From The Origin

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.


by Jerrold W. Grossman (OU)

How much mathematics should a public school teacher of this subject know? This is a question to which the Michigan Department of Education (MDE) provides a partial answer: It requires all K-12 teachers desiring mathematics certification to pass a test to demonstrate their content knowledge. This Michigan Test for Teacher Certification (MTTC) is mandated by state law. I hope to argue here that their approach is flawed in at least two respects, and that we as mathematicians at the state's institutions of higher education can and should take an active role in improving the system. (For those who are tired of reading arguments for or against certain curricular projects in these pages, you will be pleased to note that this discussion is independent of such issues and is something, I hope, on which we can all agree.)

Administering the MTTC is farmed out to National Evaluation Systems (NES) of Amherst, Massachusetts. NES publishes a study guide that explains the test development process, states test objectives for each subject area (mathematics, chemistry, etc.), and gives a handful of sample test questions. In addition to the information contained in this guide, I have a response to a letter I wrote to the MDE in 1996 from a supervisor in the Office of Professional Preparation and Certification Services, who explained the philosophy of the program and the test development process.

This supervisor asserted that "the academic majors and minors that teacher preparation candidates complete should not differ significantly from majors and minors in the Arts and Sciences; similarly, the majors and minors that candidates for elementary and secondary certification complete should not be substantially different from each other." We all have a pretty good idea of what level we expect a math major's program to reach - advanced calculus with proofs, abstract algebra, etc. Surely someone teaching geometry, algebra, trigonometry, AP calculus, and other nontrivial high school mathematics should have a solid grounding in such advanced college-level mathematics. But can anyone seriously suggest that all elementary school teachers who want a minor endorsement to allow them to teach mathematics in grades 6-8 must study these subjects? Realistically, only a few elementary education math majors or minors have completed even a couple of calculus courses.

Indeed, the National Council of Teachers of Mathematics' Professional Standards for Teaching Mathematics (NCTM, 1991) lists quite different requirements depending on grade level. In brief, they recommend nine semester hours of content for grade K-4 teachers (assuming three years of college-prep high school math as prerequisite), 15 hours for grade 5-8 teachers (assuming four years of college-prep high school math as prerequisite), and a full math major for 9-12 teachers (at my university, that's 42 hours).

The MDE's letter to me goes on, "The passing scores for the tests are set at the minimum level of competence that is required for a teacher to be successful in a classroom." If we are serious about the content knowledge we expect of those teachers certified up to grade 12, then this level must, as argued above, be fairly high. To pretend that all those we will accept to teach up to grade 8 can (or should) achieve this level is absurd. Thus the test must necessarily become meaningless for assessing real competence for high school teachers.

The information provided about the test development process revealed a glaring omission: Apparently no mathematicians were involved. The tests are developed by "teams comprised of practicing Michigan teachers and teacher educators." The teams begin by writing down detailed objectives covering all areas of K-12 mathematics (a comprehensive list about four pages long, which seems consistent with the NCTM's published national standards), and validate these objectives by surveying current teachers and teacher educators. Surely these are appropriate constituents to involve in the process. But wouldn't it also make sense to ask members of university mathematical sciences departments their opinions at this point? Part of the goal of K-12 mathematics instruction is to prepare students for college mathematics, and we can give valuable advice on what sort of preparation these students should have. Next the teams construct test questions to measure attainment of these objectives. The questions are evaluated by teachers and teacher educators (again, not by mathematicians, apparently), reviewed for possible bias, field-tested, and adopted. The State Board of Education determines the passing score, but it is not made clear how this level is set. The passing score is the same for everyone, whether the person is seeking a major or a minor, whether certification is at the elementary (K-8) or secondary (7-12) level.

The study guide provides a sample of ten multiple-choice questions in mathematics, and on the whole they are good problems. Some could be answered by any intelligent layperson (respond that the best way to communicate average monthly temperature over a period of five years is a line graph, rather than as raw numerical data or pictographs). Others require technical knowledge that one would acquire in a precalculus course, combined with a healthy dose of problem-solving ability (find the area of a regular dodecagon circumscribed around a circle of area A). Probability and statistics are emphasized, as well as the more traditional areas of the high school mathematical sciences curriculum. One of the questions appears to be geared toward the objective of "understanding the definite integral and its application to the problem-solving process" by "using algebraic and geometrical techniques to approximate the area under a curve"; but in reality it just tests common sense and function notation (more on this question below).

When I think about the students I teach who get elementary education math minors (the content requirements are a statistics course, a year-long "math for elementary teachers" sequence, precalculus, and a course on applying computer programming languages to appropriate mathematical content), I know that most of them cannot hope to make a really respectable showing on the kinds of questions in the study guide, but in fact even a lot of the weaker students pass. When I think about the students I teach who get real mathematics majors in preparation for high school teaching certification, I am encouraged to see their problem-solving skills being challenged by some of the questions, but wonder why they are not expected to know any more than the better students they teach would be expected to know.

Unfortunately, some of the questions have errors or are not well focused or well written. It is here that input from knowledgeable mathematicians and statisticians is crucial if these tests are to have credibility. In one question we are given a graph of a function passing through the origin and increasing in an irregular but monotonic way to about (3,150). The vertical axis is temperature in degrees Fahrenheit and the horizontal axis is time in minutes. We are told that this graph "represents an electric oven's internal temperature" and are asked to choose, from four options, an expression that estimates the area under the graph, because an engineer wants to use it "to estimate the amount of electrical energy consumed by the oven." Unless this oven is sitting outdoors in a Michigan winter, something seems wrong with the data, not to mention the physics. Another question asks us to find the probability that at least one of three frogs captured in a field is female, under the sole assumption that "capturing a male frog and capturing a female frog are equally likely." Does the composer of the problem understand that an assumption of independence (which is unlikely to be true in this setting) is needed in order to compute the intended answer (7/8)? The state supervisor in her letter to me admitted that the concerns I raised "were correct, [but] these questions did not represent a serious or significant problem for the candidates or the testing program."

Perhaps if the Michigan Council of Teachers of Mathematics (MCTM) and the Michigan Section-MAA meet with the MDE, we can effect an improvement in this unsatisfactory situation. First we must persuade the department to set more realistic standards - higher for the secondary teachers, and encompassing a more limited syllabus for those being certified only to grade 8. Second, we must insist that professional mathematicians and statisticians be made a part of the test construction process. Please contact me if you have relevant information on this issue or wish to get involved.

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