Scott Kim on TED.com

For years I have been using Scott Kim's Inversions to teach the concepts of mirror and rotation symmetry in Geometry class. The students love combining art and math and are able to challenge themselves while having fun.

I have never met another teacher who uses Scott Kim's book (ok, we all like to think we are unique) so was delighted to see that he now has teacher resources, relating to the Inversions book, on his website scottkim.com. I had developed my own methods for teaching the concepts and it is interesting, and helpful, to see exercise developed by the artist.

The video passes too quickly through Kim's examples to truly appreciate them (I paused the video numerous times and I was familiar with most of the slides) so I recommend visiting the website where one can view galleries of the artist's work.

The assignment I use is the following: The students must make two drawings, one using rotation symmetry and one using mirror symmetry. The may use their name (unless they have a naturally symmetric name) or a term relating to mathematics. To give the students inspiration, I show them former students' work and Mr. Kim's book. The students are not allowed to plagiarise the examples but can incorporate the design ideas in their own work. I make use of the windows in the classroom to help the students trace elements of their drawings. For anyone interested in further information in teaching symmetry can send me an email.

Scott Kim takes apart the art of puzzles | Video on TED.com

Not About Math - Phonics and Letters

Methods change over the years. There was 'phonics' then 'whole language'.

Our 7-year old recently came home talking about phonograms (my spell check doesn't even recognize that word). "A the four letter A, E the two letter E, A the two letter A," were the phrases that were being chanted around the house and I didn't have a clue what she was talking about. But then, isn't the internet great!

In case any of you are feeling oldish and out of the loop, here is an excellent site that gives a thorough explanation of phonograms and also, how to write the alphabet (the modern way).

Phonogram Page

The Mathematics of Metaphore: x = y

I have routinely and successfully used metaphor in teaching mathematics especially when describing how the brain works. Every once in a while I find a student who resists: "Why are you talking about learning to keep your balance on a bicycle? I thought we were talking about studying for the chapter 5 test?"

I found the following video, newly added to the TED.com repertoire. James Geary is amusing as he describes the importance of metaphor in human thought.

It Makes Me Want to Shout!

"Learning to Teach to Bridge the Achievement Gap". This is the currently the biggest buzz phrase in education. It is powerfully loaded, politically correct, eduspeak. But what does it really mean?

'Closing the gap is this generation’s civil rights issue, said Charles Weis, superintendent of schools in Santa Clara County. “We know what needs to be done; we know how to do it,” Dr. Weis said at a rally for educational improvement. Yet, he added, “educators are notoriously bad at adopting others’ good ideas".'

So what are some of those good ideas? Well here is one being used in a school district in Santa Clara County:

'On a recent day in Martha Borg’s third-grade classroom, she asked three students to join her at a small semicircular table. While the other students worked in their notebooks or with tutoring programs on computers, the three sounded out some new words — like “natural,” “environment” and “tortoise” — and then read aloud as Mrs. Borg moved from one to the next.'

That is the way that reading was taught 40 years ago. It appears to be working once again. We didn't use computers, instead we practiced our math algorithms (but that was before it was called drill and kill). And guess what? Students don't know their math facts anymore, especially fractions, but at least they haven't suffered sudden death due to practicing their multiplication facts and perhaps more of them are learning to read again.

Those educators who are "notoriously bad at adopting others’ good ideas", by Superintendent Weis' standards anyway, might just be the ones who, having practiced in the field for years, know what works. In this instance, the "new" method is indeed the "old" method which was sloughed-off some time ago in favor of some new fad in reading education.

"Small groups are a part of all classes at Anderson, and students get small-group time with the teacher. The groupings cluster students of similar skills, as determined by another practice the new leadership introduced: continual assessment. The assessments then guide individual instruction."

The bold and italicized type is mine. My goodness! Another example of the reversal of a longtime trend of integrating students of different skill levels. What is being said here? That teaching small groups of similarly skilled students is effective. Don't all of us of a certain age remember reading groups 1, 2 and 3? But then they changed the names of the groups (red, yellow, and blue) so the students wouldn't feel bad if they were in a lower level group, except that the plan didn't work and all the students knew which group was reading more difficult material. Sorting students into ability groups was called "tracking", a reviled practice now, and there has been a movement to abolish it for the last 30 years.

Some of the rationale to abolish tracking was good. Too many students who had the ability to do well in school were often placed in the lower tracks. This generally correlated with race and class. Conversely, students who should never have been put into an "honors" class were placed there due to their place in society and the influence of their parents.

Teacher training programs claimed that multi-level learning groups were more effective. Of course research was cited and examples given. The efficacy of the able students helping the less-able students to master the material was touted as an idea. But if a person thinks about it for more than 2 seconds, the grand plan doesn't make very much sense in terms of efficiency or outcomes.

It is ironic that a type of program that was tossed because it was unfair is being re-introduced, under a different name of course, to "help close the achievement gap".

It makes one want to shout! I hope Arne Duncan is listening.

Test Scores and Family Income

The NYT had an interesting article about the strong correlation between family income and SAT scores.

As a society that has become more systematized, we might want to start looking at what systems middle-income and wealthy people are using at home to support their kids in school and use that information to help educate lower-income families to provide that effective support for their kids too.

Parents: Examine Your Math Attitude

Many years ago a friend of mine made a huge deal about how she hated math when she was a kid and how she couldn't even balance her checkbook. Not surprisingly, her daughter, who was in the second grade, wasn't doing well in math even though she did well in all of her other subjects.

My friend was a brilliant woman, college educated with a masters degree. Yet, she claimed to know math only to about a 3rd grade level (balancing a check book requires elementary school level math). I warned her that she was most likely passing her 'attitude' about math to her daughter.

Over the years my friend kept me informed of her daughter's progress in school. After our chat, her daughter started excelling in math, took the advanced, honors math classes in high school and finished by taking A.P. Calculus as a senior.

Parents are likely to pass their fears to their children. If your loathed and detested math when you were a youngster and you are raising kids you might want to read this article from Parenting Journals.

The moral to the story is that if you want your child to succeed in math, and you detest math, then you need to at least fake it when your kids are of formative age.

More Vindication

The main policy of the district where I work: Close the achievement gap.

I have done this before with a forward thinking Principal. Double the time that students take math and the achievement gap narrows.

From an article in the New York Times:

‘No Child’ Law Is Not Closing a Racial Gap

'Where we see the gap narrowing, that’s because there’s been an emphasis on supplemental education, on after-school programs that encourage students to read more and do more math problems,” Dr. Hrabowski said. “Where there are programs that encourage that additional work, students of color do the work and their performance improves and the gap narrows.”'

But then there is the problem that most math teachers are familiar with:

'But Dr. Hrabowski said said that educators and parents pushing children to higher achievement often find themselves swimming against a tide of popular culture.

“Even middle-class students are unfortunately influenced by the culture that says it’s simply not cool for students to be smart,” he said. “And that is a factor here in these math and reading scores.”'

I love it when what I read confirms what I know.

URL Change

Hello Readers,
Please note that the url to my blog has changed.
It is now:
http://www.mathmethinks.com/

A Personal Odyssey

I felt vindicated when I read Thomas Sowell's autobiography.

Any teacher who considers rigor to be important, who has been tormented by students who want the good grade but are too lazy to work for it, who has felt pressured to compromise his or her standards by 'politically correct' but misguided administrators, who has been bullied by parents who think their children are little gods and goddesses, will appreciate Sowell's perspective on education.

The mediocre and poor teachers in the system, who are soft on rigor and give easy 'As', are the educators that no one complains about. It is a pity that so many people, probably the majority, would rather get a meaningless 'A' than really learn something but have to work for it. It is delightful to read about an educator who had the courage to stand by his principals instead of bend to the pressure of just about everyone.

Daniel Tammet Can Recite 22,514 Digits of pi From Memory

From Scientific American: Daniel Tammet can recite 22,514 digits of pi. He holds the European record for such a feat. So, I guess there is a competition? The article covers an interview with Mr. Tammet in which he discusses memory and I.Q. tests.


As I peruse the internet I am fascinated with the many different ways people spend their time. How long did it take Daniel Tammet to memorize that many digits of pi?

Typing pi to 22,514 digits in a continuous (no spaces) stream, in a Word document with smaller than average margins, using Times New Roman, size 12 font, single-spaced, would take 5 or 6 pages of paper.

An interesting relation: Suppose Mr. Tammet can recite 2 digits per second (a reasonable estimate). Then it would take approximately 3.13 hours to recite 22,514 digits. In other words, it would take close to pi number of hours to recite pi to 22,514 digits.

Arabic Tiling or Tessellation

This was created in Geometer's Sketchpad, put could just as well be created with a compass and straight-edge.

1. Start with a circle. Draw lightly if using a pencil, because the circle will be erased later. Make certain the center point is marked.


2. Using the same diameter, draw circles around the original one, using the intersections as the center points. Each outer circle intersects the center of the original circle.



3. Continue until six circles are drawn around the original.

There are a couple of interesting asides:

Six circles, with a constant diameter, fit perfectly, overlapping at the center points, around a center circle with that same diameter.

Also, six circles of the same diameter, fit perfectly around a center circle with that diameter.




4. Draw sets of parallel lines through the intersection points. Again, if using a pencil, draw lightly because the lines will have to be erased.



5. Continue, drawing the parallel lines in the 3 possible directions.



6. Erase the original circle design. Continue drawing parallel lines in the three different possible directions. Use the intersection points as guides.



7. More of the same.



8. For my design, I continued until I could draw a hexagon, whose sides were the width of 6 rhombuses.



9. Using the lines, starting from the center, draw a pattern of stars and hexagons. I chose to put a star at the center, but it could easily have been a hexagon.



10. Erase (delete) the lines and start coloring the stars...



11. ... and hexagons.

12. Finished (points were also deleted).

Social Interaction and Cell Phones

A Quick Plug for TED - I love this site

The TED website is a great source of talks by interesting and inspirational people. TED stands for Technology, Education, Design. The video production is minimal, and that is probably an overstatement. However, the focus is, therefore, entirely on the presenter and is, in a way, refreshing.


Social Interaction and Cellular Devices



In this presentation, Renny Gleeson, talks about the way we socially interact with cellular devices. I dedicate this to all teachers who struggle with monitoring cell phones in the classroom. Compared to the examples that Mr. Gleeson presents, with the exception of the guy on the motorcycle, our students have these people beat in creative, sly methods of using the cell phone.

A brief list of a dozen ways that students can be sly using the cell phone. All of these examples are actual accounts*.

1 Backpack on lap - the backpack is on the lap with a compartment, the one containing the cell phone, unzipped allowing free access to texting. When caught the student just acts like the teacher is nuts. (Backpacks on laps should be restricted.)

2 Under the desk - one of the most common methods of texting. This one can be detected quite easily because the students don't realize that even though THEY cannot see the phone, the teacher can. When caught, act like the teacher is seeing things and claim she is nuts.

3 In the desk - used in middle and elementary schools where the administrators made the mistake of ordering desks with 'cubbies'. This is much more effective than the 'under the desk' method, even though the students are usually younger. Fortunately younger students don't always think to claim that the teacher is nuts.

4 In the front hoodie pocket - teachers should be suspect of all students wearing hoodies with front pockets. If they don't, they are nuts.

5 In the saggy, baggy pants pocket - teachers should be suspect of all students wearing saggy, baggy pants, especially since they are outmoded and no teenager concerned about his image would be caught wearing them. Any student with hands in the pockets for a sustained duration of time should be suspect, otherwise the teacher is nuts.

6 Switch the cell phone for an mp3 player when caught - then when sent to the dean, blame the teacher for allowing mp3 players in class on special occasions and claim to be confused about the classroom rules. The dean thinks the teacher is nuts for allowing mp3 players on special occasions.

7 Carry two phones then when discovered using the phone swap the working phone for one that doesn't work - then when sent to the dean, claim the teacher is nuts.

8 The calculator excuse - when caught in math class, claim to be using the calculator function. Act like the teacher is nuts when she doesn't buy the logic.

9 What time is it? - when caught, in any class, claim to be needing to check the time (even though at least 90% of the time a student can see the classroom clock). Then act like the teacher is nuts for not buying it.

10 Claim that you were using the cell phone as a 'translation' device - take the Spanish teacher's cell phone out of her purse and then when caught and sent to the assistant principal, and in the presence of your father who has been called to the school, claim that the teacher always lets the students use her phone as a translation device, and of course that is why you are justified to have removed it from her purse. Act like you are the only sane one and everyone else is nuts.

11 Feign ignorance - when caught, by the teacher, while using the cell phone, the student feigns complete innocence and claims the teacher is completely nuts, because she obviously imagined things. When using this method, a student will try to turn the phone off so the teacher can't check the texting history.

12 Actually use the cell phone for an mp3 player - used on that special occasion when the mp3 player is allowed (see #6).

*These methods work because, unlike my generation, the students are able to text without looking at the key pad. I haven't found a correlation between a students grade in class and their skill in using the cell phone.

Origami and Mathematics - Robert Lang Gives a TED Talk

In his TED talk, Robert Lang discusses folding beyond paper cranes, taking origami to new heights using mathematics.



Visit Robert Lang's website to see his paper art. The site provides folding patters for pieces such as this 'Flying Katydid'.

Most Students Fear Math

Study: Most colleges students fear math,

so the headline reads.

Ok, let's forget the grammatical error (from a publication that claims, as a subtitle, "100 years of journalistic excellence").

The headline had me laughing. My inner sarcastic voice was saying, "Duh!"

I wonder how much the study cost? I wonder if a straw poll of the university math professors could have given a close result? If I were a person that bets, I'd put money on the professors having the same results with a poll put in their boxes and no funds lost.

According to this research, done at the University of Granada in Spain, 6 out of 10 students have math anxiety and the anxiety is reported more in the Health Sciences than in Technical Education. (I think the ratio is probably higher in the U.S. - if I were a betting person, I'd put money on it.) And I'm certain that I'd rather have an engineer, than the doctor, figure out the mathematically correct dose of medication for my child.

If anyone wants to know this kind of information, they should just ask a math teacher.

Students Sue Prosecutor in Cellphone Photos Case

Good for the students. When prosecutors, adults, take absurd situations to an extreme, ultimately hurting the people they have a duty to protect, they should be punished.

I was outraged when I first heard about the prosecutors intent while listening to the radio and am glad to hear that the girl involved is taking action. Shame on the prosecutor.

Read about the story here:
NY Times article.

Make a Virtual Kaleidoscope

I stumbled upon this program. It is an early lesson on symmetry and also reminiscent of the toy kaleidoscope of yore.


These designs are made from the same set of 'virtual objects'. As the highlighter rotates, portions of the design are illuminated creating the kaleidoscope effect.























zefrank.com :: kaleidoscope

Alice Waters - The Edible Schoolyard

Watch CBS Videos Online

What Does One Trillion Dollars Look Like?

I posted a link to a site that gave size perspective to collections of pennies. Here is a site, PageTutor.com, that gives us perspective on how big a trillion is in one hundred dollar bills.



Doubling the image, we have 2 trillion dollars. The deficit is at 1.75 trillion dollars. Can you see the little guy in the corner?

Teach a Man to Fish

"Give a man a fish; you have fed him for today. Teach a man to fish; and you have fed him for a lifetime."

—Author unknown

Education is about learning how to think. It is only when a person can truly think and analyze her own process that he or she will become an independent life-long learner with the ability to be skeptical when necessary. We call the ability to analyze one's own thoughts 'meta cognition' in education lingo.

It follows that the emphasis on any education should be about teaching students to think, in a variety of disciplines... the more the better.

If you look at our system, at least in California, you will see that we teach students how to perform on multiple-choice, standardized tests. Thus we will have a generation of students who can perform well on standardized tests. I am fairly certain though, that this type of training is not teaching the students to think.

More on that later.

The Discussion About Grades Returns

What is an academic grade? In 1950 a C meant 'average'. By today's standards, a C means 'failing'. In 1950 an A meant 'superb', 'excellent', 'exceptional'. Today an A means 'passing'.

Today's grade inflation is another incidence of the 'Lake Woebegone Effect', the syndrome where everyone's "children are above average". Thank you Garrison Keillor, for giving us the name.

Do we, as teachers, give everyone an A? Or do we give a student with average talent and performance a C and risk the wrath of the student and his/her parents? Does 'working hard' deserve an A even though the 'product' or 'outcome' is mediocre or even poor? Perhaps these are some questions that we need to ask ourselves.

If my banker 'tried' but made an error on my account, costing me money, I would complain to the supervisor and find a new bank if there continued to be errors. If the chef at my favorite restaurant tried 'really hard' but burned my steak all the same, I'd send it back. If the Verizon manager, try as she might, thought that there was no difference between two-thousandth of a dollar and two-thousandth of a cent, and then be idiotic enough to tell me that understanding this difference was 'a matter of opinion', I'd find another service provider.

I ponder and shudder: What are we doing, for the sake of our community, by telling our children they are always wonderful when they are not? Why would a student bother to try to be better when he or she is already great (but not)?

Currently, 60% of our students earning PHds are foreign born students, educated in other countries, many in Southeast Asia, where students are required to memorize their times tables and periodic tables. In 10 years that figure will be 75%.

For some more perspective on the issue of grade inflation see Student Expectations Seen as Causing Grade Disputes and an editorial response ‘A’ Is for Achievement, ‘E’ Is for Effort .

Industry Makes Pitch That Smartphones Belong in Classroom

Earlier this month I posted an audio recording called, "Verizon Math Fail". The recording features a man debating a team of Verizon customer service representatives about whether $0.002 is equal to 0.002¢. On the tape he lost the battle because NONE of the people at Verizon understood the math.

I read, on his blog, that Verizon later held in-service training to correct the problem.

So, I was astonished when I saw this headline in the NY Times this morning: Industry Makes Pitch That Smartphones Belong in Classrooms. The article begins:

"SAN FRANCISCO — The cellphone industry has a suggestion for improving the math skills of American students: spend more time on cellphones in the classroom."

I have taught in schools for the past 17 years and seen the onset, and accompanying problems, of cell phones in the classroom. Students would much rather be fiddling with cell phones than thinking about math.

Using the internet, calculators, and cell phones to do math is a passive way of learning and doesn't develop brain structure as effectively as learning to make the calculations with a pencil and paper or slide rules.

I am sure that if the Verizon employees had used slide rules in math class, rather than calculators, they would have understood the difference between two-thousandth of a dollar and two-thousandth of a cent.

When a slide rule is the tool, the student must calculate where the decimal point is to be placed. Using the slide rule is, in itself, practice in placing decimals and understanding the size of numbers. The brain develops to understand the task. This doesn't happen when a student uses a calculator.

I wonder if the people who make these decisions think that they could have learned to ride a bicycle by playing a video game of riding a bicycle?

If Verizon (and the rest of the cell phone providers, i.e. 'The Industry') think that Smart Phones are a useful tool to learn math, I suggest they provide them to their employees and let them get started.

The Fresh Air Fund is Looking for Camp Counselors

I received the following message from Sara at the Fresh Air Fund:

"The Fresh Air Fund is now accepting applications for counselors for this coming summer of '09. We hire staff members with a wide range in some pretty amazing fields. We are looking for college-aged men and women who love to work with children. I put together this social media news release which explains it all:

http://freshairfundcounselors.smnr.us/

It would be so helpful if you could post a mention on Math Me Thinks . If you are able to post, please send me the link so I can share it with the team.

We are also always looking for Fresh Air hosts for the summer to open their homes to a child, and any help would be appreciated. Thank you so much and please let me know if you have any questions."

Check our their video.

Another Story Involving Bad Math, Bad Science

I found the article Green Guru in a facebook post. It is from the publication First Things: The Journal of Religion, Culture and Public Life and was posted on February 2, 2009 by Stephen M. Barr.

Mr. Barr claims that,

"...if no one had more than two children, as the green guru would want it, the fertility rate could probably not be gotten above 1.4. In twenty generations the world population would plunge to less than 2 million."

This is 'voodoo math'. The error should be obvious by looking closely at what Barr is claiming. If the growth rate remains positive, then the population grows. When the growth rate declines, population growth slows. A population declines when the growth rate is negative. In other words, there is a big difference between the rate declining and the population declining. (Please comments.)

Mr. Barr should read 'The Population Bomb', 1968, by Dr. Paul Erlich, a Stanford University biologist, and 'The Population Explosion', 1990 by Paul and Anne Erlich. These books explain population growth in a manner that a lay person can understand. The math involved, in understanding population growth, can be understood by the average high school student.



To see how the basics of population growth works, see the illustration above. It is representative of the growth of a fictional family over a 100-year period, following the rule, mentioned by Mr. Barr, that a couple limits itself to begetting 'replacements' and thus has only two children.

The first couple, Andrew and Alana are born in 1900 and have two offspring, Brian and Bertha, around 1925. The family reproduces according the the 2-child rule for the rest of the century. Anyone familiar with rudimentary mathematics, sees the structure of '2 to the nth power' (an exponential function) as the basis for the population growth in this example. Alas, people die. The names in lavender, represent people who have perished.

In this example, there are 23 more people on the planet in the year 2000, compared to 1900. The rate of growth is 23 people per hundred years (0.23% per annum).

To compare the fictitious example to real life examples, we can look to the CIA who keeps statistics on population growth throughout the world. Their definition of Population Growth is as follows:

The average annual percent change in the population, resulting from a surplus (or deficit) of births over deaths and the balance of migrants entering and leaving a country. The rate may be positive or negative. The growth rate is a factor in determining how great a burden would be imposed on a country by the changing needs of its people for infrastructure (e.g., schools, hospitals, housing, roads), resources (e.g., food, water, electricity), and jobs. Rapid population growth can be seen as threatening by neighboring countries.

Referring to the chart below which uses 2008 statistics, we can see that, even accounting for war, the population growth rate in Afganistan is 3 times the rate in the U.S. The growth rate in the Gaza Strip is almost 4 times that of the U.S. The actual rate for the U.S. is almost 4 times the rate of our fictional family. The lowest rate, in this chart, is that of Russia. The growth rate in Russia indicates a population decline, as stated previously.

Mr. Barr states later in his article:

"Judging from remarks that colleagues have made in my presence, there are a remarkably many otherwise intelligent people who agree with the green guru that it is irresponsible to have more than two children, and who pride themselves on having stopped at two, as though they were benefiting society thereby. I suspect that this widespread attitude, based on an elementary mathematical error, may be one reason for the woefully low birthrates in economically advanced countries."

Oh really Mr. Barr! I think you might want to rethink your analysis.

Verizon Math Fail - Why I'm No Longer With Verizon

You really have to see this! I especially love it when the manager explains that her incompetence in math is a 'difference of opinion'.

Update!

The gentleman's response to Verizon.

Obama's Education Promises

Dave Murray, in his blog Head of the Class, urges those of us who are interested in education to keep track of President Obama's campaign promises regarding education.

Mr. Murray has sifted through 510 campaign promises from PolitiFact.com and found 37 that involve education. He has posted them to Head of the Class.

Thank you Mr. Murray.

Quote of the Week

Regarding the 'Dark Ages' in the last post, this editorial quote is in reference to a Texas school board's decisions regarding 'science' education.

"The lesson we draw from these shenanigans is that scientifically illiterate boards of education should leave the curriculum to educators and scientists who know what constitutes a sound education."

Amen!

The Obama Effect

Emerging from eight years of the second coming of 'The Dark Ages', our nation has hope. But the information in the article "Study Sees an Obama Effect as Lifting Black Test-Takers" in the New York Times, is misleading. The article doesn't provide enough information for us to know the circumstances that surrounded the exams, let alone how educators might use the study to help close 'the achievement gap' in schools.

The Obama effect:

"Now researchers have documented what they call an Obama effect, showing that a performance gap between African-Americans and whites on a 20-question test administered before Mr. Obama’s nomination all but disappeared when the exam was administered after his acceptance speech and again after the presidential election."

For example, were the researchers comparing students from the same neighborhood, with the same family income, whose parents had the same level of education? In other words, are all variables the same except for the fact that Barak Obama is now President of the United States?

The educators and teachers, who I know, are working overtime, on their lunch hour, after school, and in the evenings, to help close the 'achievement gap'. It would be a miracle if all that were required was for the teacher to march around the classroom with a placard of Barak Obama, chanting "remember our President", and presto! no 'achievement gap'.

A Question of Math Literacy

There is a paucity of interesting math education stories on the internet. In my search, I found this fragment from TusconCitizen.com:

Our organization, the Center for Economic and Entrepreneurial Literacy, has just released the startling results of a survey that shows our financial literacy tracks our math skills.

According to our survey, the same math problems plaguing 15-year-olds continue to vex us into adulthood. The average American cannot answer basic math questions involving percentages.

For example, 65 percent of respondents could not identify what remained if you subtracted 25 percent from eight. Another question revealed that 1 in 3 adults could not calculate 1 percent of 50,000.

Although the point of the article is pertinent to the state of math education in the nation, I can understand how so many people might not be able to identify "what remained if you subtracted 25 percent from eight".

The question is ambiguous. Does it mean "what remains if you subtract 25% of eight from eight?" Or, does it mean "what remains if you subtract 25/100 from 8?" These two questions have very different answers.

This being so, the problem seems to be with the illiteracy of the person who posed the question, and not the innumeracy of the person trying to answer it.

Guilloché Pattern Generator

Anyone who ever enjoyed playing with the Spirograph will enjoy the Guilloche Pattern Generator.

Definitions of Terms Commonly Used in Math

Continuing in the theme of finding funny math stuff on the Web, I give a partial list of definitions from In Between Meals. The see the rest of the list, visit the sight.

CLEARLY: I don't want to write down all the in-between steps.

TRIVIAL: If I have to show you how to do this, you're in the wrong class.

OBVIOUSLY: I hope you weren't sleeping when we discussed this earlier, because I refuse to repeat it.

RECALL: I shouldn't have to tell you this, but for those of you who erase your memory tapes after every test, here it is again.

WITHOUT LOSS OF GENERALITY: I'm not about to do all the possible cases, so I'll do one and let you figure out the rest.

ONE MAY SHOW: One did, his name was Gauss.

Interesting Websites About Pennies

This website discusses size (number, area, weight, dollar amount) in terms of pennies. It is a great site for kids to help understand how big is big. The Mega Penny Project

This sight, as an aside, has a section on coin stacking. It is fascinating to see what people do with their time.

Lego Kit - The Penrose Triangle

I found this on Stumble!