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Getting to the Core: Integrating Technology into Common Core Standards
The document outlines a workshop on integrating technology into Alabama's Common Core State Standards (ACCSS), emphasizing the importance of aligning educational practices with relevant skills for student success. It details the standards for English language arts, writing, speaking, listening, and mathematics, as well as the collaborative process involved in developing these standards. Additionally, it lists various technology tools to aid educators in implementing these standards in classrooms.
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Getting to the Core: Integrating Technology into Common Core Standards
- 1. Getting to theCore: Integrating Technology into Common Core StandardsDate/Time: July 11, 9-3pmLocation: RAC 311Presenter: Shawndra Johnson The Common Core standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success. We will explore what technology tools and resources can help you begin addressing these new standards in real and relevant ways.
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- 4. What are theACCSS?Alabama’s Common Core State Standards (ACCSS) for English Language Arts & Literacy in History/Social Studies, Science, and Technical Subjects (“the Standards”) are the culmination of an extended, broad-based effort to fulfill the charge issued by the states to create the next generation of K–12 standards in order to help ensure that all students are college and career ready in literacy no later than the end of high school.
- 5. College and CareerReadiness Standards: ReadingKey Ideas and Details 1. Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual evidence when writing or speaking to support conclusions drawn from the text. 2. Determine central ideas or themes of a text and analyze their development; summarize the key supporting details and ideas. 3. Analyze how and why individuals, events, and ideas develop and interact over the course of a text. Craft and Structure 4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. 5. Analyze the structure of texts, including how specific sentences, paragraphs, and larger portions of the text (e.g., a section, chapter, scene, or stanza) relate to each other and the whole. 6. Assess how point of view or purpose shapes the content and style of a text. Integration of Knowledge and Ideas 7. Integrate and evaluate content presented in diverse formats and media, including visually and quantitatively, as well as in words. (Please see ―Research to Build and Present Knowledge‖ in Writing and ―Comprehension and Collaboration‖ in Speaking and Listening for additional standards relevant to gathering, assessing, and applying information from print and digital sources.) 8. Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance and sufficiency of the evidence. 9. Analyze how two or more texts address similar themes or topics in order to build knowledge or to compare the approaches the authors take. Range of Reading and Level of Text Complexity 10. Read and comprehend complex literary and informational texts independently and proficiently.
- 6. College and CareerReadiness Standards: WritingText Types and Purposes1. Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoning and relevant and sufficient evidence. 2. Write informative/explanatory texts to examine and convey complex ideas and information clearly and accurately through the effective selection, organization, and analysis of content. 3. Write narratives to develop real or imagined experiences or events using effective technique, well-chosen details, and well-structured event sequences. Production and Distribution of Writing 4. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. 5. Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach. 6. Use technology, including the Internet, to produce and publish writing and to interact and collaborate with others. Research to Build and Present Knowledge 7. Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the subject under investigation. 8. Gather relevant information from multiple print and digital sources, assess the credibility and accuracy of each source, and integrate the information while avoiding plagiarism. 9. Draw evidence from literary or informational texts to support analysis, reflection, and research. Range of Writing 10. Write routinely over extended time frames (time for research, reflection, and revision) and shorter time frames (a single sitting or a day or two) for a range of tasks, purposes, and audiences.
- 7. College and CareerReadiness Standards: Speaking & ListeningComprehension and Collaboration 1. Prepare for and participate effectively in a range of conversations and collaborations with diverse partners, building on others’ ideas and expressing their own clearly and persuasively. 2. Integrate and evaluate information presented in diverse media and formats, including visually, quantitatively, and orally. 3. Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric. Presentation of Knowledge and Ideas 4. Present information, findings, and supporting evidence such that listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience. 5. Make strategic use of digital media and visual displays of data to express information and enhance understanding of presentations. 6. Adapt speech to a variety of contexts and communicative tasks, demonstrating command of formal English when indicated or appropriate.
- 8. College and CareerReadiness Standards: LanguageConventions of Standard English 1. Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 2. Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. Knowledge of Language 3. Apply knowledge of language to understand how language functions in different contexts, to make effective choices for meaning or style, and to comprehend more fully when reading or listening. Vocabulary Acquisition and Use 4. Determine or clarify the meaning of unknown and multiple-meaning words and phrases by using context clues, analyzing meaningful word parts, and consulting general and specialized reference materials, as appropriate. 5. Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 6. Acquire and use accurately a range of general academic and domain-specific words and phrases sufficient for reading, writing, speaking, and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when encountering an unknown term important to comprehension or expression .
- 9. College and CareerReadiness Standards: Mathematical PrinciplesNumber. Procedural fluency in operations with real numbers and strategic competence in approximation are grounded in an understanding of place value. The rules of arithmetic govern operations and are the foundation of algebra. Expressions. Expressions use symbols and efficient notational conventions about order of operations, fractions and exponents to express verbal descriptions of computations in a compact form. Equations. An equation is a statement that two expressions are equal, which may result from expressing the same quantity in two different ways, or from asking when two different quantities have the same value. Solving an equation means finding the values of the variables in it that make it true. Functions. Functions describe the dependence of one quantity on another. For example, the return on an investment is a function of the interest rate. Because nature and society are full of dependencies, functions are important tools in the construction of mathematical models. Quantity. A quantity is an attribute of an object or phenomenon that can be measured using numbers. Specifying a quantity pairs a number with a unit of measure, such as 2.7 centimeters, 42 questions or 28 miles per gallon. Modeling. Modeling uses mathematics to help us make sense of the real world—to understand quantitative relationships, make predictions, and propose solutions. Shape. Shapes, their attributes, and the relations among them can be analyzed and generalized using the deductive method first developed by Euclid, generating a rich body of theorems from a few axioms. Coordinates. Applying a coordinate system to Euclidean space connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Probability. Probability assesses the likelihood of an event. It allows for the quantification of uncertainty, describing the degree of certainty that an event will happen as a number from 0 through 1. Statistics. We often base decisions or predictions on data. The decisions or predictions would be easy to make if the data always sent a clear signal, but the signal is usually obscured by noise. Statistical analysis aims to account for both
- 10. How were theACCSS developed?Nationally:College and career readiness standards developed in summer 2009Based on the college and career readiness standards, K-12 learning progressions developedMultiple rounds of feedback from states, teachers, researchers, higher education, and the general publicFinal Common Core State Standards released on June 2, 2010
- 11. National CCS UpdatesTheStandards are intended to be a living work: as new and better evidence emerges, the Standards will be revised accordingly. To remain abreast of the latest information related to the Common Core visit:www.corestandards.org
- 12. How were theACCSS developed?State Initiative:May-June 2010: Specialist performed preliminary correlation of CCSS to current ELA and mathematics COS standardsJune 24, 2010: Provided overview and Presented Draft of the Common Core Standards to the State BoardJuly 12-15, 2010: Convened Task Forces for initial meetingAugust 25-27, 2010: Met with Task Forces to revise and make recommendations
- 13. How were theACCSS developed?State Initiative:July 12 –15 Meeting: ▫ Reviewed correlation of CCSS and Alabama Course of Study (Math and ELA) using Achieve Common Core Comparison Tool and noted gaps in correlation.▫ Reviewed Alabama standards not addressed by CCSS and identified initial decisions regarding standards and bullets to be added to CCSS.▫ Wrote first draft of grade or course standards to be added to CCSS.
- 14. How were theACCSS developed?State Initiative:August 25 –27:▫ Received staff and administrative review.▫ Reviewed and revised July draft.▫ Finalized draft of standards for placement on ALSDE Web site for public review and for submission to the State Superintendent as a recommendation for revision.
- 15. How were theACCSS developed?State Initiative:September 23: Update State Board on Review ProcessSeptember 28 – October 21: Post Standards on ALSDE Web sitePublic Meetings ▫Sept 28-Davidson High School, Mobile ▫Oct 5-Spain Park High School, Hoover ▫Oct 12-Carver High School, Montgomery ▫Oct 19-Decatur High School, DecaturOctober 2010: Present to State Superintendent for Recommendation to State BoardOctober 28: Present to State Board with Final RecommendationsNovember 18: Present to State Board for Approval
- 16. When is thisbe implemented?Alabama was one of forty-eight states that participated in the creation of the common core state standards, which build on the best state standards and go further by internationally benchmarking and backmapping them for grades K-12.The Alabama State board of Education voted in November 2010 by a margin of 7-2 to adopted the common core state standards. The standards will go into effect at the beginning of the 2012-13 school year.
- 18. Correlation Between StandardsMathematics ▫96% of Common Core State Standards Matched Alabama’s StandardsEnglish Language Arts ▫ 92% of Common Core State Standards Matched Alabama’s Standards*You can find these document on the Alabama State Department of Education’s website at: http://www.alsde.edu/html/sections/documents.asp?section=54&sort=21&footer=sections
- 19. Structure of theStandardsEnglish Language ArtsThe Standards comprise two main sections: a section with English language arts (ELA) standards for each grade and one for history/social studies, science, and technical subjects. Each grade is divided into strands. K–12 ELA standards have Reading, Writing, Speaking and Listening, and Language strands; the 6–12 history/social studies, science, and technical subjects section focuses on Reading and Writing. Standards for each grade follow the College and Career Readiness (CCR) anchor standards in each strand. Alabama-specific standards are noted by a state of Alabama symbol.
- 20. Students who arecollege and career ready…in reading, writing, speaking and language:demonstrate independencebuild strong content knowledgerespond to the varying demands of audience, task, purpose, and disciplinecomprehend as well as critiquevalue evidenceuse technology and digital media strategically and capablycome to understand other perspectives and cultures
- 21. Key Concepts: ELAReading:Text complexity and the growth of comprehension Writing: Text types, responding to reading, and research Speaking and Listening: Flexible communication and collaboration Language: Conventions, effective use, and vocabulary
- 22. Technology Tools: ReadingIntelThinking Tools:Visual Ranking: http://educate.intel.com/en/ThinkingTools/VisualRankingSeeing Reason: http://educate.intel.com/en/ThinkingTools/SeeingReasonShowing Evidence: http://educate.intel.com/en/ThinkingTools/ShowingEvidence
- 23. Technology Tools: WritingInspiration:InspiredLearning Community: http://www.inspiration.com/community/Inspiration/Kidspiration: http://www.inspiration.com/Webspiration Classroom: http://www.inspiration.com/webspirationclassroomGoogle Docs: http://docs.google.com
- 24. Technology Tools: Speaking/ListeningAvatarswith Voki: http://www.voki.com/Lesson Plans: http://www.voki.com/lesson_plans.phpDigital Storytelling with VoiceThread: http://voicethread.com/Lesson Ideas: http://voicethread.com/about/library/Podcasting on A.L.E.X.:http://alex.state.al.us/staticfiles/podcasts.php
- 25. Technology Tools: LanguageGoogleEarth:Explore: http://www.google.com/earth/index.htmlLessons: http://www.gelessons.com/lessons/Google Lit Trips: http://www.googlelittrips.com/GoogleLit/Home.htmlGoogle Docs: http://docs.google.com
- 26. Structure of theStandardsMathematicsThe organization of this course of study is based upon the eight Standards for Mathematical Practice adopted from the Common Core State Standards (CCSS) and the six Principles for School Mathematics found in the National Council of Teachers of Mathematics’ (NCTM) document, Principles and Standards for School Mathematics (PSSM).K-8 standards is presented by grade levelOrganized into domains that progress over several gradesGrade introductions give 2–4 focal points at each grade levelHigh school standards presented by conceptual theme (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability)
- 27. Students who arecollege and career ready…in mathematics:care about being precise. construct viable arguments. make sense of complex problems and persevere in solving them. look for structure. look for and express regularity in repeated reasoning. make strategic decisions about the use of technological tools.
- 28. Key Concepts: MathMakesense of problems and persevere in solving.Reason abstractly and quantitatively.Construct viable arguments and critique the reasoning of others.Model with mathematics.Use appropriate tools strategically.Attend to precision.Look for and make use of structure.Look for and express regularity in repeated reasoning.
- 29. Technology Tools: Math(Manyof the same tools already reviewed…plus)Videos and Exercises:BrainPop: http://www.brainpop.com/math/KhanAcademy: http://www.khanacademy.org/
- 30. Technology Tools: MathSoftwareTools:InspireData: http://www.inspiration.com/InspireDataSMART Notebook Math Tools: http://smarttech.com/us/Solutions/Education+Solutions/Products+for+education/Software/SMART+Notebook+collaborative+learning+software/SMART+Notebook+Math+Tools+softwareGoogle Docs: http://docs.google.com
- 31. Technology Tools: MathOnlineActivities:Exploratorium: http://www.exploratorium.edu/explore/Illuminations: http://illuminations.nctm.org/ActivitySearch.aspxExploreLearningGizmos: http://www.explorelearning.com/index.cfm?method=cResource.dspBrowseCorrelations&v=s&id=CC
- 32. FREE Common CoreAppFREECommon Core App available for download in the iTunes Store and the Android Market.Access the standards from any mobile device and have synthesized the College and Career Readiness Standards for Language Arts and included the Traditional and Integrated pathways for Math.You can download the app by simply searching for “Common Core Standards” in the iTunes App Store or the Android Market.
Editor's Notes
- #5 Alabama standards are based on work led by the Council of Chief State School Officers (CCSSO) and the National Governors Association (NGA) and build on the foundation laid by states in their decades-long work on crafting high-quality education standards. The Standards also draw on the most important international models as well as research and input from numerous sources, including state departments of education, scholars, assessment developers, professional organizations, educators from kindergarten through college, and parents, students, other members of the public, and review and recommendation from a Task Force from past course of study committees and State Board of Education appointees. In their design and content, refined through successive drafts and numerous rounds of feedback, the Standards represent a synthesis of the best elements of standards-related work to date and an important advance over that previous work. As specified by CCSSO and NGA, the Standards are (1) research and evidence based, (2) aligned with college and work expectations, (3) rigorous, and (4) internationally benchmarked. A particular standard was included in the document only when the best available evidence indicated that its mastery was essential for college and career readiness in a twenty-first-century, globally competitive society. The Standards are an extension of a prior initiative led by CCSSO and NGA to develop College and Career Readiness (CCR) standards in reading, writing, speaking, listening, and language as well as in mathematics. The CCR Reading, Writing, and Speaking and Listening Standards, released in draft form in September 2009, serve, in revised form, as the backbone for the present document. Grade-specific K–12 standards in reading, writing, speaking, listening, and language translate the broad (and, for the earliest grades, seemingly distant) aims of the CCR standards into age- and attainment-appropriate terms. The Standards set requirements not only for English language arts (ELA) but also for literacy in history/social studies, science, and technical subjects. Just as students must learn to read, write, speak, listen, and use language effectively in a variety of content areas, so too must the Standards specify the literacy skills and understandings required for college and career readiness in multiple disciplines. Literacy standards for Grade 6 and above are predicated on teachers of ELA, history/social studies, science, and technical subjects using their content area expertise to help students meet the particular challenges of reading, writing, speaking, listening, and language in their respective fields. It is important to note that the 6–12 literacy standards in history/social studies, science, and technical subjects are not meant to replace content standards in those areas but rather to supplement them. Alabama currently incorporates many of these standards into the standards for those subjects; however, future Course of Study Committees may consider adopting them as content area literacy standards. As a natural outgrowth of meeting the charge to define college and career readiness, the Standards also lay out a vision of what it means to be a literate person in the twenty-first century. Indeed, the skills and understandings students are expected to demonstrate have wide applicability outside the classroom or workplace. Students who meet the Standards readily undertake the close, attentive 3 reading that is at the heart of understanding and enjoying complex works of literature. They habitually perform the critical reading necessary to pick carefully through the staggering amount of information available today in print and digitally. They actively seek the wide, deep, and thoughtful engagement with high-quality literary and informational texts that builds knowledge, enlarges experience, and broadens worldviews. They reflexively demonstrate the cogent reasoning and use of evidence that is essential to both private deliberation and responsible citizenship in a democratic republic. In short, students who meet the Standards develop the skills in reading, writing, speaking, and listening that are the foundation for any creative and purposeful expression in language.
- #10 Ten Mathematical Principles form the backbone of these standards. Each principle is accompanied by an explanation that describes the coherent view students are expected to have of a specific area of mathematics. With this coherent view, students will be better able to learn more mathematics and use the mathematics they know. The principles pull together topics previously studied and target topics yet to be learned in post-secondary programs. Each principle consists of a statement of a Coherent Understanding of the principle, together with Core Concepts, Core Skills, and Explanatory Problems that exemplify and delimit the range of tasks students should be able to do. These standards, like vectors, specify direction and distance for students to be ready for college and careers: 1. Direction—The Coherent Understanding The Coherent Understandings attempt to communicate the mathematical coherence of the knowledge students should take into college and careers. They are intended to tell teachers, ‘This is how your students should see the mathematics in this area in order to aim them towards mastering it.’ 2. Distance—The Concepts, Skills and Explanatory Problems Collectively, these statements and sets of problems define and clarify the level of expertise students should reach if they are to be prepared for success in college and career. They are a. statements of concepts students must know and actions students must be able to take using the mathematics; and b. examples of the problems and other assignments they must be able to complete. In addition to the Mathematical Principles, the standards also contain a set of Mathematical Practices that are key to using mathematics in the workplace, in further education and in a 21st Century democracy. Students who care about being precise, who look for hidden structure and note regularity in repeated reasoning, who make sense of complex problems and persevere in solving them, who construct viable arguments and use technology intelligently are more likely to be able to apply the knowledge they have attained in school to new situations. These mathematical practices are described and tied to examples. Taken together, the explanations of the mathematical principles, the associated concepts and skills and the mathematical practices form the College and Career Readiness Standards for Mathematics.
- #22 The Reading standards place equal emphasis on the sophistication of what students read and the skill with which they read. Anchor standard 10 defines a grade-by-grade ―staircase‖ of increasing text complexity that rises from beginning reading to the college and career readiness level. Whatever they are reading, students must also show a steadily growing ability to discern more from and make fuller use of text, including making an increasing number of connections among ideas and between texts, considering a wider range of textual evidence, and becoming more sensitive to inconsistencies, ambiguities, and poor reasoning in texts. Writing: Text types, responding to reading, and research The Standards acknowledge the fact that whereas some writing skills, such as the ability to plan, revise, edit, and publish, are applicable to many types of writing, other skills are more properly defined in terms of specific writing types: arguments, informative/explanatory texts, and narratives. Anchor standard 9 stresses the importance of the writing-reading connection by requiring students to draw upon and write about evidence from literary and informational texts. Because of the centrality 5 of writing to most forms of inquiry, research standards are prominently included in this strand, though skills important to research are infused throughout the document. Speaking and Listening: Flexible communication and collaboration Including but not limited to skills necessary for formal presentations, the Speaking and Listening standards require students to develop a range of broadly useful oral communication and interpersonal skills. Students must learn to work together, express and listen carefully to ideas, integrate information from oral, visual, quantitative, and media sources, evaluate what they hear, use media and visual displays strategically to help achieve communicative purposes, and adapt speech to context and task. Language: Conventions, effective use, and vocabulary The Language standards include the essential ―rules‖ of standard written and spoken English, but they also approach language as a matter of craft and informed choice among alternatives. The vocabulary standards focus on understanding words and phrases, their relationships, and their nuances and on acquiring new vocabulary, particularly general academic and domain-specific words and phrases.
- #28 1. They care about being precise. Mathematically proficient students organize their own ideas in a way that can be communicated precisely to others, and they analyze and evaluate others’ mathematical thinking and strategies based on the assumptions made. They clarify definitions. They state the meaning of the symbols they choose, are careful about specifying units of measure and labeling axes, and express their answers with an appropriate degree of precision. They would never say “let v be speed and let t be elapsed time” but rather “let v be the speed in meters per second and let t be the elapsed time in seconds.” They recognize that when someone says the population of the United States in June 2008 was 304,059,724, the last few digits are meaningless. 2. They construct viable arguments. Mathematically proficient students understand and use stated assumptions, definitions and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They break things down into cases and can recognize and use counterexamples. They use logic to justify their conclusions, communicate them to others and respond to the arguments of others. 3. They make sense of complex problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for the entry points to its solution. They consider analogous problems, try special cases and work on simpler forms. They evaluate their progress and change course if necessary. They try putting algebraic expressions into different forms or try changing the viewing window on their calculator to get the information they need. They look for correspondences between equations, verbal descriptions, tables, and graphs. They draw diagrams of relationships, graph data, search for regularity and trends, and construct mathematical models. They check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” 4. They look for structure. Mathematically proficient students look closely to discern a pattern or stand back to get an overview or shift their perspective, and they transfer fluently between these points of view. For example, in 𝑥2+5𝑥+6 they can see the 5 as 2+3 and the 6 as 2×3 They recognize the significance of an existing line in a geometric figure or add an auxiliary line to make the solution of a problem clear. They also can step back and see complicated things, such as some algebraic expressions, as single objects that they can manipulate. For example, they might determine that the value of 5−3 𝑥−𝑦 2 is at most 5 because 𝑥−𝑦 2 is non-negative.d5. They look for and express regularity in repeated reasoning. Mathematically proficient students pay attention to repeated calculations as they are carrying them out, and look both for general algorithms and for shortcuts. For example, by paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, they might abstract an equation of the line of the form 𝑦−2𝑥−1=3. By noticing the telescoping in the expansions of 𝑥−1 𝑥+1 , 𝑥−1 𝑥2+𝑥+1 , and 𝑥−1 𝑥3+𝑥2+𝑥+1 , they might derive the general formula for the sum of a geometric series. As they work through the solution to a problem, they maintain oversight over the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.d6. They make strategic decisions about the use of technological tools. Mathematically proficient students consider the available tools when solving a mathematical problem, whether pencil and paper, graphing calculators, spreadsheets, dynamic geometry or statistical software. They are familiar enough with all of these tools to make sound decisions about when each might be helpful. They use mathematical understanding, attention to levels of precision and estimation to provide realistic levels of approximation and to detect possible errors.
- #29 Mathematically proficient students: 1. Make sense of problems and persevere in solving them. These students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. These students consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to obtain the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, ―Does this make sense?‖ They can understand the approaches of others to solve complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships. One is the ability to decontextualize, to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents. The second is the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 2010 Alabama Course of Study: Mathematics 7 3. Construct viable arguments and critique the reasoning of others. These students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. These students justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments; distinguish correct logic or reasoning from that which is flawed; and, if there is a flaw in an argument, explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until the middle or upper grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4. Model with mathematics. These students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, students might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, students might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas and can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5. Use appropriate tools strategically. Mathematically proficient students consider available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a Web site, and use these to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 2010 Alabama Course of Study: Mathematics 8 6. Attend to precision. These students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Mathematically proficient students are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. These students also can pause and reflect for an overview and shift perspective. They can observe the complexities of mathematics, such as some algebraic expressions as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8. Look for and express regularity in repeated reasoning. They notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As students work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details and continually evaluate the reasonableness of their intermediate results.