It describes each strategy and clarifies its advantages and drawbacks. Also included is a large sample of classroom-tested examples along with sample student responses. These examples can be used "as is" - or you can customize them for your own class. This book will help prepare your students for standardized tests that include items requiring evidence of conceptual understanding. The strategies reflect the assessment Standards benchmarks established by the NCTM. In addition, an entire chapter is devoted to help teachers use these assessments to arrive at their students' grades.
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“ The way I assess my students'performance in mathematics already seems to be working, so why should I change?”
Suppose that you had just completed teaching a mathematics unit on determining compound interest. You want to know whether your students grasped the effect of interest compounding over time. You might consider one or some combination of the following assessments:
♦ Give the students a written test with 10-20 compound interest questions such as: If $2,500 is invested at 8% interest, compounded annually, how much interest will it earn in 10 years?
♦ Give the students a written test with 5-10 compound interest problems such as: Suppose that you invest $5,000 in the bank for 15 years. I f , at the end of that time, you have a total of $10,000 in your account, determine the approximate average annual interest rate that your money has earned.
♦ Give the students two days to complete a short project with the following directions: Suppose that you want to put $1,000 in the bank on the first day of each year from now until you retire at age 65. Examine the effects of earning an average of 5% interest per year versus 7% interest per year. How much more interest will you earn between now and retirement if you can average 7% rather than 5% annual interest? (Interestingly, over the course of 50 years, the 7% average rate will yield over $200,000 more in interest — nearly double the amount of money — than the 5% rate!)
All three of these assessment scenarios will allow you to collect data about your students’ ability to determine compound interest. So, how do you go about deciding which assessment is most appropriate for your class? Certainly, there are several factors that are likely to drive your decision. First, the assessment should reflect the intent of the objectives stated in the course of study. Teachers in School District A may have a course of study that states, “Given the principal, time period, and the annual interest rate, the student will be able to determine the amount of compound interest for an investment/’ On the other hand, School District B's curriculum may read more broadly, such as, “The student will be able to explain the effects of compound interest and determine interest over a given period of time/7 The first objective is very specific and skill-oriented, whereas the second outcome emphasizes conceptual understanding of compound interest. Therefore, a teacher from School District A may elect to use a straightforward, 20-question test, while a teacher in District B may feel the need to use richer problems or even projects to determine whether a student has met the stated outcome. On the other hand, there is no rule that precludes a teacher in District A from using a project to assess student progress, but it is probably unrealistic for a teacher in District B to attempt to measure the conceptual outcome with a 20-question, skill-based written test.
Another factor that often motivates a teacher's decision on how to assess students is whether the teacher has emphasized skill development, conceptual understanding, or application of a mathematical process in classroom activities. It would be unfair, for example, to ask a class of students who spent four days working traditional interest problems on worksheets to demonstrate their understanding by conducting a complex project. The evaluation Standards of the National Council of Teachers of Mathematics (NCTM) describe the notion of alignment by stating that “the degree to which meaningful inferences can be drawn from... an assessment depends on the degree to which the assessment methods and tasks are aligned or are in agreement with the curriculum” (NCTM, 1989, p. 193). A stated curriculum and a teacher who emphasize conceptual development and application of a skill should assess students by using a more complex task to adequately measure success.
A high school teacher, writing for Mathematics Teacher journal and struggling with the inconsistencies between her teaching and assessment, asked the reader to “look at the most recent test that you gave to your class. How many questions asked students to investigate? Were students asked to formulate? Did they need to apply a variety of strategies? How many questions asked students to repeat memorized algorithms to solve isolated problems?” Reflecting on the changes she had made in her classroom teaching routines, she noted that, “I realized that although my instruction had shifted toward NCTM's vision, my assessment practices had not” (Murphy, 1999, p. 248). So, it is important that the assessment tasks given to our students be consistent with the way that the class was taught the mathematical content in the first place — assessment needs to mirror instruction.
Finally, teachers select assessment tasks based on their own beliefs about the use for mathematics and their understanding of the content they are teaching (see, for example, Cooney, Badger, & Wilson, 1993). So, if you value skill development and believe that the ability to compute compound interest is the cornerstone of this area of the curriculum, then this belief will drive your teaching and, ultimately, your assessment strategies. However, if you believe that application of a skill is necessary to demonstrate understanding, then you may opt to ask higher-level test questions or assign a project in your assessment of student progress.
The purpose of this book, then, is not necessarily to change your views about the nature of mathematics because you probably already think of the discipline as more than the memorization of rote procedures, or you would not be reading the suggestions provided here. At the same time, you might find it helpful to reflect on the assessments that you have tried in the past. Think about whether it was the wording of the curriculum, the need for consistency between instruction and assessment, your beliefs about and knowledge of the content, or some combination of these factors that generally drive your decisions. Then, ask yourself, “What would my classroom look like if I used more of the complex tasks and projects for assessment?” And, more importantly, “How would the students in my class develop differently if I changed the focus of my instruction and my assessment to an emphasis on higher-level thinking and processing skills?” The latter question is extremely important, as researchers such as Cuban (1993) have determined that teachers will only change if they can be persuaded that doing business differently will benefit both themselves and their students.
Let's explore some examples of how different types of assessments can be used to demonstrate a variety of levels of student thinking. These are actual assessments from middle and secondary classrooms that have been compiled over time.
EXAMPLES OF STUDENT ASSESSMENTS
SQUARE ROOT OF A NUMBER
In a large school district with 30,000 students in the Midwest, children were given a competency test in the eighth grade year to assess their understanding of key concepts in the district's course of study The objective under study was one that read, “The student will be able to determine a square root and explain its meaning.” For five years, the district assessed this outcome by asking four multiple-choice items, such as:
Find the square root of the number:
To demonstrate mastery of this outcome, a student had to answer three of the four questions correctly, and they were allowed to use a calculator. After discovering that students had shown 100% mastery of this objective for five years in a row, a central office administrator asked, “Do we really know if the students understand the concept of square roots, or can they simply compute its value properly with a calculator?” The following year, the question was changed to the following:
What number has a square root of approximately 13.42? Explain how you know.
To answer this item accurately, of course, the student had to recognize that if, then √x≈ 13.42 could be found by determining that (13.42)2 ≈ 180 . A calculator alone does not help a student to answer this question correctly, unless the student can use it as a tool and knows to square 13.42. In the next two years of the competency testing program, only 17% of the eighth graders answered this question correctly! Therefore, there were many students who could effectively use a calculator to determine a square root but who had little, if any, understanding of the concept of “square root.” And, of course, this revelation about the level at which students understood square roots did not arise until the question — the assessment item — was asked in a different way that emphasized conceptual understanding over computational ability.
SOLVING PROPORTIONS
In an Algebra 2 course made up primarily of high school sophomores, students had been working with proportions, including setting them up to model problems and finding their solutions. The teacher asked the students two questions on a simple assessment:
1.Solve:
2.Explain how you know that your answer is correct.
Not surprisingly, every student in the class was able to solve the simple proportion given in question #1 to find that x = 16. However, only a couple of students were able to adequately justify their correct answers. The answers to question #2 varied, but many students responded with a mechanical, algorithmic statement such as:
The correct answer Is x=16. Multiply the top number of the first proportion by the lower number of the 2nd proportion. This Is called “oross-multiplyingThen divide that answer by 3, the last number there Is that has not been used to figure out “x.”
An examination of this response shows a lack of conceptual understanding for what it means for a proportion to be true. Instead, the response is a mechanical description of “how to do it.” Also, the student misuses the term proportion by referring to each fraction in the equation as a separate proportion. Another student responded with the following answer:
x = 16 because I changed them around until I got the right answer because 3 Is higher than 2 and there can't be 3 parts of 2
According to this response, the student inverted the fractions in some way to find the missing value, but it's not clear exactly how the problem was solved. Note that the question of how the student knows that the answer is correct is never addressed and that the student carries a misconception that “there can't be 3 parts of 2.” However, the student apparently believes that it is possible to have 2 parts of 3.
Both of these responses are significant because asking a student to solve a proportion and prov...
