Transcript Powerpoint - College of Education

Mathematical Modeling: ESSENTIAL
for College AND Career Readiness
2016 SCCTM Conference
Greenville SC
Ed Dickey, University of South Carolina
Mathematical Modeling
• What is it?
• How does it fit the South Carolina
Portrait of the College and Career
Ready student?
• How do we as teachers address this
essential standard?
References and links provided on final slides and PPT can be freely download.
Henry O. Pollak
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Bell Labs engineer, MAA President
NCTM Lifetime Achievement Award
Brought modeling to attention in 1968
Part of “real-world applications” in NCTM
1989 Standards
• Representation Standard in NCTM 2000
Standards
Portrait of College and Career Ready
p. 9 of http://ed.sc.gov/agency/ccr/Standards-Learning/Mathematics.cfm
21st Century Knowledge and Skills
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Problem Solving
Critical Thinking
Communication
Collaboration
Self-management
From the National Research Council
FREE: http://tinyurl.com/q25wodk
Portrait of College and Career Ready
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Academic Success and Employability
Interdependent Thinking and Collaborative Spirit
Intellectual Integrity and Curiosity
Logical Reasoning
Self-Reliance and Autonomy
Effective Communication
p.10 of http://ed.sc.gov/instruction/standards-learning/mathematics/standards/
Big Data & Algorithms
• Decisions affect our lives
• .. Made by machines
• … applying mathematical
models
• Can reinforce discrimination
• “a toxic cocktail for
democracy”
https://www.amazon.com/Weapons-Math-Destruction-Increases-Inequality/dp/0553418815
Video Games
https://www.wired.com/2010/10/physics-of-angry-birds/
Food
https://s-media-cache-ak0.pinimg.com/736x/fb/d7/4b/fbd74bc2b72376141bc23d5208dbee4a.jpg
SC Process Standards
A mathematically literate student can:
• Make sense of problems and persevere in solving them.
• Reason both contextually and abstractly.
• Use critical thinking skills to justify mathematical reasoning and
critique the reasoning of others.
• Connect mathematical ideas and real-world situations through
modeling.
• Use a variety of mathematical tools effectively and strategically.
• Communicate mathematically and approach mathematical
situations with precision.
• Identify and utilize structure and patterns.
SC College- and Career-Ready
Standards for Mathematics
A mathematically literate student can:
4. Connect mathematical ideas and real-world situations through
modeling.
a. Identify relevant quantities and develop a model to describe their
relationships.
b. Interpret mathematical models in the context of the situation.
c. Make assumptions and estimates to simplify complicated
situations.
d. Evaluate the reasonableness of a model and refine if necessary.
SC Math Processes
Next Gen Science Practices
Modeling Process
Modeling..
• … begins with a QUESTION your students
want to answer
• … is used for some purpose (other than
learning more math)
• … has real life connections (not just curvefitting)
• … is based on assumptions that must be
challenged
Modeling in Common Core
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Identifying essential variables in a situation
Formulating models from those variables
Performing operations using those models
Interpreting the results of those operations
Validating the conclusions of those results
Reporting on reasoning behind conclusions
Modeling is Iterative
Blum, W. (2011) Can Modelling be Taught and Learnt? Some Answers from Empirical Research. In Trends in
Teaching and Learning of Mathematical Modeling. Springer: New York.
“Model” in SC Content Standards
• K-6: situations; shapes; concrete, physical,
number, fraction, area, visual models
• 6-8: visual models, scale models, “equations
that model”, “model proportional
relationships”, “model theoretical probability”,
model to demonstrate proof of Pythagorean
Theorem
“Model” in SC Content Standards
• 9-12: function that models, arithmetic and
geometric sequences as models, linear
models, quadratic models, model periodic
phenomena, modeling in geometry, modeling
quantities, model with vectors, conditional
probability models, simulation models,
exponential models
“Model” in SC Content Standards
• Alg 1, 2 & Precalc: “In this course, students are
expected to apply mathematics in meaningful ways
to solve problems that arise in the workplace,
society, and everyday life through the process of
modeling. Mathematical modeling involves creating
appropriate equations, graphs, functions, or other
mathematical representations to analyze real-world
situations and answer questions.”
“Model” in SC Content Standards
• Geometry: “Use of mathematical tools is important
in creating and analyzing the mathematical
representations used in the modeling process. In
order to represent and solve problems, students
should learn to use a variety of mathematical tools
and technologies such as a compass, a straightedge,
graph paper, patty paper, graphing utilities, and
dynamic geometry software.”
“Model” in SC Content Standards
• Prob & Stat: “Use of technological tools, such as handheld graphing calculators, is important in creating and
analyzing mathematical representations used in the
modeling process and should be used during instruction and
assessment. However, technology should not be limited to
hand-held graphing calculators. Students should use a variety
of technologies, such as graphing utilities, simulation
applications, spreadsheets, and statistical software, to solve
problems and to master standards in all Key Concepts of this
course.”
Modeling in Textbooks
• Dan Meyer: “Missing the Promise
of Mathematical Modeling”
• Mathematics Teacher, April 2015,
pp.578-583
• FREE from NCTM.org
Algebra 1 & Geometry Texts
Identifying Variables
• Texts GIVE it away
• In life, figuring out
what variables
matter is a BIG
part of problem
solving and
modeling.
Formulating Models
• “Mathematizing”
• Most texts provide an equation, a graph, a
table…
• All neatly organized… doing the work for you!
What’s in the Book?
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Performing the operations
Interpreting the results
Einstein nearly 100 years ago:
“The formulation of the problem is often
more essential than the solution.”
Validating Conclusions
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Bring the math back to reality.
George Box, applied mathematician
“All models are wrong, but some are useful”
Making Sense!
Compromising
Cost/Benefit
What do employers value?
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Identifying essential variables in a situation
Formulating models from those variables
Performing operations using those models
Interpreting the results of those operations
Validating the conclusions of those results
Reporting on reasoning behind conclusions
Mathematical Modelers &
Test Scores
• 1,000 first-year engineering students
• SAT Math score had little correlation with
performance on modeling activities
• Understanding models and modeling requires
special attention in mathematics education
Kartal, O. et. al. (2016). The Relationship between Students’ Performance on Conventional
Standardized Mathematics Assessments and Complex Mathematical Modeling Problems.
International Journal of Research in Education and Science, 2(1), 239-252.
http://www.ijres.net/article/view/5000159980
Mathematical Modelers &
Test Scores
• “… the predominant use of standardized tests
to influence students toward or away from
STEM professions cannot be supported…
• … if modeling is an important indicator of
capabilities needed for success in STEM
professions.”
Kartal, O. et. al. (2016). The Relationship between Students’ Performance on Conventional
Standardized Mathematics Assessments and Complex Mathematical Modeling Problems.
International Journal of Research in Education and Science, 2(1), 239-252.
Assessment
• Can Selected Response items measure the
proficiencies of a college and career ready
learner who fit our profile of a South Carolina
Graduate?
•NO!
SC READY – Item Types
Mathematics
Grades
Item Type
3–8
Selected Response (SR)
Students select their responses from a list of possible
answer options.
SC READY – Item Types
Mathematics
Grades
Item Type
6–8
Technology Enhanced (TE)
Students interact with the item (online only)
to provide their responses—e.g., drag and drop,
hot spot, graphing, etc.
Comparable SR items will replace TE items on
the paper/pencil assessments.
SC READY – Draft Test Blueprint
Mathematics
Points Possible
per
Reporting
Category
Reporting Categories
Grade 6 (60 total points)
12–15
1. The Number System
8–10
2. Ratios and Proportional Relationships
12–15
3. Expressions, Equations, and Inequalities
8–10
4. Geometry and Measurement
11–13
5. Data Analysis and Statistics
Sample Items
• Grade 5: http://ed.sc.gov/tests/tests-files/scready-files/sc-ready-mathematics-grade-5sample-items/
• Grade 6-8: http://ed.sc.gov/tests/testsfiles/sc-ready-files/sc-ready-mathematicsgrades-6-8-sample-items/
• EoC Algebra 1:
http://ed.sc.gov/scdoe/assets/File/tests/high/
eocep/AlgebraITGuide2015_12_10_15.pdf
GAIMME
Guidelines for Assessment & Instruction in
Mathematical Modeling Education
– http://www.comap.com/Free/GAIMME/index.html
Guidelines for Assessment and Instruction in
Statistics Education (GIASE)
– FREE at http://www.amstat.org/education/gaise/
GAIMME
• For TEACHERS!
• To help you incorporate
mathematical modeling into your
classrooms
• How teaching the modeling process can mature
as students move thru the different grades
• Independent of the mathematical knowledge
they may bring to bear
From NCTM
• Annual Perspectives in
Mathematics Education
• 25 Chapters with proven
strategies for implementation in
the classroom and teacher
education
http://www.nctm.org/Store/Products/Annual-Perspectives-inMathematics-Education-2016--Mathematical-Modeling/
What is mathematical modeling?
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Mathematical modeling
Modeling mathematics
Mathematics to represent reality
Models to represent mathematics
Modeling?
• Physical manipulatives (e.g. 3D solids, algebra
or fraction tiles, pattern blocks, etc.)
• Maps or blueprints
What is mathematical modeling?
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Demonstrate? “I will model how I want you to...”
Symbolic representation? y = 3x2 -15
Manipulatives?
Word problems? “Two trains left …”
Real world applications?
Is this
Mathematical Modeling?
• Waterford Middle School is planning a field trip to Roper
Mountain. The school as 198 students and 14
teachers/parents going on the trip. They will rent buses to
transport everyone. Each bus holds 40 passengers. How
many buses are needed?
• 5.3
• 5 buses (rounding off)
• 5 remainder 12
• 6 buses
Modeling Problem
• Ed wants to build a bookshelf in a corner. It
will be two feet wide. He has two five-foot
boards. How many two-foot sections can he
get from the boards?
Schoenfeld, 2013
Is this
Mathematical Modeling?
SC EOC Alg 1:
http://ed.sc.gov/scdoe/ass
ets/File/tests/high/eocep/
AlgebraITGuide2015_12_1
0_15.pdf
Is this
Mathematical Modeling?
• Wikipedia reports that 8% of all Americans eat at McDonalds
everyday.
• There are currently 323 million Americans and 14,248
McDonalds in the US.
• Is the Wikipedia claim be true?
• Create a mathematical argument to
justify your position.
• If you believe the claim is false,
determine reasonable percentage.
Shell Centre:
http://map.mat
hshell.org/dow
nload.php?filei
d=1708
Burkhardt, 2013
Sample Student Work:
Ed’s Work
Continuum between word problem
and modeling
GAIMME p.14
Are my students modeling.
GAIMME p.45
Problem Solving and Modeling
• Problem solving: take some mathematical
phenomenon, figure out what it’s about, make
sense of it (order, structure) and achieve goal
• Modeling (applied problem solving): take
some real world phenomenon, figure out
what it’s about, make sense of it, and achieve
goal tied to real world.
• BOTH: “mathematical sense making”
Henry Pollak
• “mathematical modeling” is not just a new
and pretentious name for “word problem” or
“problem solving”
• A word problem is to practice the math… and
has a solution implied…
• “Mathematical modeling demands more.”
Pollak, H (2013). What is mathematical modeling? Retrieved from
http://www.missourilearningstandards.com/files/cur-math-comcore-what-ismathematical-modeling.pdf
Footprint Problem
Imm, K.L. and Lorber, M.D. (2013) The Footprint Problem: A pathway to
modeling. Mathematics Teaching in the Middle School, 19(1), 46-54
Footprint Problem
Imm, K.L. and Lorber, M.D. (2013) The Footprint Problem: A pathway to
modeling. Mathematics Teaching in the Middle School, 19(1), 46-54
Footprint Problem
Footprint Problem
Lesh, R. & Yoon, C. (2007). What is distinctive in (our view about) models and modeling? In Modeling and
Applications in Mathematics Education: the 14th ICMI Study. Edited by Blum et. al. Boston: Springer.
Framework for Teachers’ Roles in
Mathematical Modeling
GAIMME p.39
Assessment
SOME large-scale assessments DO include
modeling:
• State Exams (PARCC, Smarter Balanced)
• International Exam (PISA, TIMSS)
• Modeling Contests (M3 Challenge, HiMCM,
ICM)
Smarter Balanced Example
PISA
TIMSS
Moody’s Mega Math Challenge
What holds us back?
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Modeling takes more time
There is often no one right answer
Our high school curriculum pathways are rigid
… and unchanged over some 60 years:
Algebra I, Geometry, Algebra II
NEW in last 20 years: Probability and Statistics
Rush to Calculus
College and Career Ready
• Calculus IS important for STEM majors
• But we have an obligation to EVERY child
• Not just math, science, and engineering
majors
College and Career Ready
Quoting Diane Briars:
• Our high schools focus on getting
people prepared for calculus . . .
• But we do little to teach statistics,
probability, and uncertainty . . .
• This is one of the biggest issues
facing U.S. mathematical sciences;
it is also a big problem in terms of
national competitiveness.
NCTM Task Force
• Announced on October 25:
http://tinyurl.com/gpjqdae
• Pathways through High School Mathematics:
Building Focus and Coherence
NCTM Task Force
• Address the purpose of high school mathematics
and include guiding principles such as access, equity,
and empowerment;
• Define math curricular pathways leading to college
pathways and career readiness, as well as active
participation in our democratic society; and
• Provide narrative descriptions of course exemplars,
including their big ideas, that could populate the
pathways.
Modeling and Social Justice
• NCTM’s Vision: http://www.nctm.org/About/
• “We envision a world where everyone is
enthused about mathematics, sees the value
and beauty of mathematics, and is
empowered by the opportunities
mathematics affords.”
• How can stat inform?
http://www.amstat.org/publications/jse/v15n1/lesser.pdf
SAT and Income
Why not South Carolina?
• Make time for MODELING
• Authentic Assessment
• Valid data about meeting the portrait of the
SC graduate.
• Tied to the businesses and industry of SC
– Food, agriculture, tourism, military, …
– Boeing, Michelin, Amazon, Google, Colonial Life,…
Why not South Carolina?
• Research shows we are ill-serving future,
successful STEM majors by relying on INVALID
standardized test data.
• Honor our Portrait of the College and Career
Ready Student
MODEL…
in every sense of the word.
THANK YOU!
[email protected]
Powerpoint available at:
www.ite.sc.edu/dickey.html
References
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SC College and Career Ready Portrait: http://ed.sc.gov/agency/ccr/Standards-Learning/Mathematics.cfm
NRC Education for Life and Work: http://tinyurl.com/q25wodk
Weapons of Math Destruction book: https://www.amazon.com/Weapons-Math-Destruction-IncreasesInequality/dp/0553418815
Physics of Angry Birds article: https://www.wired.com/2010/10/physics-of-angry-birds/
Pizza graphic: https://s-media-cache-ak0.pinimg.com/736x/fb/d7/4b/fbd74bc2b72376141bc23d5208dbee4a.jpg
Dan Meyer NCTM Mathematics Teacher article : http://www.nctm.org/Publications/mathematicsteacher/2015/Vol108/Issue8/Missing-the-Promise-Of-Mathematical-Modeling/
Modeling in the Common Core: http://www.corestandards.org/Math/Content/HSM/
Kartel article on SAT, modeling and engineering: http://www.ijres.net/article/view/5000159980
Sample SC Ready Items:
Grade 5: http://ed.sc.gov/tests/tests-files/sc-ready-files/sc-ready-mathematics-grade-5-sample-items/
Grade 6-8: http://ed.sc.gov/tests/tests-files/sc-ready-files/sc-ready-mathematics-grades-6-8-sample-items/
EoC Algebra 1: http://ed.sc.gov/scdoe/assets/File/tests/high/eocep/AlgebraITGuide2015_12_10_15.pdf
GAIMME: http://www.comap.com/Free/GAIMME/index.html
GAISE: http://www.amstat.org/education/gaise/
NCTM APME Modeling Book: http://www.nctm.org/Store/Products/Annual-Perspectives-in-MathematicsEducation-2016--Mathematical-Modeling/
References (continued)
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Shell Centre Cat Item: http://map.mathshell.org/download.php?fileid=1708
Burkhardt Modeling Paper: http://journals.tc-library.org/index.php/matheducation/article/view/1071
Henry Pollak article on what is mathematical modeling: http://www.missourilearningstandards.com/files/curmath-comcore-what-is-mathematical-modeling.pdf
Footprint Activity in NCTM Mathematics Teaching in the Middle School:
http://www.nctm.org/Publications/mathematics-teaching-in-middle-school/2013/Vol19/Issue1/The-FootprintProblem_-A-Pathway-to-Modeling/ 3
Smarter Balanced sample Items: https://www.smarterbalanced.org/assessments/sample-questions/
PISA Items:
https://www.oecd.org/pisa/pisaproducts/pisatakethetestsamplequestionsfromoecdspisaassessments.htm
TIMSS Items: https://nces.ed.gov/timss/pdf/TIMSS2011_G8_Math.pdf
Moody’s Mega Math Challenge: https://m3challenge.siam.org/
National Academies book: https://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025
NCTM High School Pathways Task Force: http://tinyurl.com/gpjqdae
NCTM Vision; http://www.nctm.org/About/
Lesser article in Journal for Statistics Education: http://www.amstat.org/publications/jse/v15n1/lesser.pdf