COMMON CORE STATE STANDARDS (CCSS):

Challenges and Promise for the GeoGebra Community

Maurice BurkeDepartment of Mathematical SciencesMontana State University – Bozeman

burke@math.montana.edu

Outline

1. CCSS : A Quiet Revolution2. CCSS: Perspective on Technology3. Illusion or Landmark Challenge: A Brief

Historical Tour4. GeoGebra and Possibilities5. Implications for GeoGebra Community

CCSS : A Quiet Revolution

The Instigators

Predictable Reaction:Hey! What’s Up With This??

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 Districts1980’s States begin centralizing curriculum1994 Improving America’s Schools Act2002 NCLB Forces State Standards2010 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.”

http://www.mathcurriculumcenter.org/PDFS/ExecutiveSummary.pdf

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.

Today’s Map

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 plane2. 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 motions8. 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 PracticeMathematically proficient students:1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning

of others.4. Model with mathematics.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.

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.

CCSS: Perspective on Technology

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 experiment with 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

Transportation Analogy: Detroit 1906

Detroit 1920

March of Time: Calculator Evolution

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.

1970 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 8th graders 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

1994 1996 1998 2000 2001 2003 2005

All public schools 3 14 51 77 87 93 94

Access vs. Usage2007 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)

GeoGebra and Possibilities

“…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.”

Conclusion of Devil and Dan

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

“…to investigate geometric phenomena”

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?

“…to generate plots, regression functions, and…”

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

GeoGebra the Tool

• 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.

The Political Economy

• 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.

GeoGebra and CCSS Content

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.

GeoGebra and C&I Development

• 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.

GeoGebra and PreService Teachers• 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.

Bibliography

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• 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.

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• 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.

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