Teachers who follow the practice of ConceptRich 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.
Process Compared to ProductOriented Assessment
Research shows that where processoriented 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).
Formative Compared to Summative Assessment
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 teachermade 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 ConceptRich Instruction, because this form of assessment allows teachers to assess the students' thinking as the students develop their responses to welldesigned 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 selfassessment, 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 realworld 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
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 differentshaped 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 problembased activities.
Teachers can also assess learning while individual students are engaged in classroom assignments. Here, teachers must assume the role of a participantobserver—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 openended 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, pocketsized 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).
Figure 5.1. Teacher's Checklist for Cognitive Behaviors of Word Problem Solving in Geometry
Input

Comments

Collects and organizes data systematically
Understands vocabulary

Integer, fraction, percent, decimal, ratio, exponent, inverse, rate, distance, perimeter, area, surfacearea, volume, angle, radius, Understands vocabulary rectangularsolid, cylinder, pyramid, cone, sphere, primefactor (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 threedimensional geometric figures


Elaboration

Comments

Identifies and defines problems


Projects relationships


Compares

Partwhole, 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

Comments

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, surfacearea, volume, angle, radius, rectangularsolid, cylinder, pyramid, cone, sphere, primefactor (multiple), symmetry, congruency, similarity, perpendicularity, parallelism, reflection, flip, slide, turn, enlargement, mean, median, mode

Uses nonverbal expressions to communicate

Tables, graphs, symbols

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 smallgroup and wholeclass 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 openended items.
Classroom communication and observations are integral to ConceptRich 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.
Analyzing Student Homework
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
 Teach students how to organize their work.
 Vary homework assignments.
 Ensure that students understand the assignment and are sufficiently prepared for the homework assignment.
 Make sure students understand the learning value of the assignment.
 Assign homework that is not overly long.
 Give recognition to students for completion of homework assignments.
 Check homework for understanding and modify instruction accordingly.
 Be clear on how homework assignments will be evaluated.
 Have students exchange and correct homework assignments in class.
 Provide feedback quickly and routinely on individual students' progress.
 Involve parents.
(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 problemsolving notebook with weekly entries, including the following:
 A discussion of the strategies they used to solve the problem.
 A comparison of the mathematical similarities among problems.
 Possible extensions for the problem.
 An investigation of at least one of the possible extensions.
 Reflection about their feelings about a solution.
Teachers can assess students' abilities and difficulties by analyzing students' work samples and from the students' reflections as selfreported. 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.
Individual Interviews Around ProblemSolving Activities
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 problemsolving 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 threestage 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 agreedupon 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 realworld, 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 & BenHur, 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:
 Identifying student misconceptions in a particular area of mathematics.
 Determining the depth and breadth of a student's proficiency as a mathematical problem solver. For example—
 Do students define the problem they are about to solve?
 How do they evaluate the type and difficulty level of the problem?
 How do they select an algorithm?
 How effective are the metacognitive processes?
 Are the students confident about their solutions?
 Is there divergent thinking?
 Do students have a selfconcept as being able to solve mathematics problems?
 How do students learn best?
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 smallgroup interviews provide opportunities for students to ask questions that they may not otherwise ask in a largegroup 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 setup that relieves the student's anxiety:
 Use a quiet and comfortable space.
 Calm the student by telling her that the evaluator (teacher) will help her succeed.
 Indicate that the purpose is NOT to evaluate the student, but to evaluate the effectiveness of instruction.
 Answer all questions the student may have before the interview.
 Explain the interview process.
 Ask the student to help set up the interview.
 Praise cooperation.
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
 Ask openended questions.
 Wait patiently for responses.
 Remain nonjudgmental to gain further insight into the student's thought process.
 Ask the student to clarify or explain surprising answers.
 Follow up with questions until student thinking is clear.
Teachers must also maintain uninterrupted dialogue, and avoid, if possible, phone calls, announcements over loud speakers, bell sounds, and so forth. Teachers should
 Reveal their interest and excitement about the student's work.
 Refrain from taking notes in the course of the interview (videotaping can help).
 Listen carefully.
Finally, teachers should conclude the interview on a high note, such as
 Telling the student how and explain why they enjoyed the interview.
 Inviting the student to come to them if she needs help.
 Promising to interview her again from time to time.
Journal Writing
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
 Problems that students want to solve
 Solution processes
 Presentation of alternative solution processes (if appropriate)
 Presentation of alternative solutions to the problem (if appropriate)
 Reflection on problemsolving strategies
 Discussion of the validity of the solution
 Definition of mathematical concepts and describing their meaning
 Identification of skills that students have developed from experience
 Reflections on the problemsolving experience
 Checklists to record such things as new learning tools and new problemsolving strategies
 Feelings students have about being able to solve the problem
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, notatall reflective student statements. However, there is no doubt that student journals could be one source of information among others.
Student SelfAssessment
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 selfassessment are rewarding. Selfmonitoring and selfassessment develop when teachers show students exactly what is meant by assessment and emphasize that assessment does not necessarily imply grading. Selfmonitoring and selfassessment 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, selfassessment that students share with teachers may also offer teachers a source of valuable assessment data. Because selfassessment 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 selfconcept as mathematics learners. In fact, student selfmonitoring and selfassessment may provide teachers with assessment information that may not be available through any other assessment tool.
Portfolio Assessment
Portfolio assessment is a method by which students demonstrate their ability to do major pieces of work that are more elaborate and timeconsuming 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 processoriented approach to assessment, portfolios can link successes and failures to performance and facilitate goal setting and selfmotivated 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:
 Constructions of paper or geoboard models that represent plane figures.
 Written definitions and descriptions of plane figures.
 Identification of plane figures in the environment.
 Classifications of plane figures.
 Records of investigations, explorations, and discussions of geometry concepts.
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:
 Asking openended questions.
 Assigning reports of group projects.
 Initiating work from another subject area that involves mathematics problem solving.
 Posing problems.
 Encouraging students to include excerpts of reflections on mathematics problem solving from their daily journal.
 Challenging students to draft, revise, and prepare final versions of their work on a complex mathematics problem.
 Asking students to assemble and include in their portfolio newspaper and magazine articles featuring mathematics problems.
 Encouraging students to include papers that show their corrections of errors or misconceptions.
 Providing checklists.
There are three types of mathematics assessment portfolios: the showcase portfolio, the teacherstudent portfolio, and the teacher alternative assessment portfolio (Columba & Dolgos, 1995).
 A showcase portfolio focuses on the student's best and most representative work. The important characteristic of this portfolio is that it features what students themselves select as representative of their work.
 A teacherstudent portfolio, otherwise referred to as a “working portfolio” or a a “working folder,” is a product of collaboration between the teacher and student. Its value as an assessment tool is in the maintenance of a record of communication with the student.
 A teacher alternative assessment portfolio is used solely as an assessment tool. This focused portfolio contains scored, rated, ranked, or evaluated work and provides a holistic assessment.
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.
PerformanceBased Assessment
Performancebased 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:
 Presents students with a problem related to what they are already doing in class.
 Observes what students are doing and saying and takes anecdotal notes about students' actions that exemplify the criteria set.
 Interviews students during or after the activity.
 Asks students to write about the problem or in response to a specific question on the problem, and then collects the students' writings.
 Scores the students' work against a set of performance criteria.
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, performancebased 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.
Performancebased 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.
Summary
Summative assessment is limited to the products of students' learning. It is timeconsuming and yields little benefit to students' learning. Only by inference does such assessment provide information on students' conceptual understanding and reasoning. Therefore, ConceptRich 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 participantobservers with small groups of students, ask openended 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 selfmonitor and selfassess their learning, while encouraging them to share their assessment; how to objectively analyze homework assignments and portfolios; and how to conduct performancebased 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.