Mathematics Connections

Common Core State Standards and the NGSS

Robert Mayes & Thomas Koballa Georgia Southern University

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

NGSS – Systems Perspective

NRC Framework has salient features Complex adaptive systems thinking Model-based reasoning Quantitative reasoning

Math centered practices - Duschl

Science and Engineering Practices Mathematical Practices1. Asking questions and defining problems 1. Making sense of problems and preserving in

solving them2. Developing and using models 2. Reason abstractly and quantitatively

4. Model with mathematics5. Use appropriate tools strategically

3. Planning and carrying out investigations 1. Making sense of problems and preserving in solving them2. Reason abstractly and quantitatively4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision

4. Analyzing and interpreting data 4. Model with mathematics5. Use appropriate tools strategically7. Look for and make use of structure

5. Using mathematics, information and computer technology, and computational thinking

2. Reason abstractly and quantitatively3. Constructing viable arguments and critique the reasoning of others4. Model with mathematics8. Looking for and exposing regularity in repeated reasoning

6. Constructing explanations and designing solutions

3. Constructing viable arguments and critique the reasoning of others8. Looking for and exposing regularity in repeated reasoning

7. Engaging in argument from evidence 3. Constructing viable arguments and critique the reasoning of others7. Look for and make use of structure

8. Obtaining, evaluating, and communicating information

3. Constructing viable arguments and critique the reasoning of others6. Attend to precision

Science Framework and CCSS-M Connections

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 model-based reasoning and quantitative reasoning involve an iterative process of analyzing, modeling, communicating, evaluating, and redesigning models to explain scientific phenomena or processes.

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

So what is a Complex adaptive system?

5th Practice: using mathematics, information and computer technology, and computational thinking in the context of science and engineering

Related paradigms for three areas

Framework’s Fifth Element (Practice)

We will present brief discussions of the three mathematics paradigms and provide examples of them1. Quantitative Reasoning2. Computational Science3. Data-intensive Science

3 Mathematics Paradigms

Indicate which of the following is most prevalent in science classrooms in your state. A. Students apply basic arithmetic to calculate and measureB. Students interpret graphs and science models to answer science

questionsC. Students create their own scientific models incorporating

mathematicsD. Students use computer simulations and models and engage in

data intensive scienceE. None of the above

QR Poll

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.

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

QI-QM ExemplarScience 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.

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

Science Model Complexity vs. Mathematical Models

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).

Expanding Toolbox

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).

Computational Sciencethe third paradigm for scientific exploration

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.

What is Scientific Computing? (computational science)

Computational Science ExemplarScience in a Box-Wind Roses (Shader, 2013)

Questions:You wish to describe and study the wind patterns in your city?

1. What are the important characteristics of wind?

2. How could you measure these characteristics?

3. 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?

http://www.weblakes.com/products/wrplot/index.html

Which of the following scientific methods do students get the most experience with in your state?

A. Theory/contentB. Experiment/inquiryC. Computational/modelingD. Data Analysis/StatisticsE. A & B

CS Poll

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 human-

computer 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

NetLogo Heat (unverified)Diffusion Model

http://ccl.northwestern.edu/netlogo/

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

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!

20

20

10

10 12

Simple Computational ModelA particle’s temperature changes at a rate proportional to the difference between its temperature and the average temperature of its neighbors.

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!

Same basic idea, but extra dimension is costly!

3D Heat DiffusionThe simple model becomes large

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.

Dynamic, visual

Allows easy variation of parameters

Forced to construct equations out of physical observations

Better understanding of orders of magnitude

How does this model help understanding?

How should computational science impact our teaching? (Bryan Shader)

• ProfoundlyComputational Thinking will be a fundamental 21st century skill (just like reading, writing and arithmetic)” –Jeanette Wing, Computational Thinking, 2007

• SystemicallySC 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.

• VerticallyCT 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.

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

Framework and CCSS-M Alignment Example

Grade 2Grade

Level

Framework for K–12 Science Common Core State Standards-

Mathematics

Grade 2 By the end of grade 2 students

should know: “Weather is the

combination of sunlight, wind,

snow or rain, and temperature in a

particular region at a particular

time. People measure these

conditions to describe and record

weather and to notice patterns over

time” (NRC 2012, p. 188).

The CCSS-M have 2nd graders solving

problems involving addition and

subtraction within 100, understanding

place value up to 1,000, recognizing

the need for standard units of measure

of length, representing and interpreting

data, and reasoning with basic shapes

and their attributes.

 

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 Tasks

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 2: Computational Thinking

Grade 5Grade Level Framework for K–12 Science Common Core State Standards-

Mathematics

Grade 5 By the end of 5th grade the expectation

for global climate change is: “If Earth’s

global mean temperature continues to

rise, the lives of humans and organisms

will be affected in many different ways”

(NRC 2012, p. 98).

The CCSS-M has fifth graders writing and

interpreting numerical expressions, analyzing

patterns and relationships, performing

operations with multi-digit whole numbers

and decimals to hundredths, using equivalent

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 real-

world problems, and classifying two-

dimensional figures into categories based on

their properties.

 

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 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 Climatehttp://spark.ucar.edu/video/dog-walking-weather-and-climate

Grade 5 Computational Thinking Tasks

Grade 8Grade Level Framework for K–12 Science Common Core State Standards-

Mathematics

Grade 8 The end of 8th grade expectation for

climate change is to understand that

human activities, such as carbon

dioxide release from burning fuels, are

major factors in global warming.

Reducing the level of climate change

requires an understanding of climate

science, engineering capabilities, and

human behavior (NRC 2012, p. 198).

The CCSS-M 8th grade standards include

awareness of numbers beyond the rational

numbers, work with radicals and integer

exponents, proportional relationships,

ability to analyze and solve linear equations

and systems of linear equations, use linear

functions to model relationships between

quantities, understand congruence and

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.

 

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 Tasks

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

Grade 8 CT-Tasks (Shader, 2013)

http://www.vets.ucar.edu/vg/categories/wildfires.shtml

Grade 12Grade

Level

Framework for K–12 Science Common Core State Standards-

Mathematics

Grade 12 By the end of high school students

should understand that climate change

is slow and difficult to recognize

without studying long-term trends,

such as studying past climate patterns.

Computer simulations are providing a

new lens for researching climate

change, revealing important

discoveries about how the ocean, the

atmosphere, and the biosphere interact

and are modified in response to

human activity (NRC 2012, p. 198).

The CCSS-M high school standards are

by conceptual categories not grade level.

The conceptual categories of Number and

Quantity, Algebra, Functions, Modeling,

Geometry, and Statistics and Probability

specify the mathematics that all students

should study in order to be college and

career ready. Functions are expanded to

include quadratic, exponential, and

trigonometric functions, broadening the

potential models for science.

 

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 Tasks

Grade 12 CT TasksCalculus 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

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 modelE. All of the above

QR Poll

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 Opportunities

QR Water Cycle Macro ScaleThe 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?

QR LP Assessment Exemplars

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 Biodiversity Communities Micro ScaleWhat happens to biomass and energy in a community? As you move up the food web are there more or less organisms?

QR LP Assessment Exemplars

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 Carbon Cycle Landscape ScaleThe 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.

QR LP Assessment Exemplars

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?

Robert MayesTom KoballaGeorgia Southern Universityrmayes@georgiasouthern.edutkoballa@georgiasouthern.edu

Thank You