LEARN NC

Problem centered math

Developed by LEARN NC with Grayson Wheatley

7 Research and strategies for problem-centered math

A bibliography of research-driven strategies for teaching problem-centered math at all grade levels.

By Libby Montagne

A large number of books and articles have been published in recent years offering research-driven strategies for problem-centered or constructivist teaching of mathematics. The following are a selection of some of the best and most complete resources for teachers. These books cover a range of grades K-12, but the ideas and strategies they provide may be applied to grade levels beyond those listed.

Elementary/General

Making Sense: Teaching and Learning Mathematics with Understanding

By James Hiegert, Thomas Carpenter, Elizabeth Fennema, Karen Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. Portsmouth, N.H.: Heinemann, 1997. 184 pages.

Making Sense is the result of a five-year collaboration of researchers from four different development projects. Although the classroom situations described in the book involve students learning multi-digit addition and subtraction for the first time, the observations and issues addressed are important in promoting mathematical understanding in any classroom. The authors assume that understanding should be the most fundamental goal in mathematics instruction. They communicate that understanding is complex and always changing, but define it as "seeing how things are related to or connected to other things we know." They suggest that students make connections through deliberate reflection and communication.

The book focuses on five common dimensions of classrooms that facilitate mathematical understanding:

  1. the nature of learning tasks,
  2. the role of the teacher
  3. the social culture of the classroom,
  4. the kind of mathematical tools that are available, and
  5. the accessibility of mathematics for every student.

The authors devote a concise and clearly written chapter to each of the five dimensions. The dimensions make up a synergistic system or framework that plays itself out differently in different classrooms. In four chapters, the authors connect the framework to four real classrooms using examples of student work, dialogue of teacher-student interactions, and author commentary. Each of the classrooms observed approaches mathematics teaching and learning a little bit differently: one uses "Cognitively Guided Instruction", one uses "Conceptually Based Instruction", one uses "Problem-Centered Learning", and one uses "Supporting Ten-Structured Thinking".

This study provides a clear, research-based example of how teaching for understanding in mathematics does not mean conforming to any single highly prescribed method of instruction. The framework defined in the study is a valuable tool for teachers who are interested in understanding how their practice affects student learning. For those who wish to read more about this teaching and learning with understanding, the study includes extensive references.

Beyond Arithmetic: Changing Mathematics in the Elementary Classroom

By Jan Mokros, Susan Jo Russell, and Karen Economopoulos. Palo Alto, Calif.: Dale Seymour Publications, 1995. 137 pages.

Beyond Arithmetic is primarily based on the authors’ previous work in developing the Investigations in Number, Data, and Space curriculum materials, but the discussion of pedagogy and theoretical underpinnings of constructivism are applicable to any problem-centered classroom or curriculum. The book is full of detailed descriptions of classroom activities, each with a rationale, classroom anecdotes, examples of student work, and transcripts of student-teacher dialogue.

The authors link these teaching and learning examples to recent research in mathematics and constructivist pedagogy. Constructivism assumes that all students are able to develop as mathematical thinkers by "doing" math for themselves. The authors discuss research findings regarding constructivism and emphasize the need for all students to become mathematically confident and competent. The content and format of the book, with real-life classroom activities and stories linked to research, allows teachers to easily connect their own stories to current educational theory and pedagogy. The connections facilitated by this book make it a useful resource for teachers who are interested in better understanding their own instruction and improving their students’ mathematical understanding

Elementary (K-6)

Developing Number Sense: Grades 3-6

By Rusty Bresser and Caren Holtzman. Sausalito, Calif.: Math Solutions Publications, 1999. 179 pages.

Two classroom teachers researched and wrote Developing Number Sense. The book suggests that a strong foundation in and ease with numbers is necessary for students to succeed in higher level math. The authors define number sense as "understanding relationships among numbers, the ability to think flexibly about numbers, understanding the effects operations have on numbers, and the ability to articulate mathematical thinking."

The book is divided into four sections: "Mental Computation," "The Basics and More," "Navigating the Number System," and "Estimation." Each section begins with a brief rationale that highlights the number sense ideas presented. Many of the learning situations in the book involve robust mathematical challenges that students investigate in pairs. Each activity includes a concise summary, a vignette that describes how the authors taught the activity in the classroom, examples of student work, and answers to one or more reflective questions. The authors approach mathematics teaching from a problem-centered perspective. The teacher’s role is that of facilitator and questioner. All student thinking and logic is valued. Students are exposed to one another’s reasoning through partnering and whole-class sharing of solutions. They are encouraged to question what they don’t understand and think flexibly about mathematics. The concrete examples, ideas and insights presented in this book facilitate understanding the value of number sense. The format of personal classroom stories, student work, and reflective summaries is accessible. Teachers can easily connect their own experiences and insights to the current big ideas and research regarding teaching and learning number sense.

Math Time: The Learning Environment

By Kathy Richardson. Menlo Park, Calif.: Educational Enrichment, 1997. 134 pages.

In Math Time, Kathy Richardson has developed a clear and concise resource to help elementary teachers analyze and redesign their classroom environment. This practical guide includes clearly written rationales as well as step-by-step instructions for designing a classroom environment that facilitates a constructivist, problem-centered approach to mathematics teaching and learning. The author emphasizes the importance of a positive and safe yet intellectually critical classroom culture where students are expected to explore and investigate their own thinking, communicate their logic, make mistakes, and attend to and question each other’s ideas.

The book emphasizes teaching and learning for "meaning" and understanding. In addition to advice for setting up and maintaining an environment that facilitates deep mathematical understanding, the book addresses such practical questions as: How long should children explore? How does one transition from exploration to concept development? And, How does one know when it is time to move on? The author includes scenarios from the classroom which enable teachers to easily connect ideas in the book to their own practice. Thoughtful analysis and rationales grounded in current research and learning theory make Math Time an incredible resource for experienced teachers as well as for new teachers in setting up and evaluating their classroom environment.

Middle Grades/Secondary

Empowering Students by Promoting Active Learning in Mathematics: Teachers Speak to Teachers By David Bock, Dorothy Buerk, Patricia Ehrich, Helen Gibson, Ann Oaks, and Marcial Standeras; Dorothy Abuerk, editor; Hadley Smith, editorial coordinator Reston, Va.: National Council of Teachers of Mathematics, 1994. 48 pages.

In Empowering Students by Promoting Active Learning in Mathematics, five teachers describe their discoveries about their own mathematics teaching and consequently their students’ learning. All of the teachers are engaged in improving and revising their classroom practices to create middle school and high school classrooms where students are actively engaged in mathematical thinking and problem solving and empowered to think about mathematics in flexible and varied ways. The teachers describe how they use writing to help students express their mathematical ideas and their perceptions and feelings about mathematics. The teachers reflect and realize that individual written student explanations often help them to better interpret their students’ mathematical understanding — and, just as importantly, their students’ mathematical misconceptions. The teachers also explain how they use grouping and cooperative strategies to evoke rich, safe mathematical communication among their students. These teacher stories offer clear, easy to read and accessible ideas and strategies from teachers, for teachers who are interested in promoting an active, problem-centered approach to learning mathematics.

Fostering Algebraic Thinking: a Guide for Teachers, Grades 6-10

By Mark Driscoll. Portsmouth, N.H.: Heinemann, 1999. 168 pages.

Fostering Algebraic Thinking is an excellent resource for those who are interested in rethinking and reflecting on their teaching of algebra. It is most useful for middle school teachers who are interested in building on their students’ mathematical understandings to develop the underlying concepts involved in algebraic thinking. Chapters in the book focus on questions:

  • What can be done to help students build on arithmetic and computational skills to develop their algebraic thinking?
  • What can be done to help students build on number sense to develop their algebraic thinking?
  • How and when should students be expected to engage in symbolic representation and manipulation in algebra?
  • How can we best foster student’s capacity to generalize about number systems and functions?
  • How can we help students understand multiple representations?

The ideas presented in the book were derived from recent research and from the author’s experience with three professional development projects. The author emphasizes algebraic "habits of mind": Doing-Undoing, Building Rules to Represent Functions, and Abstracting from Computation. The book shows that guiding questions from the teacher are essential in helping students develop these "habits of mind." The author approaches algebra as a set of conceptual tools to be understood — tools that can help students make sense of and access challenging problem situations — rather than a series of formulas and recipes to be memorized. For those who are interested in finding out more about teaching algebra and algebraic thinking, each chapter includes references.

High School (9-12)

Learning Mathematics Through Inquiry

By Raffaella Borasi. Portsmouth, N.H.: Heinemann, 1992. 234 pages.

In Learning Mathematics Through Inquiry, Raffaella Borasi focuses on a pedagogy called "inquiry mathematics." This pedagogy parallels and complements the theoretical underpinnings and instructional approach of "problem-centered learning". The book draws from previous research and the author’s own experience teaching a ten-lesson high-school mathematics course to two female students. Each chapter contains thorough and complete descriptions of important real-life mathematics activities, including: transcripts of student-teacher interactions, samples of materials, samples of student work, and analysis and reflection by the author, who is also the teacher and researcher. The book also includes students’ reflections on their own learning experience. The author emphasizes that learning and knowing mathematics is shaped by culture, context, and values. It is a dynamic, give and take process of asking questions and searching for answers. Learning math depends on "sense making" and construction of understanding by learners. Teaching math means providing experiences and a supportive framework for students to explore and search for their own understanding. The rich descriptions of real teaching and learning experiences, along with connections the author makes to bigger issues in mathematics, make this book a valuable resource for teachers who want to know more about, are interested in trying, or are currently engaged in an inquiry-based, problem-centered approach to teaching and learning mathematics.

Improving the Learning of Mathematics

By John Backhouse, Linda Haggarty, Susan Pirie, and Jude Stratton. Portsmouth, N.H.: Heinmann, 1992. 173 pages.

Improving the Learning of Mathematics takes a very practical approach to implementing "learner-centered" instruction in a secondary mathematics classroom. The authors provide accessible ideas and techniques to help teachers begin to develop learner-centered instructional strategies. Throughout the book, the authors emphasize that learner-centered instruction, like problem-centered instruction, allows for students to create their own mathematical understanding through a process of thinking, questioning, and communicating in mathematical situations. The authors not only provide ideas for important mathematical problems and explorations; they also continuously emphasize the importance of teacher questioning. The book offers clear ways for teachers to guide students’ mathematical thinking with questions rather than answers. The book addresses student motivation, differences between learners’ and teachers’ ways of thinking, relational understanding, classroom environment and how it affects learning, skills and understanding, mathematical communication, and teaching styles. The instructional and classroom suggestions are concise and easy to understand, making this book a useful tool for teachers interested in moving towards a more learner-centered pedagogy.

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