Reasoning and Sense Making Through the Common Core State Standards for Mathematics (CCCSM) SCCTM Annual Conference Charleston, SC October 28, 2011 Ed Dickey College of Education University of South Carolina Mathematics for the 21st Century Classroom • U.S. Secretary of Education Arne Duncan at NCTM Annual Meeting, 15 April 2011 • “Curricular materials cover so much ground too superficially, failing to provide students with an understanding of the concepts that are essential for success.” • “Tests don’t always measure what’s important, or provide information back to you to help you improve.” Mathematics in the 20th Century • Abbott and Costello Division • 28 divided by 7 is 13 or 13 times 7 is 28 • Video Mindless Application of Algorithms • Learning without understanding • Learning without sensitivity to culture • Learning without consideration of technology • Learning without student engagement Mathematics for the 21st Century Societal Need for • Competitiveness (Knowledge-based global economy) • Fulfillment (sustainability in diverse society) Competitiveness in the 21st Century • Technology • New tools for understanding and visualizing numerical ideas • Web-based, hand-held, ubiquitous • Hans Rosling • “The Joy of Stats” Competitiveness in the 21st Century • Technology • New tools for understanding and visualizing numerical ideas • Computer Algebra Systems • On iPads and Smartphones • Wolfram Alpha Wolfram Alpha Competitiveness in the 21st Century • Computer Algebra Systems • Handheld Calculators • Equation of a line ( y = mx + b) with sliders using TI Nspire Calculator Fulfillment in the 21st Century • Teaching IS a gratifying profession… Fulfillment the 21st Century • Cultural Diversity Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms Cultural Diversity Cultural Diversity – Vedic Multiplication 21 x 13 video Subtraction in Mexico 1 1 963 -369 4 7 5 94 14 NCTM Affiliated Groups • TODOS advocating for equitable, high quality mathematics education for all, in particular, Hispanic/Latino students • www.todos-math.org NCTM Affiliated Groups • Benjamin Banneker Association, advocating for the mathematics education of African-American students • www.bannekermath.org Mathematics for the 21st Century • Common Core State Standards for Mathematics • An opportunity to address… –Competitiveness –Fulfillment –Reasoning –Sense Making Common Core Standards • Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA) • First significant attempt to systematically align K-12 standards across the U.S. • Building on NCTM’s standards documents from 1980, 1989, 2000, 2006, and 2009 • NCTM among groups providing feedback Common Core Standards • Different from most current state standards • Based on most recent research regarding students’ learning trajectories related to mathematics content • Includes detailed description of the way mathematics is learned and used by students (Mathematical Practice) Common Core Development • Initially 48 states and three territories signed on • Final Standards released June 2, 2010, at www.corestandards.org • Adoption required for Race to the Top funds • As of October 1, 2011, 44 states have officially adopted (plus DC, US VI, & N. Mariana I) Common Core Development • Each state adopting the common core either directly or by fully aligning its state standards may do so in accordance with current state timelines for standards adoption not to exceed three (3) years. • States that choose to align their standards to the common core standards accept 100% of the core. States may add additional standards. Benefits for States and Districts • • • • Allows collaborative professional development based on best practices Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC) Enables comparison of policies and achievement across states and districts Creates potential for collaborative groups to get more economical mileage for: – Curriculum development, assessment, and professional development South Carolina • State Board of Education adopted the Common Core for SC on July 14, 2010 • In November 2010, Mick Zais was elected Superintendent of Education and with Governor Haley has chose to not apply for Race to the Top funds • In May 2011 Senator Mike Fair introduced a Proviso in the SC Budget to prohibit the SC Department of Education from spending related to the Common Core • In October 2011, the SC Board and Department of Education continue to move toward implementation CCSSM CCSSM stands for Common Core State Standards for Mathematics Arne Duncan at the NCTM Annual Meeting • “... today’s tests don’t measure higher-order thinking skills or deep understanding of subject material. American students deserve better than the fill-in-the-bubble tests that are now common across states.” • New assessments “… are the ones that you’ve longed for. They will measure critical thinking skills and complex student learning.” Common Core - Domain • Domains are overarching big ideas that connect topics across the grades • Descriptions of the mathematical content to be learned elaborated through clusters and standards Common Core - Standards • Standards are content statements. An example content statement is: “Use properties of operations to generate equivalent expressions.” • Progressions of increasing complexity from grade to grade Common Core - Clusters • May appear in multiple grade levels in the K-8 Common Core. There is increasing development as the grade levels progress • What students should know and be able to do at each grade level • Reflect both mathematical understandings and skills, which are equally important Characteristics • Fewer and more rigorous • Aligned with college and career expectations • Internationally benchmarked • Rigorous content and application of higherorder skills. • Builds on strengths and lessons of current state standards. • Research based Coherence • Articulated progressions of topics and performances that are developmental and connected to other progressions • Conceptual understanding AND procedural skills stressed equally NCTM states coherence also means that instruction, assessment, and curriculum are aligned Focus • Key ideas, understandings, and skills are identified • Deep learning of concepts is emphasized – That is, time is spent on a topic and on learning the topic well. This counters the “mile wide, inch deep” criticism leveled at most current U.S. standards. Math Common Core Resources • http://www.nctm.org/standards/mathcommoncore/ Assessment • Partnership for the Assessment of Readiness for College and Career (PARCC) • Smarter-Balanced Assessment Consortium (SBAC) • South Carolina participates in BOTH PARCC SBAC PARCC or SBAC • Mathematical Practices must be assessed • Assessments will formative and other assessments that address reasoning and sense making • ... and Secretary Duncan’s concern measuring critical thinking skills and complex student learning. CCSSM • Word Cloud Learning Trajectories • Descriptions of children’s thinking and learning in a specific mathematical domain, and • a related conjectured route through a set of instructional tasks designed to engender those mental processes or actions hypothesized to move children through a developmental progression of levels of thinking, • created with the intent of supporting children’s achievement of specific goals in that mathematical domain. • Clements and Sarama, 2004 Learning Trajectories or Progressions • Teachers use ordered set of instructional experiences and tasks • Students think and learn through a developmental progression of levels to reach goal • Van Hiele Levels in Geometry (since 1970s) • Formative Assessments more recently Adding Fractions • As expressed by Hung-Hsi Wu, in Fall 2011, American Educator • In the past… • Memorize steps and mimic process without attention to understanding • Adding whole numbers is “combining”… how is adding of fractions combining things? Adding Fractions Learning Trajectory • In CCSSM, learning trajectories for adding fraction spans grades 3-5 • In grade 3, students learn to think of a fraction as a point on a line. • Unit fractions, like 1/6 and copies of the unit like 5/6 Adding Fractions, Grade 3 • Equivalent fractions 6 2 32 • is equal to or 15 5 3 5 • When each of the 5 segments is divided into 3 equal segments, the unit segment has 3 x 5 =15 equal segments • 2/5 is the same point as 6/15 Adding Fractions, Grade 4 • Adding fractions is joining two parts of same whole • Two segments joined end to end 2 5 2 5 3 3 3 • 2 copies of the red segment followed by 5 copies, so combining 2 + 5 copies, gives you 7 3 Adding Fractions, Grade 5 1 5 8 6 6 1 8 5 • Equivalent Fractions: and 68 8 6 Learning Trajectories • This 5th grade student got over 80% on a preassessment involving multiplication • Is s/he ready for new 5th or 6th grade concepts like multiplication of decimals (e.g. 2.5 x 0.78)? Japan • Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies. Distributive Property • Grade 3 Multiplication • Grade 4 Properties of Operations FOIL • The verb “to FOIL” • Memorize vs Understand • Trajectory for understanding whole number multiplication to algebra: • x(a + b) • x(a + b + c) • (x + y)(a + b) • (x + y)(a + b + c) Binomial Multiplication • • • • • • • • Understand logic (multiply all possible pairs) Don’t memorize rules or tricks: UNDERSTAND Vedic Method Breaks down for 97 x 86 Distributive Vertical Grid Tile CCSSM Mathematical Practices • Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students. • Similar to NCTM’s Mathematical Processes from the Principles and Standards for School Mathematics. • Practices MUST be assessed Mathematical Practices • Expectations that begin with “understand” are especially good opportunities to connect practices to content. • “Students who lack understanding of a topic may rely on procedures too heavily.” • Understanding standards (intersection of content and practice) “are intended to be weighted toward central and generative concepts.. That most merit time, resources, innovative energies, and focus…” 8 CCSSM Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. Geometry Lesson from Japan Make Sense / Critique Reasoning Find the area of the shaded region: 8m 5m 2m 8m 3m 2m 10 m Reasoning 8m 5m 2m 8m 3m 2m 10 m Learning Trajectories • Area is invariant under transformations Formulas • • • • Memorize versus Understand Circumference and Area of a Circle All dependent on understanding π Ratio of Circumference and Diameter Geometer’s Sketchpad Circumference • If C/d = π • C = d π or 2r π Area • Depends on circumference and radius • Area of a circle, how to get the formula..flv 8 CCSSM Mathematical Practices 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Reasoning and Precision • Why is -2 x -3 = 2 x 3? • Inductive approach: • Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and then • (-2)(-3) = 6, again by adding 3 • Is this sound reasoning? Multiplying Negative Numbers • It does not generalize … what is (-5/11)(-4/3)? • How do we KNOW (-1)(-3) is 3 and not INFER? • There is a deductive approach that can prove that (-1)(-1) = 1 and if so, then all other cases follow. Deductive Reasoning • • • • • • (-1) (- 1) + (-1) = (-1) (-1) + 1(-1) = [ (-1) + 1 ] (-1) = [ 0 ] (-1) = 0 If (-1)(-1) + (-1) = 0, then (-1)(-1) = 1 Look for and make use of structure • What is the result when you add… • Even + Even ? • Odd + Odd ? Mod 2 Arithmetic • Modular arithmetic used in encryption codes.. Vi Hart’s Binary Hand Dance Mathematical Structures Vi Hart Blog at vihart.com • Base 2 • Modular arithmetic • Mathematical food • And other engaging ideas about mathematics • … BUT now back to the Common Core High School Conceptual Categories • The big ideas that connect mathematics across high school – such as Functions or Probability and Statistics • A progression of increasing complexity • Description of mathematical content to be learned elaborated through domains, clusters, and standards High School Pathways • The CCSSM Model Pathways are two models that organize the CCSSM into coherent, rigorous courses • The CCSSM Model Pathways are NOT required. The two sequences are examples, not mandates High School Pathways • Four years of mathematics: – One course in each of the first two years – Followed by two options for year three and a variety of relevant courses for year four • Course descriptions – Define what is covered in a course – Are not prescriptions for the curriculum or pedagogy High School Pathways • Pathway A: Consists of two algebra courses and a geometry course, with some data, probability and statistics infused throughout each (traditional) • Pathway B: Typically seen internationally that consists of a sequence of 3 courses each of which treats aspects of algebra, geometry and data, probability, and statistics. NCTM President Michael Shaughnessy • An Opportune Time to Consider Integrated Mathematics March, 2011 • “Students need to see mathematics as an integrated whole, with connections across the content domains... • …the United States will never show well in international comparisons of mathematics performance as long as other countries have an integrated mathematics, and we take a “layer cake” approach. • … we have an unprecedented opportunity… to integrate the content of our secondary mathematics…” Promising, Opportune… but Perfect? Problem areas: • CCSSM has never been field tested • Can the assessments address understanding and measure the Practices? • How to accommodate exceptional learners? • Learning trajectories require careful vertical articulation Not Perfect … Problem areas: • Too little technology particularly in K-8 • No statistics in K-5 • How can this be 21st Century competitive? • Piling on in Grade 6 Ideal (according to Ed) • CCSSM as standards and not mandated curriculum • Give districts choices for implementation (avoiding a lock-step approach) • Assessment includes parts addressed by teachers at the local level (as in Europe) • Reward success, don’t punish non-success • TRUST teachers to do the work we hire them to do Others “Get it” NCTM Position on Teacher Evaluation (Oct 2011): • “Although evidence of student learning can and should be considered in the evaluation of teachers, it should be only one factor among many. • “Comprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice: – Lesson planning – Lesson implementation and instruction – Demonstration of content knowledge and pedagogical content knowledge – Classroom culture – Professionalism” Others “Get it” Calif. Gov. Jerry Brown October 2011 education bill veto: • “… nowhere mentions character or love of learning….It does allude to student excitement and creativity, but does not take these qualities seriously because they can’t be placed in a data stream.” • “teachers are forced to curb their own creativity and engagement with students as they focus on teaching to the test.” Ideal • We don’t buy a dog, then bark for it. • Invest in the best pre- and in-service teacher development…. • Then get out of the way and let teachers do what these very intelligent professionals were educated to do • It works in Finland… • … and it can work in South Carolina! 2012 Institutes High School Reasoning & Sense Making: July 24-26, Los Angeles, California K-8 Algebra Readiness Institute: July 31- August 2, Atlanta, Georgia SCCTM Additional Information • For grades preK-8, a model of implementation can be found in NCTM’s Curriculum Focal Points www.nctm.org/cfp • For the secondary level, please see NCTM’s Focus in High School Mathematics: Reasoning and Sense Making www.nctm.org/FHSM Citations • Brown, Jr., Edmund G (2011. Veto Message Letter for California State Senate Bill 547. Retrieved from http://gov.ca.gov/docs/SB_547_Veto_Message.pdf • CCSSO/NGA. (2010). Common core state standards for mathematics. Washington, DC: Council of Chief State School Officers and the National Governors Association Center for Best Practices. Retrieved from http://corestandards.org/ • Clements, D., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81-89. • Clements, D., & Sarama, J. (2009). Learning and Teaching Early Math: The learning Trajectories Approach.New York: Routledge. Citation • Daro, Phil, Frederic Moser, and Tom Corcoran( (2011) Learning Trajectories in Mathematics. Retrieved from http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%2 0in%20math_ccii%20report.pdf • Dojinsha, Kyoiku. Mathematics Workbook (Grade 1 to 6). Global Education Resources, http://www.globaledresources.com/ • National Council of Teachers of Mathematics. Position Statement on Teacher Evaluation. Retrieved from http://www.nctm.org/about/content.aspx?id=31267 • Shaughnessy, J. Michael. An Opportune Time to Consider Integrated Mathematics. NCTM Summing Up, March 2011. Retrieved from http://www.nctm.org/about/content.aspx?id=28655 • Wu, Hung-Hsi (2011) Phoenix Rising: Bring the Common Core State Mathematics Standards to Life. American Educator, Fall 2011. Retrieved from http://www.aft.org/pdfs/americaneducator/fall2011/Wu.pdf Web Resources • • • • • • • • • Wolfram Alpha: http://www.wolframalpha.com TODOS: http://www.todos-math.org Banneker: http://www.bannekermath.org Common Core: http://www.corestandards.org Math Common Core Resources: http://www.nctm.org/standards/mathcommoncore/ PARRC: http://www.parcconline.org/ SBAC: http://www.k12.wa.us/smarter/ South Carolina Common Core: http://ed.sc.gov/agency/pr/standards-andcurriculum/South_Carolina_Common_Core.cfm Vi Hart Blog: http://vihart.com Videos • Abbott and Costello show 13 x 7 = 28: http://www.youtube.com/watch?v=rLprXHbn19I • Hans Rosling and 200 Countries: http://www.youtube.com/watch?v=jbkSRLYSojo • TI Nspire Sliders for y = mx + b: http://www.youtube.com/watch?v=fiD0vBjLN5E • Vedic Multiplication: http://www.youtube.com/watch?v=46oviWU-sQY • Area of a Circle: http://www.youtube.com/user/mathematicsonline#p/a/u/1/YokK p3pwVFc • Binary Hand Dance: http://www.youtube.com/watch?v=OCYZTg3jahU Thank you… go forth to reason and make sense! [email protected] www.ite.sc.edu/dickey.html

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