Promoting Equity through Problem Solving: Results from Two Decades of Mathematics Instructional Reform in the United States Professor Sarah Lubienski

Fulbright Scholar at the Centre for the Advancement of Science Teaching and Learning (CASTeL) Dublin City University On sabbatical leave from the Department of Curriculum and Instruction, University of Illinois

Introduction

My Background – Past work • Studies of math teaching, learning and equity • Large-scale analyses of national data (only basic analyses discussed here)

Introduction

My Background – Past work • Studies of math teaching, learning and equity • Large-scale analyses of national data (only basic analyses discussed here) Current situation 2010 Fulbright DCU Sept-Dec, Current research – U.S. reform led by the National Council of Teachers of Mathematics (NCTM) – Project Maths in Ireland – Equity

Anyon (1980) found social class differences in how U.S. students were being educated: Upper-class schools: • Education for executives • Emphasis on creativity, problem-solving, critical thinking Working-class schools: • Training for obedient workers • Emphasis on basic skills, following instructions, obeying authority

NCTM Reforms

In response to concerns about equity and the changing needs of society, the U.S.

National Council of Teachers of Mathematics

released: 1)

Curriculum and Evaluation Standards

(1989). 2)

Teaching Standards

(1991) 3)

Assessment Standards

(1995) 4)

Principles and Standards for School Mathematics

(2000)

NCTM Reforms

Less

: • Teacher lecture • Memorization of rules with little understanding • Repetitive drill

More

: • Student understanding of concepts • Mathematical reasoning, communication • Learning

through

problem solving • Balance among 5 curricular strands (Geometry, Msmt., Algebra/Functions, Number, Probability/Stats), • “Mathematical power for all”

Project Maths

Emphasis on:

• Student understanding of concepts • Mathematical reasoning, communication • Real world examples & applications • Problem solving skills • Statistics/Probability, Geometry/Trigonometry, Number, Algebra, Functions.

Project Maths versus NCTM Standards

Similar goals Similar rhetoric/rationale--competing internationally, changing workforce needs Some differences: • Secondary school focus of Project Maths • Syllabus “levels” (foundation, ordinary, higher) • Implementation plan • NCTM as teacher organization has no power, no money, no role in the government • Lack of coherence in U.S. reforms – professional development, assessments, curriculum frameworks

NCTM Implementation

Despite NCTM’s lack of authority & resources, the

Standards

have had a remarkably large impact on: • Textbooks – directly and indirectly • State standards • Assessments • Teacher education and professional development Still -- changing teacher practice and consistently detecting effects with students has been difficult. “Surface changes” easier to obtain than substantive change.

What has happened to U.S. student achievement in mathematics since the first NCTM Standards?

Studies of problem-centered textbooks —when used as intended —have been promising: Increase in conceptual understanding Similar levels of procedural fluency Advantages compounded over time (less summer loss) (Riordan & Noyce, 2001; Schoenfeld, 2002; Senk & Thompson, 2003, Tarr, et al, 2008)

What has happened to U.S. student achievement in mathematics since the first NCTM Standards?

Our “National Test”: National Assessment of Educational Progress (NAEP)

Administered sporadically at 4th, 8th, and 12th grades Main NAEP (vs. Long-Term-Trend NAEP)

Main NAEP Results 1990-2009

Achievement significantly increased at 4th & 8th grades. Because of recent framework change, 12th grade trends are less clear.

330 310 290 270 250 230 210 190 170 150 1990 NAEP Mathematics Achievement by Grade, 1990-2009 1992 1996 2000 2003 2005 2007 12th 8th 4th 2009

s.d.≈ 33 pts

Backlash

Despite Successes

• Critics rightly charge that most research has been conducted by pro-reformers

Backlash

Despite Successes

• Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem

NCTM’s Increased & Decreased Emphases

(A few examples from list for Grades 9-12)

INCREASED EMPHASIS DECREASED EMPHASIS

• Functions that are constructed as models of real-world problems • Formulas given as models of real world problems • The connections among a problem situation, its model as a function in symbolic form, and the graph of that function • The use of factoring to solve equations and to simplify rational expressions • The use of calculators and computers as tools for learning and doing mathematics • Paper-and-pencil graphing of equations by point plotting

Backlash

Despite Successes

• Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem • Initial drafts of some NCTM-aligned curricula may not have emphasized symbolic procedures enough (revisions were made – e.g.,

Core Plus

)

Backlash

Despite Successes

• Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem • Initial drafts of some NCTM-aligned curricula may not have emphasized symbolic procedures enough (revisions were made – e.g.,

Core Plus

) • “Mathematically Correct” organization fanning backlash flame with mathematicians and parents

Backlash

Despite Successes

• Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem • Initial drafts of some NCTM-aligned curricula may not have emphasized symbolic procedures enough (revisions were made – e.g.,

Core Plus

) • “Mathematically Correct” organization fanning backlash flame with mathematicians and parents • Long-Term Trend NAEP did not show same increases as Main NAEP

Long Term Trend NAEP Results Mathematics – Age 9, 13 & 17

From NAEP 2008 Long Term Trend report, p.29 s.d.≈ 33 pts

The Promise of the NCTM Reforms

We are convinced that if students are exposed to the kinds of experiences outlined in the Standards, they will gain mathematical power. This term denotes an individual's abilities to explore, conjecture, and reason logically…and the development of personal self-confidence.

National Council of Teachers of Mathematics Curriculum and Evaluation Standards, 1989, p. 5

Sounds good… Do all benefit as intended?

NAEP Mathematics Achievement by Race/Ethnicity, 1990-2009, Grade 4 330 310 290 270 250 230 210 190 170 150 1990 1992 1996 2000 2003 2005 2007 2009 4th Grade White 4th Grade Hispanic 4th Grade Black

NAEP Mathematics Achievement by Race/Ethnicity, 1990-2009, Grade 4 & 8 330 310 290 270 250 230 210 190 170 150 1990 1992 1996 2000 2003 2005 2007 2009 4th Grade White 4th Grade Hispanic 4th Grade Black 8th Grade White 8th Grade Hispanic 8th Grade Black

Percentage of students agreeing "Learning math is mostly memorizing facts” (2000)

Take a guess…

Percentage of students agreeing "Learning math is mostly memorizing facts” (2000)

Percentage of students agreeing "Learning math is mostly memorizing facts” (2000)

Pat has 3 fish bowls. There are 4 plants and 5 fish in each bowl. Which gives the total number of fish?

A)3 + 5 B)3 X 4 C)3 X 5 D)3 + 4 + 5

Pat has 3 fish bowls. There are 4 plants and 5 fish in each bowl. Which gives the total number of fish?

A)3 + 5 B)3 X 4

C)3 X 5

67% White & Asian Correct 40% Other Black, Hispanic, Am. Ind.

D)3 + 4 + 5

22% White students 43% Black students For more item-level analyses, see: Lubienski, S. T., & Crockett, M. (2007). NAEP mathematics achievement and race/ethnicity. In P. Kloosterman and F. Lester (Eds.) Results from the Ninth

Mathematics Assessment of NAEP.

Girls’ and Boys’ Math Achievement

• K-12 math achievement differences are small – usually about .1 s.d. • Differences in confidence and attitudes toward math tend to be larger than achievement disparities.

• 56% of 4 th grade boys vs. 43% of girls agree “

I am good at math.

”

1990 –2009 NAEP Scores by Gender, Grade 4

If reform-oriented instruction is implemented across all schools, will SES- and race-related achievement gaps narrow?

One Study that Points to Challenges:

Do lower- and higher-SES students react to a "reformed" curriculum and pedagogy differently?

What Led Me To This?

•

Writing and piloting new middle-school mathematics curriculum

•

My own working-class background

Instruction in this classroom

• Connected Mathematics Project Trial Materials • Launch-Explore-Summarize model of teaching through problem solving

Two Themes

1) Authority/direction 2) Contextualization/abstraction

The first theme arose in relation to students’ experiences with the open nature of the problems and whole-class discussions.

The second theme arose specifically in relation to the contextualized nature of the “real world ” problems.

Comparing Traditional Math with Problem-Centered Math

Lower-SES Students

Sue : I like the other way better…Some of the questions in the books I don’t understand at all, they are really confusing… Lynn : I think I’m better with number problems than with problem solving like we're doing.

Nick: In 5th grade when we had those math books I did better than I am now…[The old books gave] specific directions… Dawn : I don’t like this math book because it doesn’t explain EXACTLY!

Higher-SES students

Rebecca: This year were doing stuff that I like...Before, we just sat there with 100’s of problems on a page... Guinevere: It's a lot easier [with the new curriculum]... I guess our family's, just, we are word problem kind of people.

Christopher: Before we just learned about how to do problems and not how it has to do with real life. . . Sometimes the directions are unclear, but I just go and do what I think it's trying to say... I just try my best to figure out the directions, so If I get it wrong it's just because of directions and not because I did the problem wrong.

"Do you participate much in class discussions? Why or why not?" (From final survey)

Higher-SES Guinevere: Yes, because I need to get my point across.

Samantha: Yes, because I want other people to understand my ideas. I like arguing.

Benjamin: Yes, because I like to let people know what I'm thinking.

"Do you participate much in class discussions? Why or why not?" (From final survey)

Higher-SES Guinevere: Yes, because I need to get my point across.

Samantha: Yes, because I want other people to understand my ideas. I like arguing.

Benjamin: Yes, because I like to let people know what I'm thinking.

Lower-SES Rose: Yes. If I know what I'm talking about. But if I'm confused I just listen.

Sue: Sometimes, only if I know I've got the right answer.

Dawn: No because I don't like to be wrong in front of a whole group.

Carl: No, because I always feel awkward.

Do you learn from whole-class discussions?

Lower-SES Students

Sue: I learn better from just like the teacher instead of the whole group…When everyone is there they give their opinions and stuff it may not be right, and I mix those two up, and it just confuses me.

Higher-SES Students

Rebecca: Yeah, I think it helps me learn more things instead of just like doing it on your own, I can know everybody’s opinions and take it into consideration.

Do you find it confusing when you have all those different opinions out?

Not really, some of em aren’t true, and some of em are, and I can figure out which ones are true and which ones aren’t and stuff.

Abstraction from Contextualized Problems

Two brief examples  The Pizza Sharing Problem -- Intended to teach division interpretation of fractions  Popcorn Box Problem -- Intended to teach unit pricing ideas

Pizza Problem

Imagine that you entered a pizza parlor and saw pizzas being served at two tables of your friends, one table with 10 people and the other with 8 people. Problem: If you like the two groups of friends equally well, which table would you join and why?

Table 1 Table 2

Approaches to the Pizza Problem

Sue

You don't know how much the people at the table already ate, and — like say you came five minutes after they served everyone, and say like two pizzas were gone at both tables, and then there's only two on both of them.

Rose

If you divided them into 4 pieces each, then… there would be 4 people without seconds at table 1 and 4 people without seconds at table 2.

Samantha

I took 4 divided by 11 because you have to add more people if you’re going to join the table, and you get 36% of the pizza.

Popcorn Box Problem

Which is the better buy?

More higher-SES st udent s appro ached t he problems and discussions wit h an eye t oward t he in t ended, over-arching, math emati cal ideas.

They seemed mor e fam iliar with what Cooper and Dunne (2 0 00 ) describe as “ th e peculiar ways boundaries are drawn betw een sc hool math emat ics and everyday knowledge .”

Class Cultural Differences (An Oversimplified, Dangerous Chart)

Working Class Jobs traditionally involve: Obedience, conformity.

In child-rearing, parents tend to: Emphasize obedience to authority, rules.

Show/tell how to solve problems. Encourage communicating and reasoning in a more contextualized manner.

Middle Class Jobs traditionally involve: Creativity, autonomy.

In child-rearing, parents tend to: Emphasize reasoning, intellectual curiosity.

Guide problem-solving with questions. Encourage decontextualized uses of language and reasoning.

So is rote learning better for low-SES students?

No. This would be a repeat of past problems.

Other Research on Reform-Oriented Instruction

• Again, most evidence suggests that reform-oriented instruction helps U.S. students’ problem solving skills & conceptual understanding (e.g., Senk & Thompson, 2003; Riordan & Noyce, 2001). • A study in Brazil found that reform-oriented math instruction increased scores for both rich and poor, but widened the gap between them (Franco, Sztajn & Ortigao, 2007).

• Research in England suggests that working-class students can benefit from reform-oriented math instruction (Boaler, 2002), but contextualized problems can pose unexpected difficulties for working class students (Cooper & Dunne, 2000).

Will similar patterns occur in Ireland?

I don’t know. The point is that schools must pay attention to equity when implementing instructional reforms.

What Should Teachers & Schools Do? 1) More carefully monitor whether all students “abstract” intended mathematics from problem explorations 2) Teachers may need to “wean” students away from teacher direction/validation.

3) Consistently push mathematical meaning (What does 30 ÷ 5

mean

?).

4) Strive for equitable outcomes -- which might require “unequal treatment” ( Protect low-SES students’ interests).

Some Initial Hypothesis

(correct me if I’m wrong)

Some Initial Hypothesis

(correct me if I’m wrong)

Ireland as a relatively small country has – • strength -- geographical proximity, centralized decision making • challenge -- garnering necessary people/resources for massive curriculum, instruction, and assessment reform efforts

Some Initial Hypothesis

(correct me if I’m wrong)

Ireland as a relatively small country has – • strength -- geographical proximity, centralized decision making • challenge -- garnering necessary people/resources for massive curriculum, instruction, and assessment reform efforts Textbooks are a potential pitfall in the Project Maths implementation strategy.

Some Initial Hypothesis

(correct me if I’m wrong)

Ireland as a relatively small country has – • strength -- geographical proximity, centralized decision making • challenge -- garnering necessary people/resources for massive curriculum, instruction, and assessment reform efforts Textbooks are a potential pitfall in the Project Maths implementation strategy.

High-stakes exams pose both a challenge for change and leverage point.

Some Initial Hypothesis

(correct me if I’m wrong)

Ireland as a relatively small country has – • strength -- geographical proximity, centralized decision making • challenge -- garnering necessary people/resources for massive curriculum, instruction, and assessment reform efforts Textbooks are a potential pitfall in the Project Maths implementation strategy.

High-stakes exams pose both a challenge for change and leverage point.

There are endemic challenges to teacher change, regardless of the system of reform: teacher knowledge, beliefs, need to control classroom...

For more information:

Professor Sarah Lubienski, [email protected]

Fulbright Scholar at the Centre for the Advancement of Science Teaching and Learning (CASTeL) Dublin City University On sabbatical leave from the Department of Curriculum and Instruction, University of Illinois

Which of the following is usually measured in feet?

A) The thickness of a coin B) The length of a paper clip

C) The length of a car 75% White/Asian, 55% others

D) The distance between New York City and Chicago

Working with Scales

On the road shown above, the distance from Bay City to Exton is 60 miles. What is the distance from Bay City to Yardville?

A)45 miles B)75 miles C)90 miles

D)105 miles

Grade 4: 27% Boys, 20% Girls Grade 8: 44% Boys, 33% Girls For more item-level analyses by gender: McGraw, R., & Lubienski, S. T. (2007). 2003 NAEP mathematics findings regarding gender. In P. Kloosterman and F. Lester (Eds.)

Results from the Ninth Mathematics Assessment of NAEP

(pp. 261-287). Reston: NCTM.

Long Term Trend NAEP Results White-Hispanic NAEP mathematics average scores and score gaps – Age 9

Long Term Trend NAEP Results White-Hispanic NAEP mathematics average scores and score gaps – Age 13

Long Term Trend NAEP Results White-Hispanic NAEP mathematics average scores and score gaps – Age 17

Carl has 3 empty egg cartons and 34 eggs. If each carton holds 12 eggs, how many more eggs are needed to fill all 3 cartons?

A)2

B) 3 C)4 D)6

54% White, 53% Asian Correct 30-35% Black, Hispanic, Am. Indian Correct For more item-level analyses, see: Lubienski, S. T., & Crockett, M. (2007). NAEP mathematics achievement and race/ethnicity. In P. Kloosterman and F. Lester (Eds.) Results from the

Ninth Mathematics Assessment of NAEP.

1990 –2009 NAEP Scale Scores by Gender, Grade 4

1990 –2009 NAEP Scores by Gender, Grade 8

2005 NAEP Gender Gaps by Content Strand, Grade 12

5 4 3 2 1 0 -1 -2 -3 -4 -5

12th Grade Data Algebra Geometry Number

After High School…

• African American & Latino students are less likely than white students to attend college but roughly as likely to major in math-related fields once there. (NCES, 2006)

In contrast —gender patterns:

• More women than men attend college, but women still less likely to pursue math-related fields —e.g., women earn only 12% of Engineering PhDs. (NSF, 2000) • 10 Years out of college, women earn 69% of what men make--partially due to fewer women in math-related fields. (Dey & Hill, 2007).

ECLS-K Analysis Lubienski, S. T. & Robinson, J. P. (2009). Who is shortchanged in elementary school? A longitudinal study of math and reading gender gaps in ECLS-K. Paper presented at AERA, San Diego.

Gender math achievement gaps, by wave Fall K Boys Girls Spring K Boys Girls Spring 1st Boys Girls 0 20 40 60 80 Percentile, by gender 100 Spring 3rd Boys Girls 0 20 40 60 80 Percentile, by gender 100 Spring 5th Boys Girls 0 20 40 60 80 Percentile, by gender 100 Spring 8th Boys Girls 0 20 40 60 80 Percentile, by gender 100 0 20 40 60 80 Percentile, by gender 100 0 20 40 60 80 Percentile, by gender 100

A Different Look at Trends:

Early Childhood Longitudinal Study Kindergarten Class of 1998-99 (ECLS-K)

• Nationally representative sample of 22,000 students.

• Same students followed from K-8 th grade. • Extensive parent surveys, some teacher/school data.

• Students assessed in math in K (fall & spring), 1 (fall & spring), 3, 5, and 8.

Some ECLS-K Findings

• Free Lunch is a weak SES variable. Especially important to use additional SES measures if not controlling for prior achievement.

• Parent education level, occupation, income, number of books in the home, number of children at home, degree expected of child, music lessons, and mother’s age at first birth are some SES-related variables that go beyond lunch eligibility in predicting achievement.

• SES-related gaps widen K-5 (including during summers).

• Gaps between African American and white children widen K-5, but gaps between Latina/o and white children do not (after adjusting for SES). (Fryer & Levitt, 2004; Reardon & Robinson, 2007; Lubienski & Crane, in press)

So Why do Gaps in Students’ Achievement Gains during School?

• What can large-scale data tell us about similarities and differences in teachers, teaching, and students’ attitudes/beliefs about mathematics?

Similarities in Instruction (NAEP)

Similarities in Students’ Beliefs & Attitudes (NAEP)

1990 –2007 NAEP Scale Scores by Gender, Grade 4

1990 –2007 NAEP Scale Scores by Gender, Grade 8

Lubienski & Robinson, 2009

Math Scores by Gender & Race/Ethnicity, 2007

2007 NAEP Gender Gaps by Content Strand, Grades 4, 8

10

Number Data

0

Male/Fem ale 4th Grade Male/Fem ale 8th Grade Algebra

-10

Geometry Measurement

Long Term Trend NAEP Results White-Black NAEP mathematics average scores and score gaps – Age 9

Long Term Trend NAEP Results White-Black NAEP mathematics average scores and score gaps – Age 13

Long Term Trend NAEP Results White-Black NAEP mathematics average scores and score gaps – Age 17

Other Contributors to Achievement Disparities: Teacher Qualifications/Experience (2007 NAEP) • White 8 th graders slightly more likely than Black or Hispanic 8 th graders to have a math teacher who is fully certified, and who majored in math education. • Black and Hispanic 8 more likely to have a new teacher (0-4 years experience). th graders almost 50%

Some Concluding Thoughts

• Remarkable NAEP gains since 1990. But seems easier to raise all students up 25 points than to close a 25-point gap. • Fact that gaps between African American and White students grow while in school highlight potential importance of school related disparities (e.g., new teachers).

• Teachers: Girls’ compliance ≠ understanding. • We need to make sure that measurement & complex problem solving are taught well with

all

students.

Math Scores by Race/Ethnicity & Free/Reduced Lunch Eligibility, 2009 2009 NAEP Scores White 4 th 8 th Lunch 236 24% No Lunch 253 68% Lunch No Lunch 276 21% 298 71% 12 th (2005 data) Lunch No Lunch 142 12% 159 77% Hispanic 225 72% 237 23% 263 67% 275 28% 129 54% 136 39% Black Asian/PI American Indian 219 71% 242 31% 220 66% 232 26% 262 60% 236 32% 256 65% 285 33% 259 61% 271 31% 309 57% 278 35% 121 50% 148 31% 132 44% 168 57%

NAEP Mathematics Achievement by Race/Ethnicity, 1990-2009, Grade 4, 8 & 12 330 310 290 270 250 230 210 190 170 150 1990 1992 1996 2000 2003 2005 2007 2009 4th Grade White 4th Grade Hispanic 4th Grade Black 8th Grade White 8th Grade Hispanic 8th Grade Black 12th Grade White 12th Grade Hispanic 12th Grade Black