The Brothers Marshmallow

Yes. it's time to start making lists again over at Marshmallow Fight blog. We need your list of top ten favorite books. The rules are at the other site. You can get there by clicking here.    Do your best - ten years after you are gone we'll have forgotten most of the things about you except these lists - so make them great - they are your legacy. Let's try to get wrapped up in the next four weeks, so the deadline is  the end of March for those who believe in deadlines. For those who think of them more as guidelines, I'll be at your house March 28 with a well charged TASER. We'll see how those guidelines feel to you then. ENJOY going through your book cases and Kindle lists. See you in a month. Some of you won't see me until you stop shaking - the second time.     

Just arrived from Amazon:   

Forever?

In my new tutoring job, I am working with Calculus students who are not doing as well as they (or their parents) would like. Our most recent topic has been infinite series and I am reminded of how much I loved teaching infinite series. The concepts involved in series are some of the least intuitive in mathematics, but some of the results are the most amazing in mathematics as well. A series of numbers is a sum: 1 + 1/2 + 1/4 + 1/8 + ... We are especially interested in sums that converge. If you look at the sum of the series in the last line after each new term is added, you see a pattern. After the first term, the sum is 1; after you add the second term the sum is 1 1/2. After the third term is added, 1 3/4 ( 1 + 1/2 + 1/4); 1 7/8 after 1/8 is added, and so on. The sum keeps getting closer and closer to 2, but will never quite reach it. Mathematically we say the sum after infinitely many terms is 2.
What makes an infinite series interesting is that functions have infinite series representations. I don't know if you gave it any thought in your high school trig class, but when you typed sin (1) into your calculator, your calculator did a series to figure it out. My students always assumed that the calculator had a table to look up the values, but if you have done anything with computers, you know how much memory tables take up, A calculator would need to have tables that would include sin 1, sin 1.1, sin 1.11, sin 1.111, and so forth. That's a lot of values to remember. More importantly,  there is the question of where do the values in the table come from.
Luckily the sine function has an infinite series:

Those exclamation points are factorials: 3! means 3 X 2 X 1 = 6;   5! means 5X4X3X2X1 = 120;   7! means 7 X 6 X 5 X 4 X 3 X 2 X 1= 5040, and so on.
So sin 1 = 1 - 1/6 + 1/120 - 1/5040 + 1/362880 - ....  Your calculator just keeps adding terms following this pattern until it doesn't see any changes in the first twelve or so decimal places. Then it shows the answer as a decimal rounded to 8 decimal places (depending on your calculator).  Instead of a table with lots and lots of values, it just needs to remember a formula.

If you have invented a fast new super-computer and you would like to compare its speed to the old ones, one way to do it is to use an infinite series to find pi to lots of decimal places. Yes, that's how someone figures out the first fifty digits of pi:

                          

A really easy series to use is  pi =  4/1  -  4/3  +  4/5  -  4/7  +  4/9  -  4/11  +  ..., but this series converges pretty slowly, so it takes a lot of terms to get an accurate estimate for pi (90 terms just to get to 3.14).  So over the years mathematicians have come up with more complicated infinite series to compute pi.   

Using series and a little guidance, my calculus students would generate a surprising formula attributed to Leonhard Euler (pronounced Oiler), a highly regarded Swiss mathematician of the middle 1700's. A few days ahead of this, I would ask my students to list what they thought were the most important numbers that lay at the foundation of mathematics. Every year, the group would come up with the same five numbers 0, 1, pi, i (the imaginary unit - the square root of -1),  and e (about 2.718 - the base of natural logarithms - a huge topic in calculus). Then we would derive Euler's formula using infinite series: 

e ^iπ  +  1  =  0;  all five of the numbers they picked in one simple formula.  Isn't math amazing?

Pop Science Loses Again

In nearly 40 years of teaching, I have been assaulted many times with the newest fad. It has been especially painful to see math teachers and supervisors who have not a clue about how real science works. They seem to follow one basic premise when evaluating new tools and techniques: "Well, that sounds good - I'm sure it works." And off we would go on another wild goose chase to improve schools. Somehow, we never caught on to the fact that there is only one way to improve schools: better teaching. And becoming a better teacher is a time-intensive individual process that takes years. 

New Yorker Magazine has an article this month entitled "GroupThink" that debunks one of the biggest fads of the late twentieth century: brainstorming. Brainstorming was a technique propounded by Alex Osborne, an advertising executive at BBDO, a long-time New York agency (they did commercials for Lucky Strike cigarettes on the Jack Benny Radio Show in the 1940's) that has been cited numerous times for its creative work in advertising. Osborne's book Your Creative Power included a chapter on organizing for creativity which introduced the idea of brainstorming. From the New Yorker:   

The most important of these, Osborn said—the thing that distinguishes brainstorming from other types of group activity—was the absence of criticism and negative feedback. If people were worried that their ideas might be ridiculed by the group, the process would fail. “Creativity is so delicate a flower that praise tends to make it bloom while discouragement often nips it in the bud,” he wrote. “Forget quality; aim now to get a quantity of answers. When you’re through, your sheet of paper may be so full of ridiculous nonsense that you’ll be disgusted. Never mind. You’re loosening up your unfettered imagination—making your mind deliver.”   

That makes a lot of sense, doesn't it. Everybody likes to get lots of positive feedback. Unfortunately it doesn't work. In 1958, a team at Yale University had 48 students broken up into groups and given a set of creative puzzles. The control group of 48 students were given the same puzzles, but told to work independently. The results?  

The solo students came up with roughly twice as many solutions as the brainstorming groups, and a panel of judges deemed their solutions more “feasible” and “effective.” Brainstorming didn’t unleash the potential of the group, but rather made each individual less creative.   

In fact according to a Wikipedia:

Research from Michael Diehl and Wolfgang Stroebe (in 1987) demonstrated that groups brainstorming together produce fewer ideas than individuals working separately. Their conclusions were based on a review of 22 other studies, 18 of which corroborated their findings.  

A 2003 research study in San Francisco working with 265 female undergraduates came up with similar results, but specifically showed that the best way to get creativity going in work groups is to remove the rule against negative feedback. As the study's author Charlan Nemeth puts it, “While the instruction ‘Do not criticize’ is often cited as the important instruction in brainstorming, this appears to be a counterproductive strategy. Our findings show that debate and criticism do not inhibit ideas but, rather, stimulate them relative to every other condition.”  

So another "right answer" falls by the wayside, the victim of research and the scientific method. All too often we want to jump on the latest bandwagon when we would be best served by waiting to see what really works. My friend and education colleague John said it best, "We're ahead by being behind."  

Up In The Sky - It's a Bird, It's a Plane.

Just something else to get you to go outside on a cold night. NASA's website for International Space Station viewing.  On the left side of the screen, it asks you for your country of viewing. That would be the US for most of you, but my blog stats show that 10 people from the Netherlands and 5 from Russia have viewed this blog. (Probably either high or drunk on vodka.) On the next screen you click on your state, then find the town closest to you on the next screen.Here are the next three sightings for my area in Illinois:   

 Sun Feb 12/07:07 PM    2      38      10 above SW      38 above SSW

 Mon Feb 13/06:10 PM    4     29     10 above SSW    18 above E 

 Mon Feb 13/07:47 PM  < 1    17     17 above W        17 above W

The first number in each row (after the date and time) is the duration of the sighting. For these three instances, they are 2, 4, and less than 1 minute, so I won't have to stand outside for very long. The next number is the elevation; 90 (degrees) would be directly overhead. From the Office of Naval Research website:     

If you make a fist and extend your arm straight out in front of you, your fist measures about 10° across your knuckles. Now find the object whose height you want to measure. Starting with the bottom of your fist on the horizon, stack your fists one on top of the other until you reach the object. If it took 4 fists, the object's altitude is 40°. Four and a half fists equal 45°. For smaller distances, you can use the knuckles on your index finger. Between your fingertip and first knuckle equals about 2° between your first and second knuckles equals about 3° and between your second knuckle and last knuckle equals about 4°.

       

 

The next set of numbers tells you where to look for the station to come into view. For example, on my Sunday list above it says 10 above SW. So I will be looking at 10 degrees (one fist) above the horizon towards the South West and if I see it come into view I can follow it until it disappears 38 degrees above the horizon South South West (half way between South and Southwest). The typical path the station travels is shown on the diagram below. The website suggests viewing through binoculars to get a better look at the spacecraft.  

So, good luck. Hope you get a chance to see the space station as it races overhead. We'll leave the light on for you.  

U Can't Touch This?

We all like to reminisce about our childhoods. A website called Retro Gifts allows you to reminisce about your childhood by buying gifts based on your birth year. For example, if you were born in 1951 like I was, you would see on their list:     

A Roy Rogers Lunchbox                               Matchbox Cars                                         

Sea Monkeys                                             and a Tonka truck

They didn't all come out in 1951, but they would be the sort of toys someone born in 1951 might have received as gifts over the years. And as an added service, each of the objects listed has a link to Amazon just in case your mom threw out these things while you were away at college. For my boys, born in the early to mid 80's, the website lists:  
 

I don't think I've seen slap bracelets for a while.  But we have been cleaning the basement over the last few weeks and came across the Teddy Ruxpin bear and the pogs in boxes.  I'm still using the original Nintendo set to play Dr. Mario (best score ever - 1, 275, 400 points). So I know why these are on the list. But somebody out there had to be buying MC Hammer pants to get them on the list, right? Confession is good for the soul.  Was it you, Scott? Or maybe Brian? You can tell us.    

Universal Music?

I have read a lot recently about the pentatonic scale, that is, the scale of notes that are fifths from each other. A fifth, I have learned, means that the two notes are 4 spaces apart on the staff.    

For example, from C to G would be a fifth. (Please fix anything here that is wrong, Mary or Nate, in the comments). If you start at C and keep going up by fifths, you would get C - G - D - A - E. If you keep going, you get the famous circle of fifths, which is used for making chords and transcribing music to different keys. The five notes listed above form the pentatonic scale. There is some scientific discussion about this scale being a sort of universal scale that is hard-wired into our brain. 

Wikipedia lists the following users of pentatonic scales:       

Pentatonic scales are very common and are found all over the world, including Celtic folk music, Hungarian folk music, West African music, African-American spirituals, Gospel music, American folk music, Jazz, American blues music, rock music, Sami joik singing, children's song, the music of ancient Greece and the Greek traditional music and songs from Epirus, Northwest Greece, music of Southern Albania, folk songs of peoples of the Middle Volga area (such as the Mari, the Chuvash and Tatars), the tuning of the Ethiopian krar and the Indonesian gamelan, Philippine Kulintang, Native American music, melodies of Korea, Malaysia, Japan, China and Vietnam (including the folk music of these countries), the Andean music, the Afro-Caribbean tradition, Polish highlanders from the Tatra Mountains, and Western Impressionistic composers such as French composer Claude Debussy.  

The emphasis seems to be on folk music from many different regions of the world. I think of folk music as being people big into music, but not necessarily into music theory as taught in schools. In much of what I am reading, the emphasis seems to be that the pentatonic scale just sounds right, beginning with children.    

The website Classics for Kids shows how this appears in children's music education:   

The music used in the classroom is based on the children's own heritage with a combination of folk and composed music. Here in America, our classrooms are multicultural. Orff philosophy embraces the folk music of all cultures. They are almost all in the universal pentatonic scale. (Five note scale - separated by whole steps.) C Pentatonic would include the pitches C D E G A.

I know Mary has been extensively trained in the Orff School of Music Education, so I am interested to hear what she thinks.    


At the World Science Festival in June of 2009, scientists and musicians came together to discuss "Notes and Neurons: In Search of a Common Chord". From the festival's website:   

We don’t know much about the human brain on music. Do people instinctively know the sound patterns of the pentatonic scale? Is there a base level of musical knowledge in all of us, just waiting to be tapped? Or is the pentatonic scale simply so common in Western music that it has become ingrained in all of our minds? Improvisational genius Bobby McFerrin uses audience participation to demonstrate the power of the pentatonic scale—or at least the audience’s familiarity with it.

Here is the Bobby McFerrin video:   

World Science Festival 2009: Bobby McFerrin Demonstrates the Power of the Pentatonic Scale from World Science Festival on Vimeo.


Notice his comment: "It doesn't matter where in the world I am; when I do this exercise, the audience always gets it." 

More on this later as we look at Leonard Bernstein's six lectures at Harvard University on "The Unanswered Question".