Research on Motivation
In summarizing 20 years of research in the area of motivation in mathematics education, James A. Middleton and Photini A. Spanian (2002) found five main factors that influence motivation:
First, motivation or lack of motivation is learned. In primary grades many children report that mathematics is their favorite subject. However, educators follow grouping practices, which convince students in the middle grades that only the fast students can succeed and success is an innate ability. These poor attitudes extend to high school and college with students attempting to avoid the subject where they feel they are poor learners.
Second, motivation hinges on students' perception of their abilities to succeed or fail. Researchers have found that if students are successful about 70% of the time they will continue to be engaged.
Third, intrinsic motivation is better than a reward. Thus mathematics teachers must find strategies that interest the student in significant, rich mathematical thinking.
Fourth, inequities are influenced by how different groups are taught to view mathematics. Past studies have indicated that teachers expect greater success from boys studying mathematics and expect more boys to study higher level mathematics. The gender gap on national assessments has been closing, and the trend is for more girls to take more advanced mathematics courses in high school, but a gap still exists. The same can be said about differences across racial and ethnic groups.
Fifth, teachers do matter. "Students in inquiry-based classrooms are less likely to believe that the teacher's way ... leads to success" (Middleton & Spanian, 2002). Instead they come to believe that success comes from working hard to understand the mathematics. Teaching concepts within a context has the advantages of (1) piquing students' interest, (2) stimulating their imaginations, and (3) giving functional mathematics knowledge useful in applications.
The good news from this set of studies is that you can influence students to do, learn, and enjoy mathematics. Your best practice is to know your students-their interests and how to challenge and encourage them.
Additional Research on Learning
Kloosterman (1996) reports on a study in which he taught two classes of youngsters to solve nonroutine problems. To encourage the students to think for themselves, he refused to tell them whether their answers were correct. Soon some students refused to work, presumably because he was not doing what a teacher should do. However, after a while the students adapted and began to work very diligently to double-check their problems to ensure correctness. Kloosterman (pp. 135-137) lists the following beliefs about learning mathematics that are necessary to consider and debate:
- "Mathematics is computation." For students to be willing to persist on nonroutine problems, they must not believe that the hallmark of a good mathematician is to perform computations quickly, but rather appreciate the benefit of the ability to attack and solve problems new to their experience, even when it takes time to work one problem.
- "Mathematics requires proof." For students to become comfortable with making conjectures, they must go beyond the concept of mathematics as simply structured, deductive proof and instead perceive it as requiring mathematical justification.
- "Mathematics is useful." For students to recognize the application of mathematical procedures in solving real-world problems, they must perceive numerical operations as tools, not as goals in themselves.
- "Mathematics topics are integrated." For students to try a variety of approaches to problem solving, they must envision an integration of mathematical concepts and processes instead of a series of unrelated topics, each with its own solution method.
- "Mathematics consists of clearly defined problems." For students to learn to estimate and make reasonable assumptions, they cannot expect all mathematical problems to include only exactly what is needed in the problem.
A group of exemplary high school teachers conducted action research by alternatively teaching units with either direct instruction or guided discovery methodology to examine students' preferences and beliefs about learning mathematics. The teachers had been trained in constructivist techniques in an extensive 3-year professional development program. Three of the teachers primarily taught the innovative curriculum, Interactive Mathematics Program (IMP). A fourth teacher who had previously taught this curriculum began a new position in a very traditional school with a traditional mathematics curriculum. The teachers collaborated to rewrite the IMP materials into direct instruction format and the traditional curriculum into discovery lessons. They varied the order of presentation methods of several units of coursework and surveyed the students to find their preferences in modes of instruction. Half of the students had a strong preference for guided discovery, one-fourth had no preference, and one-fourth preferred direct instruction. The statistically significant results showed that students were more likely to feel positive about their mathematics class and more likely to rate themselves highly on their mathematics ability when taught by guided discovery. These results held when gender, grade level, course grade, or previous curriculum experience were considered. A majority of students enjoyed some teaching and learning with each pedagogy, that is, they liked variety. The students had remarkably mature thoughts on the research. One said, "The teachers should decide which method to use based on the concepts in the curriculum" (Callis, 1997, p. 57).
During our teaching experiences in constructivist classrooms in quite different educational settings, we found the secondary-level students very interested in knowing why we chose to teach in this mode. From time to time we would briefly explain basic beliefs of the characteristics of the subject and beliefs about effective learning that led to curricular decisions. The students were very receptive to these ideas. Some students, especially the high achievers, were nervous at first. It seemed they felt it a little unfair that the "learning game" had changed. They knew how to "win" in the competitive, teacher-dominated classroom and were not sure they would still succeed in the constructivist culture. This uncertainty always passed within a few weeks, however. A student may half jokingly say, "Please just give me the answer!" when you answer his question with a question. Try responding by smiling and saying, "I wouldn't insult your intelligence that way, but I will give you a clue if you choose:'
Students are motivated to model behavior that leads to outcomes they value. Mathematics students ask, "When are we going to use this?" Speakers from the community or videos such as Futures by Jamie Escalente—real-life hero of the film Stand and Deliver—help motivate students by presenting role models attesting to the importance of problem solving and skills in the world of work.
However, mathematics is not only a means to an occupation but a worthy human-made universe to explore for its own sake. Sometimes one author (LH) would answer, "It doesn't matter. We learn it because it is so beautiful to see how everything fits together." I got a few strange looks in response, but the students did think about what I said.
Incorporating contexts from students' personal experiences adds interest to the course of study. In the affective domain, teachers must model respect for students to lay the foundation for a community of learning in which contributions of all class members are honored. Knowing that a teacher is interested in them as human beings can motivate students to persist even when outside pressures might distract them. Enthusiastic teachers who display a love and knowledge of mathematics, have expectations of student success, and utilize wide-ranging problem-solving techniques are those who motivate students to achieve academic excellence.
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