*Principles and Standards for School Mathematics* (NCTM 2000) proposes that mathematics instruction provide opportunities for students to engage in mathematical inquiry and in meaningmaking through discourse. Mathematics teachers are encouraged to build on student discoveries in designing subsequent instruction. Natural consequences of using an inquiry-based approach to teaching include the emergence of unexpected mathematical results and the articulation of novel and different strategies by students. Anticipating the potential for such occurrences, *Professional Standards for Teaching Mathematics* (NCTM 1991) urges all teachers to remain flexible and responsive to student ideas in their instruction: Help students make connections among various solutions, tie student ideas to important mathematical structures, and extend student inquiry by posing questions and tasks that challenge their initial interpretations of problems or their false generalizations.

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- Author or Editor: Manouchehri Azita x

### Azita Manouchehri

### Azita Manouchehri

Imagine that a colleague says to you, “The new Principles and Standards [NCTM 2000] tells us that we should expect our students to ‘act as mathematicians.' That is a tall order for my students.” How would you respond? Is the same true for your students? Have you found any successful approaches for engaging students in working as young mathematicians?

### Azita Manouchehri

The *Curriculum and Evaluation Standards for School Mathematics* (NCTM 1989) recommends that greater attention be devoted to increasing students’ awareness of real-life applications of mathematics at all levels of education.

### Azita Manouchehri and Dennis St. John

The vision to transform mathematics classrooms into learning communities in which students engage in mathematical discourse is a remarkable hallmark of the current movement, led by the National Council of Teachers of Mathematics, to reform mathematics education (NCTM 1991, 2000). According to NCTM, “the discourse of a classroom—the ways of representing, thinking, talking, agreeing and disagreeing—is central to what students learn about mathematics as a domain of human inquiry with characteristic ways of knowing” (NCTM 1991, p. 34). Indeed, both the Principles and Standards for School Mathematics (2000) and Professional Standards for Teaching Mathematics (1991) recommend that teachers of mathematics provide opportunities for children of all ages to participate in mathematical discourse.

### Azita Manouchehri and Douglas A. Lapp

The nature of question posing and categorization of types of questions in relation to the desired purpose. Suggestions for obtaining richer information of what students understand are discussed.

### Azita Manouchehri, Pingping Zhang and Jenna Tague

With the publication of the National Council of Teachers of Mathematics' Curriculum Standards document in 1989, nurturing students' *mathematical thinking* secure a prominent place in the discourse surrounding school curriculum and instructional redesign. Although the standards document did not provide a definition for mathematical thinking, the authors highlighted processes that could support its development, including problem solving, communicating ideas, building and justifying arguments, and reasoning formally and informally about potential mathematical relationships. Less articulated were ways that mathematical thinking may be supported toward the development of proving and prooflike reasoning among students (Maher and Martino 1996).

### Azita Manouchehri and Akram Almohalwas

In recent years, the use of case-based tasks in the pedagogical preparation of teachers has gained considerable popularity in teacher education (Broudy 1990; Merseth 2003).

### Manouchehri Azita, Ozturk Ayse and Sanjari Azin

In this article we illustrate how one teacher used PhET cannonball simulation as an instructional tool to improve students' algebraic reasoning in a fifth grade classroom. Three instructional phases effective to implementation of simulation included: Free play, Structured inquiry and, Synthesizing ideas.

### Azita Manouchehri and Mary C. Enderson

The NCTM's *Professional Standards for Teaching Mathematics* (1991) has directed attention to “discourse” in the mathematics classroom. This document recommends that mathematics instruction should promote students' discourse by orchestrating situations in which each individual's thinking is challenged and by asking students to clarify and justify ideas. “Discourse,” as described by the *Standards* document, highlights the way in which knowledge is constructed and exchanged in the classroom (Ball 1992). Teaching mathematics from the perspective of developing mathematical discourse requires building a new vision for mathematics classrooms and poses a major challenge for mathematics teachers at all levels. This challenge was recognized by D'Ambrosio (1995). She identified the need to build environments in which students construct a “personal relationship” with mathematics as one of the most important requirements for promoting and sustaining the type of discourse envisioned by the reform movement. In such environments, students engage in authentic mathematical inquiry; act like mathematicians as they explore ideas and concepts; and negotiate the meanings of, and the connections among, those ideas with others in class (D'Ambrosio 1995).

### Azita Manouchehri, Mary C. Enderson and Lyle A. Pugnucco

The study of geometry in grades 5-8 should incorporate opportunities for students to engage in exploring and analyzing geometric shapes to conjecture about geometric relationships through data collection and model construction, according to the *Curriculum and Evaluation Standards for School Mathematics* (NCTM 1989). In this fashion, students will develop an intuitive understanding of geometric concepts and learn to reason formally and informally. Moreover, it is hoped that through such processes, students will formulate relevant definitions and theorems. The *Standards* document also encourages the use of computer technologies in middle school mathematics instruction. This suggestion was based on the assumption that interactive environments provided by appropriate geometry software have the potential to foster students' movement from concrete expetiences with mathematics to more formal levels of abstractions, nurture students' conjectuting spirit, and improve their mathematical thinking. Although the NCTM's visions for the geometry curriculum and for methods of teaching geometry in the middle levels are certainly attractive, many teachers are concerned about what software is useful for the middle school population, how such software can be used in instruction. what issues are associated with their use, and what the consequences are of learning and teaching mathematics within such environments.