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Table of Contents
Formal school assessment traditionally consists of criterion-referenced and norm-referenced tests that are confined to the form of paper-and-pencil, multiple-choice items. Such tests show only whether students can recognize, recall, and apply specific knowledge to solve simple problems. They constitute proxy measures from which the school leadership and policymakers try to make inferences about the students' knowledge and abilities. The analysis of the test data is focused on the products of learning, not the processes of learning and reasoning. Although they provide some comparative data, criterion-referenced and norm-referenced tests do not provide teachers the feedback they need to revise instruction and improve learning. To the contrary, these tests encourage rudimentary instruction. Experienced teachers who struggle to meet the standards of the formal achievement tests cover what they predict the tests will measure. Because teachers typically teach what tests measure (O'Day & Smith, 1993), professional developers and leaders must promote alternative assessment practices that focus on students' conceptual understanding and problem-solving abilities.
Formal school assessment and typical quizzes and tests are not congruent with the level of conceptual understanding and strategic competencies that are central to Concept-Rich Instruction (Corbett & Wilson, 1991; Shepard & Smith, 1988; Smith & Cohen, 1991). Teachers need to measure students' progress more frequently and less formally than the standards-based academic achievement measures and even the more-frequent, teacher-made quizzes that better reflect instructional objectives. These measures are limited in what they can assess in a short time. They challenge the student to solve simple rather than complex problems, and to reproduce and repeat rather than create original work. All these measures provide teachers only limited assessment of students' ability to solve problems and find new applications for the mathematics they learn. Teachers must use additional assessment tools that inform the processes of learning.
Teachers who follow the practice of Concept-Rich Instruction find alternative assessments that help them identify developmentally appropriate content, recognize student misconceptions, evaluate the meaning students make of what they learn, see whether instruction is effective in altering misconceptions, and distinguish diverse learning needs. These teachers seek assessment tools that help them choose and alternate instructional techniques effectively.
Research shows that where process-oriented alternative assessment is in place, teachers tend to be more flexible and responsive to their students' learning needs (Spinelli, 2001). Furthermore, research shows that teachers who use this type of assessment regularly keep expanding their practices with a greater range of possible choices and strategies (Larrivee, 2000).
Teachers cannot measure directly the processes that underlie reasoning and learning; they can only infer these processes from dialogues with students (Hiebert & Carpenter, 1992). In making such inferences about students' reasoning and learning problems, teachers must align the mathematics knowledge that they target with the situation, types of student responses, and characteristics of the student or group of students. Research shows that teachers who learn to do so tend to create new and better knowledge from their own experiences in teaching than do teachers who rely on traditional assessment (Stein, 2004).
Assessment can provide formative and summative information on student learning. Formative assessment provides feedback on teaching and learning, and summative assessment indicates what students have learned. Summative assessment tools include state and national achievement tests that students take at the end of a year and teacher-made tests given at the end of a unit of study. Formative evaluation involves authentic and dynamic assessment practices. Assessment that is authentic and dynamic mirrors the priorities and challenges of Concept-Rich Instruction, because this form of assessment allows teachers to assess the students' thinking as the students develop their responses to well-designed academic challenges (Wiggins, 1990). Assessment is authentic when teachers directly observe how students analyze, synthesize, and apply what they have learned in a substantial manner to solving complex problems. It is dynamic when it shows not only what the student knows, but also how the student learns.
Formative assessment may include interviews with individuals and small groups of students, student journals, student self-assessment, portfolios, performance assessment, and surveys. Because the various methods of assessment provide different types of information, teachers must learn to employ and integrate the information they gather from them.
Formative assessment may take different forms, but it usually includes a task and a rubric by which performance is evaluated. The task involves meaningful problems from real-world contexts. The rubric is always based on the teacher's understanding of the specific characteristics that make up good performance and identifies milestones of learning. The rubric also guides students to better develop the skills and understanding that are necessary to perform well.
Classroom communication and observations vary in form and purpose. Teachers can observe students as they discuss and debate mathematical ideas and solutions to problems, they can watch students as they model and explain their solutions on the board, and they can interact with individual students while they are engaged in classroom assignments. For example, observations are particularly well suited for assessing students' concept and skills of measurement. Teachers can observe students measure length, width, height, weight, capacity, volume, area, time, and temperature using standard units. They can examine whether students understand the importance of a point of reference for measurement. They can engage students in comparisons by asking which ball is heavier, which stick is longer, which of two different-shaped containers holds more water, whether a wrapping paper of a certain area can completely cover a given box, and so forth.
When the level of students' receptive and expressive language is appropriate, classroom discussions may reveal important information about students' understanding and ability to apply their mathematics knowledge. The revelation is even more acute when students see themselves not just as responders to questions, but also as posers of questions. In the case of students with limited English proficiency, teachers must rely more heavily on assessment tools that are focused on activities rather than classroom dialogue. Teachers who learn to listen to the informal language that students use in the classroom while they develop concepts and skills can identify mathematical preconcepts and misconceptions that always crop up in these contexts.
If teachers create a classroom culture in which the most struggling students feel comfortable exposing their thinking in front of their peers, teachers can then engage their students in problem solving, watch them model solutions on the board, ask questions, and listen to the students' answers. In this environment teachers can effectively probe the reasons behind students' actions, behaviors, and language. In the absence of this desirable environment, teachers must resort to assessment in the context of individual problem-based activities.
Teachers can also assess learning while individual students are engaged in classroom assignments. Here, teachers must assume the role of a participant-observer—they are part of and live in the learning community, but maintain a neutral posture. They may encourage students and praise their work; however, if they want to assess learning, they should not ask leading questions. To assess learning teachers must ask only open-ended questions and maintain a neutral posture regarding students' answers. Teachers must always avoid disturbing students when they are working intently.
Classroom observations can help teachers assess their students' understanding and ability to apply mathematical concepts, their ability to solve problems, their ability to communicate mathematically, and their ability to work with others. Any classroom observation is valuable if it is articulated by, and is limited to, specific goals and is void of extraneous information—to the extent that teachers collect and manage their assessment for future reference.
Teachers can use several effective and efficient means for collecting observation information. They can use note cards or a small, pocket-sized tape recorder for dictating observations; use a video camera; and develop and use checklists of desired concepts and actions. Figure 5.1 presents an example of a checklist that a mathematics teacher prepared based on her knowledge of Feuerstein's classification of cognitive functions (Feuerstein, Rand, Hoffman, & Miller, 1994).
Input
Comments
Collects and organizes data systematically
Understands vocabulary
Integer, fraction, percent, decimal, ratio, exponent, inverse, rate, distance, perimeter, area, surface-area, volume, angle, radius, Understands vocabulary rectangular-solid, cylinder, pyramid, cone, sphere, prime-factor (multiple), symmetry, congruency, similarity, perpendicularity, parallelism, reflection, flip, slide, turn, enlargement, mean, median, mode
Tables, graphs, symbols
Understands nonverbal expressions
Names geometric figures (including solids)
Is reflective (not impulsive) when performing a complex task
Is precise
Measures, estimates
Analyzes the properties of and relationships within two- and three-dimensional geometric figures
Elaboration
Identifies and defines problems
Projects relationships
Compares
Part-whole, relative terms, concrete and symbolic representations, units of measurement
Coordinates several variables simultaneously
Problem solving, coordinate geometry, comparison of geometric shapes by several variables
Finds causal relationships
Plans ahead
Forms hypotheses and tests them logically
Uses logic to reach valid conclusions
Inference, induction (generalization), deductive
Output
Considers communication from the receiver's point of view
Clear responses
Uses vocabulary properly to communicate mathematics ideas
Integer, fraction, percent, decimal, ratio, exponent, inverse, rate, distance, perimeter, area, surface-area, volume, angle, radius, rectangular-solid, cylinder, pyramid, cone, sphere, prime-factor (multiple), symmetry, congruency, similarity, perpendicularity, parallelism, reflection, flip, slide, turn, enlargement, mean, median, mode
Uses nonverbal expressions to communicate
Presents data in an orderly way
Communicates complete ideas
Teachers do not have to observe each student every day. Rather, to the extent possible, teachers should designate a time for each student to be observed and focus on that particular student during that time. Furthermore, because teachers use additional assessment methods, they may want to limit their classroom observations to the assessment of progress in particular areas of mathematics with particular students.
In conjunction with other forms of classroom communication, such as small-group and whole-class discussions, teachers can use surveys to collect academic, as well as affective, information. In particular, teachers can use surveys periodically to assess changes in students' affective mathematical dispositions, attitudes, efficacy, and anxieties. Such surveys may consist of Likert scale ratings and open-ended items.
Classroom communication and observations are integral to Concept-Rich Instruction, and when appropriate, teachers should share their observations with students to alter students' misconceptions. When teachers do so, new learning begins. Hence, by its nature, classroom observation is a method of dynamic assessment.
Homework is assigned independent practice. It is generally designed to reinforce classroom learning, teach students to independently apply newly acquired skills and knowledge, and develop their study skills and personal responsibility. Homework may help students to review and practice what they have learned, prepare them for the next day's class, and engage them in investigating topics more fully than classtime allows. Research shows that where homework is routinely assigned and evaluated, students tend to have higher achievement (LaConte, 1981; Lindsay, Greathouse, & Nye, 1988; Walberg, Paschal, & Weinstein, 1985).
Research on effective homework practice shows that teachers should
(England & Flatley 1985; Good & Grouws, 1979)
Besides additional learning experiences, homework assignments provide teachers with important data that they can use to diagnose students' learning problems. In fact, school reformers at Harvard Project Zero, the Annenberg Institute for School Reform, and the Coalition for Better Schools argue that analyzing student work is key to improving teaching and accountability. They argue for refocusing professional development on reflective examination of authentic student work, rather than on test scores and grades, as representations of student learning (Allen, 1998; Blythe, Allen, & Powell, 1999).
To best assess their students' work, teachers may require that students record not only their solutions to home and classroom assignments, but also maintain a problem-solving notebook with weekly entries, including the following:
Teachers can assess students' abilities and difficulties by analyzing students' work samples and from the students' reflections as self-reported. In the case of students who cannot complete their homework or students who complete their work incorrectly, teachers must look closely at the homework and may have to compare it to other samples of the students' work to find the reasons and adjust instruction accordingly.
What researchers know today about student errors comes from studies that focus on the ways individual students process information. Researchers often study errors through clinical interviews. This method of research is important because it helps identify general issues of learning and learning problems as well as what constitutes effective instruction. However, it is also important because it provides a model for an alternative assessment strategy that teachers can use in their classrooms. The particular significance of interviews is that they can reveal to teachers the differences between their and their students' values, concepts, guiding theories, and problem-solving strategies in doing math. If teachers consistently apply what is learned from student interviews in the classroom, they will improve their instructional practices.
Researchers use the clinical interview as a three-stage methodology for constructing and testing hypotheses regarding individuals' alternative conceptions. In the first stage, the interviewer formulates hypotheses of a particular reasoning and tests these hypotheses through probes. In the second stage, several researchers independently analyze the recorded interviews and arrive at agreed-upon hypotheses. These hypotheses are then subjected to further testing in the third stage. It is a rigorous process that must meet the scientific standards of validity and reliability, and researchers do not apply it lightly. However, teachers could use clinical interviews as an alternative assessment tool if they know their students, know the typical misconceptions, use the method repeatedly with some students, and follow a set of guidelines such as the one listed below.
Teachers can design interviews that help identify the learning needs of individual students. The interviews allow teachers to follow their students as the students model mathematical concepts and skills and communicate them mathematically; to learn about student misconceptions and guide students toward more complex ideas; to investigate whether students have appropriately generalized a concept; and to find out whether students can apply concepts to new problems. For example, a teacher can use an interview to assess the understanding of place value. He may ask a student to model number names with place value blocks and a place value mat. The student may name 502 as 500 and two ones; 50 tens and two ones; or five hundreds, zero tens, and two ones. Or, the teacher may find out that the student does not understand the role of zero as a placeholder. In this case, the teacher can use probing questions to guide the student toward further learning and help the student attend to misunderstandings. The teacher may also ask the student to model and explain the process for adding 57 and 34. At the end of the interview, the teacher may engage the student in solving a real-world, nonroutine word problem and watch as the student applies the concepts of place value through modeling and explaining the process she performs.
Because a heavy classload may prohibit them from using this tool with all students, teachers may conduct individual interviews with a selective few of their students to sample their class's progress with problem solving (Long & Ben-Hur, 1991). Teachers may want to assess the learning problems of students who previously tested poorly or students who perform poorly in class, and perhaps compare their performance with the performance of students who are highly proficient. When appropriate—and to save time—teachers could use the procedure with small groups of students. In this case, teachers ought to prepare specific assessment goals for the individuals as well as for the group.
Interviews may target a variety of goals:
Interviews may also provide opportunities to assess students' ability to communicate using mathematics as a language. At the same time, teachers should know that students frequently know more mathematics than they can communicate verbally (Siegler, 2003), and therefore teachers should always probe when they want to properly evaluate students' mathematical knowledge.
In addition, individual and small-group interviews provide opportunities for students to ask questions that they may not otherwise ask in a large-group setting.
For interviews to be effective, teachers must prepare ahead of time. They must choose and analyze problems in terms of the National Council of Teachers of Mathematics (NCTM) standards they may want to use, mathematics concepts and operations, and the cognitive behaviors that the problems challenge.
The effective interview begins with a set-up that relieves the student's anxiety:
Once the interview starts, teachers must be flexible. They must be ready to alter tasks and offer just enough help. The student should always complete at least some of the tasks and be pleased with her accomplishment. Teachers must ask questions that do not lead. Rather, they should
Teachers must also maintain uninterrupted dialogue, and avoid, if possible, phone calls, announcements over loud speakers, bell sounds, and so forth. Teachers should
Finally, teachers should conclude the interview on a high note, such as
Journal writing encourages students to monitor, review, and reflect upon their learning experiences. Thus, it helps students develop concepts, skills, and strategies for solving a variety of new problems. As a result of the reflective process that is involved in writing journals, students learn to view mathematics as more than just an exercise in getting the right answer. At the same time, journals can help teachers assess students' reflections of their own capabilities, attitudes, and dispositions and evaluate their abilities to communicate mathematically through writing.
To benefit the most from student journals, a teacher must develop together with the students a purpose for journal entries. For example, journals entries could include
Students should be encouraged to use in their journals language such as the following:
“This was possible because ... Alternatively ...”
“The problem here, I believe, was that ...”
“While it may be true that ...”
“On the one hand ...”
“In thinking back ...”
“On reflection ...”
“I guess that this problem has made me aware of ...”
Because the focus of journal entries may differ from time to time, teachers should always encourage students to keep their records in an orderly notebook and review, relate, and compare current journal entries to previous ones.
Journals are most valuable as learning tools if students discuss their records and teachers have opportunities to reinforce or intervene in the process by probing, suggesting new directions for reflection, challenging misconceptions, or questioning the efficiency strategies. Journals are also most valuable if students share them with each other. At the same time that journals facilitate learning, journal entries provide insight into how students are developing as problem solvers and how teachers might enhance their development.
Although journal writing provides opportunities for student reflection, there are difficulties associated with using journal writing as evidence of learning. Journal entries are not easy to analyze for assessment purposes. There is always the possibility that the reader's perceptions and expectations may alter the authentic meaning of the personal, reactive, emotive, and, at the time of writing, not-at-all reflective student statements. However, there is no doubt that student journals could be one source of information among others.
There is general agreement that students' ability to monitor and assess their own learning is important, and that this ability must be cultivated in the classroom. Students do not learn to monitor and assess their own work in classrooms that place a premium on obtaining the correct answers. Students do so only if they are unafraid to risk exposing their errors and misconceptions, and if the outcomes of self-assessment are rewarding. Self-monitoring and self-assessment develop when teachers show students exactly what is meant by assessment and emphasize that assessment does not necessarily imply grading. Self-monitoring and self-assessment develop when teachers help students identify the criteria that guide their monitoring and assessment (e.g., a rubric).
For example, Ms. G. assigned a project to group 5 to calculate how many pennies would have to be stacked on top of each other from the ground up until the pile reached the middle of the St. Louis Gateway Arch. Then students had to calculate the amount of money these coins were worth in dollars and estimate the volume of a container that could carry the coins from the bank to the site. She gave the group the necessary dimensions of the arch and assigned an individual responsibility for each group member. Individual responsibilities might include leader, facilitator, recorder, reporter, and timekeeper. Because the solution the group arrived at did not meet the required conditions, the teacher encouraged the students to alter their plans and try again. After completing the task, the students had to record which concepts and which strategies they used to solve the problem. Each student had to assess whether or not the solution met the required conditions. Eventually the students had to evaluate how they worked as a team.
In addition to its important function as a tool of learning, self-assessment that students share with teachers may also offer teachers a source of valuable assessment data. Because self-assessment that is guided by specific criteria is still subjective, it provides experienced teachers access to their students' awareness of their misconceptions and weak strategic competencies and to their self-concept as mathematics learners. In fact, student self-monitoring and self-assessment may provide teachers with assessment information that may not be available through any other assessment tool.
Portfolio assessment is a method by which students demonstrate their ability to do major pieces of work that are more elaborate and time-consuming than short exercises. The Assessment Standards for School Mathematics recognizes this form of assessment as a good example of integration of instruction and assessment activities. As a process-oriented approach to assessment, portfolios can link successes and failures to performance and facilitate goal setting and self-motivated learning. Although more subjective than traditional testing, portfolios indicate student choices and interpretations and may reveal how students think and why.
Teachers can make valid inferences about the progress in students' understanding of concepts and skills from examinations of dated work samples in students' portfolios. For example, a portfolio that contains a student's work samples in plane geometry might include the following:
The portfolio may show initial sketches and records of improper identification of designated plane figures. Later records in this portfolio may indicate the student's better understanding of the geometry concepts, and the latest records may show complete understanding.
Teachers may guide their students' work on developing their portfolios through the following strategies:
There are three types of mathematics assessment portfolios: the showcase portfolio, the teacher-student portfolio, and the teacher alternative assessment portfolio (Columba & Dolgos, 1995).
When teachers give students the opportunity to choose the portfolio contents, the students' choices give teachers insights into students' interpretation of their work, their dispositions toward mathematics, and their mathematical understanding. Furthermore, because the portfolio contents are developed over time, teachers can learn not only the current state of students' learning, but also the individual student's learning patterns.
Portfolio assessments may provide teachers and students with valuable insights into students' learning progress if they contain accurate and detailed accounts of the students' work and if they are maintained over time. Both these conditions are hard to meet, as portfolio assessment consumes more time than other forms of assessment.
Performance-based assessment involves individual or group projects around a mathematical problem. The problem may take from a half hour to several days to solve, and the students' activities are often videotaped or audiotaped. The goal is to assess both the process and product of the student's or group's solution of the problem.
The teacher guides the assessment along the following steps:
As part of, or in addition to, the above, the teacher may ask students to discuss strategies they used to solve the problem, compare problems, propose possible extensions for the problem, and reflect upon their feeling about their experience with the problem. Over time, performance-based assessment should also reveal whether students value mathematics, are confident in their ability to use mathematics to solve problems, and are learning to communicate mathematically.
Performance-based assessment provides teachers information on their students' thinking and understanding, like the other forms of alternative assessments that have been discussed. It should also help students gain insight into their own learning and understanding of mathematics. It is important that teachers encourage students to monitor their learning and evaluate their strategies and their current levels of understanding. Feedback should occur continuously, but not intrusively, as part of instruction. Teachers should carefully consider if formal or informal feedback will be most constructive.
Summative assessment is limited to the products of students' learning. It is time-consuming and yields little benefit to students' learning. Only by inference does such assessment provide information on students' conceptual understanding and reasoning. Therefore, Concept-Rich Mathematics Instruction promotes using a variety of formative assessment methods that can better reveal the state of students' learning. Because formative assessment is contextualized in learning, it does not consume instructional time without yielding direct benefits to students. It is authentic and dynamic, and therefore it is constructive to classroom learning and teaching.
Formative assessment not only targets the acquisition of new concepts and skills, it also identifies students' interests in mathematics, the meaning that students find in concepts, students' preconceptions and misconceptions, levels of strategic competency, and students' ability to communicate mathematically. It also reveals changes in the students' affective dispositions, their attitudes and anxieties, and their sense of responsibility for learning. Formative assessment also provides opportunities for teachers to test the effects of new instructional strategies.
However, formative assessment is not simple to perform with any regularity. To conduct formative assessment, teachers must learn how to systematically and continually collect and organize data while managing time and resources. They must learn how to generate opportunities for assessment and use a variety of tools. They must learn to act as participant-observers with small groups of students, ask open-ended questions, listen, and remain nonjudgmental as students reflect on their actions. Teachers must learn how to use interviews with individual students to assess learning difficulties; how to challenge students to self-monitor and self-assess their learning, while encouraging them to share their assessment; how to objectively analyze homework assignments and portfolios; and how to conduct performance-based assessment to find meaningful signs of progress in students' learning.
Teachers can conduct formative assessment in the course of classroom activities; in private meetings with individual students, parents, or other teachers; and while analyzing student artifacts after school. But they must learn to organize the various formative assessment data and integrate these data with information generated from different summative assessment tools. The combination of formative and summative assessment data will help teachers to better understand how students progress and what they need to learn more effectively to meet new learning goals.
Students benefit from formative assessment as well. Most of the tools and methods of formative assessment involve students' reflections and heightened awareness. In the process of active engagement in the assessment of their knowledge and skills, students may find new meanings; recognize their misconceptions; and find out that there are different representations, strategies, or points of view than they originally considered. They may learn how to learn and may become more involved in, and take more responsibility for, their own learning.
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