COMMON CORE STATE STANDARDS
(CCSS):
Challenges and Promise for the
GeoGebra Community
Maurice Burke
Department of Mathematical Sciences
Montana State University – Bozeman
burke@math.montana.edu
Outline
1. CCSS : A Quiet Revolution
2. CCSS: Perspective on Technology
3. Illusion or Landmark Challenge: A Brief
Historical Tour
4. GeoGebra and Possibilities
5. 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
1980’s
1994
2002
2010
16000 Independent School Districts
States begin centralizing curriculum
Improving America’s Schools Act
NCLB Forces State Standards
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/s
ummer2002/
“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/Executive
Summary.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 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:
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.
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
No LCD - Thermopaper
“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.”
March of Times: Calculator Evolution
1967
1970
1972
First electronic handheld calculator
invented.
First production Announced in Tokyo by
Canon Business Machines.
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
1986
1996
2007
Last slide rule is manufactured in US.
Casio introduces the first graphing
calculator.
TI introduces the first calculator (TI-92) that
contains a CAS (Derive) and dynamic
geometry (Cabri). Not linked.
TI introduces first calculator with multiplelinked documents, applications, symbolic
spreadsheet and dynamic variables. (TINspire-CAS)
Dynamic Geometry 20-Year Explosion
•
•
•
•
1985-86
1988-89
1991
1995
Geometer Supposer (Schwartz)
Cabri-géomètre (Laborde)
The Geometer's Sketchpad (Jackiw)
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
All public
schools
1994
1996
1998
2000
2001
2003
2005
3
14
51
77
87
93
94
Access vs. Usage
2007 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. FerriniMundy & 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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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.
a
•
•
•
•
•
•
•
•
•
•
Manoucherhri, A. (1999). Computers and school mathematics reform: Implications for teacher education.
Journal of Computers in Mathematics and Science Teaching, 18(1), 31 – 48
Mishra, P., & Koehler, M.J. (2006). Technological pedagogical Content Knowledge: A framework for teacher
knowledge. Teachers College Record. 108(6) 1017-1054.
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics.
Reston, VA. http://www.nctm.org
Niess, M. L. (2005). Preparing teachers to teach science and mathematics with technology: Developing a
technology pedagogical content knowledge. Teaching and Teacher Education, 21, 509-523.
Niess, M.L. (2008). Guiding pre-service teachers in developing TPCK. In Handbook of Technological
Pedagogical Content Knowledge (TPCK) for Educators. (Eds) Routledge, New York, pp. 223-250
Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper S. R., Johnston, C., Browning, C., Özgün-Koca,
S. A., & Kersaint, G. (2009). Mathematics teacher TPACK standards and development model. Contemporary
Issues in Technology and Teacher Education [Online serial], 9(1). Retrieved from
http://www.citejournal.org/vol9/iss1/mathematics/article1.cfm
Niess, M.L., Sadri, P., & Lee, K. (April, 2007). Dynamic spreadsheets as learning technology tools:
Developing teachers’ technology pedagogical content knowledge (TPCK). Paper presented at the meeting
of the AERA Annual Conference, Chicago, IL.
U.S. Department of Education (2007). Office of Planning, Evaluations and Policy Development; Policy and
Program Studies Services, State Strategies and Practices for Educational Technology: Volume IISupporting mathematics Instruction with Educational Technology, Washington, D. C.
U.S. Department of Education (2008). Office of Planning, Evaluation and Policy Development, Policy and
Program studies Service, National Educational Technological Trends Study: Local-level Data Summary,
Washington D.C.
Yoder, A.J. (October, 2000). The relationship between graphing calculator use and teacher’s beliefs about
learning algebra. Paper presented at the Mid-Western Educational Research Association, Chicago, IL.