Placement Tests:  The Shaky Bridge Connecting School and College Mathematics [1]
Sheldon P. Gordon
Farmingdale State University of New York
gordonsp@farmingdale.edu

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.  These interfaces between the 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; however, we can also think of other plates such as those comprising elementary school mathematics, upper division mathematics, and graduate level mathematics.

For decades, the two secondary and collegiate mathematical plates were quite stable.  The underlying magma on which they rode had been solidified, so that for long there was relatively no movement.  The interfaces between the two plates were quite smooth.  We in the colleges knew what the high school curriculum consisted of, having passed through it ourselves, and were able to count on its invariance in developing our own curricula at the college level;  similarly, those in the schools, having passed through the college curriculum, knew what their students should expect when they went on to college and so knew how to prepare the students.

Over the last two decades, the NCTM Curriculum Standards have been transforming the school curriculum in very dramatic ways.  We in the colleges have all heard about the Standards, but few of us 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 different approach to mathematics that provides students with very different content and very different teaching and learning environments:  

• There is a major emphasis on conceptual understanding of fundamental concepts and mathematical reasoning, not just routine manipulation;
• There is a greater emphasis on geometrical and numerical ideas as a balance to purely symbolic ideas;
• There is an emphasis on realistic problems, which tend to be considerably more substantial, than artificial template problems whose solutions are to be memorized and regurgitated;
• There is an emphasis on mathematics via discovery, not mathematics as a collection of facts and procedures to be memorized;
• There is an emphasis on the routine use of technology in the teaching, the learning, and the application of mathematics;
• There is an emphasis on writing and communication and working collaboratively.

Simultaneously, the Standards call for the early introduction of many new mathematical ideas into the curriculum, 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. 

Obviously, something has to go to make room for all these new emphases.  In the process, therefore, 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 at the calculus, below calculus, and post-calculus levels, as called for by the MAA in its CUPM Curriculum Guide 2004 and by AMATYC in its Crossroads Standards.  In practice, however, things are somewhat different.  The secondary school mathematics curriculum plate has shifted and the smooth interface that we in the colleges have always expected is no longer there.  As a consequence, 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.  In turn, we have repeatedly expanded our remedial (or developmental) offerings to “bring the students up to speed”.  At many institutions, remedial programs dominate the mathematics offering in terms of number of sections, number of students enrolled, and the institutional resources expended.

Several years ago, Richard Riley, Secretary of Education in the Clinton Administration, challenged the mathematics community to address the problems of articulation in mathematics education between high schools and two and four year colleges. Riley called for this national initiative, through the National Research Council, because of the growing breakdown in the once smooth transition between high school and college mathematics, as well as the differences between mathematical experiences in different colleges when students transfer from one institution to another.

Placement Tests: The Bridge between School and College Mathematics

In practice, for most students, the bridge between school mathematics and college mathematics are the placement tests that are used to determine how much students know and which course they should take.  Over the last decade or two, placement tests have changed in terms of how they are administered – most are now given electronically, are scored electronically, and students are advised electronically.  But, unfortunately, the reality is that at almost every college in the country, the placement exams used have basically the same focus as the ones used more than 20 years ago – testing the degree to which students have mastered traditional algebraic skills. 

Is it reasonable to continue giving the same, traditional placement exams?  Certainly not!  A large,  and growing, number of students coming in from high schools have been exposed to very different mathematical ideas and emphases, yet 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 seem to place lower and lower 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
• a Standards-based high school preparation leading to traditional college offerings
• a Standards-based high school preparation leading to reform college offerings.

The first of these scenarios should present no major transition problems, either to the students or to the institutions.  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.  The fourth scenario should likewise present no major transition problems.  (Of course, students can still encounter significant mathematical problems, but that is another issue altogether.)  

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 they 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.

Unfortunately, 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 all sorts of different mathematics backgrounds.  Instead, at most schools, incoming students are presented with a single placement test to determine which courses they are “ready” to take.  There are two widely used, standardized placement tests, one (ACCUPLACER) developed by the College Board and the other (Compass) by ACT.  Both tests are based on the traditional school curriculum and are designed to assess students’ ability at algebraic manipulation. (Of course, many mathematics departments use home-grown tests, but these also typically focus on the traditional high school curriculum.)  All of these placement vehicles are fine for the first scenario listed above, 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 are automatically moved on to a higher level component testing college mathematics readiness (i.e., precalculus); those students who do poorly on the algebra level are automatically moved down to a lower level component testing arithmetic and introductory algebra ability.  The intermediate algebra portion of this test covers 12 topics in an adaptive manner:

1. Square a binomial.
2. Determine a quadratic function arising from a verbal description, e.g., area of a rectangle whose sides are both linear expressions in x.
3. Simplify a rational expression.
4. Confirm solutions to a quadratic function in factored form.
5. Completely factor a polynomial.
6. Solve a literal equation for a given unknown.
7. Solve a verbal problem involving percent.
8. Simplify and combine like radicals.
9. Simplify a complex fraction.
10. Confirm the solution to two simultaneous linear equations.
11. Traditional verbal problem - e.g., age problem.
12. Graphs of linear inequalities.

Now picture what happens to students who have come through a Standards-based high school curriculum.  Such a student has likely developed an appreciation for the power of mathematics based on understanding the concepts and applying them to realistic situations. But, this type of traditional placement test clearly ignores much of what that student has learned in the way of non-manipulative techniques, of conceptual understanding, and of contextual applications. So, what happens when such students take a traditional placement test, which is designed only to determine how many manipulative skills students have retained?  Is it surprising that many such students end up being placed into developmental mathematics because their algebraic proficiency is seemingly very weak?  This is certainly unfair to students if they were never exposed to some of those skills or if the emphasis on those particular skills was lower than in the past to make time for more important mathematics or if the students’ experience in mathematics has led them to think of mathematics as something considerably more important, more practical, and more intellectually demanding than squaring a binomial.  The result is that many students are placed one, two or even more semesters behind where they likely should be placed based on the amount of mathematics they took in school. 

For instance, at the author’s previous institution, one of these two placement tests was first used about 20 years ago.  The day before classes started that year, it was discovered that over 140 incoming students who had had some calculus in high school had been placed into our arithmetic course!  The placement test program had apparently dug deep enough to discover that these students were weak at manipulating certain kinds of fractions or had some other comparable deficiencies and had “placed” them accordingly.  The department chair, working with an associate dean responsible for placement, “resolved” the problem by renormalizing the scores, so that acceptable numbers of students would be placed into higher level (albeit still developmental) courses and I believe this solution is still in effect today. 

However, this is not an isolated incident. The author has spoken with many high school teachers from different parts of the country who have complained that many of their best students – students who got grades of 4 and 5 on the AP calculus exam – have been placed into precalculus, college algebra, or developmental algebra when they arrived on campus.  Some of these teachers have been compiling data on all of their AP students to track how each one has been placed and this situation is fairly common.

Furthermore, the two standardized tests and most of the home-grown tests deny students use of technology, even though that had been an integral part of their mathematical experience in high school. (Supposedly, some of the national placement tests will soon allow students to use any standard calculator, including most graphing calculators.)

It certainly seems unreasonable to take students who have completed two, three or even four years of high school mathematics and place them into low level developmental courses because their algebra skills are weak.  That weakness is perhaps because those skills may not have been emphasized or perhaps because those skills have grown rusty due to a long lay-off since the last math course in high school.  All too often, both 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 dropout rate from first year algebra to second year algebra is now on the order of 10-15% [Usiskin].) 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.

Now picture what can happen with students who took traditional mathematics courses in high school and who are going into reform courses.  On the basis of traditional placement tests, the students’ level of manipulative skills may well be assessed as high enough to place them into 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. (We would not dream of putting a student coming out of elementary algebra into a linear algebra course; the student might have the necessary skills, but he or she would need to develop a much higher degree of conceptual ability.) 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 the slope means in a practical situation.  For that matter, the author 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 = x^{^0.75} 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, the test-makers have been under pressure to develop a new generation of national placement tests that are more aligned to Standards-based courses.  That would certainly be a huge step in easing the transition problems, assuming that they react to the pressure and 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 them 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 about these issues or even the specific nature of the placement tests that are used in their institution.  Particularly in schools that use the national tests, the entire placement testing operation is typically conducted by an individual or group completely outside of the mathematics department.  It might be instructive to consider some of the dynamics and implications of this arrangement.

As one example, the author is aware of a moderate-sized, private college where, for many years, the placement tests used were all homegrown and quite traditional in nature.  Several times each year, including a week-long period during the summer, the mathematics faculty were expected to make themselves available for placement testing.  (The English faculty were likewise expected to be available during the same periods.)  They personally administered the tests, scored them by hand or by Scantron, and individually advised the students about which course or courses they should or had to take. The last, by the way, is a rather important feature.  Nevertheless, both the math and English faculty resented the time they had to spend on this, particularly since the rest of the faculty did not have this responsibility.  To compound things, the administration, in its desire to make things as simple for the students as possible, insisted on having the placement test offered on many different days, at all hours of the day and evening instead of requiring all the students to show up on a single day or particular time.  Finally, none of the math faculty were satisfied with the quality of the incoming students, who were consistently placed into lower and lower mathematics courses, all emphasizing traditional skill development, over time.

A year or two back, the administration was contacted by a representative of one of the two national testing agencies with the promise that their product would solve all the problems at the school.  They presented data to show how widely used their placement test was, how easy it was to use (administered by computer, graded by computer, and immediate advice to the students about which course to take), and how effective the technology was in identifying students’ mathematical (and presumably grammatical) weaknesses.  Once the administration was sold on this, the mathematics faculty (and likewise the English faculty) were presented with comparable presentations and agreed to adopt the test to get the long-term onus off their backs.  Of course, the promise of a professionally designed test that is thoroughly tested and validated and used at so many other institutions was very reassuring.  However, one implication of this is that the entire placement process is now completely outside the purview of the mathematics department and will likely remain that way indefinitely.  And, once it is not an active function of the department, it will quickly become less of a focus of the faculty.  If there are any issues of poor placement (which have already arisen), it can be taken care of (as it was at the neighboring institution described previously) by tweaking the levels for admission to various courses.  Certainly, this will not address the inherent issues of the very poor match that such a test has between the high school curriculum in New York State and the college curriculum at this school.

Moreover, 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 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 has certainly heard 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 completely misplaced.  If nothing else, whenever the word “placement” is uttered by a speaker at a talk at an NCTM conference, many in the audience will immediately interject with their personal anecdotes about how poor the system is.  However, there is another dynamic here that bears considering.  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, there is another dynamic in effect that precludes any changes.  The people at these companies who are aware of 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 is no real 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.  As mentioned, 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.

The author is reminded of a story related a decade or so ago by a person who was at the time serving on the AP Calculus committee.  She had recently returned from visiting potential colleges with her daughter.  During a presentation on the mathematics offerings at one highly regarded public university, she asked the speaker about the kinds of technology students were expected to use in freshman calculus.  The speaker was apparently blindsided by this totally unexpected query and, in a very flustered manner, responded that he guessed that if they wanted something, students could use slide rules!  The parent immediately took her daughter’s hand and left.

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.   Unfortunately, no single high school has the ability to affect the procedures at a college or university.  However, if groups of neighboring high schools can pool their data and bring them to the attention of the local colleges, that might be effective in raising consciousness about placement issues.  What might be even more effective is for local (county-wide, say) or regional (state-wide) NCTM affiliates to present such a case to appropriate mathematics departments.  Any responsible chairman of a mathematics department would be willing to sit down with representatives of such a group to discuss a topic of serious concern (if nothing else, all colleges are under pressure these days to address administrative concerns about enrollment and establishing connections with local and regional high schools is often a priority).

However, in addition to merely presenting a collection of horror stories, it would probably be a good idea to also present some concrete suggestions about ways to address the problems. For instance, it is probably a good idea to show the chairman some non-routine problems that students are now doing in high school mathematics or to show him or her 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.  The author is aware of one large scale study [Davis & Perunko] conducted some 20 years ago at a large two year college where about 18 different factors, including placement test score, SAT or ACT score, age, last math course, and years since last math course were all studied in terms of being effective predictors of student performance.  They 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 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.

The author is 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!  They have collected information on the kinds of questions that their students will be expected to face on the national tests and prepare them accordingly.  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 the 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 post-secondary mathematics plate is hopefully shifting in the same directions, 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 DUE-0442160. 

 References

CUPM Curriculum Guide 2004, Mathematical Association of America, Washington, DC, 2004.

AMATYC Crossroads Standards, American Mathematical Association of Two Year Colleges, Memphis, TN, 1995.

Davis, Ronald and Marie Perunko, private communication.

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 American, Washington, DC, 2006.

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