IMPLEMENTING THE "PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS": What DoWe Have and Where Do We Go from Here?Author(s): Glenda Lappan and Susan N. FrielSource: The Arithmetic Teacher, Vol. 40, No. 9 (MAY 1993), pp. 524-526Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195848 .

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lAAPLEMENTING THE PROFESSIONAL STANDARDS FOR TEACHING /W'THEIW'TICS

What Do We Have and Where Do We Go from Here? Getting to know something is an adventure in how to account for a great many things that you encounter in as simple and elegant a way as possible. And there are lots of ways of getting to that point, lots of different ways. And you don't really ever get there unless you do it, as a learner, on your own terms. All you can do for a learner enroute to their forming a view of their own view is to aid and abet them on their own voyage. The means of aiding and abetting a learner is some- times called "curriculum," and what we have learned is that there is no such thing as the curriculum. For in effect, a curriculum is like an animated conversation on a topic that can never be fully defined, although one can set limits upon it.

-Jerome Bruner (1992)

the mathematics teaching community we have collectively

examined the K- 12 curriculum and have set limits and expectations on it. We have also examined the ways of "aiding and abet- ting" students on their journey to make sense of mathematics. The two documents in which our vision for curriculum and teaching are articulated are milestones in the history of education on this continent. The NCTM's Curriculum and Evaluation Standards for School Mathematics and the Professional Standards for Teaching Math- ematics (1991) give school mathematics programs across the continent a common philosophy and framework within which local schools can reflect their own special needs and goals.

The vision articulated in the two standards documents promotes several interrelated components of a powerful

Prepared by Glenda Lappan Michigan State University East Lansing, Ml 48824-1050 Edited by Susan N. Friel University of North Carolina Chapel Hill, NC 27599-3345

The Editorial Panel welcomes readers' responses to this article or to any aspect of the Professional Stan- dards for Teaching Mathematics/or consideration/or publication as an article or as a letter in Readers' Dialogue.

524

mathematics education for students, includ- ing (1) students actively "doing mathemat- ics"; (2) mathematics as thinking and sense-making; (3) powerful, but changing, mathematical content; and (4) a belief that all students can learn and appreciate math- ematics. The implications of this vision of mathematics and mathematics learning for teacher education and professional devel- opment are major. We are talking about the need to begin at ground level and build a teacher-support system that can help teach- ers in changing their beliefs and their prac-

Manipubtives and "fun" are not ends

in themselves.

tice to support more powerful mathematics and mathematical thinking for students. Bruner (1992, 6) describes the need for reform:

We need a surer sense of what to teach to whom and how to go about teaching it in such a way that it will make those taught more effective, less alienated, and better human beings. . . . What we need is a reform movement . . . with deeper convictions about what kind of people we want to be. Then we mount the kind of community effort that can truly address the future of our educational process - an effort in which all of the resources of intellect and compassion that we can muster, whatever the price, are placed at the disposal of the schools.

We claim that the need expressed here is exactly that for which the NCTM's Cur- riculumandEvaluationStandards for School Mathematics (1989) and the Professional Standardsfor Teaching Mathematics ( 1 99 1 ) stand. We hope that those who read the

documents see this kind of commitment to all children and teachers in both documents.

The Professional Teaching Standards identifies four major aspects of what a teacher must be able to do to teach in ways that support the vision in the Curriculum and Evaluation Standards - (1) choose worth- while mathematical tasks, (2) orchestrate classroom discourse, (3) create a safe and constructive environment for learning, and (4) analyze students' understanding and the contribution of his or her own practice to students '

learning. Let us briefly revisit these ideas.

Worthwhile Mathematical Tasks No other decision that teachers make has a greater impact on students' opportunity to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages the students in studying mathematics. Here the teacher is the architect, the designer of the curriculum.

To develop more productive notions about mathematics, students must have op- portunities actually to be involved in doing mathematics - to explore interesting math- ematical situations, to look for patterns, to make conjectures, to look for evidence to support their conjectures, and to make logi- cal arguments for their conjectures.

In considering a mathematical task a teacher judges how well the task represents the embedded concepts and procedures, how likely the students are to bump into the mathematics in the course of investigating the problem, how well the task represents what is entailed in doing mathematics, and what skill development the task will or can support. With what mathematics does the

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task surround the students? "Defensible reasoning about the mathematics of a task must be based on a thoughtful understand- ing of the topic at hand as well as of the goals andpurposesofcarryingoutparticularmath- ematical processes" (NCTM 1991, 27). Prob- lems should not be chosen because they are "fun" or use a manipulative that is available in the classroom. They must offer the poten- tial for students to engage in sound and significant mathematics as a part of accom- plishing the task. This statement is not meant to imply that manipulatives and fun are to be avoided. It is meant to imply that manipulatives and "fun" are not ends in themselves, they are tools to be used thought- fully to help students engage in, and make sense of, mathematics.

Classroom Discourse The professional standards document de- scribes discourse as "the ways of represent- ing, thinking, talking, agreeing, and dis- agreeing" (NCTM 1991, 36) as a group of students and a teacher strive to make sense of mathematics.

Given the mathematical goals that are important for students who will grow to adulthood in the twenty-first century, many mathematics educators and researchers are calling for considerable change in patterns of classroom discourse. For many math- ematics educators discourse is closely re- lated to the notion of building in the class- room a community of learners who engage in the study of mathematics in ways that are closer to what mathematicians do than is represented by many current classrooms. If a community of learners is to be created with the goal of helping all students make sense of mathematics, then the standards of com- munication in the classroom must promote mathematical ways of thinking, reasoning, knowing, and communicating.

Teachers play many roles in the class- room that have a very great influence on what sense students make of mathematics. The Mathematical Sciences Education Board (MSEB) (1991) describes roles that the teacher plays in helping students learn mathematics:

• a role model who demonstrates the mul- tiple paths, the false starts, the thinking, and the effort entailed in solving prob- lems;

• a questioner who challenges students to

MAY 1993

make sure that what they are doing is reasonable and purposeful and that stu- dents can defend their answers and ideas;

• a moderator who poses questions to con- sider but leaves much of the decision making to students working alone or in groups; and

• an interlocutor who supports students by encouraging them to reflect on their ac- tivities and to explore mathematics on their own.

Discourse is clearly at the heart of each of these roles of the teacher.

Classroom Environment What students learn is fundamentally con- nected to how they learn it. The environ- ment in which students learn affects their view of what mathematics is, how one learns it, and perhaps of more importance, their view of themselves as learners of math- ematics.

MSEB (1991) talks about environments that are supportive of the vision of math- ematical power for all students as environ- ments that do the following: • encourage students to explore; • help students to verbalize their math-

ematical ideas; • show students that many mathematical

questions have more than one right answer; • teach students through experience the

importance of careful reasoning and dis- ciplined understanding; and

• build confidence in all students that they can learn mathematics.

We may be wasting precious instructional time.

One aspect of classroom organization that effects discourse is the use of group work, in particular, of cooperative groups. But again, let us not go down the road of radical pendulum swings to take the posi- tion that we have reformed mathematics teaching and learning if only we put our students in groups to do their mathematics.

On seeing a classroom in which students are in groups, talking animatedly with manipulatives on the table, one is tempted to smile in satisfaction that the job of reform of mathematics education is done. However, let us be ever vigilant that unless a powerful, compelling mathematical question, prob- lem, or situation is underlying the animated discussion of the students in groups using manipulatives, we may be wasting a great deal of our precious instructional time. The interplay of tasks, discourse, and environ- ment with students' learning is the key to judging how well things are working and is the object of the last area of analysis in the professional teaching standards document.

Analysis How well is the system that the teacher has created working? Are the tasks engaging the students? Are they effective in helping stu- dents learn mathematics? Do they stimulate the richness of discussion that students need to develop mathematical power? Is the classroom discourse fostering learner inde- pendence? Curiosity? Mathematical think- ing? Confidence? Disposition to do math- ematics? Is the classroom environment encouraging the kind of engagement that reaches every student and supports their mathematical development? Reflective teachers regularly askthemselves these kinds of questions.

Analysis also included the regular assess- ment of students' progress for the purpose of making instructional decisions. This broad view of analysis is supportive of the concern in the Curriculum and Evaluation Stan- dards (NCTM 1989) for the mathematical development of students to include learning to value mathematics and to have confi- dence in themselves as learners of math- ematics. Assessing students' performance on skill-level items not only is insufficient but sends the wrong messages about what is important in the mathematics curriculum. The teacher needs to examine all aspects of the mathematical development of students, including how the tasks, discourse, and en- vironment are working to build mathemati- cal power for all students.

The assessment of students' performance is only one part of the evidence that the reflective teacher gathers to guide decisions about instruction. David Moore, a statisti- cian at Purdue University, says that the teacher shapes the environment for learning

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through setting tasks, encouraging open dis- cussion and group problem solving, and insisting that students express clear conclu- sions from their work (a number or a graph is not a conclusion) orally or in writing. This approach clearly calls for the teacher to look carefully at the complex set of activities and interaction in the classroom with an eye on whether and how students are learning.

So here we are! We have a vision, but how do we go about realizing this vision?

How We Realize the Vision The kinds of change called for by the vision in the standards documents are so funda- mental and pervasive that they seep into every aspect of our society. The current curriculum, expectations, and teaching practices are failing with so many of our students in mathematics that we have a responsibility to rethink what we are about. To accomplish change on a large scale, all the stakeholders - students, teachers, par- ents, school administrators, business, in- dustry, professional mathematicians, politi- cians, and others - need to understand the issues and the direction of reform and give their support to the effort. This undertaking sounds so overwhelming that we could just throw up our hands and say that it can't be done. However, the stories that are creeping out about what teachers and schools and communities all over the continent are ac- complishing are enough to make us uncom- fortable with complacency. We can start down the road to reform with the expecta- tion of success if we remember some funda- mental principles:

• Change of any behavior or expectation that has become the custom or habit is very hard and takes a sustained effort over a long time.

• Parents and the community in general want students to succeed and will support change if they see that the goals are in the best interest of their students and if they feel a part of the process.

• Change can begin anywhere on any scale - from a single classroom to a whole school, or even a whole state or province. Perhaps some change can even begin at the national level.

Schools and teachers are using various approaches to change. Many of the restruc-

526

turing efforts are finding different ways of affording teachers' time for the study and development that are necessarily a part of change. Here are some examples:

<ff* In Holt High School in Holt, Michigan, the teachers and administrators have negotiated with the teacher unions and the parents to have students report to school at 1 1 :00 a.m. every Wednesday.

Ç^The delayed opening allows the teach- ers time to work, study, and plan to- gether to implement changes in the cur- riculum, teaching practices, and the ways in which students are assessed. The school day has been lengthened by fif- teen minutes and class periods adjusted so that the net effect on students' class time is negligible.

c^In many schools, schedules have been changed to give teachers in particular areas common planning times.

<5* Some schools have a mathematics of- fice where teachers have desks and

We have a responsibility to

rethink what we are about.

equipment with which to do curriculum development. This proximity encour- ages interaction about teaching.

<&* In Portland, Michigan, the middle school was recently given a national "A+ for Breaking the Mold" award. It has taken eight years of dedicated work and effort on the part of the staff with the support of the board of education and the parents in the community to establish an exem- plary middle school mathematics pro- gram. The Portland School District tar- geted the middle school as a place from which to influence the entire K-12 mathematics education of students in the district. Over the eight years the teachers have attended Eisenhower- grant-funded summer in-stitutes, peer coaching sessions, weekend con- ferences, and after-school and evening meetings. In 1984, Mary Bouck, the

leader of the effort, was a part of a National Science Foundation research project on coaching as a strategy for changing teachers ' beliefs and practices. In the summerof 1987, Bouck and teach- ers at her school reviewed a copy of the working draft of the NCTM's Curricu- lum and Evaluation Standards docu- ment. From that point on, the curricu- lum standards have been a part of the guiding vision for change in the school and in the district.

Stories like these are happening all over ♦ the country. Teachers just like the reader

and the author are finding both ways to get started and the will to continue for long enough to see a changed mathematics pro- gram become the norm in the school. The obstacles are formidable. We can stay at the stage of reciting why we cannot change, or we can begin slowly and deliberately to wield whatever influence and skill we have to give change a push. New curriculum materials are being developed at all levels, new tests are being designed, schools are experimenting with new structures, and the National Board for Professional Teaching Standards is moving toward a certification process to help raise teaching to a profes- sional status. Other disciplines are follow- ing the lead of mathematics to articulate standards for curriculum, teaching, and evaluation. The question is, Will we be the ones that push these efforts by building the climate and support for change over time, or will we be the impediments who are locked in our own paradigms for teaching and learning and afraid to try new ideas? Some of our efforts will fail, but we can learn from these stumbles as we continue to move- forward. These are truly promising times in mathematics education. The door of the mathematics classroom has been opened, and it is unlikely that we could get it closed even if we wanted to!

References

Brunei·, Jerome. "Science Education and Teachers: A Karplus Lecture." Journal of Science Education andTechnology 1 (1992):5-12.

Mathematical Sciences Education Board (MSEB).'Toward National Standards for Math- ematics Assessment." Background paper for par- ticipants of the National Summit on Mathematics Assessment, Washington, D.C., 1991.

National Council of Teachers of Mathematics. Cur- riculum and Evaluation Standards for School Math- ematics. Reston, Va.: The Council, 1989.

. Professional Standards for Teaching Math- ematics. Reston, Va.: The Council, 1991. Щ

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