Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Transcript

1 Placement Tests: The Shaky Bridge Connecting School and College Mathematics Sheldon P. Gordon Farmingdale State University of New York [email protected] The earth's crust is composed of a series of large plates floating on the underlying molten magma. These plates are constantly shifting and, as they bump into one another, one plate often rides up one on top of the other near their edges. The interfaces between plates form fault lines in the earth and the resultant pressures that build up along the interfaces eventually release to form earthquakes, as we are too often reminded in vivid news reports from around the world. The mathematics curriculum can be viewed in much the same way as being composed of a series of such plates. Two of the largest “mathematical plates” are the secondary curriculum and the college curriculum.For decades, these two mathematical plates were quite stable. We in the colleges knew what was in the high school curriculum and those in the schools knew what their students should expect when they went on to college. Over the last two decades, the NCTM Curriculum Standards have been transforming the school curriculum in very dramatic ways. College faculty have all heard about the Standards, but few have paid great attention to them and far fewer have ever read them. Yet, the Standards are having an ever-increasing impact on what is taught in the high schools and how it is taught. The Standards call for a fresh approach to mathematics that provides students with very different content and very different teaching and learning environments with increased emphasis on conceptual understanding, geometrical and numerical ideas as a balance to purely symbolic ideas; realistic problems, which tend to be considerably more substantial than artificial template problems; mathematics via discovery, not mathematics as a collection of facts and procedures to be memorized; the routine use of technology in the teaching, learning, and application of mathematics; and an emphasis on writing, communication and working collaboratively. Simultaneously, the Standards call for the early introduction of many new mathematical ideas, particularly statistical reasoning and data analysis, matrix algebra and its applications, and some probability. Overall, they impose a higher level of expectation on the students. 1 2 Obviously, something has to go to make room for all these new emphases. The Standards call for a diminished emphasis on formal algebraic manipulation. No longer do students spend literally months factoring polynomials in every conceivable setting. Instead, it is expected that students understand the notion of the roots of an equation, that they can factor simple expressions to find the roots, and that they can determine the roots of more complicated equations graphically and numerically and then use these roots as needed. Is this a fair trade-off? I believe that, in principle, most of us in the colleges will welcome students with such backgrounds. Most of these changes are completely compatible with the spirit of mathematical discovery and research; they are also completely compatible with the spirit of the reform movement in collegiate mathematics. In practice, however, things are somewhat different. The secondary school mathematics curriculum plate has shifted and the smooth interface that we have always expected is no longer there. Thus, we decry the fact that incoming freshmen appear to have poorer manipulative skills and less of the information that we have always considered important for success in college level mathematics. Based on what we infer from dealing with these students and based on our own high school experiences, we typically conclude that either the students are academically worse or that the high schools are completely at fault. Placement Tests: The Bridge between School and College Mathematics In practice, for most students, the bridge between school and college mathematics is the placement test that is used to determine how much students know and which course they should take. Unfortunately, at almost every college in the country, the placement exam used has basically the same focus as the ones used more than 20 years ago – testing the degree to which students have mastered traditional algebraic skills. With a large and growing number of students having been exposed to very different mathematical ideas and emphases, we continue to assess their ability and knowledge on the basis of a curriculum that is rapidly (we hope) disappearing. It is little wonder that so many students place so low on these exams despite having had two, three or four years of high school mathematics. It may not be that they have failed to learn what they were taught, but rather that they were taught other things instead. Again, it is the smooth transition from school to college mathematics that is breaking down. In particular, we have the following four scenarios: a traditional high school preparation leading to traditional college offerings a traditional high school preparation leading to reform college offerings 2 3 a Standards-based high school preparation leading to traditional college offerings a Standards-based high school preparation leading to reform college offerings. Neither the first nor the fourth scenario should present major transition problems. Students are placed into courses offered in the same spirit as their high school experiences and the level of the courses should be comparable to the students’ level of previous accomplishment. However, the second and third scenarios can present significant transition problems, especially to the students. In one case, students arrive on campus, presumably with strong manipulative skills, and suddenly are faced with the expectation that they have to think deeply about and fully understand the mathematics, and that they cannot succeed just by memorizing procedures by rote. In the other case, students arrive on campus expecting to expand on their understanding of mathematical concepts, to apply mathematics to more sophisticated realistic problems, to use technology, and to work collaboratively in teams. When they are faced with courses that focus almost exclusively on skills and the expectation that they need to memorize procedures by rote, the effect is comparable to running into a brick wall. In practice, things are not quite this clear cut. Very few institutions can be selective enough to choose students with any single type of mathematical background. Thus, most schools need to think through how to deal with students having widely different mathematics backgrounds. Instead, incoming students are presented with a single placement test to determine which courses they are “ready” to take. Two widely used standardized placement tests, the College Board’s ACCUPLACER and ACT’s Compass, are based on the traditional curriculum and assess students’ ability at algebraic manipulation. These placement vehicles are fine for Scenario 1, but what of the other three scenarios? For instance, one of the two national placement tests typically starts with a component measuring a student’s ability in intermediate algebra. Students who do well automatically move on to a higher level component testing college mathematics readiness (i.e., precalculus); those who do poorly on the algebra level are moved to a lower level testing arithmetic and introductory algebra skills. The intermediate algebra portion of this test covers 12 topics, including squaring binomias,simplifying rational expressions, factoring polynomials, and simplifying and combining like radicals. Students who have come through a Standards-based curriculum are likely to have developed an appreciation for the power of mathematics based on understanding concepts and applying them to realistic situations. But standard placement tests clearly ignore much of what that student has learned in the way of non-manipulative techniques, of conceptual 3 4 understanding, and of contextual applications. So, when such students take such a placement test, it is not surprising that many end up being placed into developmental mathematics because their algebraic proficiency is seemingly very weak. Those skills may not have been emphasized or perhaps have grown rusty since the last math course in high school. This is certainly unfair to students who may be placed one, two or even more courses below where they should be based on the amount of mathematics they took in school. I have spoken with many high school teachers from different parts of the country who complain that many of their best students – students who scored 4 and 5 on the AP calculus exam – have been placed into precalculus, college algebra, or developmental algebra in college. Some of these teachers have compiled data on all of their AP students to track how each one has been placed and this situation is fairly common. In part, this may be because the two national tests and most home-grown tests deny students use of technology, even though that had been an integral part of their mathematical experience in high school. All too often, courses and textbooks assume a blank-slate philosophy, presuming that the students have never seen anything previously. That is not likely the case and will be less the case in future as the reported percentages of students who continue on to successive mathematics courses in high school increases. (Historically, the drop-out rate was on the order of 50% each year; recent evidence indicates, for instance, that the drop-out rate from first year algebra to second year algebra is now on the order of 10-15% [1].) On the flip side, for the last decade or more the fastest growing component of college mathematics enrollment has been at the developmental level. (Although this seemingly contradicts the information on school mathematics, that is likely a function of the placement tests used.) It seems that a better solution would be for departments to rethink some of the “remedial” courses they offer to see if they are reasonable based on the overall mathematical backgrounds of the students. Students who took traditional mathematics courses in high school and who enter reform courses may well be assessed, on the basis of traditional placement tests, as possessing manipulative skills strong enough to succeed in courses that are well above the level of their conceptual abilities. If they have never had to understand the mathematics they have apparently mastered and have never been expected to read a mathematics textbook, these students may well be overwhelmed by the intellectual expectations of a reform course. For instance, just because a student is able to calculate the slope of a line does not mean that he or she has any idea of what 4 5 the slope means in a practical situation. For that matter, I typically includes a problem on tests in precalculus and college algebra in which the students are presented with an array of functions – some as formulas, some as graphs, and some as tables – and asked to identify which of the functions are linear, which are exponential, which are power, and so forth. It is always distressing to see how many will seemingly randomly decide that something like y = x0.75 or an exponential curve is a linear function! But standard placement tests never seek to test whether a student knows what a line is; they only test whether the student can find the equation of a line or merely find the slope of the line through two points. Reportedly, test-makers have been under pressure to develop a new generation of tests that are more aligned to Standards-based courses. That would certainly be a huge step in easing the transition problems, assuming that the colleges eventually adopt such tests. However, the process of developing, testing, and validating such tests is a long-term undertaking and we probably cannot expect to see such products available in the immediate future. Unfortunately, departments in institutions that depend exclusively on such tests – most likely because of the ease of administering themon-line to large numbers of students – probably can’t do much until then. The Dynamics of Placement Testing Many of you are likely wondering: Why is this state of affairs happening? The reality is that almost all mathematics faculty members in most colleges and universities are oblivious to these issues or even to the specific nature of the placement tests used in their school. The entire placement testing operation is typically conducted outside of the mathematics department. It might be instructive to consider some of the dynamics and implications of this arrangement. First, most college faculty tend to resent time spent administering and interpreting placement tests, particularly over the summer prior to the start of classes. They are therefore impressed by the promise of a professionally designed test that is thoroughly tested and validated and used at so many other institutions.. Second, college administrators are also seduced by the same promises, as well as by data presented by representatives from the placement companies that show how widely used their placement test is, how easy it is to use, and how effective the technology is in identifying students’ mathematical weaknesses. Third, the questions used on these placement tests tend to be closely guarded secrets. Even when members of a mathematics department are interested in seeing specifics on what is 5 6 expected of students, they are often not provided with any details. Consequently, the faculty members who might be able to recognize the educational problems typically do not have the information on which to make an informed decision. The placement test industry certainly hears many complaints from high school teachers, and likely from NCTM itself, about the poor match between Standards-based curricula and traditional college curricula and many of the horror stories about individual students who have been so badly misplaced. However, the placement test industry sells its products exclusively to the colleges and universities; complaints from the schools have little or no impact because they are not the paying customers! Of course, the testing industry also hears complaints from some of us in the colleges and universities and there are personnel at these companies who understand the issues fully. However, the people at these companies who understand the issues and problems tend not to be the senior personnel who, in the final analysis, make the corporate decisions. And those individuals get only limited feedback about the problems with their products; but, they do get a lot of feedback from the sales representatives and most of that is very positive feedback. The catch is that the sales reps are in contact almost exclusively with college administrators, who tend to be quite satisfied with a product that is easy to administer and apparently effective to use. So, because the people who make the financial decisions at the colleges are happy, the senior personnel at the testing companies are more than happy to keep from rocking the boat. Considering the major costs associated with developing, validating, and marketing new versions of placement tests, if there does not appear to be a pressing need to do anything, why should they? What Can You Do? We live in a technological age in which most educated consumers will go to an appropriate website to research the characteristics of, and other consumers’ experiences with, $15 toasters or $150 IPods. However, comparable information on the characteristics and experiences of $150,000 college educations, and the doors that they either open or slam shut, are not available. Unfortunately, placement decisions, which can effectively slam shut doors leading to careers in virtually every quantitative field today, are not made until after students arrive on campus. That seems grossly unfair. 6 7 Reportedly, there are many high school mathematics teachers who maintain information on the kinds of technology used or banned on various campuses and use that information to advise their students on where to apply or where to avoid. It certainly seems reasonable that comparable information could be gathered and used for advisement regarding placement procedures at colleges that many of a school’s graduates typically attend. If nothing else, the placement practices are likely more significant to a student’s overall collegiate experience from a mathematical perspective than the use of technology is. Furthermore, this is also the kind of action that can potentially exert very effective pressure on the colleges and therefore on the testing industry to make significant changes in the placement tests used. Moreover, as mentioned above, many high school math teachers have begun compiling data on placement incidents related to their graduates. While no single high school has the ability to affect the procedures at a college or university, groups of neighboring high schools can pool data and bring them to the attention of the local colleges, thereby raising consciousness about placement issues. What might be even more effective is for local or regional NCTM affiliates to present such a case to appropriate mathematics departments. Any responsible chairman of a mathematics department should be willing to sit down with representatives of such a group to discuss a topic of serious concern. However, rather than just presenting a collection of horror stories, it would probably also be a good idea to present some concrete suggestions about ways to address the problems. For instance, it is likely a good idea to show some non-routine problems that students are now doing in high school mathematics or to show examples of portfolios that students have produced to illustrate the differences between traditional and modern mathematics curricula and pedagogy. It might also be worth pointing out that some mathematics departments have a placement scheme that utilizes the number of years of high school mathematics that a student has taken and his or her ACT or SAT score in conjunction with a placement test to decide on the appropriate course. Other departments take the number of years since the student’s last math course into account in placement decisions. I am aware of one large scale study conducted some 10 or 15 years ago at a large two year college where about 18 different were all studied in terms of being effective predictors of student performance. The study found that about 12 of the factors were statistically significant and so developed a multivariate regression formula for placement based on all the relevant factors. The suggestion that such factors be taken into 7 8 account, in addition to using only scores on a placement test, might be a reasonable compromise. Most importantly, once such a dialogue has been established, it is reasonable to hope that effective changes can occur down the road. But, a dialogue is a two-way conversation and high school teachers should also be prepared to hear a list of horror stories about how poorly prepared many incoming freshmen are. I am aware of one high school that has developed a proactive strategy to circumvent placement problems for its graduates. In addition to offering the usual SAT-preparation workshops for its college-bound students, this school also offers a placement-prep workshop during lunch based on the kinds of questions that their students will be expected to face on the national tests. This way, they do not subvert their teaching philosophy, which is strongly influenced by the NCTM Standards, but they do get their students ready to pay the toll imposed to cross the bridge to collegiate mathematics. Clearly, if we can ease the mathematical transitions of students, we would make things better for all of us. The teachers will be under less pressure to give in to the perceived necessity of providing a collection of arcane manipulative skills to students who have little or no use for them. The students will be better served when they arrive on campus. Enrollment in “remedial” courses may actually diminish because many of the students being placed there may not really need remediation; enrollment in college-level mathematics offerings might even increase. The students will be happier, the faculty at both levels will be happier, and the administrators will be happier. In conclusion, it is evident that the secondary school mathematics plate has shifted dramatically and will continue to shift even further. The college mathematics plate is hopefully shifting in the same direction, so that at some schools, there will continue to be a relatively smooth transition. However, many other schools may want to recall what occurs when the earth's plates shift in different directions: an earthquake is not a pleasant experience. And, the last place you want to be in an earthquake is on a bridge that spans the fault line. Acknowledgment The work described in this article was supported by the Division of Undergraduate Education of the National Science Foundation under grants DUE-0310123 and DUE0442160. An extended version of this article is available at ……. 8 9 References 1. Usiskin, Zalman, High School Overview and the Transition to College, in A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus, Nancy Baxter Hastings, et al, editors, MAA Notes #68, Mathematical Association of America, Washington, DC, 2006. 9 10 Abstract Placement tests are the bridges between school and college mathematics. But standard placement tests focus exclusively on traditional algebraic skills and are not aligned with Standards-based curricula or even reform curricula in the colleges. The article discusses various issues on the subject and presents suggestions for how high schools can lobby for change to get better placement for their graduates. 10