by
The release of the National Council of Teachers of Mathematics' (NCTM) Principles and Standards for School Mathematics (PSSM) last year marked a major milestone in mathematics education. The PSSMs combine, and expand on NCTM's three previous standards documents, which dealt with curriculum and evaluation, teaching, and assessment. Though these three works were each hugely influential, the PSSMs look to capture the attention of still a wider audience, and as such, be even more influential then the three documents out of which it was born. The PSSMs should be more accessible to a wider audience because of their use of four distinct "grade bands," Pre-K-2, 3-5, 6-8, and 9-12. This segmenting of the standards means that teachers at any level will be able to quickly reference the material that is relevant to them, rather than fishing through pages and pages in search of what they are looking for.
In 1995 the Third International Mathematics and Science Study (TIMSS) showed America that something was wrong in our middle schools. TIMSS results showed American fourth graders among the best in the word in mathematics, while by the end of high school American students ranked near the bottom internationally. Wanting to do something about this, in 1996 a group of governors and corporate CEOs founded the group Achieve Incorporated. Achieve Inc. has been working since then to help their partner states write standards benchmarked against those of the best countries in the world. More recently, Achieve formed the Mathematics Achievement Partnership (MAP) with mathematicians and mathematics educators and together wrote the document Foundations for Success: Mathematics for the Middle Grades. This document attempts to develop a middle grades mathematics curriculum that more closely parallels the curriculums of the higher performing countries who participated in the TIMSS. A major feature of Foundations for Success is it's call for students to encounter algebra and geometry in middle school, opening the doors to more advanced mathematics courses in high school.
Though these two documents were produced by two different groups, and with two different ends in mind, both acknowledge the significance of assessment in meeting their goals. Assessment is so important to NCTM that one of their six Principles is devoted to it. Assessment is so critical to Achieve's Expectations that they are in the process of writing their own assessment materials specially designed to measure students' progress through their curriculum.
Assessment is an incredibly important facet of mathematics education, and it is an issue that has been the focus of some debate. It is widely believed that traditional pencil-and-paper assessment, which is still the norm, is an inefficient method of measuring what students know and can do. Often the results of this type of assessment give an incomplete or faulty impression of a student's knowledge. Over time, a number of alternative methods of assessment have been developed, tested, and eventually implemented by more progressive teachers with varying degrees of success. Examples of these alternative methods of assessment include, but are by no means limited to, Portfolio Assessment, Performance Testing, Project-Based Assessment, and Observational Assessment. In the paper that follows, each will be explained, and potential benefits and drawbacks will be discussed. Also, each will be analyzed in terms of how well it aligns with the NCTM Principles and the Achieve Expectations.
Performance testing is the very general name given to tests that use direct measures of learning rather than indicators that suggest that learning has taken place. (Boritch & Tombari 1997) Performance tests ask students to analyze, problem solve, experiment, make decisions, measure, cooperate with others, present orally, or produce a product. This last facet, "produce a product," would indicate that portfolio assessment and project-based assessment are both examples of performance testing, but because they are both so important and widely used they will be considered individually in the pages that follow.
If you have ever watched a gymnast perform, or a figure skater skate, then you have seen a kind of performance testing. The athletes go through their routine and try to impress the judges with their abilities. The judges then score the athletes in accordance with a predetermined scoring system. Performance testing of students in an academic setting is very much the same. A carefully designed scoring rubric is shared with the class, and students are asked to demonstrate a mastery of certain skills. This mastery might be evidenced through an oral report, or some other performance test, and it might be done individually or in groups.
Performance testing is a great way to get at the problem solving, communication, and connections standards. Having students work in groups will force them to communicate with one another about the mathematics being assessed. The different group members may bring different problem solving strategies to the table that would have to be shared and evaluated, accepted or rejected. A well designed performance test will feature strong connections between different mathematical topics, and to real world applications of those skills. It seems as if performance testing is exactly what the NCTM had in mind when they wrote those standards.
Performance testing is also a great way to implement the curriculum, learning, and teaching principles. When writing assignments, teachers will have to consider what mathematics constitutes "worthwhile tasks." To complete the projects students will have to engage in the higher order thinking and analysis that we now know to be so important to the learning of mathematics. As students present their work, teachers can assess students' understanding of the material, making corrections when necessary. Teachers could also observe their students for clues as to what type of learners they are, and adjust future instruction accordingly.
Though teachers might make the use of technology in a performance test mandatory, its likely that students would do so even if it was optional. It seems to be a students first inclination is to go to the internet for research or ideas. Also with the rising popularity of dynamic geometry software for investigation, the programs are not restricted to geometry class, but could be integrated into any mathematics class. By the same token, computer algebra systems, or the more powerful hand held calculators could be used to generate graphs or data sets for a number of mathematics classes, and in a number of different settings. It seems that when students are not confined to paper-and-pencil tests, but are allowed to express their understanding as they see fit, they will often include technology in some way.
Portfolio Assessment is a method of assessment in which students build portfolios containing diverse materials cataloging their mathematical growth throughout the school year. A portfolio might include homework assignments, projects, journal entries or other writing assignments, or traditional tests and quizzes. In a way, students try to make a case for themselves by collecting evidence that they have achieved the goals set forth by their teacher at the beginning of the year. Different students may have approached these goals through different means, and so each portfolio will be different, showing the uniqueness and individuality of each student.
Historically, teachers who adopt portfolio assessment are often initially very enthusiastic, but as the endeavor wears on they realize that portfolios have three serious drawbacks. (Trice 2000) The first is that portfolios are often quite bulky and require a lot of classroom space. The second and third are that portfolios demand a great deal of time to assess, and are often difficult to interpret. Despite these considerable turnoffs to the typical teacher, portfolio assessment is one of the most promising methods of authentic assessment developed in recent years. From the standpoint of the NCTM Principles, portfolio assessment is the most well-rounded. Of the six Principles: Equity, Curriculum, Teaching, Learning, Assessment, and Technology, portfolios are most associated with assessment, but hold great implications for all six.
Portfolios are a great way to foster equity in the classroom. With traditional assessment, all students are held to the same standard and are evaluated by the same means, but much research points to differences in students learning styles and methods of expression. (Armstrong 1994; Boritch & Tombari 1997) Portfolio assessment gives non-traditional learners a chance to express what they know through their strengths, be those strengths writing, art, music, or any other creative avenue a student might choose. This freedom to choose the works that best represent their growth seems to be very beneficial, especially to minorities and disadvantaged students who typically struggle under traditional assessment. (Cole, Coffey, & Goldman 1999)
The curriculum, Teaching, and learning principles are also well addressed through portfolio assessment. Because portfolios can be so bulky, teachers will have to consider what the most important topics to be assessed are. Also, to keep the portfolios varied, teachers will have to vary their assignments in order to give their students diverse materials to include in their portfolios. This diversification of classwork would have to be coupled with varied teaching styles, so as to better model what is expected from the students. Also, while students consider what pieces to include in their portfolios, they will have to be thinking about their learning process, and reflecting on the mathematics learned. This refection can lead to deeper understanding and appreciation of the topic.
Portfolios often spur students to use technology. From simply typing their written responses to questions using word processing programs, to including investigations in dynamic geometry software, students have countless ways to express themselves with technology. Students might use statistical software to generate graphs or charts, or might create an electronic portfolio with PowerPoint and submit just a disk to their teacher rather than a bulky binder; that would definitely solve the space problem with portfolios. The fact of the matter is that technology integration is still a rather novel idea for many teachers and students alike. When building a portfolio students will be looking to include interesting and diverse items, and often will turn to technology to find them.
In project based assessment, rather than complete a typical pencil-and-paper test or quiz to demonstrate mastery of procedures, students are asked to complete projects that are often quite involved. These projects frequently come at the end of a unit, and serve to connect all of the mathematics learned into a cohesive whole. Also, these projects are usually grounded in real world situations, and stress applications of the mathematics learned. Project-based assessment is the assessment of choice for many new reform curricula, including the popular Interactive Mathematics Program, or IMP.
Instruction in a classroom using project-based assessment is different from the instruction you would find in a traditional setting. Project-based instruction moves away from the paradigm of teacher centered lessons that are short and isolated, to a different model of instruction centered around the students and focusing on long-term, interdisciplinary lessons that integrate real world situations and applications. Project-based instruction covers the same curriculum recommended in the NCTM standards, and also does a good job of addressing the problem solving, communication, and connections standards.
To complete their unit projects, students will have to build bridges connecting the mathematics to its applications. Without a deep understanding of the mathematics they are learning, students will not be able to make those required connections. Through these projects students are fostering the skills that they will need in the workplace: communication, team work, and self confidence.
When assessing students’ projects, teachers will get a glimpse into each students’ understanding of the topic. How each student goes about solving each problem give the attentive teacher clues about that student’s strengths and weaknesses, which is information that could be used in the future to design better instruction for that student. Not only does project-based assessment give teachers insight into their students, but also to the curriculum itself. Students reactions may show that the assignment was too challenging, so in the future the teacher can make sure the students are better prepared. All in all, a reflective teacher can learn a lot from his or her students’ projects.
Assessment is not synonymous with grading; assessment is a much broader term. In their 1995 standards document on assessment, the NCTM defined the term as "the process of gathering evidence about a student's knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes." (NCTM 1995, 3) It is this affective area, the student's "disposition toward" mathematics that is often overlooked during assessment. Observational assessment, though, is a way to get at students' thoughts and feeling about mathematics informally, and without intrusion.
Observational assessment is a kind of informal assessment of students in which teachers observe their students in the classroom setting, monitoring for certain skills or behaviors, and jotting them down when they are evidenced. In this way teachers are able to assess not only their students academic growth, but also their emotional growth. Typically a teacher might be looking for mastery of some topic, but just as important are the affective traits such as demonstrating a valuing of mathematics, or a positive attitude in the face of difficulty.
In his article on observational assessment Doug Clarke (1994) gives three questions to keep in mind when observing students:
Of course answers to these questions could be quite lengthy, but the idea here is to jot down quick notes that serve as brief reminders of important events in a student's learning process. Also, with a class of thirty students, the teacher could spend the whole period trying to write out notes about each student, instead, it has been recommended that teachers use self-adhesive labels to record notes about just a few students each day. Over time these note will build up, giving a detailed history of each student. (Hopkins 1997; Vincent & Wilson 1994)
Observational assessment can lead to a greater equity in the classroom for students who might otherwise go unnoticed. It is too often that quiet students, who neither excel nor cause disruptions, become all but invisible in class. They receive neither admonishment nor praise, but also do not get attention from the teacher about their academic growth. Teachers who use observational assessment, though, will quickly find these "invisible" students when they notice that after a few weeks they still have no comments recorded about them. (Clarke1994) Once identified, teachers can make a special effort to connect with these students and make sure they are getting the same level of attention as the rest of the class.
The NCTM's teaching principle calls for teachers to get to know their students as learners and to continually seek to improve, both of which can be achieved through careful reflection. It would be impossible to remember all of the important events of a school year, or even a semester, but the data collected through observational assessment makes this reflection easy. Teachers can create a short list of traits for each of their students to gain insights into their learning styles. Also, teachers will have more meaningful information to share with parents during conferences. Observational assessment also provides teachers with a lot of information on which to base instructional decisions. If a majority of the class was having difficulty understanding a certain topic then the teacher will know to try a different instructional method.
Teachers seeking to meet the Achieve expectations may also find observational assessment useful. The expectations lend themselves well to a kind of checklist observation. When a student demonstrates mastery of a certain procedure for solving a problem, it can be checked off. When a student makes a comment that implies a deeper conceptual understanding of a major concept, that could be checked off. In this way teachers can track each student's progress through the expectations, giving added assistance where needed.
Mastery learning is less an assessment method in and of itself, but rather a philosophy of education. Though many aspects of Mastery Learning can be traced to the ancient Greeks, the pioneering work in the current philosophy was done by Harvard University professor John B. Carroll. Intrigued by Carroll's ideas, University of Chicago professor, and noted educational psychologist, Benjamin Bloom continued the research, and popularized the ideas during the 1960's.
The main idea behind mastery learning is that all students can attain mastery of a topic given appropriate instruction and ample time spent on the material. A typical unit in a mastery learning setting involves the initial instruction, followed by an assessment. Each assessment has a criterion level, or cut off score, that all students are expected to score above. Those students who do are often given enrichment assignments while those who do not are given feedback, and re-taught through diversified instructional methods, or correctives. Then, when prepared, these students take a second assessment and hopefully meet or exceed the criterion score. If students' grades do not improve to the appropriate level after the second assessment, often the rest of the class moves on and those students complete additional work outside of class to eventually meet the standard. (Guskey 1997)
The NCTM's equity principle calls for "reasonable and appropriate accommodations...to promote access and attainment for all students." (NCTM 2000, 12) Mastery Learning seems to be exactly what they had in mind. Mastery Learning does away with the idea that a student's aptitude measures his or her maximum level of achievement and replaces it with the notion that a student's aptitude is instead that student's personal rate of learning. Intrinsic to the idea of Mastery Learning is this idea that all students can learn, given appropriate accommodations. In Mastery Learning, no topic is deemed too difficult, and no student is given up on. This is at the heart of the equity principle; its not that all students are treated the same, but that all students are given the same opportunities.
In the affective domain, all students tend to think positively about themselves and their academic abilities during their first years of school. As years go by, the students in the bottom third in terms of achievement begin to form feeling of inadequacy, and are more likely to quit school sooner. For these students that have been subjected to repeated failures, are there any instructional methods that will provide them with the same learning opportunities as their higher achieving classmates? Its likely that, because their attitudes towards learning have been so soured by past experiences, hat there are not. But through Mastery Learning all student progress with the same history, and with the same levels of understanding. In this way, the playing field is leveled and all students are likely to foster positive attitudes and a desire to continue their educations.
Mastery Learning is also well aligned with the NCTM's Teaching Principle. Teachers using Mastery Learning have to be constantly reflecting on their instruction, and analyzing why students may or may not have achieved the criterion score on each assessment. During the feedback and correctives phase of instruction teachers are given a chance to explore their students learning styles, and adapt their pedagogical practices to suit them. With each assessment they give teachers have this opportunity to revise instruction, and through this revision become better teachers.
Mastery Learning also seems a good way to meet the Achieve expectations. A major problem that achieve sees with the current middle grades curriculum is that it is often simply a rehashing of content learned in elementary school. Through traditional instruction perhaps this review is necessary, as students had not learned the topics before, but with Mastery Learning one can be confident that all students have mastered the topics taught, and this review would not be necessary. Instruction through Mastery Learning could set the stage for the rigorous curriculum prescribed by Achieve in the middle grades. Clearly students would not be successful with this curriculum without a strong background; Mastery Learning gives them that.
To not completely ignore the merits of a traditional paper-and-pencil test, traditional assessment is included in this discussion of alternative assessments. A major outcome of mathematics instruction should be the ability to compute and use algorithms of arithmetic and algebra to solve problems. These rote skills are important and are easily and quickly assessed through traditional assessment methods. Another important outcome of assessment is for teachers to gain feedback about their instruction. Again, the results a traditional paper-and-pencil test can often speak volumes about how well a teacher is teaching, or at least how well his or her students are understanding the instruction.
Do proponents of alternative assessment methods mean to eliminate traditional methods of assessment? Of course they do not. They do, however, advocate the diversification of assessment. They recognize that no one method of assessment is going to meet all of the varied needs of all of our varied students. In the name of equity, we must offer students more than one way to show what they have learned. They also point to some of the inadequacies of traditional assessment. For example, if a student leaves an exam question unfinished, what have we learned about his or her understanding of that problem? Was it left incomplete because the student did not even know how to begin, or because insightful, but ultimately fruitless attempts were made? Often even if a student does answer the question, right or wrong, we are able to glean just as much about their understanding. (Kuhs 1994)
Some authors do believe that the traditional pencil-and-paper test could be modified to be an effective assessment. (Hopkins 1997; (Manon 1995) Simply replacing the usual limited response questions with open ended questions that require insight and higher order thinking is a quick way to get at student understanding without too drastic of a change in pedagogy. For example, the question "Find the average of the following numbers: 5, 12, 49, 10, and 9" could be replaced with "Give an example of a list of five numbers whose average is 20. Explain." In the latter students do not necessarily need the algorithm that they needed in the former, but they do need to have a deeper understanding of the concept of average. By including the word "explain" in the question, the teacher has asked the students to share their thinking, and in so doing they give the teacher an opportunity to assess not just their dexterity of computation, but their conceptual understanding.
The Achieve expectations seem solely concerned with raising the bar of mathematics education in the middle schools. Little is spoken of conceptual understanding or application of mathematical ideas. The Achieve document reads more like a checklist of skills than a curriculum. As such, traditional testing methods are perfectly suited for assessing students in the Achieve program. For demonstrating facility with computational algorithms, or mastery of basic skills and topics, traditional assessment is still the quickest and easiest.
Many different methods of assessment exist, each with their own strengths and weaknesses, each serving different populations of students differently. Assessment is such an integral part of mathematics education (and education in general) that the predominant group of mathematics educators felt compelled to write an entire book urging the nation to evaluate their assessment methods and consider adopting their Standards. In the years since NCTM wrote the Assessment Standards for School Mathematics assessment has become no less important, nor has it become any less difficult.
If the vision is for mathematics assessments that will "help teachers better understand what students know and make meaningful instructional decisions" ((NCTM 1989) then teachers will have to move towards assessment methods that integrate written, oral, and performance formats, and incorporate calculators, computers and manipulatives into the process. (Cain & Kenney 1992) But what assessment method incorporates all of those things? None of them do, that is why we will have to do what the NCTM has been calling for all along which is to use multiple sources of assessment information. (NCTM 1995) While its true that traditional assessment does not give the complete picture of a student's mastery of mathematics, neither do any of the alternative assessment methods mentioned here when used in isolation. Teachers must work to find an appropriate balance of assessment methods to suit their needs. This balance will likely change from class to class, and student to student, but through experimentation teachers should be able to achieve an appropriate harmony of traditional and alternative, written and oral, formative and summative, assessments to fit any situation.
Armstrong, T. Multiple intelligences in the classroom. Alexandria, VA.: Association for Supervision and Curriculum Development, 1994
Boritch, G. D. & Tombari, M. L. (1997). Educational Psychology: A contemporary approach. New York, NY: Longman.
Cain, R.W. & Kenney, P.A. (1992). A joint vision for classroom assessment. Mathematics Teacher, 85, 612-15.
Clarke, D. & Wilson, L. (1994). Implementing the assessment standards for school mathematics: valuing what we see. Mathematics Teacher, 87, 542-45.
Cole, K., Coffey, J. & Goldman, S. (1994). Using assessments to improve equity in mathematics. Educational Leadership, 56, 56-58.
Guskey, T.R. Implementing mastery learning. New York, NY.: Wadsworth, 1997.
Hopkins, M. H. (1997). Getting real: implementing assessment alternatives in mathematics. Preventing School Failure, 41, 77-84.
Manon, J. R. (1995). Implementing the assessment standards for school mathematics: the mathematics test: a new role for an old friend. Mathematics Teacher, 88, 138-41.
Kuhs, T.M. (1994). Implementing the curriculum and evaluation standards: portfolio assessment: making it work for the first time. Mathematics Teacher, 87, 332-35.
National Council of Teachers of Mathematics (NCTM). Principles and standards for school mathematics. Reston, VA. : NCTM, 2000.
National Council of Teachers of Mathematics (NCTM). Assessment standards for school mathematics. Reston, VA. : NCTM, 1995.
National Council of Teachers of Mathematics (NCTM). Curriculum and evaluation standards for school mathematics. Reston, VA. : NCTM, 1989.
Trice, A.D. A handbook of classroom assessment. New York, NY.: Longman, 2000.
Vincent, M.L. & Wilson, L. (1996). Implementing the assessment standards for school mathematics: informal assessment: a story from the classroom. Mathematics Teacher, 89, 248-50.
Asturias, H. (1994). Implementing the assessment standards for school mathematics: Using students' portfolios to assess mathematical understanding. Mathematics Teacher, 87, 698-701.
Clarke, D. (1995). Implementing the assessment standards for school mathematics: quality mathematics: how can we tell?. Mathematics Teacher, 88, 326-28.
National Council of Teachers of Mathematics (NCTM). Professional standards for teaching mathematics. Reston, VA. : NCTM, 1991.
Schloemer, C.G. (1997). Implementing the assessment standards for school mathematics; some practical possibilities for alternative assessment. Mathematics Teacher, 90, 46-9.
Schloemer, C.G. (1993). Aligning assessment with the NCTM's curriculum standards. Mathematics Teacher, 86, 722-25.
Schulman, L. (1996). New assessment practices in mathematics. Journal of Education, 178, 61-71.