RESEARCH REPORT:RESEARCH REPORT:

“PROBLEM SOLVING AS A “PROBLEM SOLVING AS A MEANS TOWARD MATHEMATICS MEANS TOWARD MATHEMATICS FOR ALL: FOR ALL: AN EXPLORATORY LOOK THROUGH A AN EXPLORATORY LOOK THROUGH A CLASS LENS”CLASS LENS”

DEC 7, 2010DEC 7, 2010

Alan Mitchell, Urban Teacher ResidentCUNY-Hunter College, School of Education

MATH 635, Problem Solving - Prof. Sandra Clarkson

AGENDA

Overview Study Background Example Results Summary of Results Epilogue: Researcher’s Caveats

OVERVIEW Dr. Lubienski studied 7th-graders' experiences with a

problem-centered curriculum, analyzing dependency on Socio-Economic Status (SES)

Abstract - Summary of Results“Higher SES students tended to display confidence and solve

problems with an eye toward the intended mathematical ideas”

“Lower SES students preferred more external direction and sometimes approached problems in a way that caused them to miss the intended mathematical points”

Ref. Sarah Theule Lubienski, “Problem Solving as a Means Toward Mathematics for All: An Exploratory Look Through a Class Lens”, National Council of Teachers of Mathematics (NCTM), www.nctm.org, Journal for Research in Mathematics Education (JRME), July 2000, Volume 31, Issue 4, Pages 454 - 482

STUDY’S BACKGROUND:THE CONNECTED MATH PROJECT (CMP)

NCTM argued that problem solving should become the focus of math in school

NSF funded CMP, a middle school curriculum development to create problem-centered materials aligned with NCTM “reform” standards

CMP emphasizes problem solving as a means to learn math content and processes, not as an end in itself

Key:•NCTM – National Council of Teachers of Mathematics•NSF – National Science Foundation

STUDY’S BACKGROUND:THE RESEARCHER

From working class family – believes math can help break lower SES cycle of poverty

Held masters in math and secondary math teaching certificate

Was completing doctoral program, studying the theories underlying the NCTM reform

In third year with CMP, functioning in dual role for a year as CMP pilot teacher and researcher

STUDY’S BACKGROUND:THE SCHOOL AND CLASS

Schoolsocio-economically diverse school in declining

neighborhood of medium sized Midwestern city84% Caucasian, 11% African-American, 3% Hispanic

American, 2% Asian American Class

30 7th-graders, dedicated to mathstudied 18: equally male:female and lower:higher SESmost closely followed a targeted subgroup of eight: a low

and high achieving male and female from each SES level

STUDY’S BACKGROUND:THE CURRICULUM AND RESEARCH

Curriculum eight units, each containing several “Investigations” 1-4 problems per Investigation Investigations followed by homework and assessments “Launch, Explore, Summarize” instructional model

Research studied dependency on SES of students’ experience with two aspects

of problem-centric curriculum: openness (problems having no obvious solution, taking hours to weeks for

groups to solve) context (problems arising from a motivational situation, often based on

the real world) three sets of interviews, various surveys, student work, teaching

journal entries, and daily audio recordings over one school year

EXAMPLE RESULTS: PIZZA PROBLEM

Even high-achieving lower-SES failed to discover or employ unit ratio principles• e.g., “tables offer same opportunity because if pizzas

are cut into fourths then all but four people get seconds at either table”

EXAMPLE RESULT:HOMEWORK-ASSESSMENT CORRELATIONS

• Assessment averages correlate with homework completion, except for lower-SES females

• lower-SES and higher-SES females equally diligent on homework, but assessment results quite different

• lower-SES and higher-SES females more diligent than males on homework

EXAMPLE RESULT:STUDENTS’ ASSESSMENT OF TOP-3

• No lower-SES ranked themselves among top-3, even two females who were ranked by others

• Every higher-SES mentioned by others also ranked themselves• Two higher-SES (one male, one female) were only ranked by themselves

SUMMARY OF RESULTS:OPENNESS

4 consistently preferred CMP, all were higher-SES 6 consistently preferred Traditional, 4 were lower-SES sample of lower-SES experiences

8 of 9 often complained problems were confusing or too hard none said CMP was easier than Traditional difficulty often attributed to vocabulary, sentence structure prefer specific teacher direction (especially girls) - would quit when stuck

sample of higher-SES experiences often offered suggestions (e.g., provide a glossary) several said CMP was easier than Traditional often would think harder, persevere when stuck four who really liked to figure-out problems were higher-SES

SUMMARY OF RESULTS:CONTEXTUALIZATION

lower-SES categorically known to be more contextualized in orientation, so CMP

seemingly would be a match study found them less likely to deduce a general pattern or proof, so

CMP as means to learn math content and processes often failed higher-SES

categorically known to be less contextualized in orientation study found them able and inclined to pull back from context and

analyze intended mathematical ideas

EPILOGUERESEARCHER’S CAVEATS

Data and literature on lower-SES suggest they have the most to gain from problem-solving instruction

Study raises questions about problem-solving as a means for lower-SES to learn other mathematical concepts and skills improved both lower-SES and higher-SES understanding of math, but

increased the gap in performance direct, algorithmic mode of instruction might provide a relatively level

playing field