By Way of IntroductionAuthor(s): John LeBlancSource: The Arithmetic Teacher, Vol. 32, No. 6 (February 1985), pp. 2-3Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41192546 .

Accessed: 14/06/2014 09:36

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 195.34.78.242 on Sat, 14 Jun 2014 09:36:24 AMAll use subject to JSTOR Terms and Conditions

BvCDciyoF Introduction

The description of the goals of ele- mentary school mathematics is often made using a two-dimensional model. One dimension includes the content objectives, the other, the process ob- jectives. Content objectives include the topics in mathematics that are studied, whereas process objectives are the behaviors, the processes that students should develop and use as the content objectives are studied and mastered. One such model used in the 1981-82 National Assessment of Edu- cational Progress (NAEP 1981) is shown in figure 1 . As an alternative to this framework, some groups have identified a set of basic-skill areas to be studied in elementary school math- ematics. The ten areas listed by the National Council of Supervisors of Mathematics (1977) are as follows: (1) problem solving; (2) applying mathematics to everyday situations; (3) alertness to the reasonableness of results; (4) estimation and approxima- tion; (5) appropriate computational skills; (6) geometry; (7) measurement; (8) reading, interpreting, and con- structing tables; (9) using mathemat- ics to predict; and (10) computer liter- acy. One important goal not specifically mentioned in either of these two schemes is that of mathe- matical thinking, yet the development of mathematical thinking in students should be a central goal of any mathe- matics program.

The nature of mathematical think- ing can be viewed from different per- spectives. From one perspective mathematical thinking can be thought of as different from thinking in general because it involves terms with very

Fig. 1 . Objectives framework for the Second National Assessment

CONTENT

Numbers Variables Shape, Measure- Other and and size, ment topics

numera- relation- and tion ships position

PROCESS

Mathematical knowledge

Mathematical skill

Mathematical understanding

Mathematical application | I | | |

precise definitions; relationships be- tween and among numbers and sym- bols; and concepts represented by drawings and figures.

From another perspective mathe- matical thinking stresses the mental activity or methods used in studying mathematics. In such a view, mathe- matical thinking might take place in the following activities:

1. Developing or following a proce- dure to effect a given result (algo- rithmic thinking)

2. Looking for a pattern or an under- lying model that will organize or generalize some data

3. Seeing the need for and using infor- mal and formal methods of verify- ing a hypothesis

4. Applying heuristics in problem- solving situations

5. Using induction in establishing re- lationships

6. Using formal logic, including well- defined terms (and, or, if-then) and arguments involving syllogisms, analogies, and so on.

Finally, mathematical thinking can be associated with reasoning in the study of any aspect of mathematics. In this view one can find opportunities for developing mathematical thinking in students in every lesson of the mathematics program. By encourag- ing students to use precise language, to explain reasons for an answer, and to reflect on their methods of solving problems, we can foster not only their mathematical thinking but also their positive attitudes toward such think- ing.

The articles in this issue provide some examples of how mathematical thinking can be developed when teaching specific content or process- es. The issue opens with a strong statement by Herbert that manipula-

2 Arithmetic Teacher

This content downloaded from 195.34.78.242 on Sat, 14 Jun 2014 09:36:24 AMAll use subject to JSTOR Terms and Conditions

tives offer interesting ways to encour- age students' thinking (p. 4). Burns discusses the importance of question- ing in helping students develop and sharpen their thinking (p. 14), where- as Suydam's "Research Report" mentions the kinds of questions to ask (p. 18). Sawada suggests ways of nur- turing thinking by capitalizing on stu- dents' stories (p. 20). Harvey and Bright discuss classroom games and how they foster thinking skills (p. 23)» With several examples, from counting numbers to understanding concepts of fractions, Hatfield encourages the no- tion that instructional technology can be used as a basis for making mathe- matics a social activity guided by an adult (p. 27). Robert Reys highlights the central role of estimation in decid- ing what type of answer is required when checking its reasonableness (p> 37). Related to that process, mental computation not only helps visual thinking skills but, as Barbara Reys suggests, can encourage good number sense (p. 43). Three instructional

guidelines for developing and imple- menting a problem-solving program that promotes the development of mathematical thinking are advanced by Charles (p. 48). Burger highlights some misunderstandings that young students have about geometric figures and proposes several activities to clar- ify their thinking (p. 52). Carpenter discusses how students can capitalize on the relationship between mathe- matical concepts and procedures to foster understanding (p. 58). 4 'Let's Do It" this month focuses on ways of fostering students' think- ing by having them examine patterns (p. 7). The "IDEAS" section stresses the use of newspapers as sources of mathematical activities, and "Prob- lem Solving: Tips for Teachers" dis- cusses logical reasoning.

John LeBlañc Chairman y Editorial Panel

Bibliography Avital, Shmuel M., and Sara J. Shettle worth.

Objectives for Mathematics Learning. Bulle- tin 3. Toronto: Ontario Institute for Studies in

Education, 1968. Begle, Edward G., and James W. Wilson.

"Evaluation of Mathematics Programs.'* In Mathematics Education, Sixty-ninth Year- book of the National Society for the Study of Education, part 1, edited by Edward G. Be- gle. Chicago: University of Chicago Press, 1970.

Educational Leadership. Theme Issue: Think- ing Skills in the Curriculum. Alexandria, Va.: Association for Supervision and Curriculum Development, September 1984.

National Assessment of Educational Progress. Mathematics Objectives, 1981-82 Assess- ment. Denver, Colo.: Education Commission of the States, 1981.

National Council of Supervisors of Mathemat- ics. "Position Paper on Basic Skills." Arith- metic Teacher 25 (October 1977): 19-22.

Pikaart, Len. "A Simplified Taxonomie Model for Teachers of Mathematics in Elementary School." In Report of the TTT Project in Science and Mathematics: Tri-University Project in Elementary Education. New York: New York University, 1971.

Pikaart, Len, and Kenneth J. Travers. "Teach- ing Elementary School Mathematics: A Sim- plified Model." Arithmetic Teacher 20 (May 1973):332-42.

Wilson, James W. "Evaluation of Learning ih Secondary School Mathematics." In Hand- book on Formative and Summative Evalua- tion of Student Learning, edited by Benjamin S. Bloom. New York: McGraw-Hill Book Co., 1971. W

From the File

j Fraction^

FRACTIONS, PERCENTS, AND MESSAGES The following activity serves the dual purpose of

reinforcing the idea of a fraction and motivating the learner to do the mathematics to find the rnystery word. To introduce this exercise, explain that each word is a unit and each letter in a word is an equal part of the word. For example:

The letter "s" is 1/6 of "school." The letters "fr" are 2/6 or 1/3 of "friend."

Then the class proceeds to find different fractions of given words. The following are sample problems: 1 . Take the first 1/5 of "ocean." _o

Take the last 1/5 of "shock." _k What word have you spelled? ok

2. Make up a fraction exercise to spell your first name.

3. Take the first 2/4 of "manher." man Take the first 1/2 of "manner." man

Why do we obtain the same word? In order to make the practice moré appealing, the

exercise can now take the following form: Take-

• the first 1/2 of "vase," va • the first 2/3 of "cat," ca

• the first 2/5 of "times," Ji • the first 2/3 of "one." on

What do we look forward to at the end of school? vacation

To keep the students interested, you can make up new messages to match each season and events in the school and community. It is also effective to personalize this activity by spelling such messages as "Happy birthday, Katie" or "Jake is great."

Percents can also be reviewed effectively in the message activity since percents are fractions. Thus, the teacher can reinforce the percent concept in terms of the word model. For example:

100% means the entire word. 10% means 10/100 or 1/10 of the letters in the word. And so on.

For example: The first 20% of "mouse" _m The last 80% of "bathe" athe The first 75% of "mate" mat The last 66 2/3% of "tic" _jç The last 10% of "kilometers" _s What word have you spelled? mathematics

From the file oř Miriam A. Leiva, University of North Carolina at Charlotte, North Carolina

-Readers are encouraged to send in two copies of their classroom-tested ideas for "From the File" to the Arithmetic Teacher for review.-

February 1985 3

This content downloaded from 195.34.78.242 on Sat, 14 Jun 2014 09:36:24 AMAll use subject to JSTOR Terms and Conditions