term paper

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Secondary Education Mathematics and Performance Assessment Tasks

Melissa A. Ellis

Kutztown University

Abstract

During the research of performance assessment tasks, I looked at numerous books,

articles, and Internet sources to gather the overall consensus of what performance

assessment tasks are. PATs are a form of testing that requires students to perform a

task rather than taking a formal exam. The following are the main aspects of my

research focus: what is Performance Assessment and PATs, why use PATs,

theories and research behind PATs, pros and cons, types, suggestions and

constructing PATs, and rubrics. The reasons for researching PATs was to get a

better understanding of the overall process of creating and incorporating alternative

assessment in mathematics curriculum. This research illustrates the importance and

concerns about the importance for teachers to use PATs in their classroom.

PERFORMANCE ASSESSMENT TASKS 2

Secondary Education Mathematics and Performance Assessment Tasks

Throughout education, there has been continuous debate on what is the best method for

assessment. Assessment serves a variety of purposes in today's classroom (Appalachia

Educational Lab [AEL], & Virginia Education Association [VEA], 1992). For the students,

assessment aids learning and measures knowledge. For the teachers, it provides diagnosis of

student learning and data for making instructional decisions. Lastly, for the administration and

public, it indentifies the effectiveness of the program and displays the overall achievement (AEL,

& VEA, 1992). This is where the debate begins. Does either traditional or authentic assessment

fulfill the purpose of the student, teacher, administration, and public better? Authentic

assessment or performance assessment, asks students to analyze, apply, and sometimes

synthesize what they have learned (Blaz, 2008, p. 56). Performance assessment provides students

with tangible products that can serve as evidence of student achievement (Ronis, 2007, p. 22).

What is a Performance Assessment Task?

According to The Office of Technology Assessment of the U.S. Congress, performance

assessment is "any form of testing that requires a student to create an answer or a product that

demonstrates his or her knowledge or skills (Mahler & Chriest, n.d.)." In other terms, the product

is the student's outcome of a performance assessment task, and the skill is what the student

learned through classroom instruction. Performance assessment tasks (PATs) are activities that

students undertake to show what they know and can do (Ronis, 2007). A PAT can involve a

group of students, or an individual student. It is a mathematical task that can take only a half an


hour to several days to complete and solve (Darling-Hammond, 1993). These tasks are measured

by a given criterion or standards that are applied by the teacher explaining the purposes of

evaluation. With the criteria, the students' quality of performances can be evaluated. Students

performing in a given task can apply their mathematical knowledge and skills to complete a

high-level mathematical task (Posamentier, Smith, & Stepelman).

All performance assessment tasks in mathematics classrooms' should be correlated with

the goals, objectives, and content of the curriculum (Posamentier et al.). During a PAT, students

are provided with little intervention or instructions from the teacher. This is why the task should

be understood by the student and a concept they can achieve. The task should also be important,

realistic, and have merit. Students should be able to display mathematics as a process allowing

them to show their conceptual understanding and thinking within a mathematical problem

(Posamentier et al.). A PAT is effective, in the way that it promotes motivation, critical thinking,

and is related to real-life situations. The best task a student can be offered is one that involves

open-ended tasks, more than one methodology, and other mathematical extensions and questions

(Posamentier et al.).

Assessment tasks can come in many forms. Students can be presented with a problem

related to what they are already doing in class and listening to the responses (Darling-Hammond,

1993). The evaluation can also be completed by observing what the students do and say while

working on a performance task. Another assessment is interviewing the given student during and

after a task. Overall, PATs in mathematics classrooms teach students real-life skills, when they

will use mathematical concepts, and the importance of quality (Blaz, 2008).


Why Should Secondary Mathematic Classrooms Use PATs?

Unlike traditional assessment, performance assessment tasks require students to actively

demonstrate what they know (Sweet & Zimmermann, 1992). According to Education Research

Consumer Guide, performance assessment can be a more valid indicator of students' knowledge

and abilities. They will also provide improvement in instruction and increase in students'

understanding of what they need to know and be able to do (Sweet & Zimmermann, 1992).

Traditional assessment measures fundamental facts and concepts in mathematics. However,

PATs require students to use the fundamental facts and concepts learned during instruction.

PATs measure the methods, procedures, and analysis skills in mathematics (Slater & Ryan,

1993).

The most important factor of performance assessment tasks is the presence of

scaffolding. Scaffolding in PATs provides the following: clear directions, motivation and

purpose, students staying on task, students using time efficiently, clear expectations based on the

rubric, and no surprises or disappointments. The presence of scaffolding in a PAT takes the form

of narrowing down the topic and incorporating some creativity for the students (Blaz, 2008).

Another four important purposes of performance assessment tasks are keeping track, checking

up, finding out, and summing up (Maurer, 1996). As a teacher, it is important to gather

information about a student to see what the student has accomplished during the school year.

This is a form of monitoring or keeping track in that the PAT is designed to provide the teacher

with accountable information (Maurer, 1996). Checking-up is the formative function of the PAT.

It provides the teacher with information as to where the student is in achieving their


understanding about the curriculum. Depending on the students' understanding, the teacher uses

this information to change the focus of the instructional program. The reason for the change in

focus is to ensure that the students are gaining and understanding what is needed in the

curriculum. The third purpose for performance assessment tasks is to find out and discover how

the student has built meaning from the instruction. This is where the teacher differentiates the

curriculum to meet the needs of every student. The reason for differentiated curriculum is simple,

each student takes new knowledge and relates it to old knowledge or experiences differently

(Maurer, 1996). Summing up is provide judgment information to the student, parent, or

community. This is known as feedback from the teacher. The report of a PAT is the

accountability purpose. It is the final word on whether a student has been able to demonstrate

understanding (Maurer, 1996).

The National Council of Teachers of Mathematics (NCTM) addresses the Assessment

Principle. The book Principles and Standards for School Mathematics, states that:

Assessment should be more than merely a test at the end of instruction to see how

students perform under special conditions; rather, it should be an integral part of

instructions. Assessment should not merely be done to students; rather, it should also be

done for students, to guide and enhance their learning.... It is important that assessment

tasks be worthy of students' time and attention. Activities that are consistent with (and

sometimes the same as) the activities used in instruction should be included. When

teachers use assessment techniques such as observations, conversations, and interviews

with students, or interactive journals, students are likely to learn through the process of

articulating their ideas and answering the teacher's questions (The National Council of

Teachers of Mathematics [NCTM], pp. 22-24).


NCTM addresses the purpose of performance assessment tasks, stating that students learn to

articulate their ideas and answer teacher's questions. Therefore, by incorporating effective PATs

students will learn how to think and be more interactive during classroom instruction.

What Theory and Research Has to Say

According to the American Association for the Advancement of Science in Washington,

DC:

Studies that have looked closely at performance assessments find that, if the criteria is

clear and that examples are available to show levels of competency, performance

assessments are highly consistent across different evaluators (Kulm & Malcom, 1991).

Moreover, clear indication of what is expected of students improves student performance.

There are some indications at the K-12 levels that students perform inconsistently from

one PAT to the next (Shavelson, Baxter, & Pine, 4(4): 347). This suggests that students’

grades will be most reliably determined from a number of Performance Assessment with

other forms of assessment.

This excerpt shows that performance assessment tasks are effective when used with other forms

of assessment in curriculum. If there is any inconsistency in performance, the PAT should be

reevaluated.

Cognitive Learning Theory (CLT) defines knowledge as being constructed, while

learning is a process of creating personal meaning from new information and prior knowledge

(Maher & Chriest, n.d.). CLT shows assessment implies divergent thinking, multiple links and

solution, critical thinking skills, prior knowledge, and applying information to new situations

(Maher & Chriest, n.d.). When students work on PATs, they are using critical thinking skills,

using prior knowledge, building on their knowledge, finding multiple solutions, and using

application. Cognitive Learning Theory and performance assessment both mention the


importance of the social components in group work, which is, taking a variety of roles, and

considering group products and processes. CLT states that people perform better when they

know the goal, see models, and know how their performance compares to the standard (Maher &

Chriest, n.d.). PATs emphasize what the goal is and what the quality of the work should be. Both

CLT and PATs, address a variety in learning styles, attention spans, memory, developmental

paces, and intelligences (Maher & Chriest, n.d.). For certain PATs, students have the choice of

how to show mastery of a concept. Students all learn at different paces. PATs give the students

the opportunity to work at their pace, use personal experiences, attention span, memory, and

learning styles in their individual task. Furthermore, PATs incorporates Cognitive Learning

Theory research by giving students real-world opportunities to apply and adapt new knowledge,

self-evaluate, and engage in problem solving a nonlinear progression of discrete skills (Maher &

Chriest, n.d.).

Well-designed assessment tasks not only assess student understanding but teach concepts

and require students to explain and communicate their solutions (Morrison, McDuffie, &

Akerson, 2002). American Educational Researchers, Fuchs, Karns, and Katzaroff , found that

students in performance assessment-driven instruction classes demonstrated stronger problem

solving skills than comparison groups that were not performance assessment-driven (Morrison et

al., 2002). This corresponds with research by the Office of Educational Research and

Improvement. They researched and stated:

Research suggests that learning how and where information can be applied, should be a

central part of all curricular areas. Also, students exhibit greater interest and levels of

learning when they are required to organize facts around major concepts and actively


construct their own understanding of the concepts in a rich variety of contexts (Sweet &

Zimmermann, 1992).

They also observed that performance assessments created by teachers familiar with particular

state, district, or school curriculum, allowed the curriculum to "drive" the test, rather than be

weighted down by testing requirements that disrupt instruction (Sweet & Zimmermann, 1992).

The Pros and Cons of PATs

Performance assessment tasks provide students and teachers with countless advantages

when talking about assessing and learning. PATs emphasize having multiple correct answers and

creative solutions, ongoing throughout the unit of study, and provide an accurate picture of

student achievement (Ronis, 2007). Diane Ronis, author and professional educator states that

PATs allow teachers to develop meaningful curricula, create an environment where each child

has the opportunity to succeed, and provides multiple sources of evaluation that give an in-depth

view of student progress (Ronis, 2007). Meanwhile, Measurement and Assessment in Teaching

expresses PATs clearly communicate instructional goals that involve complex performances in

natural settings in and outside of school, plus, provide instructional targets (Miller, Linn, &

Gronlund). The book also addresses that performance assessment provides freedom to respond in

own one's own way, which, enables students to display originality (Miller et al.). This minimizes

students guessing. When grading, performance assessment encourages teacher feedback verses

ranking students' performance (Blaz, 2008). Overall, students are more accountable for their

own work, skills, knowledge they acquire, and self-reflection on their growth of performance by

actively participating in PATs (Blaz, 2008).


On the other hand, PATs address fewer learning objectives than other forms of

assessment and are typically inappropriate for measuring student knowledge of facts (Maher &

Chriest). In addition to objectives, PATs are more difficult to achieve consistent scoring results,

and data simplified to a single number (Ronis, 2007). Most assessment tasks require pre-

assessment activities, which prepare students for the task. This takes too much time (Maurer,

1996).

A major downfall for performance assessment tasks is the bias that still emerges in

grading. Bias and reliability of ratings raises a lot of issue with professionals. Teachers must pay

careful attention to the learning outcomes that the task is intended to assess and to the scoring

rubrics that will be used in rating the performances (Miller, et al.). Going along with the

professional aspect of PATs, it is difficult to determine comparability between states. Each state

implements his or her own assessment tasks and interprets them differently (Maurer, 1996).

Overall, one of the main concerns with teachers is time. Performance assessments require

a greater expense of time, planning, and thought from students and teachers. The teachers who

use PATs must also pay close attention to technical and equity issues to ensure the assessments

are fair to all students (Sweet & Zimmermann, 1992). Students also spend an enormous amount

of time creating, for example, portfolios. Some professionals point out that most colleges and

universities still use test scores and grades for admission, so portfolios would need to somehow

translate into a grade (Blaz, 2008).

Types of Performance Assessment Tasks


When deciding on the best type of performance assessment task, as a teacher, there are a

few decisions. Overall, there are two paths to take when creating or choosing a PAT: restricted-

response performance task or extended-response performance task. The freedom provided by

extended-response performance tasks enable students to display such important skills as problem

solving, planning, organization, integration, and creativity (Miller et al.). This type of PAT has

multiple goals that might be assigned at the beginning of a term or unit of study and used long

term (Ronis, 2007). On the other hand, restricted-response performance tasks are generally easier

to score and require less time than extended-response performance tasks. However, extended-

response performance tasks are better suited when measuring higher-order skills (Miller et al.).

One of the most comprehensive means of evaluating a learner's growth and progress

using a PAT is the use of portfolio creations. NCTM states that portfolios use problem solving,

communication, reasoning, and connections. They also are very specific in that they collect

information at the course level about the program standards (Maurer, 1996). Portfolios are

organized, a purposeful collection of documents, artifacts, records of achievement, and

reflections (Ronis, 2007). Artifacts are work students complete throughout the course. Whenever

a teacher leaves comments or observations on a test, paper, or assignment, the comments are

creating records of achievement. The most important purpose of a portfolio is to measure the

students' progress. However, the purpose is not for only the teacher to evaluate progress.

Reflections are the student's personal thoughts on his or her own learning throughout the course

(Ronis, 2007). This provides students with motivation, explicit examples, and engagement in

their own education. The reflections can include descriptions about how they feel about


mathematics and/or an autobiography of their experience with mathematics. Portfolios also have

goal-setting sheets, journal entries, standardized tests or information on such tests, students'

choice of materials, projects, diagrams, pictures, reports, real-world applications, and problems

created by the students (Blaz, 2008). In creating portfolios and using self-monitoring tools,

teachers and students are working together to monitor progress more effectively. Therefore, as

students reread their reflections or journal entries, they may see that what was once confusing or

frustrating is now trivial. It also, empowers students next time they struggle to understand new

concepts and acquire new skills (Posamentier et al.).

How to create a Performance Assessment Task

When first creating a performance assessment task, start with an idea and carefully

construct the learning goals for the instructional unit (Slater & Ryan, 1993). Then decide if the

idea supports learning and assessment for these goals. Once the goals are created, clearly define

the knowledge and skills students need to apply. The students will also have to demonstrate these

skills in solving a problem (Slater & Ryan, 1993). Then determine criteria or standards at which

students will be judged. This will indicate the levels of competence (Slater & Ryan, 1993). Also,

determine which components of the task need to be analyzed. Make sure to design a task that can

have multiple entries, solutions, and exit points.

To determine all of the criteria and standards of a PAT, it is necessary to establish the

purpose of the PAT. There are three types of assessment purposes, which include diagnostic

purposes, instructional purposes, and monitoring purposes. Diagnostic purposes include what

students know about how to solve certain types of problems (Maher & Chriest, n.d.). Do students


know how to evaluate findings or know how to use information learned (Maher & Chriest, n.d.)?

Instructional purposes are often indistinguishable from a learning activity. Lastly, monitoring

purposes are to judge the level of competency students achieved in doing mathematics.

Before completing the construction of a PAT, determine whether the criterion reflects the

most valued elements of student performance. This can be done by testing the idea, converting

the idea into a prompt, considering embellishments, consider what a student will need to know,

test the prompt out, and revise according to your experience (Darling-Hammond, 1993). After

revisiting the PAT, you will be able to determine if it emphasizes depth of knowledge and gain

mastery of skills (Maurer, 1996). After distributing the PAT, either directly observe students or

have students complete a structured student-answer sheet to evaluate various components of the

task (Slater & Ryan, 1993). In form of a rubric, teachers can match student performance with

criteria and determine which level most closely matches their performance. Make sure to also

provide written feedback of their performance.

It is important that there in a clear, logical set of activities that students are expected to

follow. Students must be aware of the expectations before starting on the performance

assessment task. The instructions must be specific, simple, not containing any complex sentence

structures, or difficult vocabulary. This will help make sure students have the prerequisite skills

and abilities needed to make the response (Miller et al.). In creating instruction, make sure to

avoid any racial, ethnic, or gender bias. Furthermore, it is important to create instructions and a

task that are in the appropriate reading level of all students (Miller et al.).


Other steps professionals should following while creating PATs are to make sure the task

is a meaningful task in context. What real issues, problems, themes, and student interests can

help determine this context (Maher & Chriest, n.d.)? When creating a PAT, determine how

much the task will allow for student choice. Other choices to consider are: will there be student

choice in using outside resources, choice in working individually, with a partner, or in a group

(Maher & Chriest, n.d.). After creating the performance assessment task, it is important to not

only test out the task, it is important to create multiple responses. When creating multiple

responses, make sure each one shows a different performance level. They should follow the

rubric that you created. This can minimize bias and inconsistent grading.

How to create a rubric

In 1995, the NCTM released a document called Assessment Standards for School

Mathematics, to help reform traditional thinking about assessment in mathematics. It stated with

the understanding that:

All students are capable of learning mathematics, and highlighted six assessment

standards: assessment should reflect the mathematics that students need to know and be

able to do, enhance mathematics learning, promote equity, be an open process, promote

valid inferences about mathematics, and be a coherent process.... The key to these

standards is that assessment be used not to separate or rank students as examinations

often do, but as a tool to improve instruction as well as report the results of that

instruction (Posamentier, et al., pp. 159-163).

In doing so, the decision on the best way to improve instruction and report the results of

instruction is done by deciding what is the best use of rubric.


Rubrics usually fall between one of two categories: analytic and holistic (Posamentier et

al.). A rubric that breaks the task into specific assessment categories and evaluated on each

separately is called an analytic rubric. In mathematics, teachers could grade students on their

conceptual understanding, mathematical reasoning and problem-solving strategies, and their

communication. For example, a student could receive a "Satisfactory" rating in problem-solving

strategies but an "Unsatisfactory" in communication (Posamentier, et al.). On the other hand, a

holistic rubric considers the product as a whole. In this case, students could be graded on a five

point scale (zero through four).

In creating the goals and objectives in the rubric, the teacher may decide on concepts,

mathematical reasoning, and communication skills. In this analytic rubric, the teacher identifies

each category understanding. For example, the description for communication as unsatisfactory

is identified as "Usage of mathematical terms is incorrect.” The description for satisfactory is,

"mathematical terminology and notation is used effectively (Posamentier, et al.)." This step is

describing the levels of performance for each criterion. It is important to organize your rubric

and determine whether it is an analytic or a holistic model. Teachers may want to create rubrics

that reflect the model used in high stakes assessments to better prepare their students for success

on those types of tests (Posamentier, et al.). Then test the rubric that was created. There may be

flaws which need revisions. In creating an effective PATs, it is crucial that you create different

levels of student work in anticipating common student errors and use the rubric to assess the

examples. Finally, be able to correctly categorize the errors of students' work.


Depending on the PATs, it may be more necessary to create number-based rubrics on

performance. For example, here is a generic rubric that represents a scale from one to six, with

six being the highest:

6 - Demonstrates conceptual understanding; is complete and goes beyond what is

expected; presents clear rationale; presents specific, relevant details as evidence;

represents exemplary achievement.

5- Demonstrates understanding; is complete; presents rationale; presents supporting

evidence; represents commendable achievement

4- Demonstrates some understanding, is fairly complete, presents a somewhat flawed

rationale, presents supporting evidence that lacks detail, represents adequate achievement

3- Attempts to show understanding but is unclear; is incomplete; presents a flawed

rationale; lacks supporting details; represents some evidence of achievement.

2- Demonstrates obvious misconceptions; is sorely incomplete; presents no evidence or

rationale; represents no examples; represents limited evidence of achievement

1- Demonstrates no understanding; shows no real attempt; presents a restatement of the

question; represents no evidence of achievement (Lim, 1997, p. 63).

This rubric can be applied to any topic in Mathematics by adding on the specific goals and

objectives to each level. For certain PATs, a separate rubric may be necessary for an

explanation. Here is a generic rubric that can be applied to any topic in Mathematics:

Answer: 4 - correct, 3 - almost correct or partially correct, 2 - incorrect but reasonable

attempt, 1 - incorrect with no relationship to the problem, and 0 - no answer

Explanation: 4 - complete, clear, logical, 3 - essentially correct but incomplete or not

entirely clear, 2 - vague or unclear but with redeeming features, 1 - irrelevant, incorrect,

or no explanation (Miller et al., p. 272).

This rubric shows that a teacher can grade the student on his or her final product (answers to

problems) and their explanation throughout the process of the task.

The next type of rubric can be organized in a way that the teacher is assessing the

students individually and as a whole (group). The following are categories for a generic rubric

that can be integrated to mathematics. These assessments are done by observations of the teacher


during and post activity. The categories are: student learning styles (do the individuals), student

ideas (do the individuals), communication (do the students), cooperation (do the groups),

manipulative (individually or within group, do the students) (Darling-Hammond, 1993). Each of

these categories signifies whether it is assessing the student individually or all together. To

assess individual students on learning styles, the rubric can include the following questions: Do

the individuals consistently work alone or with others, try to help others, in what ways, succeed

in asking for and getting needed help, and from whom, stick to the task or become easily

distracted, and become actively involved in the problem (Darling-Hammond, 1993)? All of these

questions evaluate the students’ progress within the group.

Other possible questions a teacher can have to evaluate the individual student within a

group are: Do the individuals try to explain their organizational and mathematical ideas, their

arguments with evidence, consider seriously and use the suggestions and ideas of others, attempt

to convince others that their own thinking is best (Darling-Hammond, 1993)? The majority of

questions a teacher can use to evaluate a students' performance in communication can be done by

observing the students in the classroom. The questions consist of whether or not the student is

staying on task, communicating with group members, making reports to the whole class, making

a group consensus as well as their own, and synthesizing and summarizing their own or group's

thinking. The teacher may also grade the students on their cooperation as a group. These

questions include: whether they divided work among the members, agree on a plan or structure

for completing the task, taking time to ensure they all understand the task, use time in a

productive way, provide support for each member, think about recording, and allow for


development of leadership. It is important to also evaluate if individuals and the group choose

and use appropriate manipulatives, share concrete objects, and appear not to need the actual

objects to be able to visualize within themselves.

No matter the method for creating a rubric, a teacher should help students understand

rubrics when giving assignments, so that students become familiar with and try to accomplish

what is required of them (Posamentier, et al.).

Reflection

After researching performance assessment tasks, I have concluded that they are a great

method for assessing students' full understanding of classroom instruction as long as the tasks

follow the objects and standards. In my opinion, the most effective way to incorporate PATs is as

a summative assessment. Allowing students to inductively work on a task after learning a few

lessons or the whole unit best measures the students' conceptual understanding. The length of my

PATs would vary throughout the course, but the best length to assess students understanding

after a few lessons is a one-to-two day PAT. This allows for students to work with classmates or

individually. It also allows for the elimination of classroom instruction, which, I feel is good for

student motivation. By observing students in the classroom, I gathered that they enjoy taking

control of the class by working on projects or tasks. Students are motivated by real-life

application and incorporating real-life application to every PAT, I feel students' conceptual and

critical thinking skills will evolve more effectively. PATs also provide feedback for the students

and the teacher. After administering and collecting a PAT from the students, I will look for


concepts that the class had difficulty with. Before giving a formal exam, I would have the chance

to go over any concept with the class as a whole before giving a unit exam. Overall, my research

topic was insightful. I became aware of the downfalls and how to construct a PAT that avoids

limitations.

References

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1, 2011 from the Prince George's County Public Schools at

http://www.pgcps.org/~elc/developingtasks.html

Kulm, G. & Malcom, S. M. (1991) Science Assessment in the Service of Reform. American

Association for the Advancement of Science, Washington, DC.

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Upper Saddle River, NJ: Pearson.

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Assessment. Washington, DC: Office of Educational Research and Improvement.

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Implementing Science and Mathematics Performance Assessment Tasks. Retrieved

November 29, 2011 from ERIC.

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Performance Tasks, Exhibitions, and Documentation. Retrieved November 29, 2011

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