Developing Adolescents’ Procedural Fluency and Strategic Competence in Mathematics

1. Developing Adolescents’ Procedural Fluency and Strategic Competencein Mathematics Brian Bottge University of Kentucky bbott2@uky.edu

2. In 20 minutes or less • The need • The strategy • The results

3. 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

4. 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)

5. 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

6. 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

7. 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)

8. I would think you’d be used to failing by now…

9. 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)

10. 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”

11. 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)

12. A cow is tied to a post in the middle of a flat meadow. If the cow’s rope is several meters long, which of the following figures shows the shape of the region where the cow can graze?

13. 30% Correct

14. 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)

15. Instruction for middle school studentsshould be . . . (NMSA, 2010) • Relevant • Challenging • Integrative • Exploratory

16. Typical Math Instruction for Low-Achieving Students (Hand, 2010; Knapp & Turnbull, 1990) Focused on Skill Deficiencies Withhold Interesting Content + Students who are . . .

17. 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.

18. 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)

19. Math Concepts in EAI Problems • Fraction of the Cost + Applied Problems (Hovercraft) • Interpret three-dimensional drawings • Draw to scale • Measure 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 distance • Graph variables and predict values based on line of best fit • Compute decimals • Construct data tables

20. 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

21. 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

22. EAI in action

23. Students with MLD HLM 3-Level Model of Treatment Effects on Fractions Computation Test (Bottge et al., in press) ES = 1.14 Informal ES = 0.81 Formal over Informal

24. Findings – Students with MLD (Bottge et al., in press) ES = 1.16 Informal ES = 0.03 Formal over Informal

25. Funding Sources: 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 22004-2008 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 32009-2013