Learning and Mathematics

Table of Contents


  1. Introduction . . . Klotz, Renninger
    A series of discussions inspired by recent work on how students learn mathematics -- ideas that form the basis of the NCTM Standards. This series is intended to be an informative and sometimes provocative overview of the current research and thinking of some key researchers in mathematics education and educational psychology. Much of the work in this area indicates that the "traditional" classroom needs to be changed if more effective learning (i.e., learning that is more conceptual and less formulaic) is to take place. These summaries include implications for classroom practice.

  2. Metacognition . . . Schoenfeld
    Newsgroup discussions - 6 July 1994
    Schoenfeld wrote this chapter in response to a challenge from mathematicians to explain "what metacognition is, why it's important, and what to do about it -- all in clear language that we can understand." Schoenfeld's explanation describes metacognition, or reflecting on how we think, through a discussion of how a problem was solved and what it was about, or where and why a difficulty occurred in the process of problem solving. He also proposes some ways metacognition could be used in the classroom.

  3. Mathematics in Everyday Situations . . . Lesh
    Newsgroup discussions - 11 July 1994
    Richard Lesh, of the Educational Testing Service (ETS) in Princeton, New Jersey, believes that if students are provided with everyday situations for practicing and learning the important uses of mathematics, they will develop such skills as "making inferences, evaluating reasonableness of results... [and] using references to 'look up' what they need to know."

  4. Common-Sense Questions . . . Polya
    Newsgroup discussions - 16 August 1994
    Polya describes a four-stage approach to mathematical problem- solving. He bases his approach on common-sense questions that would naturally occur to an experienced problem-solver. Polya claims teachers should pose these questions to students in as natural and unobtrusive a way as possible, the goal being to encourage independence and internalization of this framework.

  5. Mathematics as an Ill-Structured Discipline . . . Resnick
    Newsgroup discussions - 25 September 1994
    Resnick states that mathematics is traditionally taught as a "well- structured discipline"; i.e., students learn that for certain types of problems there are particular rules, such as formulas, to be discovered, and that following these rules will allow them to arrive at (a single) appropriate answer. Resnick argues, however, that students with this mindset miss opportunities to conceptualize mathematics and find meaning in their learning. She says that in order to foster the development of more meaningful, flexible, and inventive problem solving, mathematics should be taught as an "ill-structured discipline": a domain that invites more than one rigidly defined interpretation of a task.

  6. Cognitive Apprenticeship . . . Collins, et al.
    Newsgroup discussions - 11 October 1994
    Allan Collins, of BBN Laboratories, and John Seely Brown and Susan E. Newman, both of the Xerox Palo Alto Research Center, describe and illustrate a non-traditional way to think about the roles of teachers and learners ("traditional" here connotes an active teacher/passive student relationship, usually with the teacher lecturing at the front of the class while students sit at desks in rows and listen, take notes, and occasionally answer questions.) Collins, Brown and Newman's ideas reflect a kind of thinking about the nature of learning that has been influential in the development of the NCTM standards.

  7. Interactive Learning . . . Brown, et al.
    Newsgroup discussions - 4 November 1994
    Interactive learning -- learning in which students and their teacher share ideas and take turns leading discussions -- provides students with a model of the way experts work together to learn and understand. Interactive learning also challenges students to develop their own capabilities. Brown, Campione, Reeve, Ferrara, and Palincsar argue that it is necessary to reconsider the traditional roles of teacher and student (where the teacher lectures at the board and students sit passively at their desks, taking notes), and to give serious consideration to the quality of learning possible when classroom learning involves small group work.

  8. Acquisition of Arithmetic . . . Ginsburg
    Newsgroup discussions - 20 January 1995
    Ginsburg draws heavily on the idea of assimilation -- the incorporation of new ideas into an existing body of knowledge -- to explain how children acquire or misacquire arithmetical skills and concepts. He looks at both the informal, concrete understanding of basic concepts that children acquire before entering school and the abstract, formal concepts and computations they are expected to learn in the classroom.

  9. Knowing, doing, and teaching multiplication . . . Lampert
    Newsgroup discussions - 8 February 1995
    Magdalene Lampert, of the Institute for Research on Teaching at Michigan State University, advocates incorporating students' intuitive knowledge about mathematics into classroom lessons and encourages putting new concepts into familiar contexts so that students may more readily relate to the problems being investigated. Lampert describes experimental lessons and explains her view of the teacher's role in the classroom: to help students make explicit their ideas about analyzing and solving problems, to act as referee in arguments about the reasonableness of competing ideas, and "to sanction students' intuitive use of mathematical principles as legitimate."

  10. Language and Mathematics . . . Cocking & Chipman
    Newsgroup discussions - 24 February 1995
    Cocking and Chipman examine the mathematical ability of language minority -- particularly bilingual -- students, attempting to identify linguistic and cultural variables that might explain why their mathematical ability falls increasingly behind that of students who speak English as their primary language ("majority students"). First Cocking and Chipman investigate the relation between language and math ability; then they look at external influences on performance such as teacher competencies and attitudes and parental attitudes and support. The focus is primarily on Hispanic students, with some support from data on Native Americans.

  11. Math Horizon . . . Ball
    Newsgroup discussions - 15 March 1995
    Deborah Ball of Michigan State University examines the challenge of creating classroom practices for third graders of diverse racial, ethnic, and socioeconomic backgrounds in the spirit of current reform, with ideals involving student engagement in authentic tasks. Using her own elementary school mathematics classroom, the author presents three dilemmas -- of content, discourse, and community -- that arise in trying to teach in ways that are "intellectually honest." Ball frames and responds to these dilemmas, providing a view of underlying pedagogical complexities and the conditions needed in order to work toward current educational visions.

  12. Learning Fractions . . . Mack
    Newsgroup discussions - 29 March 1995
    Nancy K. Mack of the University of Pittsburgh considers how informal knowledge (such as dividing a pizza) can be used to enhance formal knowledge (such as one's understanding of fractions). Informal knowledge in this context means applied knowledge, whether correct or incorrect, developed by the individual and used to solve problems in real- life situations. Mack also explores how formal algorithmic and procedural knowledge may interfere with the use of informal knowledge.

  13. Strategy Acquisition and Application . . . Siegler & Jenkins
    Newsgroup discussions - 13 April 1995
    Robert S. Siegler and Eric Jenkins of Carnegie-Mellon University discuss how children acquire and apply strategies by looking closely at a small group of students over a long period of time. Strategies differ from algorithms in that they are generated by the student and are a nonobligatory, goal-directed procedure. Anything that does not accomplish a goal or accomplishes an unintended goal is not a strategy.

  14. Mathetics . . . Papert
    Newsgroup discussions - 8 June 1995 and 3 November 1995 (repost)
    In his book, The Children's Machine, Seymour Papert examines the art of learning, a topic that he contends has been widely ignored by educational researchers and practitioners. He introduces the concept of 'mathetics,' which he defines as the art or act of learning, and discusses the issues that surround mathetics -- in the school setting, theoretically, and in light of his own experiences. He presents a series of case studies that build upon and draw from his discussion of mathetics, and which demonstrate the utility of computers in promoting flexible, personal, and connected learning. He also includes a more theoretical discussion of instructionist versus constructionist viewpoints, as well as a defense of concrete knowledge and thought in the face of educational trends that favor abstract reasoning. Overall, Papert stresses support for personal variation in learning styles, and for the increased acceptance by schools of the ability of children to learn without assistance.

  15. Understanding . . . Greeno and Riley
    Newsgroup discussions - 20 November 1995
    James G. Greeno of Stanford University and Mary S. Riley of San Diego State University examine why younger children seem to lack the ability of older children to solve mathematical word problems. Greeno and Riley distinguish between the ability to do the computation required for problem completion and the ability to identify the question posed by a problem. They dispute the hypothesis that older children's greater facility in solving mathematical word problems results from greater knowledge of possible strategies. Instead, they argue that younger children possess the relevant conceptual knowledge but cannot effectively create a mental representation of the necessary information.

  16. Writing Math . . . Countryman
    Newsgroup discussions - 10 December 1995
    In her book Writing to Learn Mathematics , Joan Countryman, the Head of Lincoln School in Providence, Rhode Island, explores the relationship between math and writing and provides a comprehensive description of the approach she takes to teaching math in middle and high school. Countryman stresses the idea that the use of writing exercises in math classes leads to both a better understanding of the material and heightened math communication skills. Furthermore, she believes that writing about math leads to a less restrictive view of mathematics - instead of a series of formulas and rigid answers, the students come to see mathematics as a process and a dialogue to which they too can contribute.

  17. Not Dumb . . . Tobias
    Newsgroup discussions - 30 January 1996
    Sheila Tobias addresses a concern that has been widely felt across the country for years: the shortfall in the number of students who go on to become scientists. She notes the importance of high school mathematics in preparation for studying science in college, and suggests that in order for students to go on to become scientists, more emphasis needs to be placed on early and continuous exposure to mathematics. She makes the claim that science must open its doors to the ranks of the 'second tier', those who for one reason or another have decided not to pursue a career in science.

  18. Teaching . . . Hiebert & Wearne
    Newsgroup discussions - 14 February 1996
    James Hiebert and Diana Wearne, of the University of Delaware, describe in this article an experiment they conducted in which they compared text-based instruction with conceptually based instruction in a series of lessons on place value and related concepts. Their findings indicate that students need to be able to make links between different representations (or forms) of the same concept.

  19. Reasons for Studying . . . Nolen
    Newsgroup discussions - 3 March 1996
    Susan Bobbitt Nolen, of the University of Washington, argues that in order to foster meaningful learning and effective study strategies, teachers should reduce the emphasis on competition for grades and teacher recognition and instead encourage learning for its own sake. Although mathematics is not specifically adressed in the article, the findings and suggestions included in this paper have important implications for math learning and instruction. This link is particularly important in that Nolen tackles the issue of finding the underlying cause of student motivation and learning, as opposed to attempting to promote involvement in a particular task or lesson.

  20. Metacognition . . . Schoenfeld (repost)
    Newsgroup discussions - 20 March 1996
    Schoenfeld wrote this chapter in response to a challenge from mathematicians to explain "what metacognition is, why it's important, and what to do about it -- all in clear language that we can understand." Schoenfeld's explanation describes metacognition, or reflecting on how we think, through a discussion of how a problem was solved and what it was about, or where and why a difficulty occurred in the process of problem solving. He also proposes some ways metacognition could be used in the classroom.

  21. Small group interactions . . . Yackel, Cobb, & Wood
    Newsgroup discussions - 18 April 1996
    An experiment in which all instruction in a second grade mathematics classroom was replaced by small group problem-solving strategies for an entire school year. Work was not graded, nor was there a set amount of work to be completed; rather, the teacher gave report card grades based on her knowledge of the children, and the students were allowed to spend as much time as they needed to discuss each problem and arrive at a solution. The goal of the project was to foster collaborative learning and build conflict resolution skills, as well as to provide for other learning opportunities that do not arise in more traditional classrooms.

  22. Project-Based Learning . . . Blumenfeld, P., Soloway, E., Marx, R., Krajcik, J., Guzdial, M., & Palincsar, A.
    Newsgroup discussions - 16 May 1996
    Blumenfeld and her colleagues at the University of Michigan describe project-based learning and the benefits of using long-term projects as part of classroom instruction. The authors believe that projects have the potential to foster students' learning and classroom engagement by combining student interest with a variety of challenging, authentic problem-solving tasks. In their discussion of the essential components of project-based learning, the authors pay close attention to the design of projects with regard to classroom factors and teacher and student knowledge. After considering the possible challenges that face teachers using projects in their classrooms, the authors go on to describe how technology may be used as a support system by teachers and students involved in long-term projects.

  23. Classroom Conversation . . . Nicholls, J. & Hazzard, S.
    Newsgroup posting - 11 June 1996
    In their book Education as Adventure , John Nicholls and Susan Hazzard explore students' understandings of the nature and point of school learning and the idea that students are valuable critics of the classroom curriculum and their own learning. The authors describe the events and experiences of a second grade class, taught by Hazzard and observed by Nicholls, which focused on incorporating students' initiative, collaborative efforts, and innate curiosity and enthusiasm into classroom situations and learning activities. The authors stress the importance of conversation in the classroom as a means of both inviting and responding to students' own thoughts about school learning and education. When students and teachers are involved in dialogue to define and understand learning, Nicholls and Hazzard believe that education becomes an exciting and meaningful journey of discovery.

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