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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 [email protected] [email protected]