What Is Math But Solving Problems? (Part 2)

Click to try the Wolfram Alpha Computational Engine
Click to try the Wolfram Alpha Computational Engine


Conrad Wolfram, a British technologist and businessman, and founder of Computer-Based Math (www.computerbasedmath.org), defines math as consisting of 4 steps:

  1. Posing the right questions.
  2. Translating a problem from the real world into a math formulation.
  3. Computation.
  4. Translating the results back into the real world.

The problem with the current methods of Math education, he asserts, is that we spend 80% of the time teaching students computation (step 3), and we teach them how to do it by hand. Thus we have little time left teaching them steps 1,2 and 4.

I think Step 1 is very important and should be stressed above all the others. Posing the right questions means that the students must first understand the problem. Without this understanding it would be almost impossible to solve the problem.

I have seen too many test papers filled with all sorts of calculations but it was obvious that the student didn’t understand why he was doing those calculations. There would be an answer but it would be so obviously impractical or unrealistic and the student did not even bother reviewing his solution or asking why that was so. This is what happens when there is too much focus on computation and getting at the answer, but not enough focus on understanding and translating the results to reality.

It is like producing students who are experts at changing tires, but they will change the tires even if the problem is an oil leak or an overheated engine. They don’t know how to ask the right questions. They don’t know what the real problem is.

Wolfram proposes that we should begin teaching students at an earlier age to use computers for calculations, which can do them so much faster, more accurately, and at several more orders of difficulty. If we do that, then we have more time to focus on the other steps.

The obsession with making students calculate by hand is eating up a lot of time, and actually kills the interest of the general populace. No wonder a lot of people say “I hate Math” or “Math hates me.” They have the mistaken notion that math equals calculation instead of it being an approach to understand and solve real-world problems like “where do I invest my money so that it gives the best returns” or “which life insurance policy is most advantageous for me and my family?” or even “how do I win this poker game?”

Incidentally, I have a friend who is very good at math and also very good at poker. He was able to build his house from his winnings in poker-playing. Now, isn’t that an interesting and successful application of Mathematics?

It is understandable that in the development of math education, there was a huge focus on computation — because there weren’t any computers back then and the only way you could get to the results was to compute by hand. But that is not the case today.

A complex algebraic equation that may take several minutes or even an hour to compute by hand can be done in seconds by software like Mathematica (invented by Conrad’s brother, Stephen). Instead of teaching students the how of solving such an equation, teachers can instead focus on the why — on what it means in real world, and why it is important, and why it matters.

Wolfram also discusses one of the most common objections to this approach, which is that students must “learn the basics” first and that is why there is so much focus on computation. But what exactly do we mean by learning the basics? He comes up with this analogy.

Do people need to understand the mechanics of a car in order to learn how to drive it? Well, maybe in the early days of cars, it was necessary to have some knowledge of how an engine works and so on, because there was less automation and you had to do a lot of things manually just to even start the car.

These days, there is so much automation that you don’t even need a key to start the car, or learn manual transmission. Just push a button and step on the pedal and away you go. So now we have millions of people who can drive cars without really understanding how they work, but they know how to get from point A to point B, which is really what driving a car is all about.

With computers, we have the ability to teach our kids to handle complex mathematical equations without really doing the nitty-gritty work of solving them by hand. Instead of being disinterested or intimidated because of the long calculations, they will instead be more focused on the implications of the problem and how the results matter in real life, which is really what mathematics is all about.

Here is a video of Conrad Wolfram’s original talk back in 2010:

Originally published in Sunstar Davao.

Email me at andy@freethinking.me. View previous articles at www.freethinking.me.

What Is Math But Solving Problems? (Part 1)

Photo Credit: Neil Tackaberry Flickr via Compfight cc
Photo Credit: Neil Tackaberry Flickr via Compfight cc

There is a problem with math education today. I wrote an entrance exam for applicants of our company, a drugstore chain in the city. The exam was designed to test basic knowledge and real-world problem-solving skills. Part of the exam are questions like, if this medicine costs 4.75 each and a customer buys 18 tablets, how much should he pay? The most difficult question involves the item having a discount, and then asking how much change the customer should get if he pays with a large bill.

These are questions high school graduates should have little to no difficulty in solving (and yes, we tested it on some college students we know and they managed to solve them correctly). Yet we have many applicants who are supposedly college graduates who cannot answer the questions correctly. In fact, less than 50% of our applicants manage to pass the entire exam.

The problem, I believe is that math education has been too much focused on making students learn the how and not enough of the why. There is too much focus on skills and not enough on the purpose. A question most students ask about math is “What’s the use of this in real life?” and it’s a question math teachers brush aside or answer with some vague and useless reply like, “Oh it’s very useful. You’ll understand when you get to college.”

That is sad and unfortunate. Math teachers should pay more attention to that question. It should not be taken lightly. Answering that question satisfactorily can turn a disinterested student into an eager lifelong learner.

There is a Filipino saying that goes, “Kung gusto, may paraan. Kung ayaw, maraming dahilan.” Meaning, if you want something, you will find many ways of doing it, but if you don’t like to do something, you will also find many excuses not to.

Most students don’t understand why they’re doing math so they end up despising it because it is “useless” and a “waste of time.” They learn skills without knowing their purpose and thus easily forget them. The key is to get students to know WHY they’re doing something, and then they will become interested, and not only remember how to do it, but find even better and more innovative methods of doing so.

In our house, I am the designated Math tutor and I always have a hard time with my 12-year old son. Previously, I thought that he was just not as capable as his siblings. But recently, he developed a keen interest in playing with Rubik’s Cube. He followed some tutorial videos on Youtube and he can now solve the entire cube very quickly. I myself have never managed to solve more than one face of the cube at a time and I had to ask him to teach me. Then I told him, you know you’re already doing Math with this. It’s all about understanding what the blocks look like now, then how you want it to look, and then taking the necessary steps to get there.

“What is Math but solving problems?” said Dr. Norman Quimpo, a professor at the Ateneo de Manila University. So simple, so true, yet it is an assertion that many math teachers fail to grasp. They spend so much time teaching students to compute by hand that they have little time left teaching them how to understand the problem and how to understand the answers.

When I was teaching algebra, for example, I liked to stress that solving for x does not mean you have answered the problem. Sometimes, the answer to the problem is not the answer to the equation. And sometimes, you can even solve the problem without solving for x. One of my pet peeves is having teachers who stress only one way of solving a problem. That is such a narrow-minded approach. Math is not about knowing the ‘proper way’ to solve a problem because there is no such thing as a proper way. Rather it is about understanding a problem and then finding a solution to it and the solution may be more ingenious than you think and should in fact be celebrated rather than marked as wrong.

I remember in grade school, when my brilliant classmate Anthony Montecillo, proposed an alternate solution to a problem that the teacher had given. Instead of insisting on her method, our teacher invited Anthony to go to the board and explain his solution, which turned out to be faster and more intuitive than the “standard” method. Our teacher then praised the solution and dubbed it and said something like, “Oh, we should include this in our math books and call it the Montecillo method.”

Oh, if all teachers could be like that.


Originally published in Sunstar Davao.

Email me at andy@freethinking.me. View previous articles at www.freethinking.me.

Adjusting the Sails

Photo Credit: PralineB Flickr via Compfight cc
Photo Credit: PralineB Flickr via Compfight cc

One of my favorite John Maxwell quotes is this: The pessimist complains about the wind. The optimist expects it to change. The leader adjusts the sails.

The reason I like this quote is that it captures a fundamental behavior that most people exhibit. Whether they are optimists or pessimists, non-leaders only react to a situation, whereas leaders act on it and can even turn seemingly hopeless predicaments around to their advantage.

In my more than 2 decades of being in different types of organizations, I have observed that all of them have their share of whiners, fans, and leaders. And true to the Pareto Principle, only 20% belong to the latter group.

Whiners are people who complain about anything. In a small group tasked to do an activity, for example, they complain about the rules, about the time limit, about the difficulty of the situation, and so on.  

In society, whiners complain about the politicians, religion, the weather, and so on, but they do little else. These are the people who seem to live their lives 24/7 on Facebook and Twitter. You see them ranting about this or that, or arguing with someone else almost every hour.

Whiners are also the masters of blame. They point their fingers left and right, and front and back. Everything is everyone else’s fault. “There are killings left and right. This is your fault Dutertards! Trump won. This is your fault evangelicals and protestants!”

In the meantime, one has to wonder what else they are doing with their lives besides being miserable on social media — which they self-righteously think is actually doing something to help the groups they think they are fighting for.

The fans are people who clap and cheer, but again also do little else. They loudly agree with what the leader says, but when they are asked to produce, they find an excuse to not do it, or they perform poorly. These are also times when they turn into whiners, complaining about this or that and giving different reasons why they cannot come up with the expected results.

Leaders are few because they are people who are willing to buckle down and do the hard work. And let’s face it, who wants to do the hard work, right? But leaders roll up their sleeves and do it anyway because they know what matters is not your talk but your walk. What matters is not just what comes out of your mouth, but what comes out of your actions. In fact, many leaders are too busy doing what needs to be done that they have little time for social media and all of the drama that goes with it.

A couple of weeks back, I had a chance to reconnect with Rev. Arnel Tan who was the officiating minister in a child dedication where my wife was the ninang (godmother). He is an old friend whom I have known since my teens. He is also a fellow columnist in Sunstar. We had a brief laugh over a mixup that happened a few weeks back. Someone at the proofreading or layout department was probably too stressed or overworked that he or she ran my column with Arnel’s name and photo. It was good that it wasn’t any of my anti-religion articles or Arnel might have gotten into hot water for “writing” such a piece.

When I submitted the article the following week, I included a reminder for the editorial team to use the photo of the bald guy.

Anyway, Arnel told me something interesting they have going on in the church. He told me that on certain days, I think it was Thursday and Friday, they open the church for medical missions and feeding programs. They have volunteer doctors and nurses over to help anyone who comes into the building needing help.

What was particularly interesting for me was when Arnel mentioned that he explicitly told his junior pastors, “When people come in, do not try to convert them and make them protestants. Just help them with what they need.” This, I think, is a very enlightened approach to the diverse mix of religions and cultures we have in our city.

This is one leader we have who knows how to adjust the sails.

Originally published in Sunstar Davao.

Email me at andy@freethinking.me. View previous articles at www.freethinking.me.

Parang Pero Hindi

Photo Credit: Aramisse Flickr via Compfight cc
Photo Credit: Aramisse Flickr via Compfight cc

There are some teachers I’ve had whose names I cannot remember, and there are some teachers whom I will never forget. Orlando “Orly” Darang belongs to the latter group. I heard the news that he passed away on All Souls Day.

When I was a high school freshman, I would sometimes go to the library and look at the old yearbooks. The teachers photos were usually accompanied by short descriptions. Mr. Darang’s description, year after year, usually began with “Tall, dark, and dark,” so that is how I first became acquainted with him.

He became my Filipino teacher in junior year and I found out that he was indeed tall, dark, and dark, with strong, muscular forearms to boot. He was a bit scary that first day of class, and he sternly warned us to always bring our copy of Noli Me Tangere, or face the consequences.

We later learned what the consequence was a few weeks later when one of our classmates forgot his book. He had to lift his chair over his head while Mr. Darang proceeded to lecture and discuss the lesson like nothing was going on. I don’t think I ever forgot to bring my book after that. I mean, if my house were burning, I would probably rush in to grab my Noli de Tangere so I wouldn’t have to face Mr.  Darang’s icy stare the next day.

As the weeks and months went by, the ice began to melt and we began to see his softer side. His one pet peeve was when students began their answers with “Parang” and he would be quick to interrupt with his famous line, “Parang pero hindi?” which usually stopped a student dead cold, much to Mr. Darang’s visible amusement. A typical dialogue would go like this:

Mr. D: “Ano ang ibig sabihin ng sinabi ni Padre Damaso?”

Student: “Sir, ganito kasi iyan. Parang…”

Mr. D: “Ah parang pero hindi?”

Student: “…”

(This dialogue gets lost in translation so I won’t even bother).

You would think that we would have learned to stop saying that word but no, almost every class there was someone who would inevitably say it only to realize too late when he or she was met with the dreaded “parang pero hindi?”

One day, one of my classmates, Raul, did the unthinkable and managed to parry that interjection. When he said “Parang…” and Mr. Darang promptly interjected with, “Parang pero hindi?” Raul didn’t miss a beat and answered, “Hindi sir. Parang pero oo…” and proceeded to give his answer, which no one heard or remembered because everyone was laughing, including Mr. Darang.

There was another time when Mr. Darang walked into the classroom and wrote on the board the words “oten” and “puke” in big, bold letters. There was some nervous tension in the room. Some girls giggled and some boys howled, and the word “bastos” could be heard muttered all around. Mr. Darang promptly addressed the issue asking why these words were considered as “bastos.” Was it simply because they were in Filipino? What if he had written “penis” and “vagina” on the board? Don’t we use those terms in biology class? So why does it sound clinical and scientific in English while sounding vulgar in Filipino? That was one of the classes which really made me think about the power of language and words beyond their dictionary meanings.

He was not all fire and brimstone and was actually very approachable as I would find out when it was composition-writing time. My closest friends know that Filipino is my “hatest” subject. I was a voracious reader of English books and I spoke Chinese and Bisaya at home. The only place I ever spoke Tagalog was in school so I always had a rough time understanding and forming words.

I told Mr. Darang of my difficulty and he understood my problem. He would gently coach me as I asked him how to say this, or how to say that, or what this or that word meant. I learned that underneath his seemingly hard exterior was someone who wanted to help. He had the heart of an educator who wanted his students to excel and perform their very best.

Today, more than twenty years after, I remember the life of a man who was one of those who shaped and molded me into what I am today, who is certainly much more than what I portrayed him to be in this meager piece. And as I pause and reflect on the brief intersection of his life and mine, it seems that I am just waking up from a hauntingly, beautiful dream.

Parang isang munting magandang panaginip…parang pero oo.

Paalam, sir.


Originally published in Sunstar Davao.

Email me at andy@freethinking.me. View previous articles at www.freethinking.me.

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