Mathematics Connections
Common Core State Standards and
the NGSS
Robert Mayes & Thomas Koballa
Georgia Southern University
NGSS – Systems Perspective
 The systems perspective is represented in four recent NRC
reports




Taking Science to School (NRC, 2007)
A Framework for K-12 Science Education (NRC, 2012a)
Education for Life and Work (NRC, 2012c)
Discipline-Based Education Research (NRC, 2012b)
 emphasis on deeper learning that connects the “what” of
science with the “how” and “why.”
 push toward an integration of conceptual, epistemic, and
social competencies within science education and beyond
 Two agendas:
 STEM workforce development - next generation of scientists
 Scientifically literate citizens that can make informed
decisions on grand challenges facing their generation
http://www.nap.edu/catalog/9975.html
NGSS – Systems Perspective
 Framework has three implications that set new course
for STEM education conceptualized through climate
sciences and engineered systems (Duschl, 2013).
 science education should be coordinated around three
dimensions - crosscutting concepts, core ideas, and
practices
 the practices should represent both science and
engineering
 the alignment of curriculum, instruction and assessment
should be implemented through the development of
learning progressions that function across grade bands
 NRC Framework has salient features
 Complex adaptive systems thinking
 Model-based reasoning
 Quantitative reasoning
Math centered practices - Duschl
Science Framework
and CCSS-M Connections
Science and Engineering Practices
1. Asking questions and defining problems
Mathematical Practices
1. Making sense of problems and preserving in solving them
2. Developing and using models
2. Reason abstractly and quantitatively
4. Model with mathematics
5. Use appropriate tools strategically
1. Making sense of problems and preserving in solving them
2. Reason abstractly and quantitatively
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
3. Planning and carrying out investigations
4. Analyzing and interpreting data
5. Using mathematics, information and computer technology,
and computational thinking
4. Model with mathematics
5. Use appropriate tools strategically
7. Look for and make use of structure
2. Reason abstractly and quantitatively
3. Constructing viable arguments and critique the reasoning of
others
4. Model with mathematics
8. Looking for and exposing regularity in repeated reasoning
6. Constructing explanations and designing solutions
3. Constructing viable arguments and critique the reasoning of
others
8. Looking for and exposing regularity in repeated reasoning
7. Engaging in argument from evidence
3. Constructing viable arguments and critique the reasoning of
others
7. Look for and make use of structure
3. Constructing viable arguments and critique the reasoning of
others
6. Attend to precision
8. Obtaining, evaluating, and communicating information
Reasoning Hierarchy
(Duschl and Bismack, 2013)
The interdisciplinary nature of science, as demonstrated by experts requires socially constructing
knowledge about the interactions between various disciplines, in order to explain the physical,
human, and created worlds.
• This means understanding the relationships of natural and human systems, their ever-changing
nature, and how they influence and are influenced by other systems. To internalize this
understanding means to internalize a systems thinking approach toward viewing and analyzing
phenomena and processes.
• When scientists use models they engage in model-based reasoning, which involves the
development and use of varying forms of representations and the subsequent feedback and
redesign of the model (Lehrer & Schauble, 2002, 2006). This type of reasoning is critical to the
“doing” of science, as it incorporates analyzing, explaining, and communicating the world
around us – a foundation to the function of science.
• “Mathematics in all its forms is a symbol system that is fundamental to both expressing and
understanding science. Often, expressing an idea mathematically results in noticing new
patterns or relationships that otherwise would not be grasped” (NRC, 2007). Both modelbased reasoning and quantitative reasoning involve an iterative process of analyzing,
modeling, communicating, evaluating, and redesigning models to explain scientific phenomena
or processes.
So what is a Complex adaptive system?
 Complex adaptive systems (John Holland and Murray Gell-Mann, Santa Fe Institute) are made
up of diverse multiple interconnected elements which have the capacity to adapt – to change
and learn from experience.
 Examples: stock market, ecosystems, the cell, social systems, energy resources
Framework’s Fifth Element (Practice)
 5th Practice: using mathematics, information and computer technology,
and computational thinking in the context of science and engineering
 Related paradigms for three areas
3 Mathematics Paradigms
 We will present brief discussions of the three
mathematics paradigms and provide examples of
them
1. Quantitative Reasoning
2. Computational Science
3. Data-intensive Science
QR Poll
 Indicate which of the following is most prevalent in science
classrooms in your state.
A. Students apply basic arithmetic to calculate and measure
B. Students interpret graphs and science models to answer science
questions
C. Students create their own scientific models incorporating
mathematics
D. Students use computer simulations and models and engage in
data intensive science
E. None of the above
Quantitative Reasoning
 NSF Culturally Relevant Ecology, Learning Progressions, and
Environmental Literacy project has the goal of refining and
extending current frameworks and assessments for learning
progressions leading to environmental science literacy and
associated mathematics that focus on carbon cycling, water systems,
and biodiversity in socio-ecological systems. QR Theme Team
focuses on mathematics and statistics applied in environmental
science.
 Quantitative Reasoning in Context (QRC) is mathematics and
statistics applied in real-life, authentic situations that impact an
individual’s life as a constructive, concerned, and reflective citizen.
QRC problems are context dependent, interdisciplinary, open-ended
tasks that require critical thinking and the capacity to communicate
a course of action.
This project is supported in part by a grant from the National Science Foundation: Culturally Relevant Ecology,
Learning Progressions, and Environmental Literacy (DUE-0832173) which we refer to as Pathways.
QR Framework
 We propose a quantitative reasoning framework that has four key
components:
 Quantification Act (QA): mathematical process of conceptualizing
an object and an attribute of it so that the attribute has a unit
measure, and the attribute’s measure entails a proportional
relationship (linear, bi-linear, or multi-linear) with its unit
 Quantitative Literacy (QL): use of fundamental mathematical
concepts in sophisticated ways for the purpose of describing,
comparing, manipulating, and drawing conclusions from variables
developed in the quantification act
 Quantitative Interpretation (QI): ability to use models to
discover trends and make predictions, which is central to a person
being a citizen scientist who can make informed decisions about
issues impacting their communities
 Quantitative Modeling (QM): ability to create representations to
explain a phenomena and revise them based on fit to reality
QR Framework
QR Cycle
QA-QL Exemplar
QR STEM Project (Wyoming) – QR in energy and environment context.
Professional development text on project coming soon.
QI & QM
 We think of the difference between being a consumer of
information (QI) and the creator of the information (QM)
 Or as the difference between being a scientifically and
mathematically literate citizen and a scientist or mathematician.
QI-QM Exemplar
Science System Model (Box Model)
Quantitative Interpretation:
1. What are the variables in the
carbon cycle?
2. Which are flow processes and
which are storage areas?
3. What are the attributes of
deforestation that make it a viable
variable in this model?
4. What are the measures associated
with the variables?
5. What is the balance of CO2
entering and leaving the ocean?
6. What other questions would you
ask your students? Do they
require quantitative accounts?
Quantitative Modeling: Have your
students research and develop a box
model.
Science Model Complexity vs. Mathematical Models


Great variety and complexity in science models for students to interpret
Science graphs often have more than two variables on the same coordinate plane and embed
variables in graphs – this is not common in mathematics
Science Model Complexity vs. Mathematical Models
 Great variety and complexity in science models for students to interpret
 Maps, colors, relative size to represent embedded variables
Expanding Toolbox
 Historic paradigms of science: experimental science and theoretical
science
 Due to increasing computing capabilities, two new paradigms have
arisen
 Computational science (scientific computing)
 Scientific computing focuses on simulations and modeling to provide both
qualitative and quantitative insights into complex systems and phenomena that
would be too expensive, dangerous, or even impossible to study by direct
experimentation or theoretical methods (Turner et al. 2011)
 Data-intensive science (data-centric science)
 The explosion of data in the 21st century led to the invention of data-intensive
science as a fourth paradigm, which focuses on compressed sensing (effective
use of large data sets), curation (data storage issues), analysis and modeling
(mining the data), and visualization (effective human-computer interface).
Computational Science
the third paradigm for scientific exploration
 Computational Thinking integrates the power of human
thinking with the capabilities of computational processes and
technologies.
 The essence of computational thinking is the generalization of
ideas into algorithms to model and solve problems.
 Computational Thinking is not about getting humans to think
like computers. But to use human creativity and imagination to
make computers useful and exciting (Wing 2006).
What is Scientific Computing?
(computational science)
 It is not Computer Science. The goal of scientific computing is to improve
understanding of a physical phenomena.
 It does not replace Experiment and Theory, rather it complements these
methods.
 It is “both the microscope and telescope of modern science. It enables
scientists to model molecules in exquisite detail to learn the secrets of
chemical reactions, to look into the future to forecast the weather, and to look
back to a distant time at a young universe.”
–Lloyd Fosdick et. al, An introduction to High-performance
scientific computing, 1996.
Computational Science Exemplar
Science in a Box-Wind Roses (Shader, 2013)
Questions:
You wish to describe and study the
wind patterns in your city?
1.
2.
3.
http://www.weblakes.com/products/wrplot/index.html
What are the important
characteristics of wind?
How could you measure these
characteristics?
How might you be able to
illustrate these characteristics
in a diagram?
4. How does a wind rose illustrate
characteristics of wind?
5. What would a wind rose look like
in your city?
CS Poll
 Which of the following scientific methods do students
get the most experience with in your state?
A.
B.
C.
D.
E.
Theory/content
Experiment/inquiry
Computational/modeling
Data Analysis/Statistics
A &B
Data-intensive Science
the fourth paradigm for scientific exploration
The explosive use of personal data, new data collection technologies (such as
lidar), the capabilities and speeds of modern personal and super computers has
resulted in a wealth of information and data. Simulations of complex models are
generated on a 24/7/365 basis and involve multiple scales.
Consists of four main activities at all scales:
• Capture: New technologies allow capture of larger data sets, over wider time, spatial and
physical scales. There is an ongoing need to make this more effective: compressed
sensing.
• Curation: Where and how do we store the data to make it useable?
• Analysis and Modeling: How do we mine the data? How can we make inferences
without seeing all the data? Can we make models that explain the data?
• Visualization: How does one grock large data sets? How can we make the humancomputer interface more effective?
http://www.ted.com/playlists/56/making_sense_of_too_much_data.html
A Toy Problem (Shader, 2012)
Heat diffusion on a plate
http://ccl.northwestern.edu/netlogo/
NetLogo Heat (unverified)
Diffusion Model
NetLogo is a multi-agent
programmable modeling
environment. It is used by tens of
thousands of students, teachers
and researchers worldwide. It
also powers HubNetparticipatory
simulations. It is authored by Uri
Wilensky and developed at
the CCL. You can download it free
of charge.
This is governed by the heat equation:
How do we translate this into something computable
(just using +,-,*,/) ?
We approximate by thinking of the plate as a grid of points
Simple Computational Model
20
10
12
10
20
A particle’s temperature changes at a rate
proportional to the difference between its
temperature and the average temperature of its
neighbors.
Average temperature of P’s neighbors is 15,
which is 3 more than P’s temperature.
If constant of proportionality is 1/3, then P’s
updated temperature will be
13=12+ (1/3)*3.
• For each time step and each particle in the grid we have to do 4 additions,
1 multiplication, and 1 division. That is 6 operations, but let’s use 5 to
keep the calculation simpler.
• A plate modeled by a 100 by 100 grid would take 50,000 operations per
time-step.
• To run until stable temperature on wood would take about 100 steps; a
total of about 5 million operations!
This is just a toy problem
To have high level of accuracy with model, we might need a grid
much finer than 100 by 100.
Making grid 10 times finer in each direction requires multiplying
the number of operations by 10*10=100.
To get the same accuracy, we need the time-steps 100 times
smaller.
Even with a toy problem, we’re up to
operations!
3D Heat Diffusion
The simple model becomes large
Same basic idea, but extra dimension is costly!
A 1,000 by 1,000 by 1,000 grid cube takes
7 trillion operations
to determine the temperatures of the particles after 1,000 time steps.
How does this model help
understanding?
 Dynamic, visual
 Allows easy variation of parameters
 Forced to construct equations out of physical
observations
 Better understanding of orders of magnitude
How should computational science impact our
teaching? (Bryan Shader)
• Profoundly
Computational Thinking will be a fundamental 21st century skill (just like
reading, writing and arithmetic)” –Jeanette Wing, Computational Thinking,
2007
• Systemically
SC has symbiotic relationship with Math, Science and Engineering . CT
requires abstraction, the ability to work with multi-layered and
interconnected abstractions (e.g. graphs, colors, time). CT draws on ``real
world’’ problems.
• Vertically
CT must be developed over many years, and starts at Pre-K
• Wisely
Incorporate programming at appropriate times, tie with
theory, emphasize quality vs. quantity in experiences (Shader, 2012)
How should data-intensive science inform our
teaching? (Bryan Shader)
• Need to provide basic information literacy skills so that students can be
productive members of the 21st century workforce, and adapt to a
increasingly data-dominant world. How is data-mining done? How are
inferences drawn from large
data-sets? What are the pros/cons of models? How can one digest
data?
• Need to make learning authentic. Wealth of resources to connect content
areas to ``real world’’ problems.
• More depth, less breadth. Project based?
• Will need to change the way we “see” and sense data. 3D, color graphics,
different scales. Thus, there is a need to give students experience with
multiple interpretations.
• Need to provide interdisciplinary understandings (integrated curricula)
How does data-intensive science inform our
teaching? (Bryan Shader)
• Must help develop new intellectual tools and learning strategies in our
students: e.g. the importance of different scales, the understanding of complex
systems, how does one frame and ask meaningful questions?
• New experiences needed
Collecting and interpreting data from sensors
Mining data
Massive collaboration
Interdisciplinary synthesis
From science to policy inferences
Use of scientific computing, data gather tools
Visualization
• Statistics, statistics, statistics. But make it data-driven, and have the focus be
on understanding.
Framework and CCSS-M Alignment
Example
 Core Science area: Earth and Space Science – Earth and
Human Activity
 Core Concept: global climate change
 Quantitative Concept: change
 Exemplars of science tasks accomplished at end of grades 2,
5, 8, and 12.
 Exemplars from Computational Science
Grade 2
Grade
Framework for K–12 Science
Level
Grade 2
Common Core State StandardsMathematics
By the end of grade 2 students
The CCSS-M have 2nd graders solving
should know: “Weather is the
problems involving addition and
combination of sunlight, wind,
subtraction within 100, understanding
snow or rain, and temperature in a
place value up to 1,000, recognizing
particular region at a particular
the need for standard units of measure
time. People measure these
of length, representing and interpreting
conditions to describe and record
data, and reasoning with basic shapes
weather and to notice patterns over and their attributes.
time” (NRC 2012, p. 188).
Grade 2 Tasks
 Weather tasks: Involve students in observing television weather
reports followed by drawing pictures of and describing things they
believe make up the weather. These experiences will enable students
to construct their own definitions of weather and list variables that
make up weather, such as rain, sunshine, and wind.
 Involve students in collecting and measuring rain to the nearest
centimeter for each month of the school year for their community.
Ask students to draw pictures representing rain by month; this may
be a bar graph or a dot chart using M&M candies.
 Using visual data displays, student could answer questions about
specific weather variables: Which month was the wettest? The
driest? Conclude by having students link their findings to the context
of the local environment through such questions as these: What do
you think happened to plants in the months with low rainfall? What
other weather conditions interact with the amount of rain to affect
plant life?
Grade 2: Computational Thinking
 Basic understanding of algorithms:
 Describe the steps taken to make a PBJ sandwich
 How can one person sort a collection of items by their weight?
 How can a group of people sort a collection of items by their weight?
 An appreciation for parallel vs. serial processing
What is parallel processing? See http://nwsc.ucar.edu/young-scientists
NCAR-Wyoming Supercomputing Center
http://nwsc.ucar.edu/facility/visit
Grade 5
Grade Level
Framework for K–12 Science
Common Core State StandardsMathematics
Grade 5
By the end of 5th grade the expectation
The CCSS-M has fifth graders writing and
for global climate change is: “If Earth’s
interpreting numerical expressions, analyzing
global mean temperature continues to rise, patterns and relationships, performing
the lives of humans and organisms will be operations with multi-digit whole numbers and
affected in many different ways” (NRC
decimals to hundredths, using equivalent
2012, p. 98).
fractions to add and subtract fractions,
multiplying and dividing fractions, converting
measurement units within a given
measurement system, measuring volume,
representing and interpreting data, graphing
points on the coordinate plane to solve realworld problems, and classifying twodimensional figures into categories based on
their properties.
Grade 5 Tasks
 Climate Change task: Have students consider
data on state, national, and international
annual temperature changes. Students could
be asked to examine Climate Central’s
national map on temperature change.
Questions: What percentage of states has
warmed more than 0.2 degrees each decade
over the past 40 years? How much has the
state you lived in warmed?
 Have students examine data for the state in
which they live. Direct students to one of the
red points on the graph representing Georgia
and ask them to interpret what it means.
What does the general trend of the scatter
plot of points indicate?
 Ask students to measure the temperature each
day for a week to the nearest 0.1 degree.
What can you say about natural flux in daily
temperatures and how it relates to the annual
average temperature? If the temperature
continues to increase at the current rate, what
will the average temperature be in 20 years?
What potential impact does this warming
trend have in your state?

http://www.climatecentral.org/news/the-heat-is-on/
Grade 5 Computational Thinking
Tasks
 How does a computer represent numbers?
--Base two arithmetic
 What good are those bar codes on products?
–Error detection
 Average behavior, patterns in
randomness
Examplars
–NetLogo Mousetrap
http://ccl.northwestern.edu/netlogo
–Weather vs Climate
http://spark.ucar.edu/video/dog-walking-weather-and-climate
Grade 8
Grade Level Framework for K–12 Science
Common Core State StandardsMathematics
Grade 8
The end of 8th grade expectation for
The CCSS-M 8th grade standards include
climate change is to understand that
awareness of numbers beyond the rational
human activities, such as carbon
numbers, work with radicals and integer
dioxide release from burning fuels, are exponents, proportional relationships,
major factors in global warming.
ability to analyze and solve linear equations
Reducing the level of climate change
and systems of linear equations, use linear
requires an understanding of climate
functions to model relationships between
science, engineering capabilities, and
quantities, understand congruence and
human behavior (NRC 2012, p. 198).
similarity, the Pythagorean Theorem, solve
real-world problems involving volume of
cylinders, cones, and spheres, and use
statistics to investigate patterns of
association in bivariate data.
Grade 8 Tasks
 Climate Change task: Extend the
discussion of the Georgia warming data.
Provide students with the data for
average annual temperature per year for
the state in a table, then have them plot
the data and construct a scatter plot. Use
the plot to address questions such as:
What is the trend of the data in this
scatter plot? Is it decreasing or
increasing? Estimate a line of best fit for
the data that represents the trend.
 Have students write out the equation of
the estimated line of best fit and use the
linear model to predict temperatures for
future years. Conclude by helping
students relate this back to the science
context: What variables can we control to
reduce or stabilize the temperature trend?
Grade 8 CT-Tasks (Shader, 2013)
 The strengths/weaknesses of models
 NetLogo fire model: What affects the spread of a
wild fire?
 Does the simulation always give the same result for
the given initial conditions?
 What things stay the same for each simulation?
 What things can’t be predicted? Idea of ensembles.
 Do small changes in conditions have small changes
in outcomes?
 What things would you have to incorporate to make
this a more natural model? NCAR fire model
http://www.vets.ucar.edu/vg/categories/wildfires.shtml
Grade 12
Grade
Framework for K–12 Science
Level
Grade 12
Common Core State StandardsMathematics
By the end of high school students
The CCSS-M high school standards are
should understand that climate change by conceptual categories not grade level.
is slow and difficult to recognize
The conceptual categories of Number and
without studying long-term trends,
Quantity, Algebra, Functions, Modeling,
such as studying past climate patterns. Geometry, and Statistics and Probability
Computer simulations are providing a specify the mathematics that all students
new lens for researching climate
should study in order to be college and
change, revealing important
career ready. Functions are expanded to
discoveries about how the ocean, the
include quadratic, exponential, and
atmosphere, and the biosphere
trigonometric functions, broadening the
interact and are modified in response
potential models for science.
to human activity (NRC 2012, p.
198).
Grade 12 Tasks
 Climate Change task: Revisit the scatter plot
of state temperature data, this time ask
students to provide a power function model or
exponential model for the data. Rich
discussions of which function is the best
model for the data would engage students in
exploring error and best-fit concepts.
 Carbon dioxide as a mitigating factor in
global climate change can be explored in
more depth. For example, provide data on
historic trends in atmospheric carbon dioxide.
Ask students to quantitatively interpret the
trends in the graph as naturally occurring
cycles. The claim has been made that today
the Earth is experiencing just a phase in a
natural cycle of carbon dioxide change.
Students could be challenged to interpret the
data for evidence that supports this claim.
Questions : How were the data collected? Are
the data reliable? What are likely causes of the
fluxes in atmospheric carbon dioxide?
Grade 12 CT Tasks
Calculus ideas: rates/areas
1 What affect does an
upwind turbine have
on a downwind
turbine?
2. What do these graphs tell
us?
3. How might you estimate
the total amount of energy
generated by each turbine.
Basic programming
skills
Credits: Jay Sitaraman, University of Wyoming
QR Poll
 Which of the following presents the biggest mathematical barrier
for your students in your state?
A. Ability to identify variables in science context, understand the
attributes of the variables that make it important to the context,
and work with appropriate measures
B. Ability to measure, reason proportionally, calculate, and
understand large/small numbers
C. Ability to interpret a scientific table, graph, equation, or system
model to answer a real-world question
D. Ability to create a model from data, then test and refine the model
E. All of the above
QR Opportunities
 Pathways Project: Join us for our national data
collection on QR ability of students from grades 6 to
12. Contact Robert Mayes for more information.
 QR Assessment Exemplars
 QR in Energy and Environment: a PD and teacher
resource for teaching QR in science from QR STEM
project
QR LP Assessment Exemplars
QR Water Cycle Macro Scale
The pie chart below describes where water goes on a school grounds when it
rains. If 15 centimeters of rain falls on the school yard in one day, how could
you determine how much would runoff?
a. What are reasonable dimensions for a school yard?
b. Say reasonable dimensions are 300 meters x 200
meters. How can you determine how much rain falls
on your school yard?
c. Can you express the amount of rain in m3?
d. So how much water runs off the schoolyard? Can
you provide a common sense estimate of how much
water this is?
e. Say then that 135,000m3 of water is runoff from the
playground from the 15cm rain. Where does this
runoff go?
QR LP Assessment Exemplars
QR Biodiversity Communities Micro Scale
What happens to biomass and energy in a community? As you move up the
food web are there more or less organisms?
a. Below are pyramids of energy and biomass for a system. What do the pyramids tell
you about biomass and energy in the community?
b. What percent of energy and biomass is lost at each step in the pyramid?
c. What happens to the biomass that a consumer does not eat, such as beaks or bones?
d. Bacteria are living single-celled organisms shaped like spheres, rods, or spiral twists.
A bacteria is about 10-6 of a meter in length. Just how small is that? How many
would fit end-to-end in an inch?
QR LP Assessment Exemplars
QR Carbon Cycle Landscape Scale
The following is a box model of global carbon dioxide movement between
2000 and 2005. The numbers represent billions of tons (gigatons) of carbon
dioxide per year. Explain what you see in the diagram.
a. What do the boxes (pictures) and arrows
mean to you? What does the arrow labeled 8
represent?
b. Can you explain what the box with plants,
animals, and soils has to do with carbon
movement?
c. What is the net flux (change) in CO2 with
respect to the atmosphere? Is it increasing or
decreasing? By how much?
d. Is it a concern that CO2 is increasing in the
atmosphere (Science Qualitative)? Why?
What would we have to do to balance the
exchange of carbon dioxide with the
atmosphere?
Thank You
Robert Mayes
Tom Koballa
Georgia Southern University
rmayes@georgiasouthern.edu
tkoballa@georgiasouthern.edu