COMMON CORE STATE STANDARDS (CCSS): Challenges and Promise for the GeoGebra Community Maurice Burke Department of Mathematical Sciences Montana State University – Bozeman firstname.lastname@example.org
Outline • CCSS : A Quiet Revolution • CCSS: Perspective on Technology • Illusion or Landmark Challenge: A Brief Historical Tour • GeoGebra and Possibilities • Implications for GeoGebra Community
We only blinked our eyes! • NGA, CCSSO, and Achieve launch Common Core State Standards Initiative Spring, 2009 • Forty-Eight States, Two Territories, and District of Columbia Join Common Core Standards Initiative June 1, 2009 • Draft K-12 Common Core State Standards Available for Comment March 10, 2010 • K-12 Common Core State Standards Released for Adoption by States June 2, 2010
The Revolution: Merger Mania? Pre-1980 16000 Independent School Districts 1980’s States begin centralizing curriculum • Improving America’s Schools Act 2002 NCLB Forces State Standards 2010 CCSS Adopted by 48 States????
Intellectual Foundations A Coherent Curriculum:The Case of Mathematics By W. Schmidt, R. Houang, & L. Cogan American Educator, Summer 2002 http://www.aft.org/newspubs/periodicals/ae/summer2002/ “Curricula in the U.S. are a ‘mile wide and an inch deep.’ Here's the research behind the rhetoric.”
Race to the Money (DOE)“The feds are NOT involved.” • Race to the Top Moneys (RTTT) – extra points given to proposals from states which adopted by August 2, 2010. • RTTT is funding proposals to radically alter standardized assessments. Two consortia of states will likely be funded to create common sets of assessments aligned with CCSS.
What’s in them? 100 page document http://www.corestandards.org/ Contents: - Standards for Mathematical Practice • K-8 Standards divided by grade level and then by content domains • High School Standards divided into five content domains: Number and Quantity, Algebra, Functions, Geometry, Statistics and Probability
Grade 3 » Number & Operations—Fractions Develop understanding of fractions as numbers. 1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
High School » Geometry - Congruence Experiment with transformations in the plane 2. 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… Understand congruence in terms of rigid motions 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Eye Catchers • Emphasis on unit fractions and number lines in elementary. • No mention of function in Grades K-7 • Function separated from Algebra in high school and Grade 8 • Primacy of transformational approach to geometry, including proofs • Healthy dose of statistics and probability • Mathematical Practices – A Tall Order
Standards for Mathematical Practice Mathematically proficient students: • Make sense of problems and persevere in solving them. • 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 repeatedreasoning.
Sources for Practices Standards Adding it Up: Helping Children Learn Mathematics. National Research Council, Mathematics Learning Study Committee, 2001. Cuoco, A., Goldenberg, E. P., and Mark,J., “Habits of Mind: An Organizing Principle for a Mathematics Curriculum,”Journal of Mathematical Behavior, 15(4),375-402, 1996.
Standard 5: Use appropriate tools strategically Mathematically proficient students consider the 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 … They are able to use technological tools to explore and deepen their understanding of concepts.
Technology in Content Standards : K-8 • Grades K-6: Not mentioned. • Grade 7: Directly mentioned twice. • Grade 8: Directly mentioned twice.
William McCallum, Math Editor of CCSS • There is such a large variation in opinion that the main guide for using technology in K-8 is provided in the Mathematical Practices Standard 5. • Emphasis: It is a standard! The touchstone, when in doubt. • Technology is not to be downplayed because it is not mentioned everywhere. Avoided the design that just repeated words like “using technology appropriately.”
Technology in Content Standards : High School • Directly mentioned ten times: Complicated algebraic manipulations, complicated graphs, calculations with transcendental function values, finding area under normal curve, and transformations of function graphs and geometric figures. • Emphasized in the introductions to each content domain and significant implied use.
Number and Operation “Calculators, spreadsheets, and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. They can be used to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with non-integer exponents.” P. 58.
Algebra “A spreadsheet or a computer algebra system (CAS) can be used to experimentwith algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.” P. 62.
Function “A graphing utility or a computer algebra system can be used to experiment with properties of these functions and their graphs and to build computational models of functions, including recursively defined functions.” P. 67.
Geometry “Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.” P. 74.
Statistics and Probability “Technology plays an important role in statistics and probability by making it possible to generate plots, regression functions, and correlation coefficients, and to simulate many possible outcomes in a short amount of time.” P. 79.
Illusion or Landmark Challenge: A Brief Historical Tour
Press Release, Japan, April 14, 1970 “Canon Inc., in close collaboration with Texas Instruments Inc. of the United States, has successfully developed the world’s first pocketable, battery-driven, electronic print-out calculator with full large-scale integrated circuitry.” No LCD - Thermopaper
March of Times: Calculator Evolution 1967 First electronic handheld calculator invented. • First production Announced in Tokyo by Canon Business Machines. 1972 Hewlett-Packard introduced the HP35, the first scientific calculator that evaluated the values of transcendental functions such as log 3, sin 3, and so on.
March of Times: Calculator Evolution 1975 Last slide rule is manufactured in US. 1986 Casio introduces the first graphing calculator. 1996 TI introduces the first calculator (TI-92) that contains a CAS (Derive) and dynamic geometry (Cabri). Not linked. 2007 TI introduces first calculator with multiple-linked documents, applications, symbolic spreadsheet and dynamic variables. (TI-Nspire-CAS)
Dynamic Geometry 20-Year Explosion • 1985-86 Geometer Supposer (Schwartz) • 1988-89 Cabri-géomètre (Laborde) • 1991 The Geometer's Sketchpad (Jackiw) • 1995 TI-92 incorporates the alliance between TI and Cabri (Voyage 200 offers Sketchpad) • 2003 Cabri Junior placed on TI83 and TI84 • 2002-06 GeoGebra (Hohenwarter) • 2007 TI-Nspire multiply links Dynamic Geometry with CAS-Spreadsheet-Data Analysis tools. • ????? GeoGebra 4.0
Calculator Access • In 1986, 5% of all 7th graders could use calculators for mathematics tests. • In 1990, 33% of all 8thgraders could use calculators for mathematics tests. • In 1996, 70% of all 8th graders could use calculators for mathematics tests. • In 2007, 75% of all 8th graders could use calculators for mathematics tests.
Percentage of Instructional Classrooms with Internet Access. NCES - 2006
Access vs. Usage • DOE Office of Planning, Evaluation and Policy Development reports only 10% of 4th and 8th graders in classrooms where teachers used technology at least once a week to study mathematics concepts. 2008 DOE “National Educational Technological Trends Study: Local-Level Data Summary”: Very few teachers (< 3%) use technology to support advanced instructional practices such as inquiry and solving real-world problems.
Where We Are Today • Dynamic geometry software used on limited basis partly due to the lack of multiple computers in classrooms. • Teachers use calculators for graphing functions and numerical calculations (no more trig and log tables). • When used, calculators and computers are not used for inquiry but for demonstrations, checking answers or validating theorems given or proven, drill and practice. (CITE, Vol. 9, #1, 2009)
Many Rationales – Research Results • Lack of Imagination. Kaput (1992) • School curriculum organized to meet the needs of paper-and-pencil work rather than instrumented techniques, whose needs are not recognized. Artique (2005) • Tech tools are not part of the canon. They lack institutional status. “…even techniques for managing the graphic window, that would be very useful for students and mathematically meaningful, have no official status in French secondary teaching” Lagrange (2005)
Teacher beliefs about the nature of math and the learning of math marginalize technological approaches. Yoder (2000), Cooney and Wiegel (2003), Kastberg & Leatham (2005) Inadequate professional development on instructional technologies and resources that integrate them into lesson content. Ferrini-Mundy & Breaux (2008) Lack of research proving its value. National Mathematics Advisory Panel (2008)
“…to help understand…complex number algebra.” • Use GeoGebra CAS to experiment with polynomials P(x) and observe the results of substituting complex conjugates a+bi and a-bi for x. • Generalize to a conclusion in the theory of equations.
Some Conclusions • If P(x) is a polynomial with real coefficients then P(a+bi)=conjugate of P(a-bi) • If P(x) is a polynomial with real coefficients and P(a+bi)=r, where r is a real number, then P(a-bi) = r. • Particularly, if a complex number z is a root of a polynomial P(x) with real coefficients, then conj(z) is also a root of P(x).
“…understand how algebraic manipulations behave.” • Explore dividing Polynomial P(x) by linear terms x-a and look for patterns in the quotients and remainders. • Generalize to a major result: P(a) is the remainder when P(x) is divided by (x-a) • Use this generalization to argue that a is a root of P(x) if and only if x-a is a factor of P(x).
“…to build computational models of …recursively defined function.” Devil: “Daniel, I need some money and I know of a fabulous investment opportunity.” Daniel: “What’s that got to do with me?” Devil: “If you put $1000 into the “WIA” I have set up for you, I will double the amount of money in your account by the end of the first day. My commission for that day will be 10% of your initial investment, or $100. It will be deducted as the “Devil’s Due” for that day, leaving you $1900 in your account at the end of the first day! On each successive day, I will double the amount in your account and double the commission to be placed in the Devil’s Due for that day. But you need to promise to stick with my schemes for at least 30 days so that I can build up some capital of my own. You could be a rich man, Daniel. What do ya say?” Daniel:“Hand me a spreadsheet.”