«Articulation Issues: High School to College Mathematics Bernard L. Madison, University of Arkansas Susan Forman, Bronx Community College The ...»
Articulation Issues: High School to College Mathematics
Bernard L. Madison, University of Arkansas
Susan Forman, Bronx Community College
The Mathematical Education of Teachers II (MET II) (CBMS, 2012) chapter on high
school teachers opens by noting that the double discontinuity experienced by prospective
high school mathematics teachers and described by Felix Klein (1908) still exists today.
As stated in MET II, this double discontinuity consists of the jolt experienced by the high school student moving from high school to university mathematics, followed by a second jolt moving from the mathematics major to teaching high school (p. 53). MET II addresses ways of smoothing the second jolt, but both jolts and extensions will be considered here, as they are the essence of the articulation issues between school and college mathematics.
This article is being written as the Common Core State Standards in Mathematics (CCSSM, 2010) are being implemented in almost all states in the U.S. Consequently, there is little evidence beyond speculation as to how CCSSM will impact the transition from school to college mathematics. The impact of the new standards and the associated assessments will not be evident for several years, but inertia in the K-20 education system will likely prevent major changes.
Readiness for College Mathematics As in most state standards that preceded CCSSM, and in college admissions testing by ACT and the College Board, readiness for college mathematics has been a much discussed and sought after goal. For example, ACT has a benchmark score of 22 on the mathematics test for readiness for college mathematics (approximately 560 on SAT mathematics reasoning). This definition of college readiness is narrow, focusing only on the likelihood of making a grade of C or better in the first degree-credit bearing college mathematics course, often college algebra. CCSSM aims at college and career readiness, built on the principle that all students will meet the standards laid out for high school mathematics as with the NCTM 2000 standards. The assessments associated with CCSSM are being developed by two consortia of states to be implemented in the 2014school year. The CCSSM comprehensive assessment results, scheduled for the last part of grade 11, should therefore be available for consideration for the entering class of college students for 2016-2017. Some higher education institutions and systems in the states that signed on to CCSSM have agreed to use the CCSSM assessment results as a measure of readiness for college level mathematics, analogous to many institutions currently using the results of the ACT mathematics score or the SAT mathematics reasoning score. The agreements to use the CCSSM assessment scores in this way typically have been made at upper administrative college and universitylevels and not by mathematics departments. How mathematics faculties will accept and react to CCSSM assessments is largely unknown. In any event, validation of these assessment results as a reliable measure of readiness for college mathematics is likely half a dozen years or more away at this point.
However, the practice of placing entering college students in one of several possible entry-level mathematics courses is not likely to change with the implementation of CCSSM (as outlined above). Whether a student is prepared for calculus, for precalculus, for college algebra, or some other entry-level course likely will not be determined by CCSSM assessments. Therefore, placement programs will still be necessary.
What do we know? We know that for the near future, placement programs will continue to be needed and will probably not change.
What would we like to know? We would like to know how the implementation of CCSSM will affect the mathematical preparation of entering college students and what the CCSSM comprehensive assessments will tell us about these students. In addition, what changes might be needed in current placement tools to better align outcomes with the mathematics students will be expected to learn in high school.
What resources are available? College Board and ACT each offer placement testing systems, Accuplacer and COMPASS, respectively. The MAA placement tests are offered by Maplesoft and consist of the traditional tests in basic algebra, advanced algebra, elementary functions, trigonometry, and calculus readiness. The newest (2010) test is the Calculus Concept Readiness (CCR) instrument, and another new test, Algebra and Precalculus Concept Readiness (APCR), is being field tested in 2013 -2014. Other placement testing systems are available from publishing companies, and some, e.g.
ALEKS, have tutorial systems included.
Including Remediation1 in College Algebra One of the continuing articulation issues is that of requiring developmental courses in colleges and universities. For example, in Arkansas, state regulations require that students who have an ACT mathematics score less than 19 must complete a developmental mathematics course prior to enrolling in a degree-credit bearing mathematics course. This results in more than 40% of the entering college students being placed in developmental mathematics courses, namely courses that are prerequisites for college algebra. Since ACT sets as a benchmark an ACT mathematics score of 22, that creates a band of scores 19-22 that ACT does not believe indicate college readiness, yet Arkansas policies say otherwise. In response to this, some institutions (e.g. University of Arkansas, Fayetteville) have created an alternate college algebra course with more class time and more support for students with ACT scores of 19-22. This has proved far more efficient and effective than placing these students in a developmental course and then expecting them to finish college algebra.
We are adhering to the distinctions between remedial and developmental as
outlined, for example, by Ross (1970). Remedial instruction provides instruction in prerequisite material that is not a part of the course’s objectives. Developmental courses have specific learning objectives that are required of subsequent courses, e.g. college algebra.
Some placement examination systems (e.g. ALEKS) have instituted ways to provide learning tutorials to move the student from one placement level to a higher level. Often the purpose is analogous to the college algebra with support scheme above, namely to move the student from a developmental placement to one that is degree credit bearing.
What do we know? We know that developmental mathematics courses in college are minimally successful in moving traditional age college freshmen from being unsuccessful in mathematics to being successful.
What would we like to know? We would like to know better ways to improve learning of mathematics in high school. Better yet, we would like to know how to better motivate students to learn mathematics, especially algebra, the first time they study it.
Calculus and Precalculus Attrition from the algebra-precalculus-calculus sequence is known to be significant and affects the number of students in the science, technology, engineering and mathematics (STEM) pipeline. As these courses overlay the intersection of high school and college mathematics, clarity and consistency in content and cognitive demands are needed for good articulation and realization of the potential for understanding in the next course and beyond. Although the content of precalculus courses may be the same, the cognitive demands of courses can be very different. Recently, the content and cognitive demands of algebra, precalculus, and calculus courses have been studied. As Carlson, Oehrtman, and Engelke (2010) point out, “there is now substantial research on what is involved in learning key ideas of algebra through beginning calculus. However, a cursory examination of the commonly used curricula suggests that this research knowledge has had little influence on precalculus level curricula” (p. 114). (See also, Tallman & Carlson, 2012.) Currently, an NSF-funded project of the Mathematical Association of America (MAA) is aimed at using the research results pointing to conceptual understanding needed to succeed in algebra, precalculus, and calculus to construct placement tests to measure this essential understanding. The first of these tests, the Calculus Concepts Readiness (CCR), is being used by some institutions to test for calculus readiness. Preliminary results indicate that many beginning calculus students do not have strong understandings of fundamental concepts, the major one being that of a function, especially viewed as a process. Of the twenty-five multiple choice precalculus items on CCR, students in calculus 1 at major universities on average get fewer than half of them correct. Moreover, the results are similar when CCR was administered to a sample of a couple hundred high school mathematics teachers, mostly teachers of algebra and precalculus. A second test, Algebra and Precalculus Concept Readiness (APCR), based on the same research, is nearing completion.
The fact that many students in the calculus 1 courses in college are achieving passing grades without having strong understandings of seeming essential precalculus concepts suggests that the research results may be wrong. However, it more likely points to the lack of cognitive demands of the calculus courses themselves. This latter point is supported by the results of Tallman and Carlson (2012) from examining a sample of 150 final examinations from college and university calculus 1 courses. They determined that about 15% (slightly different for different kinds of institutions) of the examination items required understanding of concepts or applying understanding of concepts. That means that 85% of the items required recall of information or recall and application of a procedure. Interestingly, a similar examination of Advanced Placement (AP) calculus free-response items found that approximately 40% of the items required applying conceptual understanding. The fact that performance on the AP examination has a different scale for determining grades dilutes this comparison and does not necessarily indicate that the AP calculus students exhibit stronger conceptual understanding of calculus than the university calculus students.
The above points strongly to weaknesses in the algebra-precalculus-calculus sequence caused by lack of cognitive demand. Surely, these weaknesses cause difficulties in subsequent STEM courses, indicating a lack of articulation between school and college or within colleges themselves. Another place that this lack of articulation surfaces is within the mathematics major. Many mathematics majors experience a jolt when they move from the more methodological calculus courses into the more
courses in algebra and advanced calculus. In fact, many (if not most) mathematics departments have instituted bridge courses (e.g. introduction to proof) to soften this jolt. Computer-based homework systems that provide testing using multiple choice items, often used in courses up through the calculus sequence, can aggravate this jolt, as the cognitive demand of such computer managed courses is often well below that of a junior-level course in abstract algebra or advanced calculus.
What do we know? Strong evidence suggests that the algebra-precalculus-calculus sequence, whether in high school or college, is not meeting its potential for use in subsequent courses or in preparing students, particularly mathematics majors, for smooth transitions to more abstract and advanced study.
What would we like to know? We would like to know how to influence schools and colleges to offer precalculus and calculus courses that are more cognitively demanding.
Differing Systems and Pedagogies High school mathematics classrooms often differ from college and university classrooms.
Most high school mathematics classes are in interactive classrooms with less than 30 students. Many incorporate collaborative learning situations, frequently with inquirybased instruction. Contrast that with a lecture style university classroom with more than 100 students, sometimes many more than 100. This system of large lecture-style classes, present in many large universities, is not only different from the system in most high schools, but it is also inconsistent with what research in learning theory tells us that is most effective for long-term retention and transfer, which provides another instance where research results are not significantly changing classroom practices.
These differences will potentially increase with the use of online courses and degree programs in colleges and universities. The potential of delivering high-quality instruction by expert teachers to all corners of the world is indeed attractive, but many questions remain about promoting interaction and keeping the cognitive levels of grading high.
Some of these questions are raised in the following section on what is known about how people learn best.
Ignoring How People Learn Best for Long-Term Retention and Transfer The expanded edition of How People Learn (Bransford, Cocking, & Brown, 2001) reported research results on learning and how these results can improve teaching and learning. Subsequent to the publication of How People Learn, Diane Halpern and Milton Hakel (2003) reported the results of a consensus agreement among 30 experts on the science of cognition in “Applying the Science of Learning to the University and Beyond.” They summarized the findings by giving ten basic laboratory-tested principles (listed in brief below) needed for enhancing long-term retention and transfer. In the opening paragraphs Halpern and Hakel (2003) write, ”We have found precious little evidence that content experts in the learning sciences actually apply the principles they teach in their own classrooms. Like virtually all college faculty, they teach the way they were taught. But, ironically (and embarrassingly), it would be difficult to design an educational model that is more at odds with the findings of current research about human cognition than the one being used today at most colleges and universities” (pp. 37-38).
So, many of us in collegiate mathematics are unaware of or ignoring the research results on what concepts students need to understand to be successful in calculus, and we are seemingly joined in this by our high school colleagues. However, high school classroom practices are much more in tune with the ten Halpern and Hakel (2003) principles than are most college classrooms.