From the Origin provides a forum for lively discussion of issues of importance to the mathematical community. The Michigan Section-MAA Newsletter solicits opinion pieces for publication in this column from anyone in the Michigan mathematical community. In addition, comments on pieces published in earlier issues are welcomed.
Items for From the Origin should be submitted to the editor by the beginning of October to be considered for inclusion in the December issue and by the beginning of February for the April issue. Main opinion pieces should be at most 1800 words long, and responses at most 400. The editors reserve the right to shorten responses, if necessary, in order to fit as many as possible within the available space.
This issue we have two important items for discussion, so we have expanded the section to accommodate them. The first raises the issue as to whether we should allow all calculators, including those with computer algebra capability, on the MMPC, or try to restrict usage to simpler models (the current policy), or ban them altogether. The second issue is how colleges may need to adjust their placement policies to recognize that entering students may not all be getting the same flavor of mathematics in their high school classes.
CALCULATORS AND THE MMPC:
TIME TO REVISIT AN ISSUE?
by Paul Eenigenburg (WMU)
There was once a time when calculators were forbidden on exams and students were expected to solve all problems by hand! Gradually, curricula and teachers began to make more significant use of these machines in the K-12 classroom. It seemed appropriate to many that tests should reflect this increased usage.
Consequently, several years ago the Michigan Section-MAA asked me to chair an ad hoc committee to study whether calculators should be permitted on the Michigan Math Prize Competition (MMPC). After some deliberation, including feedback from local MMPC coordinators, we made the following two-part recommendation (subsequently approved by the Executive Committee): (1) with the exception of machines with QWERTY keyboards (e.g., TI-92), calculators are permitted on part 1 (a 40-question multiple choice test); (2) no calculators are allowed on part 2 (a 5-problem written test). After operating under these guidelines for several years, the advent of the TI-89 calculator has forced another look. Having the symbol-processing capability of the TI-92 but no QWERTY keyboard, the TI-89 would be permitted under current policy. However, the similar capabilities of these calculators caused many to feel that the TI-89 should fall in the same class as the TI-92. Hence, the Executive Committee decided that the TI-89 would be banned for this year, and the ad hoc committee would re-examine the policy for subsequent years.
One possible outcome is to have no restrictions at all. Students would be able to use the machine to which they are accustomed, and exam proctors would not have to distinguish between prohibited calculators and those that are allowed.
On the other hand, is an uneven playing field created when some students' machines have algebraic facility while others do not? Also, will a "no restrictions" policy result in students' bringing computers to the exam site?
It is interesting to note College Board's policy on the AP Calculus exam: the TI-89 is permitted, but not the TI-92. For them, exam question security is a large issue, and machines with a QWERTY keyboard permit fairly easy transcription of test items. This is not an issue for the MMPC; the tests are public property after they are administered, and we encourage their use to prepare for future tests.
As our committee deliberates these issues, we encourage your input. Simply e-mail me at firstname.lastname@example.org and I will convey your comments to the committee.
Editor's note: If you would like to express your opinion on this issue in the Spring issue of the Newsletter, please copy your comments to me at email@example.com, too.
BUILDING BRIDGES: PLACEMENT TESTS IN AN ERA OF REFORM
Chris Hirsch (WMU)
Pat Shure (UM-Ann Arbor)
Roger Verhey (UM-Dearborn)
A special Building Bridges session, entitled "Placement Tests in an Era of Reform" and moderated by Roger Verhey, was held at the annual Section meeting last May. This session built on the two previous Building Bridges sessions. These sessions are an attempt to establish an ongoing dialogue between high school and college level instructors on issues surrounding the transition from high school to college, and in particular the chilling effect of college placement tests on efforts to implement NCTM Standards-based curricula. This is a serious issue nationwide as evidenced by the panel discussion, "Placement Tests in an Era of Reform", chaired by Chris Hirsch at the national NCTM conference last April. As the audience heard at that conference, college placement schemes take a variety of forms, but many colleges include tests as one component of their placement procedure. Frequently these tests are very traditional multiple-choice algebra tests which have been in use for many years. They differ substantially both in content and in spirit from what is being taught in many of the newer high school courses. Few college math departments are aware of the depth of the mismatch or how totally unfamiliar such a test would look to one of these incoming students. This is discouraging to our students and provides fuel for critics who claim that "reform curricula do not prepare you for college math".
Part of the impetus for the reform movement in high school math was the recognition that students were not developing good thinking skills because so much of their time was spent doing unnecessarily complicated algebra which often took the form of algebra for its own sake. The first speaker, Chris Hirsch (WMU), began the meeting by outlining some of the goals of the CORE-PLUS Mathematics Project. He framed the CORE+ materials from the point of view of their actual content and the style of questions which the students are used to seeing. He explained that students are accustomed to seeing algebra in the context of real-world situations and problems rather than as straight manipulative exercises. Students are allowed to use graphing calculators to support their thinking, and they are seldom put under extreme time pressure to complete their tasks. Furthermore, much of the curriculum they study (e.g., statistics, discrete mathematics, modeling) never appears on placement tests. He called on colleges to recognize that students completing NCTM Standards-based curricula know considerably more mathematics than what is measured by current placement tests, and their deeper conceptual understandings of functions and modeling may compensate for some skill shortcomings. He suggested that during this transition period, placement test results might be considered advisory.
Pat Shure (UM-Ann Arbor) was the second speaker, and she started by briefly explaining placement on her campus. Since they offer only one course which precedes calculus, and since a majority of their students are heading for calculus, it is easier for them to do placement than it is for a college that offers a variety of starting courses. A combination of SAT score, overall high school GPA, and a placement test score is used to determine whether or not a student appears to need additional work before taking calculus. Students then meet with an academic advisor and decide which course to elect. Because of the questions brought up at previous Building Bridges sessions and the steady decline in algebra placement test scores over the last ten years, UM-Ann Arbor is currently reexamining its entire procedure to see whether it still predicts calculus grades accurately.
How does the placement issue look from the college vantage point? We are only beginning to see students who have completed any reform high school curriculum, and we certainly have not had a chance to assess their strengths, although we believe these strengths will become apparent soon. It seems likely that the time they have spent developing problem-solving skills will serve them well in beginning calculus. However, a persistent problem remains. Across the country, many college instructors, particularly those who teach calculus, find that incoming students from all backgrounds are being sabotaged by their inability to reliably perform what we think of as easy arithmetic and algebra.
In order to demonstrate her concerns more concretely, Pat Shure showed a series of examples of student work taken from recent UM exam papers which showed how students working through larger problems were tripped up by symbolic errors. It is important to notice that what they were having difficulty with is certainly not complicated algebra. They had to solve equations like 2 = x1/3 and x (x+1) = 6, or recognize that (1/2)x is the same as 2 -x . Thus it was clear that even the supposedly well-qualified University of Michigan students were often stumbling at this beginning level.
How do we resolve this dilemma? We believe it would be useful for instructors at the high school and college levels to get together and think about a list of skills which are really fundamental and which students should know well. To begin this discussion, here is a first attempt at such a list.
FUNDAMENTAL UNDERSTANDINGS (essential to begin calculus):
- Can they move easily between graphs, equations, and function values and understand that: if a point lies on a graph, then its coordinates satisfy the equation that led to the graph; if a problem gives linked values for the two variables, then these values give you a point on the graph; graphs are read from left to right, and a function's increase or decrease is reflected in its graph's rise and fall?
- Do they have good fluency with function notation?
- Can they move flexibly (simplifying as they go) between equivalent algebraic forms, especially those involving exponents, the distributive law, and fractional expressions?
- Do they understand the relative size of numbers well enough to see the affect of adding, multiplying, dividing, etc. a quantity by numbers that are large or small (for example, 1/ little = big )?
- Do they have the ability to work with a variety of letters as variables? They should know how to solve uncomplicated equations with both numerical and literal coefficients.
- Can they recognize and name the common functions: linear, quadratic, cubic, polynomial, power, exponential, rational, trigonometric?
- Do they know and use common terms like: domain and range, asymptote, numerator, natural logarithm, hypotenuse, radius, coefficient, etc.?
THINGS WHICH NEED TO BE MEMORIZED:
- Can they sketch the common graphs from memory including the correct shape, intercepts, asymptotes, and any key values?
- Can they quickly compute simple powers and roots in their heads? They will also need to use the symbols pi and e freely.
- Do they know some formulas: the distance formula, the Pythagorean Theorem, the quadratic formula, the areas and volumes of everyday figures?
The group at this session expressed interest in pursuing a dialogue between high schools and colleges focused on expanding and refining some such skills list. If we reach some agreement about where we are heading, it will be easier to set up placement procedures that reflect our joint expectations. Building Bridges sessions were also held at the MCTM annual meetings in October. This year's sessions focused on "A Fourth Year of High School Mathematics for College-Bound Students" and "Basic Algebraic Thinking Skills and Manipulation Skills Needed for College Mathematics".
Although we have made some progress in Michigan, it is clear that the problem is national in scope. The MAA plays a key role in placement because many colleges use some portion of the MAA battery of placement tests. Furthermore, only national organizations can address the dilemma posed by the multitude of tests facing high school teachers: state tests, national tests, textbook tests, school board requirements, etc. We would plead with the MAA to tackle this problem immediately.
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