Mathematics around the World

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  • Mathematics around the WorldAuthor(s): Mitzi SmythSource: The Arithmetic Teacher, Vol. 30, No. 8 (April 1983), pp. 18-20Published by: National Council of Teachers of MathematicsStable URL: .Accessed: 12/06/2014 21:03

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  • Mathematics around

    the World By Mitzi Smyth

    Chinese kindergarten in Singapore Children using Attribute Blocks in Sydney, Australia

    Why do our children, who live in one of the most affluent countries in the world, have such low mathematics scores? Why do so many children dislike mathematics? Do other coun- tries of the world have the same prob- lems? If not, how are they achieving success?

    With a curiosity to know the an- swers to these questions, I took a sabbatical and traveled around the world looking at schools. I visited 125 primary schools in thirty-two coun- tries. After a long search, I did find some answers. I discovered that there is a simple, relevant way to teach mathematics so that all students will

    Mitzi Smyth teaches kindergarten in the Crest- view Elementary School in Vista, California.

    enjoy the many facets of the subject; that there is a way to teach mathemat- ics that would produce better under- standing, help students achieve higher scores, stimulate more creativity in children's thinking, and appropriately prepare children in mathematics for the space age.

    The search for a more meaningful approach to mathematics was not sati- ated until I reached Australia and New Zealand - almost the last leg of my trip. Up until that time, I had found many similarities and differ- ences among the countries I visited - in class size; school starting ages; the types of manipulatives used, if manip- ulatives were used at all; and instruc- tional style, whole class versus small group or individualized. But no matter what these similarities and differences

    were, students, teachers, and admin- istrators felt the same underlying frus- tration. They faced the same basic problems we do. They were all trying different methods and approaches, but without the success they wanted to achieve.

    Because I had no planned itinerary other than the order of the countries I was to visit, I did not make any ad- vance arrangements to observe schools. Except for schools in major cities such as Helsinki. West Berlin, Paris, Athens, Singapore, and Syd- ney, Australia, where I had to clear with the departments of education, I took potluck - if I saw a school, I would go in and ask to observe.

    In the long run this method provid- ed an excellent cross-section of every type of school imaginable, including

    18 Arithmetic Teacher

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  • schools for the deaf, blind, and autis- tic. These impromptu visits also gave me an opportunity to see the schools in their normal classroom situations. Schools ranged from the most sophis- ticated in England and northern Eu- rope, with all kinds of instructional materials, to the most primitive in remote villages high in the Himalayas and in central Burma, with virtually no materials other than textbooks. But in all areas, each country was striving to uphold the highest teaching standards; all were placing a high pri- ority on education.

    The starting age of children varied from four and a half years in the British schools, to seven years in the Scandinavian countries. The degree of pressure placed on students some- how correlated with the starting age. The children who entered school at the earliest age seemed to be the ones who were subjected to the greatest pressures in school.

    I visited seven of our schools - both military-dependent and American. In each school I was struck with the excessive amount of pressure we place on our young. In the Scandina- vian schools, where children begin school at seven, education starts slowly and for some reason the teach- ers, administrators, and parents seem to understand the value of this. By sixth grade, the Scandinavian stu- dents had either caught up with our sixth graders or had surpassed them.

    Class size varied anywhere from an average of 15 to 19 students in the Scandinavian classrooms, to 40 to 60 in the Mideastrn countries. Regard- less of the size of the classes, children were working on the same page or same lesson at the same time. The exception was the British schools, where there was grouping and individ- ualization.

    An exciting new idea to me was the booklets in which the students worked. I soon discovered that all the countries except the United States use them. The booklets are of grid (graph) paper - the younger the child, the larger the squares. The stu- dents' work is all done in one book in consecutive order - no loose papers. Teachers and parents can see the progress the students are making, and

    the children can see the coherency and the improvements in their work.

    Another eye-opener was the real- ization that students in the rest of the world do not begin regrouping in sub- traction until around the fourth grade or nine years of age. The concept is considered too difficult until then - yet we struggle with this from the second grade on.

    Counting on fingers was a universal trait. I never observed a classroom in which mathematics was being taught that I didn't see counting on fingers - no stigma was attached to this. The only difference was the speed in which the children counted. Some could have competed with calcula- tors, they were so fast.

    The use of brightly colored manipu- latives was prevalent in the more af- fluent countries of Europe. Unifix cubes and Cuisenaire rods were the foremost manipulatives used for counting and exploring number rela- tionships. The use of attribute blocks to develop spatial awareness was common throughout Europe. In the Scandinavian countries, attribute blocks were the first manipulatives used in mathematics education.

    When I reached Australia I ob- served for the first time, in isolated areas, several new exciting approach- es. All were a hands-on or discovery method. For the first time I saw a sequentially planned program based on the developmental learning stages of children. Beginning in kindergar- ten, the students worked in groups with lots of manipulatives. Children were allowed to explore and experi- ment with number relationships for as long as needed to learn a concept. The programs were only in their second year of operation, but the children were enthusiastic and the teachers were encouraged.

    When I arrived in New Zealand I became even more excited about "maths." A new dimension had been added. There the children begin their counting, sorting, matching, and ex- ploring number relationships with bot- tle caps, seeds, nuts, stones, shells - items from their own environment, things they are familiar with.

    In Auckland, New Zealand the mathematics department was in their

    third year of a three-year program for their junior years (K, 1, 2). My excite- ment grew as I witnessed this ap- proach. The program was divided into four departments - logic, number, measurement, and shape/space. The students are taken through undirected experiences, directed experiences, studying numbers one to nine, opera- tions, extending operations, and nota- tion. (Surprisingly, this program was patterned after one in California by Mary Baratta-Horton called "Mathe- matics Their Way.")

    Pencil and paper are not used until such time as the children have fully grasped number relationships through extensive use of manipulatives - it is usually about two years before they are ready to push aside their manipu- latives. Textbooks do not appear on the scene until at least third grade. Students work at tables, often stand- ing, or on the floor, using all sorts of teacher-made and commercially pre- pared materials.

    During my last two hours in New Zealand I observed a multibase pro- gram in action. The students were seven-year-olds, beginning their third year in this special program. Clusters of students were seated on the carpet- ed floor with their multi-base boards and wooden base blocks. The stu- dents were working in different bases from a laminated card that provided the problems for each child to solve. To show me that the children under- stood what they were doing, the teacher had them move to another base and do the same problem. It was incredible - seven-year-olds doing multibase work and actually under- standing it. The teacher also brought out abacuses; the children were able to work the same problems on this device. The teacher pointed out that these children understood place value and could apply their understanding in word problems. In the process of do- ing base work they had learned their addition facts by rote without going through the normal process of memo- rizing by oral and written repetition.

    What I Learned After having had time to consider all that I observed, I have come to sever- al conclusions. In the United States

    April 1983 19

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  • Children using Unifix Cubes in New Plymouth, New Zealand

    we seem to operate under the illusion that the smarter the child the more pressure and work needs to be put on that child. In reality, this is not true. In the first place, children's nervous systems are not developed for rigid school training until about seven years of age. We need to proceed much more slowly, giving children plenty of time to understand concepts as they go - not jumping over neces- sary skills before the skills are learned. Few seem to want to spend the time it takes to let children under- stand mathematics and its processes, but until we do, we will always have the same problems in mathematics education and will continue to turn children away from mathematics.

    If we want to achieve the success we are striving for, we need to allow children enough time to assimilate, through repeated experimentation and exploration, the basic mathematical concepts. Children need to have more experiences in counting, recounting, sorting, ordering, seriating, discover- ing patterns, and exploring spatial re- lationships - first through manipula- tives from their own environment, then to teacher-devised materials, and

    on into more models such as Unifix cubes. Once children understand the concepts, they themselves will push away the manipulatives and reach for the pencil and paper; they will then be ready to record their work and, in time, to take on textbooks.

    One way to speed up this process would be to decrease class size, there- by allowing more time for each stu- dent. This would allow primary-grade teachers to develop thoroughly the necessary prenumber ideas. Then middle-grade teachers might not have to spend so much time trying to re- teach the students their basic addition and multiplication facts. Classes above the third grade could be larger and more time could be spent devel- oping decimal and fractional con- cepts, as well as problem solving.

    How can we improve the mathe- matics education of our children? I see the answer as one of allowing adequate time for hands-on experi- ences; assigning smaller class loads in grades K, 1, and 2; taking pressures off the youngsters; educating the par- ents, teachers, and administrators in the developmental learning stages of children; and providing mathematics

    specialists to work with teachers and to organize and coordinate mathemat- ics programs.

    We are in the space age of mathe- matics. We need to help our children learn to think mathematically. We need to give them the proper amount of time, proper tools, and in proper sequence. The children will probably amaze us with their creativity and productivity if they are provided with such a learning environment. But best of all, they may be turned on to math- ematics. It goes back to the old Chi- nese proverb: I hear, I forget; I see, I remember; I do, I understand, m

    Authors Sought The Editorial Panel of the Arithmetic Teach-

    er, the Council's journal for elementary school teachers of mathematics, is looking for prospec- tive authors for its regular features

    * 'IDEAS" and "Let's Do It." Those selected will be expected to write monthly articles for volume 32, which begins with the September 1984 is- sue.

    For more detailed information write to Man- aging Editor, Arithmetic Teacher, 1906 Associ- ation Drive, Reston, VA 22091.

    20 Arithmetic Teacher

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    Article Contentsp. 18p. 19p. 20

    Issue Table of ContentsThe Arithmetic Teacher, Vol. 30, No. 8 (April 1983), pp. 1-52Front MatterBy Way of IntroductionOne Point of View: Being a Real Teacher [pp. 4-4]Readers' Dialogue [pp. 5, 52]TEACHINGMaking Middle School Mathematics Exciting [pp. 6-7]

    Let's Do ItUsing Geometry Tiles As a Manipulative for Developing Basic Concepts [pp. 8-15]

    Mathematics around the World [pp. 18-20]Teacher Preparationa Coordinated Approach [pp. 21-22]Ideas [pp. 23-28]Challenge: For Able Students [pp. 24-24]What's Going On [pp. 29-29]Put Your Students in the Picture for Better Problem Solving [pp. 30-33]From the File [pp. 33-33]Another Use for Geoboards [pp. 34-37]Areas of Polygons on Isometric Dot Paper: Pick's Formula Revised [pp. 38-40]Reviewing and ViewingNew Books for PupilsReview: untitled [pp. 42-42]Review: untitled [pp. 42-42]

    New Books for TeachersReview: untitled [pp. 42-42]Review: untitled [pp. 42-42]

    EtceteraComputer SoftwareReview: untitled [pp. 42, 52]

    NCTM AFFILIATED GROUP OFFICERS [pp. 44-46]Books and Materials [pp. 46-51]From the FileBack Matter


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