Math Teachers at Play #4, Krypto, Updates (Odds and Evens),...

1) The latest "biweekly" edition of our new Carnival, Math Teachers at Play, is currently being hosted by Misty over at Homeschool Bytes. I enjoyed the "step by step" approach, progressing from primary math activities like Candy Math through middle school posts like Division of Fractions Conceptually to secondary articles like Ten 16th Century word problems. I contributed a post on Function Questions for the SATs. I'm not including any direct links to these articles. Go to Misty's site to enjoy this carnival!

2) I'm putting a lot of effort into the SAT Math Tips feature in the sidebar (readers of my feed won't get to see this of course unless they visit the site). I realize a good part of the country takes the ACT but the suggestions may be applicable to any standardized test and the math content is valid for anyone who wants to use it.

3) The Read More feature I recently instituted has some bugs as I'm sure you noticed. It doesn't work in RSS or Atom feeds of course and it sometimes doesn't work properly even on this site. I'm a coder at heart but I'm not sure it's worth all the effort. Clearly the Blogger developers are not interested in making things easy for us. You may see Read more... even if there is nothing else to see or it may not work all! Too late for me to migrate to Wordpress at this point... Please be patient with me here as I work through this.

4) The Math Problems of the Day in the sidebar are of good quality and are challenging but I'm seeing more repetition of questions. I will evaluate this and decide if I want to keep this feature.

5) No, I haven't forgotten about the next MathNotations contest I promised for April or May. Stay tuned...

6) Ever hear of Krypto or 24 (the game not the TV show!) or the more well-known "Four Fours Game"? I played many years ago and have always enjoyed the challenge of this tantalizing arithmetic game. I've used it effectively in upper elementary and middle school classrooms when I was a math staff developer to reinforce order of operations and basic fact recall (no calculator allowed!). Ok, so I'm now driving you crazy with another KenKen-like game!!
Click on Read more to learn more. (if this works!).




Here is a sample play of Krypto:

Suppose you're dealt the following five number cards:
5, 9, 4, 11, 1.
A 6th or objective card is turned up, say 2.
Using some or all of the four basic arithmetic operations and the 5 numbers exactly once, produce the objective number. This one is straightforward using only addition and subtraction:
11 + 5 - 9 - 4 - 1 = 2. Krypto! You've Won!

It is possible to be dealt an "impossible" hand for which there is no solution or there could be many solutions for the same hand! The original game did not allow the use of parentheses but you could choose any variation you wish, including reducing the number of cards to 4. An ordinary deck could be used modifying the face cards to be 11,12, etc.

If you want to play the online version from mphgames, go here for instructions and play. Your browser must be java-enabled but most are. For more background on the game and a discussion of the underlying combinatorial mathematics and a discussion of the computer program which generates it, look here.

ENJOY KRYPTO WITH OR WITHOUT YOUR STUDENTS!!

...Read more

CoM 18th Edition -Sooo big, Sooo good!

Enjoy the 18th CoM at jd2718. Jonathan has put together a delectable sampling of over 50 outstanding posts from a wide variety of math blogs:
Research Level - 9; Other Campus (politics, teaching) - 6; HS level - 6; Recreational (any level) - 8; Elementary - 6; Math Ed in general - 5; Modeling/economics - 3; Computer Science - 4; Humor - 3; History - 2. They come from: Greater NY - 5; Midwest - 7; California - 7; New England - 4; Mid-Atlantic - 3; Unknown US - 9; Canada - 1; Britain - 5; Other Europe - 3; Australia/NZ - 2; No idea - 3

Congratulations, Jonathan, for a Marathon of Math!
Look ahead to the next edition over at Good Math, Bad Math.

Seventeenth Edition of the Carnival of Mathematics: "At Seventeen, I Learned the Truth"

[This edition of the Carnival is dedicated to our 12-year old Golden retriever, Teddy, who passed away on 9-15-07. He was everyone's 'best friend' and gave far more than he received. When my mom passed away recently, my 4-year old grandson was told that she became a star in the sky. When Teddy died, he comforted us by saying, "Now, Teddy's a star too and Nana will be able to walk him."]


What was the age of the Dancing Queen in an ABBA classic? (ABBA is one of my favorite palindromic groups).
Does anyone remember the title of Janis Ian's classic 60's song about teenage angst and includes the phrase that is part of the title of this post?

As it is becoming a tradition to begin our Carnival with comments on the cardinal number of our edition, I will choose a few more serious facts about 17, one of which I alluded to in the post inviting submissions to this carnival (anyone recall the pdf reference?). The sources here include the Wikipedia article on this number.

From MathWorld, 1/17 = 0.0588235294117647... (repeating). Note that the length of this repeating decimal is 17-1 = 16, which is maximal. I've always been fascinated by such full repetend primes...
17 is the third Fermat prime, as it is of the form . Since 17 is a Fermat prime, heptadecagons can be drawn with compass and ruler. This was proved by Carl Friedrich Gauss. Now you know my pdf allusion: Pierre de Fermat...

17 is known as the Feller number, after the famous mathematician William Feller who taught at Princeton University for many years. Feller would say, when discussing an unsolved mathematical problem, that if it could be proved for the case n = 17 then it could be proved for all positive integers n. He would also say in lectures, "Let's try this for an arbitrary value of n, say n=17." In the class I teach on Wednesday nights, I naturally asked a young lady to give me her favorite number. I kid you not!

I could go on , but this takes away from the excellent submissions, so without further ado, I present to you exactly SEVENTEEN wonderful posts, nominations and favorites of my own choosing. By the way, there is no specific structure or bifurcation. Join me now as we take a pseudorandom walk through a slice of the math blogosphere...

(1,2) Denise over at letsplaymath, sends us two interesting posts:
Egyptian geometry and other challenges and Alex’s puzzling papyrus.
Denise described these fascinating articles:
"...two posts about ancient Egyptian math. The target audience for these was middle school to high school teachers and their students, especially homeschool families (who often try to coordinate their topics of study across the curriculum). The mathematical level is high school geometry or lower."
(3) Denise also included Math Quotes VII (too bad it wasn't XVII!).

(4) Vlorbik recommends that we visit Isabel from God Plays Dice who brings us Why g ~ π².
Don't miss the astute comments, particularly, John Armstrong's concise summary of Isabel's fascinating post:
"What Isabel is pointing out is that at least one definition of "meter" over the years has been designed so that the numerical value of g coincides with the number pi^2."

(5) Vlorbik also links us to a relatively new blog from a brilliant and modest mathematician, William Gowers, who is contributing his insights to mathematics pedagogy. His post, "How should vector spaces be introduced?" reveals his wisdom and humility. Read the comments! Thanks, Vlorbik, for these excellent nominations!

(6) David Eppstein from LiVEJOURNAL brings us into the realm of combinatorics with Not the multinomial coefficients, a 3-dimensional view of binomial coefficients that is not simply an extension of Pascal's triangle. Some combinatorial background is needed here to fully appreciate this but the presentation is clear and engaging.

(7) John Armstrong of the Unapologetic Mathematician brings us a fascinating post on Newton Fractals, a description of Newton's Method and its behavior for f(x)=x³-1, requiring only differential calculus as a background. John develops the root search in the complex plane. By iteration and suitable coloring, John is able to suggest how a fractal is formed. His ability to clearly explain this tells me how very fortunate his students are.

(8) Mathematics in a Jack Reacher Novel is a post in Jeffrey Shallit's blog, Recursivity. Jeffrey discusses extensions of the mathematics suggested in a novel by Lee Child. The main character, Jack Reacher, displays some math-savant characteristics but Jeffrey enriches this with some fascinating problems involving numbers and their digits. Try the exercises and see if you match the results discussed in the comments!

(9) Jonathan from jd2718 offers us Puzzler puzzled. This is Jonathan's specialty, devising variations on famous number puzzles that require considerable thinking outside the box. In this case, one of his students had difficulty with interpreting the language of the original question and suggested another take on the problem. Now we have two great problems to work on!

(10) Mark Dominus from the Universe of Discourse offers The Missing Deltahedron , an advanced geometry article describing some research Mark is doing. Marc is following up his analysis of the convex deltahedra, which are the eight polyhedra whose faces are all congruent equilateral triangles. The problem is that there is a break in the pattern in the number of faces and it seems to upset the harmony in the universe!

(11) Meeyauw shows us an application of geometry to knitting. She mathematically builds on another blogger's post on knitting some of Escher's classics and even found a link to 'crocheting the hyperbolic plane'!

(12) Maria at Homeschool Math Blog brings us Number Rainbows, useful for primary math teachers. The idea is that you connect two numbers with an arc if they add up to a particular number, such as 13. Using different colors to connect these pairs makes an attractive rainbow pattern and may help students learn their addition and subtraction facts. Mathmom, in her comment, suggests an extension to Gauss' method of addition.

(13) Speaking of Mathmom, enjoy her insightful and provocative Calculator Rant. Perhaps the AP Calculus Committee set the gold standard a few years ago by developing a 2-part test, one with and one without the graphing calculator. This approach is definitely filtering down through the grades and Mathmom informs us that this has been adopted for some time by MathCounts.

(14) Andreas, from Figuring Out Computer Science, brings us a discussion of the The Most Important Problems in Computer Science. He sent out queries to some experts in the field and the main theme is that of complexity vs. simplicity, including the P=NP problem. Great quotes here but my favorite is "...simplicity is the ultimate sophistication." (da Vinci)

(15) Jacob from Winter's Haven, was one of the first to submit an article for this carnival:
Pure Math, Applied Math and A Priori Proofs. He presents a thoughtful and logical case for why scientists cannot prove natural phenomenon using only a priori, i.e., theoretical, mathematical arguments, in the absence of empirical corroboration. He explains that mathematics only provides possible models of reality which then need experimental verification. Nice...

(16) Marc has published his original research: New Prime Formula Helps Investigate Prime Numbers. After reading my post on Fallout From the Sieve of Eratosthenes, he thought I might be interested in his discovery and he is helping me to work through the details. Since the unsolved problems related to primes have been a motivating force for so many mathematicians, I know you will enjoy reading this significant find.
And speaking of primes... Coincidentally with my discovering Marc's research, I also found the accounts of the new algorithm developed by 3 mathematicians from India, which is considered a significant breakthrough in primality testing. You can find the fascinating details here and here.

(17) And last but certainly not least, Kurt over at Learning Computation threw Greg Muller's 'hat into the ring' by nominating Greg's engaging Hat Guessing Puzzles, The Revenge. Greg's blog is The Everything Seminar and if this is a sample of his writing, I plan on returning. The hat-color-guessing puzzle is itself is a fun exercise in logic for all of us, but Greg shows us how it is related to error-correcting codes in computer science.

Well, there it is. Exactly XVII excellent offerings. Enjoy them!

By way of disclaimers, I sincerely apologize for any omissions, invalid links, or errors of commission I may have made. Just let me know and I'll correct them. If you sent in your submission after 12 AM on Fri 9-21-07, it might not have made it into this Carnival, but I will post an addendum and errata if needed.

BTW, if you're wondering why I've made no mention of the Prof. Steen interview and the ongoing commentary, well, I think I just did...

Addenda/Errata:
Correction has been made in the link to #10: The Missing Deltahedron. Other links should be working as well. Sorry for any inconvenience caused by this.

Invitation to Carnival of Math Edition 2^(2^2)+1

While we're waiting for Part 2 of Prof. Steen's interview, please don't forget that MathNotations is hosting the seventeenth edition of the Carnival of Mathematics on 9-21!
I've already received 3 submissions and I'm hoping for many more. In addition, I have found at least one website of original math research I will be sharing with you. There may be some other surprises and even an exciting discovery in mathematics to share as well.

I'm considering dedicating this Carnival to a close friend of mine. I'll only give you his initials in lowercase: pdf
[You have only 30 seconds to solve the riddle!]

As I requested in Edition X and Kurt did in Edition XVI, I am asking math bloggers to submit one or two of their recent posts by either:
(1) Emailing me at dmarain at gmail dotcom (preferred!)
(2) Filling out the blog carnival form

If emailing, pls include the exact phrase Carnival of Math in the subject line and include the following:

(a) title of and permalink to your post
(b) name of your blog and link to home page
(c) an overview of the mathematical content, particularly if highly technical or advanced
(d) your general target audience (e.g., "This posting was designed for ...")

Don't forget that you can also recommend a couple of articles from some of your favorite math bloggers whose light might be hidden under a bushel and deserve recognition. I may do the same.

Since a carnival is supposed to be marked by merrymaking, let's have some fun with this! I'm not as creative as some other hosts, but I'll try to make it interesting.

Please try to send your submissions by Wed 9-19 so that I have the time to do justice to them - thanks!

CoM Edition XVI - Comments & Opinions - Now up at Learning Computation

Kurt kept us in suspense but this latest edition was worth waiting for. Incisive and droll reviews of many excellent math blogs divided into 3 sections; The Good, the Bad and the Ugly. You'll have to read it to determine his classification system. Kurt also referred us to some some excellent posts that he personally found and wanted to share with all of us. Thanks, Kurt, and I hope you're feeling better! The next edition will be right here on 9-21.

Some more thoughts on splitting the Carnival...

You may want to read the comment I posted on John Armstrong's, The Unapologetic Mathematician. It's the 23rd psalm, oops, I meant comment, listed there and it does represent my strong feelings about where we are in all of this 'bifurcation' business. I'm hoping we can move on now.

The next edition is coming out next week over at Learning Computation and I will be hosting the following one on 9-21. I will have several suggestions for the format of submissions at that time, so I'm hoping you will follow this thread over the next few weeks. Actually, Kurt over at Learning Computation has anticipated most of what I will be requesting so please read his article on the Carnival thoroughly - it's very thorough and well-written. He also has some fascinating information on the mathbloggers.com site and Social Rank. Why should we be surprised that there are computer science whizzes trying to develop new tools for ranking blogs and websites. Why would they do it other than for commercial gain? Because they can, I suppose! Why did a teenager in Glen Rock NJ spend countless errors trying to unlock the secrets of the iPhone? Because he could! (earning instant fame and a new car in the process). The opportunities for technophiles to instantly change the world are now endless...

Carnival of Math Edition XV - A Monu'mental' Effort by John Kemeny

Outrageously funny, bizarre at times, brilliantly clever...
There's no other way to describe the latest edition of the Carnival of Mathematics over at John Kemeny's website. From the wildly amusing rating system for classifying math posts to the over-the-top graphics, I'm amazed that John was able to get this published on time. Well worth waiting for and it leaves little doubt that the CoM is alive and well. And, oh, by the way, John absolutely did justice to each of the submissions. Enjoy it!

The only problem for me (and every other host) is that I have to follow John's act in a few weeks and this is one tough act to follow.

Congratulations, John and in the true spirit of your blog entitled, a mispelt bog, let me add that this edition of the Carnival was mind-blogging!

Carnival of Math to the 2nd Power?

John Kemeny is hosting the 15th Edition of the CoM on 8-24 and you can submit your articles here. However, there are no more Carnivals scheduled at this time.

In the most recent edition of the Carnival of Mathematics, Vlorbik reported that there has been a call for change in the structure of the carnival. This idea was first raised by yours truly in the Tenth Edition of the Math Carnival and recently again by Michi over at his blog. Look here for the ongoing dialogue I've been having with John Armstrong and Michi (read the comments). John feels strongly that the Carnival has been heavily weighted toward math education of late and I agree. Since that is my main area of interest, I'm not overly upset about this but I do agree that the research-advanced mathematics side has taken a hit. I made a proposal that we split into a Carnival of Math Ed (K-14 oriented) and a Carnival of Research Mathematics (John's and Michi's idea). Although previous comments have suggested many are satisfied with the status quo, John feels that we already have a 'de facto' separation. I suggested we take the current temperature of the water by inviting comments. I volunteered to be the home base for the Carnival of Math Ed site but I would welcome support from others, particularly the most frequent contributors to the math ed blogosphere who have the time to take on this project. John volunteered to be the moderator of the Research group. He also suggested that we run the Carnival monthly and I agree. This would enable more to contribute and make the day of the each Carnival more of a significant event. If you're wondering why we seem to be ignoring Alon, the original architect of the Carnival of Mathematics, that is not the case. I am deeply appreciative of the monumental efforts made by Alon but he recently has decided to drop from the blogosphere and, presumably, the Carnival as well. I know he would want someone to pick up the reins and keep this moving. Alon, if you read this, I hope you will share your thoughts. There will be some technical problems to overcome involving the submission of posts and registering the 2 Carnivals of Math with the main Carnival site, but I'm sure we can work this out.

If I receive very few replies to this proposal, I may move forward on my own or with John , since we are willing to accept this challenge. Waiting to hear from you...

The 12th Carnival of Math on Friday the 13th!

Of course, there would be more harmony in the universe if this had been the 13th Carnival but enjoy the 12th Carnival nevertheless over at Vedic's Math Forum.
Vedic has selected an interesting assortment of posts covering a broad range of math. For example, Knot Homology in Algebraic Topology, Key Concepts in PerCents, Math Mnemonics, Observing Objects Near the Speed of Light, Sudoku and Graph Theory, An Interesting Way to Calculate the Square Root of 2, a discussion about finding Next Gen Math Teachers and a couple of recent posts from this blog. Enjoy! A great way to beat the summer doldrums!

The next Carnival (#13!) will be hosted on 7-27 over at Polymathematics.

Carnival of Math: Tenth Edition!

Welcome to the Tenth Carnival of Mathematics

After reading other Carnivals of Math and hosting this one, I'm wondering if anyone feels there might be a need to have two Carnivals of Mathematics:
A Carnival of Mathematics for middle and secondary grades and
A Carnival of Advanced Math for undergraduate and graduate level as well as for the research mathematicians out there.
Another approach might be to have both a Carnival of Math Education (a category into which my blog might naturally fall) and A Carnival of Mathematics.
I may be way off here and outvoted by the vast majority of readers who may simply prefer to pick and choose the posts that interest them, but there does seem to be a clear demarcation between these categories (in the non-algebraic sense of course!). The Carnival may need to reach a critical mass before this would be practical but I'd be interested in your comments here.

With the above in mind, I will begin with math blogs that focus on middle and secondary grades...

jd2718 has a wonderful variation on the Four Fours Puzzle. He has an uncanny knack for taking a good problem and adding enough complexity to it to boggle the mind! This problem is still open-ended and waiting for more ideas...

Alane over at Math Notes demonstrates divisibility tests for 7 and 11 and provides easily understood explanations for why these rules work. Her other post introduces students to perhaps their first mathematical proof, the classic "irrationality of √2" by contradiction. To assess their understanding of indirect proof one might modify the problem to "Show that √3 is irrational."

In Patty Paper Trisection, Denise, at Let's Play Math, challenges her readers to prove that Math Trek's origami trisection referred to in Carnival #9 really works. Denise has an engaging writing style that invites her readers to challenge themselves. She sees math problems as puzzles, a view shared by many who have a passion for our subject. This post is designed for students and teachers in grades 7-12 as well as homeschoolers.

Murray Bourne over at squarecirclez offers us a practical application of semi-log graph paper in plotting the dramatic increase in the ranking of You Tube in just a year and a half. The vertical scale is equally spaced, marked in powers of 10 -- logs base 10 to the rescue! Students will eat this up! He also is promoting a fascinating change in standard math notation (thanks, Murray, for promoting the name of my blog!)

Mark D from the Universe of Discourse shares a recurrence form for binomial coefficients that is far more efficient than the traditional factorial definition. He suggests that this ancient relation (published nearly 1300 years ago) has not gotten the recognition it deserves. Since the last student project in my BC Calculus class focused on efficient formulas for approximating pi (Ramanujan's formula in particular), your post fit right into the discussion.

Vlorbik on Math Ed has a fascinating post on Textbooks and Notations.
He contends, and I concur, that current texts over-stress natural language (as in spelling out the meanings of symbols in English) for set-theoretic formulas, conditional probability in particular. My philosophy has always been to introduce concepts and formulas in colloquial language to which students can relate, then move on to the formal symbolism of mathematics as early as possible. Students need to appreciate the efficiency of symbolic notation and how it provides a universal language for mathematical discourse, not subject to interpretation! Once again, great justification for the name of my blog! I knew there was a method to my madness (aka, dumb luck!).

On the technical research side of mathematics we have a couple for you to digest...

The Unapologetic Mathematician writes about the importance of category theory for undergrad math majors. Categories have become significant in contemporary mathematics. For background, read the Wikipedia article on category theory.

Michi at Michi's Blog presents a technical piece in the area of homological algebra (If only I could recall anything Professor Dyer was trying to teach me in algebraic topology 40 years ago!). The posting deals with combinatorics and coding of a very important tool for his current research - looking at extended algebraic structures in group cohomology.

Michi also recommended Terry Tao's blog. I particularly enjoyed his Advice on Mathematical Careers.

And now for something completely different...

A monthly feature, Who's Counting?, on ABC News.com is authored by the internationally recognized mathematician and author John Allen Paulos. His specialties are statistics and logic but he is also well-known for the popularizations of mathematics he has written (Innumeracy, etc.). He is a Professor of Math at Temple University and he knows how to make math interesting and meaningful. Read through some of his articles from the past 2 years. There's considerable food for thought in these articles and enough material there for projects for Statistics/AP Stat classes for every month of the school year! Not to mention that it makes for fascinating reading. He is a gifted writer who weaves a beautiful web.

I want to personally thank all of our contributors who were considerate about replying by June 13th! Further, those who responded to my gmail account provided some fascinating insights about their passion for mathematics. I felt right at home...

There are so many outstanding math bloggers out there. I can never do justice to all of them. This Carnival is just the tip of the iceberg. One that I've recently discovered is Mathematics Weblog. The author has concisely summarized all of the Carnivals to date and his discussion of math humor is worth the read (he reviews books like Comic Sections and Mathematics Made Difficult). If the books are as humorous as their titles, they're worth looking at!

I haven't yet mentioned any of my recent postings on this Carnival as I wanted to celebrate others' blogs, not mine. However, I'm working on a way to introduce and develop recursive functions and linear recurrence relations for grades 7-12. It will be entitled --
"Take any number, Add Three, Divide the Result by -1. Now Repeat this!" I hope you will look for it and share your comments as we approach our summer break (for me, a more permanent break!). Also, all of those 'beautiful' mortgage formulas I've been alluding to in the series of posts on applications of exponential functions are now displayed as screenshots from the TI-84. Those posts have received many visits and I'm not sure if it's more for the math or more for mortgage advice (believe me, you don't want advice from me on that!).

Update: Submitted late but I decided to add it on 6-15:
An interesting brain teaser for the frontal lobes on SharpBrains. To solve it you need to analyze balance-scale relationships among 3 quantities (spades, clubs and diamonds). Some might try this mentally using logic, others may want to set up algebraic equations. Have fun!

Would you believe, another couple of late additions that I discovered in my web travels--
Best of the Web - Math Blogs
(Of course my blog didn't make the cut!!)

Not Even Wrong - A Random Collection of Stuff (a nice summary of some technical math blogs from a Columbia math professor I believe)
There's no end to this so I had better stop...


Stay tuned for our next Carnival on June 29th over at Grey Matters.

Carnival of Math Edition X Coming on June 15th

Reading Carnival of Math #9 over at jd2718 has been a uniquely lexicographical experience! I am not that creative! For an excellent synopsis of all previous Carnivals of Math, visit the Mathematics Weblog.

Invitations are now open for you to submit 2 or 3 of your favorite posts from the past few weeks either by:
(1) Emailing me at dmarain at gmail dotcom (spelled out to discourage those mean little spam robots)
(2) Filling out the blog carnival form

Those of you who have had the opportunity to visit this blog know that I am an educator (soon to be retired) who is deeply committed to developing conceptual understanding of mathematics in our K-14 students (I've been in classrooms at all these levels in my career) and challenging our students to delve beneath the surface of this beautiful subject. While I invite technical math bloggers to contribute, I particularly welcome submissions from like-minded educators who are trying to shine a light on the vast darkness of education. If only Escalante had a blog...

If emailing pls include the exact phrase Carnival of Math Submission in the subject line.
Also, to do justice to each of your posts I would greatly appreciate some personal background and a description of each submission including
(a) a brief overview of the mathematical content, particularly if highly technical or advanced
(b) your general target audience (e.g., "This posting was designed for ...")
(c) Your name (optional of course, but it adds a personal touch when I refer to you as Evelyn over at...)
(d) From where did your passion for math originate? (Mine came from my dad and a book by Ivan Niven that my mom bought for me as a teenager). Also, what part of mathematics do you enjoy teaching or pursuing the most. (For me, it's number theory and developing problem-solving strategies and enrichment activities in math).

PLS TRY TO SUBMIT YOUR LINKS BY WED JUNE 13TH. As those in education know, at this time of the school year, there is a time crunch. I'll do my best to include all of your submissions but I will definitely include all who meet the above deadline. Also, feel free to provide links to other of your favorite math blogs (in the email). Many excellent math bloggers may not even be aware of the Carnivals. This will also help me if there is a low number of submissions at this time of the year.

'Carnival' of MathNotations 5-25-07

New feature...

I apologize for stealing the 'Carnival' term - I couldn't resist it. If the Carnival people want equal time, here's my plug for one of the best: Visit the home page of the Carnival of Mathematics - we are all indebted to Alon Levy (I hope I got that right!) for developing and maintaining this wonderful venue for the math-minded out there who may not have time to discover or visit all of the outstanding math blogs on the web. Considering that I'm hosting the Carnival the week of June 15th, I thought this attempt could also be a dry run...

It's easy to miss some interesting dialogue when we jump from site to site on the superhighway. I watch how my students and my children surf the net- if they don't see what they want on a page within 5-10 seconds, they're gone!

Since some of the most interesting 'stuff' often takes place off-post in the comments section, I thought it would make sense to summarize some of this at the end of the week. I'll probably tire of this after a couple of weeks, or choose to do this every other week like the real Carnival of Mathematics, but I'd like to try it anyway.

I apologize in advance to my other faithful readers and expert commentators for any errors of omission. Their contributions enrich this blog and their continual support inspire me to maintain these efforts. I will add more as I sift through my posts from the past two weeks. For now here are some highlights from the past 8 days.

• A series of posts (Part I and Part II) this past week have focused on developing meaning and application for exponential functions (and their relation to geometric sequences and series) designed for Advanced Algebra classes. These extensive classroom investigations use the concepts of mortgages, stressing the ideas behind those esoteric formulas. There are full-blown activities with open-ended questions and data analysis using the graphing calculator. Screen shots of data tables from the TI-84 enhance the activity. Further related activities to follow...
  
• A set of Thinking Outside The Box problems came up earlier that stimulated some great comments and ideas from Denise (author of the excellent site letsplaymath) and others. This has led to an ongoing discussion of the sticky problem regarding the smallest positive integer having a given number, N, of factors. The problem has been solved if N is prime, or if N is a semiprime (N = pq, where p,q are distinct primes), but it's not clear that we have a general solution. Someone out there must know the algorithm for finding this number - pls share it! I haven't yet posted answers or detailed solutions to some of the questions, so stay tuned.
  
• TC, another of my dedicated readers, posed the pi = 2 'proof' introduced first in the comments here and continued in the comments section of Part I of the mortgage problems. I challenged my Calculus class to explain the fallacy in this limit problem and one student, Matthew, came up with a pretty nice solution. This problem could easily be a post in and of itself with many possible extensions. TC presented an excellent 'curvature' argument and I added some thoughts of my own.
• Eric Jablow, our math/computer science expert, suggested 3 excellent algorithms (bisection method, method of false position and Newton's Method) for approximating roots of an equation. These are worth sharing with our algebra and precalculus students as well as those in Calculus and Computer Science. These comments are found under the post entitled Catching Up and Preview of Coming Attractions.
  

Pls give me your feedback. If you think the title is too presumptuous, say so! If you think it's a waste of time to summarize only a week or so of posts and comments, let me know that too. If you have something nice to say, I'll accept that as well! Signing off for now...

Notes on April 6th Carnival of Mathematics

A recent post in MathNotations received a favorable comment by Charles Daney in the 5th edition of the Carnival of Mathematics. This is particularly gratifying because the post relates to one of my students.

This week's Carnival pays special tribute to one of the 20th century's most outstanding mathematicians, Fields Medal recipient, Paul J. Cohen, who passed away on March 23rd, the date of the previous Carnival. Dr. Daney's tribute to the 'beautiful mind' of Dr. Cohen is sincere, passionate and touching. This is one mathematician's way of expressing gratitude to someone whose inspirational teaching affected his life personally. I am willing to wager that each of us who has pursued mathematics as a career in some capacity has had their life altered in some way by an inspiring educator. In addition to the major influence of my father, a great teacher in his own way, I was blessed to have many other outstanding math teachers, but my watershed experience came in Dr. Silvio Aurora's general topology class at Rutgers over forty years ago. He showed me what it means to think mathematically and to appreciate its beauty. Thank you, Dr. Daney, for bringing that flood of memories back to me.

Carnival of Mathematics Selection

I just want to thank Prof. Jason Rosenhouse for selecting MathNotations for the March 23rd edition of the Carnival of Mathematics. Click on the link in the sidebar to see many other excellent math blogs selected by Jason as well as links to earlier Carnivals. This sidebar is continually updated - enjoy the many other outstanding math blogs linked there.

Here is Jason's description of two of the posts you may have recently read:

Dave Marain of MathNotations has a fascinating post on the evolution of a difficult standardized test math problem. Math and evolution? My kind of post! See also his post on Geometry and Reasoning. Much food for thought.