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COMMON CORE STATE STANDARDS (CCSS): Challenges and Promise for the GeoGebra Community

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COMMON CORE STATE STANDARDS (CCSS): Challenges and Promise for the GeoGebra Community

Maurice Burke

Department of Mathematical Sciences

Montana State University – Bozeman

- CCSS : A Quiet Revolution
- CCSS: Perspective on Technology
- Illusion or Landmark Challenge: A Brief Historical Tour
- GeoGebra and Possibilities
- Implications for GeoGebra Community

- 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

Pre-1980 16000 Independent School Districts

1980’sStates begin centralizing curriculum

- Improving America’s Schools Act
2002NCLB Forces State Standards

2010CCSS Adopted by 48 States????

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 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.

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

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.

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.

- 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

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.

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.

CCSS: Perspective on Technology

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.

- Grades K-6: Not mentioned.
- Grade 7: Directly mentioned twice.
- Grade 8: Directly mentioned twice.

- 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.”

- 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.

“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.

“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.

“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.

“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.

“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

“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

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.

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)

- 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

- 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

- 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.

- 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)

- 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)

GeoGebra and Possibilities

- 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.

- 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).

- 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).

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.”

The pattern on the spreadsheet suggests a formula for the amount in Daniel’s WIA at the end of day n:

WIA(n)=2^(n-1)*(2-.1n)(1000)

If m = earnings multiplier, p = percent of initial investment, and s = initial investment, then

WIA(n)=m^(n-1)*(m-p*n)*s

Zoom imitates similarity transformation.

Which theorems about angle measure are obvious to the naked eye?

Use Zoom to see.

How can you make this rigorous?

Do regressions really give us best fits by the least squares criterion? Explore various functions to see if you can do better than the exponential regression in GeoGebra. Use the following data set:

{(1,1) ,(2,7), (2,4), (3,2), (5,6)}

Explore other regression options.

Implications for GeoGebra Community

- Impressive progress in developing a multiply linked environment that incorporates all of the technologies mentioned in CCSS. But there is a still much to do to achieve the multipurpose tool that is as user friendly as the GeoGebra geometry component.
- The symbolic spreadsheet could become a standard tool in educational environments. This could give George a headache.

- If Standard 5 is taken seriously, schools will likely seek seek a unified technology package that is affordable, stable, regularly updated, and easy for teachers to use.
- PD will have to be provided in the use of the technology to achieve CCSS goals, including exposure to curriculum materials that clearly benefit from the technology.

The emphasis on transformations at the high school level and number lines at the elementary level can be supported by GeoGebra. But this may prompt the need to develop new tools within the existing GeoGebra structure such as a number line tool, and tools to make the study of transformations easier to manage when using GeoGebra.

- There is a need to develop a wide range of lessons that illustrate mathematical experimentation and exploration with technology.
- There is a need for research into teacher practices that affect student outcomes in contexts where experimentation and exploration are frequent.

- Research has consistently pointed to the teacher as the critical determinant of the success or failure of technological change in the classroom. What TPACK is needed for success and how does this TPACK develop? GeoGebra can play a crucial role here by allowing teachers to experiment without large investments.
- GeoGebra community must develop strategies for training future teachers and reaching out to colleges and universities.

It is not about the economy, stupid, it’s about people.Investing money into technology is one thing, but it is not enough to empower people to be the best they can be in their professions. This requires time. Thank you, Markus, Michael, Yves, and George and ……..! Thank you for the immense time you have devoted to empowering us.

- Artique, M. (2005). The integration of Symbolic Calculators into secondary education: some lessons from didactical engineering. In D. Buin, K. Ruthven, & L. Trouche (Eds.) The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument (pp. 231-294). Dordrecht: Kluwer Academic.
- Association for Mathematics Teacher Educators (2009). Mathematics Teacher TPACK Standards and Indicators. http://www.amte.net
- Blume, G. W., & Heid, M. K. (Eds.) (2008). Research on Technology and the Teaching and Learning of Mathematics, Volumes 1 &2. Charlotte, NC: Information Age Publishing.
- Common Core State Standards Initiative (2010). http://www.corestandards.org
- Cooney, T. & Wiegel, H. (2003). Examining the mathematics in mathematics teacher education. In Bishop, A., Cements, M.A., Keitel, C., Kilpatrick, J., Leung, F. (Eds.), The Second International Handbook of Mathematics Education, Part Two (pp. 795-822). Dordrecht: Kluwer Academic.
- Drijvers, P. (2000). Students encountering obstacles using CAS. International Journal of Computers for Mathematical Learning, 5(3), p.189-209.
- Ferrini-Mundy, J., & Breaux, G.A. (2008). Perspectives on research, ploicy, and the use of technology in mathematics teaching and learning in the United States. In Blume, G. W., & Heid, M. K. (Eds.). Research on Technology and the Teaching and Learning of Mathematics, Volume 2 (pp. 427-448). Charlotte, NC: Information Age Publishing.
- International Society for Technology in Education (2008). National Educational Technology Standards and Performance Indicators for Teachers. Eugene, OR.
- Kaput, J. (1992). Technology and mathematics education. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-556). New York: MacMillan Publishing.
- Kastberg, S., & Leatham, K. (2005). Research on graphing calculators at the secondary level: Implications for mathematics teacher education. Contemporary Issues in Technology and Teacher Education 5(1). Retrieved from http://www.citejournal.org/vol5/iss1/mathematics/article1.cfm .
- Lagrange, J. B. (2005). Using symbolic calculators to study mathematics. In D. Buin, K. Ruthven, & L. Trouche (Eds.) The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument (pp. 113-135). Dordrecht: Kluwer Academic.

- Manoucherhri, A. (1999). Computers and school mathematics reform: Implications for teacher education. Journal of Computers in Mathematics and Science Teaching, 18(1), 31 – 48
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