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Teacher-developed mathematics performance assessments in the context of reform-based professional development.

One of the disappointments associated with the mathematical reform movement is the increasing mismatch mismatch

1. in blood transfusions and transplantation immunology, an incompatibility between potential donor and recipient.

2. one or more nucleotides in one of the double strands in a nucleic acid molecule without complementary nucleotides in the same position on the other
 between the improvements made in curriculum and instruction and prevalent assessment modes (Firestone fire·stone  
1. A flint or pyrite used to strike a fire.

2. A fire-resistant stone, such as certain sandstones.

Noun 1.
 & Schorr Schorr (Hebrew: שור‎) is a surname and may refer to:
  • Bill Schorr
  • Daniel Schorr (b.
, 2004; Niss, 1993). Despite early calls for change in assessment practice, such as the 1989 Curriculum and Evaluation Standards (NCTM NCTM National Council of Teachers of Mathematics
NCTM Nationally Certified Teacher of Music
NCTM North Carolina Transportation Museum
NCTM National Capital Trolley Museum
NCTM Nationally Certified in Therapeutic Massage
), recent research into the teaching and learning of mathematics that has provided detiled consideration of its "socially situated nature," has not focused to the same degree on mathematics assessment (Morgan Morgan, American family of financiers and philanthropists.

Junius Spencer Morgan, 1813–90, b. West Springfield, Mass., prospered at investment banking.
, 1998). Therefore interest has increased in matching assessment methods to developments in curriculum.

There is a pressing need to assess a much wider range of abilities than has been the case heretofore, including problem posing and solving, representing, and understanding. Traditional mathematical assessment has frequently relied upon the ability of students to display behavior Display behavior is the tendency of living things to express actions or formations, it is thought, for competitive advantage. Among animals
Animals may use display behavior for different purposes including threat, courtship and direct competition for example.
 that matches their assessor's expectations rather than on any underlying understanding (Morgan, 1998). These traditional assessments communicate that mathematics is an endeavor that involves determining a quick answer using a preexisting pre·ex·ist or pre-ex·ist  
v. pre·ex·ist·ed, pre·ex·ist·ing, pre·ex·ists
To exist before (something); precede: Dinosaurs preexisted humans.

, memorized method (Bell, 1995; Clarke Clarke   , Arthur Charles Born 1917.

British writer, scientist, and underwater explorer noted for his stories of space exploration. His works include 2001: A Space Odyssey (1968).
, Clarke & Lovitt, 1990; Hancock & Kilpatrick Kilpatrick is an Irish and Scottish surname. The name refers to:

  • Ben Kilpatrick (1877–1912), American outlaw in the Old West, member of Butch Cassidy’s Wild Bunch gang
  • Carolyn C. Kilpatrick (b.
, 1993) thus failing to represent the true complexity of mathematics (Galbraith, 1993; Izard Iz´ard

n. 1. (Zool.) A variety of the chamois found in the Pyrenees.
, 1993; Wheeler, 1993). In contrast, assessment data that provide direct information about improving the learning experience increase legitimate mathematical learning that is thorough and connected (Black & William, 1998; NCTM, 1995). The measurement of de-contextualized technical skills should be replaced with measures that reflect what is known about what it means to know and do mathematics, i.e., that capture the degree of acquisition of both conceptual and procedural knowledge Procedural knowledge is the knowledge exercised in the performance of some task. See below for the specific meaning of this term in cognitive psychology and intellectual property law.  and the connections between them, and that assess the solving of worthwhile problems, the communication and justification of conjecture CONJECTURE. Conjectures are ideas or notions founded on probabilities without any demonstration of their truth. Mascardus has defined conjecture: "rationable vestigium latentis veritatis, unde nascitur opinio sapientis;" or a slight degree of credence arising from evidence too weak or too , and the representation of mathematical thinking in multiple ways (NCTM, 2000). As Ridgeway A ridgeway is a road or path that follows the highest part of the landscape. Roads and pathways
  • One of the best known ridgeways is the Ridgeway National Trail, also known as The Ridgeway Path
 (1998) states, "As an issue of policy, the implementation of standards-based curricula should always be accompanied by the implementation of standards-based assessment A standards based test is one based on the outcome-based education or performance-based education philosophy. [1] Assessment is a key part of the standards reform movement. The first part is to set new, higher standards to be expected of every student. . In fact, incremental Additional or increased growth, bulk, quantity, number, or value; enlarged.

Incremental cost is additional or increased cost of an item or service apart from its actual cost.
 change in assessment systems will foster concurrent improvement in professional and curriculum development" (p. 2). The 1989 Standards states, "As the curriculum changes, so must the tests. Tests also must change because they are one way of communicating what is important for students to know.... In this way tests can effect change" (pp. 189, 190). Both the Assessment Standards (NCTM, 1995) and the Principles and Standards (NCTM, 2000) state that assessment tasks communicate what type of mathematical knowledge and performance are valued (p. 22). Therefore, standards-based assessment complements standards-based instruction (Dunbar & Witt, 1993).

Paralleling reform in mathematics curriculum and instruction have been calls to authenticate (1) To verify (guarantee) the identity of a person or company. To ensure that the individual or organization is really who it says it is. See authentication and digital certificate.

(2) To verify (guarantee) that data has not been altered.
 student assessment in all subject areas. Terms such as "Authentic Assessment Authentic assessment is an umbrella concept that refers to the measurement of "intellectual accomplishments that are worthwhile, significant, and meaningful,"[1] as compared to multiple choice standardized tests. ," "Alternative Assessment," and "Performance Assessment" have become banners to rally focused efforts to change paradigms about the nature and purpose of assessment. According to according to
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

 McMillan (2004), a performance assessment is "... one in which the teacher observes and makes a judgment about the student's demonstration of a skill or competency COMPETENCY, evidence. The legal fitness or ability of a witness to be heard on the trial of a cause. This term is also applied to written or other evidence which may be legally given on such trial, as, depositions, letters, account-books, and the like.
 in creating a product, constructing a response, or making a presentation. They possess several important characteristics:
 1. Students perform, create, construct, or produce something
 2. Deep understanding and/or reasoning skills are assessed.
 3. They involve sustained work.
 4. They call on students to explain, justify, and defend.
 5. Performance is directly observable.
 6. They involve engaging ideas of importance and substance.
 7. There is a reliance on trained assessor's judgments for scoring.
 8. Multiple criteria and standards are pre-specified and public.
 9. There is no single correct answer.
10. The performance is grounded in real-world contexts and constraints"
    (p. 213).

Although defined in many ways, performance assessment that is designed according to the above criteria provides many benefits that are closely tied to instruction, including the integration of instruction and assessment and the tying of assessment to real-world challenges and reasoning processes, thus helping instruction target more important outcomes, providing an alternative to traditional assessment, and authenticating the assessment process (Wiggins, 1993). In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently
, performance assessments conceptualized in this manner are therefore legitimately alternative and authentic in nature. Mathematics educators have joined in calling for the use of performance assessments that incorporate the aspects identified by McMillan in mathematics as both a means to align align (līn),
v to move the teeth into their proper positions to conform to the line of occlusion.
 assessment with new reform curricula (Firestone & Schorr, 2004; Shepard, 2000) and as a means to improve the links between teaching practice and assessment (Pelegrino, Chubowksy, & Glaser, 2001).

Although performance assessments provide benefits heretofore unexperienced via more traditional assessment procedures, they are not without limitations. They are usually expensive in terms of both the amount of time required and the materials needed to administer them. In addition, the results obtained from them can be subjective, an issue of inter-rater reliability Inter-rater reliability, Inter-rater agreement, or Concordance is the degree of agreement among raters. It gives a score of how much , or consensus, there is in the ratings given by judges. . Finally, those results often provide an inadequate basis for generalizing across tasks.

Statement of Purpose

The purpose of this study was to describe the principles that guided the teaching of elementary teachers in the development and administration of reform-oriented, grade level-specific performance assessments in the context of a professional development project, and to assess the internal consistency In statistics and research, internal consistency is a measure based on the correlations between different items on the same test (or the same subscale on a larger test). It measures whether several items that propose to measure the same general construct produce similar scores.  of the ratings of student performance obtained from the teachers in the course of providing that teaching.

Assessment Development

As an integral part of a two-year professional development program, 85 elementary teachers representing virtually the entire faculties of three schools and an additional smattering of teachers from fifteen other schools--all in central Utah--were involved in the development and implementation of a performance assessment system. We reasoned that this involvement would not only proide a natural context to enhance their understanding of appropriate assessment practice, but also serve to accelerate and enhance the acquisition of fundamenal notions of mathematics education reform.

We designed the assessment creation task to call for adequate attention to the interplay in·ter·play  
Reciprocal action and reaction; interaction.

intr.v. in·ter·played, in·ter·play·ing, in·ter·plays
To act or react on each other; interact.
 among cognitive processes Cognitive processes
Thought processes (i.e., reasoning, perception, judgment, memory).

Mentioned in: Psychosocial Disorders
, content categories, and task levels (Dunbar & Witt, 1993). Teachers worked in grade level teams to create two assessments: (a) a number sense assessment, and (b) an operation sense assessment. (Two examples of assessment instructions that were developed appear in Appendices ap·pen·di·ces  
A plural of appendix.
 A and B.) These two topics were chosen because of the need to assess important mathematics (NCTM, 2000; Dunbar & Witt, 1993; Morgan, 1998). Number and operations are the cornerstone cornerstone

Ceremonial building block, dated or otherwise inscribed, usually placed in an outer wall of a building to commemorate its dedication. Often the stone is hollowed out to contain newspapers, photographs, or other documents reflecting current customs, with a view to
 of the entire mathematics curriculum internationally (NCTM, 2000; Reys & Nohda, 1994). A worthwhile mathematical task, i.e., an engaging word problem incorporating the intended mathematics (NCTM, 1991), was written for each assessment in such a way as to allow for the incorporation of varied levels of numerical numerical

expressed in numbers, i.e. Arabic numerals of 0 to 9 inclusive.

numerical nomenclature
a numerical code is used to indicate the words, or other alphabetical signals, intended.
 complexity. The number sense assessments for each grade levels of numerical complexity. The number sense assessments for each grade level are similar in that they each call for demonstrating number comprehension comprehension

Act of or capacity for grasping with the intellect. The term is most often used in connection with tests of reading skills and language abilities, though other abilities (e.g., mathematical reasoning) may also be examined.
 in a real-life context. For example, the fifth-grade assessment, based on U.S. geography, begins with a simple task:

I went on a trip to see some of the wonderful tourist attractions Noun 1. tourist attraction - a characteristic that attracts tourists
attractive feature, magnet, attractor, attracter, attraction - a characteristic that provides pleasure and attracts; "flowers are an attractor for bees"
 in the United States United States, officially United States of America, republic (2005 est. pop. 295,734,000), 3,539,227 sq mi (9,166,598 sq km), North America. The United States is the world's third largest country in population and the fourth largest country in area. , like the Grand Canyon Grand Canyon, great gorge of the Colorado River, one of the natural wonders of the world; c.1 mi (1.6 km) deep, from 4 to 18 mi (6.4–29 km) wide, and 217 mi (349 km) long, NW Ariz. , the Black Hills, and the Florida Everglads. I have flown ______ miles in my travels. What does that number mean?

This general, open-ended question A closed-ended question is a form of question, which normally can be answered with a simple "yes/no" dichotomous question, a specific simple piece of information, or a selection from multiple choices (multiple-choice question), if one excludes such non-answer responses as dodging a  (NCTM, 2000), "What does that number mean?" is a part of the number sense assessments of all grades. Its open-endedness is designed to elicit e·lic·it  
tr.v. e·lic·it·ed, e·lic·it·ing, e·lic·its
a. To bring or draw out (something latent); educe.

b. To arrive at (a truth, for example) by logic.

 an intitial umprompted response. Subsequent questions are then asked as needed as needed prn. See prn order.  to further probe the nature of the students' number sense, such as, "Can you draw a picture of that number?" "Can you represent that number in expanded form?" "How many groups of ______ are in ______?" "What number is 100 (or 10, or 1,000) less than ______" etc. A sample set of responses for a three-digit task appears in Figure 1.

The operation sense assessments for each grade level were based upon the operation most emphasized in the state curriculum for that grade: e.g., subtraction subtraction, fundamental operation of arithmetic; the inverse of addition. If a and b are real numbers (see number), then the number ab is that number (called the difference) which when added to b (the subtractor) equals  in third grade, multiplication multiplication, fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N.  in fourth grade, etc. The fifth-grade operation sense assessment, for example, begins with the following worthwhile mathematical task:


I have ______ pieces of candy candy: see confectionery.

Sweet sugar- or chocolate-based confection. The Egyptians made candy from honey (combined with figs, dates, nuts, and spices), sugar being unknown.
 that I am going to put into bags of ______. How many bags will I have?

A sample set of responses for a two-digit division task appears in Figure 2.


Note that numbers of varying numerical complexity can be inserted in the blanks depending upon the child's estimated level. These levels were based upon a "Hierarchy of Numerical Complexity" relative to number and the four operations that was also developed, which appears in Table 1.

Multiple levels allows for obtaining data that is developmental in nature (Pegg, 2003; Wilson, 1999). In order to estimate the level at which the assessments would be administered, quick inventories were designed. The number sense inventory simply calls for the reading of numerals of varying sizes and appears in Figure 3. The operation sense inventory calls for the solving of simple exercises of varying complexity in the operation associated with the grade. Since the fifth-grade curriculum focuses on divsion, the fifth grade inventory consists of various division exercises and appears in Table 2.

Note that both inventories are quite procedural in nature. It seemed logical to assume that procedural performance would be a good way to obtain a quick, rough estimate of the level at which the assessments would be administered as long as a child's instructional experience includes the development of solid connections between the learning of concepts and procedures (NCTM, 2000). If the child's performance in the initial stages of the assessment would warrant an adjustment in level, the teacher would make that adjustment by re-administering the task with numbers of greater or lesser complexity. This assumption was born out as the teachers implemented the assessments in their classrooms. Those whose instructional programs promoted conceptual-procedural connections found the inventories to produce accurate level estimates. If the teachers tended to promote less conceptual thinking Conceptual thinking is problem solving or thinking based on the cognitive process of conceptualization --is a process of independent analysis in the creative search for new ideas or solutions, which takes as its starting point that none of the accepted constraints of , the inventory results tended to provide level estimates that had to be adjusted. This latter situation served to inspire teacher change as the students' responses consistently revealed that procedural knowledge does not necessarily imply an underlying conceputal understanding. For example, one child's inventory revealed a procedural knowledge of multiplying mul·ti·ply 1  
v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies
1. To increase the amount, number, or degree of.

2. Mathematics To perform multiplication on.
 two-digit numerals by one-digit numerals, such as 12 x 3. When she was presented with a worthwhile task that incorporated 12 x 3, she had absolutely no idea as to how to solve it, let alone solve or represent it in multiple ways. In fact, this particular student could not comprehend a multiplication situation as simple as 2 x 3, without at least some minimal assessor support.

Seven criteria were selected as standards by which student performance would be judged: five analytical analytical, analytic

pertaining to or emanating from analysis.

analytical control
control of confounding by analysis of the results of a trial or test.
 criteria based upon the NCTM "Process Standards" (2000) as suggested by Dunbar & Witt (1993) and two holistic Holistic
A practice of medicine that focuses on the whole patient, and addresses the social, emotional, and spiritual needs of a patient as well as their physical treatment.

Mentioned in: Aromatherapy, Stress Reduction, Traditional Chinese Medicine
 criteria as suggested by the "Learning Principle," also part of the Principles and Standards document. (We recognize that this suggestion by Dunbar & Witt was written seven years prior to the Principles and Standards document. However, the same five fundamental processes were key components of the predecessor to the Principles and Standards, namely the Curriculum and Evaluationh Standards (NCTM, 1989). These criteria serve, as stated by Morgan (1998) "... to provide a language that teachers and students can use both to help students to display the behaviors that will lead to success in the assssment process and critically to interrogate (1) To search, sum or count records in a file. See query.

(2) To test the condition or status of a terminal or computer system.
 the assessment practices themselves" (paragraph 3). The five analytical criteria were:

1. Problem solving problem solving

Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error.
 -- accurately solving a worthwhile task using multiple strategies,

2. Communicating -- explaining problem solving strategies clearly,

3. Reasoning -- justifying those strategies in a mathematically sound manner,

4. Representing -- showing or modeling mathematical ideas in multiple ways, and

5. Connecting -- explaining the connections between strategies and/or representations.

The two holistic criteria were:

1. Conceputal -- demonstrating an overall understanding of the mathematics involved with solving the task, and

2. Procedural -- demonstrating a knowledge of the rules or algorithms The following is a list of the algorithms described in Wikipedia. See also the list of data structures, list of algorithm general topics and list of terms relating to algorithms and data structures.  involved with solving the task.

The criteria were used to design a four-point rubric RUBRIC, civil law. The title or inscription of any law or statute, because the copyists formerly drew and painted the title of laws and statutes rubro colore, in red letters. Ayl. Pand. B. 1, t. 8; Diet. do Juris. h.t.  with its scoring hierarchy based upon the degree of assessor prompting required in order for a student to experience success in the assessment. The incorporation of prompting as a factor in distinguishing rubric levels results in a blurring of the line between instruction and assessment in harmony with current assessment philosophy (McMillan, 2004). The rubric appears in Table 3.

Both assessments also included an instruction guide and suggested questions or prompts to insure Insure can mean:
  • To provide for financial or other mitigation if something goes wrong: see insurance or .
  • Or you may be looking for ensure or inshore.
 that opportunities were provided for students to express themselves verbally as well as in written form (Dunbar & Witt, 1993; Glaser, Raghaven, & Baxter, 1992), and to insure that students were invited to display behavior that addressed all analytical criteria, i.e., the Process Standards (Mewborn & Huberty, 1999). In addition, the questions associated with the number sense assessment were analyzed an·a·lyze  
tr.v. an·a·lyzed, an·a·lyz·ing, an·a·lyz·es
1. To examine methodically by separating into parts and studying their interrelations.

2. Chemistry To make a chemical analysis of.

 to insure that the key componets of number sense were addressed (NCTM, 2000; Sowder, 1992). In like manner, the operation sense questions were analyzed to insure that the key components of operation sense were addressed (NCTM, 2000). In this way we became confident that important mathematical knowledge was assessed (Dunbar & Witt, 1993; Morgan, 1998; NCTM, 2000) and were provided with evidence that the interpretations associated with the assessment possessed construct validity construct validity,
n the degree to which an experimentally-determined definition matches the theoretical definition.
 (Messick, 1989). A form for students to record their work and a teacher recording form were also developed and appear in Figures 4 and 5.

Besides its use as a guide in estimating the level of numerical complexity at which the assessments should be administered, the "Hierarchy of Numerical Complexity" was then intended to be used after assessment administration to record the actual level of numerical functioning relative to both tasks. An additional determination of level regarding place value comprehension was designed for the number sense assessment, in cases where the numerals being examined were at least two digits in size, based upon the Ross Ross , Sir Ronald 1857-1932.

British physician. He won a 1902 Nobel Prize for proving that malaria is transmitted to humans by the bite of the mosquito.
 Five-Stage Place Value Understanding Model (Ross 1990, 1999). Those levels are:

1. Interprets a numeral numeral, symbol denoting anumber. The symbol is a member of a family of marks, such as letters, figures, or words, which alone or in a group represent the members of a numeration system.  as the whole number it represents, but assigns Individuals to whom property is, will, or may be transferred by conveyance, will, Descent and Distribution, or statute; assignees.

The term assigns is often found in deeds; for example, "heirs, administrators, and assigns to denote the assignable nature of
 no meaning to individual digits

2. Recognizes place value names ("ones," "tens") but attaches no meaning to the digit A single character in a numbering system. In decimal, digits are 0 through 9. In binary, digits are 0 and 1.

digit - An employee of Digital Equipment Corporation. See also VAX, VMS, PDP-10, TOPS-10, DEChead, double DECkers, field circus.
 in those places

3. Interprets digits by their "face value," e.g. that the "2" in 25 means 2 of something but not necessarily 2 tens

4. Recognizes digits represent groups of the particular place value, e.g. that the "2" in 25 means 2 tens or 20, but the understanding is limited and performance is unreliable

5. Recognizes digits represent groups of the particular place value, e.g. that the "2" in 25 means 2 tens or 20, and the understanding is complete and performance is reliable

Additional problem-solving labels for the strategies used in the operation sense assessment were based upon the Cognitively Guided Instruction Overview
Cognitively Guided Instruction is an instructional method most often found in elementary math programs. Centered around the belief that all children come to school with informal or intuitive math knowledge, CGI involves learning with manipulatives or through the
 single digit and multidigit invented algorithms (Carpenter, Fennema, Franke, Levi, & Empson, 1999). When the assessment is administered, single digit strategies would be labeled as either Direct Modeling, Counting, Memorized Fact, or Derived Fact. Multidigit would be labeled as either Direct Modeling with Tens and Ones, Incrementing, Combining, or Compensating.


Three limitations associated with performance assessments that were discussed previously were addressed in our development efforts. First, the issue of expense with regard to materials was managed by the creation of tasks that allowed for the use of mathematics manipulatives commonly used by the teachers. The issue of expense with regard to time was managed in two ways. Upper grade teachers developed a response sheet that actually listed the task as well as prompts that paralleled many of the questions that would normally be asked in a one-on-one interview. In this way, all students could respond to the performance assessment simultneously with a record of their work being completed in written and pictorial form. The teacher would then review these records and invite students to respond to futher probing and prompting questions should the written record warrant it. A fifth-grade teacher reported being able to accomplish the assessments for his entire class of 20 in about two hours in this way.


Inasmuch as in·as·much as  
1. Because of the fact that; since.

2. To the extent that; insofar as.

inasmuch as

1. since; because

 younger children do not often possess sufficient independent writing ability to allow for simultaneous assessment administration, lower grade teachers determined to use paraprofessionals and preservice university students who were already a part of the school culture to free them up from some of their other instructional responsibilities so that they could deliver the assessments one-on-one.

A second limitation of performance assessments relates to the issue of subjectivity or lack of inter-rater reliability (Linn linn  
n. Scots
1. A waterfall.

2. A steep ravine.

[Scottish Gaelic linne, pool, waterfall.]
 & Baker, 1996). This is an issue of consistency in the scoring process that is well understood, however, and can be easily controlled with appropriate training of raters (Dunbar & Witt, 1993). Videotaped assessment administrations conducted by the instructor were shown to the teachers in order to deepend their comprehension of assessment procedures. In addition, the teachers each scored these videotaped assessments and the teachers' scores were used to promote scoring consistency. The data obtained from this video training will be discussed in further detail in the next section.

A third limitation relates to the inability of performance assessment results to provide a basis for generalizing across tasks. For example, if a fifth-grade student performed well on a division operation sense task that does not necessarily mean that he or she could perform well on all division tasks. This limitation was addressed by the consistent instruction to teachers that the performance assessments that were developed serve as only one component of a more comprehensive assessment system. That is to say, each teacher was educated in multiple forms of assessment-observations, interviews, conferences, opened-ended questions, portfolios, student self-assessment, constructed response items, selected-response items, and short-answer items (NCTM, 2000; Stenmark, 1991)-so that data obtained from the performance assessments would only partially constitute a more complete collection of information about student mathematical achievement.

Inter-rater Reliability Analysis Procedures

The instructor in the program administered the performance assessments to six different children, one in each of the grades K-5. The teachers then viewed and scored videotaped administraitons of those assessments. Following the presentation of each videotaped performance assessment, teachers recorded their scores, and then a discussion was conducted in an effort to deepen deep·en  
tr. & intr.v. deep·ened, deep·en·ing, deep·ens
To make or become deep or deeper.


to make or become deeper or more intense

Verb 1.
 understanding of the scoring criteria. The scores were subsequently analyzed using Cronbach's Alpha Cronbach's (alpha) has an important use as a measure of the reliability of a psychometric instrument. It was first named as alpha by Cronbach (1951), as he had intended to continue with further instruments. . Cronbach's Alpha is a test for a model's or survey's internal consistency and is sometimes referred to as a "scale reliability coefficient coefficient /co·ef·fi·cient/ (ko?ah-fish´int)
1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities.

" (Moffatt, 2005, p. 1). Multiple ratings of the same performance are analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development.

 to a test or survey in which several items purport To convey, imply, or profess; to have an appearance or effect.

The purport of an instrument generally refers to its facial appearance or import, as distinguished from the tenor of an instrument, which means an exact copy or duplicate.

PURPORT, pleading.
 to measure the same factor or attribute. In that sense, determining the degree of consistency between ratings compares to computing computing - computer  the degree of internal consistency (or relability) between items. One method for determining the degree of internal consistency is referred to as "split-half" in which a correlation coefficeint is computed between scores obtained from half of the items that measure an attribute and scores obtained from the other half. Cronbach's Alpha is mathematically equivalent to the average of all possible split-half estimates, although that is not exactly how it is computed (Trochim, 2005).

The first analysis involved computing the alpha for all ratings for all assessments in each of the four cohorts of teachers, as well as an overall alpha for all cohorts combined. As shown in Table 4, an extremely high degree of consistency existed among raters.

A second analysis was performed in which the degree of consistency among all raters for each of the two assssments in each grade level was computed, the results of which are displayed in Table 5. It is not surprising that somewhat lower alpha coefficients were obtained when compared to those in Table 4 because only ten percent of the ratings used for obtaining the coefficients in Table 4 were used to compute To perform mathematical operations or general computer processing. For an explanation of "The 3 C's," or how the computer processes data, see computer.  each of the coefficients in Table 5. With that statistical fact kept in mind, they do indicate a comfortable degree of reliability overall.

We also intended that similar statistical analyses would be conducted in order to determine the degree of consistency among level determinations-both numerical complexity levels as well as place value comprehension levels and problem solving labels. However, there was complete agreement among all raters with regard to these level and label determinations which rendered a more sophisticated statistical analysis of no value. It is clear that the overall statistical analyses reveal a very high rate of correlation among teachers' ratings, with one caveat. Cronach's alpha ignores differences associated with rater rat·er  
1. One that rates, especially one that establishes a rating.

2. One having an indicated rank or rating. Often used in combination: a third-rater; a first-rater. 
 means due to generosity Generosity
See also Aid, Organizational; Kindness.

Abbé Constantin

self-sacrificing priest; curé of Longueral. [Fr. Lit.: The Abbé Constantin, Walsh Modern, 105]


takes interest in Paul. [Br. Lit.
 or leniency le·ni·en·cy  
n. pl. le·ni·en·cies
1. The condition or quality of being lenient. See Synonyms at mercy.

2. A lenient act.

Noun 1.
 errors. We intend to conduct more thorough statistical analsyses using Many-Facets Rasch Modeling Rasch models are used for analysing data from assessments to measure things such as abilities, attitudes, and personality traits. For example, they may be used to estimate a student's reading ability from answers to questions on a reading assessment, or the extremity of a person's  (Linacre, 1989, 2003) in order to further investigate the presence of such errors.


We learned that the incorporation of thorough, extended work in performance assessmnt provides viable support for professional development in reform pedagogy. Our efforts also appear to be founded upon sound psychometric psy·cho·met·rics  
n. (used with a sing. verb)
The branch of psychology that deals with the design, administration, and interpretation of quantitative tests for the measurement of psychological variables such as intelligence, aptitude, and
 principles as demonstrated by the high degree of correlation among teacher ratings. If the ideals of the mathematics reform movement are to achieve widespread adherence adherence /ad·her·ence/ (ad-her´ens) the act or condition of sticking to something.

immune adherence
, then there must be a synchrony synchrony /syn·chro·ny/ (-krah-ne) the occurrence of two events simultaneously or with a fixed time interval between them.

atrioventricular (AV) synchrony
 of improvement efforts in the areas of curriculum, instruction, and assessment (Morgan, 1998). Improvements in each of those areas will have concurrent efforts for the other two. Educating teachers in the designing and implementing of performance assessments provides a natural context in which reform-based assessment philosophy and research can be fostered.


Bell, K. N. (1995). How assessment impacts attitudes toward mathematics held by prospective elementary teachers. (Doctoral dissertation dis·ser·ta·tion  
A lengthy, formal treatise, especially one written by a candidate for the doctoral degree at a university; a thesis.


, Boston Univeristy, 1995). Dissertation Abstracts International, 56, 09-A.

Black, P. & William, D. (1998). Assessment and classroom learning, Assessment in Education, 5, 7-74.

Carpenter, T. P., Fennema, E., Franke, M., Levi, L. & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, N.H.: Heinemann.

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Appendix A

5th Grade Level Number Sense Instructions:

1. Administer the Number Sense Inventory to the child in order to estimate the level at which you should present your worthwhile mathematical task.

2. Present the worthwhile mathematical task at an estimated level of number complexity. Encourage the student to solve the task in any way she/he chooses. Provide manipulatives and invite the student to record responses on the Student Recording Form as needed.

I went on a trip to see some of the wonderful tourist attractions in the United States, like the Grand Canyon, the Black Hills, and the Florida Everglades. I have flown ______ miles in my travels. What does that number mean?

3. Ask additional questions as needed to prompt and probe student thinking.

Clarifying Questions:

1. How many groups of ______ are in ______? (P.S.)

2. How many ones (tens, hundreds, etc.) are in ______? (P.S.)

3. What is one (ten, hundred, etc.) less than ______? (P.S.)

4. Group this number another way. (P.S.)

5. Is the number ______ higher or lower than this number? (P.S.)

6. What is the place and the value of the digit ______ in this number? (P.S.)

7. Is this number big or small? What about compared to ______. (P.S.)

8. How do you know that ... (referring to above questions)? (Communicating)

9. Why do you think that ... (referring to the above question)? (Reasoning)

10. Show this number in another way. (pictures, manipulatives, numerals, expanded form) (Reasoning)

11. How does this picture (or manipulatives) match with these numbers. (Connecting)

4. Adjust the complexity of the number involved in the task if necessary at any time during the assessment--up or down.

5. Score and record the level according to the Hierarchy of Numerical Complexity.

6. Score and record the number comprehension level according to the following hierarchy:

1: Interprets a 2-digit numeral as the whole number it represents, but assigns no meaning to individual digits

2: Recognizes place value names ("ones," "tens") but attaches no meaning to the digit in those places

3: Interprets digits by their "face value," e.g. that the "2" in 25 means 2 of something but not necessarily 2 tens

4: Recognizes digits represent groups of the particular place value, e.g. that the "2" in 25 means 2 tens or 20, but the understanding is limited and performance is unreliable

5: Recognizes digits represent groups of the particular place value, e.g. that the "2" in 25 means 2 tens or 20, and the understanding is complete and performance is reliable

7. Score the overall performance according to the rubric.

Appendix B

Fifth-grade Whole Number Operation Sense Instructions:

1. Administer the Operation Sense Inventory to entire class to determine the level at which you should present your worthwhile mathematical task.

2. Present the worthwhile mathematical task at an estimated level of number complexity. Encourage the student to solve the task in any way she/he chooses and to use the Student Recording Form as needed.

Worthwhile mathematical task:

There are ______ pieces of candy. We need to put them into ______ bags. How many pieces of candy will be in each bag? (Real-life application) (Partitive par·ti·tive  
1. Dividing or serving to divide something into parts; marked by division.

2. Grammar Indicating a part as distinct from a whole, as some of the coffee in the sentence

3. Ask additional questions as needed to prompt and probe student thinking and communication

a. What type of a problem is this? Which operation would you use to find the answer? (Operation Sense) (Problem Solving)

b. Explain how you solved the problem. (Communication)

c. Solve the problem in a different way. Explain or show me. (Relationship between operations) (Connecting)

d. What would happen to the numbers in the question if you multiplied mul·ti·ply 1  
v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies
1. To increase the amount, number, or degree of.

2. Mathematics To perform multiplication on.
 them? (If student multiplied, ask what would happen to the numbers if they divided them.) (Relative effects of operations) (Reasoning)

e. I have ______ pieces of candy that I am going to put into bags of ______. How many bags will I have? (Measurement) (Multiple definitions of operations)(Real-life application)

f. Show this problem as a fraction. (Representing)

g. Solve this problem using pictures, manipulatives, etc. (Connecting)

4. Adjust the complexity of the number involved in the task if necessary at any time during the assessment--up or down.

5. Score and record the numerical level according to the Hierarchy of Numerical Complexity.

6. Score and record the problem solving level according to the following list:

1. Direct Modeling

2. Counting Strategy

3. Memorized Fact

4. Derived Fact

5. Direct modeling with ones and tens

6. Incrementing

7. Combining with ones and tens

8. Compensating

9. Standard Algorithm algorithm (ăl`gərĭth'əm) or algorism (–rĭz'əm) [for Al-Khowarizmi], a clearly defined procedure for obtaining the solution to a general type of problem, often numerical.  

7. Score their overall performance according to the rubric.

Damon L. Bahr

Brigham Young University Brigham Young University, at Provo, Utah; Latter-Day Saints; coeducational; opened as an academy in 1875 and became a university in 1903. It is noted for its law and business schools.  

Richard R Sudweeks

Brigham Young University
Table 1. Hierarchy of Numerical Complexity

Level  Number Sense       Addition            Subtraction

A      Rote counting      Joining sets        Separating sets
B      One-to-one         Single digit        1 digit - 1 digit = 1
       correspondence     addends & sum       digit 5 - 3 = 2
                          3 + 2 = 5
C      Single digit < 5   Single digit        2 digits - 1 digit = 1
                          addends & double    digit (decomposing)
                          digit sum           13 - 5 = 8
                          3 + 9 = 12
D      Single digit > 5   Multiple single     2 digits - 1 digit = 2
                          digit addends       digits (no decomposing)
                          3 + 2 + 4 = 9       27 - 5 = 22
E      2 digit > 15 < 20  2 digits + 2
                          digits no
                          32 + 24 = 56
F      2 digit > 9 < 16   2 digits + 2        2 digits - 1 digit = 2
                          digits with         digits (decomposing)
                          composing           27 - 9 = 18
                          32 + 29 = 61
G      2 digit > 20       3 2-digit addends   2 digits - 2 digits = 2
                          with composing      digits (no decomposing)
                          32 + 25 + 46 = 103  36 - 24 = 12
H      3 digit            3 digits + 3        2 digits - 2 digits = 1 or
                          digits varying      2 digits (decomposing)
                          composing           32 - 18 = 14
                          391 + 467 = 858
I      3 digit, zeroes    3 3-digit addends   3 digits - 2 or 3
       in ones or tens    with composing      digits = 1,2,or 3 digits
       places             323 + 257 + 469     (decomposing involving 1
                                              zero) 406 - 178 = 228
J      4 digit            3 4-digit addends   3 or 4 digits - 2 or 3
                          with composing      digits (1 decomposing)
                          3235 + 2579 + 4696  1469 - 635
K      5 digit            4 digits + 4        4 or 5 digits - 4 or 5
                          digits varying      digits (2 alternating
                          composing           decomposes) 4628 - 1809
                          4625 + 1856
L      6 digit                                Variable digit number (2
                                              consecutive decomposes)
                                              631 - 253
M      6 digit, zeroes                        Variable digit number (3
                                              consecutive decomposes)
                                              54363 - 14581
N      7 digit                                Variable digit number
                                              (decomposing involving 2
                                              or more zeroes)
                                              4001 - 1376
O      Whole #'s &
P      Whole #'s
Q      Whole #'s &

Level  Multiplication                 Division

A      1 digit x 1 digit = 1 digit    1 digit / 1 digit = 1 digit
       2 x 3 = 6                      8 / 2 = 4
B      1 digit x 1 digit = 2 digits   2 digit / 1 digit = 1 digit
       (composing) 2 x 6 = 12         12 / 2 = 6
C      10 x single digit 10 x 3 = 30  1 or 2 digits / 1 digit = 1 digit
                                      with remainder 7 / 3 = 2 r 1
D      10 multiple x single digit =   10 multiple / 1 digit = 10
       2 digits 20 x 3 = 60           multiple (no decomposing)
                                      (60 / 2 = 30)
E      10 multiple x single digit =   3 digit 10 multiple / 1 digit (no
       3 digits (composing)           decomposing) 120 / 4 = 30
       30 x 4 = 120
F      2 digits x 1 digit (no         2 or 3 digits / 1 digit = 2 digits
       composing) 13 x 2 = 26         (no decomposing) 65 / 2 = 32 r1
G      2 digits x 1 digit             2 or 3 digit 10 multiple / 1
       (composing) 14 x 3 = 42        digit = 2 digits (decomposing)
                                      150 / 7 = 21 r3
H      10 multiple x 10 multiple = 3  2 or 3 digits / 1 digit = 2 digits
       digits (composing)             (decomposing) 78 / 3 = 26
       20 x 20 = 400
I      2 digits x 2 digits (no        3 digits / 1 digit = 3 digits
       composing) 13 x 12             432 / 2 = 216
J      2 digits x 2 digits (one       3 digits / 1 digit = 3 digits
       composing on first row)        (zero in quotient) 432 / 2 = 216
       23 x 14
K      2 digits x 2 digits (one       2 or 3 digits / 10 multiple
       composing on second row)       60 / 20 = 3
       43 x 24
L      2 digits x 2 digits (two
       composes) 23 x 65
M      2 digits x 2 digits (larger
       digits) 67 x 98

Ask the child to perform tasks A and B, then ask the child to read the
remaining numerals until she/he is not successful.

A.  How far can you count?
B.  Count these blocks (up to 10)
C.          4
D.          7
E.         18
F.         12
G.         68
H.        492
I.        709
J.      3,579
K.     24,683
L.    147,836
M.    305,284
N.  3,548,921
O.         73.9
P.        697.34
Q.         28.108

Figure 3. Number Sense Inventory

Table 2. 5th Grade Operation Sense Inventory--Whole Number Division

  8 / 2   12 / 2   7 / 3
 60 / 2  120 / 4  65 / 2
150 / 7   78 / 3  60 / 20

Table 3. Performance Assessment Rubric

Rubric level          Problem Solving  Communicating    Reasoning

4 Independent         Can solve the    Can clearly      Can clearly
  Understanding       problem in two   explain the      justify the
                      ways             problem solving  problem solving
                      independently    strategies       strategies
3 Understanding with  Can solve the    Can clearly      Can justify all
  minimal help        problem in two   explain all but  but one part of
                      ways with        one part of the  the problem
                      minimal help or  problem solving  solving
                      one way          strategies       strategies
2 Understanding with  Can solve the    Can explain      Can justify
  substantial help    problem at       portions of the  portions of the
                      least one way    problem solving  problem solving
                      with help        strategies       strategies
1 Little              Cannot solve     Cannot explain   Cannot justify
  understanding       the problem      the strategies   the strategies
                      even with help   even with help   even with help

Rubric level          Representing      Connecting         Procedural

4 Independent         Can represent     Can independently  Can solve the
  Understanding       the problem in    connect            problem using
                      at least two      representations    a procedure
                      ways              or strategies      independently
3 Understanding with  Can represent     Can connect        Can solve the
  minimal help        the problem in    representations    problem
                      two ways with     or strategies      procedurally
                      minimal help or   with minimal help  with minimal
                      one way                              help
2 Understanding with  Can represent     Can connect        Can solve the
  substantial help    some of the       representations    problem
                      problem with      or strategies      procedurally
                      help              only with          with
                                        substantial help   substantial
1 Little              Cannot represent  Cannot connect     Cannot solve
  understanding       the problem even  representations    the problem
                      with help         or strategies      procedurally
                                        even with help     even with

Rubric level          Conceptual

4 Independent         Can show thorough understanding of the problem and
  Understanding       of the associated mathematics independently
3 Understanding with  Can show some understanding with minimal help
  minimal help
2 Understanding with  Can show some understanding with substantial help
  substantial help
1 Little              Cannot show understanding even with help

Table 4. Cohort and Combined Alpha Coefficients

Cohort    N   Cronbach's Alpha

A         21  .992
B         19  .998
C         28  .993
D         17  .947
Combined  85  .978

Table 5. Grade Level Alpha Coefficients

            Cronbach's Alpha
Grade Level   Number Sense  Operation Sense

Kindergarten  .932          .865
First         .711          .633
Second        .909          .846
Third         .806          .925
Fourth        .928          .881
Fifth         .915          .910
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Date:Jan 1, 2008
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