http://goo.gl/HWXq9

If you think it’s ludicrous to suggest that you can un-mix a liquid, then watch the entirety of this 2-minute video!

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http://www.youtube.com/watch?v=yVkdfJ9PkRQ

These pendula of staggered lengths are set swinging, and the result is fascinating.  At first, the weights seem to trace out sine waves, but they quickly start to cycle through a wide range of behaviors.

At times the pendula appear highly organized, and at times almost chaotic, as they cycle through various patterns.  Amazingly, the weights eventually return to their initial state!

A really beautiful mathematical demonstration.

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In general, the packing problem entails trying to find the most efficient way to pack a certain kind (or kinds) of objects into a certain fixed space.  Packing probelms are, generally speaking, very challenging because every packing problem is unique.  There isn’t a good, efficient procedure that solves them all.

Here is yet another example of problems with packing problems.  After shedding a bunch of CD cases, I thought I’d try to pack them up in a box.  Here was my first attempt.

I got 49 CDs in the box, but there was a bit of unused space left over.  I couldn’t fit a CD into that unused space, but I thought maybe I could rearrange everything to make some of that space usable.

So I tried again.

The number of CDs in this new arrangement differed by one.  While I can compare which of these packings is more efficient, the problem is comparing all possible packings!  There are a lot of options to consider.

As useless as they are, I ended up having a lot of fun with these CD cases.  I made some parallelepipeds with them and used them to demonstrate Cavalieri’s Principle!

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www.howtofoldit.org

The website has several videos and cool animations that demonstrate some of the basic ideas in mathematical folding, like the one-cut problem, the map puzzle, and folding polygons into convex polyhedra.

There are also a few folding patterns available for download, just in case you’d like to produce a turtle with one cut!

And for more resources on math and origami, check out my fun with folding page!

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I mean, isn’t a kid going to have less overall hair than an adult?  I understand that an adult head may have less coverage percentage-wise, but the larger head-radius more than makes up for the lower proportion of hair.  I’ve got to believe that the price-per-hair is considerably higher for a child than an adult under this pricing system.

Of course, I would not be so naive as to expect understandable pricing structures from hair stylists.

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http://www.bbc.co.uk/podcasts/series/maths

Narrated by mathematician Marcus du Sautoy, these podcasts begin with Newton, Leibniz, and Calculus, and cover other great personalities in mathematics like Euler, Fourier, Ramanujan, Poincare, and, of course, Gauss.

Each podcast is about 15 minutes in length, and all 10 are freely available for download.

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When it’s not opened all the way up, the two layers of screen overlap in the middle.  Depending on the angle you are looking from, you can see some cool images.

At this angle, for example, I see a contour map of a function of several variables.

I wish I understood where the curves come from!

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http://weusemath.com/careers

About 40 or so careers are profiled here, like Mathematical Biophysicist, Foregin Exchange Trader, and Psychometrician.  Each profile includes a salary range, educational requirements, job description, and potential employers.

Another nice potential answer to “When am I ever going to use this?”

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http://www.freebookcentre.net/

In addition to hundreds of math e-books on topics such as Linear Algebra, Geometry, Applied Mathematics, and Probability Theory, there are also selections of free books on many Computer Science topics, Chemistry, Physics, Languages, and much more!

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For example, the probability of rolling a 6 on a six-sided die is 1/6, but if it is known that the number showing is greater than 3, then the conditional probability that a 6 is rolled is 1/3.

There are many applications of conditional probability, but a recent “Math Encounter” from the Museum of Math made me aware of an application of conditional probability that all of us see on a regular basis:  Google search autocomplete.

Suppose I type in the search term “under”:

Here, Google is trying to autocomplete my search query.  In essence, Google is trying to guess the next word I’m going to type.  How does it make its guess?  It computes a conditional probability!

Google has a lot of data on when words follow other words.  When I enter “under” into the search bar, Google looks for the word/phrase with the highest conditional probability of being next.  Here it turns out to be “armour”; the word with the second highest conditional probability is “world”, and so on.

Naturally, as more information is provided, the conditional probabilities change.

 A fascinating, and perhaps surprising, application of a powerful mathematical idea!

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I have always found great value and pleasure in writing.  It is a valuable skill, a necessary tool of scholarship, and a powerful creative outlet.  And now I see its value as a math teacher.  The more my students write, the more useful and interesting we all find it.

In this post, I’ll discuss some strategies for getting students writing in math class.

How to Get Students Writing in Math Class

1)  One-page solutions

Have students choose a good math problem they like and write up a solution in one page.  Have them narrate their process and explain the choices they make.  They can also write about mistakes someone else might make, or offer an alternate solution if possible, or suggest a new problem that’s related but slightly harder.  Problems from math competitions (like the New York Math League or the American Mathematics Competition) are usually great places to start.

2)  Create New Questions

Have students choose a good math question and write up three new questions based on the original.  If the original problem asks to find the sum of consecutive integers from 1 to 10, some new questions might be “What is the sum of the consecutive integers from 1 to 100?”, “What is the sum of consecutive integers from 1 to n?”, and “What is the sum of the squares of the integers from 1 to 100?”  Make sure that the students understand that the assignment is to create the question, not answer it; not being required to answer the question takes the pressure off, and it frees them to be more creative.  Again, math contest questions are great places to start.

3)  Mathematical Poetry

Have students write short mathematical poems, like haikus, quatrains, limericks, or even sonnets.  Ask them to explain a math concept, pose a math problem, or describe a geometric object in verse.  And keep it short!  Insisting that the students only use 17 syllables or four lines to describe their mathematical topic not only forces them to refine their thoughts, but it also makes the assignment so easy that it’s nearly impossible not to do it!

4)   Writing Prompts

Have students respond to prompts like:  “What is your favorite number?”; “What’s the best shape?”; “Describe a mathematical epiphany you’ve had”; “Write an original math joke”; or “Write a mathematical aphorism”.  Create your own prompts, or better yet, have students suggest them!

This is an adaptation of the second part of a piece I wrote for the Celebration of Teaching and Learning‘s Edblog.  You can read the entire piece here.

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http://goo.gl/cnMwj

This question is based on the complex methods companies like Apple use to reduce their tax bills.  How much more would Apple have to pay if it were taxed like an individual?

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Of course, one reason this is a such a nice triangle is because it is a right.  We can easily see that the side lengths satisfy the Pythagorean Theorem

Another reason I like this triangle so much is because it plays a part in another of my favorite triangles:  the 13-14-15 triangle!

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http://www.stellarium.org/

Preloaded with 600,000 stars, all you have to do is set your coordinates and start gazing!

You can search by name for various heavenly bodies, map the constellations, and even get close-ups of observable planets!

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The video of this device in action, from Robert Howsare, is definitely worth a look.

An interesting challenge would be to come up with the parametric equations of these curves based on the angular velocity of the two turntables, much like what can be done with actual harmonographs.

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http://goo.gl/bGsSc

Lehrer discusses the results of a study from the 1980s in which psychologist Paul Sackett attempted to measure the speed of supermarket cashiers.  A short “check-out test” was developed which involved scanning a small number of items.  The test was administered, and resulted in a list of the fastest cashiers.

What is interesting is that when Sackett compared the results of the test with long-term data collected by the electronic scanning systems, there was a surprisingly weak correlation between the results of the speed test with the data from regular usage.  That is to say, there was no real connection between being fast on the test and being a fast on a day-to-day basis.

Sackett’s misconception, and perhaps one held by many, is that there is a natural correlation between maximum performance (that on a short test) and typical performance (that is, under normal, day-to-day circumstances).   Tests like the SAT, the GRE, and other high stakes tests, are tests of maximum performance.  Our educational system relies on these  more and more, but are we sure they measure what we assume they measure?

Lehrer points out that individual success is determined more by character traits like perseverance and self-control,  but of course, it’s hard to capture that in a timed, multiple choice exam.

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I have always found great value and pleasure in writing.  It is a valuable skill, a necessary tool of scholarship, and a powerful creative outlet.  And now I see its value as a math teacher.  The more my students write, the more useful and interesting we all find it.

In this post, I’ll first address the question “Why Write in Math Class?”.

Why Write in Math Class?

There are infinitely many good reasons to write.  I’ll offer three that have been on my mind lately.

1)  Writing is a fundamental mathematical skill

Many people might not be aware of it, and many might not admit it, but good writing is a fundamental mathematical skill.  A proof isn’t a proof unless others understand it, and that can’t happen if it isn’t written clearly and concisely.  Also, it’s great when we find the right answer in a math problem, but as most teachers know, it’s usually more important to understand the problem-solving process than to get the right answer.  Good writing skills help narrate and record that process, and make that process available for teachers and peers to understand.

2)  Writing is an indispensible professional skill

I’ve had many different jobs in my life, and being a good writer made me more effective at all of them.  Whether designing technology systems, meeting with clients or consultants, talking through project specifications, or working on a team, being able to document and communicate effectively about the process gave me an edge.   Writing about mathematical ideas and procedures can be hard, but it’s great training for thinking and communicating about the kinds of open-ended problems students will face in the real world.

3)  Writing helps me understand my students better

By regularly interacting with my students through writing, I get to know them in a significantly different way than through their work on exams and homework.  Through various writing activities, I can develop a better sense of what kinds of math problems they like, what kinds of problem-solving techniques they are most comfortable with, and of course, what kinds of ideas are difficult for them to consume.  Getting a different look at how my students think mathematically is incredibly valuable as a teacher, and it can be extremely fun, too!  Giving students the chance to think and write creatively about math almost always produces something unexpectedly wonderful!

This is an adaptation of the first part of a piece I wrote for the Celebration of Teaching and Learning‘s Edblog.  You can read the entire piece here.

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http://darkwing.uoregon.edu/~wmnmath/People/index.html

In addition to resources about historical figures, this list contains references to many living mathematicians.

And here is another nice collection of biographies of female mathematicians from the University of St. Andrews.

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http://goo.gl/cALpO

The interview is broken up into different topics like the Hausdorff Dimension, economics and mathematics, fixed points, and the birth of fractals.  In addition, Mandelbrot talks about his personal, academic, and professional life.  It’s an interesting window into a profoundly important person.

The website WebOfStories.com also offers clips of interviews of other scientists, like phsicists Murrary Gell-Mann and Freeman Dyson and Biologist Francis Crick.

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The New York Times framed the issue as requiring the student to use a technique outside the normal curriculum; the problem is worse than that.  The NYS education commissioner dismissed the error as a “typo”‘; the error can not be considered a typo.  The chancellor of the NY Board of Regents decreed that anyone who claims the tests are invalid is just pushing back against teacher evaluations; no one who understands mathematics can claim that this question is valid.

The problem starts with a trapezoid of sides 5, 16, 13, and 28.  After asking the student to find the perimeter of the trapezoid, the problem then states

A new trapezoid is formed by doubling the lengths of sides AB and CD.  Find the perimeter of the new trapezoid.

 And here’s where the trouble begins.

1)  Is this a right trapezoid?

The Schoolbook piece assumes that the trapezoid is a right trapezoid, i.e., that angle ADC is a right angle.  Nowhere in the problem is it stated or indicated that the trapezoid is right.  And even if we know that it is right, 5th graders are not expected to know the Pythagorean Theorem.

2)  Why does BC change while AD remains constant?

The Schoolbook piece also assumes that as AB and CD are doubled, the length of AD remains constant while the length of BC changes.  Thus, in order to find the new perimeter, the student must find the new length of BC (using the Pythagorean Theorem).

This is the critical error in the construction of this problem:  the test authors don’t seem to understand the subtleties of scaling figures.

Doubling AB and CD doesn’t specify a unique new trapezoid.  BC could change while AD remains constant; AD could change while BC remains constant; AD and BC could both change.  (It is interesting to note that it is impossible to double AB and CD while keeping both AD and BC constant).

Was the original intent to tell the students, or have them assume, that the angles stayed the same?  If so, the resulting figure could not exist.

Was the original intent to tell the students, or have them assume, that AD was also supposed to be doubled?  If so, this still doesn’t specify a unique new trapezoid (unless the angles also remain constant).

The concept of this problem is fundamentally flawed, and it demonstrates a real lack of mathematical understanding on the part of those who created, edited, and screened it.

What’s worse, education officials pretend that this is just a ‘typo’, and that this is no reason to question that validity of these tests.

If the consistent appearance of erroneous math questions on state exams year after year doesn’t constitute legitimate criticism of the validity of these exams, then what possibly could?

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http://mathinsight.org/curl_idea

If you interpret a vector field as the flow of a fluid, then you can interpret the curl as a measure of the tendency to rotate at a given point.

One way to think of this is to imagine a tiny sphere, or paddle-wheel, fixed at a point in space, and then consider how that object would rotate if subjected to the flow of fluid as given by the vector field.

This write-up and series of animations from MathInsight.org are very useful in attaching some intuition to this complex idea.

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http://www.jasondavies.com/necklaces/

Combinatorics is the mathematics of counting things, and one of the classic “advanced” counting problems is this:  given a certain number of beads of various colors, how many different bracelets can you make?

The problem may seem easy enough, but it becomes quite difficult when you start to understand what “different” really means.

For example, if you turn one bracelet into another by rotating it, then those two bracelets aren’t different.  Even more complicating is that if you can obtain one bracelet from another by flipping it over, then they are also the same!

This visualization can really help develop a sense of the complicated symmetries at work here.

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While genuine public discourse about educational policy and philosophy should be a good thing for us all, it’s all too easy to lay the “accountability” at the feet of teachers and ignore the many other factors that contribute to student “performance”, some of which may be even more fundamental to student success.

For example, it turns out that if we provide students with healthier, more nutritious meals, they will perform better and miss less school.

http://www.guardian.co.uk/education/2011/apr/10/school-dinners-jamie-oliver

Test scores up.  Absenteeism down.  Lifetime income substantially raised.  All by replacing industrial, highly-processed cafeteria food with the real thing.

I always liked Jamie Oliver.

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No calculators allowed!

Which of following is the largest?

                                     (a)                            (b)  

                                     (c)                    (d)  

There are a couple of nice ways to solve this without using a calculator, and plenty of good ways to extend this question.

The real challenge is to come up with your own version of this problem!

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http://goo.gl/dCBy2

This question is based on rapid population growth in African countries.  How long will it take Nigeria’s population to quadruple?

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http://tedxnyed.com/2012/speakers/patrick-honner/

I will be giving a short presentation on creativity and mathematics.  I am very excited to be a part of the event, and to see other presenters like Christopher Emdin, Frank Noschese,  Bre Pettis, and many others!

The event will be live-streamed.  More information can be found at the TEDxNYED website.

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http://goo.gl/WoXu3

Windswept is a giant billboard covered in little aluminum arrows that twist and turn in the wind.  The arrows can be thought of as bits of data, namely, the direction of the wind that point.  Taken together, and dynamically, they give a sense of the complicated nature of swirling and changing winds.

Just like this wonderful wind map, this sculpture represents mathematical ideas in a beautiful and thought-provoking way!

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