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Hands-on Learning: Using Math Tools in Afterschool Danette Parsley, Principal Consultant dparsley@mcrel.orgHeather Martindill, Sr. Consultant hmartindill@mcrel.orgMcREL

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What are Math Tools?

Mathematical tools can be defined broadly to include concrete materials (i.e., manipulatives), symbols, pictures, drawings, and technology

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Personal Knowledge of Tools Inventory

CountersCuisenaire rods/ Integer barsBase 10 blocksUnifix cubesTangramsPattern blocks

Paper MoneyGeoboardsDiceRulers & Tape MeasuresJudy ClocksCalculators & Computers

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Common Math ToolsCounters

Cuisenaire Rods/Integer Bars

Unifix Cubes

PaperMoney

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Common Math Tools (cont.)Tangrams Pattern

Blocks

JudyClock

Dice

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Common Math Tools (cont.)Base 10Blocks

Geoboards

Calculators & ComputersRulers &

Tape Measures

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Personal Knowledge of Tools Inventory

CountersCuisenaire rods/ Integer barsBase 10 blocksUnifix cubesTangramsPattern blocks

Paper MoneyGeoboardsDiceRulers & Tape MeasuresJudy ClocksCalculators & Computers

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Counters & Color Tiles

PatterningEstimating numbersKeeping count during a survey and forming simple graphsSolving number riddles/ acting out word problemsAnalyzing factors and multiplesIllustrating even and odd

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Cuisenaire RodsWhole numbersFractionsMeasurementRatioArea & perimeterSymmetry & congruenceThree-dimensional geometryPatterns Functions

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Unifix Cubes

CountingPatternsNumber comparisonsAddition & SubtractionFractionsBeginning MeasurementProbability

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Paper Money

Solving money related problems (word problems, addition, subtraction)Showing the day’s date using different combinationsExploring place value

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Tangrams

Forming simple shapesExploring geometric propertiesCombining shapes to form another shapeIdentifying congruent and similar shapes

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Pattern BlocksNaming and sorting shapesInvestigating relationships between shapesAnalyzing characteristics of shapesExploring patternsModeling and comparing fractionsInvestigating congruence, symmetry, tessellation, area, perimeter, angle measure

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Dice

ProbabilityRandom number generatorPermutations & combinationsGames

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Judy Clocks

Telling timeSolving time-related problems

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GeoboardsCountingNaming, forming, comparing and making generalizations about geometric shapesIllustrating and verifying well-known formulas for finding area and perimeterNaming, forming, comparing, and generalizing about anglesExploring fractions

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Base 10 Blocks

Exploring and discovering number relationshipsSolving problems that require addition, subtraction, division or multiplication of multi-digit numbersMeasuring in metric unitsModeling area of volume

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Rulers & Tape Measures

MeasurementScaleRelationship between units of measurement (metric & standard)

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Calculators & Computers

ComputationComplex problem solvingGraphingProgramming/GamesVirtual Manipulatives

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Phases of Mathematical Understanding

AbstractConcrete Transitional

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Why does the practice of using math tools work?

Tools can help students see connections among objects, symbols, language, and ideas.

Tools help students think flexibly about mathematics, use creativity to solve new mathematics problems, and explore mathematics with less anxiety.

Given the active and social aspects of afterschool, it is an ideal environment for allowing students to explore, test, build, think, talk, connect, and reason with mathematical tools.

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“We remember 10% of what we hear, 30% of what we see, and

90% of what we do.”- Jean Piaget

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Activity 1

Using the tools on your table, solve the following problem. Be prepared to share how you used the tool to solve the problem with the large group.

Draw a 7 X 7 square on graph paper. Show how to cut the square into the least number of smaller squares. Your cuts must be along the lines of the graph paper.

Problem from : Techniques of Problem Solving: Problem Deck AAA #37© 1980 Dale Seymour Publications.

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2 x 2 2 X 2

3 X 34 X 4

3 X 3

2 X 2

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Activity 2

In groups of three, select a tool you would like to use to solve the problem presented to your table.Each group at your table will work with a different tool.After each group solves the problem, debrief at your table.

Why did you choose that tool?How did you solve the problem with the tool you chose?

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What do you do if you want to use math tools in afterschool?

Determine what concepts and skills students are learning by talking to the day school teacher.Develop activities that students will enjoy.Provide an assortment of tools.Allow students to discuss how specific tools contributed to their thinking.

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Implementation Tips

Plan for the use of mathematical toolsEnsure students have access to a variety of toolsExplore the appropriate use of mathematical toolsPlan for frequent intervention and dialogueCommunicate how the tools help children learn

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Technology: Virtual Manipulatives

Sample of Tangrams online using The National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html

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Further Details on the use of Math Tools in Afterschool

http://www.sedl.org/afterschool/toolkits/

Afterschool Training Toolkit developed by the National Partnership for Quality Afterschool Learning

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“I hear and I forget. I see and I remember. I do and I

understand.”- Confucius

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National Partnership for Quality Afterschool Learning © McREL 2006

Hands-on Learning: Using Math Tools in Afterschool Resources

(These are just a few of the online resources available) Sites with printable templates for Manipulatives http://mason.gmu.edu/~mmankus/Handson/manipulatives.htm Provides templates for making your own manipulatives. http://math.donnayoung.org/ Provides templates for fractions, money, clocks and much more. http://mathcentral.uregina.ca/RR/database/RR.09.98/loewen2.html Provides manipulative templates & activities for upper grades. Sites with activities using manipulatives http://teacher.sholastic.com/max/index.htm Scholastic’s Max’s Math Adventures Provides activities with supplemental student pages many of which contain manipulatives. http://illuminations.nctm.org/ National Council of Teachers of Mathematics Includes activities, lessons and web links for mathematics activities using various tools. http://www.susancanthony.com/Resources/base10ideas.html Includes some ideas for using Base 10 blocks with students. Sites with Virtual Manipulatives http://matti.usu.edu/nlvm/nav/vlibrary.html National Library of Math Manipulatives http://www.arcytech.org/java Base 10 Blocks, Pattern Blocks, Integer Bars/ Cuisenaire Rods, Fraction Bars http://standards.nctm.org/document/eexamples/chap4/4.2/ Geo Board http://www.aaa.math.com/students/tools.html Provides links to online calculators, especially good for upper grades http://www.emints.org/ethemes/resources/S00000511.shtml These sites explain how to create tessellations with polygons using rotation and reflection. Also includes information about artist M.C. Escher and examples of his tessellation artwork. There are several interactive games for manipulating patterns online.

National Partnership for Quality Afterschool Learning © McREL 2006

Hands-on Learning: Using Math Tools in Afterschool References for Math Tools

Ball, D. (1992). Magical hopes: Manipulatives and the reform of math education. American Educator, 16 (1), 14-18, 46-47.

Carpenter, T.P., Fennema, E., Frank, M.L., Levi, L., & Empson, S.B. (1999). Children's Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann, Reston, VA: NCTM.

Clements, D. H., & McMillen, S. (2002). Rethinking "concrete" manipulatives. In D. L. Chambers (Ed.), Putting research into practice in the elementary grades (pp. 252-263). Reston, VA: National Council of Teachers of Mathematics.

English, L., & Halford, G. (1995). Mathematics education: models and processes. Hillsdale, NJ: Lawrence Erlbaum.

Fuson, K. C. (1992). Research on whole number addition and subtraction. In D.A. Grouws (Ed.), Handbook of research on teaching and learning (pp. 243-275). Old Tappan, NJ: Macmillan.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: teaching and learning mathematics with understanding. Portsmouth, NH. Heinemann.

National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

Sowell, E. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20, 498-505.

Stigler, J. W., & Barnes, R. (1988). Culture and mathematics learning. In E. Z. Rothkropf (Ed.) Reveiw of research in education, 15 p. 253-306. Washington, D.C.: American Educational Research Association.

Van de Walle, J. A. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.