Using Literature to Teach Factorials

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Fig. 4 Sara’s picture storyFig. 5 Austin’s tree storythe same pattern, and each ends with a question.Furthermore, we can see understanding of factorialsthrough the use of sequential numbers beginningwith “1,” in Polly’s use of “each,” and in Tom’s use of“both.” In contrast, the use of such phrases as “onthose 2 planets” in figure 2 does not clearly showthat Kim understands the multiplicative relationshipinherent in a factorial problem. As a next step inteaching, these students would benefit from opportunitiesto examine factorials in other settings anduse them to solve other problems. They could beginan explicit study of combinations and permutations,including the factorial-based calculation proceduresassociated with these concepts. For Kim, theteacher will need to consider her level of understandingas other work samples are examined.What can we learn from the writing samples infigures 4 and 5? Sara ends her story with a questionand still uses a sequence of numbers beginningwith “1.” However, her story does not clearly revealher understanding of the sequential relationshipamong the numbers. For example, Sara starts withone table with a picture on it. Her story continuesby talking about the picture, although the use of “1table” suggests that the table is the beginning of themultiplicative sequence. She simply lists a numberof items in the picture; they increase in number, butthere is no sense of the “each,” which would conveyan understanding of the multiplicative as we saw inPolly’s story. Sara needs additional exposure to storieslike Anno’s Mysterious Multiplying Jar and additionaltime to discuss and explore factorials in thiscontext. The teacher might also examine her understandingof multiplication as repeated addition; thisrecognition is a precursor to understanding exponentsand factorials as different shorthand systemsfor some forms of multiplication, and Sara may notyet understand these concepts sufficiently.Similarly, Austin shows misunderstanding aboutfactorials and the task at hand. He does not end hisstory with a question, nor does he use such termsas each to convey the repeated multiplication of factorialrelationships. Another clue is his switch from“two trees in the hole” to “3 branches on the tree.”He recognized the need for a sequence of numbersbeginning with 1, but he did not seem to understandthe fact that the numbers suggest multiplicationeach time. Like Sara, Austin needs additionalpractice with factorials in contexts similar to thisone and may need an assessment of his understandingof prerequisite skills. For these two students,using manipulatives to model simpler situationsinvolving factorials might precede rereadingthe book. These students could also be taught thetechnique of drawing tree diagrams to illustratefactorial relationships.This teacher has an excellent opportunity to differentiateinstruction. By examining these students’work, she can divide her class into instructionalgroups on the basis of their current understandingof this topic. Polly, Kim, and Tom can move to moreadvanced study. Kim may need additional support.Sara and Austin need additional practice beforethey are ready to progress.ConclusionWE BELIEVE THAT THE LESSON DESCRIBED INthis article demonstrates the power and potential ofusing literature to support mathematical thinking inmiddle school mathematics programs through engagement,integration, and sensitivity. Harste(1999) states, “Nobody becomes literate without activeengagement in the process.” Although Harstewas referring specifically to reading, we believe thatthe same is true for mathematics: active engagementis essential for learning. Students were activelyengaged because this experience made teachingand learning factorials interesting, comprehensible,and meaningful. An integral part of the engagementwas blurring the disciplinary boundaries between literatureand mathematics, as well as creating a seamlessintegration between reading and writing. Similarly,the teacher was actively engaged because thisexperience enabled her to be an inquirer or teacher-464 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
esearcher. She collected and analyzed dataand, on the basis of that analysis, could besensitive and responsive to the individualneeds of students, which is, after all, whatgood learning and teaching are all about.ReferencesAnno, Masaichiro, and Mitsumasa Anno. Anno’sMysterious Multiplying Jar. New York: PhilomelBooks, 1983.Campbell, Jay. R., et al. NAEP 1994 Reading ReportCard for the Nation and States. Washington,D.C.: Educational Testing Service for theNational Center for Educational Statistics, Officeof Educational Research and Improvement,U.S. Department of Education, January1996.Goodman, Ken. “Who’s Afraid of Whole Language?Politics, Paradigms, Pedagogy, and thePress.” In Defense of Good Teaching: WhatTeachers Need to Know about the “ReadingWars,” edited by K. Goodman. York, Maine:Stenhouse Publishers, 1998.Harste, Jerome. Letter to authors, 10 June 1999.Kentucky Department of Education (KDE). “KentuckyCore Content.” Frankfort, Ky.: Divisionof Curriculum Development, 1999.Miller, Karen E., Jennifer E. Nelson, and MaryNaifeh. Cross-State Data Compendium for theNAEP 1994 Grade 4 Reading Assessment:Washington, D.C.: Educational Testing Servicefor the National Center for EducationalStatistics, Office of Educational Research andImprovement, U.S. Department of Education,1995.Mullis, Ina, et al. Executive Summary of theNAEP 1992 Report Card for the Nation andthe States: Data from the National and TrialState Assessments. Washington, D.C.: EducationalTesting Service for the National Centerfor Educational Statistics, Office of EducationalResearch and Improvement, U.S. Departmentof Education, 1993a.Mullis, Ina. NAEP 1992 Report Card for the Nationand the States: Data from the National and TrialState Assessments. Washington, D.C.: U.S. Departmentof Education, 1993b.Smith, Frank. “Just a Matter of Time.” Phi DeltaKappan (April 2001).Williams, Paul L. “NAEP 1994 Reading: A FirstLook.” Rev. ed. Washington, D.C.: EducationalTesting Service for National Center for the EducationalStatistics, Office of Educational Researchand Improvement, U.S. Department ofEducation, October 1995. VOL. 8, NO. 9 . MAY 2003 465
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Page 3: contents of one mysterious jar that

Using Literature to Teach Factorials

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