 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.
 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.
 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."
 CommonSense Questions . . . Polya
Newsgroup discussions  16 August 1994
Polya describes a fourstage approach to mathematical problem
solving. He bases his approach on commonsense questions that would
naturally occur to an experienced problemsolver. 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.
 Mathematics as an IllStructured 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 "illstructured discipline": a domain that invites more than
one rigidly defined interpretation of a task.
 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 nontraditional 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.
 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.
 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.
 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."
 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.
 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.
 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.
 Strategy Acquisition and Application . . . Siegler & Jenkins
Newsgroup discussions  13 April 1995
Robert S. Siegler and Eric Jenkins of CarnegieMellon 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, goaldirected procedure. Anything that does not
accomplish a goal or accomplishes an unintended goal is not a
strategy.
 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.
 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.
 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.
 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.
 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 textbased 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.
 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.
 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.
 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 problemsolving 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.
 ProjectBased 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
projectbased learning and the benefits of using longterm 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
problemsolving tasks. In their discussion of the essential components
of projectbased 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 longterm projects.
 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.
