Programs that provide readymade, worked-out solutions to teaching problems should not expect that teachers will see themselves as in control of their own learning. In a teacher preparation program, teachers clearly cannot learn all they need to know about the mathematics they will teach, how students learn that mathematics, and how to teach it effectively. Consequently, some authorities have recommended that teacher education be seen as a professional continuum, a career-long process.

They need to be able to adapt to new curriculum frameworks, new materials, advances in technology, and advances in research on student thinking and teaching practice.

### Introduction

They have to learn how to learn, whether they are learning about mathematics, students, or teaching. Teachers can continue to learn by participating in various forms of professional development. But formal professional development programs represent only one source for continued learning.

We consider below examples of four such program types that represent an array of alternative approaches to developing integrated proficiency in teaching mathematics. For example, prospective elementary school teachers may take a mathematics course that focuses, in part, on rational numbers or proportionality rather than the usual college algebra or calculus.

Such courses are offered in many universities, but they are seldom linked to instructional practice. The lesson depicted in Box 10—1 comes from a course in which connections to practice are being made.

### Issue Archives

The prospective teachers stare at the board, trying to figure out what the instructor is asking them to do. After calculating the answer to a simple problem in the division of fractions and recalling the old algorithm—invert and multiply—most of them have come up with the answer, It is familiar content, and although they have not had occasion to divide fractions recently, they feel comfortable, remembering their own experiences in school mathematics and what they learned.

But now, what are they being asked? The instructor has challenged them to consider why they are getting what seems to be an answer that is larger than either of the numbers in the original problem and. Confused, they are suddenly stuck. None of them noticed this fact before.

## Mathematics Education Theses and Dissertations | Mathematics Education | Brigham Young University

Can you come up with an example or a model that shows what is going on with dividing one and three fourths by one half? The prospective teachers set to work, some in pairs, some alone. The instructor walks around, watching them work, and occasionally asking a question. Most have drawn pictures like those below:. I have two pizzas.

## Suggested Thesis/Project Formats, MS in Math Education

My little brother eats one quarter of one of them and then I have one and three quarters pizzas left. My sister is very hungry, so we decide to split the remaining pizza between us.

We each get pieces of pizza. I have cups of sugar. Each batch of sesame crackers takes cup of sugar. How many batches of crackers can I make? And another pair has envisioned filling -liter containers, starting with liters of water.

## Suggested Thesis/Project Formats, MS in Math Education

After about 10 minutes, the instructor invites students to share their problems with the rest of the class. One student presents the pizza situation above. Most students nod appreciatively. When a second student offers the sesame cracker problem, most nod again, not noticing the difference.

The instructor poses a question: How does each problem we heard connect with the original computation? Are these two problems similar or different, and does it matter? Through discussion the students gradually come to recognize that, in the pizza problem, the pizza has been divided in half and that the answer is in terms of fourths —that is, that the pieces are fourths of pizzas.

In the case of the sesame cracker problem, the answer of batches is in terms of half cups of sugar. In the first instance, they have represented division in half, which is actually division by two; in the second they have represented division by one half. The instructor moves into a discussion of different interpretations of division: sharing and measurement. After the students observe that the successful problems— involving the sesame crackers and the liters of water—are measurement problems, she asks them to try to develop a problem situation for that represents a sharing division.

In other words, could they make a sensible problem in which the is not the unit by which the whole is being measured, but instead is the number of units into which the whole has been divided? For homework, the instructor asks the students to try making representations for several other division situations, which she chooses strategically, and finally asks them to select two numbers to divide that they think are particularly good choices and to say why.

In this excerpt from a university mathematics course, the prospective teachers are being asked to unpack familiar arithmetic content, to make explicit the ideas underlying the procedures they remember and can perform. Repeatedly throughout the course, the instructor poses problems that have been strategically designed to expose concepts on which familiar procedures rest.

A second principle is to link that work with larger mathematical ideas and structures. For example, the lesson on the division of fractions is part of a larger agenda that includes understanding division, its relationship to fractions and to multiplication, and the meaning and representation of operations. Moreover, throughout the development of these ideas and connections, the prospective teachers work with whole and rational numbers, considering how the mathematical world looks inside these nested systems.

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The overriding purpose of a course like this is to provide prospective teachers with ample opportunities to learn fundamental ideas of school mathematics, how they are related, and how students come to learn them. But the course is not about how to teach, nor about how children learn. It is explicitly and deliberately a sustained opportunity for prospective teachers to learn mathematical ideas in ways that will equip them with mathematical resources needed in teaching.

Teachers do not learn abstract concepts about mathematics and children. In the programs, teachers look at problem-solving strategies of real students, artifacts of student work, cases of real classrooms, and the like.

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Furthermore, the teachers in these programs are challenged to relate what they learn to their own students and their own instructional practices. They learn about mathematics and students both in workshops and by interacting with their own students. Specific opportunity is provided for the teachers to discuss with one another how the ideas they are encountering influence their practice and how their practice influences what they are learning.

The workshop described in Box 10—2 forms part of a professional development program designed to help teachers develop a deeper understanding of some critical mathematical ideas, including the equality sign. The program, modeled after Cognitively Guided Instruction CGI , which has proven to be a highly effective approach, 41 assists teachers in understanding how to help their students reason about number operations and relations in ways that enhance the learning of arithmetic and promote a smoother transition from arithmetic to algebra.

Several features of this example of professional development are worth noting. Although they begin by considering how children think, the teachers. At the workshop, the teachers share their findings with the other participants. These findings, which surprised most teachers, have led them to begin to listen to their students, and a number of teachers have engaged their students in a discussion of the reasons for their responses.

The discussion generates insights about how children are thinking and what teachers can learn by listening to their students. The task is to decide whether the sentence is true or false. Sometimes the decision requires calculation e. The teachers work in small groups to construct true and false number sentences they might use to elicit various views of equality. Using these sentences, their students could engage in explorations that might lead to understanding equality as a relation. The sentences could also provide opportunities for discussions about how to resolve disagreement and develop a mathematical argument.

The teachers work together to consider how their students might respond to different number sentences and which number sentences might produce the most fruitful discussion. Used by permission of the authors. The teachers also begin to ponder how notation is used and how ideas are justified in mathematics.