Mathematics Instruction Essay

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Mathematics Teaching

Learners Studying Basic Mathematics To Enable Helping Their Children With Their Education

The work of Jackson and Ginsburg (2008) reports on a series of algebra classes involving a group of African-American mother and elementary-aged children, who are low income and who "had limited and negative formal experiences with algebra." (p. 10) The women in the study who arrived to the algebra classes are reported to have had "well-informed view of what algebra was -- a disconnected body of knowledge that they did not understand -- and corresponding views of who could 'do' algebra." (Jackson and Ginsburg, 2008, p. 11) The study resulted in the women's views of who could 'do' algebra being changed "through genuine, intellectual inquiry into the mathematics of algebra." (Jackson and Ginsburg, 2008, p. 11) It is reported that questioning "is recognized as a critical instructional tool in the teaching of mathematics for understanding." (Jackson and Ginsburg, 2008, p. 18) Solomon, Lawson and Croft (2008) relate that while many learners "may be successful in mathematics" that they "nevertheless see themselves as existing only on the margins of the practice, or as lacking stability in it -- in this sense, they have what is called a fragile identity." (p. 1) Swan reports that research has shown that "many students view mathematics as a series of unrelated procedures and techniques that have to be committed to memory. Instead, we want them to engage in discussing and explaining ideas, challenging and teaching one another, creating and solving each other's questions and working collaboratively to share methods and results." (2006, p.162) Reported as the second aim is the development of more "challenging, connected, collaborative orientations towards their teaching." (Swan, p. 162) Jackson and Ginsberg reports that the women in the study in their role as parents were striving to "support their children's mathematics learning" and that the women held the view that the "algebra they explored grew out of their children's assignments and in turn would feed back into the children's work as they felt better able to support that work." (p. 26) As the women persisted with their learning of algebra the subject is reported to have become "intellectually challenging, accessible and pleasurable." (2008, p. 26) Two implications are stated from the research: (1) the role of the elementary mathematics curriculum was central to the academic and social accomplishments of the algebra sessions; (2) the importance of framing parents as learners. (Jackson and Ginsburg, 2008, p. 26-27) Wiley (2008) reports on Latina mother's participation in a community mathematization project and not only this but also participating as curriculum designers in a collaborative project with university researchers. It is reported that "the crux of this study lies in the process through which the mothers moved from one level of mathematical understanding to a more advanced level that could be communicated mathematically." (Wiley, 2008, p. 35) There were reported to be three critical components that "facilitated this process to varying degrees" including: (1) social networks; (2) agency; and (3) mathematical sense-making. (Wiley, 2008, p. 35) It is reported that the attitudes of parents and perceptions about mathematics education are reported "in the context of a community mathematics program designed to provide an opportunity for parents and children to learn together. Within the venue of this program, Mexican immigrant parents' voices revealed the importance of their own experiences as learners and how these experiences shape their perceptions of mathematics education in the U.S." (Allexsaht-Snider and Marshall, 2008, p. 9) It is reported that the attitudes of parents are revealing of the types of knowledge in mathematical practices which are valued and how the value of thee practices is differed across contexts. Specifically stated is "for these parents, whose formal mathematics experiences in Mexico valued memorization and the application of algorithms, conversations with researchers showed the importance of mutual respect and a shared understanding in building bridges between formal and informal networks of support." (Allexsaht-Snider and Marshall, 2008, p. 9)

III. CRITIQUE OF LEARNING THEORIES

Weinstein (nd) states that many perspectives exist concerning mathematics learning. Specifically stated is that the beliefs teachers hold about mathematics "vary widely and those beliefs affect their teaching philosophies." (Weinstein, nd, p. 670) Ernest reports that the teaching practice in mathematics is dependent upon key elements including the following stated elements: (1) the mental contents or schemas of the teacher and the teacher's beliefs about mathematics and its teaching and learning; (2) the social context of the teaching situation and the constraints and opportunities presented; and (3) the level of thought processes and reflection of the teacher. (Ernest, 1989, p. 1)

Ernest writes that there are three philosophies of mathematics including the instrumentalist view "that mathematics is an accumulation of facts, rules and skills to be used in the pursuance of some external end." (1989, p.1) The second stated view is the Platonist view which holds that mathematics is "a static but unified body of certain knowledge" and that mathematics is not created but instead is discovered. (Ernest, 1989, p. 1) The third view is the "problem solving view of mathematics" which holds that mathematics is "a dynamic, continually expanding field of human creation and invention, a cultural product." (Ernest, 1989, p. 1)

Swan (2006) states that traditional methods of teaching mathematics fail to promote "transferrable learning that endures over time or that may be used in non-routine situations" and additionally serve to "demotivate students and undermine confidence'. (p. 162) According to Bandura the pros of the theory of behaviorism in learning is that it is scientific and emphasizes "observable, measurable phenomena" and utilizes a methodology that is rigorous. The negative aspects of the theory of behaviorism in learning is that it "ignore the things that make humans 'human' such as cognitions, emotions and free will. (nd, p. 2) According to Bandura psychodynamic explanations of behavior are flawed in that "they are based on inferred drives and needs" which are impossible to test. In addition, Bandura states that they "ignore conscious cognitions" and "ignore situational influences." (nd, p. 3) Piaget held that "the growth of knowledge is the result of individual constructions made by the learner's understanding" and held that the "current state of knowledge is temporal changing as time passes as knowledge in the past has changed, it is not a static instance; it is a process." (Kim, 2005, p. 9) From the constructivist view knowledge is formulated "out of sensual and perspective experience of the learner in which learning is internalized through personal experience rather than the experiences of others." (Kim, 2005, p. 9) Constructivists teaching characteristics include: (1) student being invited to ask questions and share ideas; (2) ideas invented by students are encouraged and accepted; (3) student leadership, seeking of information, cooperation and presentation of ideas is encouraged; (4) instructional strategies are modified based upon the student's experiences, interest and thoughts; (5) printed materials are used as well as are experts to provide more information; (6) the constructivist teacher allows and encourages free discussion; (8) students are encouraged to prediction causes and effects in regards to specific events or cases; (8) students are encouraged to test their own ideas; (9) the ideas of students are invited prior to being presented with learning materials and instructions; (10) students are encouraged to challenge the ideas and concepts of others; (11) cooperative teaching strategies are utilized through interactions with students and respect as well as sharing of ideas and learning tasks; (12) students are encouraged to respect other's ideas. (Kim, 2005, pp. 9-10) However, the traditional classroom is different in that curriculum is presented "part to whole with emphasis on basic skills" and there is "strict adherence to fixed curriculum which is valued highly." (Kim, 2005, p. 10) The curriculum in the traditional classroom is greatly dependent on textbooks and workbooks and students are viewed by the teacher as "blank slates onto which information is etched by the teacher." (Kim, 2005, p. 10) In the traditional classroom the behavior of teachers is reported to be "didactic" in nature and involves the dissemination of information to students and the teacher requires that students provide the 'correct answer to validate student learning'. (Kim, 2005, p. 10) Student learning is assessed separately from teaching and almost completely through testing and students work alone rather than collaboratively." (Brooks and Brooks, 1993, p. 17 cited in Kim, 2005, p. 11)

III. CRITIQUE OF TEACHING APPROACHES

Ernest reports that a recent position "in the philosophy of mathematics is fallibilism which emphasizes the practice of mathematics and the human side of mathematics." (1991, p. 1) Fallibilism is reported to view mathematics as "the outcome of social processes." (Ernest, 1991, p. 1) The absolutist view of mathematics is one that is "an objective, absolute, certain and incorrigible body of knowledge, which rests on the firm foundations of deductive logic." (Ernest, 1991, p.1) According to Ernest the practice of teaching mathematics is dependent on certain elements including the "mental contents or schemas" of the teacher as well as the "social context of the teaching situation"…[continue]

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