What is the point of teaching students if they're not learning? Here a general education course operates from a learning-theoretic mold; mathematics education instructors become involved to help students construct their own learning.
Background and Purpose
Creating a general education course that is successful in developing quantitative reasoning is a reoccurring issue for most mathematics departments. A variety of approaches currently flourish, ranging from offering practical problems to viewing the course as linked to sociological, historical, or "awareness" issues. For many of us, such a course is a major source of a department's credit hours, yet the mathematics community has not been as aggressive about looking into new models for these courses as they have been, for example, about calculus reform. Moreover, general education courses in mathematics are usually not found to be enjoyable courses, either by students or by instructors. Students are often thrust into these courses because they need a single mathematics course and often have failed to place into a higher-level course. During a typical course of this nature, the instructor quickly proceeds through various skills, like Bartholomew shedding his 500 hats, providing exposure to a range of "interesting" topics rather than helping students develop in-depth understanding.
Illinois State is a midsize multipurpose university of 20,000 students, drawing primarily from the state. As members of the mathematics department who are doing research on undergraduate learning and who are interacting with RUMEC , we began to implement learning-theory ideas to design our general education course, "Dimensions of Mathematical Problem Solving." In order to do this, we asked ourselves: What does it really mean for students to "learn?" What do we most want them to achieve from such a course? And, how can this course offer students an enriching, positive experience in mathematics? The professional literature [3, 5, 8] has stressed the need to promote active learning and teacher-student interaction. For example, the report Reshaping College Mathematics  speaks to "the necessity of doing mathematics to learn mathematics," "the verbalization and reasoning necessary to understand symbolism," and "a Socratic approach, in which the instructor works carefully to let the students develop their own reasoning" (pp. 111-112). This mode of instruction has been used by us and other mathematics education instructors who have worked with developing reasoning and problem-solving skills of students in our courses. Using this shared-knowledge base about teaching undergraduates, the pedagogy for this new education course developed.
Even though our students have had three or four years of high school mathematics, they still believe that doing mathematics means memorizing and identifying the appropriate equation or formula and then applying it to the situation. We wanted a course that would generate substantial improvement in their reasoning and problem-solving skills rather than exposing them to more topics to memorize and to not understand. The philosophy guiding the "Dimensions" course can best be summarized by the following paragraph from the syllabus of the course.
It is important that you realize that you cannot understand mathematics by observing others doing mathematics. You must participate mentally in the learning process. This participation includes studying the material; working with others; struggling with non-routine problems; using calculators for solving and exploring problems; conjecturing, justifying, and presenting conclusions; writing mathematics; listening to others; as well as the more typical tasks of solving problems and taking examinations. The emphasis in this course will be on ideas and on understanding and reasoning rather than memorizing and using equations or algorithms.
Mathematical topics are selected from number theory, discrete mathematics, algebra, and geometry. Special attention is given to topics from the 6-10 curriculum for which students have demonstrated a lack of 1) conceptual understanding and 2) reasoning and problem-solving skills. By focusing on material that students have presumably covered, such as proportional reasoning and algebraic generalizations, we can emphasize the development of reasoning and problem-solving skills as well as conceptual understanding. Guiding us are the following course principles.
Many believe that proper assessment means collecting data for evaluations for grades. These data present only a snapshot or limited view of in-depth understanding. In "Dimensions," we have designed a course in which assessment is an ongoing, formative process. A guiding principle for our pedagogy is that what and how our students feel about learning mathematics affects what and how they learn mathematics.
To measure attitudes, at the beginning of the course we administer the "Learning Context Questionnaire."  This questionnaire is based on the works of William G. Perry Jr. [2, 6] and requires students to select a response from a six-point Likert scale (strongly agree to strongly disagree) on 50 items, some of which are "I like classes in which students have to do much of the research to answer the questions that come up in discussions," and "I am often not sure I have answered a question correctly until the teacher assures me it is correct."
We also administer a second survey which we developed, "Survey on Mathematics," that asks students more about their beliefs of what it means to learn mathematics, such as
1. How would you respond to someone in this class who asks, "What is mathematics?"
2. What does it mean to understand mathematics?
3. How do you best learn mathematics?
4. Explain the role of reasoning when doing a problem that uses mathematics.
5. Describe the role of a mathematics teachers AND describe the role of a mathematics student.
6. Describe a memorable mathematical experience you have had.
On the Learning Context Questionnaire, a low score (received by about 65% of our students) describes students who look to authority figures for correct answers or for making decisions. These students do not believe they have the ability to work out solutions by themselves, suggesting immature cognitive development. About 25% of our students fall into the second category. These students realize there may be different perspectives to situations; however they are not able to differentiate or to evaluate among perspectives. They still tend to look to authority figures for help in finding solutions and have little confidence in their own ability to find solutions. Only 10% of the students in these classes are at the upper end of the cognitive development scale. These students take a more critical look at situations and rely on their own ability to find solutions to problems.
The results of the Survey of Mathematics are analyzed both by noting clusters of similar responses for each question and by making connections among individual student's responses over all questions in order to better understand students' perspectives on learning mathematics. For example, responses to the first question typically indicate that students often think of mathematics as "the study of how to find an answer to a problem by using numbers and various theories." Some note that in mathematics there is only one correct solution. About 20% indicate that mathematics and understanding are related, such as "mathematics deals with understanding and computing numbers." However, question two provides additional insights into what students mean by understanding. Students feel that being able to explain their answer or being able to solve equations and perform procedures means to understand. For question #5, most students state that teachers need to go through procedures step by step, allow students time to practice, and show specific examples, "I learn the best when a teacher explains step by step the answer to a question." Few students state, as one did, "...to really learn you have to actually sit down and try to figure it out until you get it." Thus, students perceive that the role of the teacher is to show, tell, explain, and answer questions. Their perceived role of a mathematics student is to listen or to pay attention, ask questions, and some even state "to take notes." There is probably no better place to bring to student attention these fundamental misconceptions of our mutual roles than in a general education course.
Use of Findings
What we find from both the Learning Context Questionnaire (LCQ) and our own survey about how students anticipate learning mathematics forms our general approach as well as day-to-day decisions about our instruction. We do not lecture students on how to think about solving problems, but guide them to develop their own mathematical processes. Because the majority of our students are at the lower end of the scale on the LCQ, we recognize the need to help them realize that they are ultimately responsible for their own learning and "sense-making."
Class periods are spent analyzing varieties of solutions by having students use the board or transparencies to present their solutions. In addition, students must provide complete explanations about their reasoning and learn to include appropriate representations (pictorial, algebraic, graphical, etc.). Grading on written assessments is based on correctness of reasoning, understanding of the situation, and completeness and quality of explanations. However, we also informally assess by comparing students' mathematical performance with that suggested by their beliefs to evaluate how the students' problem-solving and reasoning skills are developing.
During a class session, we first determine if a strategy used by the student is comparable to a strategy used in class, looking for whether the student has merely mimicked the class example or was able to generalize. An important component of our assessment is the degree to which students' explanations match the students' use of numbers and symbols: often the answer is correct, but the explanation does not match the symbols. In other words, we look for cognitive growth on the part of the student. Our ongoing assessment (both verbal and written) assists us to identify misconceptions that may have arisen as a result of classroom discourse.
The course as now developed and taught is labor-intensive. We continually search for content-rich problems. In addition, the instructor contributes a lot of effort to encourage students to volunteer and to respectfully respond to other students' reasonings and explanations. It is not surprising that with such a format we initially meet with resistance and frustration from students. It becomes important to resist the temptation to tell students how to do a problem. Likewise, we must reassure students that we are aware of their frustration but that making sense out of mathematics for themselves is a worthwhile and rewarding experience. Unfortunately, we find we cannot have all students understand every topic, so we focus our attention on those students who are willing to spend 6-10 hours each week struggling to understand and make sense of the mathematics; it is these students who influence our instructional pace.
As part of our ongoing course development, the weekly planning sessions for instructors provide an opportunity for instructors to share. These sessions become mini staff development sessions with all instructors learning from each other.
In spite of the many obstacles and demands this course creates, it is rewarding that we are able to collaborate with students in their learning. For many of our students, this course experience has allowed them for the first time to make sense out of mathematics. We see our course as a process course, and, as such, we feel we are able to offer students the best possible opportunity for improvement of their lifelong mathematical reasoning and understanding processes. Simultaneously, we put ourselves in an environment in which the drudgery of low-level teaching is replaced by the excitement of being able to dynamically reshape the course as our students develop their mathematical reasoning, understanding, and problem-solving skills.
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