Unformatted text preview: Rigor, Relevance, and Relationships
by Design in High School Mathematics
by Eric Robinson, Margaret Robinson
NC Raising Achievement and Closing Gaps Conference
March 27, 2007 Session Purpose:
Session
To move forward the North Carolina Raising
To
Achievement and Closing Gaps Commission’s
mission to assist “schools and school systems
mission
in identifying and developing programs and
strategies to raise achievement and close
gaps.”
gaps.” Not just about doing things
better, but doing better things!
better, Session Overview:
Session
• Part I: Design components needed in
Design
curriculum programs to address rigor,
rigor
relevance, and relationships
relevance and relationships
• Part II: Evidence
– Realization of design principles
– Effectiveness Part I
Part
Design A look at the terms:
look
Rigor
Exposing students to challenging class
work with academic and social support
Relevance
Relevance
Demonstrating how students will use
their learning
Relationships
Relationships
Building caring and supportive
connections with students, parents,
and communities Rigor
Rigor
Exposing students to challenging class work Rigor
Rigor
Exposing students to challenging class work Deep mathematical understanding
Deep Rigor
Rigor
Exposing students to challenging class work Deep mathematical understanding
Deep
that allows students to:
1.) see the connections between bits
of mathematical knowledge; 2.) apply
mathematical thinking to formulate
and execute problemsolving
strategies; 3.) apply mathematics in
novel situations. Rigor
Rigor
Exposing students to challenging class work Deep mathematical understanding
Deep
that allows students to:
1.) see the connections between bits
of mathematical knowledge; 2.) apply
mathematical thinking to formulate
and execute problemsolving
strategies; 3.) apply mathematics in
novel situations . Design elements
Design
•
•
•
• Connections
Mathematical thinking
Problemsolving
Flexible and fluent Rigor
Rigor
Exposing students to challenging class work
Relevance
Demonstrating how students will use their learning Deep mathematical understanding
Deep
that allows students to:
1.) see the connections between bits
of mathematical knowledge; 2.) apply
mathematical thinking to formulate
and execute problemsolving
strategies; 3.) apply mathematics in
novel situations; 4.) see and use
mathematics in real world situations. Rigor, Relevance
Rigor,
Relationships
Relationships
… building…connections…with students… Deep mathematical understanding
Deep
that allows students to:
1.) see the connections between bits
of mathematical knowledge; 2.) apply
mathematical thinking to formulate
and execute problemsolving
strategies; 3.) apply mathematics in
novel situations; 4.) see and use
mathematics in real world situations;
5.) communicate and collaborate Rigor, Relevance
Rigor,
Relationships
Relationships
… building…connections…with students… Deep mathematical understanding
Deep
that allows students to:
1.) see the connections between bits
of mathematical knowledge; 2.) apply
mathematical thinking to formulate
and execute problemsolving
strategies; 3.) apply mathematics in
novel situations; 4.) see and use
mathematics in real world situations;
5.) communicate and collaborate Design elements
Design
•
•
•
• Connections
Mathematical thinking
Problemsolving
Flexible and fluent
• Mathematically Model
Mathematically
• Communicate
Communicate
• Collaborate
Collaborate Relationships Relevance Social Need Knowledge
Knowledge
of Teaching
and Learning
and Mathematics
Mathematics
Rigor What does mathematics as a
discipline say?
discipline
Mathematics is a way of
Mathematics
thinking about, understanding,
explaining, and expressing
phenomena.
phenomena.
Mathematics is about inquiry and
Mathematics
insight. Computation is (usually) a
means to an end.
means Body of Knowledge Method of
Thinking,
Reasoning, and
Explaining
Explaining Collection of Skills
Collection
and Procedures
and Language MATHEMATICAL PROCESSING
MAKE & TEST
CONJECTURES
WONDER SEARCH FOR USE INDUCTIVE
REASONING PATTERNS
DRAW
ANALOGIES
FORMULATE
ALGORITHMS GENERALIZE
RESULTS/
ALGORITHMS CREATIVITY CHOOSE A
STRATEGY/METHOD IMAGINATION CHOOSE
REPRESENTATIONS MATHEMATICALLY
MODEL
DETERMINE WHAT’S
KNOWN & WHAT’S
LOGICALLY
NEEDED
DEDUCE RESULTS/
ALGORITHMS
JUSTIFY SPECIAL
CASES
EXPLORE
EXAMPLES ABSTRACT
PROPERTIES
POSE
PROBLEMS Mathematical IMPLEMENT
ALGORITHM/
PROCEDURE/
FORMULA Reasoning INTUITION Words such as conjecture, show,
explain, justify, prove, abstract, and
generalize are central components
of a rigorous mathematics program
of
─that students need to do. Relationships Relevance Social Need Knowledge
Knowledge
of Teaching
and Learning
and Mathematics
Mathematics
Rigor What does the research on learning
suggest?
suggest?
We learn new knowledge by attaching it
We
to our current knowledge.
We tend to learn by proceeding from the
“concrete” to the “abstract.”
“concrete”
There are multiple learning styles. Contextualized development of
content
content
Context: An environment in which
Context:
mathematics is developed or
mathematical understanding is
augmented.
A context should be a familiar and
engaging environment for the
student. 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? ...
. ..
..
.
.
. Real World Mathematical Model Clearly identify situation Abstract
Build math model Apply Real World
Conclusions Mathematically
Modeling Compute
Process
Deduce Revise Pose wellformed question Mathematical results Interpret
Mathematical
Conclusions Welcome to Lineville! 1.) Where would you build the fire
1.)
station if there were only two
houses? Explain.
houses? 2.) Where would you build the fire
2.)
station if there were only three
houses? Explain.
houses? . . . .. 1 1 5 4 5 3.) Where would you build the fire station of there were 4 houses? 5
3.)
houses? Explain.
houses?
4.) Make a conjecture about the location of the fire station if there were n
4.)
houses. Can you justify your conjecture?
houses. Background includes some linear modeling,
Background
some Euclidean and coordinate geometry, and
the mean of a quantitative data distribution.
the
The mathematical content for this unit includes
geometry (using a nonEuclidean 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).
in
Integrated topics include algebra, geometry, and
precalculus.
precalculus. Contextualized development of
content
content
Context: An environment in which
Context:
mathematics is developed or
mathematical understanding is
augmented.
A context should be a familiar and
engaging environment for the
student. Contextual Development
Contextual
• 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
• Realworld contexts add value to
mathematical content Making Connections: integrating
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 Relationships Relevance Social Need Knowledge
Knowledge
of Teaching
and Learning
and Mathematics
Mathematics
Rigor Relevance and relationships:
Relevance What about all students?
What
all Curricular
Curricular
Objectives
• Create mathematically literate citizens
• Prepare students for the workplace
• Prepare students for further study in
Prepare
disciplines that involve mathematics
disciplines
• Prepare students to be independent learners
• Provide an appreciation of the beauty, power,
Provide
and significance of mathematics in our
culture
culture 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 openended situations
and problems that are not wellformulated
• Be able to work with others
─ Henry Pollack Call for better things. Consider:
Call
• Updating, refocusing, and resequencing content within state
sequencing
guidelinesor change them
guidelinesor
• Incorporating concepts and methods
Incorporating
from statistics, probability, and
discrete mathematics
discrete Closing the Gap: Methods of
Closing
addressing equity in curriculum:
addressing
• Students feel at home in the
Students
curriculum
curriculum
• Students see a reason for doing
Students
problems
problems
• Students are actively involved in their
Students
learning
learning
• Students are respected and feel
Students
personally validated
personally ..more on addressing equity:
• Problems that allow multiple
Problems
approaches
approaches
• Problems that are openended
– Students make (mathematical) choices
Students
choices • Problems that allow investigation and
Problems
response at multiple levels
response
• Different gradations of problems
• Verbalization and varied representation
• Reading Curriculum designed to raise achievement
and close gaps with rigor relevance and
relationships should include:
relationships
Mathematical connections, thinking and reasoning,
problemsolving, 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. Not addressed in depth in this
presentation:
presentation:
• Topical content
– But should include data analysis and
But
statistics
statistics • Technology Part II
Evidence Secondary Mathematics curriculum programs with
these design elements :
• Contemporary Mathematics in Context (CorePlus
Mathematics Project; CPMP) (Glencoe/McGraw
Hill, Publisher) [2:304:00 PM, Cedar B, Billie Bean]
• Integrated Mathematics: A Modeling Approach
Using Technology (SIMMS IM) (Kendall Hunt,
Publisher) [2:304:00 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) (IT’s About
Time, Publisher)
Links to all at http://www.ithaca.edu/compass Does this approach raise
Achievement?
Achievement? Achievement Goal:
Achievement
Deep understanding of mathematical
Deep
concepts and processes that
includes the ability to use
mathematics effectively in realistic
problemsolving situations
problemsolving A growing body of evaluation evidence
suggests that it can*
suggests Cumulatively, the summary of evidence below
Cumulatively,
stretches from field test results from the early
1990’s to district adoptions in the 2000’s. It cuts
across urban, suburban, and rural districts and
ethnically and culturally diverse populations.
Measurement instruments and research designs
vary.
vary. On Evaluation of Curricular Effectiveness:
Judging the Quality of K12 Mathematics
Evaluations
Evaluations
─National Research Council (2004)
On average, the evaluations in this
On
subgroup had reported stronger patterns
of outcomes in favor of [these curricula
and their K8 counterparts]…than the
evaluations of commerciallygenerated
curricula.
curricula.
─this result is not sufficient to establish the
this
curricular effectiveness of these programs as a
whole with absolute certainty.
whole A short list of summary references:
short
• Senk, S. L. and Thompson, D. R. (Eds.) StandardsSenk,
Standardsbased school mathematics curricula? what are they?
based
what do students learn?; Lawrence Erlbaum
Associates (2003)
Associates
• Harwell, M.R., Post T.R.,Yukiko M., Davis J.D., Cutler
Harwell,
A.L., Anderson E., Kahn J.A., “Standardsbased
A.L.,
mathematics curricula and secondary students’
performance on standardized achievement tests,
Journal of Research in Mathematics Education
(January, 2007)
(January,
• 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)
America • On standardized tests that measure
On
quantitative thinking, reasoning and
realistic problemsolving ability,
students in all five curricula mentioned
above most often do significantly
better than their traditional
counterparts.
counterparts.
– Instruments included subtests from NAEP,ITEDQ
– Senk and Thompson; Mary Ann Huntley, Chris L. Rasmussen,
Roberto S. Villarubi, “Effects of StandardsBased Mathematics
Education: A Study of the CorePlus Mathematics Project Algebra
and Functions Strand;” Journal of Research in Mathematics
Education, May 2000, Vol.31 • On tests that included measures of updated or
On
nontraditional mathematical science content
(including statistics) students from several of
these programs who were tested scored above
their traditional counterparts
their
– 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 WisconsinMadison, April, 1997 ; Senk and Thompson • Students from these programs generally
Students
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, SAT9
ACT, – Merlino, J. & Wolf, E. (2001).Assessing the Costs/Benefits of an
NSF“ StandardsBased" Secondary Mathematics Curriculum on
Student Achievement. Philadelphia, PA: The Greater
PhiladelphiaSecondary Mathematics Project :
http://www.gphillymath.org/StudentAchievement/Reports/Assess
CostIndex.htm ; Schoen and Hirsch; Senk and Thompson • Results on achievement with regard to
Results
symbol manipulation within first editions of
these programs are mixed.
these
– Schoen, H.L. Hirsch, C.R. ; Huntley, et. al. ibid Do Programs with these design
principles close the achievement gap?
• There is growing evidence that when
There
changing to such a program, the lowest
achievers will realize the largest gains.
achievers
– Merlino & Wolff; Harwell, et. al. ; Webb, N. L., & M. Dowling (1996),
Impact of the Interactive Mathematics Program on the retention of
underrepresented students: Crossschool analysis of transcripts
for the class of 1993 for three high schools. Project Report 962.
Madison: University of Wisconsin–Madison, Wisconsin Center for
Education Research (WCER) • Data about CPMP, IMP and MMOW suggest that
Data
students at the high achievement levels are well
served through programs with these design
elements
elements
– Abeille and Hurley Final Evaluation Report of MMOW curriculum (2001)
at http://www.comap.com/highschool/projects/mmow/FinalReport.pdf ;
Harwell et. al., Merlino and Wolff. • Students in these programs take more
Students
mathematics courses (including AP courses).
mathematics
– Kramer, S. L. (2003). The joint impact of block scheduling and a
standardsbased 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
Students
better attitude toward mathematics
better
– Clarke, D., et al. (1992). The other consequences of a problembased
mathematics curriculum, Research Report No. 3. Mathematics
Teaching and Learning Centre, Australian Catholic University;
Schoen and Prickett (1998) Students perceptions and attitudes in a
standardsbased high school mathematics curriculum, paper
presented to the American Educational Research Association; Senk
and Thompson Secondary Mathematics curriculum programs with
these design elements : Evaluations:
• Contemporary Mathematics in Context (CorePlus Mathematics Project;
CPMP) (Glenco/McGraw Hill, Publisher) http://www.wmich.edu/cpmp/evaluation.html [2:304:00 PM, Cedar B,
Billie Bean]
• Integrated Mathematics: A Modeling Approach Using Technology (SIMMS
IM) (Kendal Hunt, Publisher)
http://www.montana.edu/~wwwsimms/others.html [2:304:00 PM, Imperial A,
Gary Bauer] • Interactive Mathematics Program (IMP) (Key Curriculum Press, Publisher) • Mathematics: Modeling Our World (ARISE) (COMAP, Publisher)
http://www.comap.com/highschool/projects/mmow/introduction.htm • MATH Connections: A Secondary Mathematics Core Curriculum (MATH
Connections) (IT’s About Time, Publisher)
http://www.itsabouttime.com/htmls/mc/mccasestudies.html http://www.mathimp.org/ Links to all at http://www.ithaca.edu/compass Relationships:
Relationships:
Building caring and supportive connections with
Building
students, parents, and communities
students,
(addressed to administrators, teachers, staff)
*Success depends on relationships:
• The development of a common belief system for
all constituencies
all
•Support for and engagement of teachers in a
Support
strong, ongoing curriculum centered
professional development program
professional
• Support for programs from administrators
Support
• Recognition of the needs of administrators and
parents
parents
•Implementation with fidelity
•And atmosphere of communication and
And
cooperation ...
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 Math, Mathematics education, high school mathematics, Deep mathematical understanding

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