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# Effects of Engagements in Non-Decimal Numeration Systems on the Understanding of Place Value - PowerPoint PPT Presentation

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

Aaron Trocki

November 14, 2011

Stage Setting the Understanding of Place Value

• 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 the Understanding of Place Value

• 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 the Understanding of Place Value

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 the Understanding of Place Value

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 the Understanding of Place Value

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 the Understanding of Place Value

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 the Understanding of Place Value

• Sample search terms

• Non-decimal

• Numeration systems

• Alternate bases

• Place value

• Pre-service

• Conceptual understanding

Reporting the Results the Understanding of Place Value

Two basic categories emerged

Studies Done with Children

Studies Done with Pre-Service Teachers

Reporting the Results the Understanding of Place Value

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 the Understanding of Place Value

Diedrich & Glennon (1970)

• Test Dines’ “Mathematical Variability Principal”

• Researcher guest taught

• Posttest treatment effect found but not at a statistically significant level

• Speculated that student interest in mathematics increased

Reporting the Results the Understanding of Place Value

Senger (1997)

• Qualitative in nature

• Students created their own non-decimal number systems

• Understanding on difference between number and number symbol increased

Reporting the Results the Understanding of Place Value

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 the Understanding of Place Value

• 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 the Understanding of Place Value

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 the Understanding of Place Value

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 the Understanding of Place Value

What do these studies tell us?

Finishing with the Findings the Understanding of Place Value

• 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 Understanding of Place Value

• 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 the Understanding of Place Value

• 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 the Understanding of Place Value

• 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