Assessment is critical to the education process.  The nationwide implementation of No Child Left Behind (NCLB) will make understanding assessment a critical skill for teachers in every discipline.  My thesis is that NCLB will have minimal impact on the teaching of mathematics.  Unlike language arts, math is a subject that leads the student to one correct answer.  Mathematics is a subject that can be adequately assessed using standardized testing alone.

What is Authentic Assessment in Mathematics?

In order to test students effectively in mathematics, authentic assessment must be defined.  In the past, memorizing and solving set problems defined authentic mathematics assessment.   Kaplan (as cited in Harac, 2005, p. 4) states

For all the talk about ‘math wars’ in this country, most public school classes focus more on rote memorization than on theory.  …[Kaplan], after 40-plus years as educators, believe that learning the mathematical process is an end in itself.  The approach that they and their Math Circle instructors employ…is simple:  Present an abstract concept as a puzzle, then let the students wrestle with it and come up with their own solutions.  Not only is the process fun, the Kaplans believe, but it can be applied to other subjects as well.

 

Many other mathematics educators agree with Kaplan.  The purpose of mathematics education is to teach students to solve puzzles using mathematical logic not just to get the right answer to the mathematical problem.  Thus, the definition of authentic mathematics assessment has changed.

If the goal of mathematics education was to derive a single right answer, then standardized testing would accurately assess student performance in mathematics.  But authentic math assessments, “[r]equire more than simply answering a specific equations.  They involve developing as mathematical model that can be used to describe, explain, manipulate or predict the behavior of a variety of systems that occur in every day situations" (Lesh & Lamon, 1992, p. 17).  In fact, educators want to assess mathematic ability with a student portfolio that captures multiple examples of mathematical tasks.  Those tasks should be valuable individually and collectively for learning math.  The activities should be designed to cover the use of math in real situations with only the tools that would be available in that situation (Lesh & Lamon, 1992).  The situation should “encourage students to engage their personal knowledge, experience, and sense-making abilities" (Lesh & Lamon, 1992, p. 17).  This real-world assessment shows the student how to use math rather than just how to get a right answer.

In the view of the Mathematical Sciences Education Board (as cited in Lesh and Lamon, 1992, p. 24), “[m]athematics is the science and language of pattern.…[used] to employ knowledge of patterns for various practical purposes.…Facts, formulas and information have value only to the extent that they support effective mathematical activity."  Business and industry would concur with the view of the Mathematical Sciences Education Board.  Business and industry need people who can solve problems using mathematical tools. The formulas, mathematic theorems and algebraic equations are only tools to solve problems (Foreman & Steen, 1999).  In

many real-life problem solving and decision-making situations, an overwhelming amount of information is relevant, but this information often needs to be filtered, weighted , simplified, organized or interpreted before it is useful.  Sometimes needed information may not be provided, yet a decision may need to be made anyway and made within specified time limits, budget constraint, risks and margins for error (Lesh & Lamon, 1992, pp. 22-23).

 

With the structure of mathematical logic, one can use facts and assumptions to develop multiple alternatives and conclusions that ultimately solve business challenges.

To meet the needs of the real-world, mathematics assessment should address higher order understanding of complex processes.  With that ability, students would be able to use their math skills to gather, analyze, and interpret data.  To support that goal assessment should, "recognize and reward more than a single type and level of correct responses” (Lesh & Lamon, 1992, p. 17).  Math assessment should focus on the development of mathematical models, not just getting the right answer (Lesh and Lamon, 1992).  Being able to construct math models will show the student how to use their mathematical education beyond balancing a checkbook.  A mathematic education should build each student's mathematical skills.  A student should develop mathematical power as described by National Assessment Governing Board (2003, p. 35)

Mathematical power is characterized as a student’s overall ability to gather and use mathematical knowledge through exploring, conjecturing, and reasoning logically; solving nonroutine problems; communications about and through mathematics; and connecting mathematical ideas in one context with mathematical ideas in another context or with ideas from another discipline in the same or related contexts.

 

            A single measurement does not properly assess mathematical ability.  Power develops as the student progresses, by stages, to problem solving.  Educators must first assess students understanding of basic mathematic concepts.  Then they must assess a student's knowledge of mathematic procedures.  Finally educators assess problem solving.  The National Assessment Governing Board recommends students in these stages be

assessed on their ability to reason in mathematical situations, communicate perception and conclusion drawn from a mathematical context, and connect the mathematical nature of a situation with related mathematical knowledge and information gained from other disciplines or though observation (National Assessment Governing Board, 2003, p. 35).

 

Is Standardized Testing a Panacea?

 

Standardized testing is best for problems with convergent solution sets.  Standardized testing does not work as well for divergent solution sets.  According to Mark Goldberg, standardized

test questions remain narrow and often trivial, largely for ease of scoring, which holds down the cost.…[T]he kinds of problems that can appear on a standardized test are, of course, quite limited in form and complexity, the student is allocated only a minute or two to complete each one (Goldberg, 2005, p. 389).

 

Bracey agrees that “[t]he chief disadvantage of multiple –choice questions is that they usually test small samples of knowledge out of context” (Bracey, 2005, p. 13).  Assessment for mathematic knowledge is more complex than standardized testing.  To assess authentic mathematic knowledge, you need to be able to assess the student's ability to reason mathematically.  Standardized testing does adequately assess the student's ability to reason mathematically.

In addition, the standardized testing format “ requires students to answer many questions in a short period of time.  Students who stop to think about a question are in trouble:  They won’t finish the test” (Bracey, 2005, p. 14).  But Lesh and Lamon assert that a good math question takes anywhere from five to fifty minutes to set up the problem.  To set up a problem accurately, you must take time to think.

Kohn also condemns standardized testing saying, “Standardized-test scores often measure superficial thinking…[students are] ‘superficially’ engaged if they just copied down answers, guessed a lot, and skipped the hard parts…as a rule, it appears that standardized-test results are positively correlated with a shallow approach to learning” (Kohn, 2000, p. 1).  Superficial learning does not support authentic math assessment defined as

activities [that] generally require more than simply answering a specific question.  They involve developing a mathematical model that can be used to describe, explain, manipulate or predict the behavior for a variety of systems that occur in everyday situation (Lesh & Lamon, 1992, p. 18).

 

Since the short-term emphasis is on test scores to ensure compliance with NCLB requirements, teachers must, to some extent, teach testing.  For students to succeed at standardized testing, teachers must ensure the learning objectives covered on the standardized test are adequately covered in class.  Also, teachers must take time to address test taking strategies to further assist their students in test preparation.  The time used for test preparation is time away from content area studies.  In essence, we trade test preparation for “…time to train students in how to approach a problem or think through an issue” (Goldberg, 2005, p. 389).

Goldin observes that

School boards, administrators, and teachers are reluctant to devote class time to conceptual understanding, exploratory activity, construction of mathematical meanings, or mathematical invention, because they (often correctly) perceive these activities as untested and irrelevant to the test scores that are the short-term bottom line in mathematical achievement.  In the long run, or course, the damage shows up in many forms including low scores, even on the standard test items, because students have not developed adequate conceptual foundations (Goldin, 1992, p. 66).

 

The time required to teach conceptual understanding of math concepts is time well spent on building student mathematical prowess.  If a student successfully grasps the complexities of mathematical concepts their mathematical achievement will increase along with the student’s test scores.

How good are the test results from standardized tests?  A good test has reliable responses.  A good test is equally difficult every year.  Even if your test is reliable and fair every year,  “many external things can get in the way- from fire drills to unexpected noise” (Goldberg, 2005, p. 391).  Kohn goes further to say

Noninstructional factors explain most of the variance among test scores when schools or districts are compared.  As study of math results in the 1992 National Assessment of Educational Progress found that the combination of four such variables (number of parents living at home, parents’ educational backgrounds, type of community, and poverty rate) accounted for a whopping 89 percent of the differences in state scores.  To the best of my knowledge, all such analyses of state tests have found comparable results, with the numbers varying only slightly as a function of which socioeconomic variables were considered (Kohn, 2000,    p. 1).

In the end, all of the elements that contribute to student success on standardized testing are not within the control of the educator.

One element of testing that is tightly controlled is the NCLB testing standards.  Surprisingly, when you explore the details of NCLB guidelines, you find a lack of common standards among the fifty states.  “Every state sets its own standards for what students should know and be able to do, and the rigor of these standards varies widely” (Hall, 2004, p. 2).  For instance, the state can choose to require students to be proficient or meet standards or reach partially proficient levels to be successful on achievement testing (Hall, 2004). 

The inequities identified in the definition of success extend to the percentage of students meeting the requirements.  For instance, the math standard for Texas is 33% proficient while in Florida the standard is 38% proficient (Goldberg, 2005).  As with any statistically based analysis, the number in the sample analyzed affects the analysis accuracy.  Since NCLB does mandate subgroup composition, it is possible to adjust the size of the subgroup (Goldberg, 2005).  Thus state-testing scores can be improved by changing the subgroup population not only by improved student test scores. 

Conducting Authentic Mathematics Assessment

Assessment of students serves many purposes.  It is a tool for teacher and student.  For students, “Assessment activities often are the primary sources from which students discern what teachers really value and what teachers really want them to know” (National Assessment Governing Board, 2003, p. 35).  For teachers, the goal of assessment goes beyond just assigning a grade for the student’s performance.  Teachers also use assessment to understand how well students grasp concepts and what methods students using to solve problems.  In Mathematics, educators want “…to obtain information concerning students’ mathematical thinking….[T]o determine the methods students use to solve problems (Ginsberg, Lopez, Mukhopadhyay, Yamamoto, Willis, & Kelley, 1992).  More important is to understand what concepts students don’t understand.  As Ginsberg et al. wrote,

Classroom assessment techniques are essential because good teaching requires sensitive assessment.  Effective teaching requires knowledge of what students know and do not know, how students approach problems, and how they react to educational activities (Ginsberg et al., 1992, p. 280).

 

Standardized testing does not provide this level of insight on student progress.  While the test producers are providing data, the data is not necessarily useful in structuring classroom activities to address topics the student does not understand.  For standardized testing results to be useful, the test results should show the teacher where the students are having difficulty understanding mathematics. Ideally the results would provide the teacher a specific area of weakness for each student.  Then the educator could structure the lessons to address areas of student weakness.  Unfortunately, “the only information [the teacher] got back was the percentages in each group of students who failed…Because the data weren’t broken down, the information was essentially useless to a classroom teacher"(Goldberg, 2005, p. 340).

            Multiple-choice questions do not easily test the ability of a student to develop a mathematical model and then develop a solution using that model.  Multiple-choice questions ask only for the solution.  In other words, multiple-choice questions can emphasize recognizing the answer and not actually developing or calculating the answer.  Schwarz asks, "Do we really want to say to students that being able to recognize an answer to a question is a sufficient level of expertise to attain?” (Schwartz, 1992, p. 433).  In the end, an educator wants students to be able to develop a solution even if the answers are not available.

Schwartz is also concerned that another

more destructive, intellectual consequence of using multiple-choice test for serious assessment purposes.  The implicit message conveyed to students is that all issues worth discussing and examining can be reduced to a selection among four or five alternatives.  This can be presumed to be true no matter how much subtlety or nuance may be involved.  As a result, I believe, we have come to a be a public that thinks mathematics (and science and history and much else) is an intellectual domain in which question necessarily have answers and that the best answers can be briefly stated (Schwartz, 1992, p. 433).

 

Conclusion

 

Prior to completing the research for this paper, I assumed mathematics was an area of study that was not impacted by the national trend of mandatory standardized testing.  While it is feasible to grade math knowledge using standardized tests, it is not the optimal method.   Not only do the flaws in the standardized testing standards negatively impact math assessment so does the actual test administration.  Even more damaging to my original thesis is the fact that true math assessment must assess the student’s ability to reason and solve problems.  These skills are not adequately assessed with standardized, multiple-choice tests.

REFLECTION:

This paper opened my mind to other ways to test mathematics.  I had forgotten that numerical answers are important, but the real learning is the language of math--or as I think of is logic.  Math does teach you a structured way to solve problems in real life not just to get "the answer".  After all, computers can get an answer, but structuring a problem remains a human endeavor.

I also learned that NCLB is not as structured as I thought.  When standards for success are defined and tested differently in each state, it is difficult to evaluate progress accurately.

Download the paper (57KB)