Changing One's Practice: Teacher Readiness - A teacher’s approach to teaching mathematics reflects her beliefs about what mathematics is as a discipline (Hersh, 1986). If she characterizes mathematics as involving correct answers and infallible procedures consisting of arithmetic operations, algebraic procedures, and geometric terms and theorems, chances are, her instructional approach will likely emphasize the presentation of mathematical concepts, procedures, facts, and theorems with a focus on student practice and memorization.
The meaning and context associated with many of these theorems and procedures may be relegated to the fringes of her curricular focus. On the other hand, if she views mathematics as an active, creative endeavor involving inquiry and discovery, she will likely emphasize activities that involve students in generating and uncovering meaning and making connections. She will view her role as a facilitator, challenging students to think and to question their findings and assumptions. Ernest (1988) outlines three conceptions of mathematics, each of which prompts a
different emphasis in instruction:
First of all, there is a dynamic, problem-driven view of mathematics as a continually expanding field of human creation and invention, in which patterns are generated and then distilled into knowledge. Thus mathematics is a process of enquiry and coming to know, adding to the sum of knowledge. Mathematics is not a finished product, for its results remain open to revision (the problem-solving view).
Secondly, there is a view of mathematics as a static but unified body of knowledge, a crystalline realm of interconnecting structures and truths, bound together by filaments of logic and meaning. Thus mathematics is a monolith, a static immutable product. Mathematics is discovered, not created (the Platonic view).
Thirdly, there is the view that mathematics, like a bag of tools, is made up of an accumulation of facts, rules and skills to be used by the trained artisan skillfully in the pursuance of some external end. Thus mathematics is a set of unrelated but utilitarian rules and facts (the instrumentalist view).
Each of these views perceives the essence of mathematics differently. The instrumentalist view sees mathematics as a set of tools. Teachers with an instrumentalist view can be expected to stress rules, facts, and procedures in their classes. Their classes tend to be teacher-directed and emphasize routine drill and practice. The Platonic view sees math as a body of knowledge.
Teachers who ascribe to the Platonic view of mathematics focus on the interrelationships, underlying concepts, and internal logic of mathematical procedures. The problem-solving view focuses on the process of inquiry. Teachers with a problem-solving view tend to be more learner-focused and constructivist in their teaching style, actively involving students in exploring mathematical concepts, creating solution strategies, and constructing personal meaning in a problem-rich environment (Thompson, 1992).
Students’ beliefs about the nature of mathematics are greatly influenced by their teacher’s beliefs. Surveys of student beliefs about mathematics reveal that most students think there should be a ready method for solving problems and that that method should quickly lead to an answer (Schoenfeld 1989, 1992). Schoenfeld (1992) cites a 1983 survey conducted by the National Assessment of Educational Progress (NAEP) in which half of the students who responded agreed that “learning mathematics is mostly memorizing facts.” Three-quarters agreed that “doing mathematics requires lots of practice in following rules,” while 90 percent agreed with the statement, “There is always a rule to follow in solving mathematical problems.” Students holding such beliefs may not even attempt to solve a problem that involves too much complexity or does not appear to offer a clear-cut algorithmic approach.
Furthermore, Schoenfeld (1992) notes that most students believe that all problems have an answer; that there is only one right answer and one correct solution method; and that ordinary students cannot expect to understand mathematics but can merely memorize and apply mathematical procedures in a mechanical fashion. These beliefs largely develop out of the experiences students have in mathematics classes and from the attitudes and beliefs passed on by their teachers.
A problem-solving approach to teaching mathematics helps broaden students’ perception of mathematics from a rule- and fact-based discipline to one that involves inquiry, uncertainty, and creativity. But first, the teacher must make his own paradigm shift, and this requires him to come face-to-face with deeply held personal beliefs about teaching and learning, and to face his own propensity for risk and initiative (Dirkes, 1993). Many teachers feel unprepared to take a problem-solving approach to teaching mathematics.
Few teachers learned math themselves in this way. Even if they encountered problem solving in their college methods courses, once in the classroom, they often conform to the conventional methods that hold sway in most schools. Being an agent of change, when one is surrounded by deeply ingrained beliefs about teaching and learning, is a difficult role to perform. Teachers today are often caught between daily pressure from colleagues, parents, and others to uphold tradition in the classroom, and pressure from policymakers to employ standards-based practices (with the conflicting expectation that students will perform highly on standardized tests that measure basic skills, not performance of standards-based material).
A teacher’s path to change must begin with an acknowledgment of her previous experience. She will build on her past experiences by reflecting on them in light of new ideas about effective teaching strategies (Richardson, 1990). Broadening teachers' conceptions of the nature of problem solving and its potential as an instructional tool requires that they, too, engage in solving open-ended problems. This means spending time solving a wide variety of problems and reflecting on their attempts to solve them. Changing one’s practice is further facilitated when effective teaching techniques are modeled in the classroom by a practitioner who is skilled in problem-solving instruction.
This modeling should be followed by a discussion among the teachers about the selection and use of strategies. Modeling and discussion provide concrete illustrations of the teachers' role in teaching problem solving (Richardson, 1990). Reading literature on the theory and practice of problem-solving instruction can also influence teachers to make changes in their practice (Thompson, 1989). “Examining research inquisitively and skeptically,” writes Ball (1996), “teachers can seek insights from scholarship without according undue weight to its conclusions.
They can use the broadly outlined reforms as a resource for developing inspired but locally tailored innovations.” The truth is, teachers are constantly making changes to meet the changing needs of their students and to try out ideas they've heard from other teachers. Teachers establish their own voice of authority in defining what takes place in the classroom. The notion of authority plays a critical role in conceptualizing and advancing mathematics teacher change (Wilson & Lloyd, 2000). Teachers themselves must be involved in making judgments about what change is worthwhile and significant (Richardson, 1990).
In pursuing reform goals, teachers often feel anxious about their effectiveness and knowledge. Moving in the direction of math reforms means confronting up close the uncertainties, ambiguities, and complexities of what “understanding” and “learning” might really mean. When we ask students to voice their ideas in a problem-solving context, we run the risk of discovering what they do and do not know. Those discoveries can be unsettling when students reveal that they know far less than the teacher expected or far more than the teacher is prepared to deal with (Ball, 1996). Inquiry- and problem-based teaching requires qualities beyond mathematics knowledge and skill. Personal qualities, such as patience, curiosity, generosity, confidence, trust, and imagination, matter a great deal. Interest in seeing the world from another's perspective, enjoyment of humor, empathy with confusion, and concern for the frustration and shame of others are other important qualities that can help a teacher create a learning environment that fosters students’ problem-solving abilities (Ball, 1996). “As teachers build their own understandings and relationships with math, they chart new mathematical courses with their students. And, as they move on new paths with students, their own mathematical understandings change,” Ball writes.
The meaning and context associated with many of these theorems and procedures may be relegated to the fringes of her curricular focus. On the other hand, if she views mathematics as an active, creative endeavor involving inquiry and discovery, she will likely emphasize activities that involve students in generating and uncovering meaning and making connections. She will view her role as a facilitator, challenging students to think and to question their findings and assumptions. Ernest (1988) outlines three conceptions of mathematics, each of which prompts a
different emphasis in instruction:
First of all, there is a dynamic, problem-driven view of mathematics as a continually expanding field of human creation and invention, in which patterns are generated and then distilled into knowledge. Thus mathematics is a process of enquiry and coming to know, adding to the sum of knowledge. Mathematics is not a finished product, for its results remain open to revision (the problem-solving view).
Secondly, there is a view of mathematics as a static but unified body of knowledge, a crystalline realm of interconnecting structures and truths, bound together by filaments of logic and meaning. Thus mathematics is a monolith, a static immutable product. Mathematics is discovered, not created (the Platonic view).
Thirdly, there is the view that mathematics, like a bag of tools, is made up of an accumulation of facts, rules and skills to be used by the trained artisan skillfully in the pursuance of some external end. Thus mathematics is a set of unrelated but utilitarian rules and facts (the instrumentalist view).
Each of these views perceives the essence of mathematics differently. The instrumentalist view sees mathematics as a set of tools. Teachers with an instrumentalist view can be expected to stress rules, facts, and procedures in their classes. Their classes tend to be teacher-directed and emphasize routine drill and practice. The Platonic view sees math as a body of knowledge.
Teachers who ascribe to the Platonic view of mathematics focus on the interrelationships, underlying concepts, and internal logic of mathematical procedures. The problem-solving view focuses on the process of inquiry. Teachers with a problem-solving view tend to be more learner-focused and constructivist in their teaching style, actively involving students in exploring mathematical concepts, creating solution strategies, and constructing personal meaning in a problem-rich environment (Thompson, 1992).
Students’ beliefs about the nature of mathematics are greatly influenced by their teacher’s beliefs. Surveys of student beliefs about mathematics reveal that most students think there should be a ready method for solving problems and that that method should quickly lead to an answer (Schoenfeld 1989, 1992). Schoenfeld (1992) cites a 1983 survey conducted by the National Assessment of Educational Progress (NAEP) in which half of the students who responded agreed that “learning mathematics is mostly memorizing facts.” Three-quarters agreed that “doing mathematics requires lots of practice in following rules,” while 90 percent agreed with the statement, “There is always a rule to follow in solving mathematical problems.” Students holding such beliefs may not even attempt to solve a problem that involves too much complexity or does not appear to offer a clear-cut algorithmic approach.
Furthermore, Schoenfeld (1992) notes that most students believe that all problems have an answer; that there is only one right answer and one correct solution method; and that ordinary students cannot expect to understand mathematics but can merely memorize and apply mathematical procedures in a mechanical fashion. These beliefs largely develop out of the experiences students have in mathematics classes and from the attitudes and beliefs passed on by their teachers.
A problem-solving approach to teaching mathematics helps broaden students’ perception of mathematics from a rule- and fact-based discipline to one that involves inquiry, uncertainty, and creativity. But first, the teacher must make his own paradigm shift, and this requires him to come face-to-face with deeply held personal beliefs about teaching and learning, and to face his own propensity for risk and initiative (Dirkes, 1993). Many teachers feel unprepared to take a problem-solving approach to teaching mathematics.
Few teachers learned math themselves in this way. Even if they encountered problem solving in their college methods courses, once in the classroom, they often conform to the conventional methods that hold sway in most schools. Being an agent of change, when one is surrounded by deeply ingrained beliefs about teaching and learning, is a difficult role to perform. Teachers today are often caught between daily pressure from colleagues, parents, and others to uphold tradition in the classroom, and pressure from policymakers to employ standards-based practices (with the conflicting expectation that students will perform highly on standardized tests that measure basic skills, not performance of standards-based material).
A teacher’s path to change must begin with an acknowledgment of her previous experience. She will build on her past experiences by reflecting on them in light of new ideas about effective teaching strategies (Richardson, 1990). Broadening teachers' conceptions of the nature of problem solving and its potential as an instructional tool requires that they, too, engage in solving open-ended problems. This means spending time solving a wide variety of problems and reflecting on their attempts to solve them. Changing one’s practice is further facilitated when effective teaching techniques are modeled in the classroom by a practitioner who is skilled in problem-solving instruction.
This modeling should be followed by a discussion among the teachers about the selection and use of strategies. Modeling and discussion provide concrete illustrations of the teachers' role in teaching problem solving (Richardson, 1990). Reading literature on the theory and practice of problem-solving instruction can also influence teachers to make changes in their practice (Thompson, 1989). “Examining research inquisitively and skeptically,” writes Ball (1996), “teachers can seek insights from scholarship without according undue weight to its conclusions.
They can use the broadly outlined reforms as a resource for developing inspired but locally tailored innovations.” The truth is, teachers are constantly making changes to meet the changing needs of their students and to try out ideas they've heard from other teachers. Teachers establish their own voice of authority in defining what takes place in the classroom. The notion of authority plays a critical role in conceptualizing and advancing mathematics teacher change (Wilson & Lloyd, 2000). Teachers themselves must be involved in making judgments about what change is worthwhile and significant (Richardson, 1990).
In pursuing reform goals, teachers often feel anxious about their effectiveness and knowledge. Moving in the direction of math reforms means confronting up close the uncertainties, ambiguities, and complexities of what “understanding” and “learning” might really mean. When we ask students to voice their ideas in a problem-solving context, we run the risk of discovering what they do and do not know. Those discoveries can be unsettling when students reveal that they know far less than the teacher expected or far more than the teacher is prepared to deal with (Ball, 1996). Inquiry- and problem-based teaching requires qualities beyond mathematics knowledge and skill. Personal qualities, such as patience, curiosity, generosity, confidence, trust, and imagination, matter a great deal. Interest in seeing the world from another's perspective, enjoyment of humor, empathy with confusion, and concern for the frustration and shame of others are other important qualities that can help a teacher create a learning environment that fosters students’ problem-solving abilities (Ball, 1996). “As teachers build their own understandings and relationships with math, they chart new mathematical courses with their students. And, as they move on new paths with students, their own mathematical understandings change,” Ball writes.
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