Critical Issue: Providing Hands-On, Minds-On, and Authentic Learning Experiences in Mathematics

ISSUE: Learning does not mean simply receiving and remembering a transmitted message; instead, "educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding" (Mathematical Sciences Education Board, 1989, p. 58). When educators begin to see learning as knowledge construction, they change their thinking about curriculum, instruction, and assessment, developing more powerful approaches to connecting thinking and mathematics and designing more mathematically significant instructional learning experiences. Such learning experiences are:

• Hands-on, involving students in really doing mathematics - experimenting first-hand with physical objects in the environment and having concrete experience before learning abstract mathematical concepts
• Minds-on, focusing on the core concepts and critical thinking processes needed for students to create and re-create mathematical concepts and relationships in their own minds
• Authentic, allowing students to explore, discover, discuss, and meaningfully construct mathematical concepts and relationships in contexts that involve real-world problems and projects that are relevant and interesting to the learner

OVERVIEW: The national call for reform in mathematics teaching and learning can seem overwhelming, because it requires a complete redesign of the content of school mathematics and the way it is taught. The basis for reform is the widespread belief that the United States must "restructure the mathematics curriculum - both what is taught and the way it is taught - if our children are to develop the mathematical knowledge (and the confidence to use that knowledge) that they will need to be personally and professionally competent in the twenty-first century" (Mathematical Sciences Education Board, 1990, p. 1). Simply producing new texts and retraining teachers will not be sufficient to address the major changes being recommended.

How will this transformation in mathematics teaching and learning occur? Realizing a new vision of school mathematics will require public acceptance of a realistic philosophy of mathematics that reflects practice and pedagogical experience. A new vision for teaching and learning in mathematics cannot be properly and collaboratively agreed upon without first addressing the following fundamental questions:

1. What is mathematics?
2. Which mathematics should be taught?
3. How do people learn mathematics?
4. How can mathematics be taught effectively?

Although few mathematicians and teachers spend much time thinking about these philosophical questions, answers will be necessary to help clarify for both educators and the public what mathematics is really about - what it studies, how it operates, what it is good for (Romberg, 1988). Several of the Action Options listed below suggest ways for individuals and teams to consider these questions and answers.

The brief responses to the four questions raised above represent a considerable change from the descriptions of mathematics classes drawn from the National Science Foundation (NSF) case studies in the early 1970s, and even though the observations described in the study were made almost 20 years ago, much of the routine described continues today (NCTM, 1989; National Research Council, 1989). There are many obstacles to making significant change in mathematics teaching and learning in schools, including the beliefs that students and teachers bring to the classroom and the assumptions held by administrators, parents, and society about mathematics, curriculum, teaching, and learning. In order to change perspectives, school improvement teams need to spend time learning together, answering the questions above and exploring a variety of resources that will give direction on how mathematics can be taught and learned to enhance the development of mathematical power.

GOALS:

• All students are actively engaged in meaningful, hands-on, minds-on, and authentic, learning experiences in mathematics. The NCTM (1989) has carefully articulated five general goals for all students:
1. They learn to value mathematics
2. They become confident in their ability to do mathematics
3. They become mathematical problem solvers
4. They learn to communicate mathematically
5. They learn to reason mathematically
• All teachers are involved in ongoing learning and professional development and working to improve their practice by providing hands-on, minds-on, and authentic learning experiences for all students.
• All administrators are exercising instructional leadership in articulating a unifying vision of mathematics teaching and learning. They also support and encourage mathematics teachers who are working together and with other teachers and community members to help provide mathematically significant learning experiences for all students.
• All parents and community members are meaningfully engaged in helping the school community rethink the learning and teaching of mathematics; making decisions about the instructional approach and program; providing resources for supporting this new vision of teaching and learning; and modeling this new vision in their direct interactions with students.

ACTION OPTIONS:

All goals point to one primary goal: all students will gain "mathematical power." The NCTM (1989) defines mathematical power as "an individual's abilities to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods to solve nonroutine problems." This goal challenges students, teachers, administrators, and parents and community members to create and implement a comprehensive plan for the improvement of mathematics education.

Students:

• Take advantage of opportunities to engage actively in learning mathematics.
• Develop skills and processes for problem solving, reasoning, communicating, and connecting concepts within and outside of mathematics.
• Learn to work with others, and value the diverse ideas of your peers.

Teachers:

• Develop positive attitudes toward and genuine interest in mathematics.
• Examine and continue to learn about how children learn mathematics.
• Encourage students to write about mathematics and make reading and children's literature an integral part of mathematics.
• Offer activities that encompass various learning styles and instructional formats to stimulate learning for all students.
• Engage students in an active process of learning in which they create and discover mathematical concepts.
• Use a variety of assessment alternatives to gain information about what students understand and how they feel about mathematics in order to help them learn.
• Understand a comprehensive mathematics curriculum and build upon your own mathematics content knowledge.
• Examine your current practice and take advantage of opportunities to learn new instructional strategies linked to effective practice - e.g., participate in workshops, attend professional meetings, read professional journals, and conducting classroom research.
• Seek opportunities to work with other teachers in learning communities and support each other in developing new curricular ideas and instructional approaches.
• Establish high standards for all students and be certain that every effort is made to provide learning opportunities for each of them.
• Create a positive classroom environment that allows students to discuss mathematics and make sense of mathematics in cooperative learning situations.
• Facilitate learning by posing open-ended questions, asking students to clarify and justify their ideas, and encouraging students to seek assistance from one another.
• Engage students in the use of manipulative materials and active mental involvement to support their learning of mathematics.
• 
  Marilyn Burns, a Mathematics Consultant, talks about how children learn with manipulatives. Excerpted from the video Mathematics With Manipulatives: Six Models, produced by Focal Point Productions, Inc. for Cuisenaire Company of America, Inc. (QuickTime slide show, 485k).
  • Video
  • Video with captions (audio description available)
  • Text transcript (fastest download)

• 
  Gloria Mitchell, First and Second grades teacher in the Bellevue School District, talks about the importance of manipulatives. Excerpted from the video Mathematics With Manipulatives: Six Models. The video was produced by Focal Point Productions, Inc. for Cuisenaire Company of America, Inc. (Audio comment, 153k). A text transcript is available.
• Teach children specific mathematical content and how to apply mathematical ideas in the context of problem-solving situations.
• Seek ways to relate mathematics learning to other disciplines to form mathematical connections.
• Use technology to enhance classroom experiences.

Administrators:

• Become informed about the changes necessary for reform in mathematics teaching and learning.
• Establish performance review systems for the ongoing improvement of staff that connect and model research and best practice in mathematics teaching and learning.
• Provide the leadership and resources to support teachers in their collegial interactions and professional development.
• Garner the support and engagement of parents and community members in reforming mathematics education.

Parents and Community Members:

• Hold high expectations for children in mathematics achievement, expect schools to do the same, and support schools in their efforts to meet these expectations.
• Provide learning opportunities in the home, allowing students to participate in activities that stimulate reasoning and problem solving.
• Ensure that all expected homework is completed, providing support, assistance, and encouragement as needed and appropriate.
• Create partnerships between schools and community facilities and businesses that will enhance mathematics teaching and student learning.

IMPLEMENTATION PITFALLS: Educators will need sufficient resources and time for professional development to design and implement this new problem-based, concept-oriented curriculum. They will need time for meaningful and ongoing conversations about the NCTM Curriculum and Evaluation Standards for School Mathematics (1989) and what the Standards mean for their operating vision of learning and teaching in mathematics.

If teachers have to continue to rely on worksheets and textbooks to "deliver" instruction in mathematics and "cover the curriculum," they will not be able to move toward the new vision of the math classroom. To transform classrooms into learning communities of active and collaborative mathematical inquiry, teachers will need to access and be able to use instructional materials for thought-provoking activities and projects, software for simulation and modeling, and resources in the community for authentic learning experiences.

To implement reforms that engage all students in meaningful mathematics learning, teachers will need to learn a new role as a facilitator and coach in the classroom, expand their knowledge base in mathematics, develop new curricular and instructional strategies, and change their expectations for students. These changes will require ongoing and intensive professional development that allows teachers to interact with their colleagues and that is based at their school and linked to its organizational development.

Community members and parents will need to be "on board" with the reform. A broad base of community members must understand the rationale for the reform and be actively engaged in formulating and implementing the substance of the reform; without this kind of support and involvement, the reform effort itself will be subject to criticism and misunderstandings about its message and intent.

DIFFERENT POINTS OF VIEW: One point of view that opposes the call for reform holds that the existing system is adequate, that change is not necessary, and that the standards movement and mathematics education reform in general are similar to "new math" and will eventually go away.

Many view mathematics narrowly - as arithmetic - and argue that the new math curriculum is too ambitious and that we should focus instead on ensuring that all students have basic computational skills, while some students are put on a track toward calculus. They reason that students need to be comfortable and proficient with computation before they can advance to the sort of mathematical problem solving that the NCTM and other reformers would stress.

ILLUSTRATIVE CASES:

The Algebra Project

The Quasar Project

Cognitively Guided Instruction (CGI)

School-Business Partnership: The Science and Mathematics Network of Central Ohio

CONTACTS:

Midwest Consortium for Mathematics and Science Education, North Central Regional Educational Laboratory, 1120 Diehl Road, Suite 200, Naperville, IL 60563-1486, (630) 649-6500, fax (630) 649-7600 , Internet e-mail: info@ncrel.org, WWW: http://www.ncrel.org/msc/msc.htm

Eisenhower National Clearinghouse for Mathematics and Science Education, The Ohio State University, 1929 Kenny Road, Columbus, OH 43210-1079, 614-292-7784, Fax 614-292-2066, Internet e-mail: info@ENC.org; For materials and resources that support equity in the science classroom, connect to the ENC Catalog of Curriculum Resources. and use the keyword "equity" in the search.

Association for Supervision and Curriculum Development, 1250 N. Pitt Street, Alexandria, VA 22314, 703-549-9110

The EQUALS Project, Lawrence Hall of Science, University of California, Berkeley, CA 94720. 510-652-1823, Fax 510-643-5757, e-mail: equals@maillink.berkeley.edu, WWW: http://equals./lhs.berkeley.edu

Mathematical Sciences Education Board, National Academy of Science, and National Research Council, 2101 Consitution Avenue, N.W., Washington, DC 20418, 202-334-2000

The Math Solution/Mary Burns Associates, 150 Gate 5, Suite 101, Sausalito, CA 94965, 415-332-4181

National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091-1593, 703-476-2970, e-mail: InfoCentral@nctm.org, WWW: http://www.nctm.org

National Parent Teacher Association, 700 North Rush Street, Chicago, IL 60611-2571, (312) 787-0977, Fax: (312) 787-8342

References

This Critical Issue summary was researched and written by Cathy Cook, North Central Regional Educational Laboratory.

Date posted: 1995