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