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

Aaron TrockiNovember 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 FrameworkResearch 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 FrameworkResearch 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 FrameworkDemonstrate 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 emerged1. Studies Done with Children2. Studies Done with Pre-Service

Teachers

Reporting the ResultsKavett (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 ResultsDiedrich & 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 ResultsSenger (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 ResultsMcClain (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 ResultsRadin (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 ResultsKhoury 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 ResultsZazkis (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