Effects of engagements in non decimal numeration systems on the understanding of place value
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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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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 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)


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


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)


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)


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)


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


Mapping the Method

  • ERIC and Google Scholar

  • Sample search terms

    • Non-decimal

    • Numeration systems

    • Alternate bases

    • Place value

    • Pre-service

    • Conceptual understanding


Reporting the Results

Two basic categories emerged

Studies Done with Children

Studies Done with Pre-Service Teachers


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


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


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


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


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


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


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


Finishing with the Findings

What do these studies tell us?


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


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


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)


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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