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

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.

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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?

Teaching mathematics effectively, requires an appreciation of the big picture – of how the broad issues of mathematics fit into our daily lives. How you see math, and how deeply you see the patterns below the surface, will influence your teaching and the quality of the instruction you deliver.  Ed Tech 4 Math is dedicated to exploring how educational technology can support high quality teaching. 

Las Meninas (Spanish for The Maids of Honor) is a 1656 painting by Diego Velázquez (1599 – 1660), the leading artist of the Spanish Golden Age. The work’s obscure and mysterious composition provokes questions on illusion and reality creating a somewhat vague relationship between the viewer and the figures portrayed. This is the very reason why Pablo Picasso was almost obsessed with Las Meninas creating tens of interpretations of this composite work of art.

I have a very strong reason to place Las Meninas and Picasso’s interpretation of it right at the beginning of the very first article in this blog which is supposed to be on mathematics: Imagine you happened to see Picasso’s version of Las Meninas first. Would you be able to “see” what it really reveals? In other words, would you be able to see the Las Meninas that really lies behind? Let us set aside the symbols and analogies for a moment and ask ourselves the ultimate question: Can we manage to see the big picture without getting lost in the details? Let us keep in mind that life is never as crystal clear as Las Meninas and things often appear as complicated as the way Picasso sees it.

We are the inhabitants of a world in the age of information flooded by bits and pieces of knowledge which grow with an ever increasing pace that the word exponential is simply not enough to express. Despite the level of technology that has been attained, the critical part of the job still remains to be performed by the humans: decide on what information to make use of and what to discard. Believe me, it is harder a job than it sounds!

 All in all, those who can see the Las Meninas behind the Picasso will be the ones who survive in the big picture situations and manage to move forward in life; those who fail to do so will unfortunately remain insignificant.

Once again, this blog is dedicated to mathematics and technology, with a special emphasis on how we can employ technology in the mathematics classroom in order to enhance the students’ learning process. However, we will not hesitate to share instances where wisdom has improved the way people think and changed their lives, as well as society, for the better. Therefore, if you have any relevant ideas, comments, suggestions, or experiences that you think are worth being published in this blog, please contribute! Let us join our forces to advance humanity! After all, is this not the definitive objective of mathematics and technology?