Cool Math Games: how to integrate math and computers
Cool K-12 Math Games, Lesson Plans and Math Avoidance Strategies in Mathematics.
#cool math games #make money #get a job #plan to work #math #calculate
MISS USA CONTESTENTS SPEAK ABOUT MATH
Reform Math / Fuzzy Math
Reform math says kids should explore and understand concepts like place value before they become fluent in the standard way of doing arithmetic. Critics say it fails to stress basic computational skills, leaving students unprepared for higher math. Stanford University mathematician James Milgram calls the reform math-inspired standards a "complete mess" - too advanced for younger students, not nearly rigorous enough in the upper grades. And teachers, he contends, are largely ill-prepared to put the standards into practice. "You are asking teachers to teach something that is incredibly complicated to kids who aren't ready for it," said Milgram, who voted against the standards as part of the committee that reviewed them. "If you don't think craziness will result, then you're being fundamentally naive."
44 States have started using the Common Core learning standards, which has made math - adding, subtracting, multiplying and dividing become as complicated as calculus. The standards for elementary math emphasize that kids should not only be able to solve arithmetic problems using the tried-and-true methods their parents learned, but understand how numbers relate to each other. Everyone is stumped by unfamiliar terms like "rectangular array" and "area model." They wrestle with division that requires the use of squares, slashes and dots. They rage over impenetrable word problems.
Simple arithmetic isn't so simple anymore, leading to plenty of angst at home. Even celebrities aren't immune: The comedian Louis C.K. took to Twitter recently to vent about his kids' convoluted homework, writing that his daughters went from loving math to crying about it. SEE OKLAHOMA CORE CURRICULUM TEST GRADE 8 PRACTICE AND SAMPLE TEST WORKBOOK PDF
Long Division Style
Meyer Levin School for Performing Arts Classes 622 and 624 perform a Gangnam Style parody teaching long division with decimals.
Poor "Gut Sense" Of Numbers Contributes To Persistent Math Difficulties
Journal Child Development
Having a poor "gut sense" of numbers can lead to a mathematical learning disability and difficulty in achieving basic math proficiency. This inaccurate number sense is just one cause of math learning disabilities, according to the research led by Dr. Michele Mazzocco of the Kennedy Krieger Institute. Heightened interest in the nature and origins of these learning difficulties has led to studies to define mathematical learning disability (MLD), identify its underlying core deficits, and differentiate children with MLD from their mathematically successful counterparts. The study findings suggest that an innate ability to approximate numbers, an intact ability present in human infants and many other species, contributes to more sophisticated math abilities later in life, while a less accurate ability underlies MLD. Additionally, the findings reveal that a poor number sense is not the only potential source of math difficulties, reinforcing that a 'one size fits all' educational approach may not be the best for helping children who struggle with math.
"Some children have a remarkably imprecise intuitive sense of numbers, and we believe these children have math learning disability, at least in part, due to deficits in this intuitive type of number sense," said Dr. Mazzocco, Director of the Math Skills Development Project at Kennedy Krieger. "But other students who underperform in math do so despite having an intact number sense. This demonstrates the complexity of determining precisely what influences or interferes with a child's mathematical learning. Difficulty learning math may result from a weak number sense but it may also result from a wide range of other factors such as spatial reasoning or working memory.
Math and Reading Help for Kids
Homework Help, Tutoring, testing and Parenting Advice Your guide to math, reading, homework help, tutoring and earning a high school diploma. Parents, students and teachers searching for Kids' Educational Math and Reading Games found the following articles and tips relevant.
Math Tutor films on YouTube for free.
Khan Academy Resource Overview
Sal Khan produced low-budget math films on YouTube for free.
Khan aims to demote the institution of "school" to just one of many educational options. Is a solution to some of education’s most intractable problems?
Marion Brady ~ I can speak with authority only for one experienced teacher: Myself.
I haven’t the slightest doubt that the
Number One way that most of us know what we know is what Jonah was demonstrating — autonomous, firsthand, curiosity-driven, wide-ranging, self-directed, trial and error, immediate feedback, personal experience.
Number Two in efficiency is learning through shared experience and the dialogue that ordinarily accompanies it.
The Number Three way we learn — from “delivered information”— is a distant third in teaching-learning efficiency.
Khan Critiqued Thumbs Down 7/2012
This is the background for the following post on the kind of teaching that the Khan Academy offers. It was written by Marion Brady, veteran teacher, administrator, curriculum designer and author!!
Two associate professors from Grand Valley State
University in Michigan critiqued one of the Khan videos — on negative and positive integers — in a satirical video that pointed out problems in the specific lesson and, more broadly, made fun of the entire Khan enterprise, Education Week reported. The academy pulled the suspect video and replaced it. Educator Dan Meyer and Ed Week opinion blogger Justin Reich, noting that there are errors in some of the Khan Academy videos, have started a contest inviting readers to critique the academy lessons. You can see how to participate here.
need to make money or plan to have a job?
How much money will you need to earn?
This hourly rate calculator to give you a guide based on your costs, number of billable hours and desired profit. It is a simple tool for you to play with. Remember your hourly rate should always take into account factors like market demand, industry standards, skill level and experience - things that unfortunately we can't put into a calculator! Use these calculations as a guide and then modify to suit your circumstance and conditions. It will take you about 5-20 minutes to complete depending on how much attention you give each calculation.
USA Today - Money - Calculators
Offers a host of financial calculators found under the following topics: Mortages, Autos, Credit, Retirement, and Savings. Answers questions like how much you can borrow, what your loan payments will be, how depreciation will effect the value of your car, should you convert your IRA to a Roth IRA, how much it will take to save for a car or home, and much more!
cool Math Games
Cool Math Games
Pizza Math did you know that the volume of a pizza with radius z, and height a, is, itself, pizza!
XKCD A webcomic of romance, sarcasm, math, and language. Change girls attitutdes towards computers.
Getting Really Young Girls Into Math
This is a great web site with lots of hands-on activities
Visual Math Learning - excellent resource
cool math Games:
Geometry Step-by-Step from the Land of the Incas provides an eclectic mix of sound, science, and Incan history in order to raise students' interest in Euclidean geometry. Visitors will find geometry problems, proofs, quizzes, puzzles, quotations, visual displays, "scientific speculation", Cuzco, Machu Picchu, Lost City of the Incas, Nazca Lines, the Quipu, the Lord of Sipan, Caral: the oldest civilization in the Americas, and more. On July 7th, 2007, Machu Picchu was voted as one of New Open World Corporation's New Seven Wonders of the World. Site created and maintained by Antonio Gutierrez.
- I would like to inform you about a new free on-line animated geometry illustration of: "Generalizing Van Aubel Theorem using Duality by Michael de Villiers" in 15 steps
- De Gua's Theorem, Pythagorean theorem in 3-D
- Interactive Mindmap of "Principles and Standards for School of Mathematics, National Council of Teachers of Mathematics NCTM" Principles and Standards for School Mathematics describes a future in which all students have access to rigorous, high-quality mathematics instruction, including four years of high school mathematics.
- RSS Feed
Along with their substantial instructional resources, the American Mathematical Association of Two-Year Colleges (AMATYC) has worked to place valuable mathematical materials online to aid both teachers and students. Find links to materials that can be used to teach students about the fundamentals of geometry, including plane motion, polyhedras, symmetry, and tessellations.
For math fluency, use xtramath.org
This is a non-profit and completely free. I have seen amazing success in automaticity with basic facts even with kids that have not yet learned them. It follows the more recent research about using known facts and introducing unknown to it. The program is designed for independence and even only 1 computer.
8th grade mathematics and science lessons
Get free Math courses from the world's leading universities. You can download these audio & video courses straight to your computer or mp3 player.
53 public use lessons collected as part of the TIMSS video studies on timssvideo.com. Users must register on the site to access the videos, but registration is free. In addition to the 53 full-length videos of from seven countries, the site also provides full English-translation subtitles for each lesson, a searchable transcript, and a set of resources collected with each lesson such as scanned text materials and teacher commentaries. The site also includes a discussion forum where users can share ideas for how they are using the site.
Adults Learning Mathematics
The Adults Learning Mathematics (ALM) organization is "an international research forum bringing together researchers and practitioners in adult mathematics/numeracy teaching and learning in order to promote the learning of mathematics by adults." The site contains sections such as "For teachers", "For PhD students", and "For policy makers". The "For teachers" area includes links to the ALM newsletters and their resources, which include downloadable posters and factsheets. Moving along, the "For PhD students" area includes information for those doctoral students that are seeking to incorporate pedagogical techniques for teaching math to adults in their research. Additionally, the "For policy makers" area includes helpful documents such as "High stakes assessment: Assessing numeracy for Nursing in
two recent projects".
- A Full Series of Undergrad Math Courses from the University of Colorado (Requires Registration) – Complete Video Archive
- Abstract Algebra - Multiple Formats – Benedict Gross – Harvard
- Core Science Mathematics – YouTube – SK Ray, IIT
- Differential Equations – YouTube – iTunes – MIT – Arthur Mattuck
- Linear Algebra – YouTube – iTunes – Gilbert Strang, MIT
- Introductory Probability and Statistics for Business - iTunes - Video Feed - Audio Feed - Web Site – Fletcher Ibser, UC Berkeley
- Introduction to Statistics – iTunes – Fletcher Ibser, UC Berkeley
- Single Variable Calculus - YouTube – iTunesU – David Jerison, MIT
- Multivariable Calculus – YouTube – iTunes – Dennis Auroux, MIT
- Sets, Counting, and Probability - Multiple Formats – Paul Bamberg, Harvard
- The Calculus Lifesaver – Download Videos – Adrian Banner, Princeton
cool math games:
- Math Practice Site
Practice makes perfect, and IXL makes math practice fun. With unlimited math questions in more than 1,000 topics, students improve their skills and confidence and always have new challenges to meet. Click a grade below to get started!
- The math work sheet site
create an endless supply of printable math worksheets. The intuitive interface gives you the ability to easily customize each worksheet to target your student's specific needs. Every worksheet is created when you request it, so they are different every time.
- Practice/reinforcing concepts (and useful for students who need repetition to get something down).
- National Library of VirtualManipulative
- Search for the Greek God and Goddess - Teaching Ratios
Math - Grades 5, 6, 8
- The State of State Math Standards 2005
- Volume / Area / Length / Weight / Temperature Converter
Cool K-12 Math Games, Lesson Plans and
Math Avoidance Strategies in Mathematics.
Finding Math Hard? Blame Your Right Parietal Lobe
Source: University College London
Scientists have, for the first time, induced difficulties with mathematics (dyscalculia) in subjects who normally find math easy. The study, which finds that the right parietal lobe is responsible for dyscalculia, potentially has implications for diagnosis and management through remedial teaching.
Dyscalculia is just as prevalent in the population as dyslexia and attention deficit hyperactivity disorder -- around 5% of the population is affected. However, dyscalculia has not been given the same attention as other disorders and the underlying brain dysfunction causing dyscalculia is still a mystery. It is hoped that this study will provide a better understanding of the condition and lead to better diagnosis and treatment.
Dr Roi Cohen Kadosh, of the UCL Institute of Cognitive Neuroscience, said: "This is the first causal demonstration that the parietal lobe is the key to understanding developmental dyscalculia. Most people process numbers very easily -- almost automatically -- but people with dyscalculia do not. We wanted to find out what would happen when the areas relevant to maths learning in the right parietal lobes were effectively knocked out for several hundred milliseconds. We found that stimulation to this brain region during a maths test radically impacted on the subjects' reaction time.
"This provides strong evidence that dyscalculia is caused by malformations in the right parietal lobe and provides sold grounds for further study on the physical abnormalities present in dyscalculics' brains. It's an important step to the ultimate goal of early diagnosis through analysis of neural tissue, which in turn will lead to earlier treatments and more effective remedial teaching."
Functions Grapher [Macromedia Flash Player]
Discussing functions can be a tricky endeavor, but having a handy interactive way to talk about functions can relieve a great deal of stress. As part of the Mathematical Sciences Digital Library, this Functions Grapher application is designed to let users enter one or two functions. After doing so, users can trace along either one with coordinates shown dynamically changing at all times. The application was created by Professor Barbara Kaskosz of the University of Rhode Island, and it can be used by students in algebra, pre-calculus, or calculus courses. Of course, educators may wish to use it in their classrooms for illustrative purposes and they can also pass along to students who might find the very idea of functions and their operation a bit puzzling.
AMSER (the Applied Math and Science Education Repository)
A portal of educational resources and services built specifically for use by those in Community and Technical Colleges but free for anyone to use.
AMSER is a non-profit organization and is funded by the National Science Foundation (NSF) as part of the National Science Digital Library (NSDL). We are a fairly new project, and are constantly adding new resources to our library. (About 200-300 per week) The resources we add to our site come from other online digital collections, staff (area expert) additions, as well as from faculty from around the country. The goal of AMSER is not only to provide a site where instructors and students can find high-quality and authoritative web resources but to provide tools to use these resources as well. We provide a folder system (which allows you to share resources online), bulletin service (to update you with subject specific resources), saved searches, rating and comments systems and more. Because we are new, we would appreciate any feedback for improvements that could be made to the site to make it more useful for you, as well as any suggestions for online material that we could add.
Collection of 521 lessons for preK-12 math educators
Avoidance Strategies in Mathematics
STUDENTS MAY BE LEARNING MORE ABOUT AVOIDANCE STRATEGIES THAN ARITHMETIC IN MATH CLASS; STUDY SHOWS WHAT TEACHERS CAN DO
WASHINGTON - "Please don't call on me" can be a pervasive thought by students who are not doing well in math class. By early adolescence, it is common for some students to become experts in avoidance strategies -- avoiding asking for help when they need it, withdrawing effort and resisting novel approaches to learning -- in order to deflect attention from low ability. This type of behavior can cause students to fall further behind academically and may\ eventually lead some to drop out of school. But new research shows that teachers that emphasize learning rather than performance may help prevent this self-destructive behavior. The findings appear in the current issue of the Journal of Educational Psychology, published by the American Psychological Association (APA).
In their study involving 1,092 sixth-grade students in 65 sixth-grade classrooms in four ethnically and economically diverse school districts in three Midwestern states, Julianne C. Turner, Ph.D. of the University of Notre Dame and co-authors surveyed the students to determine use of avoidance strategies. The study also included the use of trained observers who watched and audiotaped nine of the students' teachers while they taught their math classes.
The researchers found that students reported using fewer avoidance techniquesin classrooms perceived as emphasizing learning, understanding, effort and enjoyment. In those classrooms, teachers helped students who had problems understanding, gave them opportunities to demonstrate new competencies and provided substantial motivational support for learning. Teachers in these "mastery-oriented" classrooms made sure their students did not feel inadequate or ashamed when they did not understand. "By modeling their own thinking processes, these teachers demonstrated that being unsure, learning from mistakes, and asking questions were natural and necessary parts of learning," according to the authors. By contrast, "students reported higher incidences of avoidance strategies in classrooms in which teachers devoted little attention to helping students build understanding and in which motivational support was low."
The teacher observations, say the authors, provided valuable insight into how teaching methods affect avoidance behaviors. For instance, in a classroom where students used more avoidance strategies, the teacher placed greater emphasis on getting an answer correct, with little discussion about the important concepts in a lesson and little explanation of why an answer was correct. If a student did not know the answer, the teacher would ask another student and did not usually stop to explain the answer. "Because the teacher typically did not respond to mistakes and misunderstandings with explanations or allow students to explain their strategies, his students may have felt vulnerable to public displays of incompetence and adopted more avoidance strategies," explained the researchers.
In classrooms where students used fewer avoidance strategies, the teachers tended to model, hint and elicit support from other students to help their students learn. In those classrooms, the students were active participants in instructional discourse that stressed understanding and explanation. "Perhaps because they knew their teachers and peers would help, students in these classrooms did not seem to need to adopt avoidance strategies to appear able to others," said the authors.
Classrooms with students who reported using avoidance strategies less also had teachers that used math-related humor as part of their lessons. Humor may lessen tension and encourage students to view their math classes as more enjoyable, say the authors.
Full text http://www.apa.org/journals/edu/press_releases/march_2002/edu94188.html.
Lead author apa.org/
Famous Curves Index
Throughout history, there have been many famous curves. In this case, the famous curves profiled here have names such as rhodonea, right strophoid, and the Kampyle of Eudoxus. These curves belong to the world of the mathematical sciences, and they are offered up for teachers and the generally curious by the staff at the School of Mathematics and Statistics at the University of St. Andrews. Visitors can scroll through the complete list of curves (there are over eighty here), and click on each one for an illustration and a listing of the equation that would create such a curve. The site is rounded out by an interactive map that lets users learn about the birthplaces of famous mathematicians from Leibniz to Babbage.
As Math Scores Lag, a New Push for the Basics
SEATTLE For the second time in a generation, education officials are rethinking the teaching of math in American schools.
The changes are being driven by students lagging performance on international tests and mathematicians warnings that more than a decade of so-called reform math critics call it fuzzy math has crippled students with its de-emphasizing of basic drills and memorization in favor of allowing children to find their own ways to solve problems. At the same time, parental unease has prompted ever more families to pay for tutoring, even for young children. Shalimar Backman, who put pressure on officials here by starting a parents group called Wheres the Math?, remembers the moment she became concerned. When my oldest child, an A-plus stellar student, was in sixth grade, I realized he had no idea, no idea at all, how to do long division, Ms. Backman said, so I went to school and talked to the teacher, who said, We dont teach long division; it stifles their creativity.
Across the nation, the reconsideration of what should be taught and how has been accelerated by a report in September by the National Council of Teachers of Mathematics, the nations leading group of math teachers. It was a report from this same group in 1989 that influenced a generation of teachers to let children explore their own solutions to problems, write and draw pictures about math, and use tools like the calculator at the same time they learn algorithms. But this fall, the group changed course, recommending a tighter focus on basic math skills and an end to mile wide, inch deep state standards that force schools to teach dozens of math topics in each grade. In fourth grade, for example, the report recommends that thecurriculum should center on the quick recall of multiplication and division, the area of two-dimensional shapes and an understanding of decimals.
The Bush administration, too, has created a panel to study research on teaching math. It is expected to issue recommendations early next year. Here in Washington, Gov. Chris Gregoire has asked the State Board of Education to develop new math standards by the end of next year to bring teaching in line with international competition, and a year later to choose no more than three curriculums to replace the dozens of teaching methods now in use. Ms. Gregoire, a Democrat, also wants new math requirements for high school graduation. In Utah and Florida, too, state education officials are re-examining their math standards and curriculum. Grass-roots groups in many cities are agitating for a return to basics. Many point to Californias standards as a good model: the state adopted reform math in the early 1990s but largely rejected it near the end of the decade, a turnaround that led to rising math achievement.
The Seattle level of concern about math may be unusual, but theres now an enormous amount of discomfort about fuzzy math on the East Coast, in Maine, Massachusetts and Pennsylvania, and now New Jersey is starting to make noise, said R. James Milgram, a math professor at Stanford University. Theres increasing understanding that the math situation in the United States is a complete disaster.
Schools in New York City use a reform math curriculum, Everyday Mathematics, but some parents there, too, would like to see that changed, a step they are advocating through NYC HOLD, a group of parents and teachers that has a Web site with links to information on math battles nationwide.
A spokesman for the New York City Department of Education said that Everyday Mathematics covered both reform and traditional approaches, emphasizing knowledge of basic algorithms along with conceptual understanding. He added that research gathered recently by the federal Department of Education had found the program to be one of the few in the country for which there was evidence of positive effects on student math achievement.
The frenzy has been prompted in part by the growing awareness that, at a time of increasing globalization, the math skills of children in the United States simply do not measure up: American eighth-graders lag far behind those from Singapore, South Korea, Hong Kong, Taiwan, Japan and elsewhere on the Trends in International Mathematics and Science Study, an international test.
Parental discontent here in Washington State intensified after the announcement in September that only 51 percent of 10th graders passed the math part of state assessment tests, far fewer than showed proficiency in reading or writing. Math is on absolutely everybodys radar in the state right now, said Ms. Backman, whose Wheres the Math? group drew hundreds of parents and math teachers last month to a forum on K-12 math. Many parents and teachers remain committed to the goals of reform math, having children understand what they are doing rather than simply memorizing and parroting answers. Traditional math instruction did not work for most students, say reform math proponents like Virginia Warfield, a professor at the University of Washington. It produces people who hate math, who can't connect the math they are doing with anything in their lives, Dr. Warfield said. Thats why we have so many parents who see their children having trouble with math and say Honey, dont worry. I never could do math either.
In Asian cultures, she added, the assumption is that everyone learns mathematics, and of course, parents will help with mathematics.
But even many of those who admire the goals of reform math want their children to have more drills. My mother is a high school math tutor, and her joke is that this math is whats kept her in business, said Marcy Berejka, who each week brings Ben, 8, and Dana, 6, to Kumon, a tutoring center based in Japan that has more than a dozen franchises in the Seattle area. Theres a lot thats good in the new curriculum, but if you don't memorize the basic math facts, it gets harder as math gets more complicated.
The states superintendent of public instruction, Terry Bergeson, a supporter of reform math, said in an interview: I came through the reading wars years ago, and now were right in the middle of that with mathematics. It comes back to balance. Of course you need to know your math facts, but you also have to understand what youre doing. The whole country has been in denial about mathematics, and now were sort of at a second Sputnik moment.
In part, the math wars have grown out of a struggle between professional mathematicians, who say too many American students never master basic math skills, and math educators, who say children who construct their own problem-solving strategies retain their math skills better than those who just memorize the algorithm that produces the correct answer. After Dr. Milgram of Stanford appeared at a Wheres the Math? meeting, Dr. Warfield, an expert on teaching math educators, wrote in a newsletter that when Dr. Milgram told parents to fight for change, it was implicit in the instructions that mathematicians who do not agree are classified as mathematics educators (a rung or two below the night custodian). The battle here has left many parents frustrated, confused and not sure if they should trust their childrens schools to give them the skills they need. Many have already voted with their feet, enrolling their children in math tutoring. State Representative Glenn Anderson, a Republican member of the House education committee who has fought for a more rigorous curriculum, said state data showed that Washington residents spent $149 million on tutoring and other education support services in 2004, more than three times the $44 million they spent 10 years earlier. Kumon, which has a global clientele of more than four million children in 43 countries, focuses on drilling children on basics. Students work their way through hundreds of assignments that move in incremental steps from tracing numerals all the way through differential calculus.
Every week for five years, Tove Burrows has brought her son, Petter, 13, to the Kumon Center in Mercer Island to turn in the worksheets he has done at home, sit down to new drills and pick up a set of assignments for the week ahead.
If the math curriculum in the schools were different, I would not be doing Kumon, said Ms. Burrows, whose son is an A student at Islander Middle School. But I want to make sure hes mastered the basics, and in school they dont spend enough time on basics to get that mastery. On Mercer Island, an affluent suburb of Seattle that had the states best scores on the 10th-grade test, the pendulum has begun to swing toward emphasizing computational skills, especially in high school. Were looking at texts that have more numbers and less language, said Lisa Eggers, president of the Mercer Island School Board, who at one point sent two of her three children to Kumon. And were one of the few districts where the math scores are going up.
Even so, seeking outside math help is common in the district, with almost 100 students leaving the high school for math and going instead to nearby private academies for one-on-one tutoring, for which the school give will give them credit.
John Harrison, principal of Mercer Island High School, estimates that as many as 10 percent of his schools 1,400 students are getting outside math help. Its not surprising that math is so important in Seattle, with so many people earning their living at Microsoft or Boeing, Mr. Harrison said. Our kids do very well on the state tests, compared to the state averages, but even here, math proficiency is less than reading and writing.
Teaching Math as Narrative Drama
October 3, 2010 By Katherine Mangan Waco, Tex.
When Edward B. Burger presents a math challenge to his class at Baylor University, he paces the aisles and pairs students together. "I want to hear chattering," he says. Before long, students are laughing and shouting out answers. He dashes to the chalkboard to scribble them down, creating long rows of numbers topped with running stick figures. Mr. Burger, 46, who is visiting from Williams College, keeps up a rapid-fire banter with his students, whom he calls by name.
He is here this semester as a recipient of Baylor's annual Robert Foster Cherry Award for Great Teaching, which came with $215,000 in cash and $35,000 for Williams's math department. The 12-member committee that culled more than 100 nominations from around the country was impressed with his string of teaching awards, his multimedia textbooks and videos for secondary schools, and his televised analysis of the math behind the 2010 Winter Olympics.
Mr. Burger was younger than the students he's teaching at Baylor when he discovered how much fun teaching math could be. Armed with a lesson plan and a conviction that he could cut through his classmates' collective fog, he asked his high-school teacher if she'd step aside and let him teach two classes. "She agreed, and at the age of 17, I stood up in front of a precalculus class of about 40 students who looked at me like I was the biggest nerd in the world," says Mr. Burger.
He earned a doctorate from the University of Texas at Austin and began teaching night classes at Austin Community College at age 22. "What I was trying to do was to take really complex, intricate, abstract ideas of mathematics and make them come to life for these students," he says. He began encouraging students to be creative and take risks, and even bases a portion of their grades on "the quality of their failure." He judges that quality, he says, "by the size of the risk they've taken and the amount of insight they have generated from their mistakes."
At age 24, he received a tenure-track position at Williams, where he is also a professor of social responsibility and personal ethics. The most important issue, he says, is what students will retain from his class 10 years later. "If we are in the business of transforming lives and can't give a good answer to that question," he says, "we're failing." To demonstrate the concept of infinity to a class of mostly liberal-arts students at Baylor, he sketches a trough that he describes as containing an infinite number of Ping-Pong balls, which are falling into a barrel, 10 at a time, as a hypothetical student reaches in and plucks balls out at shorter and shorter intervals. "Soon you'll be working faster than the speed of sound, than the speed of light. You black out, regain consciousness, approach the barrel, look inside. My question to you is, 'What's inside? What is in the barrel?'"
The students pair up at their desks and compare guesses. "It has to be infinity," one says. His partner responds, "He's trying to trick us.... Maybe the answer's zero." Mr. Burger writes these and other guesses, which he draws out of more-hesitant students, on the board. He tells the class to come back on Tuesday for the answer.
Adam Telatovich, a sophomore math major, says some of his favorite lessons in Mr. Burger's number-theory class follow that pattern. "He starts out with a big picture, describing these really far-out problems, and says this is what we're going to work up to. Then he builds up suspense and leaves the punch line for the next class. When the class is over, we're disappointed."
Lance L. Littlejohn, chairman of the department of mathematics at Baylor, describes Mr. Burger as "a teaching phenomenon": well organized, articulate, and engaging.
At Williams, when his students arrive for the first day of class, they sometimes tell him that they've already had him in a course. That's because he wrote an online, multimedia math textbook used in many classrooms nationwide. California just started a pilot program in which middle-school students are given iPads to read his textbook and watch his lecture series.
Mr. Burger, who once planned to go to law school, discourages students from zeroing in too early on a career. "The whole point of higher education is to mess things up and challenge basic assumptions about how you look at the world and fit into it," he says. "If you don't allow your education to challenge those assumptions, there's no point in it." He advises students to choose their careers by finding things they would do on their own for fun. "On good days," he says, "I almost feel it's criminal to accept money for what I do."
This fall he's stimulating discussion about good teaching across the Baylor campus by helping to organize weekly lunch discussions for faculty members. He will also speak this week to a regional meeting of K-12 math teachers and plans to visit local public schools to meet with math teachers and students. "He's wonderful in the college classroom," says Heidi J. Hornik, a professor of art history and chair of the Cherry award committee, "but he also reaches deep down into the academic system to make math exciting for everyone." By the way, the answer to the question about the number of Ping-Pong balls in the barrel: zero.
- Teachers Work
- NCTM's Illuminations site (lesson plans, applets, and more) mostly middle school
- AP Central (free registration, for upper level math)
- Quia's Shared Games/Quizzes (these are free for review, you must pay for an account where you actually test students online and record their grade)
- APlusMath - this may be too elementary but depending on your students' skill level, you may find the straight forward and easy to use drills and games here helpful
- MathForum - the one stop shop for math teachers. I really like the problem of the week where students must come up with creative answers to a challenging problem.
- CoolMath - Okay, ignore the juvenile graphics!! This is actually a good place for high school math practice including pre-calculus and Algebra II.
- High School http://www.cchs165.jacksn.k12.il.us/SalukiMath/EYH2003/EYH.htm
Internet Public Library: Calculation & Conversion Tools
Calculation & Conversion Tools
Websites that assist with mathematical calculations, including online calculators and conversion dictionaries and tables
Offers conversions covering 27 different categories including: distance, temperature, power, weight, volume, energy, speed, currency, clothing/shoe size, computer storage, and cooking.
Directory of online calculators, browsable by a hierarchical subject/function directory. "There are calculators for finance, business, and science. There are ones for cooking, hobbies, and health. Some solve problems, some satisfy curiosity, and some just for fun."
This site lists conversion relationship between U.S. customary units and SI (International System) units. It gives conversion factors for commonly needed values -- area, length, pressure, force, and volume -- as well as gage standards for things such as sheet metal, wire, and pipes and measures of hardness, stress, and temperature.
Conversion of Units
A forms based utility that converts values in one unit to any other.
Convertworld is a unit conversion website. Currency can be calculated for 36 different countries. The date cane also be changed to reflect the most current value of the currencies. Other conversion areas are: length, temperature, weight, data storages, area, energy, pressure, speed, volume, and time. The site is available in English, German, and Portugese.
"This utility allows you to convert dollar values between any two years to adjust for inflation." Uses the Consumers Price Index from the U.S. Bureau of Labor Statistics.
"Units conversion offering a very extensive range of categories and units. This interactive facility not only converts between units but also includes descriptions of usage and definitions of many of the units."
The Inflation Calculator
At this site, you can enter any dollar amount and compare
its value during any two years between 1800 and 1998. For
example, you can compare the worth of $100 in 1880 to its
worth in 1950.
The Medical Algorithms Project
"A Medical Algorithm is any computation, formula, survey, or look-up table, useful in healthcare. We have collected over 1600 algorithms (taken from the biomedical literature, including research journals and textbooks) spanning major medical domains, organized into 43 chapters. To ensure the widest possible audience, the algorithms have been implemented in an Excel workbook which you can freely download to run on your Windows or Macintosh computer. You will need MS Excel, and should be familiar with running spreadsheets." There is also Spanish mirror site linked from this site.
"megaConverter.com is an ever-growing set of weights, measures and units conversion/calculation modules. megaConverters allow users to discover things like how many seconds old they are, the difference between a gallon in the USA and a gallon in the UK, how many nanometers in an inch, how many quarts in a chaldron, and what the heck a nebuchadnezzar is."
This site offers a multilingual currency converter which contains the exchange rates for 164 currencies. Travelers can print out a cheat sheet: "a multi-value table of an exchange rate to facilitate calculations while you travel." The site also offers historical charts of exchange rates and forecasts for global markets.
Converts just about any unit of measure to any other with over 5,000 units offered for conversion in areas including acceleration, angles, area, astronomy, cooking, clothing, density, energy, finance, frequency, distance, light, temperature, torque viscosity, volume and weight.
Time Zone Check
"The coolest way to find out the time in other parts of the world."
The Universal Currency Converter
"The Universal Currency Converter allows you to perform interactive foreign exchange rate conversion on the Internet. Type the amount of source currency in the input box. You may include commas and a decimal point. Select the source and destination currencies using the scrolling selection boxes."
Why Johnny Can’t Add Without a Calculator
Technology is doing to math education what industrial agriculture did to food: making it efficient, monotonous, and low-quality.
When Longfellow Middle School in Falls Church, Va., recently renovated its classrooms, Vern Williams, who might be the best math teacher in the country, had to fight to keep his blackboard. The school was putting in new “interactive whiteboards” in every room, part of a broader effort to increase the use of technology in education. That might sound like a welcome change. But this effort, part of a nationwide trend, is undermining American education, particularly in mathematics and the sciences. It is beginning to do to our educational system what the transformation to industrial agriculture has done to our food system over the past half century: efficiently produce a deluge of cheap, empty calories.
I went to see Williams because he was famous when I was in middle school 20 years ago, at a different school in the same county. Longfellow’s teams have been state champions for 24 of the last 29 years in MathCounts, a competition for middle schoolers. Williams was the only actual teacher on a 17-member National Mathematics Advisory Panel that reported to President Bush in 2008.
Williams doesn’t just prefer his old chalkboard to the high-tech version. His kids learn from textbooks that are decades old—not because they can’t afford new ones, but because Williams and a handful of his like-minded colleagues know the old ones are better. The school’s parent-teacher association buys them from used bookstores because the county won’t pay for them (despite the plentiful money for technology). His preferred algebra book, he says, is “in-your-face algebra. They give amazing outstanding examples. They teach the lessons.”
The modern textbooks, he says, contain hundreds of extraneous, confusing, and often outright wrong examples, instead of presenting mathematical ideas in a coherent way The examples bloat the books to thousands of pages and disrupt the logical flow of ideas. (For instance, the standard geometry book for Fairfax County, which is used in schools around the country, tries to explain what a mathematical point is by analogy to pixels on TV screens, which are not in fact point-like.) Teachers at other schools in the county have told him that they would rather use the old books, too, but their principals would kill them. Other teachers have told me the same about new technologies—they, like Williams, think the technologies are ineffectual, but lack his courage to oppose them. According to an October 2011 report, 89 percent of high school math teachers think their students are ready for college-level mathematics. But only 26 percent of post-secondary teachers think the students are ready once they get there. This shortfall in mathematical preparation for college-bound students has existed for a long time, but it is being exacerbated by the increased use of technology. College-level math classes almost never use graphing calculators, while high-school classes invariably do. College professors want their students to understand abstract concepts; technology advocates claim their products help teach students such abstractions, but in practice they simply don’t. Take the Promethean, one of the two interactive whiteboards the school uses. When I asked a Longfellow science teacher what she could do with the Promethean she couldn’t do on the blackboard, the first thing she showed me was a music video featuring a Rube Goldberg machine. She did not intend this ironically. The second thing she showed me was a drawing of an electric circuit in which wires connect a light bulb to a battery. When the circuit was closed, the bulb lit up. This drawing goes to the heart of the technological disconnect. Her students like it when the bulb lights up, she says, because it reminds them of a video game. But this shortcut is dangerous. Learning how to visualize—as required when an electric circuit is drawn on a blackboard—is vital for developing the ability to think abstractly. You also have to make students manipulate real circuits with real batteries, with real wires that connect them and sometimes break. Showing them a toy circuit in computer software is an unhappy middle ground between these two useful teaching exercises: You neither learn how to trouble-shoot in the real world, nor do you think clearly about how electrons work.
Math and science can be hard to learn—and that’s OK. The proper job of a teacher is not to make it easy, but to guide students through the difficulty by getting them to practice and persevere. “Some of the best basketball players on Earth will stand at that foul line and shoot foul shots for hours and be bored out of their minds,” says Williams. Math students, too, need to practice foul shots: adding fractions, factoring polynomials. And whether or not the students are bright, “once they buy into the idea that hard work leads to cool results,” Williams says, you can work with them. Educational researchers often present a false dichotomy between fluency and conceptual reasoning. But as in basketball, where shooting foul shots helps you learn how to take a fancier shot, computational fluency is the path to conceptual understanding. There is no way around it.
The fight between those who seek a way around hard work (a “royal road to geometry,” in Euclid's famous phrase), and those who realize that earned fluency is the only road to understanding goes back millennia and became particularly acrimonious in America in the last half-century in the so-called math wars. On one side are education researchers like Constance Kamii, at the University of Alabama, who argues that teaching children to add and subtract is harmful. This camp says it has insights into the way children learn that warrant departure from traditional ways of teaching math. On the other side is the consensus of working scientists and mathematicians as well as teachers like Williams, who notes that it took very smart adults thousands of years to develop modern mathematics, so it makes sense to teach it to students rather than get them to “discover” it themselves.
What is new to this fight is the totalizing power of technology. A 2007 congressionally mandated study by the National Center for Educational Evaluation and Regional Assistance found that 16 of the best reading and mathematics learning software packages—selected by experts from 160 submissions—did not have a measurable effect on test scores. But despite this finding, the onslaught of technology in education has continued. The state of Maine was the first to buy laptops for all of its students from grades seven to 12, spending tens of millions of dollars to do so, starting with middle schoolers in 2002 and expanding to high schools in 2009.
The nation is not far behind. Though no well-implemented study has ever found technology to be effective, many poorly designed studies have and that questionable body of research is influencing decision-makers. Researchers with a financial stake in the success of computer software are free to design studies that are biased in favor of their products. (I’m sure this bias is, often as not, unintentional.) What is presented as peer-reviewed research is fundamentally marketing literature: studies done by people selling the software they are evaluating.
For instance, a meta-analysis of the effectiveness of graphing calculators from Empirical Education Inc. reports a “strong effect of the technology on algebra achievement.” But the meta-analysis includes results from a paper in which “no significant differences were found between the graphing-approach and traditional classes either on a final examination of traditional algebra skills or on an assessment of mathematics aptitude.” In that same paper, calculators were marginally helpful on a tailor-designed test. The meta-analysis included the results of the specially made test, but not the negative results from the traditional exam.
Take this gem from researchers at SRI International. They say that standardized tests don’t capture the “conceptual depth” students develop by using their software, so the “research team decided to build its own assessments”—and, of course, they did relatively well on the assessments they designed for themselves. Another example: A recent study by the Educational Development Center compared students who took an online algebra 1 class with students who took nonalgebra eighth-grade math. The online students did better than those who didn’t study algebra at all (not exactly surprising). But the online students weren’t compared with those who took a regular algebra class.
Despite the lack of empirical evidence, the National Council of Teachers of Mathematics takes the beneficial effects of technology as dogma. There is a simple shell game that goes on: Although there is no evidence technology has been useful in teaching kids math in the past, anyone can come up with a new product and claim that this time it is effective.
I tried using one such product, Cognitive Tutor from Carnegie Learning, which claims to be “intelligent mathematics software that adapts to meet the needs of ALL students.” One problem asked me to calculate the width of a doorframe, given the frame’s height and a diagonal measurement of the door. After 30 seconds’ work with pen and paper, I submitted my answer: 93.7cm. But Cognitive Tutor wouldn’t accept it. It wanted me to go through an elaborate and cumbersome series of steps to get its answer: 93.723. This isn’t teaching math—it’s teaching how to use a particular software package. The supposed “real-world applications” don’t even reflect the real world. Show me a tape measure that allows you to measure to one-hundredth of a millimeter.
Though serious empirical research fails to show any beneficial effects of technology, it also doesn’t demonstrate any harm. The emphasis on technology is in part damaging because of its opportunity cost, both in effort on the part of policymakers and in terms of money. It also distracts from the real problem: teachers who don’t understand enough about math or science. This has been a problem for a long time.
A report earlier this year from Michigan State University showed that K through eight teachers with no math specialization (the vast majority—more than 90 percent of K through six teachers and more than two-thirds of sixth- to eighth-grade teachers) got only half the questions right on a base-line test meant to see whether they knew the material they wer supposed to be teaching.* The good news is that most teachers are aware of their own limitations: Only about 10 percent of the nonmath specialization K through eight teachers said they were “confident to teach all topics” in math.
Hung-Hsi Wu, a math professor at UC-Berkeley (and another member of Bush’s math panel), has been running three-week classes for elementary and middle school teachers every summer for the last dozen years. His “students” must wrestle with deep mathematical questions that both pertain directly to simple math and are poorly understood by most teachers. Why does (-2)x(-3)=6? The answer isn’t straightforward, and Wu takes several pages to give it. If you don’t understand it, though, you don’t really understand multiplication. But Wu has only been teaching about 25-30 teachers a summer—there is money for new technology but little for comprehensive teacher training. Meanwhile, the new technology makes it easier than ever for teachers to avoid learning their subject. Promethean, the “interactive whiteboard” company, advertises as a selling point the fact that teachers can share lesson plans online. But drawing up a lesson plan is itself educative: A teacher who plans his own lecture is forced toward mastery of the material, but one who downloads a PowerPoint presentation doesn’t have to know anything beyond how to download the presentation. It is a mirage of efficiency: empty calories.
The real shortfall in math and science education can be solved not by software or gadgets but by better teachers. Programs like Wu’s can make more teachers more like Williams. That’s where efforts should be focused, not on imagined technological solutions, which obscure more than they reveal.
In this, the new Common Core standards for math, which were adopted with lightening speed by 45 states and Washington, D.C., fall short. They fetishize “data analysis” without giving students a sufficient grounding to meaningfully analyze data. Though not as wishy-washy as they might have been, they are of a piece with the runaway adaption of technology: The new is given preference over the rigorous.
Computer technology, while great for many things, is just not much good for teaching, yet. Paradoxically, using technology can inhibit understanding how it works. If you learn how to multiply 37 by 41 using a calculator, you only understand the black box. You’ll never learn how to build a better calculator that way Maybe one day software will be smart enough to be useful, but that day won’t be any time soon, for two reasons. The first is that education, especially of children, is as much an emotional process as an imparting of knowledge—there is no technological substitute for a teacher wh cares. The second is that education is poorly structured. Technology is bad at dealing with poorly structured concepts. One question leads to another leads to another, and the rigid structure of computer software has no way of dealing with this. Software is especially bad for smart kids, who are held back by its inflexibility.
John Dewey, the father of American education reform, defined miseducative experiences as hose that have “the effect of arresting or distorting the growth of further experience.” “Growth,” he wrote, “depends upon the presence of difficulty to be overcome by the exercise of intelligence.” The widespread use of computer technology is inimical to the exercise of intelligence. I fear this is no more than shouting into the wind, but resist it while you can, because once it gets locked in—as our food system is, to monocultures and antibiotics in factory farms—it will be even tougher to get away from.
Correction, June 25, 2012: This article originally misidentified the university that carried out a study of teachers’ knowledge of math. It was Michigan State University, not the University of Michigan. (Return to corrected sentence.)