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Algebra and the Underprepared Learner

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Algebra and the Underprepared Learner
By Timothy Stoelinga and James Lynn
education.uic.edu/ruepi
ABOUT THE AUTHORS
Timothy Stoelinga
isa Senior Program Associate at theLearning SciencesResearch Institute atthe University of Illinois at Chicago. He is also a partof the UIC Office of High SchoolDevelopment.
James Lynn
is the Visiting Director of the Office of HighSchool Developmentat the University of Illinois at Chicago.He is also a part of the Learning Sciences Research Institute at theUniversity of Illinois at Chicago.
policy
BRIEF
UIC Research on Urban Education Policy InitiativeJune 2013 Vol. 2, Book 3
EXECUTIVE SUMMARY
Algebra acts as a gatekeeper forhigh school graduation and post-secondary success. Students whopass Algebra 1 by the end of ninthgrade are more likely to takeadvanced mathematics courses,graduate from high school, andsucceed in college. Yet persistentinequities in access to rigorousalgebra due to issues of placement, preparation, andquality of instruction have keptthe gate closed for a largeproportion of students,particularly minority and low-income students. In response,“Algebra for All” policies have beenimplemented whereby all studentsare required to take Algebra 1 by adesignated grade level—typically eighth or ninth grade. While suchpolicies are on target in theirintention to increase the numberof students who successfully complete Algebra 1 in a timely way, evidence also shows that fortoo many students, these policies
by themselves
have neitherincreased mathematicsachievement nor advanced greateropportunity. Rather, they oftenresult in the watering down of Algebra 1 content and significantly increase the number of students who fail the course. Theseconsequences are concentratedamong underprepared students, whom the policies were designedto serve in the first place. As such,the worthy goals of Algebra for Allmay only be realized when arigorous approach to Algebra ismaintained for
all
students, and when necessary systems are inplace to prepare and support allstudents to be successful. TheCommon Core State Standards forMathematics (CCSS-M) now provides clearer and morerigorous expectations for thealgebra content all studentsshould learn, but the articulationof such standards is only a starting point. Algebra policy, therefore,should include provisions forequitably maintaining this level of rigor for all students, whileproviding a system of supports to:(1) better prepare students tosucceed
before
taking Algebra 1;(2) enhance learning opportunities for underpreparedlearners
during
Algebra 1; and (3)enhance teaching capacity tosupport all learners, particularly those who are underprepared tosucceed in Algebra 1.
2
UIC Research on Urban Education Policy Initiative
INTRODUCTION
For students who are underpreparedto succeed in Algebra 1, passing thecourse can represent anoverwhelming challenge. Yet,success in Algebra 1 is no less crucialfor these underprepared studentsthan it is for students who are betterprepared to succeed. A dilemmaemerges: policies that promotesuccessful completion of Algebra by all students are weighed againsttheir potential to increase failurerates for underprepared students,and the consequences failure canhave on these students’ academictrajectories. In the current context of secondary education and in light of the relationship of Algebra to collegeand career readiness, however, wehold to the position that all students
should
take algebra in a timely manner.
WHY IS SUCCESSFULCOMPLETION OF ALGEBRA 1IMPORTANT FOR ALLSTUDENTS?
Education policy nationwidecontinues on a trend toward raising mathematics requirements for highschool graduation. In 2013, 42 states,including Illinois, requiredsuccessful completion of at leastthree years
1
of mathematics, and 16of these states required completionof four years. These numbers willlikely increase in the next few yearsas pending legislation is enacted inseveral states. By contrast, in 2001only 28 states required three years ormore of mathematics for graduation,and only 4 states required four years.
2
Whether or not Algebra 1
3
isexplicitly stated as a courserequirement (it is in 23 states), thecompletion of the course—and inmany cases the passing of a relatedend-of-course exam—tends to bethe critical step in meeting theseincreasingly rigorous graduationrequirements. In addition, 45 states,also including Illinois, have adoptedthe Common Core State Standardsfor Mathematics (CCSS-M)—writtenspecifications of what studentsshould know and be able to do inmathematics in various grades. Thealgebra standards in CCSS-Mprovide a clear and coherentarticulation of algebra studentsshould learn—in elementary gradesas well as in high-school Algebra 1and Algebra 2 courses.
4
Incomparison to previous statestandards, CCSS-M has increasedthe rigor of Algebra 1 by positioning some content earlier in the overallsequence of Algebra topics.Together, these policy trends raiseboth the stakes and the expectations
policy
BRIEF
The algebrastandards in CCSS-M provide a clear and coherentarticulation ofalgebra studentsshould learn—inelementary gradesas well as in high-school Algebra 1and Algebra 2courses.
1 The term “years” is used for clarity in place of Carnegie Units, in which the data was srcinally presented. One Carnegie Unit generally equates to credit received for successful completion of atwo-semester, credit-bearing course in secondary school.2 National Center for Educational Statistics,
Digest of Education Statistics
, http://nces.ed.gov/programs/digest/d01/dt153.asp; Kyle Zinth and Jennifer Dounay, “Aligned to the Research:Science and Mathematics Graduation Requirements,”
State Notes: Mathematics and Science
, www.ecs.org/clearinghouse/74/52/7452.pdf.3 In some settings, a sequence of Integrated Mathematics replaces the traditional high schoolmathematics sequence. This Integrated sequence is currently being proposed by the IllinoisState Board of Education as one model for implementing CCSS-M in grades 9-12. BecauseIntegrated Mathematics 1 includes a concentrated focus on many of the concepts similar tothose found in Algebra 1, the issues discussed in this brief can be similarly applied to IntegratedMathematics 1.4 National Governors Association Center for Best Practices, Council of Chief State School Officers,
Common Core State Standards—Mathematics
(Washington D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010).
for Algebra 1, with the intention of increasing students’ preparednessto take more advancedmathematics courses and helping students obtain the skills needed tosucceed in college and the workplace.In light of these trends, Algebra 1retains its role as a gatekeeper forhigh school graduation and post-secondary success, and the urgency of passing through continues tointensify.
5
High school algebra is widely considered a key step along the path to college and careerreadiness. Because of increasedgraduation requirements, failing Algebra 1 puts students atsignificant risk of not completing high school. In Chicago PublicSchools (CPS), students who earned5 credits and failed no more thanone course in ninth grade were over3.5 times more likely to graduatefrom high school in four years thanstudents who did not achieve thisbenchmark.
6
Thus, the widespreaddifficulties faced by many studentsin passing Algebra 1 establishes it asa critical link related to success ratesin high school.
THE CHALLENGE OF ALGEBRA
Yet, in CPS and elsewhere, Algebra 1continues to generate the highestfailure rate of any high schoolcourse.
7
The reasons for this arecomplex and difficult to isolate, butseveral themes have emerged fromongoing research in mathematicseducation that can provideguidance in the design of policy and practice. With respect to coursecontent, Algebra 1 has historically represented an importanttransition point in the learning of mathematics, requiring the use of generalized models, mathematicalabstractions, and understandingsof variables and symbols, all of which are particularly challenging for many students.
8
Simply stated,content associated with Algebra 1 isnotoriously difficult compared withthe number and operationsconcepts concentrated in earliergrades. Research has also indicatedthat many eighth and ninth gradestudents who are required to take Algebra 1 are also underpreparedand need more support to succeedbecause of weak foundations inprerequisite concepts.
9
As many of these students enter their firstalgebra course, they experienceearly, reinforcing patterns of failure, which can lead to the belief they will not be able to earn a highschool diploma. Without effectiveforms of intervention and support,these patterns of failure can causestudents to fall further behind andeventually drop out of school. But what kinds of support are neededfor under-prepared students tosucceed in this high-stakes course?How can algebra policy helpprovide these supports as a way tonot only increase graduation rates,but to truly help prepare studentsfor college and post-secondary success?To investigate these questionsamidst these current challenges,this brief examines evidence relatedto algebra policies and their effectson students, particularly those whoare underprepared to succeed.First, the policy landscape of algebra is examined. This sectionfocuses on both the mathematicscontent that students are requiredto learn and research on the effectsof these requirements. Second, thisbrief analyzes additional researchon mathematics education andoffers three principles that beardirectly on improving students’success in algebra: (1) Studentsneed systematic exposure toalgebra beginning early in theireducation and extending throughhigh school; (2) underpreparedstudents need targeted, structuredsupport to succeed in a rigorous Algebra 1 course; and (3) increasing students’ success requiresenhanced teaching capacity thatneeds to be addressed in teacherpreparation programs as well as in
Algebra and the Underprepared Learner
3
policy
BRIEF
5 Clifford Adelman,
The Toolbox Revisited: Paths to Degree Completion from High School through College
(Washington, DC: U.S. Department of Education, 2006).6 Elaine M. Allensworth and John Easton,
What Matters for Staying On-Track and Graduating in Chicago Public High Schools: A Close Look at Course Grades, Failures, and Attendance in the Freshman Year
(Chicago, IL: Consortium of Chicago School Research at the University of Chicago, 2007).7 Chicago Public Schools Department of Evaluation, Research, and Accountability,
Two-Yearm Course Taking Patterns and Pass Rates of CPS High School Students in Math and Science
(Chicago, IL: Chicago Public Schools, 2003).8 E.g., Dietmar Kuchemann, “Children’s Understanding of Numerical Variables,”
Mathematics in School
7, no. 4 (1978): 23-26; Carolyn Kieran,“Concepts Associated with the Equality Symbol,”
Educational Studies in Mathematics
12, no. 3 (1981): 317-26; Sigrid Wagner and Sheila Parker,“Advancing Algebra,” in
Research Ideas for the Classroom, High School Mathematics
, ed. Patricia S. Wilson (New York, NY: Macmillan, 1993): 119-39.9 Elaine Allensworth and Takako Nomi, “College-Preparatory Curriculum for All: The Consequences of Raising Mathematics GraduationRequirements on Students’ Course Taking and Outcomes in Chicago” (Paper presented at the Second Annual Conference of the Society forResearch on Educational Effectiveness, March 2009, Arlington, VA), www.educationaleffective-ness.org/conferences/2009/conference.shtml; TomLoveless,
The Misplaced Math Student: Lost in Eighth-Grade Algebra
(Washington, D.C.: The Brookings Institution, Brown Center on EducationPolicy, September 2008).
in-service professionaldevelopment. Finally, this brief poses a set of recommendations forimproving mathematics educationpolicies and practices in algebra.
THE ALGEBRA POLICY LANDSCAPE
In order to graduate from highschool, Illinois currently requiresthat students must complete threecredits in mathematics, including Algebra 1 and a course ingeometry.
10
This requirement is theresult of legislation enacted in 2005, which increased overall graduationrequirements in core disciplines with the intention of ramping upthe academic preparation of Illinoisgraduates. Algebra 1 is theintroductory course in a typicalsecondary mathematics sequenceof Algebra 1, Geometry, and Algebra2. Most students take Algebra 1 inninth grade, though increasing numbers of students take it ineighth grade, and some as early assixth grade. Guidance from theIllinois State Board of Education(ISBE) on implementing thegraduation requirements citesevidence that high school students who take rigorous courses are moreprepared to graduate, succeed incollege, and participate in the workforce.
11
One response to this requirementhas been to delay underpreparedstudents’ enrollment in Algebra tobeyond ninth grade in order toprovide more coursework in pre-algebra skills. Evidence, however,shows that this approach does not work. Students typically continueto struggle learning the same pre-algebra skills from the middle-grades curriculum, taught using thesame approaches. Consequently,they continue to fall further behindand eventually disengage frommathematics altogether.
12
Anotherapproach has been to slow the paceof algebra for underpreparedstudents by stretching thecurriculum over a two-year spanacross ninth and tenth grades. While this approach does moveunderprepared students forward inthe high school curriculum, it doesso at the cost of setting them back by a full year, rather than allowing them to catch up to their peers.
ALGEBRA FOR ALL:INTENTIONS ANDCHALLENGES
To further ensure access tochallenging mathematics for allstudents, some state and districtpolicies require that Algebra 1 betaken by a specified grade level—typically ninth grade, but in somecases eighth grade—as a measureto ensure students’ preparednessfor more advanced mathematics.
13
Such districts notably include largeurban districts, such as Chicago,Philadelphia, Los Angeles,Baltimore, and Milwaukee, wherelarge inequities exist in the numberof minority and low-incomestudents taking advancedmathematics classes. Research onthe effects of these policies,however, points to both positiveand negative consequences.
14
Onthe positive side, Algebra for All hasallowed more students to enrolland successfully complete Algebra1, which in turn opensopportunities for challenging coursework in mathematics, andincreases the likelihood of graduation, college enrollment, andpostsecondary success.
15
On thenegative side, these policies do notprovide for the supports needed by underprepared students to succeedin Algebra 1. A policy that has been in place inCPS since 1997 mandates that allstudents take Algebra 1 by the endof ninth grade. The policy’s aim is toraise the bar on mathematics for allstudents on the premise thatramping up to a college-preparatory curriculum levels theplaying field and improvesachievement, particularly among minority and low-income students.
10 Illinois State Board of Education,
State Graduation Requirements
(105 ILCS 5/27-22, 27-22.05, 27-22.10), November 2012 Guidance Document(Springfield, IL: November, 2012).11 Illinois State Board of Education,
State Graduation Requirements
.12 Allensworth and Nomi, “College-Preparatory Curriculum for All”; Adam Gamoran, Andrew Porter, John Smithson, and Paula White, “Upgrading High School Mathematics Instruction: Improving Learning Opportunities for Low-Achieving, Low-Income Youth,”
Education Evaluation and Policy Analysis
, 19, no. 4, (1997): 325-338; Jeannie Oakes,
Keeping Track: How Schools Structure Inequality
(2nd ed.) (New Haven, CT: Yale University Press, 2005).13 E.g., Chicago Public Schools Department of Policy and Procedures, “Chicago Public Schools High School Graduation Requirements,” (Chicago, IL:Chicago Public Schools, 2012).14 Allensworth and Nomi, “College-Preparatory Curriculum for All”; Matthew Rosin, Heather Barondess, and Julian Leichty,
Algebra Policy in California: Great Expectations and Serious Challenges
(Mountain View, CA: EdSource, Inc., 2009).15 Adelman,
The Toolbox Revisited
; Allensworth and Easton,
What Matters for Staying On-Track
; Chicago Public Schools Department of Evaluation,Research, and Accountability,
Two-year Course Taking Patterns
.
policy
BRIEF
4
UIC Research on Urban Education Policy Initiative

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