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Procedia - Social and Behavioral Sciences 93 (2013) 151 - 157

3rd World Conference on Learning, Teaching and Educational Leadership (WCLTA-2012)

Survey of mathematics practices with concrete materials used in

Brazilian schools

Aleandra da Silva Figueira-Sampaioa*, Eliane Elias Ferreira dos Santosb, Gilberto

Arantes Carrijoc, Alexandre Cardosoc

Federal University of Uberlandia, P.O. Box 593, CEP 38408-100, Uberlandia, Minas Gerais, Brazil aCollege of Electrical Engineering, Postgraduate Course. Current Address: College of Business and Management

bSchool of Elementary Education cCollege of Electrical Engineering

Abstract

The objective of this study was to conduct a survey on teaching practices employing concrete materials used by mathematics educators. The study was carried out at eleven middle schools affiliated with Brazilian public universities. Data was collected by means of electronic questionnaires. Twenty nine concrete materials are used by Brazilian teachers. Most of these materials are designed for specific educational purposes. However, some have been adapted for content outside of the material's original scope. Most teaching materials deal with various mathematical topics. © 2013TheAuthors.PublishedbyElsevier Ltd.

Selection and peer review under responsibility of Prof. Dr. Ferhan Odaba§i Keywords: Concrete materials; teaching practices; mathematics; K-12 schools.

1. Introduction

Many mathematics teachers search for alternative ways to help teach content and promote authentic and critical learning. To this end, many teaching practices are based on constructivist principles and aim at building meaning through experience and investigation (Piaget, 1960; Flavell, 1963).

The objective of this study was to conduct a survey of teaching practices that involve using concrete materials at some of the best schools in Brazil.

2. Methods

Research was conducted at K-12 schools affiliated with Brazilian public federal universities. These schools are considered centers of excellence among Brazilian public schools. The teachers have exclusive contracts and are required to work 40 hours per week in two daily shifts and are forbidden to perform any other paid work. The

* Aleandra da Silva Figueira-Sampaioa. Tel.: +55-34-3239-4707; fax: +55-34-3239-4706 E-mail address: aleandra@fagen.ufu.br

1877-0428 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer review under responsibility of Prof. Dr. Ferhan Odaba§i

doi:10.1016/j.sbspro.2013.09.169

effectiveness of their didactic and pedagogic strategies is reflected in the high performance of their students in national assessments (Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira, 2011). Only two of the thirteen schools solicited (from 11 states) chose not to participate in the study.

Data was collected via an electronic questionnaire that was emailed to teachers of 6th to 9th grade mathematics (11 to 14 year old students). During the study (August to December, 2010), 36 of approximately 49 teachers (73%) answered the questionnaire.

The questionnaire listed concrete materials used in teaching. Teachers were asked to indicate which of these materials they had experience with and to link them to the corresponding mathematical content categories of the National Curriculum (Secretaria de Educaçâo Fundamental, 1998): numbers and operations (arithmetic and algebra), space and shape (geometry), quantities and measurements (arithmetic, algebra and geometry), and information processing (statistics, probability and combinatorics). The teachers also answered closed questions about their professional experience.

3. Results and discussion

3.1. Professional profile

Seventy-five percent of the teachers surveyed had graduate degrees (56% masters and 19% doctoral). It can be argued that these mathematics educators have distinguished professional profiles compared to teachers at other public schools. Eleven percent of the teachers had less than 2 years of teaching experience, 3% had 5 to 6 years, 8% had 7 to 9 years, 22% had 10 to 14 years, and 56% had 15 years or more. The vast majority (78%) had more than 10 years of teaching experience. This group of teachers has extensive experience and maturity as educators.

3.2. Teaching practices with concrete materials

The teachers indicated that they used 29 of the 32 concrete materials listed in the questionnaire for teaching mathematics. The materials were divided into three groups: most used, intermediate use and least used. Although a calculator was not listed, 8% of the teachers mentioned its use in activities involving irrational numbers. In the "most used" group, 33% of the concrete materials are used by more than 50% of Brazilian teachers. Most materials are in the "intermediate use" group. Approximately 37% of materials are used in activities developed by 22 to 42% of the teachers. The remaining 30% were classified as "least used" and were mentioned by 3-17% of the teachers.

It can be seen that specific educational goals dictate the choice of concrete materials. Target content was one of the key selection factors, while the same concrete material can be used to teach more than one type of mathematical content. Considering the thematic categories of the National Curriculum, 73% of the concrete materials were used to teach content from the numbers and operations category, 57% from the space and shape, 57% from the quantities and measurements thematic category, and only 7% from the information processing category.

The double-pan balance is the concrete material teachers use most to develop content for the numbers and operations category. Forty-eight percent of the professors are using this tool to help students construct procedures to solve first degree equations and inequalities based on the principles of equivalence. According to educational proposals (Warren & Cooper, 2005; Gardete & César, 2006), the double-pan balance is very useful in educational activities that teach procedures used to solve first degree equations.

Three other concrete materials that are widely used to develop content from the numbers and operations category are Montessori golden beads, squared paper and paper folding (Table 1). Forty-one percent of the professors use Montessori golden beads to work with the number system, natural numbers, rational numbers and related operations and properties. When the Italian physician and educator Maria Montessori designed Montessori golden beads, she intended to use it in activities that teach number writing and create connections between

Aleandra da Silva Figueira-Sampaio et al. / Procedia - Social and Behavioral Sciences 93 (2013) 151 symbols and the number of objects indicated (Szendrei, 1996).

Tablel. Percentage of teachers using concrete materials for teaching mathematics

Category Usage % (n=29)

Concrete Material Usage % (n=36) d s anns s io sea e ape h S d с d ts § g J 1 tr

br be В a Io a e c a p S ■.а з ns aa ue ffS

Rulers/compasses/protractors 2' 3 86 14 38 24

Geometric solids 2 86 - 34 28

Squared paper 1 2' 3 83 38 38 14

T 1, 2,3 Tangram 81 17 7 10

String 3 78 3 38 31

Double-pan balances 1 67 48 - 14

Paper folding 1 2 3 64 24 26 3

Sticks/straws/popsicle sticks 2 3 58 3 17 10

Packages 2 3 53 7 21 17

Montessori golden beads 1, 3 50 41 - 10

Geoboard 2, 3 42 - 21 3

Abacus 1 39 17 - -

Ethyl vinyl acetate - EVA 2 39 7 3 3

Dominos 1 33 10 - -

Math dominós 1 33 10 3 -

Rubiks cube 1 31 - 3 -

Cylindrical packages and marbles 1 31 - 3 3

Color tiles 1 31 10 3 -

Polyminoes/geometric puzzles 2 31 3 - -

Cardstock 1, 2, 3 28 14 7 3

Paper ruler (Metro de papel) 3 22 - - 14

Algebraic blocks (Blocos algébricos) 1 17 3 3 -

Cuisenaire rods 1, 2, 3 17 10 - 3

Cylindrical packages and beans 1 17 - 3 -

Egg cartons 1 14 3 - -

Algeplan (Algeplan) 1 8 3 - -

Piet Heins soma cube 2 8 - - -

Calculators 8 10 - -

Four color dominos (Dominó das quatro cores) 1 2 3 6 - - 3

Matix 1 3 3 - -

Recommended for: 1Numbers and operations; 2Space and shape and 3Quantities and measurements (original Portuguese denomination)

Thirty eight percent of the teachers use squared paper to explore rational numbers, equivalent fractions, comparison and operations, divisibility, primes, prime factorization, greatest common divisor - GCD, least common multiple - LCM, algebraic expressions, and 1st degree equations. Twenty four percent use paper folding to teach rational numbers in fraction form, fraction equivalence, comparisons and operations, and polynomial factoring. According to Cramer et al. (2009), the educational goal of using paper folding is to introduce the part-whole model to concepts of fractions, unit, order and equivalence and addition and subtraction of fractions at the concrete level.

The use of other concrete materials from the numbers and operations category is still limited despite their positive attributes and potential contributions to mathematical education (Table 1). Approximately 17% of the teachers use the abacus to teach the meaning of operations and related properties of natural numbers and rules of the decimal system. The very nature of this concrete material facilitates learning of this type of content. According to Zhou and Peverly (2005), the abacus enables semi-concrete representations of numbers and is easily used by students to create mental images of material needed for building understanding of mathematical concepts.

Many materials that can be divided into equal parts can be used to introduce fractions and their operations. According to Behr et al. (1992), the principle of the unit and its division into equal parts is suitable for learning about various fractions and associated interpretations.

Brazilian teachers are using these materials with some consistency. Several concrete materials are used to understand the concept and properties of fractions including the tangram (Rodríguez & Sarmiento, 2002), Cuisenaire rods (Cramer, Wyberg & Leavitt, 2008; Kurumeh, 2010), circular cardstock or EVA (Cramer, Wyberg & Leavitt, 2008; Spangler, 2011), square cardstock or EVA (Spangler, 2011) and color tiles (Lamon, 2011).

Color tiles have been suggested to represent positive (e.g. blue) and negative numbers (e.g. green) and to solve specific integer operations (Gadanidis, 1994). Brazilian teachers have also used them to teach positive and negative and to help in the understanding of numbering system rules, natural number operations and properties, and concepts of divisors and multiples of natural numbers.

The four concrete materials most commonly used by Brazilian teachers in the space and shape category are strings (38%), squared paper (38%), rulers/compasses/protractors (38%) and geometric solids (35%). Fewer teachers adopted paper folding (28%), packages (21%), geoboard (21%), and sticks/straws/popsicle sticks (17%) (Table 1).

Strings, squared paper and rulers/compasses/protractors are widely used to teach basic concepts of 2D geometry, such as measuring and classifying angles and the construction of circles and circumferences. The National Curriculum (Secretaria de Educa^ao Fundamental, 1998) recommends that teachers explore situations in which some geometric constructions with ruler and compass are necessary, such as visualizing and applying the properties of shapes and constructing other relationships.

Geoboard can help students understand the mathematical aspects of space and measurement to approach 2D shapes, translation, rotation, reflection, similarity, counting, right angles, classification, scale, position, congruence, area and perimeter (Scandrett, 2008). Solving geometric problems is more successful when students develop exploratory activities with geoboard (Cotic et al., 2010). This tool is widely used to construct and explore relationships of geometric shapes (Furner & Marinas, 2011). Brazilian teachers mainly use geoboard to study 2D shapes and the properties and elements of polygons, circumference and circles.

The art of paper folding helps develop concepts used in the study of 2D shapes, points, lines, and planes (Boakes, 2008). Using paper folding as a teaching tool is not new. Initial ideas are attributed to the German educator Friedrich Froebel who used paper folding to aid the understanding of basic geometry in early childhood education. Specifically, teachers use paper folding to explore consecutive, adjacent, complementary, supplementary, and vertex angles, the sum of angles within triangles and quadrilaterals, properties of isosceles and equilateral triangles, angles of parallelograms (rectangles, diamonds, squares), trapezoids and other geometric figures. Sticks/straws/popsicle sticks are used by Brazilian teachers to build empirical understanding of angles, relative positions of two lines in a plane, and elements and properties of 2D geometric figures.

Spatial visualization involves constructing and manipulating mental representations of objects in two and three

dimensions and seeing these objects from different perspectives (National Council of Teachers of Mathematics, 2000). As such, hands-on exploration of objects and geometric shapes is recommended to develop understanding of geometric relationships and shape structures (Boakes, 2008). In response to this recommendation, the teachers surveyed introduce the elements and properties of non-planar geometric figures through hands-on use of geometric solids and packages. To work with the concepts and properties of spatial geometry, Rancan and Giraffa (2011) challenged participants to fold paper into various geometric solids.

Other concrete materials are used much less frequently. Only 3% of the professors mentioned using algebraic blocks, magic cubes, cylindrical packages, color tiles, math dominos, and EVA to develop geometric knowledge. Of these materials, only EVA is suggested as a teaching resource in the educational proposals of the space and shape category (Table 1). Even without usage guidelines, the teachers surveyed were able to use other materials to develop content-specific activities for non-planar geometric figures.

Brazilian teachers use EVA so that students can construct 2D geometric figures and explore the elements that comprise them. Color tiles could replace EVA because it is cheaper and has similar characteristics and functionality in activities focused on teaching and learning about 2D geometric figures.

Although some materials are used more than others, all materials included under the quantities and measurements thematic category are used by Brazilian teachers: strings (31%), rulers/compasses/protractors (24%), packages (17%), squared paper and metro paper (14%), sticks/straws/popsicle sticks, Montessori golden beads and tangram (10%); only 3% used paper folding, geoboard, four color dominos, and Cuisenaire rods (Table 1).

Both string and ruler/compass/protractor materials are adopted to teach length and surface measurements. Research by Nunes, Light and Mason (1993) and Brososky and Neufeld (1994) shows that these materials function as measuring instruments whether standardized or not. According to Clements (1999), a sequence of activities which starts by measuring length with nonstandard units (e.g. a string) and then incorporates standard units of measurement (e.g. a ruler) allows the construction of and reflection on the meaning of measurement.

Packages found at the supermarket or at home have different shapes (rectangles, circles, squares, etc.) and dimensions and can be helpful in understanding units of measure. Therefore, packages are useful in activities that require measurements and that develop concepts of spatial geometry such as measurements of volume, capacity and mass.

According to Kamii and Clark (1997), iterative unit is the ability to think of the length of a small block as part of the length of an object that can be measured by repeated placement of the smaller object. This ability is important for understanding and establishing relationships through physical action.

To this end, teachers use different materials to address geometric quantities. To explore length measurements, metro (linear) made of paper or similar material is used as linear segments such as sticks/straws/popsicle sticks. Squared paper, tangram, Montessori golden beads, and even paper metro (squared) are used to measure the surface area of geometric shapes (usually 2D). The teachers surveyed were unanimous in their selection of Montessori golden beads to aid in understanding volume measurements.

These results closely follow educational proposals. To understand the concept of area, squared paper segments (Lamas et al., 2007) and parts of the tangram (Arruda & Almeida, 2008) are recommended in investigations of surface measurements with nonstandard units. Montessori golden beads are recommended in activities involving volume deduction and volume calculation of 3D shapes (Lamas, 2008).

Tangram pieces can be used for two purposes: to create geometric shapes that can be used to explore concepts, elements and properties of 2D shapes, and for surface measurements in nonstandard units which are used to build concepts of area and equivalent shape (Brinckova, Haviar & Dzurikova, 2007). The potential to inter-relate these concepts means that this teaching resource can facilitate the development of notions of space, form, position, size and measurement. Thus, Brazilian teachers are justified in their enthusiasm about using this material in exploratory activities of geometric content and measurement.

Even though geometric solids and cylindrical packages and marbles are not found in educational proposals for the construction of concepts of quantity and units of measure, teachers do associate them with 3D geometric

figures that are used to teach volume and capacity measurements. The double-pan balance is only used for activities involving mass measurement and EVA is used to explore the formula for calculating the areas of circles and regular polygons.

For information processing category, some teachers use squared paper (21%) and rulers/compasses/protractors (10%) to organize information in tables and for the construction and interpretation of graphs (Table 1). Tables and graphs are recommended for the representation of statistical concepts and terms such as data types, variables, averages, scale, frequency polygons, and others. Tables and graphs are also used to introduce probability (Backer, 2001).

Considering all the concrete materials used by teachers in the survey, nineteen are from the numbers and operations category, fourteen from the space and shape, and thirteen from quantities and measurements. The teachers in the survey were using 79% of the materials recommended in the numbers and operation category, 71% of space and shape, and 100% of the quantities and measurements materials. In addition, some materials supported content not originally indicated in the educational proposals of the thematic category.

4. Conclusions

In search of new educational activities, teachers use almost all the concrete materials suggested in the educational proposals for teaching mathematics. These materials are useful, attractive and well established in the practices of Brazilian teachers.

Most concrete materials cover various mathematics topics from the thematic category of the Brazilian National Curriculum. Teachers link the unique aspects of each teaching material with potential contributions to target mathematics content. Therefore, teachers are concerned about adopting the right material to express mathematical relationships and represent mathematical concepts. The care teachers put forth in working with the best materials promotes active and efficient learning.

Acknowledgements

The authors are grateful for financial support from FAPEMIG - Fundafao de Amparo á Pesquisa do Estado de Minas Gerais.

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