Maryrose L Spriggs
University of Maryland
In October of 1998, The National Council of Teachers of Mathematics (NCTM) produced Principles and Standards for School Mathematics: Discussion Draft. This document proposes to consolidate the classroom aspects of three previous documents: Curriculum & Evaluation Standards for School Mathematics, Professional Standards for Teaching Mathematics and Assessment Standards for School Mathematics. The standards are intended to be guidelines for those in the field of math education.
The challenge for teachers is to implement these guidelines. If we, as members of NCTM, don't want this Standards document to sit on the bookshelves of math teachers everywhere, what can be done to help teachers to transition into meeting these requirements. It is difficult to change to a different way of teaching all at once. Shirley Frye, past president of the NCTM, on the topic of implementing the original standards in 1989, said "Change is a process of growth rather than a movement to a plateau ... change is a long term process" (Frye,1989). It is easier to make changes incrementally. The first five standards of the working draft are content standards. An individual teacher does not have very much control over content; the curriculum is usually proscribed. An individual teacher has more control over the processes used to teach. The second five standards are process standards. I believe that the process standards are the place to begin the transition to a Standards 2000 classroom.
Many teacher's editions of
high school math books have a section in the beginning entitled "How
to use this book". This section breaks the book into manageable sections
by suggesting a plan when using the text for the first time. In the
interest of promoting the use of the standards, it would be helpful to
include such a plan with the standards. It should be included before
the introduction, on a separate page, in the form of a letter to teachers.
I recommend the following language:
Dear fellow classroom teacher,
Sincerely,We are on the front lines. We are the last link in the chain from policy makers to students. Our actions have the greatest impact on the students we teach. At the same time we are asked to do more than ever before, we are also encouraged to make time for professional development, and change the way we teach. It can be discouraging. But there is hope. There are teachers out there who share our concerns. This is the key. Journal articles by Hitch (1990) and Mumme and Weissglass (1989) made recommendations after the first set of standards were written in 1989. These recommendations still apply today. Get together with colleagues and share concerns, research and lesson planning. Reflect on your own teaching practices and get feedback from fellow teachers. The second recommendation is to start with one or two standards. I propose two standards: the Problem Solving and Communication standards. Read the introduction to Principles and Standards and the Problem Solving and Communication sections for the grade band that you teach. Plan a unit with your colleagues. Get together after you teach and revise your plans based on your experiences. Share what works with others. Together we can make a difference in the way mathematics is taught.
Collaborating with Colleagues
How can teachers go about implementing the standards? There is no one answer to this question. Each teacher must find a way that he/she feels comfortable with. But experience shows that meeting goals can be a good cooperative endeavor. Leinwand (1992) suggests that teachers look to colleagues for help with "cooperatively seeking solutions to shared concerns." This can include forming a group of teachers teaching the same topic, to surveying the current literature as a group, dividing the available resources and reporting to one another what has been found. Teachers can plan a series of lessons together. It is a less intimidating task if the work is shared. The teachers can meet again after teaching the lesson and analyze their experiences. At this point each teacher knows what he/she needs to do to make the approach his/her own. Teachers can observe others teaching or have others observe them. Leinwand (1992) also suggests the professional development of teachers through attending conferences and by reading professional journals. Prevost (1993) describes his own version of the three R's: Reflect, Risk, and Revise. A teacher first reflects on his teaching practice, comes up with a plan, risks by implementing it, and finally revises the plan.
Reasons for Focusing on the Problem Solving and Communication Standards
Problem solving and communication are interdependent. Students collaborating with other students on solving problems attempt to explain their logic to one another and in turn try to understand the thought processes of their fellows (Yackel et al, 1990). As students attempt to solve problems in this way "not only is the amount of time they spend participating in problem-solving activities increased, but the nature of their problem-solving activity is itself extended to encompass learning opportunities that rarely arise in traditional instructional settings"(Yackel, Cobb & Wood, 1989, as cited in Yackel et al, 1990).
The Problem Solving Standard
The five process standards: problem solving, reasoning and proof, communication, connections and representation are equally important, but problem solving is unique in that it is "at the heart of quantitative literacy--- the use of mathematics in everyday life, on the job, and as an intelligent citizen" (Pollak, 1997). Problem solving is important because society is changing so quickly that teachers must prepare students "to learn things that no one yet knows" (Charles & Lester, 1982). It will not be enough for students to learn how to solve familiar well defined problems. In the future, the problems our current students will face will be unfamiliar and poorly defined. Many mathematicians have written about teaching problem solving such as Polya (1957) and his four steps and Schoenfeld (1985) and his four categories.
The Communication Standard
and Smith believe that "the current interest in issues of communication
is both more widespread than ever before and more central to reform efforts
than at any other time in the history of mathematics education" (1997).
The Communication Standard encompasses both oral and written communication.
If students talk about their thinking as they solve problems, the teacher
can tailor the lesson to suit the student's way of thinking.
"Teachers' knowledge of students' thinking is an important guide in planning
effective lessons" (Maher & Martino, 1992).
Teachers can learn about their students' thinking through the students'
writing as well as the students' spoken words. In fact, "students
who will not ask questions in class may express their confusion privately
in writing" (Miller, 1991). So it is possible
for the teacher to adjust the lesson during the lesson on the basis of
the students' oral comments, and it is possible for the teacher to fine
tune future lessons on the basis of students' written comments.
Charles, R. & Lester, F. (1982). Teaching Problem Solving: What, Why & How. Palo Alto, CA: Dale Seymour Publications.
Hitch, C. (1990). How can I get others to implement the standards? I'm just a teacher! Arithmetic Teacher, 37(9), 2-4.
Frye, S. M. (1989). The NCTM standards, challenges for all classrooms. Arithmetic Teacher 36,(9), 4-7.
Leinwand, S. J. (1992). Sharing, Supporting, Risk Taking: First Steps to Instructional Reform. Mathematics Teacher 85,(6), 466-470.
Maher, C., Davis, R. & Alston, A. (1992). Teachers paying attention to students' thinking. Arithmetic Teacher, 39(9), 34-37.
Miller, L. D. (1991). Writing to Learn Mathematics. Mathematics Teacher, 84(7), 516-521.
Mumme, J. and Weissglass, J. (1989). The Role of the Teacher in Implementing The Standards. Mathematics Teacher 82,(7), 522-526.
Polya, G. (1957). How to Solve It (2nd ed.). Princeton, NJ: Princeton University Press.
Pollak, H. O. (1997), Solving Problems in the Real
World. In L. A. Steen (Ed.), Why Numbers Count: Quantitative
Literacy for Tomorrow's America (pp.91-105). New York: College Entrance
Prevost, F. J. (1993). Rethinking How We Teach: Learning Mathematical Pedagogy. Mathematics Teacher 86,(1), 75-79.
Silver, E. & Smith, M. (1997). Implementing reform in the mathematics classroom: Creating mathematical discourse communities. In Reform in Math and Science Education: Issues for Teachers. Columbus, OH: Eisenhower National Clearinghouse.
Schoenfeld, A. (1985). Mathematical Problem Solving. Orlando, FL: Academic Press.
Yackel, E., Cobb, P., & Wood, T. (1989). Small-Group Interactions as a Source of Learning Opportunities in Second-Grade Mathematics.
Yackel, E., Cobb, P., Wood, T., Merkel, G., &
Battista, M. (1990). Research into Practice. Experience, problem
solving, and discourse as central aspects of constructivism. Arithmetic
Teacher, 38(9), 4-5.