Madison^s Mathematics Laboratory for PreserviceElementary School Teachers

Charles R. NeatrourAssociate Professor of Mathematics Education, Madison College

Harrisonburg, Virginia 22801

Have you given your elementary students an opportunity to seekan understanding of mathematical ideas "through doing?" I ventureto say that less than one percent of our elementary students couldreply affirmatively to this question. Too frequently classrooms aredevoid of those manipulative materials essential to individualizeddiscovery and learning of mathematical concepts.

Greater effort is being made in educational circles today to haveelementary teachers individualize mathematics instruction for theirstudents. Recent attempts to nongrade elementary programs have ledto the development of new techniques through which to master the

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task of teaching mathematics. The Nuffield Project in England, theMadison and the Minnemath Projects in the United States, and theexperimentation in mathematics learning of Dr. Z. P. Dienes (pres-ently at Sherbrooke University, Canada) are promoting and devel-oping individualized laboratory techniques and materials for teach-ing mathematics in the elementary school. Teachers everywhere arebeing advised to evaluate, revise, and experiment with individually-oriented and activity-centered techniques for teaching mathematics.

In order to orient teachers to this concept of teaching, the course^Teaching Mathematics in the Elementary School" offered at Madi-son College for preservice teachers was recently restructured to pro-vide both laboratory and lecture sessions. The laboratory sessions areactivity-centered instructional periods during which both small andlarge groups of preservice teachers work with physical materials in anattempt to see better how children learn mathematics, to becomefamiliar with those materials most easily used with the laboratoryapproach, and to actually experience the student-centered classroomenvironment.The materials and activities selected for the laboratory sessions

are of such a nature that preservice teachers can see how it is possibleto individualize the teaching of mathematics in the elementary schoolwhen the laboratory approach is utilized. Although a number of com-mercial materials are available, many of the materials are designedand constructed before or during the laboratory sessions.

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Although the author serves as instructor for the lecture sessions,his position during the laboratory sessions is more that of a coordin-ator than that of a teacher. The preservice teachers are given an op-portunity to actually become involved in a setting where laboratorymaterials and activities useful in the learning of specific mathematicalconcepts are designed, constructed, and utilized. Three films shown

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during the lecture sessions provide background for the laboratory ex-

periences: Conservation (showing the Piaget-type investigation of"how children learn^), I Do and I Understand, and Mathematics atYour Finger Tips.Among the materials and activities included in these laboratory

sessions are:(1) Geoboards to teach ideas of metric and nonmetric geometry(2) Paper folding to teach ideas of metric and nonmetric geometry(3) Multibase blocks to teach concepts of numeration(4) Cuisenaire rods to teach number concepts (including addition, subtrac-

tion, multiplication, and division of whole numbers and fractions)(5) Attribute and logic blocks to teach sets and relations of sets(6) Fractional kits, geoboards, paper folding, and Cuisenaire rods to teach

fractional concepts(7) Mirror cards, paper cutting, and tracing to teach symmetry(8) Ropes, sticks, tapes, trundle wheels, hypsometers, scales, geoboards,

blocks, etc., to teach ideas of measurement(9) Dice, coins, Cuisenaire rods to teach probability

(10) Algebraic experience materials to teach algebraic concepts (includingrelations, functions, and linear algebra)

(11) Equalizer, abacus, and number line to teach basic number concepts(12) Ruler and compass to teach construction in geometry(13) Activities including the collecting, organizing, and presenting of data

(height, weight, temperature, etc.) to teach statistics(14) Activities designing individualized assignment cards for the teaching of

mathematical concepts in the elementary school laboratory

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The listed materials and activities represent only a few of those thatcan be used advantageously in the mathematics laboratory. Numerousother commercial and teacher-fabricated materials can be used. It isanticipated that, after teachers experience initial success with lab-oratory materials and techniques, they will endeavor to create alaboratory setting in their own classrooms where individualization ofmathematics learning can take place.

LAKE BOTTOM SEDIMENTSVery shortly scientists will be able to see what underlies lakes as clearly as

they can see on land, says geophysicist Robert P. Moyer.Meyer and project specialists William Unger and Thomas Mayer, all of the

University of Wisconsin Geophysics and Polar Research Center, developed tech-niques that make it possible to profile lake bottom sediment in much more detailthan previously possible.Meyer explained how his group used new techniques and recording equipment

to study the bottom of Green Bay and the southern part of Lake Michigan.Working from boats, they sent out a short, relatively high frequency sound

pulse every 10 feet and looked for reflections from the bottom every three to fourfeet. The reflections tell the thickness and distribution of the subbottom layers.This information, plus data from sample cores of the lake bottom, gave a detailedpicture of the sub-bottom layers. The combination of techniques makes it possibleto see bottom and sub-bottom structures as small as one foot in thickness.