Expectations for faculty, current students, and incoming students
Expectations for faculty, current students, and incoming students
Professor Jack Nachman was the chairperson during 2002-2010. Click on the graphic at the right for a video of Professor Nachman's statement on succeeding in Oakland University mathematics and statistics courses.
Departmental Policies on Expectations
Click here for a statement on the responsibilities of faculty and students, as well as our grievance procedures.
Expectations — What We Would Like to See Students Bring to College
The National Council of Teachers of Mathematics ( NCTM) lists five "process standards" and five "content standards" that they consider to be the crux of K–12 mathematics education. Within this framework, our department has the following views on what knowledge, skills, and attitudes we would like to see students bring to college. The NCTM is also preparing a "focal points" document at the high school level, to complement their recently released statement at the K–8 level. See their website for more details. In addition, please see the latest curriculum guidelines prepared by the Michigan Department of Education ( content expectations).
A summary statement is provided at the end of this list.
1. Problem Solving
In our view mathematics is about solving problems, not pushing symbols around. Students need to come to college with the idea that they will be learning to enhance their ability to solve problems using mathematical ways of thinking and mathematical tools, not that a mathematics course consists of memorizing procedures to be performed on demand. Students should be exposed to problems that involve several steps — and even problems that have no clear-cut solutions — not just toy "story problems" that succumb to routine application of formulas. (In fact the routine application of formulas that have not been derived should not be stressed. Every formula should be derived or justified before being used. It is more important to be able to understand a formula than to mechanically apply it.)
The application of critical thinking to developing problem-solving skills should be emphasized. Student should be given specific problem-solving strategies as well (a la Polya's four-step process), such as looking at simpler cases and discerning patterns, drawing pictures, or working backwards; they should never be allowed to say "I don't know how to start." They should be encouraged to reflect on a problem after they have solved it, asking whether the answer obtained makes sense and thinking of ways to go beyond the specifics to generalize their ideas.
Another aspect of problem-solving is to think about mathematics algorithmically, and the study of computer programming is an excellent choice for high school students preparing for college. The computer is an excellent tool when used effectively, and we would encourage high school teachers to spend some time applying such software as spreadsheets, computer algebra systems, statistical software, or interactive geometry programs to nontrivial mathematics problems.
2. Reasoning and Proof
The most important word in learning mathematics is "why", not "how" or "what". Students should, for example, understand why the quadratic formula gives the roots of a quadratic polynomial. They should expect to justify their solutions to problems, not merely to provide answers. They should understand some simple logic (for example, what it means for an "if...then" statement to be false). They should understand the important role of definitions in mathematics and distinguish between definitions, axioms, and theorems. They should understand the role of counterexamples, and the idea that in mathematics we require a convincing argument before accepting something as true, not just some numerical evidence or the word of an authority. Mathematics should be presented to some extent as an experimental science, in which the student or mathematician investigates situations and makes conjectures, but then tries to understand why these conjectures are or are not true.
Currently students come to college thinking that there is no need to communicate ideas in a mathematics class — that what they write down is just for themselves and need not communicate anything to anyone, because all that matters is the answer. Nothing could be further from the truth. Students should be taught to organize their mathematical thoughts and communicate them clearly, orally and in writing. (For example, they should never introduce a variable for solving a problem using formal algebra without making it clear what the variable represents.) They should also be required to spell mathematical words correctly.
Of course, before writing come reading. One might argue that the development of reading skills is not the responsibility of the mathematics teacher, but such division of labor is counter-productive. Students should learn how, for example, to read a page of text, out of which they will extract the relevant information for the analysis and solution of a problem. Moreover, they should be expected to read critically assertions related to numbers, whether presented in the text or in tables or figures.
Although abstract mathematical reasoning and skills are important, we want students to think of mathematics as having application to the real world as well. We hope that teachers will try to do a lot of mathematical instruction within realistic contexts, to motivate the mathematics and give students a reason to want to know the answers. When teaching applied problems, it is important to pay attention to units and to precision of reported answers.
The crux of mathematics is abstracting — translating unclear, informal, incomplete, or badly worded problems into precise problems using the languages of algebra, geometry, and other formalisms of mathematics. This is the mathematical modeling process. In order to do this well, students must value reading in mathematics classes as much as they value it in language arts, science, or humanities classes.
We want students to use mathematical notation correctly and confidently. They should understand equivalent ways of representing the same concepts and easily move among multiple representations (for example, viewing functions as rules, sets of ordered pairs, tables, graphs, and mappings). They should be taught to avoid nonsensical representations (such as price signs one sees in stores confusing dollars and cents, or misuses of the equals sign in chains of calculations).
1. Number and operations
Students should be proficient in the basic arithmetical calculations on integers and fractions, mentally, with paper and pencil, and with calculator. They should not have to — or want to — use a calculator for problems that can be done mentally. They should understand the different kinds of numbers and their various representations (for example, being comfortable converting among fractions, decimals, and percentages). They should also have "number and operation sense" — knowing intuitively, for example, that for positive numbers, dividing by a larger number produces a smaller quotient. We find that many students are hopelessly confused by such things as division involving 0, the difference between A divided by B and B divided by A, and even adding fractions. Students should develop a good intuition for how the arithmetic operations relate to concrete situations — for example, to understand how the remainder from an integer division may provide the answer to an applied problem.
The important link between arithmetic and algebra should be emphasized. We find that students who cannot simplify arithmetic expressions or are weak in applying critical thinking to problem solving in arithmetic will show the same deficiencies in algebra.
Technology should be used appropriately, efficiently, and creatively. Students need to understand that they need to think through the solution of a problem and not try to use the calculator to replace analysis.
As stated above, algebra should be viewed as an extension of arithmetic. In arithmetic we do operations on specific numbers we know; in algebra we do the same operations on numbers we do not know (but are thereby able to discover). Students should understand deeply the use of variables to represent unknown or arbitrary quantities and how they can change, the use of the equals sign as expressing a relationship and not as a command to perform a calculation, and the concept of a function in mathematics (which is central to all advanced work).
They should have technical skills that are usually taught in courses with titles like Algebra I and Algebra II: solving equations and inequalities, isolating variables in an equation, solving systems of equations, and manipulating algebraic expressions of all kinds (fractions, exponents, logarithms), using properties like the distributive law — and avoiding incorrect symbol-pushing, such as squaring a binomial by squaring each term.
They should completely understand the way graphs in the coordinate plane represent relationships between variables. They should be able to work with ratios and proportions to solve problems (and know when proportional reasoning does not apply). Students who plan to pursue advanced work (for example, engineering majors) should have a solid understanding of trigonometry, as well.
Most students should not be encouraged to study calculus in high school; rather they should be given a solid background in precalculus areas, which will better prepare those who need calculus to study it in college. However, those students who have completed this preparation before their senior year can take AP Calculus with the intent of earning college credit by passing the national exam and going on to subsequent courses in college.
Students should have a good intuitive grasp of geometry facts (for example, that opposite sides of parallelograms are congruent, and the Pythagorean theorem) and knowledge of the vocabulary needed to express them. They should understand the idea that mathematical objects can be defined, axioms assumed, and theorems proved; and geometry provides a good vehicle for doing these things. They should confront the idea of what it means for two geometric objects to be the same or different in various ways (ideas of congruence and similarity, and in terms of measurable parameters like area) and how these notions can be understood through transformations. They should understand the ideas of symmetry.
Geometry is important because of the level of mathematical and critical thinking required, its connection to algebra via analytic geometry (which should be given more emphasis at the high school level), its relationship to trigonometry, and the visual aspect of mathematics embedded in it. This is an area of mathematics where early introduction to the importance of definitions, proofs, logical reasoning, and making connections will be highly productive.
Students should understand the idea behind measuring attributes of objects, the limitations inherent in measurements (notions of accuracy and precision), and the arbitrariness of most units of measurement. They should understand how to work with units the same way one works with factors and fractions. They should understand the effect on measurements when sizes change (for example, that area changes as the square of linear measure, and volume changes as the cube). They should know how to compute some basic derived measurements (such as the area of a circle in terms of its radius).
5. Data Analysis and Probability
Although we would not assume any specific knowledge in probability or statistics in our courses, every informed citizen (i.e., every high school graduate) should understand some basic notions in these areas, so as to be able to interpret the results of political polls or medical studies, know how unlikely they are to have success when gambling, and know what is meant by such terms as mean, median, and mode. Beyond that, developing critical thinking skills by studying these subjects would be a bonus. It would certainly be better for students to take a good probability and statistics course in high school than to rush to study calculus. A course involving probability also provides an opportunity for discussing combinatorics and other aspects of discrete mathematics, which are increasingly important in light of computer applications.
In summary, we want students to come to college thinking that mathematics makes sense, that they can solve nontrivial problems (both in pure mathematics and within real-life contexts) by reasoning and using powerful mathematical tools, and that they are expected to communicate their solutions effectively. They need to be on top of arithmetic and algebraic skills. But perhaps most important of all, students need to develop some degree of love and enthusiasm for mathematics. We would like to see a typical high school student think like a mathematician, develop his or her critical thinking, apply the critical thinking skills in solving problems, and be able to see the interconnections among different areas of mathematics and between mathematics and the real world.
To accomplish this, students should be encouraged to take as much mathematics as possible (at least learning well the content of two years of algebra and a year of geometry, so that we don't have to offer remedial courses in college). A fourth or fifth year of college-preparatory mathematics is to be strongly encouraged, such as some advanced concepts of algebra (e.g., including mathematical induction, set theory, formal logic, the binomial theorem, functional transformations, advanced work with exponential and logarithmic functions, matrices, and lots of applications to real-life situations), trigonometry, discrete mathematics, probability, and statistics. (Students should not be rushed to take calculus in high school, and studying calculus outside the AP program makes no sense at all.) It may well be that not doing a mathematics course in their senior year is likely the single most damaging choice students can make in high school with regards to their success in college.