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Effects of Engagements in Non-Decimal Numeration Systems on the Understanding of Place Value

Effects of Engagements in Non-Decimal Numeration Systems on the Understanding of Place Value. Aaron Trocki November 14, 2011. Stage Setting. Need for Highly Qualified Teachers (NCTM, 2000) Call for Conceptual Understandin g in Elementary School (Common Core, 2010)

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Effects of Engagements in Non-Decimal Numeration Systems on the Understanding of Place Value

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  1. Effects of Engagements in Non-Decimal Numeration Systems on the Understanding of Place Value Aaron Trocki November 14, 2011

  2. Stage Setting • Need for Highly Qualified Teachers (NCTM, 2000) • Call for Conceptual Understanding in Elementary School (Common Core, 2010) • “Explain why addition and subtraction strategies work, using place value and the properties of operations” (Common Core, 2010)

  3. Stage Setting • Since at least the 1960’s it has been posited that study of non-decimal numeration systems can increase foundational mathematical understanding and this notion is still evident in current research (it just won’t go away!) (include personal note here) • Purpose: Synthesize the research on place value understanding related to engagements with non-decimal numeration systems • Organization: Set Stage, Furnish Framework, Map Method, Report Results, Finish with Findings, Include Implications • Define Non-Decimal Numeration System

  4. Furnishing the Framework Research on Knowledge for Teaching • Need for content knowledge (Shulman, 1986) • Specialized Content Knowledge (Ball, 1988, 2008) Teacher Knowledge Related to Student Achievement • 1st and 3rd grade student achievement positively correlated to specialized knowledge for teaching (Hill, Rowan, Ball, 2005) • Place Value knowledge of teacher positively correlates to: • Types of activities afforded to students (contructivist, in depth, student creativity, etc.) • Student Achievement (Tanase, 2011)

  5. Furnishing the Framework Research on Pre-Service Teachers and Pre-Service Teachers’ Training • Prospective teachers’ knowledge upon entering college (Ball, 1990) • Prospective teachers’ knowledge at end of college (Ball, 1988) • 1) Build on currently held conceptions; 2) model teaching for understanding; 3) focus on connections between content knowledge and other types of knowledge (Thanheiser et al., 2010) • Teach to ‘unpack’ learning goals when analyzing mathematical tasks (Morris, Heibert, and Spitzer, 2009)

  6. Furnishing the Framework Demonstrate Importance and Difficulty in Understanding Place Value • Place value understanding is difficult to acquire even in a familiar context of dealing with money (Chandler and Kamii, 2009) • A lack of place value understanding can inhibit and slow conceptual understanding of algorithms (Cooper and Tomayko, 2011)

  7. Mapping the Method Include articles that assess the effects of non-decimal numeration system engagements Does not include simple activity descriptions such as in most practitioner articles

  8. Mapping the Method • ERIC and Google Scholar • Sample search terms • Non-decimal • Numeration systems • Alternate bases • Place value • Pre-service • Conceptual understanding

  9. Reporting the Results Two basic categories emerged Studies Done with Children Studies Done with Pre-Service Teachers

  10. Reporting the Results Kavett (1969) • 4th and 6th grades, 430 students • Trained 18 teachers in non-decimal numeration systems • Significant differences found on retention between control and experimental groups • Recommendation to teach non-decimal systems in late elementary school

  11. Reporting the Results Diedrich & Glennon (1970) • Test Dines’ “Mathematical Variability Principal” • Four 4th grade classes • Researcher guest taught • Posttest treatment effect found but not at a statistically significant level • Speculated that student interest in mathematics increased

  12. Reporting the Results Senger (1997) • 5th grade, 13 students • Qualitative in nature • Students created their own non-decimal number systems • Understanding on difference between number and number symbol increased

  13. Reporting the Results McClain (2003) • Class of Pre-Service Teachers • Hypothetical Learning Trajectory Proposed on Place Value and Multidigit Addition and Subtraction • Includes work in Base 8 • Results showed increased conceptual understanding of place value and addition and subtraction

  14. Reporting the Results Radin (2007) • Two classes of pre-service teachers • Different Instructors • Pre and Posttests and comparisons between classes • Significant differences found in systematic unfamiliar problem types but not in overall test scores

  15. Reporting the Results Khoury and Zazkis (1994) • 124 Pre-Service Teachers • Understanding multiple representations of fractions depends on understanding of place value • Tests implemented and interviews conducted • Most pre-service teachers believed the value of a fraction changed when represented in a different numeration system • Need for conceptual understanding of place value and fractions for pre-service teachers

  16. Reporting the Results Zazkis (1993) • 82 Pre-Service Teachers • Investigated problem solving strategies and systematic errors in place value and rational number understanding • Unfamiliar domain of non-decimal numeration systems • Common strategies for interpreting place value were found • Only 12 pre-service implemented a correct strategy

  17. Finishing with the Findings What do these studies tell us?

  18. Finishing with the Findings • Engagements with non-decimal numeration systems have been shown to increase student understanding of place value but results have been inconsistent and not necessarily generalizable • Student interest in and attitude toward math may improve through such engagements

  19. Finishing with the Findings • The evidence suggests that engagements with non-decimal numeration systems increases teachers understanding of place value and representations of fractions • Engagements with non-decimal numeration systems may serve as a profitable window to uncover pre-service teacher understanding or lack thereof of place value and representations of fractions

  20. Including the Implications • Engagements in non-decimal numeration systems are potentially powerful to increase understanding • Mathematics educators should consider using these engagements as many are available (include some of the 9 articles found)

  21. Including the Implications • More studies are needed to assess the effects of engagements in non-decimal numeration systems with both pre-service teachers and students • More research needs to be conducted to address where non-decimal numeration system study should be included in the learning trajectory on place value

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