The Mathematics Education of Primary-Grade TeachersAuthor(s): Joan P. Isenberg and Carol J. Altizer-TuningSource: The Arithmetic Teacher, Vol. 31, No. 5 (January 1984), pp. 23-27Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190888 .

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The Mathematics Education of Primary-Grade Teachers

By Joan P. Isenberg and Carol J. Altizer-Tuning

Mathematics is often viewed as dif- ficult subject matter by students pre- paring to be primary-grade teachers (kindergarten through grade 2). Many of them need to develop competence and confidence in mathematics in or- der to teach it in a meaningful and appropriate way. Providing for the preservice mathematics education of primary-grade teachers must be the joint responsibility of both the mathe- matics educator and the teacher edu- cator.

As school districts continue to ex- press concern about basic skills, mini- mum competencies, and declining achievement in mathematics, teacher educators must reexamine their pro- grams to ensure the preparation of the most competent primary-grade teach- ers of mathematics. Attention to the Guidelines for the Preparation of Teachers of Mathematics (NCTM 1981) and An Agenda for Action: Rec- ommendations for School Mathemat- ics of the 1980s (NCTM 1980) are critical components in this reexamina- tion process. Against this back- ground, we will discuss the general professional preparation of competent primary teachers, as well as the math- ematical content and methods used in that preparation.

Joan Isenberg teaches courses in early child- hood education and reading and language arts at early childhood levels at George Mason University in Fairfax, VA 22030. She is an assistant professor of education. Carol Altizer- Tuning teaches mathematics methods courses for elementary education majors and super- vises student teachers at Longwood College, Farmville, VA 23901.

General Professional Preparation

The general professional preparation of primary-grade teachers must ad- dress the development of young chil- dren and include cognitive as well as physical, social, and moral develop- ment. Preservice teachers must pay attention to how learning fits into the total development of the child. Recent research in human development indi- cates that individual differences in learning styles, language and thought, and the nature of the teacher/child relationship are key dimensions that relate to children's learning (Steven- son 1975). An in-depth understanding of the dynamics of such processes forms the basis of sound teaching practices. The works of Piaget, Bruner, Gagné, and Dienes have greatly influenced our understanding of how children learn. Such an under- standing must precede the implemen- tation of an appropriate mathematics program.

Knowledge of how children think combined with a skill in using that knowledge helps children develop healthy attitudes toward learning mathematics. Curriculum theorists such as Taba (1962) recommend that classroom practices teach children how to think for themselves. The classroom becomes more than a fact- finding place; it becomes a forum for thinking.

Despite the importance of this liter- ature on young children, classroom teachers report that their chief source of knowledge about teaching is their

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personal experience. They feel that teacher education programs are not helpful, and they avoid education principles, theory, and research find- ings (Howsam 1981). This pervasive attitude greatly hampers attempts to upgrade education in general and mathematics education in particular. Better preparation in mathematics education at the college level is one way to provide better classroom teaching of mathematics.

Preparing kindergarten teachers in mathematics education is often ne- glected in the professional program. Most kindergarten teachers can be certified with an endorsement in ei- ther elementary or early childhood education, and the two programs are often not the same. In general, an elementary-level endorsement to teach focuses on the elementary grades (kindergarten through grade 6) with training to teach kindergarten absorbed into the whole range of grades. Endorsement in early child- hood education focuses on the years from prekindergarten through grade 3. Differences in the mathematics prepa- ration of kindergarten teachers exist because of the differences between the two kinds of certification.

According to Castle (1978), the ap- proach to educating the elementary school teacher emphasizes mathemat- ics as a distinct discipline and a sepa- rate area of the curriculum, whereas the approach to educating the early childhood teacher focuses on the child as an area of study and encourages future teachers to help children learn mathematical concepts. The results from these two different approaches are contrasted in the classroom. Often teachers from a program for elemen- tary teachers do not appreciate the significance of the preoperational ac- tivities that young children need; teachers from a program for early childhood education often do not know when or how to move children from play and manipulative activities to elementary mathematics. Kinder- garten children can develop important mathematical concepts, and their teachers can provide activities that are important for their later growth in mathematics. Too often this opportu-

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nity is easily overlooked. All preservice primary education

majors need time to work with kinder- garten children in mathematics. Rath- er than perceiving the kindergarten year only as training and drill in prep- aration for first grade, preservice teachers who work with kindergarten children come to understand what kinds of mathematics are appropriate for kindergarten to learn. In addition, all prospective primary-grade teach- ers need time to observe, participate in, and react to the total primary mathematics program so that they can be effectively prepared in the mathe- matics that they will be teaching.

The classroom is a forum for thinking.

Overall goals The following considerations are of- fered for improving the mathematics education of future primary-grade teachers, kindergarten through sec- ond grade:

1 . Provide opportunities for field- based mathematical experiences at each primary-grade level prior to stu- dent teaching. Students need oppor- tunities to observe, plan, and interact with children at all developmental lev- els in the area of mathematics.

2. Give students experience with the materials they will use in teach- ing. If the use of concrete materials is important to children's learning and thinking, we must provide experi- ences with such materials in classes for teachers. If students have not used the materials during their own educa- tion, they will be less likely to use them in their own classrooms. Simi- larly, individual and small-group in- struction in the form of learning labo- ratories or centers may be effectively used in methods courses.

3. Rely on the strengths of your

teaching faculty. The general educa- tor and the mathematics educator must work together to equip students with techniques for teaching mathe- matics in the primary grades.

4. Encourage maximum communi- cation among teacher educators in specialized content areas. Helping students identify the commonalities of theory increases their ability to view learning as a unified whole rather than as separate and distinct bits of infor- mation. In turn, mathematics educa- tion can be further integrated into the total curriculum and can become a useful tool for practical, everyday liv- ing.

5. Place major emphasis on un- derstanding how children learn and develop mathematical concepts. Mathematicians, educators, and psy- chologists have many views on how understanding develops. Although their views are not all similar, they are complementary. There seems to be little disagreement that children should learn mathematics in a pro- gression from the use of concrete ma- terials to semiconcrete materials and finally to abstract statements. Teach- er educators have a responsibility to provide experiences for preservice teachers that illustrate various proce- dures and techniques for teaching for understanding.

6. Encourage prospective teachers to do their own thinking in the class- room. Using their own ideas will help prospective teachers help children do the same, and perhaps they will be less dependent on the commercially prepared materials for the mathemat- ics they are to teach.

Nothing positive, however, can happen in the classroom unless the teacher has a firm grasp on the con- tent and methodologies necessary to make mathematics education come alive for children growing up in to- day's complex and highly technologi- cal world. We need teachers who are prepared to educate children for to- morrow.

Arithmetic Teacher

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Mathematics Content

Four assumptions about mathematics content underlie the comments that follow:

1. A knowledge of mathematics is important in achieving economic se- curity and a personally rewarding life.

2. Most students - boy or girl, regardless of race, ethnic background, or socioeconomic status - are capable of learning the mathematics of the elementary and intermediate grades. Many of these students are also capa- ble of learning the algebra and geome- try taught at the high school level.

3. Good teachers must know more about a subject than they are required to know to teach their students.

4. The teaching of mathematics in the elementary grades is a critical factor in the future mathematical suc- cess of most students.

The study of mathematics begins in the primary grades. Even preschool children develop primitive mathemati- cal concepts from which they can proceed to more refined and complex concepts. It is vitally important that teachers of children of this age be both competent and confident in the mathematics that span these early grades.

What mathematics will primary- grade teachers be teaching? The mathematics of the primary grades can be classified into nine areas.

Prenumeration concepts Children need to be provided with experiences in which they recognize likenesses and differences of objects, sort objects by attributes, put things in some identifiable order, and find patterns.

Numbers and numeration

Children need to develop a concept of number. They must learn to recognize the numerical quality of sets of ob- jects - the "twoness" of two eyes, two hands, two apples, and so on; the "tenness" often fingers, ten toes, ten pennies, and so on. Children must also learn to recognize the numerals

associated with each number and the structure of our decimal system of numeration. The understanding of ba- sic mathematical concepts aids the development of the algorithms associ- ated with the operations of addition, subtraction, multiplication, and divi- sion. For example, an understanding of place value aids in understanding regrouping in addition and subtrac- tion.

Geometry

Primary-aged children need to be in- troduced to the simple geometric fig- ures - circle, triangle, square, and rectangle, for example. They should

The effective use of calculating devices requires a good "number sense."

be aware of the distinction between points or things inside a geometric figure and outside the figure as well as the points on the figure. They should also be able to find examples of ̂real- life" things shaped like circles, squares, or polygons.

Relations

Young children can learn to distin- guish between simple mathematical relations like the following:

7 > 3 2< 5 4 + 2 = 6.

Operations After children have developed a con- cept of number and know the numer- als that are associated with the num- bers, they can learn operations with numbers. In the primary grades, chil- dren become acquainted with addition and subtraction, simple multiplica- tion, and even some notion of divi- sion. They must learn algorithms for these operations and, even more im- portant, must learn to recognize the situations in which each operation is used. For example, if Mary and Jason put their money together to buy some

candy, what operation do you use to find out how much money they have to spend on candy?

Measurement

Before they use rulers, children need measuring experiences. Teachers can ask questions like the following: Who is the tallest child in the class? How do we find out? Which glass holds the most chocolate milk? How can we tell? They should also measure things with nonstandard units - for instance, Juan's foot, Thuan's foot, Ginny's foot.

Organization and interpretation of data

Even young children can become in- terested in collecting data about them- selves: What color eyes do the chil- dren in our class have? What color eyes do most of the students in this class have?

Problem solving

Young children can be encouraged to be good problem solvers even before they have learned to read either words or numerals. Of course, the problems must be appropriate for the children's levels of learning and interests (Castle 1978; Stevenson 1975). Children at- tack problems with unfettered minds until they have been conditioned to do otherwise. Some of their ideas for solutions may initially be unusual, but children can learn by their own trial and error to refine their guesses and test their solutions.

Children in the primary grades will need to present and test their solu- tions orally. Their spoken vocabulary may be imprecise, but if teachers lis- ten carefully to children's words and watch their body language, they can usually follow their thinking. Teach- ers must themselves know enough mathematics to recognize good math- ematical concepts and thinking when they see them in a primitive form. And having seen these concepts for what they are, teachers need to know where the children's mathematical concepts fit into the scheme of mathe- matics so they can build on the con-

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cepts the children already have and help the children sharpen their think- ing and extend their vocabularies.

Computing devices

Most people see elementary school mathematics as arithmetic - a collec- tion of number facts and operations with numbers. Teaching algorithms for the four fundamental operations becomes an end in itself. The pres- ence in the classroom of devices that can perform the fundamental opera- tions in fractions of seconds raises questions about the nature of mathe- matics and how it should be taught. Minicalculators and microcomputers will certainly reduce the need for hours of practice on such exercises as long division with four-digit divisors. But the effective use of calculators and computers will require a better "number sense" than ever before. Operators of computing devices need to know what operations are needed and to have some approximate idea about the size of the results. Children need to know when the answers the computing devices give them are rea- sonable, and they need to understand how to check their calculations. These skills rely on an understanding of elementary mathematics.

Courses for teachers in the primary grades The NCTM Commission on the Edu- cation of Teachers of Mathematics (1981) has made specific course rec- ommendations for teachers in the pri- mary grades. They suggest the follow- ing three courses, each equivalent to three hours a week for one semester - a total of nine semester hours: 1. Number systems, from the natural

and whole numbers through the rational numbers

2. Informal geometry, including mea- surement, graphing, and geometri- cal constructions, and the ideas of similarity and congruency

3. Methods of teaching the mathe- matics of early childhood and the primary grades, including diagno- sis and remediation of children's learning difficulties in mathematics

In recommending these particular courses, the commission assumes that preservice teachers have a high school background of the equivalent of two years of algebra and one year of geometry. If students in a teacher- training program do not have this mathematical background, they will need to take these courses as part of their preparation for teaching.

The commission's recommenda- tions exceed the current requirements in most teacher education programs (Dossey 1981). The standards of the past, however, are inadequate for the present and future decades. The tech- nological developments of the past decade have minimized the need for algorithmic skills at the same time that they have increased the need for sound mathematical thinking in daily life.

Methodological concerns

Teachers of mathematics in the ele- mentary school have a variety of deci- sions to make and a still wider variety of options to implement those deci- sions. Teachers begin with a group of students and some general ideas about the sequence of mathematics topics to be taught in elementary schools. Plan- ning to teach the various topics will almost certainly include decisions about things the children must under- stand, problems they must solve, and computational skills they must ac- quire.

In the course of teaching mathemat- ical topics, teachers need to -

• use teaching techniques appropriate for developing understanding, prob- lem-solving ability, and computa- tional skill;

• use textbooks and stimulate stu- dents to learn by reading the materi- al contained in those books;

• use concrete materials and visual aids when they will contribute to understanding;

• use laboratory activities to provide applications of the mathematics be- ing learned;

• provide remedial instruction for stu- dents with learning problems.

Teachers must adapt their instruction to the many different needs, interests, abilities, and learning styles of their students. They also must continuous- ly assess their efforts by using various evaluation procedures and instru- ments. Evaluation may include ob- serving students as they complete written practice activities or solve problems. It may also include diag- nostic testing using locally developed or standardized tests.

Teacher competencies

Although the foregoing discussion represents an oversimplified view of teaching mathematics, the complex nature of teaching is evident. Teach- ers need to learn to teach for a variety of objectives and to select content and learning activities that will achieve those objectives. They must offer readiness activities that aid the devel- opment of concepts as well as pro- mote the memorization of basic facts and the use of mathematics in practi- cal applications.

Teachers must develop many com- petencies to lead students to desirable achievement levels in mathematics as well as promote positive attitudes to- ward the subject. These competencies are developed not only through the study of selected topics in preservice methods courses but also in in-service training sessions. Areas of compe- tence should include the following:

1 . Knowing how children learn and how their thinking develops

2. Using manipulative materials and visual aids effectively in the de- velopment of specific concepts and skills

3. Selecting drill-and-practice ac- tivities to develop speed and accuracy in computation

4. Planning and implementing an appropriate problem-solving program that will improve students' ability to use and apply mathematics (Problem solving has been identified by NCTM as the major focus of school mathe- matics in the 1980s.)

5. Using the potential of calcula- tors and computers in the full range of mathematics instruction

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6. Assessing students' progress, di- agnosing errors, and implementing re- mediation

7. Grouping students for effective instruction

8. Adapting techniques for teach- ing mathematics to exceptional learn- ers

9. Applying strategies to help stu- dents read the language of mathemat- ics

Summary Teacher educators, state department of education personnel, mathematics supervisors, and those currently teaching mathematics in the primary grades must join together to plan courses and in-service training ses- sions that will give teachers the confi- dence and competence essential to the effective teaching of mathematics. Any program designed to improve the teaching of mathematics must inte- grate the general professional prepa- ration, the mathematics content, and the methodological concerns ad- dressed in this article at both the preservice and in-service levels of teacher education. The process is never ending.

References

Castle, K. "Suggestions for Kindergarten Mathematics Teacher Education." In Mathe- matics Teacher Education: Critical Issues and Trends, edited by Douglas Aichele. Washington, D.C.: National Education Asso- ciation, 1978.

Dossey, John A. "The Current Status of Pre- service Elementary Teacher-Education Pro- grams." Arithmetic Teacher 29 (September 1981):24-6.

Howsam, R. B. "The Trouble with Teacher Preparation." Educational Leadership 39 (November 1981): 144-47.

National Council of Teachers of Mathematics. An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, Va.: The Council, 1980.

National Council of Teachers of Mathematics, Commission on the Education of Teachers of Mathematics. Guidelines for the Preparation of Teachers of Mathematics. Reston, Va.: The Council, 1981.

Stevenson, Harold W. "Learning and Cogni- tion." In Mathematics Learning in Early Childhood, edited by Joseph N. Payne, pp. 1-14. Thirty-seventh Yearbook of the Na- tional Council of Teachers of Mathematics. Reston, Va.: The Council, 1975.

Taba, Hilda. Curriculum Development. New York: Harcourt, Brace, Jovanovich, 1962. m

Research Report

Manipulative Materials By Marilyn N. Suydam Ohio State University Columbus, OH 43212

In responses to questionnaires, most teachers indicate that they be- lieve that manipulative materials (chips, blocks, fraction pieces, etc.) should be used for mathematics in- struction. Children should be involved in the process of doing mathematics, and the use of concrete materials is integrally related to the development of meaning. As children work with objects and talk about what they're doing, they begin to see relation- ships - to learn mathematics.

Yet belief is not always translated into action. First-grade teachers re- port rather frequent use of manipula- tive materials. But teachers from grade 2 on indicate less and less use of materials.

As we are faced with widespread concern about achievement, it seems important to consider the evidence from research: • Lessons using manipulative materi- als have a higher probability of pro- ducing greater mathematics achieve- ment than do lessons in which such materials are not used. This finding presumes that using manipulatives is plausible in a lesson - they can't be used with all topics or for all pur- poses. • Achievement is enhanced across a variety of topics, at every grade level

K-8, at every achievement level, at every ability level. • Children need not necessarily ma- nipulate materials themselves for all lessons, however. Watching the teacher use the materials in a demon- stration is sometimes at least as effec- tive. This result may be because di- recting children's attention to important mathematical ideas is easier when the teacher is in control of the materials.

Only one caution is needed: not all children need to use manipulatives for the same amount of time. Prolonged use may keep some children using procedures too simple and inefficient for them. Concern for individual needs must govern the use of manip- ulative materials. Bibliography

Suydam, Marilyn N., and Jon L. Higgins. Ac- tivity-based Learning in Elementary School Mathematics: Recommendations from Re- search. Columbus, Ohio: ERIC/SMEAC, 1977. m

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ARITHMETIC TEACHER accepts classified advertising for the sale or purchase of materials and services related to mathematics and mathe- matics education, including positions wanted and positions available.

Classified ads are one column wide like this ad. Minimum depth is one inch; maximum depth is three inches. The rate for the space is $50/inch. You may have your own type set or we can have it set for you for an additional $10/inch.

January 1984 27

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