Teachers who teach math want their students to succeed in mathematics. They want students to experience high-quality, engaging mathematics instruction. There are ambitious expectations for all, with accommodation for those who need it. Of course not all learners of mathematics are successful for various reasons. Therefore, mathematical leaders and politicians started the reform movement in mathematics. Prodded by a series of critical national advisory reports and by disappointing results from international comparisons of mathematics achievement, organizations such as the National Council of Teachers of Mathematics (NCTM) formulated an agenda for reform in three volumes of professional standards: Curriculum and Evaluation Standards for School Mathematics (1989), Professional Standards for Teaching School Mathematics (1991), and Assessment Standards for School Mathematics (1995). In the year 2000, NCTM created Principles and Standards of School Mathematics (PSSM) which includes the principles and standards. In July 2001, The Mathematics Achievement Partnership (MAP) created the draft for Foundations for Success: Mathematics for the Middle Grades (FFS). These documents were created to support teachers by equipping them with the knowledge and skills they need to help raise student proficiency and understanding of the mathematics curriculum.
There are many issues worth tackling in the field of mathematics education among the “math wars.” One issue many traditionalist and reformist educators cannot agree upon is communication in mathematics. Journal writing is a popular tool teachers use to incorporate writing in their math classes. When reading, writing, discussing, and thinking is encouraged in the math classroom, students often not only focus on procedural knowledge of algorithms, but they also communicate their thoughts, explanations, or clarifications in writing. In this paper, I will describe and summarize the major arguments and research for and against the issue of communication in math and state my personal view of this issue. This paper will also include my explicit suggestions for Principles and Standards of School Mathematics (PSSM), created by The National Council of Teachers of Mathematics and Foundations for Success: Mathematics for the Middle Grades (FFS), created by Mathematics Achievement Partnership (MAP).
Many educators of mathematics believe communication is a crucial part of mathematics. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public (NCTM, 2000). When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others orally or in writing, they learn to be clear and convincing. Listening to others’ thoughts and explanation about their reasoning gives students the opportunity to develop their own understandings. Conversations between peers and teachers will foster deeper understanding of the knowledge of mathematical concepts. When children think, respond, discuss, elaborate, write, read, listen, and inquire about mathematical concepts, they reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically (NCTM, 2000).
Silver and Smith believe that "the current interest in issues of communication is both more widespread than ever before and more central to reform efforts than at any other time in the history of mathematics education" (1997). The communication in math encompasses both oral and written communication. If students talk about their thinking as they solve problems, the teacher can tailor the lesson to suit the student's way of thinking. "Teachers' knowledge of students' thinking is an important guide in planning effective lessons" (Maher & Martino, 1992). Teachers can learn about their students' thinking through the students' writing as well as the students' spoken words. In fact, "students who will not ask questions in class may express their confusion privately in writing" (Miller, 1991). So it is possible for the teacher to adjust the lesson during the lesson on the basis of the students' oral comments, and it is possible for the teacher to fine tune future lessons on the basis of students' written comments.
Within the field of education there is fresh interest in inquiry as a framework for teaching and learning, an interest that resonates with the theoretical turn toward social constructivism in both mathematics and reading education. Building on the work of Peirce (1982) and of Dewey (1933), both of whom defined inquiry as the process of settling doubt and fixing belief within a community, some educators have begun to consider ways to help students experience this more powerful, if more tentative, view of knowledge. This perspective on inquiry is especially significant for mathematics education, given the dominance of a "techniques curriculum" (Bishop, 1988) that represents mathematics as a collection of facts and procedures. Such curricula tend to reinforce myths about what it means to know and learn mathematics, myths that are popular yet dysfunctional for learners (Borasi,1992). Engaging students in mathematical inquiries, that is, inviting them to experience and appreciate first hand the ambiguity, nonlinearity, and "conscious guessing" (Lampert, 1990) associated with the mathematical thinking of professional mathematicians, is one way to demystify mathematics learning for students (Borasi & Siegel, 1994b).
Language takes on new importance in classrooms in which knowledge is regarded as a social construction, providing the symbolic resources for members of a community to negotiate meanings and representations of their world. This is quite different from the way language works in mathematics classrooms in which learning techniques is the main goal. In these classrooms, language functions as a channel through which previously established knowledge is transmitted from expert (the teacher or textbook) to novice (the students) through teacher talk and textbooks. From this perspective, reading serves as a means of receiving the expert's message (and may be perceived by the teacher as an obstacle to students' learning when appropriate reading skills are lacking), whereas writing functions as a mode of knowledge display that provides evidence that the students have achieved the desired learning outcome (Pimm, 1987). In contrast, the idea that language is a productive force that both shapes and is shaped by the community suggests new roles for reading, writing, and talking in the mathematics classroom, roles that may provide the support students need to fully experience and learn from their inquiries.
In a study done by Siegel and Judith (1998) on using reading, writing, and talking in mathematics to boost students’ understanding, their findings prove that communication in a mathematics classroom can be very beneficial. They analyzed three classroom experiences in which secondary mathematics students engaged in "inquiry cycles" on quite different topics. These instructional experiences were developed by a collaborative team of mathematics teachers, mathematics education researchers, and a reading researcher in the context of action research and teacher research. Analysis of the data led to the identification of 30 functions of reading that are specific to distinct elements of an inquiry cycle. On the basis of these findings they suggest that reading can serve multiple roles in inquiry-based mathematics classes and, in doing so, can afford students unique opportunities for learning mathematics.
Many mathematics educators are advocating methods of assessment other than frequent tests, quizzes, and daily worksheets (Williams & Wynne, 2000). In fact, the NCTM's Curriculum and Evaluation Standards states, "The assessment of students' ability to communicate mathematics should provide evidence that they can express mathematical ideas by speaking, writing, demonstrating and depicting them visually" (1989, 214). One alternative form of assessment that incorporates these standards is journal writing. Writing or journal writing in math helps students stretch their thinking and make sense of problems that can sometimes leave them confused or frustrated. When children write in journals, they examine, express, and keep track of their reasoning, which is especially useful when ideas are too complex to keep in their heads (Burns & Silbey, 1999). Although algorithms provide an efficient route to securing correct answers, children often do not understand how or why these procedures work. Merely directing children to follow the traditional rules for algorithms in a lockstep fashion is like expecting children to arrive without having made the journey (Barnes, 1995). However, when teachers encourage children to write about the process, it can be a valuable way for learners to make sense of these algorithms for themselves (Countryman, 1992). Teachers can use writing to assess their students’ understanding by evaluating their progress and recognizing their strengths and needs, foster conceptual understanding, and extend mathematical conversations in class.
Traditionalist educators are reluctant to accept the method of communications in mathematics. When traditionalists see students engaged and talking with one another, asking questions, thinking about the mathematics and mathematical relationships, they view these behaviors and infer that the basics and other important mathematics are not being taught (Stiff, 2000). They believe learning takes place when the basics and fundamental mathematical concepts are taught in the classroom. Teachers in a traditional classroom often disseminate knowledge, facts, algorithms and generally expect students to identify and replicate the fields of knowledge disseminated (Brooks & Brooks, 1993, chap. 2). Most teachers rely heavily on textbooks (Ben-Peretz, 1990). They claim students should be engaged in text-book inquires, not inquiring within the teacher and other students (Morse, 1998). Students do not learn “naturally” by reading, writing, discussing, elaborating, thinking, and inquiring mathematics (Morse, 1998). Because the emphasis is on procedural knowledge and memorization of algorithms, students often work independently completing workbooks and ditto sheets. Teachers believe independent practice rather than cooperation with groups will help students learn the mathematical concepts. When asking students questions, most teachers seek one “right” answer to the math problem and will explain why the answer is correct. In addition, traditionalists believe schooling is premised on the notion that there exists a fixed world that the learner must come to know. The construction of new knowledge is not as highly valued as the ability to demonstrate mastery of conventionally accepted understandings (Brooks & Brooks, 1993, chap. 2).
Some educators are opposed to writing and reading in the math classroom. Grass-roots groups oppose writing and reading in math because of poor math scores on the National Assessment of Educational Progress. In 1996, California fourth graders were near the bottom, outscoring only students in Mississippi, Guam, and the District of Columbia (Kantrowitz & Murr, 1997). Educators claim that students were not mastering specific skills by certain grades. Some parents were complaining that their kids were not learning rudimentary computation (Kantrowitz & Murr, 1997). These educators believe fundamental math skills are the basics in achieving success in a student’s mathematical education. Educators opposed to writing and reading in math believe basic computational skills should be learned in a math class. Many parents also agree that writing and reading should take place in English or writing classes. Some parents learn their kids can’t multiply without a calculator and spend more time writing and reading about math than doing it (Walters, 1997). They also argue that math class is a time to learn mathematical concepts and not writing, reading, or discussing.
Why Communication is Important
In my opinion, I think students should have the opportunity to construct their own knowledge when learning about mathematical concepts. I view students as thinkers with emerging theories about the world. Students should be able to work cooperatively in groups and independently to make the necessary mental constructions about a particular math concept. For example, there is great benefit to allowing students to construct their own algorithms for multiplication and division. However, this does not mean that the standard algorithms for multiplication and division cannot be taught in meaningful ways that help students integrate new knowledge or procedures with existing understandings of multiplication and division. Nor does teaching the standard algorithm mean that standard algorithms are the first or only algorithms to which students should be exposed. Certainly, teachers can foster a greater understanding of these operations by using objects as referents for numbers and demonstrating the physical manipulations associated with each operation (Stiff, 2000). When students construct their own knowledge of mathematical concepts, they need to have the opportunity to think about, discuss, extend, elaborate, verbalize, write, listen, and read in the mathematics classroom. Normally in a math class, students are not accustomed to “talk” about mathematical concepts. They are usually “taught” the concept by the teacher. Therefore, teachers need to use a number of approaches to probe students’ thinking in mathematics. Students are not natural talkers in the math classrooms. Older students will encounter more complex concepts in higher levels of math, discussing, talking, elaborating, writing, reading, and thinking about complex themes and concepts will help students to obtain deeper understanding in math. The president of NCTM, Lee V. Stiff states positively, “NCTM's Principles and Standards is not synonymous with constructivism or any other single teaching approach.” I agree with this statement. I believe students learn in a variety of ways. Communication should definitely be a part of the mathematics classroom. Using different approaches of learning to teach mathematical concepts is my method as well.
An effective educational program needs to have clearly defined goals as to what students should know, understand and be able to do, as well as a plan that effectively guides instruction to the levels defined by the standards. The Communication Standard in the PSSM clearly states the importance of communication in mathematics education early on in a child’s schooling career. On page 59, it states, “Instructional programs from pre-kindergarten through grade 12 should enable all students to:
· organize and consolidate their mathematical thinking through communication;
· communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
· analyze and evaluate the mathematical thinking and strategies of others;
· use the language of mathematics to express mathematical ideas precisely.”
On pages 59-62, the PSSM clearly explains the expectations of the Communication Principle.
Students can organize and consolidate their math thinking through communication. When students present their methods of solving problems, justify their reasoning to a classmate or teacher, and formulate a question about something that is puzzling, they gain insights into their thinking. Reflection and communication are intertwined processes in mathematics learning. Writing in math can also help students consolidate their thinking because it requires them to reflect on their work and clarify their thoughts about the ideas developed in the lesson.
Students need to communicate their math thinking coherently and clearly to peers, teachers, and others. In order for a math result to be recognized as correct, the proposed proof must be accepted by the community of math professionals. This way, students can test their ideas on the basis of shared knowledge in the mathematical community of the classroom to see whether they can be understood. In order for students to communicate their math thinking coherently in the classroom, they need to have numerous opportunities to participate in whole-class discussions or small group discussions and activities to practice.
Analyzing and evaluating the mathematical thinking and strategies of others is another important expectation. While working with others on math problems, students will gain several benefits. Students who often has one way of seeing a problem can profit from another student’s view, which may reveal a different aspect of the problem. Students can actively participate in sharing and analyzing one another’s strategies in solving arithmetic problems. The strategies can become objects of discussion and critique.
It is crucial for students to use the language of mathematics to express mathematical ideas precisely. Early on in their school careers, they need to build a connection to formal mathematical language. Teachers can foster this by using mathematical vocabulary starting in the early ages of students’ education. Beginning in the middle grades, students should understand the role of mathematical definitions and should use them in mathematical work. In high school, the language should become more pervasive. However, is important to avoid a premature rush to impose formal math language on students. They need to have an understanding of the concept in order for them to use the mathematical vocabulary in a conventional way, in their own words.
I personally think the PSSM is a well written standards and principles document. It gives a clear summary for the four sub-standards of the Communication Standards in Chapter 3. In Chapter 4, standards for k-2 grades are given; in Chapter 5, standards for grades 3-5 are given; in Chapter 6, standards for grades 6-8 are given; in Chapter 7, standards for grades 9-12 are stated. Each chapter gives expectations for teachers and students in the classroom.
On page 193, the document gives examples of what communication should look like in grades 3 through 5. On this page, an example problem is given from a fifth grade classroom.
Pretend you are a jeweler. Sometimes people come
in to get rings resized. When you cut down a ring to make it smaller, you keep
the small portion of gold in exchange for the work you have done. Recently you
have collected these amounts:
|
1.14 g |
.089 g |
.3 g |
Now you have a repair job to do for which you need
some gold. You are wondering if you have enough. Work together with your group
to figure out how much gold you have collected. Be prepared to show the class
your solution. (P. 114)
It describes the decimal activity in detail including expectations of the teacher and of the students. In this activity, the teacher presented the students with a problem-solving situation. Although they had worked with representing decimals, they had not discussed adding them. As was customary in the class, the students were expected to talk with their peers to solve the problem and to share their results and thinking with the class. The students used communication as a natural and essential part of the problem-solving process. As the groups worked, the teacher circulated among the student. Students were engaged in dialog and students continued to solve the problem through student and teacher discussions.
The example given in Chapter 5 is fairly explicit. However, it does not give examples of communication expectation for each individual grade: third, fourth, and fifth. Currently it is difficult to relate the examples to a specific grade level. K-2, 3-5, 6-8 or 9-12 spans too broad a spectrum of abilities, and the examples should be specific. The document should include an example of communication sub-standards for each specific grade in order for teachers to have a clear understanding of the Communication Standard. This way, teachers of all grades will have a “guide-line” to follow with examples for each specific grade. Because I think the PSSM is a well written document, I only have one suggestion for the Communication Standard. I agree with the suggestions and expectations given for students and teachers in the Communication Standard. By following the expectations and standards, teachers will have success in teaching the importance of communication in a mathematics classroom.
The Mathematics Achievement Partnership (MAP) has been working with its partner states to strengthen U.S. mathematics education. MAP recognizes that improving student performance depends on a comprehensive approach based on:
supporting teachers by equipping them with the knowledge and skills they need to help raise student proficiency;
measuring student proficiency on a regular basis; and
using assessment results to assist teachers and improve classroom practice.
MAP’s work is grounded in the 1995 Third international Mathematics and Science Study (TIMSS). TIMSS data show that in too many U.S. classrooms, mathematics curricula in grades six through eight simply repeat previously taught concepts and do not provide deep study in any area. The data also show that American mathematics curricula for the middle grades generally do little to advance mathematics knowledge beyond arithmetic computation. Achieve reaffirmed the TIMSS findings with its own analysis of 21 state tests of fourth and eighth grade students. It found that more than 60 percent of the eighth grade test items dealt with computation, whole number operations, and fractions – concepts that students in other countries master before the seventh grade. In top performing nations, seventh and eighth grade curricula include proportionality and slope, congruence and similarity, equations and functions, and two and three dimensional geometry – topics that most U.S. state tests address sparingly, if at all.
Following the TIMSS analysis, MAP asked mathematicians and mathematics educators to take a fresh look at the mathematics expectations for the middle grades. MAP incorporates its advisory panel’s conclusions and the TIMSS findings in Foundations for Success, a blueprint for mathematics in the middle grades that is benchmarked to international standards. This set of expectations is designed to be challenging yet realistic, and eventually attainable by all students and teachers who are given adequate support. With the goal of providing all students with a strong foundation in mathematics before they begin high school, it aims to cultivate every student. Those who master fundamental concepts in the middle grades will have the tools they need to succeed in high school, college and the workplace.
Foundations for Success offers guidelines and targets for states to provide mathematics education that is benchmarked to the best in the world. It identifies the skills and knowledge that will underlie MAP’s professional development, curriculum, and assessment tools. The expert panel of mathematics educators have worked with Achieve to develop these expectations which represent a wide spectrum of perspectives about mathematics education. In the strands algebra, geometry and data analysis, Foundations for Success represents a balanced and informed viewpoint about the necessary emphases and scope of mathematics in the middle grades. To help illustrate what the MAP expectations mean, a number of sample problems and methods for solving these problems accompany the outline of learning objectives. The primary purpose of these problems is to assist curriculum developers and teacher educators as they rethink their learning objectives for students in the middle grades.
Foundations for Success is a well written document which clearly states their expectations for middle school students. The expectations comprise four strands, each of which encompasses three primary topics:
· Number: whole numbers, rational numbers, real numbers
· Data: measurement and approximation, data analysis, probability
· Geometry: common figures, measurements, transformations
· Algebra: symbols and operations, functions, equations
Each strand contains a brief introduction to clarify its purpose; a summary to convey its scope; and a set of expectations concerning what students need to know, understand, and be able to do in each topic area. In addition, a major part of each strand is devoted to sample problems designed to help illustrate the scope, depth and meaning of the expectations. These problems show more than just procedures and skills. They demonstrate the depth of mathematical understanding and reasoning skills that students need in order to become engaged citizens and productive employees in the twenty-first century.
The sample problems are real-life problems students need to figure out. Even though communication of math thinking is not mentioned in the document, the problems itself require students to think critically. The problems also have a heavy emphasis on reasoning. In the Data Strand, problem D3 requires students to use reasoning to solve the problem.
In Canada they measure distances in kilometers. One kilometer is about 60% of one mile. Estimate this same speed measured in both meters per hour and meters per second.
In order for students to solve this problem, they need to think critically. In order to convert 500 km per hour into meters per hour, we need to know that there are 1000 meters in each kilometer. So we multiply by 1000 to convert. Notice how we can make the dimensions cancel:
speed
= km m m
500— x 1000 — = 500,000 —
h km h
To convert 500 km per hour into meters per second, we begin with the previous
result:
500 km per hour is the same speed at 500,000 meters
per hour. Dividing by 60 (because there are 60 minutes for every hour; notice
how, in the calculation, we’re multiplying by 1) gives us the speed in meters
per minute:
m 1 hour
speed= 500,000 —— x —————
hour 60 minutes
= 500,000 meters
60 minute
Dividing once again by 60 (because there are 60
seconds in a minute) give us the speed in meters per second.
500,000 meters 1 minute
speed= ———— ——— x —————
60 minutes
60 seconds
= 139 meters
second
Even though this problem does not require a student to write an explanation to the problem, there is much reasoning behind the thinking. Problems such as this one are consistent in requiring students to use reasoning and think critically.
The emphasis on reasoning is evident throughout the problems in Foundations for Success. MAP also sets high standards for students by requiring them to think critically for each math problem. In order for students to understand mathematical concepts, they need to talk, discuss, elaborate, or write their thinking for more in depth understanding. It would be beneficial if MAP add a communication strand to the Foundations for Success document. In the communication strand, they can state their expectations for communication: why is communication in math important; how critical thinking and reasoning encompasses communication in mathematics ; problems and solutions that require students to write in depth explanations about specific real-life math problems. Other strands in the document require students to use reasoning. However, when students are not required to write down or communicate their thoughts or reasoning in some form or way, they will not do it. If middle school students are required to know strands related to number, data, geometry, and algebra, why not add on a communication strand, where students are required to write or communicate their thinking? MAP can consider placing math problems related to the number, data, geometry, or algebra strand into the new communication strand. This way, students will be accountable for communicating their thoughts and thinking for the problems involving each of the strands.
"Teachers' knowledge of students' thinking is an important guide in planning effective lessons" (Maher & Martino, 1992). Teachers can learn about their students' thinking through the students' writing as well as the students' spoken words. In fact, "students who will not ask questions in class may express their confusion privately in writing" (Miller, 1991). If the Foundations for Success document is mainly for teacher and educator use, teachers need to some kind of guideline or sample questions and solutions for communication. If the emphasis of this document is on reasoning, how will teachers assess students’ understanding if students are not communicating their reasoning and thinking in some means? MAP may also consider placing the communication expectation within each of the four strands. Communication can take part in the Summary of Expectations for number, data, geometry, and algebra strands. This way the expectation to communicate students’ thinking will be incorporated in each strand. Teachers may assess students’ understanding in each strand more easily if the communication piece is added to the Foundations for Success document.
Should communications be required in the mathematics classroom? Communication is an essential part of the mathematical classroom. Students may use verbal language to communicate their thoughts, extend thinking, and understand mathematical concepts. They may also use written language to explain, reason, and process their thinking of mathematical concepts. Communication is a tool which can help students to form questions or ideas about concepts. Conversations in which mathematical ideas are explored from multiple perspectives help the participants sharpen their thinking and make connections. Students who are involved in discussions in which they justify solutions—especially in the face of disagreement—will gain better mathematical understanding as they work to convince their peers about differing points of view (Hatano and Inagaki 1991). Such activity also helps students develop a language for expressing mathematical ideas and an appreciation of the need for precision in that language. Students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically (NCTM, 2000).
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