Ed Tech 4 Math

“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!

The town of Leixlip, in County Kildare, is the birthplace of Guinness beer. But Guinness was yesterday; the company of today is Intel, which recently built a plant worth $5 billion.

Thousands of well educated workers are making the chips that drive Europe’s computers. Intel knows a good thing when it sees one; a new building is being added to its plant in County Kildare.

Do you remember the sad town of Limerick of Frank McCourt’s Angela’s Ashes (1996), where jobs were as rare as jewels? Well, today 4,000 workers in Limerick are busy building computers for Dell.

Leixlip, Kildare County, and Limerick are only three of the regions that experienced the miraculous economic boom in the Republic of Ireland, one of the wealthiest European countries, which used to be on the other extreme only two decades ago. The name “Celtic Tiger” is therefore very well deserved, owing to rapid economic growth in Ireland that began in the 1990’s: Irish economy has grown by more than 200% as of today [1] (“Celtic Tiger” is analogous to the phrase “East Asian Tigers” used to annotate South Korea, China, Singapore, Hong Kong, and Taiwan during their periods of rapid tiger growth in the 1980’s and 1990’s).

The economic growth in Ireland can be attributed to foreign investments; at present, over 1,200 multinational corporations (including Microsoft, Dell and Intel) have chosen Ireland as their base to serve the European market and beyond. Here, these companies do not only find favorable tax environments, and competitive operating costs, but they also have the advantage of a highly skilled, productive, and less expensive English speaking workforce created by a strategically planned and carefully executed long term investment in domestic education (for further information please refer to [2]). Thus, the rate of unemployment in Ireland has remarkably decreased and the persistent emigration of the qualified population has also been successfully reversed.

The educational revolution in Ireland all began in 1965, with the influential report entitled “Investment in Education” sponsored by the Organization for Economic Cooperation and Development (OECD) on education in Ireland. This report emphasized that education was “the key” to the future of Ireland’s society and economy. Although not directly recommended in the report, the government started investing on education beginning in 1967 by paying for all secondary schooling and transportation to school in addition to an intense public awareness campaign initiated on education. This measure resulted in not only a rapid rise in the level of education attained by the younger population, but also a series of noteworthy transformations in the way education was perceived by the society:

• More and more students were availing of education (particularly in second and third level education);
• students, parents and teachers increasingly demanded better education services;
• the range of services provided by the Department started growing in response to technological and economic changes and the increasing demand for second level education;
• the public started demanding a more efficient and effective use of government resources and transparency and accountability in the uses to which these public resources are put;
• there was a growing interest by the media and the general public in education policies.

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“It’s not what you are that holds you back. It’s what you think you’re not.” - Denis Waitley

Do you know how circus elephants are trained?

While training a newly caught wild elephant for a circus, its handler uses a very strong chain to tie one of its legs to a long steel pole deeply buried in the ground. No matter how hard the elephant struggles, it fails to pull away, and in time stops fighting, learning that it cannot break loose. As time passes, the long steel pole and the strong chain are replaced by a shorter pole and a less strong chain. Eventually, a flimsy rope tied to a tiny stake becomes sufficient to keep a fully grown elephant remain imprisoned at the circus.

In 1967, at a well known circus in Europe, six elephants tragically died in a tent fire; the poor creatures did not even attempt to escape from the thin ropes they were tied with!

We, human beings, too, tend to build thick and strong virtual walls around us sometimes. When confined within such walls, circumstances seem to be a lot more difficult to us than they actually are. This phenomenon is actually a predicament that one needs to get over as quickly as possible, especially after one or more unsuccessful attempts. For instance, let us imagine that a student has failed his first two algebra midterms; even though he can still pass the class with a satisfactory performance in the final exam, this is hardly the way he sees or evaluates his situation.

Correct assessment of the situations is of greater significance for those people who are expected to lead a group since their actions are likely to affect the lives of many. Here is an interesting question which is often used in training people who are newly promoted to management positions: By drawing four (not five) straight line segments without your pen leaving the paper, how can you go through all of the following nine points?

1. The line segments can be as long as you wish and you can (in fact you should) go outside of the square.

2. The line segments need not be horizontal or vertical; they can be diagonal as well.

Most of the time the trainees can solve the question after these hints: Here is one of the solutions:

All of the examples we have given up to this point have the same underlying message: “thinking out of the box” increases the number of possibilities tremendously. Whether one is a teacher, a student, or a manager, he should be closed to prejudices and open to novelties. This is how one can go beyond his boundaries; improve himself to a great extent; and set a memorable example that will inspire many others for further and greater accomplishments. 

Some men see things as they are and say “Why?” I dream of things that never were and say “Why not!!!” Robert F. Kennedy

Sooner or later we will all have to accept the fact that we need to keep up with the fast technological progress. The question that we need to ask ourselves now is: “What does it really take to ‘act accordingly’ within our area of interest which is education?

My professor, Dr. Gary Bitter, who is a true innovator, gave the opening speech at a European conference last June where he addressed an auditorium of university faculty, as well as undergraduate and graduate students, whose primary affiliation was educational technology. In part of his speech, he employed a YouTube video entitled “Did you know?” (http://www.youtube.com/watch?v=xHWTLA8WecI). I recommend that you spare six minutes of your time to watch this video if you have not already seen it in another occasion.

This video, which by all means is a mind opener,  was originally created as a PowerPoint presentation for a faculty meeting in August 2006 at Arapahoe High School in Centennial, Colorado, and later spread like a virus on the World Wide Web in February 2007; as of June 2007, more than 5 million online viewers had already watched it. Today, old and new versions of this video presentation have been seen by at least 11 million people in a countless number of occasions including conferences, workshops, training institutes, and other venues.

The Arapahoe High School’s faculty used this PowerPoint to convince the administration to invest more money on educational technology. However, the very purpose of this video goes far beyond that; it is meant to help raise awareness of the phenomenon of “change” in the ever evolving global world of today. Here are a few remarkable quotes from the video:

• The amount of technical information is doubling every two years. So, for students starting at a four year technical or college degree, this means that half of what they learn in their first year of study will be outdated by their third year of study.
• According to the data provided by the US Department of Labor, the top 10 jobs that will be in demand in 2010 did not exist in 2004, which means that we are preparing our students for jobs that don’t yet exist.
• In 2013, a supercomputer will be built that exceeds the computation capability of the human brain; by 2023, a $1000 computer will exceed the capabilities of the human brain; by 2049, a $1000 computer will exceed the capabilities of the human race.

What do these all mean? It means that “We are living in exponential times when ‘shift happens’; therefore we should anticipate, embrace and adapt change quickly and modify our courses of action accordingly.”

Sooner or later we will all have to accept the fact that we need to keep up with the fast technological progress. The question that we need to ask ourselves now is: “What does it really take to ‘act accordingly’ within our area of interest which is education? Is it sufficient to give everyone a state-of-the-art laptop computer that can access the Internet through the fast wireless network we just launched?” Unfortunately, no! The laptop computers will be of absolutely no use as long as they stay in students’ backpacks; they need to be made useful through meticulously designed and carefully planned activities. But how and who should design such activities?  The answer is simple: We, the educators, should undertake such responsibilities; but do we really have the time?

Fortunately, we don’t really need to invent or reinvent the wheel; wise people who have foreseen the near future already started building excellent educational activities that utilize the audiovisual aspects of technology and made them available to the world through the Internet. So all it takes for us is to listen to what they have to say and find out what they have to offer in order to use such activities in our classes; thus it will be possible for us to save considerable amount of time and effort that we can devote to improving and perfecting our teaching practice.

“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

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.

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