Since 1988, Massachusetts' Massasoit Community College has offered two alternative introductory algebra courses for students receiving low scores on mathematics admission tests. One alternative course provides 5 hours of instruction per week, rather than the 3 hours per week in the traditional course, while the other segments the traditional course syllabus into three 1-credit units taught by separate teachers. A study was undertaken to gather data on the characteristics of students taking alternative courses since 1990 and to compare outcomes for students in alternative and traditional courses. The study sample consisted of 436 students taking the traditional course in 1993 (Group 1), 360 students who completed the segmented modules since 1990 (Group 2), and 212 students who completed the 5-hour-per-week course since 1990 (Group 3). Study findings included the following: (1) in general, Group 2 students were less likely to fail and more likely to pass the introductory algebra course than either Group 1 or Group 3 students; (2) 223 of the Group 2 students passed and 53 failed, while 237 passed and 134 failed from Group 1 and 87 passed and 65 failed from Group 3; and (3) Group 2 students did, however, have a higher mean grade point average for all coursework (2.27) than both Group 1 students (2.26) and Group 3 students (1.91). (HAA) ED405044

Kirshner, D., Ed. (1994). Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (16th, Baton Rouge, Louisiana, November 5-8, 1994). Volume 2: Research Papers, Oral Reports, and Posters (Continued). 350pp. For Volume 1, see SE 055 636. This PME-NA proceedings volume contains the full text of 41 research papers. In addition, brief usually one-page reports, are provided for 11 oral presentations and 13 poster sessions. The full research reports are as follows: "Cognitive Analysis of Chinese Students' Mathematical Problem Solving" (J. Cai and E. A. Silver); (2) "Mathematical and Verbal Abilities in Mathematical Problem Solving by Talented Students" (S. D. Moore); (3) "One Student's Effort in Resolving His Self-Generated Measurement Problem: 'See If It Would Work With a Triangle.'" (A. Reynolds and G. H. Wheatley); (4) "Problem Solving Using Arithmetic and Algebraic Thinking" (A. R. Teppo and W. W. Esty); (5) "The Development of Children's Concept of UnitGrades 4-8" (S. J. Lamon); (6) "Interaction and Fraction Knowledge: Children's Construction of the Iterative Partitioning Scheme" (R. Tzur); (7) "Using Case Studies to Promote Instructional Change" (B. Armstrong, And Others); (8) "The Validity of Concept Maps as a Research Tool in Remedial College Mathematics" (J. Laturno); (9) "Collaborative Action Research in Mathematics Education" (A. M. Raymond); (10) "Middle School Students' Perceptions of Their Everyday Mathematics Usage" (J. O. Masingila); (11) "A Basis for Equity in Mathematics Education: An Experiment In Cultural Course Development" (N. C. Presmeg); (12) "A Case of Equity Reform In Mathematics" (B. F. Risacher); (13) "The Negotiation of Social Norms in a Mathematics Class" (S. D. Trowell and G. H. Wheatley); (14) "Exploring the Social in Social Construction" (S. R. Williams and K. M. C. Ivey); (15) "Empowering Students to Talk About Mathematics In a Seventh- Grade Classroom" (V. M. Adams); (16) "Confidence in Answer Keys Among College Calculus Students" (H. T. Barton); (17) "Enhancing Student Interest in Mathematics Using Integrative Curricula" (M. Mitchell); (18) "The Effects of Psychosocial Variables on Middle School Student Problem-Solving Achievement in Mathematics" (J. L. Nath, And Others); (19) "Justifying the Reasonableness of Answers: Processes of Middle School Mathematics Students" (S. E. Williams and J. V. Copley); (20) "A Teacher's Perception of Time in a Mathematics Classroom" (K. M. C. Ivey); (21) "Preservice Secondary Teachers' Beliefs About Mathematics and Their Expectations About Student Performance on Open-Ended Assessment Tasks" (B. W. Grover and P. A. Kenney); (22) "Changing the Mathematics Learning Environment in Relation to Beliefs, Knowledge, and Practices" (P. A. Jaberg and C. A. Lubinski); (23) "Connecting Orientation Towards Authority to First-Year Teachers' Thinking About Teaching" (B. E. Shealy); (24) "Transforming Mathematics Teaching in Grades K-8: The Role of Material Resources in Supporting Teacher Change" (L. R. Davenport); (25) "Affective Issues in Developing Mathematics Teaching Practice" (L. T. Goldsmith and L. R. Davenport); (26) "Case Studies on Empowering Secondary Mathematics Teachers in Computer-Intensive Environments" (M. K. Heid, And Others); (27) "Perceived Deficits in Middle Grades Mathematics Teaching" (M. K. Heid, and S. J. Feeley); (28) "How Middle School Teachers Adjust to Change: Classroom Testing Of Materials From An Innovative Curriculum Project" (D. V. Lambdin and R. V. Preston); (29) "The Needs of Second Career, Secondary Mathematics Teachers: How Well Are They Met by Academic Programs and Inductive Processes?" (S. A. Maxwell); (30) "Development of Classroom Social Norms and Mathematical Practices with Preservice Teachers" (B. McNeal and M. A. Simon); (31) "Teaching in an Era of Reform: Mathematics in Elementary Classrooms" (P. Sztajn and F. K. Lester, Jr.); (32) "Changing Practice: Describing Mathematics Teachers' Development Through A Modification of Perry's Scheme" (M. R. Wilson and M. P. Goldenberg); (33) "Teacher's Graphical Representations of Rate of Change" (S. B. Berenson and G. S. Carter); (34) "Distributive Flaws: Latent Conceptual Gaps in Preservice Teachers' Understanding of the Property Relating Multiplication to Addition" (S. Campbell and R. Zazkis); (35) "Elementary Teachers' Understanding and Implementation of Unitizing Operations" (T. L. Golding and M. J. Behr); (36) "Elementary School Teachers' Perceptions of Algebra: The Role of Modelling and Technology" (B. J. Pence); (37) "Learning Mathematics While Teaching" (S. J. Russell, And Others); (38) "Teacher's Changing Conceptions of the Nature Of Mathematics" (D. Schifter); (39) "The Fundamental Theorem of Arithmetic: Used and Confused" (R. Zazkis and S. Campbell); (40) "Delineating the Transformation of Subject-Matter Knowledge in Pedagogical Content Knowledge" (C. L. Ebert); and (41) "Pedagogical Content Knowledge, Curricular Knowledge and Teacher Change" (B. S. Rich, And Others). (WTB) ED383534

Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (1999). A Developmental Model of Algebra Problem Solving: Trade-offs between Grounded and Abstract Representations. Paper presented at the Annual Meeting of the American Educational Research Association (Montreal, Quebec, Canada, April 19-23, 1999). Some figures not reproduce clearly. This paper presents a developmental model of students' acquisition of competence in quantitative and algebraic problem solving. A key notion underlying the developmental model is a distinction between grounded and abstract representations. Grounded representations, like story problems, are more concrete and familiar, closer to physical objects and everyday events. Abstract representations, like symbolic equations, are concise and easy to manipulate, but are distanced from any physical objects of reference. The complementary computational characteristics of grounded and abstract representations lead to hypotheses about the order of skill acquisition. In prior research, the authors demonstrated that early in the development of algebraic competence, the advantages of grounded representations outweigh those of abstract representationsfor simpler problems, students are better at story problems than the analogous equations. This paper presents two studies that test the hypothesis that later in algebra development, the advantages of abstract representations emergefor more complex problems, students are better at equations than the analogous story problems. Includes 6 tables, 7 figures, and 16 references. (Author/WRM) ED433245

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Lacampagne, C. B., Ed., & Others, A. (1995). The Algebra Initiative Colloquium. Volume 1: Plenary and Reactor Papers. 231pp. For volume 2, see SE 056 573. This volume contains the plenary or reactor papers presented at a conference on reform in algebra held in Leesburg, Virginia, December 9-12, 1993. Papers included are: (1) "Introduction" (C. B. Lacampagne); (2) "Summary" (C. B. Lacampagne); (3) "Recommendations" (C. B. Lacampagne); (4) "The Development of Algebra and Algebra Education" (V. J. Katz); (5) "Long-Term Algebra Reform: Democratizing Access to Big Ideas" (J. J. Kaput); (6) "Algebra in the K-12 Curriculum" (G. Burrill); (7) "What Is the Appropriate K-12 Algebra Experience for Various Students?" (J. Fey); (8) "Algebra at the College Level" (M. Artin); (9) "Algebra Initiative" (V. Pless); (10) "Algebra and the Technical Workforce" (H. Pollak); (11) "Reshaping Algebra to Serve the Evolving Needs of the Technical Workforce" (S. Garfunkel); (12) "A Cognitive Perspective in the Mathematical Preparation of Teachers: The Case of Algebra" (A. G. Thompson & P. W. Thompson); (13) "Preparing Teachers to Teach Algebra for All: Preliminary Musings and Questions" (M. Enneking); and (14) "Algebra for All: Dumbing Down or Summing Up?" (L. A. Steen). Appendices include the conference agenda; Conceptual Framework for the Algebra Initiative of the National Institute on Student Achievement, Curriculum, and Assessment; and a participant list. (MKR) ED385436

Lacampagne, C. B., Ed., & Others, A. (1995). The Algebra Initiative Colloquium. Volume 2: Working Group Papers. 157pp. For volume 1, see SE 056 572. This volume presents recommendations from four working groups at a conference on reform in algebra held in Leesburg, Virginia, December 9-12, 1993. Working Group 1: Creating an Appropriate Algebra Experience for All Grades K-12 Students produced the following papers: (1) "Report" (A. H. Schoenfeld); (2) "Five Questions About Algebra Reform (and a thought experiment)" (D. Chazan); (3) "Algebra and the Democratic Imperative" (R. B. Davis); (4) "Realism(s) for Learning Algebra" (R. Hall); (5) "Algebra, The New Civil Right" (B. Moses); (6) "Issues Surrounding Algebra" (E. Phillips); (7) "Is Thinking About 'Algebra' a Misdirection?" (A. H. Schoenfeld); and (8) "Thoughts Preceding the Algebra Colloquium" (Z. Usiskin). Working Group 2: Educating Teachers, Including K-8 Teachers, to Provide These Algebra Experiences produced: (1) "Report" (A. Buccino); (2) "Educating Teachers to Provide Appropriate Algebra Experiences: Practicing Elementary and Secondary TeachersPart of the Problem or Part of the Solution?" (C. Gifford-Banwart); (3) "Educating Teachers for Algebra" (A. Buccino); (4) "Experience, Abstraction, and 'Algebra for All': Some Thoughts on Situations, Algebra, and Feminist Research" (S. K. Damarin); (5) "Educating Teachers, Including K-8 Teachers, to Provide Appropriate Algebra Experiences" (N. D. Fisher); (6) "On the Learning and Teaching of Linear Algebra" (G. Harel); and (7) "Algebra: The Next Public Stand for the Vision of Mathematics for All Students" (H. S. Kepner, Jr.). Working Group 3: Reshaping Algebra to Serve the Evolving Needs of the Technical Workforce produced: (1) "Report" (S. Forman); (2) "Algebra, Jobs, and Motivation" (P. Davis); (3) "To Strengthen Technical Education Systematically" (J. G. Greeno); (4) "Thoughts About Reshaping Algebra to Serve the Evolving Needs of a Technical Workforce" (R. Lesh); (5) "Algebra for the Technical Workforce of the 21st Century" (P. D. McCray); (6) "Some Thoughts on Algebra for the Evolving Work Force" (T. A. Romberg & M. Spence); and (7) "Algebra: A Vision for the Future" (S. S. Wood). Working Group 4: Renewing Algebra at the College Level to Serve the Future Mathematician, Scientist, and Engineer produced: (1) "Report" (J. Gallian); (2) "Some Thoughts on Teaching Undergraduate Algebra" (W. D. Blair); (3) "Toward One Meaning for Algebra" (A. Cuoco); (4) "Some Thoughts on Abstract Algebra" (S. Montgomery); and (5) "Suggestions for the Teaching of Algebra" (W. Y. Velez). Appendices include the conference agenda; Conceptual Framework for the Algebra Initiative of the National Institute on Student Achievement, Curriculum, and Assessment; and a participant list. (MKR) ED385437

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Adams, T. L. (1994). Graphing Function Problems in Teaching Algebra. 20p. Algebra students typically pay little attention to choosing appropriate domains, ranges and scales of axes for graphing functions. The purpose of this study was to investigate the effect of introducing an assignment and employing the graphing calculator to examine and alleviate students' difficulties regarding the selection of appropriate domains, ranges and scales of axes for graphing functions and to examine difficulty students might have regarding the identification, construction and definition of function. The research sample consisted of 128 college algebra students enrolled in a north-central Florida community college. There were eight classes of students: six treatment classes and two control classes. All classes were intact, thus random assignment of students to the classes was not possible. Two classes were in each of the following groups: Graphing Calculator and Assignment Group, Graphing Calculator Only Group, Assignment Only Group, and the Control Group. The analysis of covariance was used to examine mean differences on two instruments between the four groups in the study. The Domain/Range/Scale Instrument and the Identification/Construction/Definition Instrument were both administered as pretests and posttests. The results of the study suggest that the treatments were in various ways interactively and independently effective regarding the students' understanding of the concept of function. Contains 25 references. (Author/MKR) ED372942

Marsh, C. A. (1995). Middle School Students' Attitudes toward High School Foreign Language Study. 16p. A total of 283 students at Fassett Middle School in Oregon, Ohio, a suburban, middle-class city were surveyed about their attitudes toward foreign language study in high school. The participants were 151 male and 132 female students in grade 6 (87), grade 7 (104), and grade 8 (92). The objective was to determine whether or not the middle school exploratory language component of the curriculum was achieving its goal of increasing student interest in language study. Most respondents had been enrolled in language courses during their middle school years. The teacher-developed questionnaire elicited information about students' grade, gender, languages studied, plans to take foreign language(s) in high school, plans to attend college; letter grades in languages taken; attitudes concerning the importance, requirement, and enjoyment of language study; and favorite and least favorite subjects in school. As anticipated, older students and students with higher language grades had more positive attitudes about language study. Female students had more positive attitudes, but students having taken at least two foreign language courses previously did not have significantly more positive attitudes than others. Contains 28 references and the survey used. (MSE) ED392247

Masingila, J. O., & King, K. J. (1996). Middle School Students' Use of Function Ideas in Everyday Mathematics. 16pp. Paper presented at the Annual Meeting of the American Educational Research Association (New York, NY, April, 1996). This paper describes preliminary analyses of data from an ongoing project entitled "Connecting In-School and Out-of-School Mathematics Practice." During this project the authors are: (1) investigating how middle school students use mathematics concepts and processes in a variety of out-of-school situations, and (2) working with middle school teachers to develop curriculum ideas to facilitate connection-making by students and investigating whether students are connecting their in-school and out-of-school mathematics learning and practice. The majority of the paper discusses the observations of middle school students (n=6) using function ideas in their everyday activities. Analysis of data showed that students used functions ideas in many situations. (MKR) ED398066

Mayer, D. P. (1997). New Teaching Standards and Old Tests: Dangerous Mismatch? 51pp. Paper presented at the Annual Meeting of the American Educational Research Association (Chicago, IL, March 24-28, 1997). As almost every state attempts to reform mathematics instruction by implementing new teaching standards, state testing practices remain largely unchanged. Is there a mismatch between these new standards and the old tests? This question is investigated by examining whether middle school and high school algebra students taught in a manner consistent with the National Council of Teachers of Mathematics (NCTM) "Professional Standards" performed differently on three standardized algebra assessments than students taught in traditional classrooms. The data come from 94 teachers, 2,369 students, and 40 schools in 1 of the nation's largest school districts. Results indicate that a mismatch does not exist between the "Standards" and the old tests. In fact, middle school algebra students whose teachers spent more time using the NCTM teaching approach had higher growth rates than students whose teachers spent less time using the approach. However, students with higher ability levels benefited more. The growth rates of the lowest achieving students, the high school students (who were disproportionately poor and black), were not helped or hindered by the NCTM teaching approach. This study provides policymakers with evidence that traditional multiple choice tests do not directly undermine the standards movement in this one school district. On the other hand, old tests will not provide teachers of low-achieving students with any incentive to adopt the "Standards." (Contains 2 figures, 8 tables, and 50 references.) (Author) ED413348

Mizelle, N. B. (August 1999). Helping Middle School Students Make the Transition into High School. Young adolescents entering high school look forward to having more choices and making new and more friends; however, they also are concerned about being picked on and teased by older students, having harder work, making lower grades, and getting lost in a larger, unfamiliar school. For middle school students, including those who have been labeled "gifted" or "high- achieving," the transition into high school can be an unpleasant experience. This Digest discusses how educators can ease students' transition into high school by providing challenging and supportive middle school environments and by providing well-designed transition programs. These programs should include activities that provide information to students and parents, activities that provide social support, and activities that bring middle and high school educators together. Finally, the importance of parents being involved in their young adolescent students' transition from middle to high school can hardly be overestimated, and parent involvement in the transition process can be encouraged through a variety of activities, including conferences with the high school counselors and school visits. (LPP) ED432411

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Orey, M. A., & Nelson, W. A. (1994). Visualization Techniques for Examining Learner Interactions with HyperMedia Environments. 13pp. In: Proceedings of Selected Research and Development Presentations at the 1994 National Convention of the Association for Educational Communications and Technology Sponsored by the Research and Theory Division (16th, Nashville, TN, February 16-20, 1994); see IR 016 784. Methods are described for visualizing and characterizing user interactions with hypermedia systems; they have been found to be effective in several developmental and evaluation studies. Analyzing the patterns of user interaction makes it possible to evaluate the design and usefulness of such systems. Path algebras can be used to describe and compare routes users take through hypermedia systems. Directed graphs are another method of constructing a network representation of user paths using the Pathfinder algorithm. Another approach is to construct an image based on interaction data with the Toolbook program. Fifty-seven data files from a previous study were entered into the Toolbook program to generate a map of each user's interaction. Six gifted middle-school students, six teachers, and six doctoral students in instructional technology were asked to categorize the images and to associate them with course grades of the users. Although subjects could not generally associate images with grades, they were able to place them in identifiable categories, which suggests the potential of the method for depicting user paths. Nine figures illustrate the approaches. (Contains 16 references.) (SLD) ED373747

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Palow, W. P. P. L. (1999). Is Arithmetic Really Necessary for Algebra? A Case for an Integrated Curriculum. By measuring the performance of 62 students enrolled in a community college introductory algebra course, this study challenges the generally accepted assumption among mathematics instructors that mastery of arithmetic is necessary for the learning of algebra. Study subjects were 35% male, 74% Hispanic, 16% Black, 8% white, and 2% other. A pretest, consisting of 10 items designed to indicate minimum competency in computational arithmetic, was administered at the beginning of the semester, and the final exam was used as a posttest. Using analysis of variance, analysis of covariance, and a Pearson r calculation, no significant differences were found between three groups that received different types of instruction: (1) computer-managed instruction combined with individualized instruction in a math lab; (2) computer-managed instruction used in the classroom; and (3) a traditional class, used as a control group. The study concludes that the ability to perform well in computational arithmetic has little, if anything, to do with the ability to perform well in beginning algebra. As a result of this finding, it is suggested that community colleges combine the topics of arithmetic and beginning algebra into one integrated course that uses a "spiral approach." Contains 3 tables and 11 references. (CAK) ED429657

Pettitt, L. M., & Others, A. (1995). Middle School Students' Perceptions of Math and Science Abilities and Related Careers. 21pp. Paper presented at the Biennial Meeting of the Society for Research in Child Development (61st, Indianapolis, IN, March 30-April 2, 1995). Funding also received from Girls Count. A total of 162 students from a suburban Denver middle school were surveyed to determine the following: their career aspirations, whether they perceive differences in society's acceptance of certain career choices for women versus men, the relationship between their perceptions of their abilities and their desire for certain jobs, and possible relationships between jobs they would not like to hold and their endorsement of reasons for not wanting certain jobs and/or their concerns about balancing career and family. The survey revealed that, although girls and boys believe society accepts multiple career options for women and men, their own career aspirations remain fairly sex stereotyped. Girls considered themselves capable of succeeding in doctor/veterinary jobs but not science-related jobs, whereas boys showed the reverse pattern. Neither girls nor boys saw a relationship between science abilities and ability to succeed at doctor/veterinarian careers. They did, however, see some relationship between science ability and success in science-related careers. It was concluded that both girls and boys need more information about the requirements of particular career options and additional encouragement to consider the rewards of a variety of types of work. (Seventeen figures/tables are included.) (MN) ED380663

Phillips, E. (1995). A Response to a Research Base Supporting Long-Term Algebra Reform. 10pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. For related papers, see SE 057 182-183. This paper is a reaction to a plenary address, "A Research Base Supporting Long Term Algebra Reform?" by James Kaput (SE 057 182). The reactions fall into three categories: comments on Kaput's dimensions of algebra reform, a brief discussion of algebra and algebra reform from the viewpoint of a curriculum developer of the Connected Mathematics Project (CMP), and some concerns about Kaput's three stages of reform. (MKR) ED389541

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Rachlin, S. L. (1995). The Development of Problem-Solving Processes in a Heterogeneous Eighth Grade Algebra Class. 8pp. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21-24, 1995). For entire conference proceedings, see SE 057 177. The primary purpose of this study was to evaluate the application of a process approach for the teaching of algebra with a heterogeneous class of eighth-grade students. The assessment was conducted by identifying the processes used by above- average, average, and below-average algebra students in solving standard and nonstandard problems ranging across a content X process X form matrix-(integers, fractions, polynomials) X (generalization, reversibility, flexibility) X (expression, equation). Transcripts of the experimental data were coded and contrasted with normed data from earlier studies involving more homogeneous groups. (Author) ED389571

Rosman, N. J. S. (1994). Effects of Varying the Special Educator's Role within an Algebra Class on Math Attitude and Achievement. 88pp. Master of Arts Thesis, University of South Dakota. This study compared the effects of two different special educator roles within high school algebra classes containing special needs students. The special educator served as either in-room assistant or as co-teacher. The study was conducted for one chapter of algebra study over a 3-week period. No significant differences between groups were found prior to the treatment. Following the two treatments, students in the co-teaching condition scored significantly higher in achievement than did students in the in-room assistance group or the control group. Females in the co-teaching condition achieved significantly higher scores than did females in the control group. Females receiving in-room assistance also scored significantly higher than control group females. No significant findings in achievement were found for males or for students receiving special education services. Appended are forms used in the program and a brief description of the collaboration model. (Contains 61 references and 15 tables.) (DB) ED381993

Ross, J., & Faucette, D. (1994). College Algebra and Writing: A Pilot Project, Spring Semester 1994. Final Report for the Title III Literacy across the Curriculum Team. 12p. A study was undertaken at Central Arizona College to determine the effect of writing activities on student performance in College Algebra. An experimental group of eight students was established and given lists of math-related vocabulary words and problems which required written explanations. Quizzes were constructed requiring the group to respond in writing, while eight students in a control group took standard quizzes. A standard final college algebra examination served as the post-test for both groups, while the experimental group also completed a survey on their perceptions of the writing activities. The mean test score on the final examination was 84 for the experimental group and 74.3 for the control group. Conclusions drawn from the scores and from responses to the survey included the following: (1) the small sample size did not make the difference in mean test scores significant; (2) students who excel in mathematics not benefit from writing assignments but lesser prepared students benefit a great deal; (3) experimental group responses indicated that students were generally positive about writing activities, with five of the students agreeing that the assignments helped them to understand algebra, two expressing no opinion, and two disagreeing; and (4) in general, students did not understand that the writing assignments were meant to help them learn algorithms rather than show the teacher that they could communicate math concepts in writing. (Appendixes include test scores, the survey instrument, student and teacher comments, and writing samples.) (KP) ED387166

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Sanford, E. E. (1998). Interpretive Guide for North Carolina End-of-Course Tests. Algebra I, English I, U.S. History. Local Use Form 1997-1998. 26p. The North Carolina Testing Program is based on the assessment of higher level skills, the skills required to solve problems and make informed choices and decisions. This guide explains the end-of-course tests that are a component of the Testing Program and provides information to interpret them. The end-of-course tests for high school students are aligned with the state's standard course of study and administered during the last 10 days of the course. Some forms of the Algebra I, English I, and U.S. History tests have been designated as "secure for local use" and are no longer considered secure for statewide use. Designating a form of the test for local use allows school systems to retest students after interventions, deliver staff development, and conduct research studies. For each item on a "secure for local use" form for these three subjects, this document provides the curriculum goal and objective from the state's "Standard Course of Study," the related thinking skill, the correct answer, and the p- value (percentage of students who responded correctly during the 1998 administration of the test). (SLD) ED425180 You be able to order this document from the ERIC Document Reproduction Service.

Savelsbergh, E. R., Ferguson-Hessler, M. G. M., & de Jong, T. (1998). Physics Learning with a Computer Algebra System: Towards a Learning Environment That Promotes Enhanced Problem Representations. 34pp. Paper presented at Annual Meeting of National Association For Research in Science Teaching (71st, San Diego, CA, April 19-20, 1998). An approach to teaching problem-solving based on using the computer software Mathematica is applied to the study of electrostatics and is compared with the normal approach to the module. Learning outcomes for both approaches were not significantly different. The experimental course successfully addressed a number of misconceptions. Students in the experimental course found the visualization tools to be more instructive than the problem-solving tools. This course imposed a high cognitive load on the students. This paper discusses how the course can be improved and how it can be a valuable supplement to the usual teaching approach. (Contains 20 references.) (DDR) ED421346 Available from: University of Twente, Faculty of Educational Science and Technology, Dept. of Instructional Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands.

Schaufele, C., & Zumoff, N. (1996). Earth Algebra. 67p. Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through mathematical analysis of real data, students gain new perspectives on mathematics as a tool. The course incorporates group work, written reports, simplistic mathematical models, and the use of technology. An executive summary involving a project overview, statement of purpose, background and origins, project description, evaluation/project results, and conclusions is included. Appended is a final evaluation report, "Earth Algebra Project: A Final Evaluation Report" (Pamela J. Drummond). Also appended are the evaluation instruments. (Author/ASK) ED417937

Schifter, D. (1997). Developing Operation Sense as a Foundation for Algebra. 33pp. Paper presented at the Annual Meeting of the American Educational Research Association (Chicago, IL, March 24-28, 1997). This paper posits that, as educators' attention turns to elementary mathematics as preparation for algebra, it is important to begin by examining the kind of elementary mathematics content teaching that is aligned with reform. The paper calls for a close examination of students' development of operation sense. The role of children's engagement with the four basic operations in their preparation for algebra is also discussed. One method of data collection used in this study is episode writing. Six of the scenarios presented in this paper are taken from episodes written by teachers in a National Science Foundation teacher-enhancement project known as Teaching To the Big Ideas. Although the question of how elementary education can prepare students for algebra is addressed here, teachers at other levels still need to be prepared to help their students begin to make meaning of the language of algebra. Implications of this work include the idea that other notional systems such as diagrams, graphs, and tables in written and electronic forms are important for students to understand. Contains 62 references. (DDR) ED408152

Schullo, S. A. A., Burton L. (1998). Low SES Algebra 1 Students and Their Teachers: Individual and a Bi- directional Investigation of Their Relationship and Implicit Beliefs of Ability with Final Grades. 46pp. Paper presented at the Annual Meeting of the American Educational Research Association (San Diego, CA, April 13-17, 1998). Mathematics achievement and the relationship of low socioeconomic status (SES) students with teachers have been a major concern for educators. Although there are social and economic explanations of failure of low SES students, ultimate success and failure be significantly affected by differing personal beliefs about learning among students and teachers. C. S. Dweck's implicit beliefs about intelligence and mathematics were examined using a bi-directional methodology with R. D. Lang's Interpersonal Perception Method (IPM) in algebra 1. Incremental/entity beliefs, gender of students, gender of teachers, and low achievers and high achievers were subjected to an analysis of variance. Eighty low SES students and 43 male and 37 female teachers responded to the incremental beliefs that reflected abilities as changing and increasing with effort and to entity beliefs that reflected abilities as unchanging or fixed. Individual incremental female students confirmed the prediction that incremental believers would have higher final grades than the female students endorsing entity beliefs. Students matched with incremental-belief male teachers confirmed the dyadic prediction that incremental intelligence beliefs would lead to higher achievement than for a dyad endorsing entity beliefs. Dweck's personal theory integrated with the IPM examined several complex issues about real students' and teachers' experiences. Examining these experiences revealed some adaptive and maladaptive patterns that lead to math achievement for the low SES female student. Issues of methodology for examining student teacher relationships, power of projections over accuracy of perceptions (particularly for female students' expectations from their teachers), and the necessity of collecting data from both students and teachers for a clearer understanding of the issues affecting low SES students' math achievement are discussed. (Contains 1 table, 10 figures, and 44 references.) (Author/SLD) ED424289

Schwartz, W. (1996). Preparing Middle School Students for a Career. 5pp. Based on "Vocational Education in the Middle School," ERIC/CUE Digest No. 155, ED 377 314. This guide provides information about how middle schools can promote the development and education of adolescents, especially about how they can focus students' attention on career opportunities and training. It also offers families some ideas about how they can encourage their children's career awareness, and it presents some criteria for parents to consider when selecting a middle school or participating in the planning and functioning of the school their children attend. Many middle schools can help students plan for a career, even if they do not provide specific training for a vocation, by exploring interests, aptitudes, abilities and the broad scope of work. Assisting in the development of aspirations and goals and social skills and values helps in career preparation. Career planning can be integrated into the middle school curriculum through general themes in classes and through work-related skills development. Effective instructional strategies can include team teaching, exploratory programs, learning organized around key themes or concepts, and cooperative learning. Parents can encourage their children through open communication about their interests and abilities. (SLD) ED401351

Schwarz, C., & White, B. (1998). Fostering Middle School Students' Understanding of Scientific Modeling. 28pp. Paper presented at the Annual Meeting of the American Educational Research Association (San Diego, CA, April, 13-17, 1998). This paper reports on the evaluation of an 11-week curriculum created to foster seventh grade students' understanding of scientific modeling. In the curriculum, students engaged in model-oriented activities such as creating non-Newtonian computer microworlds to embody their conceptual models, evaluating their models with criteria, and reflecting on the nature of models. Various methods of assessing modeling are also discussed. The overall analysis suggests that students gained a significantly better understanding of the nature and utility of models without promoting similar gains in students' understanding of the process of creating and evaluating models. These results indicate that while modeling knowledge is difficult to obtain even from an extended model-focused curricula, progress can be made by further refining model-oriented curricula and assessments. (Contains 39 references and written modeling assessment results.) (DDR) ED424095

Shakrani, S. (1996). Eighth-Grade Algebra Course-Taking and Mathematics Proficiency. NAEPFACTS, 1, 1996 Feb. This brochure is part of NAEPFACTS, a new series that briefly summarizes findings from the National Assessment of Educational Progress (NAEP). Course-taking is generally a powerful indicator of mathematics achievement. This occurs partially because students who are more proficient tend to take more mathematics classes and, at the eighth grade, the better students are tracked into more advanced classes. The 1992 NAEP results linking proficiency to course work confirm this pattern, with eighth graders enrolled in pre-algebra and algebra courses having higher proficiency scores than students taking eighth-grade mathematics. Contains two tables for 1992: (1) National average proficiency of public and private school eighth-grade students by mathematics course-taking, and by race and gender and (2) Average proficiency of eighth-grade public school students by mathematics course-taking, and by state. (MKR) ED396915

Sharp, J. M. (1995). Results of Using Algebra Tiles as Meaningful Representations of Algebra Concepts. 8pp. Paper presented at the Annual Meeting of the Mid-Western Education Research Association (Chicago, IL, October 13, 1995). Mathematical meanings can be developed when individuals construct translations between algebra symbol systems and physical systems that represent one another. Previous research studies indicated (1) few high school students connect whole number manipulations to algebraic manipulations and (2) students who encounter algebraic ideas through manipulating physical models gain conceptual knowledge of algebra. Five high school algebra classes used algebra tiles to study operations with algebraic expressions. Results suggested no differences between groups who used or did not use manipulatives when tested with traditional chapter tests. Results of diary narrative data indicated that the majority of students stated that the tiles added a mental imagery that made learning "easier." They indicated that they found it easy to think about algebraic manipulations when they visualized the tiles. (Author) ED398080

Spahr, A. E. (1995). Predicting Graduation Status of Nursing Students Using Entering GPA and Grades in Algebra, Biology, and Chemistry. A study was undertaken at Morton College, in Illinois, to examine the relationship of entering grade point average (GPA) and grades in prerequisite support courses in algebra, biology, and chemistry to graduation in the college's nursing program. A sample was developed of 255 students admitted to the nursing program in 1990, 1991, and 1992 and regression equations were calculated to determine the relationships between these predictor variables and graduation status after 2 years. Means and standard deviations were determined, and an additional follow-up logistic regression was performed to estimate a student's odds of graduating from the nursing program based solely on GPA. Results indicated that the best predictor of graduation status was entering GPA, followed by grades in biology and grades in chemistry. Although all of these correlations were significant, the amount of variance in graduation status explained by each was very small. Moreover, when entering GPA was included in the analysis, the independent contributions made by course grades were inconsequential. However, using GPA as the sole predictor indicated that a student with a low entering GPA was almost as likely to graduate from the program within 2 years as a student with a high GPA. This indicates that other variables besides entering GPA should be examined, such as reading skill level, computational skill level, and vocational interest tests. (KP) ED386224

Sworder, S. (1997). Evaluation of Intermediate Algebra Prerequisite Enforcement at Saddleback College. 26p. In ED409052

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Teaching Middle School Mathematics: A Resource For Teachers by the Teachers of PROJECT LINCS (Lead teachers INvolved in making ConnectionS).(1994). 163pp. Cover title varies: "Teaching and Learning Middle School Mathematics TODAY. A Resource FOR Teachers BY Teachers from Project LINCS.". This resource book represents shared experiences of Project LINCS' (Lead teachers INvolved in making ConnectionS) teachers and staff in addressing the question, "Are we teaching mathematics so that all students will be empowered to use it flexibly, insightfully, and productively?" and other fundamental questions that are driving change in school mathematics. These questions affect: (1) the mathematics that middle school students learn; (2) the way the curriculum is organized; (3) the way students learn mathematics; (4) the roles of teachers; and (5) the climate and activities of our mathematics classrooms. The major portion of this resource book is devoted to sharing sample activities to assess, build on, and nurture student understanding and progress toward important mathematical goals. The teaching ideas are grouped under the major categories of algebra, alternative assessment, cooperative learning groups, geometry, number and computation sense, probability and statistics, problem solving, and writing to nurture and communicate understanding. Sample activities contain information to help teachers and students to focus, explore, disucss and reflect, and extend the activities. Each section contains references. (MKR) ED385444

Telese, J. A. (2000). School Algebra Reform: Meeting the Grade? Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 24-28, 2000). Page Length: 35. This paper reports on a case study that was conducted at five high schools from a large, urban school district located in South Texas. The purpose of the study was to gain an understanding of Algebra 1 teaching strategies. The research questions were: (1) What is the predominant mode of instruction for Algebra 1? and (2) What is the level of achievement of Algebra 1 students? The schools consist primarily of students ofMexican descent. The teachers were also of Mexican descent. Both qualitative and quantitative data were collected. Field observations revealed a teaching methodology that focused on skills and procedures facilitated through lecture and supervised practice. The instruction was heavily influence by the state's high stakes test. Student achievement reflected the instructional methodology, which focused on particular skills and processes. Students failed to complete open-ended and higher order thinking questions on a standardized measure of algebraic skills and concepts developed by a publisher. Recommendations are included for improving algebra instruction. (Contains 19 references/ASK.) (Author) ED441675

Telese, J. A. (February 1999). The Role of Social Constructivist Philosophy in the Teaching of School Algebra and in the Preparation of Mathematics Teachers. Paper presented at the Annual Meeting of the Association of Teacher Educators (79th, Chicago, IL, February 13-17, 1999). As a learning theory, constructivism describes knowledge as being in flux where an individual internally constructs knowledge through social and cultural mediation. Constructivist learning theorists contend that social activity and discourse must play important roles in order for understanding to occur. This paper describes how the social constructivist view of learning can be useful in the teaching of high school algebra and in the preparation of mathematics teachers. A brief summary of the social constructivist view as a philosophy of mathematics is presented, followed by a review of learning theory associated with school algebra. Implications for the preparation of pre-service mathematics teachers are discussed. (Contains 17 references.) (ASK) ED432469

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Walen, S. B. (1994). Students' Prejudices and Algebra Reform: Value, Ability, and Self. 22pp. For companion study, see SE 054 378. For effective educational reform, a teacher must be aware of students' views of their role in the classroom culture, and only then a teacher assist students to become aware of their own prejudices and resistance to change in the classroom structure. This paper discusses a study that investigated the changing student role, particularly from the student viewpoint, which provides an essential description of problematic areas that teachers address in their implementation of current mathematics education reform goals. Observed in the study were (n=52) students from two algebra classes of a teacher dedicated to changing her teaching methods, and in particular the classroom talk, from traditional interchanges to the type of discourse recommended by the NCTM Standards. Five students from each class agreed to serve as key informants and to participate in a journal exchange and interviews with the researcher. Discussion includes a summary of the students' views prior to analysis and analysis of factors that influenced students' classroom perceptions, such as the value of classroom algebra, mathematics as memorized facts, and effort and ability. Contains 26 references. (MKR) ED372940

Walz, K. (1998). A Visual Approach to Simplifying Square Roots Applied in Geometry and Advanced Algebra Courses. Master's Action Research Project, Saint Xavier University and IRI/Skylight. This report describes a program for increasing students' ability to simplify square roots. The targeted population consisted of high school students in a rural community in a midwestern plains state. The problem of the ability to understand the abstract algebraic process of simplifying square roots was documented through teacher-made tests, teacher observation, and poor/inadequate homework. Analysis of probable cause data revealed that early teens not have the cognitive ability to simplify square roots using an algebraic method based upon the mathematical/logical intelligence. The understanding of abstract concepts appears to be related to the way in which students perceive their own thought processes. A review of solution strategies suggested by knowledgeable others, combined with an analysis of the problem setting, resulted in selecting an intervention consisting of a visual approach to simplifying square roots. The students were encouraged to use their choice of the algebraic or the visual method. Post intervention data indicated an increase in the targeted students' ability to simplify square roots through the use of the visual method presented in this report. Additional analysis of the data indicated a preference among the students for the visual method as compared to the algebraic method. (Contains 20 references.) (Author) ED436357

Wang, K.-H. (1994). Middle School Students' Decision-Making on Solid Waste Management in Taiwan. 19pp. Paper presented at the Annual Meeting of the National Association for Research in Science Teaching (Anaheim, CA, March 1994). The purpose of the study was to determine the effectiveness of a HyperCard simulation upon student's concepts, opinions, and option-rankings on solid waste management, and to investigate what cognitive activities of the students were involved in the decision-making processes. One hundred eighth-grade students in Taiwan participated in this investigation. The subjects were randomly assigned into the computer-assisted-learning, the simulation-printed-materials, and control groups. The subjects received pre- and post-tests of achievement, opinion, and option-ranking on solid waste management. Students' decision-making behaviors were traced by the computer program for understanding the students; decision-making processes. Interviews verified the consistency of students' decision-making behavior. There were no significant differences on the achievement test score, the opinion assessment score, and the option-ranking score among the three groups. However, the subjects in the computer-assisted- learning group changed significantly on their option-ranking behaviors after the treatment. It was found that the students' information-searching behaviors were very different from each other, though the strategy used most frequently by the students for searching information is the conjunctive strategy. Interviews suggested that students made decisions by using their personal intuitive conceptions and feelings about solid waste management. Contains 16 references. (MDH) ED370775

White, J. W., & Norwich, V. H. (1997). Computer Activities for College Algebra and Precalculus. 89p. Mathematics software can be a great aid in understanding difficult mathematics concepts at all levels. This paper presents nine exercises on calculus concepts by using different software used in mathematics education. Each exercise includes instruction on how to use software in order to highlight a specific concept in mathematics. This paper also presents a technology review by the comparison of ISETL, Derive, Geometer's Sketchpad, TI-82, TI-83, TI-85, TI-86, TI-92 and CBL along with discussing the advantages and disadvantages of each system. The mathematical concepts and software used in this paper include: (1) vertices of a triangle, midpoint formula, median of a side, equation of a line, and slope using the software "The Geometer's Sketchpad"; (2) writing the equation of a line using slope-intercept form and point-slope form and writing equations of parallel and perpendicular lines to a given line and passing through a given point using the software "The Geometer's Sketchpad"; (3) visualizing trigonometric identities for sine, cosine, and tangent functions using the software "The Geometer's Sketchpad"; (4) using equations and graphs to identify families of functions including linear, quadratic, and exponential using the software "Derive"; (5) functions, vertical line test and domain, using the software "Power Point"; (6) Pascal's triangle, binomial expansion, and pattern recognition using the software "Power Point"; (7) graphing factored polynomials of degree n>1 and solving higher degree inequalities by factoring and graphing using the software "Microsoft PowerPoint"; (8) review of quadratic functions and application problems using quadratic functions using the software "Microsoft PowerPoint and Microsoft Excel"; and (9) shifting graphs including horizontal, vertical, and combination graph shifts using the software "Derive". (ASK) ED412119

Williams, D. C., Hemstreet, S., Liu, M., & Smith, V. D. (1998 Length: 8 Page(s); 1 Microfiche). Examining How Middle School Students Use Problem-Based Learning Software. In: ED-MEDIA/ED-TELECOM 98 World Conference on Educational Multimedia and Hypermedia & World Conference on Educational Telecommunications. Proceedings (10th, Freiburg, Germany, June 20-25, 1998); see IR 019 307. This study investigated the use of tools and design features as employed in a problem-based learning (PBL) environment and their effectiveness on seventh grade students' learning of science concepts. Specifically, the study examined: (1) the effect of the computer-supported PBL environment on the achievement of middle school students; (2) the effect of the computer-based PBL environment on middle school students' attitudes toward science; and (3) the relationship between students' math or reading ability and their achievement in the PBL environment. Dependent measures were achievement, degree and frequency of access to the tools/features, and attitude toward learning science. The treatment consisted of three groups: computer-supported PBL environment, paper-based PBL, and a control group. Results indicated that: both the groups that used the computer-supported and paper-based PBL significantly improved their achievement scores, while the increase for the control group was not significant; there was no significant difference between the computer-supported and paper groups; students' reading ability is a better predictor for students' achievement in a PBL environment than their math ability; and students' attitude toward learning science was not affected by the introduction of PBL. Two tables and a graph present data. Contains 14 references. (DLS) ED428738

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Zorfass, J. M. (1998). Teaching Middle School Students To Be Active Researchers. 137pp. "With Harriet Copel.". This book offers middle school educators information about inquiry-based, interdisciplinary learning, highlighting the I-Search process. I-Search offers a powerful mix of ingredients (e.g., respecting curiosity, supporting independence, and fostering creativity) to capture students' and teachers' imaginations. It makes research more learner centered and encourages students with diverse needs to collaborate. It provides a natural context for using technology, media, and materials. Based on 10 years of work with middle schools and interdisciplinary teams, this book shows how teachers and students asked real-life questions and presented their new understanding of overarching concepts. Teachers and students explored community resources, read varied materials, conducted interviews, and used new technology to keep track of their work, which culminated in I- Search papers and exhibitions. Chapter 1 defines active research and the four phases of an I-Search unit. Chapter 2 describes how teachers guide students to pose I-Search questions. Chapter 3 explains how teachers help students learn to access various materials. Chapter 4 discusses how teachers help students integrate and process information they have gathered. Chapter 5 illustrates how teachers help students represent their knowledge via written papers and exhibitions. Chapter 6 describes what is needed to successfully design, implement, and reflect on I-Search units. The appendix's 10 sections describe planning and assessment criteriafrom planning immersion activities to creating guidelines for student peer conferences, developing rubrics for assessment, and to evaluating the implementation of an I-Search unit. (Contains 61 references.) (SM) ED426050 Available from: Association for Supervision and Curriculum Development, 1703 North Beauregard Street, Alexandria, VA 22311-1714; Tel: 800-933-2723 (Toll Free) or 703-578-9600; e-mail: member@ascd.org; Web site: http://www.ascd.org (stock no. 198180, $15.95 for ASCD members, $18.95 for nonmembers). You be able to order this document from the ERIC Document Reproduction Service.

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