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Second Graders Understanding ofConstant Differenceand the Empty NumberGwenanne Salkind.EDCI 726 858May 10 2008 Introduction The NCTM Standards 2000 state thatprekindergarten through grade 2 students.should develop and use strategies for wholenumber computations with a focus onaddition and subtraction p 78 Second graders typically have difficultyunderstanding and solving two digit.subtraction problems that require regrouping Review of Literature Children can solve two digit subtraction problemsstrategically Carpenter et al 1999 Carroll Porter Representations can be powerful tools for learning. NCTM 2000 Goldin 2003 The empty number line is a visual representation thathas been used to develop conceptual understandingof subtraction strategies Bobis 2007 Klein Beishuizen Treffers 1998 . Constant difference is a powerful strategy forsubtraction because messy unfriendly problems caneasily be made friendly Fosnot Dolk 2001 p The Empty Number Line24 27 .53 27 Constant Difference Adding or subtracting the same number toboth the subtrahend and the minuend in asubtraction problem does not change the.50 25 25 49 24 25 Research QuestionsDo second grade students who weretaught using empty number lines 1 Use a constant difference strategy to.solve subtraction problems morefrequently 2 Have better mental computation skills speed accuracy 3 Have greater procedural competence . accuracy Participants Second Treatment Group 8 boys 6 girls 36 Asian 21 black 14 white .14 Hispanic 14 multi racial Control Group 7 boys 8 girls 40 Asian 33 Hispanic 13 multi racial 7 black 7 white. Similarities Differences inInstruction Both groups Two week unit 6 lessons Two digit subtraction. Constant difference Number lines Strings T F Story Problems Treatment group only Empty number lines.Strings True or False Problems12 6 15 7 16 8 Aaron is 31 yearsold Fahim is 1813 7 years old What is35 30 36 29.14 8 the difference intheir ages 29 17 30 1950 25 Sara is 43 years.51 26 32 20 33 21 old Tom is 8 years52 27 younger than Sara 30 22 29 23 How old is Tom 49 24 Example of number line used.during instruction both Examples of empty numberlines used during instruction treatment group only True or False 49 24 50 25.True or False 35 30 36 29 Data Sources Used to AnswerEach Research QuestionResearch QuestionsData Sources 1 2a 2b 3.Mental Speed Tests Written Subtraction Tests Student Interviews Student Work Samples Quantitative. Individual student scores were determined formental speed tests written subtraction tests andinterviews T tests were used to compare means betweentreatment and control groups . Qualitative Student written work samples written subtractiontests and notes from student interviews wereanalyzed for evidence of the use of the constantdifference strategy . True False equations interviews were codedaccording to students solution strategies invalidstrategy I guess G solved both sides S andused relational thinking R Mean Scores of.Pre PosttestsTreatment Controln 14 n 15Tests Pre Post Pre PostMental Speed 10 1 64 2 71 3 27 2 87.Written Subtraction 8 3 21 3 57 3 80 3 80Note There were no statistically significant differences between means Mean Scores of InterviewTreatment ControlSubtests Pre Post Pre Post.True False 10 2 43 5 86 1 43 2 86Differences 12 5 29 6 14 6 14 5 43Story Problems 3 0 86 1 71 1 57 1 14Mental Computation 4 0 71 1 43 0 57 1 14Note There were no statistically significant differences between means . Use of Constant Difference There was no evidence that a studentchanged a subtraction problem into aneasier problem using a constantdifference strategy . Students did use the constantdifference strategy to find givendifferences and to solve true falseequations Example of Using a Constant.Difference Strategy to FindGiven Differences Examples of Using a ConstantDifference Strategy to Solve True False Equations. Percent of Students Who Used a ConstantDifference Strategy on Written ClassworkPercent of Students60 Treatment GroupControl Group.40 n 15 Finding Given Solving True FalseDifferences Equations Number of Students Who Used aConstant Difference Strategy on the.True False Interview TaskNumber of StudentsTreatment Group Control Group Key Findings A high percentage of students used a.constant difference strategy to findgiven differences in both classes Only students in the who were taughtusing empty number lines used aconstant difference strategy to solve.true false equations Key Findings There were no statistically significantdifferences in mental computationspeed or accuracy between students.taught with an empty number line andthose who were not There were no statistically significantdifferences in procedural competencebetween students taught with an empty.number line and those who were not Limitations The instructional unit was too short There was not enough difference ininstruction between the two treatment. ReferencesBobis J 2007 The empty number line A useful tool or just anotherprocedure Teaching Children Mathematics 13 8 410 413 Carpenter T P Fennema E Franke M L Levi L Empson S B 1999 Children s mathematics Cognitively guided instruction .Portsmouth NH Heinemann Carroll W M Porter D 2002 Invented strategies can developmeaningful mathematical procedures In D L Chambers Ed Puttingresearch into practice in the elementary grades pp 16 20 Reson VA The National Council of Teachers of Mathematics .Fosnot C T Dolk M 2001 Young mathematicians at work Constructing number sense addition and subtraction Portsmouth NH Heinemann Goldin G A 2003 Representation in school mathematics A unifyingresearch perspective In J Kilpatrick W G Martin D Schifter Eds .A research companion to principles and standards for schoolmathematics pp 275 285 Reston VA NCTM Klein A S Beishuizen M Treffers A 1998 The empty numberline in Dutch second grades Realistic and gradual program design Journal for Research in Mathematics Education 29 4 443 464 .National Council of Teachers of Mathematics 2000 Principles andstandards for school mathematics Reston VA Author Introduction The NCTM Standards (2000) state that prekindergarten through grade 2 students should â€œdevelop and use strategies for whole number computations, with a focus on addition and subtractionâ€ (p. 78) Second graders typically have difficulty understanding and solving two-digit subtraction problems that require regrouping.