Ed Tech 4 Math

In about 6 hours Eastern Standard Time, the magnificently decorated New Year’s Eve Ball will once again descend 77 feet (23 meters) over the course of a minute₁ to symbolize the passage of time while millions of people gather together to ring in the year 2009.

There haven’t been too many New Year’s Eves that I haven’t witnessed this televised traditional event.  And as a kid, was always awed by how the ball would drop as time ticked away on the television screen.  Since the dawn of the new millennium, the New Year’s Eve ball has truly been a magnificent piece of artwork from the designers at Waterford Crystal, and I am ever more still eager to watch its brilliance descend from the heights of the New York skyline.

2009 gets a make-over

This year, the ball is a 12 foot (in diameter) geodesic sphere based on a truncated icosahedron, weighing in at 11,875 pounds.  It is covered by 2,668 Waterford crystals and illuminated with 32,256 LED lights (about 3 times more than last year). And even will all this, the ball is more energy efficient than ever, consuming only the same amount of energy per hour as it would take to operate two traditional home ovens₂.

I can’t wait to see the show that will be put on this year as the ball descends, and with the enhancement of technology is capable of producing 16 million colors and billons of kaleidoscope patterns.  It’s sure to be a spectacular show!

In the classroom

The design and new facts of the New Year’s Eve ball lends itself to a variety of math problems to be solved.  One in particular that comes to mind is in regards to the overall surface area of the ball.

The New Year’s Eve ball is not quite a sphere, it’s a geodesic sphere (think of EPCOT at Disneyworld).  Its surface is made up of many triangles, with these triangles having different dimensions because of the curvature of the sphere₃.  So, we would need to know the dimensions of every kind of triangle and how many of those different triangles to calculate the surface area.

But, if we simply considered the New Year’s Eve ball as being a sphere, we can use the formula .  After the calculations, we find the surface area is approximately 452.39 square feet.

S = 4(Pi)r².  With what we know about the 2009 ball, our radius is 6 feet.  Therefore, S = 4(Pi)6²

What if we were to fill the New Year’s Eve ball with a bunch of confetti?  How much confetti can the ball hold?

As with surface area we have a formula that we can use to fine the volume: V = 4/3(Pi)³

I’d much rather explore the volume of sphere with a 3-D interactive animation that helps me to build conceptualization of the concept.

Adaptive Curriculum offers such a resource in “Volume of a Sphere.”  In this Activity Object dynamic modeling is used to derive the formula for the volume of a sphere from the formula for the volume of a pyramid.  As the user changes the number of pyramids in the sphere they observe the relationship between the sum of the volume of the pyramids and the volume of the sphere.  The visualization of deriving this formula assists students in understanding where the formula came from and also strengthens reasoning abilities.  The final visualization reminds me a little of the New Year’s Eve ball!

For more on the New Year’s Eve ball:

• Time’s Square Alliance
• Modern Marvels: New Year’s Eve Ball Drop
• New Year’s Eve Ball Grows Bigger and Brighter

References:

1. Times Square Ball.  http://en.wikipedia.org/wiki/Times_Square_Ball  Retrieved December 31, 2008.
2. Time’s  Square Alliance.  http://www.timessquarenyc.org/nye/nye_ball.html   Retrieved December 31, 2008.
3. Geodesic dome. http://en.wikipedia.org/wiki/Geodesic_dome   Retrieved December 31, 2008

This morning, when I went online (my browser opens to MSN), and had scanned the homepage any faster, I would have missed two unique articles!

At first, the article “20 Thing You Didn’t Know About … Pencils” flashed before my eyes.   So, naturally, being the math teacher that I am, I opened the link and read about 20 things I didn’t know about pencils.  Dean Christopher (Discovery Magazine) lists some very interesting factoids about our beloved pencil.

Did you know …

• The average pencil has enough graphite to draw a line about 35 miles long
• The first American pencil factory opened in 1861 in New York City
• The word pencil derives from the Latin “penicilus,” meaning “little tail”

You can read the rest of Christopher’s pencil facts at “20 Things You Didn’t Know About … Pencils“.

And if pencils weren’t interesting enough, the next headline to catch my eyes was “ ‘Smoot’ reaches new heights in MIT.”

I can’t remember why I opened this link, but when I did, I read about the 50^th anniversary of using a “Smoot” as a unit of measurement.  In 1958, Oliver Smoot and his fraternity brothers at MIT measured the Harvard Bridge using Oliver as the unit of measurement!    They found that the bridge was approximately 364.4 Smoots long (Oliver measured 5 feet and 7 inches).  Smoot later became the chairman of the American National Standards Institute.

So, how could I not look for more information on the Smoot?

I didn’t really come across anything more than what I had already read.  But, I did find a nice article from Cross & Crescent, a publication from Smoot’s fraternity.  And I found on Google calculator that 35 feet = 6.26865672 Smoots.

Ferris Bueller said it best:  “Life moves pretty fast. If you don’t stop and look around once in a while, you could miss it.”

“It was much different in our times…” is a cliché that we often hear from our elders. But now, it is my turn to say it. When I was in high school, which corresponds to the late 1980’s, we did not have the luxury of exploiting technology in our math classes. In order to sketch the graph of a function, we had to perform a series of tedious steps which would yield a graph that we were never a hundred percent sure of, and unfortunately, we would have to count on the sketch presented in the text book. I can also remember going over a number of books and tens of graphing exercises prior to a really scary precalculus exam.

Once when I was in high school, I came upon a question similar to this one:

How many real solutions does the equation given by e^x = x^3 + 4 have?

At first sight, it really did not ring a bell and I desperately looked up the solution: Sketch the graphs of y = e^x and y = x^3 + 4 on the same set of coordinate axes and observe the number of intersection points. This was a complete awakening for me (and would later constitute a major corner stone in my math teaching career: I wrote a book, based upon how to exploit the capabilities of the graphing calculator, which was recognized by many as well as the US Department of Education).

Now, if I make the following claim, what would you think?

It is possible to solve any type of equation or inequality with a graphing calculator, whether it is algebraic, trigonometric, exponential, logarithmic, polynomial, or transcendental, in a similar way without having to perform tedious steps.

With a graphing calculator, yes, it is possible. However, it would be such a waste not to exploit other useful functions of these brilliant handheld computers that are given the modest name of “graphing calculators”. With our mathematical knowledge and what is already there within those handheld giants, the sky is the limit to what can be performed, mathematically speaking!

For instance, let us consider the following question:

What is the equation of the parabola that passes through the points (2, 0), (4, 6), (-3, 20)?

Here is one approach to the solution: Let the parabola be y = ax^{^}2+ bx + c. Plug in the points and you get the three equations given by 4a + 2b + c = 0, 16a + 4b + c = 6 and 9a – 3b + c = 20. Now you have a linear system of three unknowns and three equations that you can use to solve for a, b and c to find the correct solution: a = 1, b = – 3 and c = 2.

Now, how can you use technology to help you? Can you use it to solve the linear system for a, b and c? Yes, indeed you can. But let me propose a rather “radical” approach. Does your graphing calculator perform quadratic regression? I bet it does. I suggest that you use quadratic regression to find the equation of the parabola; since you have three points and no more, quadratic regression will exactly give you the parabola that passes through all three of these points and in one easy step. I believe this approach is much better than what I used to be doing when I was in high school since it saves lots of time avoiding the danger of making a mistake while solving the system of three equations for the three unknowns.

Consequently, in the level of civilization attained today, we do have the luxury of using technology. All it takes is to use this option smartly by combining the capabilities of technology with our knowledge of mathematics. Thus, we can not only increase the quality of our teaching, but also speed up the teaching and learning process. Here is a humble opinion of mine: There will be times when the difference between a fair math teacher and a good one will depend on how competently that teacher employs technology in the classroom. Believe me, my dear friends, those times have already arrived!

“Constructive learning” has emerged as a prominent approach to teaching during this past decade; it assumes that learners construct their own knowledge on the basis of interaction with their environment.

Have you ever thought of why we can get so excited while listening to some lectures, not wanting them to come to an end, but on the other hand cannot help getting bored in others and keep yawning all the time looking forward to the end of the lecture to arrive? I remember the film, Sister Act (1992), where Whoopi Goldberg, as Dolores Van Cartier, impersonated a nun who challenged and reorganized a church’s school choir transforming them into a modern singing group which resulted in attracting an audience to the church that was so large that the Pope himself decided to pay a visit to the convent in order to watch a special performance given in his honor. This film is a typical example for us to observe many of the elements of “constructive learning” where the teaching and learning process employs a series of meaningful interactions between Dolores and the choir members, as well as the choir and the church audience, which eventually led to many young people who preferred going to the church on Sundays to hanging out on the streets.

“Constructive learning” has emerged as a prominent approach to teaching during this past decade; it assumes that learners construct their own knowledge on the basis of interaction with their environment. In the heart of “constructive learning” lies the following assumptions: “Knowledge is physically constructed by learners who are involved in active learning; knowledge is symbolically constructed by learners who are making their own representations of action; knowledge is socially constructed by learners who convey their meaning making to others; and knowledge is theoretically constructed by learners who try to explain things they don’t completely understand” [1]. The constructivist approach honors constructive activities of the learner rather than the demonstrative behavior of the teacher. The reason why students see their physical education, fine arts, or industrial arts classes as their most interesting classes is nothing more than the fact that in these classes they actually “do something” where they are active participants in learning rather than passive recipients of information. This is the primary message of constructivism; students who are engaged in active learning are making their own meaning and constructing their own knowledge in the process.

We, the mathematics teachers, are in fact extremely fortunate to be surrounded by the tremendous possibilities of activities that we can employ in our classes so as to spice up our teaching practice. In this respect, technology plays an important role in that it provides a variety of visual tools and online activities specifically designed for the sole purpose of enhancing the students’ learning experience. For instance, let us assume that our topic of concern is the Fundamental Counting Principle as a preliminary step while teaching permutations and combinations. We can use the following illustrative problem:

A man can change how he looks by selecting one of five different hair styles; one of four different hair colors; and one of the three choices of wearing a beard, a moustache or both. How many unique appearances are possible for this man?

Instead of saying “there are in total 5 x 4 x 3 choices available for the man to change his appearance,” would it not be more meaningful to let the students change the appearance of the man through an online activity and have them realize by themselves that they need to multiply the number of choices in each step in order to find the total number of appearances that the man can select from?

Hopefully, you answered yes. 

An activity object on the "Fundamental Counting Principle" in mathematics within Adaptive Curriculum, an award winning online learning environment

In fact, such an activity object does exist within the award winning online learning environment entitled “Adaptive Curriculum” [2]. In this activity object, the students build tree diagrams and use the Fundamental Counting Principle to determine the number of possible disguises that a man can select from. With each new problem associated with this activity object, a new set of possible disguises are created while the level of difficulty gradually increases. What makes this activity object better than just using the Fundamental Counting Principle is that in the activity object, students are provided with high quality animations and practice in manipulation of the mathematical concepts by building tree diagrams.  This then leads to an understanding of how the Fundamental Counting Principle works, rather than it just being a “rule” to use. 

 The activity object we just mentioned coexists with hundreds of other math and science activity objects in “Adaptive Curriculum” giving students the freedom of playing with a number of parameters and engaging them in the event while learning the underlying mathematical or scientific concepts. This smartly engineered system is specifically built for the sole purpose of improving and speeding up the learning and teaching processes.

The World Wide Web provides many brilliant tools for enhancing the learning and teaching process.  And they are just a click away! What it takes to become a good teacher, or a good student, is therefore much easier nowadays than it was in the past: Just make use of the right tool and enjoy the progress. 

References:

[1] http://www.prainbow.com/cld/cldp.html

[2] http://www.adaptivecurriculum.com

Suppose that you have just introduced your 7^th grade students to the basic concepts in geometry and one of your students happens to ask you the following question: “Will I ever use this in my life?” Have you ever thought of what your answer would be?

“Will I ever use this in my life?” This is an extremely tricky question that each of us, as a math teacher, has come across and had to answer every now and then. Some students often use this question as a means of creating a highly controversial discussion in order to avoid the rest of the class; others really want to hear a rational viewpoint. In fact, the answer to this question is simple: No, most of the topics the students learn in math or any other class will hardly be of use in their lives. As the topics get more complicated, this becomes even more dramatic. When does a student really use the concept of probability in his life, unless of course he or she becomes a professional poker player, for instance? Or when will a student have to calculate the areas of simple or complex figures unless he or she becomes an architect?

Some of our colleagues occasionally fall into the trap set forth by this question and desperately make up exceedingly artificial real life situations, related with the current topic of concern, which often fail to be convincing (I strongly recommend avoiding responses based upon “fabrication” unless they really make sense). However, I do believe that this question deserves a logical answer just like every other question, regardless of its intention. In the way I see it, this question presents a unique opportunity to deliver a wise message which will not only attract the attention of the students making them concentrate on the class, but will also help us gain the respect of the students as well as their parents, since the students evidently “talk it through”.

Here is my answer for this question:

OK, you want to know whether you will use this concept in your lives or not. I guarantee that 99.9% of the time you will use neither this topic nor many others in your lives. Most of what you have learned here in this class will sooner or later evaporate unless you become a math teacher like me (which is very much fun, by the way). However, you still have to learn these concepts, but why? We teach you math because we want to improve your analytical abilities; in other words, we want to make you smarter so that you will earn the capacity to handle harder situations in the future whether you choose to go to college or not. We teach you history not only because we want you to have a basic knowledge of your past but also for improving your memorization skills; a good memory is extremely essential for every individual. We teach you languages and there is no doubt that you will use them in your lives especially in these times when the concept of “globalization” is so hot. We teach you sciences because we want you to understand yourselves and what it is really like to live in this world so that you will be more appreciative of your own lives. We teach you arts because we want you to discover your talents that are likely to be very handy in your lives. So, you see, everything that we teach you is meant to push you forward so that you can become more sophisticated individuals and respectable members of the society. Whether you use them in your lives or not, whatever we teach you is intended to help you serve yourselves in the best way possible. Now, let’s get back to our class…

Tomorrow, if you were asked by one of your students, “Will I ever use this in my life?” what would be your response? How would you convince your students that what they are learning in math at that very moment will be of use to them later on in life?

According to Malcolm Gladwell, innovators can be classified into two distinct groups depending on whether they are “conceptual” or “experimental”. A very typical conceptual innovator was Pablo Picasso, who had a great idea which he executed it almost perfectly for a little while, then unfortunately faded. Whereas Paul Cezanne, the experimental innovator, worked harder and harder at his expertise, which continuously improved his work.

So, have you ever thought what kind of an innovator are you?

Malcolm Gladwell is a senior writer for “The New Yorker” whose work often deals with the unforeseen implications of research in the social sciences, repeatedly and extensively using academic work, particularly in the areas of sociology and psychology. Gladwell’s first work, The Tipping Point (2000), discusses the immense consequences of small scale social events, while his second book, Blink (2005), explains how the human subconscious interprets events and how past experiences allow people to make informed decisions very rapidly; both of these works became international bestsellers.

I came to know Malcolm Gladwell not because of his books, though.  What really captivated me about Malcolm Gladwell was the opening speech he gave at the NCTM National Conference last April. In his speech, he classified innovators into two distinct groups depending on whether they are “conceptual” or “experimental”. A very typical conceptual innovator was Pablo Picasso, who had a great idea which he executed it almost perfectly for a little while, then unfortunately faded. Whereas Paul Cezanne, the experimental innovator, worked harder and harder at his expertise, which continuously improved his work. It is therefore not a surprise that Picasso created his most valuable works when he was only in his mid – twenties. On the other hand, Cezanne’s most valuable works came in his much later years and resulted in being 15 times more valuable than his earlier works.

Gladwell moved on with other examples that include the Eagles (conceptual innovators) versus Fleetwood Mac (experimental innovators), and Herman Melville (conceptual) versus Mark Twain (experimental). He continued to talk about how the American society readily embraces the Picasso models; the music industry is unfortunately not thriving the way it used to, because we are simply in search of immediate successes and we are no longer patient enough to support the experimental innovators to create the albums that will last forever. The same applies to the automotive industry in the US where excellent cars have been produced exploiting the brilliant ideas of the conceptual innovators while in the mean time the Japanese automotive industry has grown with a relatively slower, but ever increasing pace producing environmental friendly and fuel efficient vehicles whose performance are comparable to those produced in the US.

Regardless of the area of concern, it is evident, as pointed out by Malcolm Gladwell, that success is the outcome of hard work and persistence, and the American society really needs to rethink the way we educate our children, especially when it comes to teaching math.               

Gladwell gave the example of an international math contest where those countries who did well in this contest were the ones who demonstrated persistence and dedication to filling out a really long survey right before the contest, so he made the following inference: “How well a country does well in math doesn’t have to be measured with math questions.”           

Gladwell also gave the example of a very hard mathematics test administered to a group of American students and a group of Asian students. While the American students left the room giving up after a few brief attempts to solve the questions, discovering that they were indeed too hard, the Asian students never gave up until the proctors said their time was up.

All in all, I agree with Malcolm Gladwell means:  we must be persistent with our dedication and hard work until success arrives; and when it does, we must not do what Picassos do. Every achievement must constitute the basis for the next one. Eventually, when there is no one left to compete with, we must keep competing, with ourselves this time, trying to push our boundaries consistently for the better. This, my friends, is how we can advance ourselves and the communities that we belong.

Reference

http://thejosevilson.com/blog/2008/04/10/notes-from-the-nctm-malcolm-gladwell-speech/

A university professor once gave an assignment to his students at a major university in the United States in which the students had to write an essay addressing the following question: “What is risk?” One of the students responded with just one sentence and his name on a blank piece of paper: “This is risk!!!” The professor was dazzled and gave an A+ to this three word essay.

Next year, the same professor gave the same assignment to a different group of students. One of the students, familiar with what happened the last year, submitted a similar essay which received an F along with the following comment made by the professor: “It is foolish to take the same risk twice.”

Learning from experience is an important issue for everyone. No matter how many times you tell a child not to approach a hot stove, he will learn this only after he gets burned once; a Turkish proverb goes as follows: “One experience is worth a thousand warnings.”

In fact, learning from experience is absolutely essential sometimes, especially while we are teaching mathematics. Every so often, we want our students to make some very critical mistakes and we create questions with a variety of traps that our students will very likely fall into. The reason is simple. When our students make such mistakes, we will have the opportunity to bombard them with a heavy load of knowledge.

Experience does not only involve learning from mistakes; it also comprises learning from experimenting. At this point you might start arguing that mathematics is not an experimental branch of science. With all due respect, this is not true. We can conduct experiments and we are very fortunate that our experiments only require a pen, a few papers, and our brains.

Follow the link for a more thorough exploration of this idea.

(more…)

Being able to solve multiple problems at the same time using limited resources is a typical example of what we, the mathematicians, refer to as “analytical thinking” and this is one of the major goals of teaching mathematics in schools.

Suppose it is 2:00 AM in the morning and you are driving your brand new convertible sports car on a major highway trying to get to your home in the city center which is 80 miles away. Suddenly, the weather changes and it starts raining heavily with a fierce thunderstorm making the rest of your journey very unpleasant. Struggling to drive for 20 minutes, you decide to pull over as soon as you see the nearest shelter next to the highway and wait until the weather gets better to drive.

You see three people there: the first person is the doctor who once saved your life (he is lucky to be there with his car that suddenly stopped working); the second person is a patient heavily injured in the thunderstorm; and the third person is “the one” that you would like to spend the rest of your life with and unfortunately, this is your one and only chance to meet her (or him), so you really have to take this opportunity to do so.

The question is: What would you do?

Here are a few further details: there is not enough time for an ambulance, which means the only hope for the patient is driving in your car to the nearest hospital; the doctor has stopped the bleeding for a while but the bleeding is very likely to start again because the first aid kit in the doctor’s car and that in yours do not have the necessary medical equipments (besides, there might also be internal bleeding). To make things even worse, there is room for just one person in your car aside from the driver.  And, none of the mobile phones are working, neither yours, nor the others’.

This question was asked to one hundred people who applied for a management position at a major international corporation. Most of the applicants said they would take the patient and go. Some of the applicants misunderstood the underlying idea and said they would not even stop. A few of the applicants even came up with a shallow response saying they would take “the one”. In fact, only two of the applicants were wise enough to comprehend the hint in the question itself; the question was: “What would you do?” and not “Who would you take?”

The correct answer was: “I would give my car to the doctor to drive the patient to the nearest hospital, and stay there with ‘the one’ and use all of my communication skills to leave a good impression on her (or him) until the doctor would come back with the good news that the patient’s life is saved.”

You probably heard this question before, or different versions of it, because it is really not a new one. But here is a brand new question that I am asking you at this very instant: What is the purpose of this question? What do we mean to learn by asking this particular question to, say, John Doe, who seems to be a brilliant young man?

First of all, not everyone would be willing to share his brand new convertible sports car, so this question gives us an idea about whether or not John is willing to share his resources. Secondly, we understand how the stress factor is likely to affect John; will he keep sane enough and struggle to discover the best course of action, or will he simply collapse under so much stress to give up and go with the majority? Thirdly, when resources are limited, how will John respond to the situation? Remember, the mobile phones are not working while there is limited time and space for only two people in the car. These, indeed, are all what the question reveals about John.  However, in the heart of the question lies the following inquest: There are multiple problems that John must very quickly solve at the same time, he does have to save a life but he must also please himself by meeting the lady of his dreams.

Being able to solve multiple problems at the same time using limited resources is a typical example of what we, the mathematicians, refer to as “analytical thinking.” This type of thinking involves careful analysis of the situation and formulating the optimum solution, if not the best one. This, ladies and gentlemen, is what we mean to give our students by teaching them mathematics; they will probably forget the law of cosines in trigonometry, but the analytical thinking skills they acquire on the way will be theirs to keep forever.

So now, the question is how are we as educators in an ever changing, technological world, going to accomplish this task?