How Will the Common Core State Standards Affect Your Teaching? Jim Rahn www.jamesrahn.com James.email@example.com
Schedule for the Day See Page 1 of the Handout for a description of the activities we will be considering today.
Reason for Standards • Forty-five states, the District of Columbia, four territories, and the Department of Defense Education Activity concur that students deserved clear, focused learning goals wherever they might live, • and that common standards could promote collaboration across state lines in the development of • instructional materials, • assessments, and • approaches to professional development
The Common Core State Standards • provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. • are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers.
The Common Core State Standards • Will help fully prepare • students for the future and • our communities will be best positioned to compete successfully in the global economy
The MATHEMATICS Standards define • what students should understand and • be able to do in their study of mathematics.
Asking a student to understand something means asking a teacher to assess whether the student has understood it. • What does mathematical understanding look like? • One hallmark of mathematical understanding is • the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. • There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. • The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). • Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. • The Standards do provide clear signposts along the way to the goal of college and career readiness for all students.
Two Types of Standards: • Standards for Mathematical Practice • Designed to ensure that the “processes and proficiencies of mathematics” are at the heart of teaching and learning • Standards for Mathematical Content • Designed to define what students should understand and be able to do at each grade level
Standards for Mathematical Practice These Standards • describe varieties of expertise that mathematics educators at all levels should seek to develop in their students • Rest on important “processes and proficiencies” • Describe ways in which students ought to be “engaged” with mathematics
Standards for Mathematical Content • Provide a balanced combination of procedure and understanding • Provide clear signposts along the way to the goal of college and career readiness for all students
Standards for Mathematical Content • K–8 Standards • Grade specific • High School Standards • Listed in conceptual categories that portray a coherent view of mathematics • Number and Quantity • Algebra • Functions • Modeling • Geometry • Statistics and Probability
Standards for Mathematical Content • (+) Represents additional mathematics that students should learn in order to take advanced courses • N-CN (Number and Quantity – Complex Numbers) • 4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. • 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). • 9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Standards for Mathematical Content • (★) Represents specific modeling standards • F-IF (Functions - Interpreting Functions) • 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ • 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★
Standards for Mathematical Content • (★) Represents specific modeling standards • F-IF (Functions - Interpreting Functions) • 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★
Standards for Mathematical Practice • Where did the Mathematical Practice Standards originate? • What are the Mathematics PracticeStandards? • To whom do they apply? • How will student actions, teacher actions, open-ended questions change? • What do lessons look like that encompass the Standards for Mathematical Practice?
Origin of the Mathematics Practice Standards? • The foundation for Mathematical Practice Standards are • the NCTM process standards of: • Problem solving • Reasoning and proof • Communication • Representation • Connections • The strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: • Adaptive reasoning, • Strategic competence • Conceptual understanding • Procedural fluency • Productive disposition
Adaptive Reasoning • the capacity to think logically about the relationships among concepts and situations • Strategic competence • the ability to formulate mathematical problems, represent them, and solve them • Conceptual understanding • comprehension of mathematical concepts, operations and relations
Procedural fluency • skill in carrying out procedures flexibly, accurately, efficiently and appropriately • Productive disposition • habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
Make sense of problems and persevere in solving them. • Mathematically proficient students • Start by explaining to themselves the meaning of a problem and looking for entry points to its solution • Analyze givens, constraints, relationships, and goals • Make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt • Consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution
Make sense of problems and persevere in solving them. • Mathematically proficient students • Monitor and evaluate their progress and change course if necessary • 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. • Check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” • Understand the approaches of others to solving complex problems and identify correspondences between different approaches
Reason abstractly and quantitatively. • Mathematically proficient students • make sense of quantities and their relationships in problem situations • Demonstrates the ability to decontextualize and the ability to contextualize • Demonstrates the ability to create a coherent representation of the problem at hand • considering the units involved • attending to the meaning of quantities not just how to compute them; • knowing and flexibly using different properties of operations and objects.
Construct viable arguments and critique the reasoning of others • Mathematically proficient students • Understand and use stated assumptions, definitions, and previously established results in constructing arguments • Make conjectures and build a logical progression of statements to explore the truth of their conjectures • Able to analyze situations by breaking them into cases, and can recognize and use counterexamples • Justify their conclusions, communicate them to others, and respond to the arguments of others • Reason inductively about data, making plausible arguments that take into account the context from which the data arose
Construct viable arguments and critique the reasoning of others • Mathematically proficient students • Are 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 objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. • Later, students learn to determine domains to which an argument applies. • Listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Model with mathematics. • Mathematically proficient students • 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, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. • By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. • Are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.
Model with mathematics. • Mathematically proficient students • Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They analyze those relationships mathematically to draw conclusions. • 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.
Use appropriate tools strategically. • Mathematically proficient students • Consider the available tools when solving a mathematical problem • 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 • Are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems • Able to use technological tools to explore and deepen their understanding of concepts
Attend to precision. • Mathematically proficient students • Try to communicate precisely to others. • Try to use clear definitions in discussion with others and in their own reasoning. • State the meaning of the symbols they choose, including using the equal sign consistently and appropriately. • Are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. • calculate accurately and efficiently, 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.
Look for and make use of structure. • Mathematically proficient students • look closely to discern a pattern or structure • Young students 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. • recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems
Look for and make use of structure. • Mathematically proficient students • Can step back for an overview and shift perspective. • Can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. • 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.
Look for and express regularity in repeated reasoning. • Mathematically proficient students • 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 they work to solve a problem, they maintain oversight of the process, while attending to the details. • They continually evaluate the reasonableness of their intermediate results.
To Whom Do the Standards for Mathematical PracticeApply? • The 8 Standards for Mathematical Practice apply to • All states who adopt the Common Core State Standards • All schools who are within the states that adopt the standards • All teachers within those schools • These are the Mathematical Practices that must be taking place in all mathematics classrooms if students are to be mathematically proficient and prepared for schooling beyond high school and for the work force.
Changes will be taking place in our mathematics classrooms • We will no longer be able to teach as we have been taught. • The mathematics classroom will need to become an engagement rich classroom • Students will need to be actively engaged in a higher-level of thinking.
How will student actions, teacher actions, open-ended questions change? • Divide into groups. • Select one Mathematical Practice. • Think of ways student actions, teacher actions, and open-ended questions will change in this new classroom
What do lessons look like that encompass the Standards for Mathematical Practice?
Congruence G-CO • Experiment with transformations in the plane • Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Discovering Transformations • Each group will receive the following materials • Communicators • Templates • Pens • Eraser Cloths • Directions for one type of Transformation • Groups will work through the activities to understand how coordinates change when figures are translated, reflected, or rotated. • At the conclusion you will make a short presentation on ideas you now know about each type of transformation and how it might related to our video that kicked off the lesson.
Reflections on Lesson • How many and which of the standards were integrated into the lesson? • Describe specific examples • Standard 1: Make sense of problems and persevere in solving them • Standard 2: Reason abstractly and quantitatively • Standard 3: Construct viable arguments and critique the reasoning of others • Standard 4: Model with mathematics • Standard 5: Use appropriate tools strategically • Standard 6: Attend to precision • Standard 7: Look for and make use of structure • Standard 8: Look for and express regularity in repeated reasoning
The Number System 7.NS • Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. • Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. • Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. • Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. • Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. • Apply properties of operations as strategies to add and subtract rational numbers.
Review of Addition of Integers • The concept of addition was understood by starting with some discs and placing more discs on the desk. Then look for zero pairs to simplify the answer.
Review of Addition of Integers • Using the diagrams below explain what you notice when you only are dealing with sets of red or yellow discs.
Review of Addition of Integers • Using the diagrams below explain what you notice when you are dealing with sets of both red and yellow discs.
Addition of Integers • If yellow discs represent positive numbers and red discs represent negative numbers, translate each problem into a statement about positive and negative integers.
Review of Addition of Integers • If yellow discs represent positive numbers and red discs represent negative numbers, describe the discs that would be used for each problem.
Review of Addition of Integers • If you start with red discs and add red discs, what happens? • If you start with yellow discs and add yellow discs, what happens with the answer?
Review of Addition of Integers • If you start with red discs and add yellow discs, what happens?
Numbers can be Masters of Disguise • Representations that look very different can represent the same number or quantity. All represent -2 or 2 red