Stuart Beatch200 275 063ECS 410 (Peta White)January 27, 2012

When Am I Ever Going To Use This?: Authentic Assessment in Mathematics

In this writer's opinion, there is no subject affected more by standardized testing and traditional forms of assessment than mathematics. To parents and administrators alike, “there is almost always a single right answer to a mathematics problem” (National Council of Teachers of Mathematics [NCTM], 1991, p. 5). The material in mathematics is not always so simple. As its value lies in the ability to develop our students' problem solving abilities and lateral thinking skills, mathematics should be assessed in a way to bring these qualities out and emphasize the nature of the subject as it is used in the real world.

In this paper, an argument will be made against traditional forms of assessment, such as multiple-choice tests. The topic of authentic mathematics assessment will then be presented as a solution to this problem. Research and writing will be used to clarify the cons of this approach, but also support the pros, to reach a balanced understanding of this approach and why it should be utilized. Lastly, a framework will be defined to give practical applications to this approach.

Mark Wilson (1992) makes a humorous point on the matter of testing in his article, saying that if our aim within mathematics is to provide students with a mass of skills and facts, then a test of right and wrong answers can be appropriate, but not all curricula are “based on the premise that learning is a matter of absorbing and reproducing provided information” (p. 213-214). From an ethical standpoint, it seems ludicrous to teach and assess mathematics in this fashion, especially with the oft-repeated question, 'When am I ever going to use this?' This is further exacerbated by standardized tests, which “are not ... a sound basis for indicating how well students are becoming educated in mathematics” (Romberg, 1995, p. 173). The National Council of Teachers of Mathematics (1991) agrees, stating “conventional standardized tests are generally inconsistent with promoting effective mathematics teaching practices in terms of what is assessed, how it is assessed, and how the results are used” (p. 8). Traditional forms of testing are an equally flawed way to evaluate. They are often formed from short, quickly-answered, and equally assessed questions which are either correct or incorrect; based on selected concepts that occur in popular textbooks, which were judged and determined by distant experts; established with an equilibrium of questioning that is neither too easy nor too hard; and finally, is assessed through a running count of correct responses (Romberg, 1995, p. 3). It is important to point out that despite these suggested flaws, there is a place for traditional assessment, such as multiple-choice questioning. However, “they are inadequate for assessing many of our new goals. Limiting assessment to this form is comparable to a doctor's use of only a stethoscope to diagnose a serious heart problem” (NCTM, 1991, p. 6). While this may seem to be an extreme comparison to draw, it is important to remember the earlier statement that mathematics is rarely as simple as a single correct answer to a problem. If we are to educate our students in mathematical thinking, we must call upon deeper forms of assessment that can prove their learning in new and applicable ways.

To this end, it will be argued that authentic assessment is one possible solution to this problem. However, it should be noted that authentic assessment is a politically charged phrase. Thomas Romberg (1995) suggests that the phrase has two connotations, one of trustworthiness, and another of distinction from other assessment forms (p. vii). Romberg's definition implies that authentic assessments are correct. However, this further implies that other assessments, such as traditional exams, are not. This has caused some difficulty and resistance from researchers and educators, which will be addressed later.

To further this definition of authentic assessment, it can be considered to be comprised of

authentic assessment tasks. These are the specific forms of questioning and problems that we use, and should fall under the following criteria, as outlined by Archbald and Newmann, “(1) disciplined inquiry; (2) integration of knowledge; and (3) value beyond evaluation” (as quoted in Romberg, 1995, p. 9). By these three categories, authentic assessment tasks should ask students to produce new knowledge through prior knowledge, view content as a continuum rather than discreet fragments, and possess attributes that make them worthwhile beyond the classroom (Romberg, 1995, p. 10). A further definition of authentic mathematical tasks states that they should involve “(i) real mathematics, (ii) realistic situations, (iii) questions or issues that might actually occur in a real-life situation, and (iv) realistic tools and resources” (Lesh & Lamon, 1992, p. 18). Clearly, there is an emphasis being placed on realism as the central key to this approach. The NCTM (1991) corroborates this, stating that authentic assessment tasks “highlight the usefulness of mathematical thinking and bridge the gap between school and real mathematics” (p. 3). They continue, stating that these tasks, which involve pattern recognition, paraphrasing, creating models, and so on, resemble the actions and thinking of real mathematicians and create real-life applications that allow for authentic judgements of achievement (NCTM, 1991, p. 3). From this definition, a framework for its use begins to emerge. The NTCM addresses the content basis for authentic assessment and provides a foundation from which to build assessment practices in the classroom.

Despite the best efforts of educators building knowledge and literature in this area, it is most certainly not without its opponents. James Terwilliger (1997) is particularly scathing in his remarks:

The promotion of 'authentic' assessment, however well intentioned, is flawed in several respects. First, 'authentic' is misleading and confusing. The term inappropriately implies that some assessment approaches are superior to others because they employ tasks that are more 'genuine' or 'real.' ... Second, it is a mistake to deny the role of knowledge in the assessment of educational outcomes. To do so ignores a substantial body of theory and ample empirical evidence that supports the central role of knowledge in many domains of performance. (p. 27)

This author argues that Terwilliger is misinformed in both regards. Considering authentic approaches to be superior is never assumed, since traditional assessments are still useful. Considering knowledge to be the main focus of mathematics also directly contradicts Saskatchewan's Ministry of Education (2010), which states that students are to develop understandings and abilities necessary to be confident and competent in their future life and work experiences with mathematics (p. 8), not necessarily to be fluent in knowledge and processes. David Tanner (2001) presents some further problems, but ones that are not so easy to disprove. These lie mainly in that the approach is time and cost heavy, requiring consensus among the markers, and creating a conflict as to if the activity is authentic to the student or the teacher (p. 28). Tanner is accurate with his complaints. Time and money are valuable, especially in a system that demands efficiency and offers pathways that make traditional assessment simpler. The question of authenticity is also one that requires answering through personal reflection on the part of the teacher, and is not something that can be prescribed. However, the notable thing about Tanner (2001) is that he later admits a solution to be within a synthesis of both traditional and authentic assessment styles (p. 29). This is in accordance to the suggestion earlier that traditional assessment still has a place, and is the same error in judgement that Terwilliger experienced.

Though there are opponents to this approach in assessment, there are also supporters. The NCTM is one of the main organizations for mathematics teachers, and a large provider of resources and advocacy. In their 1991 publication, they give a large list of the positive aspects of adding authentic and other alternative assessments into the classroom, as self-reported by students and teachers alike:

Students: think more deeply about problems; feel free to do their best thinking because their ideas are valued; ask deeper and more frequent questions of themselves, their classmates, and their teachers; improve their listening skills and gain an appreciation for the role of listening in cooperative work; feel responsibility for their thoughts and ownership of their methods; observe that there are many right ways to solve a problem; ... feel more tolerance and respect for other

people's ideas; ... find satisfaction and confidence in their ability to solve problems.; [sic] look less to the teacher for clues about the correctness of their methods and focus less on imitating the “right” way. (p. 4)

Though it may seem excessive to quote a list of this length, it efficiently summarizes every main point that makes authentic assessment a quality approach in the classroom. It may also seem disconcerting that these positives are not professionally and quantitatively researched. However, it could further be argued that this is negated in some part by the nature and credibility of the institution that reports them.

Now that the nature of authentic assessment, its definition, and comparing the positive and negative aspects of its use has been accomplished, a framework for practical applications of authentic assessment can be produced. In Lesh & Lamon (1992), the authors provide an approach which re-forms existing learning tasks to make them more authentic, specifically through improving “practical relevance ... the coherence or fragmentation of the task ... the range of possible responses ... the extent and value of the task ... the mode of working on the task” (p. 120). As mentioned before, this approach does take a great deal of time and effort, but this is unavoidable in any attempt to move away from the preferred standard, educational or otherwise. For each mathematical activity or lesson, the following process should be followed:

First, one must decide the range and balance of types of task [sic] that the “target group” of students should be able to do. Brainstorm and search until a reasonable set of attractive possible tasks are found. Then devise a form of presentation of each task that leads students to understand what is required and how to tackle the task. Finally, try it out, observing what happens, and revise the presentation ... In the light of a sample of student responses (written or otherwise), devise a grading scheme for assigning credit, then check that this works well with those who will be doing the grading ... Imagination plus realistic empirical development is the key. (Lesh & Lamon, 1992, p. 122-123)

In assembling these tasks, it is also important to ensure that they cover a wide range of lengths, allow flexibility and autonomy for the student, are at least partially unfamiliar, are practical, are contextual (especially relating to the experience and interests of the students), and contain mathematical content, sampling a wide-range of concepts and skills from the curriculum (Lesh & Lamon, 1992, p. 126-127). Through these balanced tasks, moving away from static and disengaging materials, we are given a passageway to apply the authentic principles. This approach gives meaning to the problems, allows our students to think like mathematicians, and therein work towards assessing our students in a more authentic and valuable way.

In this paper, authentic assessment and the tasks involved in its use were discussed and suggested as an additional resource to traditional assessment practices, which are narrow in scope and ineffective in evaluating true mathematical thinking. The negative aspects of this approach were raised but dismissed by the positives and support by the NCTM. With a practical approach to this style of assessment provided, it becomes a truly versatile and important tool to use in the classroom among other practices; in our society, diversity is commonplace, and so to should it be in our teaching.

References

Lesh, R., & Lamon, S. J. (Eds.). (1992). Assessment of authentic performance in school mathematics. Washington, DC: American Association for the Advancement of Science.

Ministry of Education (2010). Foundations of mathematics 20. Regina, SK: Ministry of Education.

National Council of Teachers of Mathematics (1991). Mathematics assessment: Myths, models, good questions, and practical suggestions. Reston, VA: NCTM.

Romberg, T. A. (Ed.). (1995). Reform in school mathematics and authentic assessment. Albany, NY: State University of New York Press.

Tanner, D. E. (2001). Authentic assessment: A solution, or part of the problem?. High School Journal, 85(1), 24-29.

Terwilliger, J. (1997). Semantics, psychometrics, and assessment reform: A close look at “authentic” assessments. Educational Researcher, 26(8), 24-27.

Wilson, M. (1992). Measuring levels of mathematical understanding. In T. A. Romberg (Ed.), Mathematics assessment and evaluation: Imperatives for mathematics educators (pp. 213-241). Albany, NY: State University of New York Press.