## Research on teaching mathematics: Making subject matter knowledge part of the equation (1991)

Venue: | in Advances in Research in Teaching, Volume 2 |

Citations: | 15 - 0 self |

### BibTeX

@INPROCEEDINGS{Ball91researchon,

author = {Deborah Loewenberg Ball},

title = {Research on teaching mathematics: Making subject matter knowledge part of the equation},

booktitle = {in Advances in Research in Teaching, Volume 2},

year = {1991},

pages = {1--48},

publisher = {JAI Press}

}

### OpenURL

### Abstract

Subject matter understanding and its role in teaching mathematics are the focus of this paper. Although few would disagree with the assertion that, in order to teach mathematics effectively, teachers must understand mathematics themselves, past efforts to show the relationship of teachers ' mathematical knowledge to their teaching of mathematics have been largely unsuccessful. How can this be? My

### Citations

1076 | The Reflective Practitioner: How Professionals Think - Schön - 1983 |

28 |
The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice
- Thompson
- 1984
(Show Context)
Citation Context ...o saw mathematics as a "subject of ideas and mental processes," not a "subject of facts," emphasized problem solving and encouraged her students to make and pursue their own mathematical conjectures (=-=Thompson, 1984-=-, pp. 112-113). Alerted by these and other similar findings, some researchers have returned to press on subject matter as a critical variable in teaching mathematics. However, "subject matter knowledg... |

22 |
The knowledge, beliefs and attitudes of the mathematics teacher: a model
- Ernest
- 1989
(Show Context)
Citation Context ...arned as a proxy for teachers' knowledge, how they conceptualize and study "subject matter" varies. Some researchers examine teachers' conceptions of or beliefs about mathematics (e.g., Blaire, 1981; =-=Ernest, 1988-=-; Ferrini-Mundy, 1986; Kuhs, 1980; Lerman, 1983; Peterson, Fennema, Carpenter, and Loef, in press; Thompson, 1984). These researchers use a variety of methods to identify teachers' conceptions, includ... |

20 |
Expertise in mathematics instruction: subject matter knowledge
- Leinhardt, Smith
- 1985
(Show Context)
Citation Context ...ptions about mathematics on their teaching of the subject. Other researchers focus on teachers' understanding of mathematical concepts and procedures (e.g., Ball, 1988a; Ball and McDiarmid, in press; =-=Leinhardt and Smith, 1985-=-; Owens, 1987; Post, Behr, Hamel, and Lesh, 1988; Steinberg, Haymore, and Marks, 1985). Using interviews and structured tasks, they explore how teachers think about their mathematical knowledge and ho... |

15 | Critical variables in mathematics education; Findings from a survey of empirical literature - Begle - 1979 |

14 | Young children reinvent arithmetic: Implications of Piaget’s Theory - Kamii - 2000 |

13 | Personal knowledge: toward a post-critical philosophy - Polyani - 1962 |

11 | The scientific basis of the art of teaching - Gage - 1978 |

11 | The concept of the structure of a discipline - Schwab - 1968 |

10 |
Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education
- Ball
- 1988
(Show Context)
Citation Context ... highlight the influence of teachers' assumptions about mathematics on their teaching of the subject. Other researchers focus on teachers' understanding of mathematical concepts and procedures (e.g., =-=Ball, 1988-=-a; Ball and McDiarmid, in press; Leinhardt and Smith, 1985; Owens, 1987; Post, Behr, Hamel, and Lesh, 1988; Steinberg, Haymore, and Marks, 1985). Using interviews and structured tasks, they explore ho... |

9 |
Teachers' Thinking
- Clark, Yinger
- 1979
(Show Context)
Citation Context ...e also responsible for a variety of educational outcomes that require different approaches. In light of these features of the job, it was simplistic to seek a single most effective teaching approach (=-=Clark and Yinger, 1979-=-; Peterson, 1979). Some scholars sought to uncover optimal patterns of instruction for students with particular characteristics, or "aptitude treatment interactions" (ATI) (e.g., Brophy, 1980; Evertso... |

7 |
Teaching styles and pupil progress
- Bennett
- 1976
(Show Context)
Citation Context ...led settings, studies indicated that there was little difference on such "affective outcomes" (Peterson, 1979). Some researchers even concluded that students were more anxious in informal classrooms (=-=Bennett, 1976-=-; Wright, 1975). Critical to understanding this phase of research on effective teaching are its assumptions about 3smathematics and the goals of teaching and learning mathematics. Taking the prevalent... |

6 | When do instructional methods make a difference - Tobias - 1982 |

5 |
The effectiveness of teachers
- Medley
- 1979
(Show Context)
Citation Context ...influence on the teaching of mathematics. Driven by common sense and conventional wisdom about teaching, the earliest research compiled characteristics of teachers whom others perceived as effective (=-=Medley, 1979-=-). The second phase of research attempted to establish connections between what teachers do and what their students learn. In the most recent phase, researchers have investigated teacher thinking. Wha... |

5 | Twice as less: Black English and the performance of Black students in mathematics and science - Orr - 1987 |

4 |
Unlearning to teach mathematics. For the Learning of Mathematics
- Ball
- 1988
(Show Context)
Citation Context ...tribute to the development of a theoretical framework for the question, "What do prospective teachers bring with them to teacher education that is likely to affect their learning to teach math?" (See =-=Ball, 1988-=-a). 3 Following is a brief explanation of the substantive underpinnings of the question. It deals with four important ideas in mathematics: division, the concept of infinity, what it means for somethi... |

4 |
Problem-solving or knowledge-centred; the influence of philosophy of mathematics teaching
- Lerman
- 1983
(Show Context)
Citation Context ...hey conceptualize and study "subject matter" varies. Some researchers examine teachers' conceptions of or beliefs about mathematics (e.g., Blaire, 1981; Ernest, 1988; Ferrini-Mundy, 1986; Kuhs, 1980; =-=Lerman, 1983-=-; Peterson, Fennema, Carpenter, and Loef, in press; Thompson, 1984). These researchers use a variety of methods to identify teachers' conceptions, including interviews, questionnaires, and inferences ... |

3 |
The story of 0
- Blake, Verhille
- 1985
(Show Context)
Citation Context ...e is no number that can be multiplied by zero to equal seven. Therefore, division by zero is actually undefined--it does not fit the definition of division. 4 I borrow the term "algorithm rhyme" from =-=Blake and Verhille, 1985-=-. 5 A provocative finding was the lack of difference by level between the responses of secondary teacher candidates, who are math majors, and of elementary teacher candidates, who are not. This issue ... |

3 |
The priority of knowledge and understanding in teaching
- Buchmann
- 1984
(Show Context)
Citation Context ...e argued that researchers should focus their inquiries elsewhere (p. 53). Begle's (1979) conclusion was counterintuitive. Teaching is fundamentally tied up with knowledge and the growth of knowledge (=-=Buchmann, 1984-=-). What sense does it make to say that what teachers know about mathematics is not a significant influence on what their students learn? Yet, in spite of the weight of common sense, the empirical resu... |

3 |
Cognitive consequences of practice in culture-specific procedural skills. The Quarterly Newsletter of the Laboratory of Comparative Human
- Hatano
- 1982
(Show Context)
Citation Context ...laces mattered, but was unable to pull it together. Tacit Versus Explicit Ways of Knowing Assuming that people have conceptual knowledge of procedures which they have learned to perform is a fallacy (=-=Hatano, 1982-=-). As one of the math majors reflected when he tried to explain the basis for the multiplication procedure, "I absolutely do it [multiplication] by the rote process--I would have to think about it." C... |

3 |
Conceptual and procedural knowledge
- Hiebert, Lefevre
- 1986
(Show Context)
Citation Context ...implications are taken up in Ball (1988a). 6 Researchers are currently pursuing critical questions about the relationship between conceptual and procedural knowledge in mathematics. See, for example, =-=Hiebert and Lefevre, 1986-=-. 7 This requirement of explicit understanding holds even for teachers who do not choose to teach by telling. Facilitating students' construction of mathematical understanding, for instance, involves ... |

2 |
Teacher effectiveness in mathematics instruction (National Longitudinal
- Begle, Geeslin
- 1972
(Show Context)
Citation Context ...t positive relationships were found in fewer than 30 percent of the possible cases. No single teacher characteristic proved to be "consistently and significantly correlated with student achievement" (=-=Begle and Geeslin, 1972-=-). Begle (1979) concluded from these results that many widely held beliefs about good teaching "are false, or at the very best rest on shaky foundations" (p. 54). One of these beliefs was the notion t... |

2 |
Why the professor can't teach: Mathematics and the dilemma of university education
- Kline
- 1977
(Show Context)
Citation Context ...erms of disciplinary understandings or ideas about pedagogy? Some of what is gained through sitting in upper-level mathematics courses may in fact serve as counterproductive preparation for teaching (=-=Kline, 1977-=-). What Do Effective Teachers Do? Instead of critically appraising the reported findings, however, researchers began a new search. Driven to understand what distinguished more effective from less effe... |

2 |
Knowing, doing, and teaching multiplication. Cognition and Instruction
- Lampert
- 1986
(Show Context)
Citation Context ...this strategy because it contributes to her goal of fostering "a habit of discourse in the classroom in which work in mathematics is referred back to the knower to answer questions of reasonability" (=-=Lampert, 1986-=-, p. 317). In her teaching, Lampert tries to balance her pedagogical responsibility to make sure students learn what they are supposed to know with her commitment to helping students invent and constr... |

2 |
A Study of Four Preservice Secondary Mathematics Teachers
- Owens
- 1987
(Show Context)
Citation Context ... their teaching of the subject. Other researchers focus on teachers' understanding of mathematical concepts and procedures (e.g., Ball, 1988a; Ball and McDiarmid, in press; Leinhardt and Smith, 1985; =-=Owens, 1987-=-; Post, Behr, Hamel, and Lesh, 1988; Steinberg, Haymore, and Marks, 1985). Using interviews and structured tasks, they explore how teachers think about their mathematical knowledge and how they unders... |

2 |
Direct instruction reconsidered
- Peterson
- 1979
(Show Context)
Citation Context ...ects. While some feared that students would enjoy school less in such tightly supervised, teacher-controlled settings, studies indicated that there was little difference on such "affective outcomes" (=-=Peterson, 1979-=-). Some researchers even concluded that students were more anxious in informal classrooms (Bennett, 1976; Wright, 1975). Critical to understanding this phase of research on effective teaching are its ... |

2 | Content, time, and direct instruction - Rosenshine - 1979 |

2 |
Teachers' knowledge and content structuring in mathematics. Paper presented at the annual meeting of the American Educational Research Association
- Steinberg, Haymore, et al.
- 1985
(Show Context)
Citation Context ...s focus on teachers' understanding of mathematical concepts and procedures (e.g., Ball, 1988a; Ball and McDiarmid, in press; Leinhardt and Smith, 1985; Owens, 1987; Post, Behr, Hamel, and Lesh, 1988; =-=Steinberg, Haymore, and Marks, 1985-=-). Using interviews and structured tasks, they explore how teachers think about their mathematical knowledge and how they understand (or misunderstand) specific ideas. What counts, according to these ... |

1 |
Philosophies of mathematics and perspectives of mathematics teaching
- Blaire
- 1981
(Show Context)
Citation Context ...s or credits earned as a proxy for teachers' knowledge, how they conceptualize and study "subject matter" varies. Some researchers examine teachers' conceptions of or beliefs about mathematics (e.g., =-=Blaire, 1981-=-; Ernest, 1988; Ferrini-Mundy, 1986; Kuhs, 1980; Lerman, 1983; Peterson, Fennema, Carpenter, and Loef, in press; Thompson, 1984). These researchers use a variety of methods to identify teachers' conce... |

1 |
Teachers' cognitive activities and overt behaviors
- Brophy
- 1980
(Show Context)
Citation Context ...lark and Yinger, 1979; Peterson, 1979). Some scholars sought to uncover optimal patterns of instruction for students with particular characteristics, or "aptitude treatment interactions" (ATI) (e.g., =-=Brophy, 1980-=-; Evertson, Anderson, and Brophy, 1978; Solomon and Kendall, 1976). In 1982, Tobias wrote that even ATI studies were failing to specify one mode of instruction appropriate for students with a particul... |

1 |
Mathematics teachers' attitudes and beliefs: Implications for inservice education. Paper presented at the annual meeting
- Ferrini-Mundy
- 1986
(Show Context)
Citation Context ...xy for teachers' knowledge, how they conceptualize and study "subject matter" varies. Some researchers examine teachers' conceptions of or beliefs about mathematics (e.g., Blaire, 1981; Ernest, 1988; =-=Ferrini-Mundy, 1986-=-; Kuhs, 1980; Lerman, 1983; Peterson, Fennema, Carpenter, and Loef, in press; Thompson, 1984). These researchers use a variety of methods to identify teachers' conceptions, including interviews, quest... |

1 |
Teachers and teaching: By ten thousand high school seniors
- Hart
- 1934
(Show Context)
Citation Context ...' assessments of their best teachers, these studies reported that good teachers were enthusiastic, helpful, and strict. Students also said that the best teachers knew the subject matter better (e.g., =-=Hart, 1934-=-). Although such findings seemed intuitively valid, the early studies did not empirically test the influence of "good" teachers' characteristics on what they did or what their students actually learne... |

1 |
Elementary school teachers' conceptions of mathematics content as a potential influence on classroom instruction. Unpublished doctoral dissertation
- Kuhs
- 1980
(Show Context)
Citation Context ...ledge, how they conceptualize and study "subject matter" varies. Some researchers examine teachers' conceptions of or beliefs about mathematics (e.g., Blaire, 1981; Ernest, 1988; Ferrini-Mundy, 1986; =-=Kuhs, 1980-=-; Lerman, 1983; Peterson, Fennema, Carpenter, and Loef, in press; Thompson, 1984). These researchers use a variety of methods to identify teachers' conceptions, including interviews, questionnaires, a... |

1 |
The trouble with math: A view from everyday practice. Paper presented at the annual meeting of the American Educational Research Association
- Lave
- 1987
(Show Context)
Citation Context ...ice: the practice of mathematics. Class activities are designed to involve the students in what it means to think about and do mathematics as mathematicians do (Collins, Brown, and Newmann, in press; =-=Lave, 1987-=-). Lampert's view of her role appears to grow out of the interaction of her constructivist assumptions about learning and her disciplinary focus. With a goal of involving students in mathematical comm... |

1 |
A potpourri from the Rational Number Project. Paper prepared for the National
- Post, Behr, et al.
- 1988
(Show Context)
Citation Context ...ng of the subject. Other researchers focus on teachers' understanding of mathematical concepts and procedures (e.g., Ball, 1988a; Ball and McDiarmid, in press; Leinhardt and Smith, 1985; Owens, 1987; =-=Post, Behr, Hamel, and Lesh, 1988-=-; Steinberg, Haymore, and Marks, 1985). Using interviews and structured tasks, they explore how teachers think about their mathematical knowledge and how they understand (or misunderstand) specific id... |

1 |
Access: A framework for thinking about student empowerment (Elementary Subjects Center Series No
- Prawat
- 1988
(Show Context)
Citation Context ...cquire the mathematical skills and understanding necessary to judge the validity of their own ideas and results, in other words, to be "independent learners" of mathematics (or to be "empowered"; see =-=Prawat, 1988-=-). Lampert's pedagogy subtly blends goal and process. For example, when students give answers or make assertions, Lampert almost always comes back with, "Why do you think that?" or "How did you figure... |

1 |
Individual characteristics and children's performance in "open" and "traditional" classroom settings
- Solomon, Kendall
- 1976
(Show Context)
Citation Context ...s sought to uncover optimal patterns of instruction for students with particular characteristics, or "aptitude treatment interactions" (ATI) (e.g., Brophy, 1980; Evertson, Anderson, and Brophy, 1978; =-=Solomon and Kendall, 1976-=-). In 1982, Tobias wrote that even ATI studies were failing to specify one mode of instruction appropriate for students with a particular set of characteristics. How Do Teachers Understand Their Work ... |

1 |
The affective and cognitive consequences of an open education elementary school
- Wright
- 1975
(Show Context)
Citation Context ...tudies indicated that there was little difference on such "affective outcomes" (Peterson, 1979). Some researchers even concluded that students were more anxious in informal classrooms (Bennett, 1976; =-=Wright, 1975-=-). Critical to understanding this phase of research on effective teaching are its assumptions about 3smathematics and the goals of teaching and learning mathematics. Taking the prevalent school curric... |