CONNECTING LEARNING AND TEACHING THROUGH ASSESSMENT

CONNECTING LEARNING AND TEACHING THROUGH ASSESSMENTAuthor(s): C. Jean MoonSource: The Arithmetic Teacher, Vol. 41, No. 1 (SEPTEMBER 1993), pp. 13-15Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195870 .

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CONNECUNG LEARNING AND TEACHING THROUGH

ASSESSMENT С Jean Moon

are beginning to understand that the teacher-as-researcher role encourages a teacher to

reflect on teaching practices and students' learning, which is a promising strategy for changing the way mathematics is taught, learned, and assessed (Tobin 1989). When teachers actively participate in evaluating classroom lessons, they can begin to create new, more effective frameworks in which to help their students make mathematical connections.

Using assessment to make connections between learning and teaching requires both conceptual and reflective involvement by classroom teachers, such as thinking about the big ideas that evolve into designing a lesson. The success of the project described in this article describes some of the benefits of the teacher-as-researcher role in making assessment a true part of mathematics teaching.

Project Description In 1988 and 1990, the Exxon Education Foundation awarded several grants to the Center for Math/Science Education Re- search at the University of Wisconsin - Milwaukee (UWM), in cooperation with the Milwaukee Public School (MPS) sys- tem. These grants supported a partnership between UWM and Garfield Elementary School, an inner-city MPS mathematics and science specialty school, so that a K-3 mathematics-specialist position might be created. An assessment project to align teaching, learning, and assessment in grades K-

3 with the NCTM's Curriculum and Evalu- ation Standards (1989) followed the cre- ation of the specialist's position. One of the goals of the project was to develop assessment models that focused on communications, problem solving, and number sense as outlined in the standards for grades K-4.

The project also sought to help teachers understand the link between assessment and teaching in a manner quite different from traditional after-school or Saturday in- service sessions. The project offered guided opportunities for twelve Garfield Elemen-

Guided observations of

your own teaching can be helpful.

tary School teachers to participate in assessment by acting as researchers in their class- rooms as they identified, piloted, and re- fined assessment criteria related to communications, problem solving, and number sense in mathematics. Throughout the assessment project, each of the twelve teachers was given one release day a month; substitute-teacher costs were supported by Exxon grant funds. Further, teachers participating in the monthly meetings were awarded district in-service credit as part of a program contained within the districtwide Staff Development Academy.

During each of these release days, the teachers and I met for guided discussions and reflection, focusing on specific questions at the heart of identifying assessment criteria. These questions are addressed sub- sequently. In addition to monthly release

days, the Garfield teachers agreed to meet one day a week, before the start of the school day, fo support and discussion. Later in the project, videotaped class lessons were re- viewed.

The Garfield mathematics specialist became part of a larger network of K-3 teachers who met once a month on Saturday mornings. This network was composed of teachers from eight Milwaukee Public Schools who were interested in assessment. Funds from the Exxon grant enabled these teachers to earn a college credit from the University of Wisconsin - Milwaukee for actively participating in this network.

The project supported critical areas as defined by the Garfield teachers. The support permitted the following: · time for teachers to meet in a group; · a partnership structure with a university; · a meaningful reward by allowing district in-service credit; • a link to a larger national network through the Exxon K-3 Mathematics Project; and the backing of the group by school and district administrators. In addition, funds from the grant allowed the purchase of limited teaching materials helpful to implementing certain mathematics lessons.

The Role of Teachers in Defining Assessment The critical bridges among teaching, learning, and assessment at Garfield were con- structed as each of the K-3 teachers began toexplore a teacher-as-researcher role. They engaged in this exploration through guided observations of their own teaching and through watching videotaped classroom lessons from various school settings. They observed lessons taught by the Garfield mathematics specialist, as well. The as- sumptions contained within the NCTM's curriculum and evaluation standards sug- gests that this new role is helpful for all mathematics teachers. When teachers are given an opportunity to observe and reflect on the classroom process, two of the excit-

Jean Moon is the director of the С enter for Mathemat- ics, Science and Technology in Education at Lesley College in Cambridge, MA 02138-2790. From 1988 through 1992 she was project director of the Exxon- sponsored K-3 Mathematics Specialist and assessment projects, which served as the basis for this article.

The author wishes to express her appreciation to the Exxon Education Foundation and foundation personnel for their consistent support.

SEPTEMBER 1993 13



ing outcomes of the standards are achieved: (1) greater involvement in making instruc- tional and assessment decisions on the part of teachers, and (2) development of a sense of expert judgment in instruction and assessment.

Beginning Questions The K-3 staff at Garfield, which included teachers, the mathematics specialist, and the science-resource person, began exploring assessment by observing classroom lessons while examining several questions: • What is assessment? • How do the NCTM's evaluation stan-

dards describe assessment? • In what ways is performance-based as-

sessment different from traditional evaluation practices?

• How does the role of the teacher change in the context of performance-based assessment?

• What is the role of the student in the assessment process?

• How can assessment and instruction be integrated?

These questions were not difficult for the Garfield staff, since most had previously studied both the standards and the concept of performance-based assessment. More problematic to the group was defining, in a clear and explicit manner, what assessment might look like in a mathematics class. For example, if we were to assess problem solving in a second-grade mathematics lesson, what exactly would we do? What would problem-solving activities look like? How would we know if a student is successful?

These questions reveal a significant truth about the relationship between teaching and assessment because the behaviors being assessed affect the way in which teaching takes place: Teachers need to create and evaluate assessment activities. The connection between learning and assessment can- not occur unless teachers themselves know the abilities they are asking their students to develop. This new view of assessment is asking us what we want students to know and be able to do, which is very different from a mathematics classroom that focuses on correct answers or content, readily scored from an answer sheet, as evidence of learning. The combination of new questions and new answers, or student outcomes, forces a

broader view of evaluation and assessment. While the Garfield staff became more cog- nizant of this broader view, they continued to struggle with discerning the format of specific assessment models.

As the Garfield staff grappled with the question "What could assessment in problem solving look like?" it was helpful for them to work through the following questions in rich detail: • What is problem solving in mathematics? • How could we describe the behavior of a

child who has a well-developed sense of problem solving? What are those behaviors?

• How could we describe the behaviors of a child who does not have a well- developed sense of problem solving? What are those behaviors?

Determining student criteria is necessary in linking assessment

and instruction.

Our experience in the Exxon project showed us that these questions directed us toward the heart and soul of the assessment and the teaching process: the criteria on which to evaluate students' performance. Identified criteria represent a slice of the range of behaviors that make up a broader outcome, such as problem solving or number sense. Responses to these questions, and others like them, present a solid picture of what counts in the classroom as acceptable performance in a given mathematics ability or outcome.

The Role of Criteria in Linking Teaching and Assessment Determining criteria may be the most difficult aspect of the process of assessment; it certainly was for the staff participating in this project. We were tempted to leave the discussion and dialogue too early, before we had articulated the criteria we believed to be

most appropriate. Our discussions tended to look for quick descriptions without first completing the necessary reflective work. At some point within the process, however, it became clear to us that determined, often collaborative, work is needed to examine what is at the heart of any competent performance within a specific ability, whether that ability is mathematical reasoning, problem solving, or connection-making. This work should not be done in isolation, apart from colleagues. The moment of clarity differed among the staff, but gradually the work became more openly collaborative and, as a result, more productive in meeting project goals.

After six months of dialogue, reflection, revision, practice, and feedback, the Garfield staff identified a set of behavioral criteria for grades K-3 within the context of problem solving, communications, and number sense in mathematics. They had come to a professional agreement that these criteria reflected the behaviors they expected their students to demonstrate, the behaviors they valued as part of the problem-solving, number-sense, and communications outcomes. A sample of the problem-solving criteria developed for grades K-3 is pre- sented in table 1 . Teachers created criteria in which development could be observed over time. It is assumed that this criteria will not necessarily be achieved by students at the earlier grades. However, it was agreed that all students should demonstrate these stated abilities by the end of the third grade.

In defining the K-3 problem-solving criteria, the staff had simultaneously set up a direct connection to instruction. For example, when they identified assessment criteria related to the construction of problems from everyday life, they were making a statement about how they would be constructing their daily lessons. Lessons had to furnish consistent opportunities for students to practice constructing mathematics problems in their own language and symbols. Lessons also had to teach students that being able to talk and write using mathematical symbols and words was an important skill on which they would be evaluated. As a result, the connection between criteria in problem solving and communications became quite clear.

Throughout our work we viewed criteria as guides for learning and not as ends in and of themselves. Criteria became guides for assisting teachers and students in making

14 ARITHMETIC TEACHER



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Examples from K-3 problem solving criteria for mathematics

Illustrate, describe, and model various problems by - • using objects, illustrating with graphs, and drawing pictures; and • repeating a problem in one's own words.

Verify, interpret, and justify solution strategies by - • explaining the problem-solving process in her or his own words; • seeing relationships between problem types and solution strategies; and • checking and testing the reasonableness of a solution in appropriate units.

Construct problems from everyday life with various mathematics concepts by - • forming a mathematics problem and thinking about it; • telling in one's own words through embedding a mathematics problem in a story or

illustrating to others through pictures, graphs, charts, or written compositions; and • demonstrating an understanding of how a problem can be solved.

connections between daily mathematics lessons and understanding the whole of mathematics.

A New Set of Questions and Connections On the basis of their new understanding of criteria, the Garfield staff is now addressing a new set of questions: • What learning experiences are necessary

to offer practice for students in the identified criteria or students' behaviors ?

• How can we share this criteria with our students in a meaningful way? How can we model this criteria in the teaching process?

• How can we sample students' performance during ongoing learning experiences in a way that is informative yet manageable?

• What kind of learning experiences or lessons provide a context in which the students can most effectively demonstrate their learning?

• How can we best give feedback to students on their performance?

• How can we encourage students to assess their own work using the identified criteria and not just their subjective interpre- tations?

To facilitate answering these questions, we are now videotaping many of the K-3 mathematics classes at Garfield. By view- ing these tapes as a professional community and applying the assessment criteria to stu-

dents' performances, we are beginning to find concrete suggestions, connections, and solutions. In retrospect, the Garfield staff has taken several important steps: (1) they have begun to develop a clearer sense of what assessment means and how to imple- ment it within the mathematics classroom; (2) they have found a clear role in working together with university personnel to iden- tify a base of professional knowledge within assessment; and (3) they have received feed-

Students need to know assessment

criteria.

back from parents and students stressing the important link among teaching, learning, and assessment. According to a second- grade teacher at Garfield, "I feel the assessment criteria we developed has helped me become more aware of the 'how and why' of my math lessons. It has also provided me with the 'language' necessary to talk with parents during conference." The idea of a "language" for talking with parents refers to the ability of the teachers to use problem- solving criteria or number-sense criteria to inform parents about the goals of their mathematics instructions. The assessment criteria fortified for teachers, students, and

parents the often abstract goals critical to mathematics instruction.

Teachers need to be an integral part of defining and building assessment criteria. Until teachers can address the "how and why" of their mathematics lessons, they will be unable to model the connections essen- tial to helping students understand. Students and teachers are best informed when they have explicit assessment criteria to help guide and make connections in teaching and learning mathematics.

Bibliography Diez, Mary, and Jean Moon. "What Do We Want

Students to Know? ... and Other Important Ques- tions." Educational Leadership 49 (May 1992):38- 41.

Meek, Anne. "On Thinking about Teaching: A Con- versation with Eleanor Duckworth." Educational Leadership 48 (March 1991):30-34.

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

Schon, Donald. Educating the Reflective Practitio- ner. San Francisco: Jossey-Bass, 1987.

Tobin, Kenneth. "Teachers as Researchers: Expand- ing the Knowledge Base of Teaching and Learn- ing." In Looking into Windows: Qualitative Re- search in Science Education, edited by Marsha Lakes Matyas, Kenneth Tobin, and Barry Fraser, 1 -7. Washington, D.C.: American Association for the Advancement of Science, 1989. Щ

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