Multiple studies of the culture of the mathematics classroom have concluded that the student’s role and actions depend primarily on the view of mathematics “projected” by the teacher. In her summary of the relevant research, Nickson (1992) concluded: “The linearity and formality associated with most teaching of mathematics from published schemes or textbooks tend to produce a passive acceptance of mathematics in the abstract, with little connection being made by pupils between their work and real life. Pupils accept the visibility of mathematics in terms of a ‘right or wrong’ nature, and their main concerns seem to be with the quantity of mathematics done and its correctness…. When beliefs about mathematics differ and where views of mathematics as socially constructed knowledge prevail, pupils take on quite a different role. The messages they receive are that they are expected to contribute their own ideas, to try their own solutions, and even to challenge the teacher … (p. 110).”
• Teachers with “an integrated, conceptual understanding” of mathematics tend to organize their classrooms and learning activities that encourage students to engage and interact with the conceptual aspects of mathematics. Furthermore, the depth of the mathematics taught correlates highly with the depth of the teachers’ mathematical knowledge (Fennema and Franke, 1992).
• Warm and supportive teachers are more effective than critical teachers (Titunoff et al., 1975; Rosenshine and Furst, 1971).
• Teachers maintain student engagement at doing mathematics at a high level if they (1) select appropriate tasks for the student, (2) support proactively the student’s activity, (3) ask students consistently to provide meaningful explanations of their work and reasoning, (3) push students consistently to make meaningful connections, and (4) do not reduce the complexity/cognitive demands of the task. Student engagement in mathematical activities declines if teachers (1) remove the challenging aspects of the tasks, (2) shift the students’ focus from understanding to either the correctness or completeness of an answer, or (3) do not allow an appropriate amount of time for students to complete the task (Henningsen and Stein, 1997).
• A crucial role of the teacher is to structure “a pervasive norm in the classroom that helping one’s peers to learn is not a marginal activity, but is a central element of students’ roles” (Slavin, 1985).
• In a review of 80 research studies on grouping in mathematics classrooms, Davidson (1985) concluded that students working in small groups significantly outscored students working individually in more than 40 percent of the studies. Students working as individuals in a mathematics classroom performed better in only two of the studies (and Davidson suggests that these studies were faulty in design).
• Students working on solving a mathematics problem in small groups exhibit cognitive behaviors and processes that are essentially similar to those of expert mathematical problem solvers (Artz and Armour-Thomas, 1992).
• Learning mathematics in cooperative groups is effective, especially for younger students. When students reach high school, the research evidence is less clear, as these students exhibit stronger individual motivations, interact socially in more complex ways, and often are defensive or embarrassed about their knowledge and learning in mathematics (Steen, 1999).
• The research conclusions on the effect of cooperative learning in mathematics classrooms are quite consistent (Davidson, 1990; Davidson and Kroll, 1991; Leiken and Zaslavasky, 1999; Slavin, 1985; Weissglass, 1990):
1. Students with different ability levels become more involved in task-related interactions.
2. Students’ attitudes toward school and mathematics become more positive.
3. Students often improve their problem solving abilities.
4. Students develop better mathematical understanding.
5. The effects on students’ mathematics achievement have been positive, negative, and neutral.
• Students working in cooperative groups outperform individuals competing against each other. A meta-analysis of 800 studies on problem solving suggests that the differing factor is the generation of more problem solving strategies by cooperative groups than by individuals working competitively (Qin et al., 1995).
• Teachers can maximize mathematical learning in a small group environment by engaging students in learning activities that promote “questioning, elaboration, explanation, and other verbalizations in which they can express their ideas and through which the group members can give and receive feedback” (Slavin, 1989).
• Students solving mathematical problems in small groups invokes three features that enhance the individual student’s cognitive (re)organization of mathematics:
1. The student experiences “challenge and disbelief” on the part of the other members of the group, which forces them to examine their own beliefs and strategies closely.
2. The group collectively provides background information, skills, and connections that a student may not have or understand.
3. The student might internalize some of the group’s problem solving approaches and make them part of their personal approach (Noddings, 1985).
• Teachers trying to build and sustain mathematical discourse amongst students need to create an environment in which students build a “personal relationship” with mathematics. Three key elements need to be in this environment:
1. Students need to engage in authentic mathematical inquiries.
2. Students must act like mathematicians as they explore ideas and concepts.
3. Students need to negotiate the meanings of, and the connections among, these mathematical ideas with other students in the class (D’Ambrosio, 1995).
• Several factors influence or maintain student engagement at the level necessary to do quality mathematics. The primary factors are: high-quality tasks that build on students’ prior knowledge of mathematics, effective scaffolding on the teachers’part, an appropriate amount of time to engage in the mathematics, both teacher
and student modeling of high performance actions, and a sustained effort by the teacher to ask for explanations and meaning (Henningsen and Stein, 1997).
• Several teacher actions help establish a classroom culture that supports mathematical discourse (Yackel and Cobb, 1993):
1. Have a routine of setting norms for both small-group and large-group activities.
2. Address student and group expectations in class.
3. Insist that students solve personally challenging problems.
4. Insist that students explain their personal solutions to peers.
5. Insist that students listen to and try to make sense of the explanations of others.
6. Insist that students try to reach consensus about solutions to a problem.
7. Insist that students resolve any conflicting interpretations or solutions.
8. Capitalize on specific incidents when a student’s activity either “instantiated or transgressed a social norm” by rediscussing the classroom expectations.
• Teachers need to do more than ask questions in a mathematics classroom, as the cognitive level of the questions being asked is very important. Though the research is quite depressing in regard to teachers’ use of questioning, it is quite consistent:
1. 80 percent of the questions asked by mathematics teachers were at a low cognitive level (Suydam, 1985).
2. During each school day, there were about five times as many interactions at low cognitive levels than at high cognitive levels (Hart, 1989).
3. Low cognitive level interactions occurred about 5.3 times more often than high cognitive level interactions (Fennema and Peterson, 1986).
4. In an average of 64.1 interactions in a 50-minute class period, 50.3 were low level cognitive interactions, involved high-level cognitive interactions, and the remaining interactions were not related to mathematics (Koehler, 1986).
• Students tend not to correct their own errors because of either an unwillingness or an inability to search for errors. Most students are “just too thankful to have an answer, any answer, to even dare to investigate further” (K. Hart, 1981d).
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