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Using Technology to Enhance Student Learning in Contemporary Mathematics in Context

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mathlinkSpring 2000 ELC’s Newsletter forSecondary Mathematics Teachers Vol. 3, No. 2everyday learning corporation’sCONTENTSUsing Technology to Enhance Student Learning in Contemporary Mathematics in Context 1Mathematics Bookshelf 5Interview: Core Plus and Achievement in Science 6Core-Plus Mathematics Project Web Site:An Invaluable Resource 8ELC News 12New and Revised Products 14Suggestion Box 15The Last Word: Write for ELC’s Mathlink! 16Using Technology to Enhance Student Learning inContemporary Mathematics in Contextby Mark Thompson;Teacher; Contemporary Mathematics in Context;Kent City High School; Kent City, MichiganThe use of technology,particularly graphing calculators,is an integral partof the Contemporary Mathematics in Context (CMIC) program.It per-mits the curriculum and instruction to emphasize reasoning with multiplerepresentations (verbal,numerical,graphical,and symbolic) and to focuson goals in which mathematical thinking is central.By the time studentscomplete the first twenty pages of Course 1,they are making and reason-ing with histograms on their graphing calculators and the fun has justbegun.Students develop facility in using an incredible number of calcula-tor functions as tools.Within a short period of time,they are the pros,often showing me how to do new things.By the time students completeCourse 3,they have become quite proficient with technology,using fea-tures such as matrices,list operations,random-number generators,and afew of the sequence mode features.Course 4 students use the parametricequation capability to model motion,use the statistical distribution fea-tures to evaluate binomial and normal distributions,and utilize a Core-Plus Mathematics Project (CPMP)–developed spreadsheet program forproblem solving (to name just a few new uses of technology).CPMP hasalso developed special calculator software (downloadable from a com-puter) that enables students to explore topics or do procedures moreeasily,an opportunity not possible without complicated calculator programs.From my experience teaching the CMIC curriculum the past four years,itseems that technology serves three more specific purposes.The most fas-cinating use,in my opinion,is the way in which technology facilitatesteaching and learning by helping students make connections among dif-ferent strands of mathematics. A second powerful aspect of technologyuse is how it enables students to discover relationships and patterns,make and check conjectures,and reason about the relationships.Finally,appropriate technology is used for complex calculations and completingrepetitive procedures quickly.(Technology, continued on page 2)ISSUESinMATHEMATICS EDUCATIONelc’s mathlinkVol. 3, No.2PublisherAnna BelluominiEditorialChristine FraserElizabeth GlosniakSusan HalkoDesignFran BrownJess Schaal ProductionAnnette DavisElizabeth GabbardSilvana ValenzuelaContributing WritersDamon BlackmanJames K.LaserMark ThompsonMarcia Weller WeinholdELC’s Mathlink is publishedby Everyday LearningCorporation.ELC’s Mathlink is availableupon request to teachersand administrators who usethe secondary mathematicsprograms published byEveryday LearningCorporation.Main Offices:Two Prudential PlazaSuite 1200Chicago,IL 606011-800-322-MATH(312) 540-0210www.everydaylearning.comTo subscribe, contribute, orchange an address, write toELC’s MathlinkP.O. Box 812960Chicago,IL 60681©2000 byEveryday LearningCorporation2MAKING CONNECTIONS AMONGSTRANDS OF MATHEMATICSOne of my favorite examples of howtechnology helps students make con-nections among strands of mathemat-ics is from the Course 2 unit,Patternsin Chance, Lesson 4: “Expected Valueof a Waiting-Time Distribution.”Whileit would be very simple to inform stu-dents that the expected value of awaiting-time distribution is the recip-rocal of the probability of success oneach trial p because I said so,studentsgain a deeper mathematical under-standing of this investigation whenthey use technology.Students createlists on their graphing calculator simi-lar to Figure 1 where p = 0.3.I usuallyhave each student group do two of theprobabilities from 0.1,0.2,0.3,…,1.Then the results can be shared withthe entire class.In List 1 (L1),students enter the num-bers 1 through 25. (They even use theseq function to do this quickly if theywish to use larger tables.) In List 2(L2),they enter p(1–p)(L1–1),which pro-duces the probabilities very rapidly.List 3 (L3) is the product L1• L2,andthen the built in sum function for L3isused to get the estimated average wait-ing time (or expected value).A table of average waiting times iscompleted by the class and entered inother lists on the calculator.(Figure 2)Next,students construct a scatterplotof this new list (Figure 3) and discusswhether the shape of the graph is lin-ear,quadratic,exponential,power,orsome other kind of algebraic model.After concluding that it may be aninverse power model,they use the calculator again to complete a powerregression.(Figures 4 and 5)Having already discussed that the sumof the lists are low because they arenot infinitely long,it is easy to con-clude from the regression equationthat expected value E.V.= is agood equation for this model.This investigation would take days tocomplete by hand,yet students canfinish it in one 55-minute class periodby using technology.Students have abetter understanding of a theoreticalexpected value,and they have seen anexcellent link between statistics,prob-ability,and algebra.1p(Technology, continued from page 1)Figure 1Figure 2Figure 3Figure 4Figure 53Another situation in which technology is effectivelyused to connect mathematical ideas is in the Course 3unit,Discrete Models of Change, Lesson 3: “IteratingFunctions.”In the firstinvestigation,students usethe last answer feature ofthe calculator to iterate asimple linear function.Using the linear functionand the line y = x on agraph,they can identify thefixed point (and find itusing the intersect featureor table capabilities).Iterating the function determinesif it is attracting,repelling,or neither. A graphical itera-tion can be done using the Web feature of a sequentialgraph.(Figure 6) With more exploration,students makea connection between the slope of the function and itsbehavior when iterated.This is certainly a nice linkbetween algebraic func-tions and the discretemathematics strand.Of course,there are manyother mathematical con-nections that are made inthe curriculum thanks tofeatures available on agraphing


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