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Developing Adolescents’ Procedural Fluency and Strategic Competence in Mathematics. Brian Bottge University of Kentucky bbott2@uky.edu. In 20 minutes or less. The need The strategy The results. National Assessment of Educational Progress (NAEP). Grade 8 (Lee, Grigg, & Dion, 2007) - PowerPoint PPT Presentation
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Brian Bottge
University of Kentucky
bbott2@uky.edu
In 20 minutes or less
The need
The strategy
The results
National Assessment of Educational Progress (NAEP)Grade 8 (Lee, Grigg, & Dion, 2007)
67% with disabilities scored below Basic
26% without disabilities scored below Basic
Grade 12 (Perie, Grigg, & Dion, 2005)
83% with disabilities scored below Basic
36% without disabilities scored below Basic
Basic means students . . .
“should complete problems correctly with the help of structural prompts such as diagrams, charts, and graphs”
and includes
“the appropriate use of strategies and technological tools to understand fundamental algebraic and informal geometric concepts in problem solving” (p. 20). (Lee, Grigg, & Dion, 2007)
Expectations for Employees in Industry (ETS, 2007; NCEE, 2007; NRCCTC, 2006)
Compute whole numbers, fractions, decimals
Interpret data in graphs, tables, and formulas
Form and test hypotheses
Solve problems with fellow workers
Communicate orally and in writing
Use computers to perform tasks
Contextualized Nature of Problem Solving
Nursing (Noss, Hoyles, & Pozzi, 2002)
-rate to manually “flush” drug left in the “dead space” of a IV catheter
Automobile manufacturing (Smith, 1999)
-order a series of drill bits by diameter
Carpet laying (Masingila, 1994)
- estimate and calculate area, measure to scale
Problem Contexts, Prior Knowledge, and Mental Models
Learning involves connecting new information with previously learned, stored knowledge
From these connections, individuals form their own “mental models”
Mental models become more vivid with use
Problem for teachers: Many students have not had rich learning experiences (or not the kind valued by school), difficult for teachers to establish connections (if we keep teaching the same way)
I would think you’d be used to failing by now…
Criteria for Judging “Problem Solving” in Cognitive Science
“…a task (a) in which the student is interested and engaged and for which he wishes to obtain a resolution, and (b) for which the student does not have a readily accessible mathematical means by which to achieve that resolution” (Schoenfeld, 1989) (underline added)
“students should know what it feels like to be completely absorbed in a problem” (Bruner, 1960)
Arrange “experiences” that are engaging to students and that “live fruitfully and creatively in subsequent experiences.” (Dewey, 1938)
Procedural Teaching Approaches Lead to . . .
An “applied problem” from the National Assessment of Educational Progress (1983)
45,000 13-year-olds
An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed?
29% of the students chose“31 remainder 12”
Procedural Teaching Approaches Lead
to . . .
John had 12 baseball cards. He gave 1/3 of them
to Jim. How many did John have left?
Many 6th graders answered “11 2/3”.
Not one student thought that 2/3 of a baseball
card was odd.(Marshall, 1995)
30% Correct
Characteristics of Middle School Students (NMSA, 2010)
Curious and willing
Prefer active learning
Enjoy group work
Establish connections (concrete – abstract)
Make decisions that put them intellectually
“at risk” (oppositional behaviors)
Instruction for middle school studentsshould be . . . (NMSA, 2010)
Relevant
Challenging
Integrative
Exploratory
Students who are . . .Students who are . . .
Focusedon Skill
Deficiencies+
WithholdInteresting
Content
National Mathematics Advisory Panel (2008)
STUDENTS SHOULD
learn key math concepts (e.g., represent fractions on a number line, identify equivalent fractions)
AND
build procedural fluency (e.g., add and subtract fractions)
WHILE
learning to formulate and solve problems.
Enhanced Anchored Instruction (EAI)
Uses video-based and applied problems Situates mathematics in authentic-like, engaging contexts
Merges instruction on foundation skills (e.g., perform operations with whole and rational numbers) with problem-solving applications (e.g., design/build hovercrafts)
Taps students’ background knowledge and promotes learning transfer
Brings together unique combinations of teachers (math, technology education, special education) in multiple settings (general education, special education)
Math Concepts in EAI Problems
Fraction of the Cost + Applied Problems (Hovercraft)Interpret three-dimensional drawingsDraw to scaleMeasure lengths and convert units (feet to inches, inches to feet)Compute combinations with whole numbers and fractions
Kim’s Komet + Applied Problems (Car Derby)Compute rate given time and distanceGraph variables and predict values based on line of best fitCompute decimalsConstruct data tables
Skills and Concepts Addressed
Fraction of the Cost + Application (Hovercraft)–Compute whole numbers and fractions–Measure lengths–Interpret and make tables–Interpret 3D drawings–Draw to scale–Convert units (feet to inches, inches to feet)–Estimate and compute combinations–Calculate sales tax
Findings Meaningful – Engagement
- Most students become actively involved- Engagement reduces inappropriate behavior
Explicit – Foundations- It’s not always necessary to wait- Students see benefit of learning basic skills
Informal – Intuitions- Low achievers often have intact problem-solving skills- Teachers can uncover students’ “inert” knowledge
(De) situational – Transfer- Students make use of problem contexts- Learning and context are not separable
EAI in action
Students with MLD HLM 3-Level Model of Treatment Effects on Fractions Computation Test (Bottge et al., in press)
ES
Spre2 Spost
2 2pre, postSpreSpost
ES = 1.14 InformalES = 0.81 Formal over Informal
Findings – Students with MLD (Bottge et al., in press)
ES = 1.16 InformalES = 0.03 Formal over Informal
ES
Spre2 Spost
2 2pre, postSpreSpost
Advancing the Math Skills of Low-Achieving Adolescents in Technology-Rich Learning Environments
U.S. Department of Education, Institute of Education Sciences, Cognition and Student Learning Research Program, Goal 2
2004-2008
Evaluating the Efficacy of Enhanced Anchored Instruction for Middle School Students with Learning Disabilities in Math
U.S. Department of Education, Institute of Education Sciences, Cognition and Student Learning in Special Education Research Program, Goal 3
2009-2013
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