It is with these thing in mind that this interactive page will try to supplement the enrichment material in "Mathematics and The Human Condition". The additions will include recent developments and imaginative ideas from books and periodicals, along with comments, questions, and tips from teachers on how to bring the mathematics curriculum to life. Please send materials and reference them to pages in the book to:
firstname.lastname@example.org and I will try to scan them and include them in the web-site. Thanks, Vic Selby
RECENT ADDITION TO ENRICHMENT TOPICS:
ASILOMAR CONF. 2016, Victor Selby (enrichmath.com)
From the New Yorker (DEPT. OF MYTH-BUSTING JUNE 27, 2016 ISSUE)
Andrew Hacker, an outspoken critic of mandatory algebra education, is asked to defend his contentions at the National Museum of Mathematics.
By Rebecca Mead
A few years ago, Andrew Hacker, the political scientist, wrote an Op-Ed for the Times titled “Is Algebra Necessary?,” in which he proposed eliminating mandatory high-school math. “Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves,” he wrote. Recently, the National Museum of Mathematics, on East Twenty-sixth Street, invited Hacker to defend his assertions in a public debate with James Tanton, the mathematician at large of the Mathematical Association of America, and an educator and consultant.
Higher math, Hacker conceded, is one of humanity’s greatest accomplishments. “I would really love for everybody to appreciate mathematics—its glories, its goals,” he said. “But this isn’t being done by making people slog through polynomials year after year.”
When Tanton took the floor, he paced, ted-talk-like, and spoke in a rapid-fire Australian accent. “I have never used the quadratic formula in my personal life,” he acknowledged. “I don’t think I have ever used it in my research life. But learning the formula wasn’t the point. It was the story of quadratics. And, from that story, I know I can nut my way from most any problem to do with that subject.” Algebra II, he said, need not be too high a bar to set for most students. “The issue is: How do we teach the subject? Do we teach with beauty and joy and wonder and humanness?”
Are you interested in using the history of mathematics in your teaching but are disappointed by "sidebar" approaches that treat the history as a superficial add-on to the main discussions? Have you ever wondered whether the topics you teach might be more effectively considered by your students if you could connect them more clearly to the ideas that motivated the mathematicians who initially developed them?
Integrating six great scientific models into common core
Session Description *
This session will show the power and reality of mathematics as the language of science. These awe inspiring moments, from the use of proportions by Eratosthenes to calculate the size of the earth to Einstein's time dilation formula, from Zeno to Newton's idea of limits, will show how math is a crucial symbol system. Topics will include the first calculations of the distance to the moon and the speed of light. Participants will receive a copy of my book "Mathematics And The Human Condition".
The four great symbol systems (language, art, music and mathematics) and the ability to climb into some of the greatest minds of human history.
Review of Pi ( one idea that is lost to most people). Answering the question of Pi by placing the diameter (one unit) of a circle around the circumference. The idea of transcendental numbers.
Eratosthenes (240 B.C.E.) calculates the size of the earth. Aristarchus (235 BCE) calculates the distance from Earth to Moon. The speed of light calculated by observing the moons of Jupiter (Olaf Roemer 1676).
4) The idea of limits. Zeno’s paradox. First start with .999… being
exactly = 1. Cantor’s orders of infinity (pp. 66-72)
5) Galileo to Newton: d=16t^2 to s=32t. (p. 123-124)
6) Proof of the Pythagorean Theorem. (P. 45)
7) Einstein’s theory of Special Relativity. (pp. 126-130)
8) Games Theory, John Nash, and “The Prisoners Dilemma”.
2014 CMC North Asilomar Conference
Session Title: Mathematics: So beautiful it can't be expressed by words.
Description: Discuss the 4 great symbol systems. As with language, art, and music, we can "put on a show" with mathematics that enhances motivation for all students. From Pythagoras to the equations of the conic sections, connect the great ideas that have built civilizations. Use the nature of space to understand the "poetry" of Bucky Fuller and the applications of Design Science in our developing world. Attendees will receive a copy of my book "Mathematics and The Human Condition".
STEAM principle: Curiosity – the kind of inquisitive thinking that leads to exploring and investigating is another goal of STEM lessons. Lessons should be designed to evoke and nurture curiosity in kids, and to develop a thirst for knowledge that drives learning.
There has been a lot of talk over the last couple of years about the importance of STEM education. Science, Technology, Engineering, and Mathematics play a critical role in our country’s ability to compete in the global market. But there is an essential part of this acronym that is missing: Art. It is the spark that breathes life into STEM, and without it, innovation is dead. There would be no inventions, no discoveries, no advances in technology. Without creativity and analyzation, Benjamin Franklin would not have considered flying a kite in a storm, Steve Jobs would have just been another computer sales guy, and Ellen Ochoa would have been some girl who likes stars.
Vol. 108, No. 5 • December 2014/January 2015 | MATHEMATICS TEACHER 377
"The adoption of the Common Core State Standards (CCSS) by most of the states is a bold initiative, exciting and at thesame time overwhelming. It is asking teachers to teach in ways that cover “the sequential or hierarchical nature of the disciplinary content from which the subject matter derives” in such a way that their teaching “reflect[s] not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline” (CCSSI 2010). What a beautiful picture this paints for classrooms across the country. And we have much work to do before we get there. In this article, we use the mathematics topic of circles and the lines that intersect them to introduce the idea of looking at the single mathematical idea of relationships—in this case, between angles and arcs—across a group of problems. We introduce the mathematics that underlies these relationships, beginning with the questions we ask, the tools we use, and the language we use in that process to talk about the mathematics.
Tool 1: An inscribed angle on a circle has half the measure of the arc it intercepts.
Tool 2: Any exterior angle of a triangle is congruent to the sum of the two remote interior angles (a useful corollary to the sum of the interior angles of a triangle is 180 degrees).
AND SO IT GOES: PROVING MORE “TOOLS’”ABOUT CIRCLES WITH INTERSECTING LINES AND SEGMENTS.
Common Core is supposed to give students more depth and greater understanding, but where are the “awe inspiring” moments in the classroom?
The following is “word for word” taken from the Common Core: The use of "Transformational Geometry" takes away some of the most important ideas, such as the use of Euclid's Axioms to prove the basic theorems. This ommision takes away the important idea that mathematics starts with deductive logic and science with observation and inductive logic.
Prove geometric theorems. [Focus on validity of underlying reasoning while using variety of ways of writing proofs.]
725 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°.
The medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
“Any intelligent fool can make things bigger, more complex and more violent. It takes a touch of genius–and a lot of courage– to move the opposite direction.” -Albert Einstein
Where are the “Axioms”? How can students learn the difference between using observations that are the basis of the Scientific Method and the use of axioms to derive the theorems of mathematics.
A basic understanding of the difference between “doing science” and “doing mathematics” should be one of the foundations of any modern education. As such, ideas such as “limits” show the beauty and imagination of mathematics, while observations such as the wandering planets pose the questions that give rise to scientific theory.
From the CCSS Algebra 2 standards and examples:
Example. The Juice Can. Suppose we wanted to know the minimal surface area of a cylindrical can of a fixed volume. Here, we consider the surface area in units cm2, the radius in units cm, and the volume to be fixed at 355 ml = 355 cm3. One can find the surface area of this can as a function of the radius: ????(????)=2(355)????+2????????^2.
Note that the formulas above and below are given with no attempt to allow students to derive simple cases, such as using a compound interest example and taking limits to derive e in the fiorst place.
Students in Algebra II develop models for more complex or sophisticated situations 224 than in previous courses, due to the expansion of the types of functions available to 225 them (F-BF.1). Modeling contexts provide a natural place for students to start building 226 functions with simpler functions as components. Situations involving cooling or heating 227 involve functions that approach a limiting value according to a decaying exponential 228 function. Thus, if the ambient room temperature is 70° and a cup of tea is made with 229 boiling water at a temperature of 212°, a student can express the function describing the 230 temperature as a function of time by using the constant function ????(????)=70 to represent 231 the ambient room temperature and the exponentially decaying function ????(????)=142????^−???????? 232 to represent the decaying difference between the temperature of the tea and the 233 temperature of the room, leading to a function of the form: 234 ????(????)=70+142????^−????????.
Students might determine the constant ???? experimentally. (MP.4, MP.5) 235
Example (Adapted from Illustrative Mathematics 2013). Population Growth. The approximate United States Population measured each decade starting in 1790 up through 1940 can be modeled by the function ????(????)=(3,900,000x200,000,000)????^0.31????/200,000,000+3,900,000(????^0.31????−1),
where ???? represents decades after 1790. Such models are important for planning infrastructure and the expansion of urban areas, and historically accurate long-term models have been difficult to derive.
Wrong Answer: The case against Algebra II — Harpers
Posted on August 23, 2013 by Thom Prentice
Underwood Dudley, a number theorist who taught for many years at DePauw University, is another longtime critic of math requirements. He’s against them because he loves the subject. As he wrote in The American Mathematical Monthly in 1987:
Mathematics is so useful that there could be no civilization without it, and it is so beautiful that some theorems and their proofs — those which cause us to gasp, or to laugh out loud with delight — should be hanging in museums. And yet: “The vast majority of the human race, and the vast majority of the college-educated human race never need any mathematics beyond arithmetic to survive successfully.”
We must stop telling students lies, Dudley maintains, to the discomfiture of some of his colleagues. “We cannot justify teaching mathematics to 18-year-olds by asserting that they will find it useful,” he wrote. “We cannot claim that we are presenting beauty, either. We are, of course, but what percentage of our students can see it, however dimly?”
This session will show a few ways to empower students and give depth to what they are learning. We can move in that direction!
LET’S DEFINE REALITY AND GIVE HOPE TO THE IDEA THAT ALL STUDENTS CAN TUNE INTO MATHEMATICS BY TELLING THE STORIES THAT SHOW THE BEAUTY AND POWER OF SOME OF THE WORLD’S GREATEST THINKERS.
LET’S SUPPLEMENT THE COMMON CORE BY CREATING CONTEXT, CONNECTING IDEAS, AND ALLOWING TIME FOR STUDENTS TO ANALYZE AND WRITE ABOUT THE ROLE OF MATHEMATICS AS THE PROPELLER OF CULTURAL EVOLUTION.
STEM education seems to be focused on being fluent in the use of technology, and while that is an important skill, there seems to be an ongoing lack of emphasizing the power of mathematics as the language of science. Handing students kits to build robots or making solar cooked marshmallows is fun but misses the main point; that cultural evolution is a vital process, and understanding the ideas that fueled the modern age empowers students and gives great incentive to learn both math and science.
Number sense and fluency is really important, and I don’t think you can get that through an app or googling that or whatever because we’re challenging your mind and what your mind can do mathematically.
LET’S TALK ABOUT CODING!
The above might be within the reach of a few students...someday! But why not start:
CONNECTING CODING TO SCIENCE
Drawing of Michael Faraday's 1831 experiment showing electromagnetic induction between coils of wire, using 19th century apparatus, from an 1892 textbook on electricity. On the right is a liquid battery that provides a current that flows through the small coil of wire (A) creating a magnetic field. When the small coil is stationary, no current is induced. However, when the small coil is moved in or out of the large coil (B), the change in magnetic flux induces a current in the large coil. This is detected by the deflection of the needle in the galvanometer instrument (G) on the left.
Drawing of Faraday disk, the first electromagnetic generator, invented by British scientist Michael Faraday in 1831. The copper disk (D) rotated between the poles of a horseshoe shaped magnet (A), creating a potential difference between the axis and rim due to Faraday's law of induction. If an electrical circuit such as a galvanometer was connected between the binding posts (B) and (B') the motion induced a radial flow of current in the disk, from the axle toward the edge. The current flows into the spring contact (m) sliding along the edge of the disk, out of binding post (B') through the external circuit to binding post (B) , and back into the disk through the axle. Turning it in the opposite direction reverses the direction of current. The caption also says that passing a current from a Bunsen cell (battery) through it would cause the disk to turn, making it function as an electric motor. The labeled parts are given in the caption as: (A) inducing magnet (D) induced disk (B) binding-screw for current entering or exiting axis of disk
(B') binding-screw for current entering or exiting circumference of disk (m) rubber (sliding spring contact) for edge of disk.
A printing electrical telegraph receiver, with transmitter key at bottom right.
THE TESLA MOTER IS BUILT USING ELECTROMAGNETIC INDUCTION.
ALL DAMS AND POWER PLANTS CREATE ELECTRICITY BY USING ELECTROMAGNET INDUCTION.
“If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.”
Antoine de Saint-Exupery, author: “The Little Prince”
Engaging students in their learning is more difficult than ever. With a flood of information available for children 24 hours a day, how do we make learning at school a compelling piece of their day? Students have opportunities to learn throughout television, technology, and their community in greater ways than any prior generation. For schools, this means that business as usual and change at its current pace won't allow for excellence. We will look closely at how providing choice, voice, and authentic audience, combined with high quality instructional strategies,quality technology integration, education for sustainability, and purposeful empathy-building can bring schools results.
Content and pedagogy has to be used at the same time. So let’s talk about pedagogy!!
Giving students a “tool box” of understanding, by letting them explain the most important and beautiful of ideas, the way that mathematics was and is the language of science.
EVERY CLASS IN MATHEMATICS CAN HELP EMPOWER STUDENTS AND TURN THEM ON TO THE SUBJECT, SO LET’S GET ON WITH “THE SHOW”!
MAYA PLUGIN FOR COMPLEX 3D SHAPES
Thursday, 23 October 2014
Inspired by Buckminster Fuller and with "an affinity for pure geometric forms such as the torus and platonic solids", Liquid Buddha Studios has developed the Cosmic Geometry Tookit, a plugin for Autodesk Maya that allows artists to "create highly complex geometric forms such as toroidal structures and tetrahedral designs for 3D printing."
Included are tabs for platonic solids and Fuller's favorite, the Jitterbug. The plugin can be found on GitHub and is open for use under a Creative Commons ShareAlike license. Get it: Cosmic Geometry Tookit, GitHubRead more: http://www.3dprinterworld.com/article/additive-enlightenment-with-cosmic...
LIVING BREAKWATERS WINS 2014 FULLER CHALLENGE
Friday, 31 October 2014
The Buckminster Fuller Institute (BFI) is pleased to announce that a comprehensive climate change adaptation and community development project, Living Breakwaters has been selected as the winner of the 2014 Fuller Challenge, "socially responsible design’s highest award". The project was submitted by SCAPE / LANDSCAPE ARCHITECTURE PLLC based in New York.
"Don't fight forces, use them." —R. Buckminster Fuller
"Living Breakwaters is about dissipating and working with natural energy rather than fighting it. It is on the one hand an engineering and infrastructure-related intervention, but it also has a unique biological function as well. The project team understand that you cannot keep back coastal flooding in the context of climate change, but what you can do is ameliorate the force and impact of 100 and 500 year storm surges to diminish the damage through ecological interventions, while simultaneously catalyzing dialog to nurture future stewards of the built environment," said Bill Browning of Terrapin Bright Green, a 2014 senior advisor and jury member.
"This year’s Challenge winners deeply know that doing a physical intervention off the coastline would not be enough to create systemic change. Living Breakwaters is a project based in connections—the leadership team brings their deep expertise in technology and ecological science into the social dimension onshore in partnership with the community itself," added Sarah Skenazy, Fuller Challenge Program Manager.
The Living Breakwaters project integrates components ranging from ecologically engineered "Oyster-tecture," to transformational education around coastal resiliency and the restoration of livelihoods traditional to the community of Tottenville in Staten Island, while also spurring systemic change in regulatory pathways at the State level.
Kate Orff of SCAPE said, "We are so honored to be the 2014 Fuller Challenge recipient - Fuller was optimistic about the future of humanity and deeply believed in cooperation as the way forward. As climate change impacts threaten shoreline populations, Living Breakwaters hopefully represents a paradigm shift in how we collectively address climate risks, by focusing on regenerating waterfront communities and social systems, and enhancing threatened ecosystems." SCAPE Associate Gena Wirth added, "The project embraces people as a critical participant in a healthy urban ecosystem, and uses the regenerative power of ecology to reduce risk and grow a layered, resilient shoreline.”
DESIGN SCIENCE IN ACTION
Real-World Learning: Applying Science in Global Service
By Leah Penniman April 1, 2014 1:15 pm
Haiti Rising: The Project
The collaboration between Haiti and Tech Valley High School (TVHS) was forged in 2009 when we partnered with a team of engineers to design solar ovens. We were seeking student mastery of specific mathematics and science content – quadratic equations, insulative and conductive properties of materials, engineering design, impacts of deforestation – and desired a meaningful application. The result was a challenge to students to design, build, and test a solar oven using a parabolic trough concept. They also had to research and present on the history of deforestation in impoverished countries and the imperative to move away from wood as a fuel source. Their presentations were judged by professional engineers, and the strongest student designs were sent to Haitian nonprofits that were disseminating solar ovens.
After the devastating 2010 earthquake, we decided to forge a direct connection with the farmers of Komye, Haiti. These farmers were at the epicenter of the quake and experienced environmental and economic devastation. They identified immediate needs to end water contamination from human waste and to restore the health of the soil to make farming more productive. TVHS students set to work designing a composting sanitation system and accompanying training to support the farmers in achieving both goals. They researched low tech strategies for safely managing human waste and took into consideration local geography, climate, and material resources. I travelled to Haiti to implement the training and kick-start a community-wide composting initiative, all designed by students.
Recognizing the need is the primary condition for design."
-- Charles Eames,
A student holds an evenly dried mango slice over his head like a trophy. His feet are caked with the dry Haitian earth and his arms kissed by the equatorial sun. He and his classmates designed and tested this solar mango dryer as part of their ninth grade Environmental Analysis class, but that was months ago in New York. To see it working firsthand, to imagine its potential to enhance agroforestry, and to reduce vitamin A deficiency in this rural community – is thrilling.
It’s possible to weave together mastery and purpose in our science classes. We can and should implement projects that simultaneously engage students in rigorous scientific thinking and provide opportunities for students to make tangible contributions to their communities. At Tech Valley High School in New York, science students are engaged in projects such as urban soil remediation, invasive species tracking, sensor engineering for water quality monitoring, mapping food deserts, and quantifying carbon sinks. They master content better because they are applying their learning to initiatives that truly matter. The project that most exemplifies this interplay of mastery and purpose is Ayiti Resurrect – literally “Haiti Rising” – a 5-year, multi-disciplinary environmental service project that is changing lives in the farming community of Komye, Haiti.
THE HAWAII PROJECT
While Kilauea Volcano has been erupting since Jan. 3, 1983, the current flow that began spewing from one of its vents on June 27 has inspired innovation like never before.
Many of the ideas kicking around among government officials and scientists are still in the "what if" stages. But a few dozen students from Pahoa's Hawaii Academy of Arts and Science charter school will see their idea for a vog scrubber become reality Monday when it goes on sale at the Pahoa ACE Hardware store.
One of Logan Treaster's friends at the Hawaii Academy of Arts and Science had to move out of Puna two weeks ago because of the vog from Kilauea, which Treaster smells almost every day.
"It smells like rotten eggs," said Treaster, a 17-year-old senior.
ACE Hardware will sell the component parts for the students' vog scrubber for $100. A fully assembled scrubber put together by the students will go for $150, which means the school's science, technology, engineering and math (STEM) program will get the extra $50 for future research.
But Treaster said it's more important to do something "really nice" for people affected by vog.
"It feels good to be able to help our community," he said.
The scrubber pulls the vog out of the ambient air with a fan and neutralizes the acidity with a compound similar to baking soda.
The students have come up with three other lava-related ideas. One is similar to new technology that Hawaii Electric Light Co. used to protect six poles directly in the path of the lava that has already crossed Apaa Street in Pahoa. And after lava undermined one pole wrapped in HELCO's anti-lava technology, the students came up with another idea they think is even better. They've also come up with designs for a water-cooled bridge that would dissipate heat and allow drivers to span roads overrun by lava."We teach giving back," said Eric Clause, Hawaii Academy of Arts and Science's STEM coordinator. "I also teach the kids, ‘You can work the problem — or you can let the problem work you.'"
The Following Exercises Were Used In An Eighth Grade Geometry Class
Discussion: ERATOSTHENES ESTIMATE OF EARTH’S
CIRCUMFERENCE (CIRCA 240 B.C.E.)
Exercise: Given that a central angle of 7 degrees subtends
an arc, from Alexandria to Syene, of 500 miles,
calculate the circumference of the earth.
Calculate the radius of the earth.
Discussion: PHASES AND PERIOD OF THE MOON.
ARISTARCHUS OBSERVES DURATION OF
ECLIPSE OF MOON (CIRCA 235 B.C.E.)
Exercises: A. Explain where earth, sun, and moon are
positioned during a half-moon, full moon,
and during an eclipse of the moon.
B. Given that the center of the earth’s
shadow takes 3 hours to cross the moon,
calculate the distance (in units of earth’s
diameter) from the earth to the moon.
Convert into miles.
C. Calculate the size of the moon’s diameter.
FOR A COMPLETE HANDOUT OF THE NOTES FROM THE SESSIONS ABOVE AND BELOW, I WILL E-MAIL THEM TO ANYONE INTERESTED.
ASILOMAR MATH CONFERENCE, DEC. 2012 SESSION #416, SAT, 1:30-3:00, NAUTILUS EAST
THE STEM EFFECT: Making sense of how math shapes science. Add retention and depth to proof and key concepts in Geometry, and Algebra 2 by allowing students to recount the history of Euclid's fifth axiom and the progression of ideas from DesCartes to Reimann. Build the models that led to Einstein's General Theory of Relativity. Create an understanding of the ideas of changing axiom systems, finite-unbounded spaces, and how mathematics becomes the language of science. Attendees will receive a copy of my book, "Mathematics and The Human Condition".
This session is an extension of my presentation in 2011. My book is an enrichment supplement for teachers of Algebra 1, Geometry and Algebra 2. The goal is to integrate the history of The Scientific Method into the mathematics curriculum. Give students sufficient opportunities to broaden their horizons in other areas of mathematics, such as in various geometries.
In 1907, two years after proposing the special theory of relativity, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. At this point there occurred to Einstein, described by him as the happiest thought of my life , namely that an observer who is falling from the roof of a house experiences no gravitational field. He proposed the Equivalence Principle as a consequence: "... we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame."
After the major step of the equivalence principle in 1907, Einstein published nothing further on gravitation until 1911. Then he realized that the bending of light in a gravitational field, which he knew in 1907 was a consequence of the equivalence principle, could be checked with astronomical observations. He had only thought in 1907 in terms of terrestrial observations where there seemed little chance of experimental verification. Also discussed at this time is the gravitational redshift, light leaving a massive body will be shifted towards the red by the energy loss of escaping the gravitational field. Einstein realized his problems. If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them. Einstein then remembered that he had studied Gauss's theory of surfaces as a student and suddenly realized that the foundations of geometry have physical significance. He consulted his friend Grossmann who was able to tell Einstein of the important developments of Riemann, Ricci (Ricci-Curbastro) and Levi-Civita.
Einstein wrote: "... in all my life I have not labored nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now."
VICTOR M. SELBY— WWW.ENRICHMATH.COM
P.O.BOX 1095 PACIFIC GROVE CA. 93950 (831) 917-9159
Asilomar, Dec. 2012, Session 412
I. REVIEW OF ENRICHMENT IN ALG. 1
MATH AS THE LANGUAGE OF SCIENCE: INTEGRATING IN ALG. 1
A. Science as part of the human condition
1. Paleolithic art: Inductive and deductive reasoning
3. Alloys: Analysis and synthesis/Synergy
4. Symbol systems and “Powers of Ten”, PP. 10-16, p. 39, prob. I
Discover article, Dec.2012, pp.42-50: Kepler is observing a conical zone about
3,000 light years deep, there are more than 3,000 candidates. Extrapolating gives
a number of 150 billion in our galaxy.
B. The rise of mathematics: Functions, PP. 26-35
1. Hooke’s Law of Elasticity
b. Law of levers: p34
c. Approximations of Pi
3. The Drake equation & p.39, #I, see Discover, Dec. 2012, P. 48
II. REVIEW OF ENRICHMENT IN GEOMETRY
THE GREEKS AND THE MATHEMATICS OF SPACE
A. From Thales to Aristotle to Euclidian Space PP. 43-51.
Descartes and Coordinate Geometry: Functions (equations of straight lines
1. The power of reasoning
2. The Pythagorean Theorem, turning space into number
3. Euclid, Axioms, and Proof, P.56
B. BIG IDEAS: “Everything is moving” and “Curved Space”
1. Review: Phases of the moon, Speed of light, Intro. to “Space-time” .
A. Eratosthenes & Diameter of earth (see student work)
B. Aristarchus & 3 hour moon eclipse time, dist. to moon (student
C. Aristarchus & dist. to sun, central angle (E.S.M.) .15 degrees
D. Olaf Roemer’s observation: 16.7 min diff. in eclipse of Io. Leads to:
E. Speed of light, Michelson-Morley experiment, and Einstein’s first
“thought” experiment: time dilation, P.129
2. Sphereland, and intro. to “finite unbounded space” pp. 58-66
A. Lineland to Circleland
B. 3-D to 4-D, build a Tesseract
C. Flatland to Sphereland, geodesics, no parallels : Bernhard
Riemann (1854), proposes that our space may be the 3-D
hypersurface of a 4-D hypersphere.
D. Time as a 4th. dimension, Gamow’s analogy, 1-2-3, p.70,
186,000 mile cube existing for one second.
EXTENDING TO ALGEBRA 2
Review: analysis-synthesis, p. 101
The models that led to Einstein's General Theory of Relativity
(Additional enrichment for chapt. 17, pp. 125-6)
1. Gravitation as the effect of curved space-time
A. Definitions of curved space (cylinder ?).
Euclidian Geometry does not hold.
B. Feynman’s “hot-plate” model
C. Einstein’s equivalence principle: There is no difference between
gravitational acceleration and a ship in space accelerating.
Gravitation is not something that exists within spacetime, but is
rather an attribute of spacetime…Gravity is geometry!
Everything is “moving” and some things are accelerating.
D. Acceleration slows time and G.P.S. has to take this into account.
E. Gravitational lensing demonstrates the curvature of space
2. The universe as an “oversphere” or ?
A. The big-bang and cosmic background radiation (Images of the
Universe, P.46) and shows, in a sense, a picture of the big-bang.
B. There is no “center on a circle, sphere, or oversphere
C. Painting concentric circles on the surface of a sphere analogy to
painting concentric spheres in 4-space )Flatterland p.50-51
C. The pimple model
3. The conic sections as a result of curved space-time, chapter 17, p. 125
Arianrhod, Robyn. Einstein’s Heroes. New York: Oxford University Press, 2005.
Bronowski, Jacob. The Ascent of Man. Boston: Little, Brown and Co., 1973.
Burger, Dionys. Sphereland. New York: Harper & Row, 1965
Discover Magazine, Dec. 2012, p.48.
Einstein & Leopold. The Evolution of Physics. New York: Simon & Schuster, 1966.
Epstein, Lewis Carroll. Thinking Physics is Gedanken Physics. San Francisco: Insight Press, 1985.
Feynman, Richard. The Very Best of the Feynman Lectures. Perseus Books, 1962.
Ferris, Timothy. Coming of Age in The Milky Way. New York: Doubleday, 1988.
Gamow, George. One Two Three Infinity. New York: Viking Press, 1975.
Hey, Tony and Walters, Patrick. Einstein’s Mirror. Cambridge University Press, 1997.
Hawking, Steven. A Briefer History of Time: New York: Bantam
Pickover, Clifford A. Archimedes to Hawking. New York: Oxford University Press, 2008.
Schlling, Govert. Images of the Universe, Amsterdam: The Pepin Press.
Selby, Victor M. Mathematics and The Human Condition. Pacific Grove: Park Place Publications, 2008. (Available through www.enrichmath.com).
Stewart, Ian. Flatterland, Perseus Publishing, 2001.
Wolfson, Richard. Einstein’s Relativity, The Teaching Company, 2000.
The best way to to assess student knowledge is to give them a chance to explain the core ideas in any given discipline. This allows us the opportunity to test the depth of their understanding, the importance of which is discussed in the following excerpt.
For example, the idea of equivalent fractions should be so clear in the mind of every student that they should be able to teach that idea to others. There is a world of difference from being able to follow a set of rules for adding fractions, to being
able to explain why 2/3 is equivalent to 10/15. While the "Standards" provide teachers with a guide for the breadth of mathematical procedures necessary to advance, there seems to be no such guide to the truly crucial core ideas in mathematics. The first essential idea for algebra should be that mathematics is the language of science, and thus is the symbol system that powered the rapid evolution of our present day culture. When students can explain those connections, we are creating a truly educated population.
(a) The Honorable James B. Hunt, Jr. (Former Governor of North Carolina and Foundation Chair, James B. Hunt, Jr. Institute for Educational Leadership and Policy): http://edlabor.house.gov/documents/111/pdf/testimony/20090429JamesBHunt
Content standards must form a clear, coherent message about teaching and learning in each subject area, and we must ensure that world-class content standards form the basis of every child's education.
In 2007, the Hunt Institute began partnering with the Alliance for Excellent Education and the Council of Chief State School Officers (CCSSO) to explore the potential for a common set of content standards. Findings from the Hunt Institute's project with the National Research Council (NRC) and discussions during our 2007 and 2008 Governors Education Symposia informed this effort.
The partner organizations agreed that a common set of state standards should be fewer, clearer, and higher than our current state standards. They must be internationally benchmarked and based on evidence about the essential knowledge and skills that students need to be prepared for college and work...EdWeek reported last month that experts are siding with depth of knowledge versus breadth of knowledge--especially when it comes to the sciences.
From CUEngineering #26, Spring 2009 p. 24:
"CU-Boulder freshman engineering students at risk of failing calculus don't have to struggle alone. When these students are encouraged to talk about the subject's concepts in small discussion groups, they are more likely to pass the class.
When voluntary pre-test discussions, called oral assessments, were introduced into a special two-semester-long Calculus I class in 2003, 95 percent of the students passed the course. The national failure rate for Calculus I is about 40 percent...
Based on the success of these oral assessments, the National Science Foundation awarded a $450,000 grant to the Department of Applied Mathematics to introduce assessments into other engineering classes on the CU-Boulder campus and in math classes at CU-Colorado Springs and a local high school...
Groups of 5 students meet with a facilitator a day or two before a written exam to talk about the fundamental concepts...We find that by getting students to talk about basic concepts, two things happen. One is that by sharing solution strategies and by explaining their mathematical thinking to the facilitator and to others in the group, conceptual understanding is developed and enhanced...
...They see that it helps them to talk about the concepts because when they can put the mathematical information into their own words, they take ownership of their learning..."
There were two ways that I used these strategies in my math classes. The first, as demonstrated in the text, was liberal use of written essays on tests asking students to explain important concepts e.g. equivalent fractions, the laws of exponents, functions, axiom systems, and the scientific method. The other was to form study groups and give time in class for group work, with a particularly good student acting the role of facilitator. Since all students knew that they would have to explain concepts on the coming test, they were able to practice with others and get feedback. The reinforcement and payoff was great once confidence and true understanding took hold. Everybody now had a motivation to buy into knowing the powerful symbol system that powers science and our modern culture.
From The New Yorker Dec. 15 2008, Gladwell, Malcolm "Most Likely To Succeed"
"Eric Hanushek, an economist at Stanford, estimates that the students of a very bad teacher will learn, on average, half a year's worth of material in one school year. The students in the class of a very good teacher will learn a year and a half's worth of material...Teacher effects dwarf school effects..."
So whats going on here? Mr Gladwell goes on to suggest we install a merit pay system, but really doesn't go beyond basing merit on standardized test scores. He does indicate that finding the qualities that will translate into being a "good teacher" is not an exact science, but tries to push the idea that teachers should be given try-outs like pro football players or money managers. The "winning" teachers would be paid more.
There are two characteristics that he has found to identify the good teacher. One is called "withitness", which is defined as "a teacher's communicating to the children by her actual behavior...that she knows what the children are doing, or has the proverbial 'eyes in the back of her head'". The other comes from findings from Bob Pianta, the dean of the University of Virginia's Curry School of Education who finds that the teacher who gives "individualized feedback" to students, i.e. one who moves around the room and connects with each student to troubleshoot and give personal attention, is most effective.
ALGEBRA 1 SOURCEBOOK
Frank, Adam. "Cosmic Abodes of Life", Discover, May, 2009, pp.46-51.
Text reference: The Drake equation, pp.35-36.
"Studies of deep mines and distant stars hint that biology may be widespread across the universe-and suggest the best places to find it." This introduces an article that brings us up to date from the 1976 Viking 1 landing on Mars. Since that unsuccessful probe for life, there has been a great leap forward. "The pace of progress is staggering. Just last November new studies of Saturn's moon Enceladus strengthened the case for a reservoir of warm water buried beneath its craggy surface...the Cassini spacecraft witnessed geysers of water vapor blowing out from its surface. Now Enceladus joins Jupiter's moon Europa on the growing list of unlikely solar system locales that seem to harbor liquid water and, in principle, the ingredients for life."
The article goes on to examine the data supporting the probability that there are a huge number of Earth-like worlds orbiting many of the 200 billion stars in our galaxy, The new Kepler probe is ready to begin the grand search later in 2009.
Also, the habitable zones of the cosmos have greatly expanded with the discovery of strains of bacteria that don't use sunlight, oxygen, or food from the surface. Chris McKay, one of the pioneers of astrobiology describes one type that lives 5 miles deep in South African gold mines. "These creatures get their energy from sources we never imagined. The South African extremophile bacteria are powered by the radioactive decay of unstable atoms in the rocks. It's amazing."
Lanier, Jaron. "Jarons World", Discover, Feb. 2008, pp.21-22.
Text reference: Communication with alien intelligence; pp.27-30.
This article proposes an incredible new way of communicating with intelligent civilizations in our galaxy. The idea is to send a fleet of spacecraft to act as "gravitatational tractors" that would adjust the trajectories of objects in the Kuiper belt. This would, in turn, tilt the plane of the solar system and then change the trajectory of the sun through the Milky Way. Additional clusters of ships would be sent to 15 or so nearby stars, changing their trajectory to create a "sign" that would be recognizable to other civilizations. This message would be in place for billions of years. This "graphstellation" would be like a constellation, but would be a form of writing. Lanier and Piet Hut at the Institute for Advanced Study also advocate searching for these signs of intelligence along with the search being carried out by S.E.T.I.
Pickover, Cliffird A. Archimedes to Hawking. New York: Oxford Univ. Press, pp 74-75.
Text Reference: Spring Function, p. 30
Robert Hooke was one of the most brilliant minds of the 17th. century whose inventions include the iris diaphram in cameras and the balance wheel in a watch using springs instead of gravity to drive a clock mechanism that could keep fairly accurate time at sea. His "Law of Elasticity" (1660) states that the size of a material deformation (x) is directly proportional to the deforming force or F=-kx, where k is the spring constant in our classroom experiment.
This simple relationship was instrumental in creating the technology to test the strength of materials so that engineering decisions could be made with precision.
Kim, Scott. "Mind Games". Discover, Aug. 2007, p. 74
Text Reference: Self-Referential Statements, pp.75-76
From Douglas Hofstadter's book I Am a Strange Loop (Basic Books, 2007), Mr. Scott excerpts some key ideas: "Once a system is able to represent itself, strange and loopy things happen. Here are 11 self-referential sentences. Which are true, which are false, and which are something else entirely?" Answers provided in parentheses.
1. This sentence is true. (OTHER. The sentence can be assigned the truth value "true" or "false" without contradiction, and therefore cannot be categorized as either.)
2. This sentence is neither true nor false. (OTHER. Assigning either truth value leads to a contradiction.)
3. This sentence is five words long. (FALSE)
4. This sentence is less than seven words long. (FALSE)
5. The following sentence would be true if "seven words long" were a single word.
( FALSE. this sentence becomes true when the order of sentences is reversed.)
6. This sentence is more than seven words long. (TRUE)
7. This is the second instance of this sentence. (FALSE)
8. This is the second instance of this sentence. (TRUE)
9. The following sentence is identical to this one, except that the words "following" and "preceding" have been exchanged, as have the words "except" and "in" and the phrases "identical to" and "different from." (TRUE. This sentence becomes false when the order of sentences is reversed.)
10. The preceding sentence is different from this one, in that the words "preceding" and "following" have been exchanged, as have the words "in" and "except" and the phrases "different from" and "identical to." (TRUE. This sentence becomes false when the order of sentences is reversed)
11. Thit sentence is not self-referential because "thit" is not a word. (OTHER. Very confusing!)
Note: I would not include this sentence on a test. Have fun!
ALGEBRA 2 SOURCEBOOK
Wilkinson, Alec. "What Would Jesus Bet?" The New Yorker. March 30, 2009, pp.30-35.
Text Reference: P103, defining a "real game".
I wish I had this article available when teaching about games theory. In "The Ascent of Man" (pp.432-33) Brownoski tells of his encounters with John von Neumann and of the time he was lectured about the difference between a "real game" like poker and games like chess and tic-tac-toe which von Neumann characterized as "well defined forms of computation". When preparing students to understand the prisioner's dilemma, it's important for them to know how a champion poker player creates strategy. Wilkinson makes these points:
"Games for which a flawless strategy is known are said to be solved. Tic-tac-toe is solved; blackjack is solved; checkers is solved ." (see entry below) "Chess is not solved, and poker is not, either...Among mathematicians, chess is regarded as a game of perfect information, because nothing is hidden. If its ideal strategy were discovered, there would no longer be any reason to play it--no move could be made for which the response was not already identified. Poker is a game of imperfect information, since so much is concealed. Solving it would not overcome the disadvantage of being unable to know why your opponent is acting as he is." The article goes on in depth to tell the story of Chris Ferguson, the poker champion, whose mother was a mathematician and whose father taught game theory at U.C.L.A. and says about Chris "He learned to think about playing and strategies and what other people know about what you know. It's not important in chess, but it's important in poker. It's a rather deep game, when you get involved." While knowing the odds and much of the mathematics of when to play a hand and when to fold, the article describes Chris' method called "optimal strategy". He says it "doesn't mean 'How do I win the most?'...but "How do I lose the least?"...a player using optimal strategy also builds into his play bets that sometimes appear improbable and make it mathematically difficult for the opponent to know what to do.
This lesson becomes very clear when we played the prisoner's dilemma game in class and nobody wanted to be taken for a sucker. Then the issue of when to cooperate and when to defect also becomes very clear.
Rehmeyer J. "Check on Checkers". Science News, July 21, 2007 Vol. 172, pp.36-37
Text Reference: Game Day, pp. 102-103
It took 2 decades to completely solve the game of checkers. Johnathan Schaeffer of the University of Alberta calculated that there are approximately 500 billion billion
(5x1020) possible positions, which means that for each move there is an exact counter-move that eventually will result in a tie game if each contestant has the necessary "smarts". He started with the endgame, putting just two pieces on the board and calculating every possible outcome. Then he went on to 3 pieces and on up to 10. At this point 39 trillion positions were possible. Each step took 10 times as much work as the previous one. He then moved to the beginning stages of the game, calculating all different positions resulting from one move, then two moves, etc. until the program checked the database of endgames to find the outcome.
Since tic-tac-toe is one of our "games" on game day, it would be a quick student project to "solve" the game by calculating the total number of possible different positions. This would confirm that tic-tac-toe is not a "real game" like poker but just a form of calculation.
By the way, the estimate is that chess has about 1020 times as many positions as checkers does, Schaffer says, and may take another 100 years to solve.
Text Reference: P. 146, after showing the periodic table of elements.
It is a good idea to go back to Archimedes in Algebra 2 in order to explore his "Eureka" moment and emphasize why he is regarded as "...the greatest mathematician and scientist of antiquity and one of the four greatest mathematicians to have walked Earth-" (Reference: Pickover, Clifford A. Archimedes To Hawking, Laws of Science and the Great Minds Behind Them. New York, Oxford University Press. 2008). So a great question to connect chemistry and physics and to start some thinking is to ask: "How did anybody figure out how to weigh helium?" (It's good to bring a helium balloon to class.) After trying to convince kids that if we put the balloon on a scale, the weight would have to be "negative" since the balloon would pull the balance beam upward, we introduce the famous King's crown problem.
Archimedes' Principle of Buoyancy explains flotation and is an example of how careful observation results in a mathematical model that can be used to plan and create such things as submarines. The principle states that the force pushing up on an object placed in a fluid is equal to the weight of the fluid displaced by the object. The mathematical model is: B=wf , where B is the buoyant force and wf is the weight of the displaced fluid. If an object floats, the weight of the object must be equal to the weight of the fluid displaced. A small, dense object like a b.b. displaces a tiny amount of water so the buoyant force is smaller than the weight of the object, and it sinks.
The density of an object such as King Hieron's "Gold" crown can only be calculated if the volume is known. When Archimedes lowered himself into the bath, he observed the water flowing over the top. The larger the volume of the submerged object, the larger the volume of displaced water. This is an easy experiment to reproduce in class. The density of silver is 10.5 grams per cubic centimeter, while gold is 19.3 gr./c.c. The crown was found to have a density between these two values, and the royal goldsmith was executed. So it goes.