Seamless Assessment/instruction = Good Teaching
Lambdin, Diana V., Forseth, Clare, Teaching Children Mathematics
[Assessment serves] to help teachers better understand what students know and make meaningful instructional decisions. (NCTM 1989, 189)
Assessment of students and analysis of instruction are fundamentally interconnected. Mathematics teachers should monitor students' learning on an ongoing basis in order to assess and adjust their teaching. (NCTM 1991, 63)
Effective teachers have known it for years, though the recent assessment movement sometimes trumpets it as news:
Good teaching and effective assessment are often hard to distinguish.
Teachers use assessment to make decisions throughout the instructional process - in planning for the year, in deciding how to organize individual lessons, and even in orchestrating classroom activities.
Instructional decisions include identifying appropriate content, sequencing and pacing lessons, modifying or extending activities for students' special needs, and choosing effective methodologies. Teachers cycle repeatedly through the decision-making process as they reflect on information gathered about what students know and can do - information collected not only from such written work as assignments, quizzes, and tests but also from observing, listening to, and questioning students. Clearly, the quality of teachers' decisions is determined by how accurately they can infer what their students need from the assessment data that they have collected.
The recently published Assessment Standards for School Mathematics (NCTM 1995) identifies six standards, or criteria, to be used for judging assessment practices: mathematics, learning, equity, openness, inferences, and coherence. Since assessment and instruction are so often intertwined in the classroom, applying these assessment standards to instruction makes perfect sense, too. Experienced teachers have internalized and frequently use such questions as the following to monitor their practice:
* How does the mathematics of this assessment/instruction fit within a framework of important mathematics? (Mathematics Standard)
* How does this assessment/instruction contribute to students' learning mathematics? (Learning Standard)
* What opportunities does each student have to engage in this assessment/instruction? (Equity Standard)
* How have students become familiar with the assessment/instruction and its purposes, expectations, and consequences? (Openness Standard)
* How are multiple sources of evidence being used for drawing the inferences that lead to assessment/instructional decisions? (Inferences Standard)
* How does this assessment/instruction match specific goals? (Coherence Standard)
Uses of assessment in instructional decision making permeate the entire teaching process. At the beginning of each unit, semester, or year, teachers make long-range decisions about the content, the methods, and the frequency and types of assessments they will use. Throughout the year, they make short-range decisions about the next investigation or unit. And every day in the classroom, they make moment-by-moment decisions that affect the immediate lesson.
Using Assessment in Long-Range Planning
The first tasks for a teacher considering assessment in long-range planning are to decide what mathematics is important for students to learn and how evidence can be collected about students' progress in learning it. The state's, school district's, or perhaps a textbook's curriculum framework often guides long-range planning.
A long-range plan also includes establishing such classroom routines as deciding when to use daily warm-up or calendar activities, problems of the week, journals, or portfolios. Such routines are chosen, in part, with an eye to their potential for rendering assessment evidence about students' progress. For example, a teacher who particularly wants to document students' communication and self-reflection might ask them to keep journals. The teacher might also require that selected journal entries be included in a portfolio as part of a more diverse collection of student work. Since portfolios contain samples of work gathered over time, they present the necessary evidence for documenting students' progress throughout the year. Teachers may also use portfolios to monitor students' instruction and to guide appropriate adjustments. For example, after examining portfolios partway through the year, a teacher might note an abundance of work focusing on computation but a lack of entries requiring estimation. As a result, long-range plans might be reworked to allow more opportunities for students to work on developing number sense and estimation skills.
The key to making effective instructional decisions is to base them on valid inferences - reasonable conclusions reached by balancing evidence across a number of dimensions. Therefore, long-range instructional plans should include assessment not only of conceptual understanding and procedural skills but also of a range of other components of mathematical power, such as students' confidence as mathematics learners, their ability to make meaningful connections, and their effectiveness in communicating ideas. Furthermore, whenever instruction involves various activities, assessment practices should be aligned accordingly. For example, valid inferences require that when assigned, group work, as well as individual work, should be assessed. Valid inferences are also more likely when opportunities are given for responses in a variety of modes, such as talking, writing, graphing, or illustrating.
Equity is an equally important issue in instruction and assessment. Effective teachers make every effort to take into account the backgrounds, interests, and abilities of all their students. When tasks require background or skills that students do not have, learning is likely to be impeded and inferences drawn may be invalid. The use of context-rich or culture-dependent tasks can make assessments more meaningful to some students but less accessible to others. For example, students who enjoy sports often find sports-related problems meaningful and motivating. Yet an assessment task that involves using batting records to select a pinch hitter may not give accurate evidence of mathematical power for students unfamiliar with baseball. They may need a preassessment activity, such as working with some examples of batting records before assigning the pinch-hitter task, to acquaint them with the context. Alternatively, their teacher can make assessment more equitable by letting them select from several different tasks that assess the same mathematics, such as problems related to baseball, cooking, or travel. Gathering a collection of appropriate preassessment or alternative-assessment tasks requires careful planning and alignment with long-range goals.
Teachers also need to consider how their long-range plans for instruction and assessment align with externally imposed assessments, such as state-, province-, or district-mandated tests or portfolios. Problems arise when external agencies focus on objectives that are very different from those of the classroom. The Assessment Standards document (1995) envisions teachers working with other professionals on district, state, provincial, or national committees to craft common performance criteria. Such collaborative efforts help align external assessments with classroom assessment and instruction, as well as ensure that expectations are clear to all involved.
Using Assessment in Short-Range Planning
In short-range planning, teachers use assessment evidence to make decisions about tomorrow, or the day after tomorrow, or next week. As they plan for a unit, they review their long-range plans, recalling what outcomes they had considered important to assess and how this assessment was to be done, and they consider this plan in light of information they have gathered about their students during recent lessons. They make a conscious effort to integrate their teaching and assessing by identifying points within the unit as being indicators of specific student understanding, by asking certain questions, or by collecting certain work to monitor students' progress toward previously identified goals.
Let us look at an example involving Ernest Millichap, a fifth-grade teacher whose goals for the year included engaging students in cooperative problem solving and helping them to make mathematical connections. In an earlier unit on fractions, Millichap asked his students to explain how long each side of an equilateral triangle would be if the perimeter was seven and one-half inches. Since the students had had no prior instruction in his class on division involving fractions, they attacked the unfamiliar problem situation in very different ways. The work of three students is shown in figure 1.
Millichap was pleased with the variety of responses. Many students demonstrated a good grasp of the idea of mixed numbers, although some clearly needed more experience with fraction concepts, representations, or notation. He used this assessment in two ways. First, he used what he had learned about the range of understanding in his class to plan his subsequent instruction on fractions. He was surprised that few students thought of using 7.5 in working the problem, since calculators were readily available. Also, 7.5 [divided by] 3 can be easily divided mentally if one thinks of money and 75 [divided by] 3. He resolved to allow more opportunities for his students to relate fractions and decimals and more situations in which computations could be simplified by making appropriate choices about representations. Second, Millichap encouraged his students to discuss the various approaches they had used on this problem so that they could broaden their problem-solving repertoires by learning from each other. He also shared his reactions to some of the solutions.
It is important for teachers to discuss expectations, such as routinely conducting class discussions about specific examples of excellent and less-than-adequate student work and by presenting students with models of scoring rubrics and discussing how the rubrics would be applied. Students can also work together occasionally to develop their own scoring guidelines for an assignment involving a teacher-facilitated peer review. In planning such activities, openness in assessment and in instructional decision making become intertwined.
Planning appropriate assessments is a challenge. It is important to identify developmentally appropriate activities, to pose good questions, to listen carefully to student responses, and to follow up with questions that help students communicate what they think and can do. An understanding of how students learn specific mathematical concepts is useful, including an awareness of common misconceptions and a familiarity with strategies for helping students confront and reconsider their misconceptions. Students may need to work their way through misconceptions on their way to developing robust understandings. In the following example, Becca Abraham decided to delay intervention with a student named Carson while continuing to monitor his progress over time.
Abraham's third-grade class was just beginning a unit on division. The district goal for the end of the unit in grade 3 was that students "correctly divide one-digit numbers into two-digit numbers, with remainders." The activity posed to the class was to illustrate and explain the problem 23 [divided by] 5 using two-color counters, drawings, or cubes. The students' work for this lesson was collected at the end of the day. Carson's work is shown in figure 2.
Abraham was intrigued by Carson's incorrect response and showed it to her colleague. They discussed what she might do to help Carson, such as talk individually with him the next day to get a better sense of his thinking, to ask questions, or to offer careful guidance. The colleague suggested that Abraham resist the urge to correct Carson's misconceptions right away. She had noted in previous years that students generally developed a robust understanding of division only after they had grappled with the concept of equal partitions and leftovers in a variety of contextualized and decontextualized situations. Abraham decided to let Carson's understanding develop as the unit unfolded and to reassess it in a week or so. Carson's later work - on a problem involving the same numbers but set in a context involving money - shows a solid understanding of division [ILLUSTRATION FOR FIGURE 3 OMITTED].
This example may support making short-range instructional decisions not to correct all the work that students do and not to feel obligated to assign daily grades. Much work will self-correct if students are involved in appropriate learning experiences. Effective teachers establish criteria for performance and allow time for students to develop the knowledge and skills desired. In fact, whether to intervene must often be decided during the lesson as the teacher observes students at work.
Using Assessment to Inform Practice Moment by Moment
Moment-by-moment classroom assessment is just that - assessment done quickly to permit the teacher to make informed instructional decisions as students are working in the classroom. Observing, listening, and questioning are common methods of moment-by-moment assessment.
The ability to pose appropriate questions to students is just as important for moment-by-moment assessment as it is in the deliberate planning of lessons. Often, questioning students leads to a redirection of the lesson. The Professional Standards for Teaching Mathematics (NCTM 1991, 35) lists useful suggestions for questioning and talking with students.
The teacher of mathematics should orchestrate discourse by -
* posing questions and tasks that elicit, engage, and challenge each student's thinking;
* listening carefully to students' ideas;
* asking students to clarify and justify their ideas orally and in writing;
* deciding what to pursue in depth from among the ideas that students bring up during a discussion;
* deciding when and how to attach mathematical notation and language to students' ideas;
* deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with a difficulty;
* monitoring students' participation in discussions and deciding when and how to encourage each student to participate.
Teachers must learn to be perceptive in interpreting students' talk and to make quick decisions about the flow of instruction. Students sometimes give correct answers for the wrong reasons or incorrect answers as a result of misinterpreting questions. Often, careful probing is needed to determine what students really understand. Changing the direction of a lesson can be justified from the confusion ensuing as second graders attempt to measure their heights with metersticks or from the excited conversation resulting as fifth graders learn to explore linear functions on a graphing calculator. Similarly, a teacher might adjust plans for the week as a result of recognizing that students' responses to a problem are far richer than expected, musing, "Wow! I hadn't thought of so many ways to do this problem. I wonder if there are even more. Maybe we should spend some more time on it tomorrow."
Classroom assessment is an ongoing process, not a culminating event. Teachers orchestrate a variety of classroom activities designed to help students learn, but even in the midst of this orchestration they must constantly gather information to make decisions about when to move on, stop, or change direction and when to question, listen, and so on. Put simply, their goal is to ensure that students learn. It is perfectly natural to ask, "Is this work assessment or instruction?" Good teaching is seamless - assessment and instruction are often one and the same.
National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.
-----. Professional Standards for Teaching Mathematics. Reston, Va.: The Council, 1991.
-----. Assessment Standards for School Mathematics. Reston, Va.: The Council, 1995.
Diana Lambdin is a teacher educator in the School of Education, Indiana University, Bloomington, IN 47405. Her professional interests include problem solving, cooperative learning, assessment and evaluation, curriculum development, and issues in mathematics-education reform. Clare Forseth teaches sixth-grade mathematics and science at the Marion Cross School in Norwich, VT 05055. She has been active in helping teachers throughout her state use portfolio assessment effectively with their mathematics classes and has worked nationally on the New Standards Project.…
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Publication information: Article title: Seamless Assessment/instruction = Good Teaching. Contributors: Lambdin, Diana V. - Author, Forseth, Clare - Author. Magazine title: Teaching Children Mathematics. Volume: 2. Issue: 5 Publication date: January 1996. Page number: 294+. © 1999 National Council of Teachers of Mathematics, Inc. COPYRIGHT 1996 Gale Group.
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