This paper reports the results of a year-long intensive study of the teaching and learning that took place in a tenth-grade geometry class, which will be called the target class. The class took place during the 1983-84 academic year in a highly regarded suburban school district in upstate New York. The study included periodic observations of the target class and of eleven other mathematics classes, interviews with students and teachers, and questionnaire analyses of students' perspectives regarding the nature of mathematics. The target class was observed at least once a week, and was videotaped periodically for subsequent detailed analysis. Two weeks of instruction near the end of the course, dealing with locus and construction problems in geometry, were videotaped in their entirety. The analyses focused both on the mathematics that was learned, and on what the students learned about the mathematics -- including how and when they would use, or fail to use, the mathematics that they had studied.
This paper explored the difference between performing mathematical algorithms and truly understanding the underlying fundamental ideas of mathematics. Due to the structure of the study, they were able to quantify certain classroom behaviors; time spent in questioning, active learning time, amount of praise, and amount of feedback. The study also included the type of grouping, the size of grouping and so on. Originally, they defined “learning” as how well the students performed on achievement tests. I say originally, because through the article it illustrates how these tests fail in significant ways to measure subject matter understanding.
The article also suggests that providing students with repetitive routine exercises that can be solved out of context and no significance provided, actually causes the subject matter to seem frivolous to students. This monotonous work actually deprives the students of the opportunity to apply their learning in a context that is meaningful to them.
Brown and Burton (1978) developed a diagnostic test that could predict, about 50% of the time, the incorrect answers that a particular student would obtain to a subtraction problem -- before the student worked the problem! Teachers are still currently doing this, when we provide students with “distractors” on a multiple choice exam. If we already know what mistakes students are going to make, before they make them, we need to start changing our instructional models.
The predominant model of current instruction is based on what Romberg and Carpenter (1985) calls the “absorption theory of learning”: “The traditional classroom focuses on competition, management, and group aptitudes; the mathematics taught is assumed to be a fixed body of knowledge, and it is taught under the assumption that learners absorb what has been covered” (p. 26) This view is essentially implying that the “good” math teacher has multiple methods of covering the same outcome. Through these multiple methods the students will eventually “get it”. Unfortunately, what most math teachers fail to realize, is that through these multiple methods we are forcing the students to “get” something entirely different; resistance to change.
Math teachers need to stop testing students on algorithms and more on the understanding of math concepts. Below is an example of two problems Werthemier (1959) gave to various elementary school students.
Many of the students, who were deemed as high achievers, added the terms in the numerator and then performed the indicated division. They followed the conventional order of operations of BEDMAS, or to some PEDMAS. An idea which should not be taught to students out of context, which I have wrote about here. Even though these students calculated the correct answer, I would argue that these students do not demonstrate any depth of understanding of mathematics. To truly understand the underlying substance students should recognize that repeated addition is equivalent to multiplication and division is the inverse of multiplication.
This example truly illustrates that being able to perform the appropriate algorithmic procedures, does not necessarily indicate any depth of understanding. Also, the sad truth; virtually all standardized exams for arithmetic competency focuses primarily on algorithmic mastery, and not deep understanding of the math concepts.
This is the first part of the article summarized, more to come….
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