Central Bucks Schools Teaching Authentic Mathematics in the 21st Century

Once authentic intellectual work is the focus, performance-based assessment of the authentic intellectual work can follow. Performance-based assessments can call for either low or high levels of authentic intellectual work; thus, to promote AIW through assessments and also through daily lessons, both student and teacher assessments must reflect Newmann's three criteria.

First, it is important to understand that performance-based assessment is "a direct, systematic observation of actual student and/or teacher performances according to pre-established performance criteria" (Butler, 2001, p.12). So, the quality of the assessment provides not only the expectations for performance to the students and/or teachers in advance, but also provides a tool for ongoing feedback (p. 6). Thus, the performance-based assessment becomes authentic when it allows the assessed to know which expectations must be met, and it provides feedback on growth, which can drive instruction and assessment (p. 6). Authentic performance-based assessment reflects evaluation in the real world.

Recall the earlier example of an assessment activity: the area exercise worksheet. The activity does not challenge students to construct new knowledge, embark on a disciplined inquiry, or produce a product or presentation with value in real life. However, revising the activity and the subsequent performance to include new applications connected to the real world increases the value, but does it meet the standards for authentic intellectual work? Consider the following revision:

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  Consider how much carpet would be needed to cover a room which is ten feet by twelve feet.
  
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  Explain in a paragraph how the area calculations applied to the carpeting of a room have a real-life application.
  

The standard of authentic student work must include an in-depth inquiry using the basic knowledge and elaborated communication which has some value to the student or others beyond showing success in school, the two most important criteria among Newmann's (2001) standards (2-3). Consider the following revision:

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  You will be purchasing carpet for your bedroom in your dream house. You will need to communicate your first choice and an alternative to the builder.
  
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  You must follow these guidelines:
  
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    You have a flooring budget of $1500.00 for the room.
    
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    You must use the weekly flyers and your own room measurements to select the first and alternate choices for the builder.
    
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    You will write a memo to the builder with your choices and a scaled drawing of the room.
    
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  Be prepared to share your work with the class.
  

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  First, students construct new knowledge by creating a dream house floor plan for the room (original application) and using the measurements of their dream bedroom, the formula for area, the price of desired floor coverings for the room, and the budget to calculate the area of a space and the prices for the flooring options for the room.
  
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  Then, students participate in a sustained in-depth inquiry as they compare the costs of the flooring to the budget, make alternatives to both selections and room size, and repeat calculations to meet the budget restrictions and evaluate/select alternatives.
  
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  Finally, students elaborately communicate their work in writing and in visual representations in a memo to the builder, as well as to other students as they share their work with the whole group to look for purchasing patterns.
  

To promote AIW in the classroom, performance-based assessment tasks need to "elicit demonstrations of knowledge and skills in ways that resemble 'real life' as closely as possible" (6). If the instruction is authentic, the assessment tasks are authentic and the student work is authentic, authentic student achievement will be reached because students invest in their work and learning and see the value of the work and learning (Avery, 2000, p.1).

If teachers are not striving to meet the standards for student work evaluation in regards to AIW, then the lessons "do not allow students to use their minds and the work has no intrinsic value beyond the school setting" Newmann, F. M., Secada, W. G., & Wehlage, G. G. (1995). When observing a teacher, or assessing one's self, four standards of authentic teacher instruction to promote AIW can be evaluated in assignments, activities, assessments, and classroom structure (p. 2):

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  Higher order thinking: Is the instruction dependent on recall and recitation, or does the instruction require organization and manipulation of information to discover meaning?
  
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  Depth of knowledge: Does the instructional goal move beyond a trivial understanding and progress to a more significant and complex understanding?
  
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  Connectedness to the world: Is the instructional application dependent on school-related connections, or is the application connected to an adult real-life experience?
  
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  Substantive conversations: Does the instruction rely on teacher-directed answer sessions, or does the instruction allow for student inquiries and ideas connecting learning?
  

Teachers also have other methods to assess performance; the first is questioning. In questioning, the prompt is often teacher-directed; however, the type of performance is varied: oral or written, one or more answers, interviews, themes, surveys, and more (Butler, 2001, p. 13). The goal of questioning is for teachers to assess where a student is in terms of concept attainment and what needs to be explored yet in further instruction. The teacher can provide feedback, modify upcoming instruction to meet curriculum standards and benchmarks, and can evaluate AIW teaching strategies. In order for questioning to be valuable in promoting AIW, it is essential that questions be open-ended as opposed to closed-ended, have multiple responses, and have one correct response. The goal in questioning is to support and encourage the construction of new knowledge and in-depth inquiry. For example, in a statistics class, open-ended questions can prompt students to consider the data or logic in a new way: To what degree is the change in value A related to historic movements in value B? How has value A been affected by value D? What are the characteristics of value A which limit its relationship to other values?

Similarly, constructed responses are short answers or theme responses which vary in purpose. Constructed response tools include short answer sentences/paragraphs, diagrams/illustrations, graphic organizers, graphs/tables, and matrices. Generally speaking, constructed responses may be either open responses, in which the student constructs the response with a general "correct" response desired, or open-ended questions, in which a "correct" response is secondary to the synthesis of information or evaluation of solutions (Butler, 2001, pp. 14-19).

Open-response questions are used to assess basic knowledge and understanding; information is re-constructed from prior knowledge. Open-ended responses are used to demonstrate a deeper understanding of a concept and its real world connection (Butler, 2001). A teacher's open-ended questions encourage "originality, imagination, and creative thought" (pp. 14-19). Students can use graphic organizers to visually represent their thinking or to organize information to be explored. Thus, the mental map becomes representative of a student's knowledge of key skills and demonstrates the thinking processes which students are using to solve complex problems (Butler, 2001). Open-ended responses meet Newmann's three criteria for authentic intellectual work. For example, in our mathematics example, the use of a matrix to chart connective data would show key multiplication skills and the thinking process used to organize the data (construction of new knowledge), as well as the thinking process used to evaluate measurements, costs, and budget constraints to evaluate two purchase options (in-depth inquiry) in the real world (application outside of the classroom).

Products: Logs, journals, notebooks, portfolios, and projects

A final category of performance tasks represents alternative forms of assessment. Alternate forms of performance tasks are valuable in assessing how the students are completing their authentic intellectual work. In these alternate forms, the students' in-depth analysis is highlighted. Logs document evidence of events and show a timeline of the events (Butler, 2001, p. 20). They are valuable in terms of understanding student thinking as they document their participation during in-depth analysis and show the events that led to the conclusions or final product (Butler, 2001). Journals provide the same documentation to show student involvement; however, they differ in that students are able to document thoughts connected to the problem solving and higher level thinking tasks (Butler, 2001, p. 20). In this way, a teacher can provide feedback in the form of a substantive conversation, encouraging perseverance during in-depth analysis (Butler, 2001).

Notebooks function in two ways: either as a collection of tasks and key knowledge related to the course content or as documentation and reference of completed tasks (p. 20). The use of a notebook allows students, teachers, and parents to gauge the involvement of a student in his own learning and assess the pool of knowledge from which the student is drawing information to construct new knowledge. Each of the forms alone does not meet the standards for AIW, yet they can be used to document and encourage AIW.

Portfolios provide a similar paper trail of learning, but the achievement shown is growth-related or skills-related. It is crucial, prior to the inclusion of portfolios in the classroom, that the assessment purpose be defined (Butler, 2001). In a growth portfolio, representative work from stages of development are kept to show a growth in constructed knowledge, disciplined inquiry, and elaborated communication as the student grows (pp. 24-25). Conversely, a skills portfolio contains representations of best work associated with assessed skills (pp. 24-25). Although theoretically, both portfolios show concept attainment, the artifacts contained within are different. The value of the portfolio as an assessment tool rests in the purpose of the portfolio (Butler, 2001). Again, the portfolio alone does not meet the criteria for AIW but can show the products or presentations which resulted from AIW.

As discussed, authentic assessment begins with the standards of authentic evaluation of student work: construction of knowledge, disciplined inquiry, and value beyond school. As reflected in a matrix developed by Avery (2000), teachers must construct tasks which promote organization of information, consideration of alternatives and core content, thinking processes, elaborate written communication, and a problem to be solved (p. 1). Then, in classroom instruction, teachers must encourage and support higher order thinking, deep knowledge construction, substantive conversation, and connections to the real world to promote AIW (p.1). Finally, students must perform analysis, involving disciplinary content and elaborate written communication (p.1). Placed in a matrix, the strategies create a rubric to authentically assess both student and teacher achievement in terms of assessment tasks, instruction, and performance guided by the standards for authentic student achievement.

Avery, P. (2000). Authentic student performance, assessment tasks, and instruction. University of Minnesota Research/Practice Newsletter, Vol. 8 (1). Retrieved 05/06/07 from http://www.education.umn.edu/CAREI/Reports/Rpractice/Fall2000/avery.html.

Butler, S. (2001). Assessment vocabulary. Retrieved May 5^th, 2007 www.ncsu.edu/sciencejunction/route/professional/Assessment/assess.html.

Newmann, F. (2001). Authentic intellectual work: What and why?

Newmann, F., Bryk, A, and J. Nagaoka, J. (2001). Authentic intellectual work and standardized tests: Conflict or coexistence? Consortium on Chicago School Research.

Newmann, F. and G. Wehlage. (1993). Five standards of authentic instruction. Authentic Learning. Authentic Learning. pp. 8-12.

Newmann, F. M., Secada, W. G., & Wehlage, G. G. (1995). A guide to authentic instruction and assessment: Vision, standards, and scoring.

 

Table 1 – Accommodations for Learning

Students with physical disabilities have a difficult time navigating and inputting information into computers. A variety of peripheral devices and software are available to assist these students. The options listed below represent a subset of many accommodations/modifications that are available. Students have diverse needs and often require team consultations and short-term trial periods to properly match their needs to an accommodation's feature(s). Please refer further questions to your local or regional Assistive Technology Specialist.