Ways of Knowing in the Discipline

How Knowledge is Formally Established in the Disciplines

As disciplines of investigation, both mathematics and science are more than simply accumulated bodies of knowledge; each embodies a unique set of ideas about what it means “to know” something, and how that knowledge is generated. The Conference Board of the Mathematical Sciences (2001) recommends that teacher preparation include a specific focus on the nature of knowing and generating knowledge in the discipline of mathematics. These recommendations, and related arguments for this kind of teacher knowledge of both mathematics and science, include attention to the formal establishment of knowledge in the discipline through mathematical proof and scientific inquiry.

Cuoco (2001, 2003) suggests that a teacher needs to know mathematics “as a mathematician,” including the nature of generating knowledge in the discipline. He notes, for example, the importance of understanding how doing mathematics often involves temporarily accepting certain ideas as true without proof to determine whether they offer productive avenues of thought, later providing proofs for these ideas if they turn out to be helpful.

In their studies of the work of teaching, Ball and colleagues have also identified ways of knowing and working in the discipline as important aspects of teachers’ mathematical knowledge. Formal ways of presenting knowledge are a part of this knowledge base: “Teaching also involves using tools and skills for reasoning about mathematical ideas, representations, and solutions, as well as knowing what constitutes adequate proof.” (Ball, 2003, p. 6-7)

In science, the ways of knowing may vary among disciplines. While the same general epistemological principles hold across disciplines, some phenomena (e.g., chemical reactions) lend themselves to classical experimentation, while others (e.g., the birth and death of stars) do not. Teachers’ attempts to impart “the scientific method” to students may do more harm than good. Referring to the preparation of science teachers, NSTA states:

In general, the term “scientific method” (for the hypothetico-deductive method) should be avoided, since it may lead students to believe there is only one way to conduct scientific inquiries. Inductive studies have played a valuable role in science, as have mathematical and computer modeling. Hypotheses are not used formally by scientists in all research, nor are experiments per se the substance of all research. Candidates should study cases in which different approaches to inquiry are used in science, and should endeavor to communicate such differences to their students. (NSTA, 2003, p. 19)

For the facets of teacher content knowledge bibliography, click here. [PDF 17K]

Habits of Working and Thinking that Characterize the Disciplines

Science and mathematics each embody a unique collection of ideas about how knowledge generation arises and accrues within the discipline. The National Science Teachers Association (2003) urges explicit attention to science as a way of knowing in teacher preparation programs. Describing the successful science teacher, NSTA (2003) states:

Teachers of science engage students effectively in studies of the history, philosophy, and practice of science. They enable students to distinguish science from non-science, understand the evolution and practice of science as a human endeavor, and critically analyze assertions made in the name of science. (p. 16)

Similarly, the Conference Board of the Mathematical Sciences (2001) has recommended that teacher preparation include a specific focus on the nature of knowing and generating knowledge in the discipline of mathematics. The habits of working and thinking that characterize the disciplines and underlie the generation of knowledge are included alongside the formal means of establishing knowledge (i.e., mathematical proof, scientific inquiry).

The Conference Board of the Mathematical Sciences (2001) recommends that teacher preparation include, “[a]ttention to the broad and flexible applicability of basic … modes of reasoning” and “should develop the habits of mind of a mathematical thinker.” These habits of mind include “actions like representing, experimenting, modeling, classifying, visualizing, computing, and proving.” Wu (1999) provided more elaboration:

Content knowledge of mathematics includes the knowledge of how mathematics is usually done: the unending trials and errors, the need to search for concrete examples and counterexamples to guide one’s intuition, and the need to make wild guesses as well as subject these guesses to logical scrutiny. (Wu, 1999, p. 3)

Cuoco (2001, p. 171), too, noted the importance of teachers understanding “how mathematical results are obtained” as opposed to just “how they are presented.” He described the underlying thinking and work of mathematics as including “false starts, extensive calculations, experiments, and special cases.” Cuoco (2003) termed this kind of knowledge as knowing mathematics “as a philosopher,” including the habits of mind that characterize mathematical work.

The American Association for the Advancement of Science’s Science for All Americans (1989, p. 147) asserts that “science, mathematics, and technology are defined as much by what they do and how they do it as they are by the results they achieve,” recommending that teachers provide students with “some experience with the kinds of thought and action that are typical of those fields.” Similarly, the National Research Council’s publication Ready, Set, SCIENCE! (2008, p.19) includes “generating scientific evidence” as an important component of effective science instruction, suggesting that teachers need to highlight that “(e)vidence is at the heart of scientific practice,” and that “proficiency in science entails generating and evaluating evidence as part of building and refining models and explanations of the natural world.”

For the facets of teacher content knowledge bibliography, click here. [PDF 17K]