Mathematics Connections Common Core State Standards and the NGSS

1. Mathematics ConnectionsCommon Core State Standards and the NGSS Robert Mayes & Thomas Koballa Georgia Southern University

2. 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

3. 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

4. Math centered practices - Duschl • NRC Framework has salient features • Complex adaptive systems thinking • Model-based reasoning • Quantitative reasoning

5. Science Framework and CCSS-M Connections

6. 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.

12. 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

13. 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

14. 3 Mathematics Paradigms • We will present brief discussions of the three mathematics paradigms and provide examples of them • Quantitative Reasoning • Computational Science • Data-intensive Science

15. QR Poll • Indicate which of the following is most prevalent in science classrooms in your state. • Students apply basic arithmetic to calculate and measure • Students interpret graphs and science models to answer science questions • Students create their own scientific models incorporating mathematics • Students use computer simulations and models and engage in data intensive science • None of the above

16. Quantitative Reasoning 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. 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.

17. 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

18. QR Framework

19. QR Cycle

20. QA-QL Exemplar QR STEM Project (Wyoming) – QR in energy and environment context. Professional development text on project coming soon.

21. 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.

22. QI-QM ExemplarScience System Model (Box Model) Quantitative Interpretation: What are the variables in the carbon cycle? Which are flow processes and which are storage areas? What are the attributes of deforestation that make it a viable variable in this model? What are the measures associated with the variables? What is the balance of CO2 entering and leaving the ocean? What other questions would you ask your students? Do they require quantitative accounts? Quantitative Modeling: Have your students research and develop a box model.

23. 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

24. 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

25. 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).

26. Computational Sciencethe 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).

27. 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.

28. Computational Science ExemplarScience in a Box-Wind Roses (Shader, 2013) Questions: You wish to describe and study the wind patterns in your city? 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? http://www.weblakes.com/products/wrplot/index.html

29. CS Poll Which of the following scientific methods do students get the most experience with in your state? Theory/content Experiment/inquiry Computational/modeling Data Analysis/Statistics A & B

30. Data-intensive Science the fourth paradigm for scientific exploration • Theexplosive 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

31. A Toy Problem (Shader, 2012) Heat diffusion on a plate 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. http://ccl.northwestern.edu/netlogo/ NetLogo Heat (unverified) Diffusion Model

32. This is governed by the heat equation:

33. How do we translate this into something computable (just using +,-,*,/) ? We approximate by thinking of the plate as a grid of points

34. 20 Simple Computational Model 12 20 10 A particle’s temperature changes at a rate proportional to the difference between its temperature and the average temperature of its neighbors. 10 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!

35. 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!

36. 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. Same basic idea, but extra dimension is costly!

37. 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

38. 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)

39. 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)

40. 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.

41. 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

42. Grade 2

43. 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?

44. 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

45. Grade 5

46. 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/

47. 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

48. Grade 8

49. 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?

50. Grade 8 CT-Tasks (Shader, 2013) http://www.vets.ucar.edu/vg/categories/wildfires.shtml 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