Rigor, Relevance, and Relationships by Design in High School Mathematics

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Rigor, Relevance, and Relationships by Design in
High School Mathematics

• Eric Robinson, Margaret Robinson

NC Raising Achievement and Closing Gaps
Conference March 27, 2007
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Session Purpose
To move forward the North Carolina Raising
Achievement and Closing Gaps Commissions mission
to assist schools and school systems in
identifying and developing programs and
strategies to raise achievement and close gaps.
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Not just about doing things better, but doing
better things!
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Session Overview

• Part I Design components needed in curriculum
  programs to address rigor, relevance, and
  relationships
• Part II Evidence
• Realization of design principles
• Effectiveness

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Part IDesign
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A look at the terms

• Rigor
• Exposing students to challenging class work with
  academic and social support

Relevance Demonstrating how students will use
their learning
Relationships Building caring and supportive
connections with students, parents, and
communities
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RigorExposing students to challenging class work
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RigorExposing students to challenging class work

•  Deep mathematical understanding

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RigorExposing students to challenging class work

•  Deep mathematical understanding that allows
  students to
• 1.) see the connections between bits of
  mathematical knowledge 2.) apply mathematical
  thinking to formulate and execute problem-solving
  strategies 3.) apply mathematics in novel
  situations.

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RigorExposing students to challenging class work

•  Deep mathematical understanding that allows
  students to
• 1.) see the connections between bits of
  mathematical knowledge 2.) apply mathematical
  thinking to formulate and execute problem-solving
  strategies 3.) apply mathematics in novel
  situations .

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Design elements

• Connections
• Mathematical thinking
• Problem-solving
• Flexible and fluent

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RigorExposing students to challenging class work
Relevance Demonstrating how students will use
their learning

•  Deep mathematical understanding that allows
  students to
• 1.) see the connections between bits of
  mathematical knowledge 2.) apply mathematical
  thinking to formulate and execute problem-solving
  strategies 3.) apply mathematics in novel
  situations 4.) see and use mathematics in real
  world situations.

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Rigor, Relevance
Relationships buildingconnectionswith
students

•  Deep mathematical understanding that allows
  students to
• 1.) see the connections between bits of
  mathematical knowledge 2.) apply mathematical
  thinking to formulate and execute problem-solving
  strategies 3.) apply mathematics in novel
  situations 4.) see and use mathematics in real
  world situations 5.) communicate and collaborate
  mathematically.

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Rigor, Relevance
Relationships buildingconnectionswith
students

•  Deep mathematical understanding that allows
  students to
• 1.) see the connections between bits of
  mathematical knowledge 2.) apply mathematical
  thinking to formulate and execute problem-solving
  strategies 3.) apply mathematics in novel
  situations 4.) see and use mathematics in real
  world situations 5.) communicate and collaborate
  mathematically.

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Design elements

• Connections
• Mathematical thinking
• Problem-solving
• Flexible and fluent

• Mathematically Model
• Communicate
• Collaborate

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Relationships
Relevance
Knowledge of Teaching and Learning
Social Need
Mathematics
Rigor
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What does mathematics as a discipline say?

• Mathematics is a way of thinking about,
  understanding, explaining, and expressing
  phenomena.

Mathematics is about inquiry and insight.
Computation is (usually) a means to an end.
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Body of Knowledge
Method of Thinking, Reasoning, and Explaining
Collection of Skills and Procedures
Language
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MATHEMATICAL PROCESSING
LOGICALLY DEDUCE RESULTS/ ALGORITHMS
IMPLEMENT ALGORITHM/ PROCEDURE/ FORMULA
Mathematical Reasoning
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Words such as conjecture, show, explain,
justify, prove, abstract, and generalize are
central components of a rigorous mathematics
program
-that students need to do.
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Relationships
Relevance
Knowledge of Teaching and Learning
Social Need
Mathematics
Rigor
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What does the research on learning suggest?

• We learn new knowledge by attaching it to our
  current knowledge.

We tend to learn by proceeding from the
concrete to the abstract.
There are multiple learning styles.
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Contextualized development of content

• Context An environment in which mathematics
  is developed or mathematical understanding is
  augmented.

A context should be a familiar and engaging
environment for the student.
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From Mathematics Modeling Our World
(COMAP) Unit 1 Course 2 Welcome to Gridville!
This small village has grown in the past year.
The people of Gridville have agreed they now need
to build a fire station. What is the best
location for the fire station?
.
.
.
.
.
.
.
.
.
.
.
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Real World
Mathematical Model
Abstract
Build math model
Clearly identify situation Pose well-formed
question
Mathematically Modeling
Compute Process Deduce
Revise
Mathematical results
Apply
Interpret
Mathematical Conclusions
Real World Conclusions
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Welcome to Lineville!
. .
1.) Where would you build the fire station if
there were only two houses? Explain.
1
5
. . .
2.) Where would you build the fire station if
there were only three houses? Explain.
1
4
5
3.) Where would you build the fire station of
there were 4 houses? 5 houses? Explain. 4.)
Make a conjecture about the location of the fire
station if there were n houses. Can you justify
your conjecture?
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Background includes some linear modeling, some
Euclidean and coordinate geometry, and the mean
of a quantitative data distribution.
The mathematical content for this unit includes
geometry (using a non-Euclidean metric in the
plane), absolute value, functions and algebra
involving the weighted sum of absolute value
functions, piecewise linear functions, and
minimax solutions (choosing the minimum value in
a set of several maximum values). Integrated
topics include algebra, geometry, and
pre-calculus.
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Contextualized development of content

• Context An environment in which mathematics
  is developed or mathematical understanding is
  augmented.

A context should be a familiar and engaging
environment for the student.
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Contextual Development

• Provides cognitive glue for ideas and thought
  processes
• Provides rationale for doing mathematical
  activities, such as finding patterns, making
  conjectures, studying quadratics, etc.
• Allows development from the concrete to the
  abstract or the extension of ideas and
  structure
• Real-world contexts add value to mathematical
  content

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Making Connections integrating mathematical
topics

• Permits synergistic development and multiple ways
  of connecting old and new content
• Provides genuine opportunity to revisit topics in
  more depth
• Addresses various student strengths
• Presents mathematics as a unified discipline
• Provides access to a broader collection of
  problems and solutions

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Relationships
Relevance
Knowledge of Teaching and Learning
Social Need
Mathematics
Rigor
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Relevance and relationships

• What about all students?

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CurricularObjectives

• Create mathematically literate citizens
• Prepare students for the workplace
• Prepare students for further study in disciplines
  that involve mathematics
• Prepare students to be independent learners
• Provide an appreciation of the beauty, power, and
  significance of mathematics in our culture

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Mathematical needs of the workforce beyond
computational skills

• Understand the underlying mathematical features
  of a problem
• Have the ability to see applicability of
  mathematical ideas in common and complex problems
• Be prepared to handle open-ended situations and
  problems that are not well-formulated
• Be able to work with others

- Henry Pollack
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Call for better things. Consider

• Updating, refocusing, and re-sequencing content
  within state guidelines-or change them
• Incorporating concepts and methods from
  statistics, probability, and discrete mathematics

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Closing the Gap Methods of addressing equity in
curriculum

• Students feel at home in the curriculum
• Students see a reason for doing problems
• Students are actively involved in their learning
• Students are respected and feel personally
  validated

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..more on addressing equity

• Problems that allow multiple approaches
• Problems that are open-ended
• Students make (mathematical) choices
• Problems that allow investigation and response at
  multiple levels
• Different gradations of problems
• Verbalization and varied representation
• Reading

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Curriculum designed to raise achievement and
close gaps with rigor relevance and relationships
should include

• Mathematical connections, thinking and reasoning,
  problem-solving, modeling, and communication. It
  needs to address multiple learning styles, issues
  of equity and access, and multiple objectives.

Methods suggested in this session include the
contextual development of concepts integration
of topics, and placing mathematical methods of
thinking and reasoning at the center of the
curriculum.
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Not addressed in depth in this presentation

• Topical content
• But should include data analysis and statistics
• Technology

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Part II

• Evidence

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Secondary Mathematics curriculum programs with
these design elements

• Contemporary Mathematics in Context (Core-Plus
  Mathematics Project CPMP) (Glencoe/McGraw Hill,
  Publisher) 230-400 PM, Cedar B, Billie Bean
• Integrated Mathematics A Modeling Approach Using
  Technology (SIMMS IM) (Kendall Hunt, Publisher)
  230-400 PM, Imperial A, Gary Bauer
• Mathematics Modeling Our World (ARISE) (COMAP,
  Publisher)
• Interactive Mathematics Program (IMP) (Key
  Curriculum Press, Publisher)
• MATH Connections A Secondary Mathematics Core
  Curriculum (MATH Connections) (ITs About Time,
  Publisher)

Links to all at http//www.ithaca.edu/compass
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Does this approach raise Achievement?
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Achievement Goal

• Deep understanding of mathematical concepts and
  processes that includes the ability to use
  mathematics effectively in realistic
  problem-solving situations

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A growing body of evaluation evidence suggests
that it can
Cumulatively, the summary of evidence below
stretches from field test results from the early
1990s to district adoptions in the 2000s. It
cuts across urban, suburban, and rural districts
and ethnically and culturally diverse
populations. Measurement instruments and
research designs vary.
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On Evaluation of Curricular Effectiveness
Judging the Quality of K-12 Mathematics
Evaluations-National Research Council (2004)

• On average, the evaluations in this subgroup had
  reported stronger patterns of outcomes in favor
  of these curricula and their K-8
  counterpartsthan the evaluations of
  commercially-generated curricula.

-this result is not sufficient to establish the
curricular effectiveness of these programs as a
whole with absolute certainty.
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A short list of summary references

• Senk, S. L. and Thompson, D. R. (Eds.)
  Standards-based school mathematics curricula?
  what are they? what do students learn? Lawrence
  Erlbaum Associates (2003)
• Harwell, M.R., Post T.R.,Yukiko M., Davis J.D.,
  Cutler A.L., Anderson E., Kahn J.A.,
  Standards-based mathematics curricula and
  secondary students performance on standardized
  achievement tests, Journal of Research in
  Mathematics Education (January, 2007)
• Schoen, H.L. Hirsch, C.R. Responding to calls
  for change in high school mathematics
  implications for collegiate mathematics
  Mathematical Association of America Monthly, vol.
  110, (February, 2003)

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• On standardized tests that measure quantitative
  thinking, reasoning and realistic problem-solving
  ability, students in all five curricula mentioned
  above most often do significantly better than
  their traditional counterparts.
• Instruments included subtests from NAEP,ITED-Q
• Senk and Thompson Mary Ann Huntley,
  Chris L. Rasmussen, Roberto S. Villarubi,
  Effects of Standards-Based Mathematics
  Education A Study of the Core-Plus Mathematics
  Project Algebra and Functions Strand Journal
  of Research in Mathematics Education, May 2000,
  Vol.31

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• On tests that included measures of updated or
  non-traditional mathematical science content
  (including statistics) students from several of
  these programs who were tested scored above their
  traditional counterparts
• Webb, N. and Maritza D., "Comparison of IMP
  Students with Students Enrolled in Traditional
  Courses on Probability, Statistics, Problem
  Solving, and Reasoning," Wisconsin Center for
  Education Research, University of
  Wisconsin-Madison, April, 1997 Senk and
  Thompson

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• Students from these programs generally received
  cumulative scores as high as and often higher
  than their traditional counterparts on
  traditional items on standardized tests such as
  the PSAT, SAT, ACT, SAT-9
• Merlino, J. Wolf, E. (2001).Assessing the
  Costs/Benefits of an NSF Standards-Based"
  Secondary Mathematics Curriculum on Student
  Achievement. Philadelphia, PA The Greater
  PhiladelphiaSecondary Mathematics Project
  http//www.gphillymath.org/StudentAchievement/Repo
  rts/AssessCostIndex.htm Schoen and Hirsch Senk
  and Thompson

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• Results on achievement with regard to symbol
  manipulation within first editions of these
  programs are mixed.
• Schoen, H.L. Hirsch, C.R. Huntley, et. al.
  ibid

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Do Programs with these design principles close
the achievement gap?

• There is growing evidence that when changing to
  such a program, the lowest achievers will realize
  the largest gains.
• Merlino Wolff Harwell, et. al. Webb, N. L.,
  M. Dowling (1996), Impact of the Interactive
  Mathematics Program on the retention of
  underrepresented students Cross-school analysis
  of transcripts for the class of 1993 for three
  high schools. Project Report 96-2. Madison
  University of WisconsinMadison, Wisconsin Center
  for Education Research (WCER)

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• Data about CPMP, IMP and MMOW suggest that
  students at the high achievement levels are well
  served through programs with these design
  elements
• Abeille and Hurley Final Evaluation Report of
  MMOW curriculum (2001) at http//www.comap.com/hig
  hschool/projects/mmow/FinalReport.pdf Harwell
  et. al., Merlino and Wolff.

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• Students in these programs take more mathematics
  courses (including AP courses).
• Kramer, S. L. (2003). The joint impact of block
  scheduling and a standards-based curriculum on
  high school algebra achievement and mathematics
  and course taking. Ph. D. dissertation,
  University of Maryland Webb and Dowling Harwell
  et. al., Senk and Thompson
• Students in these programs tend to have a better
  attitude toward mathematics
• Clarke, D., et al. (1992). The other consequences
  of a problem-based mathematics curriculum,
  Research Report No. 3. Mathematics Teaching and
  Learning Centre, Australian Catholic University
  Schoen and Prickett (1998) Students perceptions
  and attitudes in a standards-based high school
  mathematics curriculum, paper presented to the
  American Educational Research Association Senk
  and Thompson

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Secondary Mathematics curriculum programs with
these design elements Evaluations

• Contemporary Mathematics in Context (Core-Plus
  Mathematics Project CPMP) (Glenco/McGraw Hill,
  Publisher) http//www.wmich.edu/cpmp/evaluation.ht
  ml 230-400 PM, Cedar B, Billie Bean
• Integrated Mathematics A Modeling Approach Using
  Technology (SIMMS IM) (Kendal Hunt, Publisher)
  http//www.montana.edu/wwwsimms/others.html
  230-400 PM, Imperial A, Gary Bauer
• Interactive Mathematics Program (IMP) (Key
  Curriculum Press, Publisher) http//www.mathimp.or
  g/
• Mathematics Modeling Our World (ARISE) (COMAP,
  Publisher) http//www.comap.com/highschool/project
  s/mmow/introduction.htm
• MATH Connections A Secondary Mathematics Core
  Curriculum (MATH Connections) (ITs About Time,
  Publisher) http//www.its-about-time.com/htmls/mc/
  mccasestudies.html

Links to all at http//www.ithaca.edu/compass
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RelationshipsBuilding caring and supportive
connections with students, parents, and
communities(addressed to administrators,
teachers, staff)

• Success depends on relationships
• The development of a common belief system for
  all constituencies
• Support for and engagement of teachers in a
  strong, ongoing curriculum- centered professional
  development program
• Support for programs from administrators
• Recognition of the needs of administrators and
  parents
• Implementation with fidelity
• And atmosphere of communication and cooperation