Tag: math

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My Love-Hate Relationship with Math (Part 4)

Part 1 | Part 2 | Part 3

Later on, I would teach math in high school (I would also teach English, but that is another story).

I guess my love-hate relationship with math helped me in relating with my students’ difficulties. I spent a lot of time on the basics, often reviewing lessons supposedly mastered in elementary, which was often not the case, except for a handful of students. I remembered my own difficulties in high school and I knew that were it not for a lucky circumstance that unlocked my understanding, I would probably be in the same boat with them.

So my goal was always to make my students understand, never mind if I was behind the prescribed curriculum. I thought a lot of it was trash anyway, unnecessary and inapplicable for high school students. I mean, seriously, let’s be honest and realistic. Who uses logarithms or proves trigonometric identities in real life?

What use was it trying to teach them how to factor the difference of two squares when they could barely add or subtract fractions? How could I discuss the Pythagorean theorem and its applications when they did not even know the difference between a square root and a cube root?

After one of my exams, a student reported to me that their elementary teacher was the proctor and he looked at my exam and exclaimed, “Why is your exam like this? I already taught you these things before!” If he had said that to my face I would have replied, “Well, had you done a better job, I wouldn’t have had to reteach all this, would I?”

I also hated memorizing stuff. I just didn’t see the point. You don’t go around in real life with everything memorized. There’s no rule against looking up references. So I had a policy that all my quizzes and exams were open-notes and books. I didn’t think it was valid for students to fail just because they forgot some part of a formula. I wanted them to analyze and think for themselves, not spend precious time memorizing. Besides, even the great Albert Einstein was once said to have looked up his own phone number in the directory, saying, “I don’t unnecessarily fill my head with things that I can always look up.”

Of course, if you waited until the exam before you opened your notes and tried to figure things out, you weren’t likely to pass either because you wouldn’t have enough time, and I always reminded them of that.

I would skip lessons that (in my view) were too esoteric, saying, “Ah most of you won’t even get to touch this in college and more so in real life. Let’s just focus on mastering the basics.”

And yes, I get that question a lot. “Why do I need to study this? Will I really use all this algebra in my life?”

To which I reply, “Well, yes, I actually use algebra in my life.”

“For what?” they’ll ask.

“Well, to teach algebra,” I would reply with a wink.

Originally published in Sunstar Davao.

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

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My Love-Hate Relationship with Math (Part 3)

Image from GeniusQuotes

Part 1 | Part 2

From that moment on, I came to love Math and everything about it seemed easy. I wondered why I was having such a hard time before when it all seemed so simple now.

Then came Junior year and Geometry which was a whole different animal. We had a teacher who wasn’t connecting with a lot of us. Fortunately, one of my best friends from my elementary years, Anthony, came back from a high school in Manila, and had already taken Geometry in his sophomore year.

So after he had explained the basic concepts of proving to me (which at first I found as puzzling as problem-solving), everything became easy. We would solve problems at the end of the chapter and compare notes with each other, while the teacher was still explaining the lesson. If both of us got it right, we could relax and do some other stuff like read a pocketbook or doodle. If one of us was wrong, we would exchange solutions and each would try to see who was wrong.

This tag team with Anthony would later be joined by Eric, my other best friend from elementary, and the trio was complete once more. We would go on to our senior year doing this with physics and trigonometry, and it didn’t really matter who our teacher was though some of my classmates found it difficult to connect with them, but our informal peer tutoring and competition made us zoom ahead of the lesson by leaps and bounds.

In fact, we weren’t paying attention one time and chatting with each other a little too loudly so our math teacher got really mad at us, called us a bunch of “smart alecks” and walked out of the class. We were silent for a few a seconds as he stormed out of the room. Someone at the back who probably also wasn’t paying too much attention asked in a bewildered voice, “Who’s Alex?” and the class erupted in laughter.

I spent most of my high school in sheer enjoyment of math, but college was another matter. I walked into my freshman pre-calculus class oozing with confidence. I listened to the first lecture and found out that pre-calculus was just a review of algebra so I relaxed and sat back and didn’t take any notes. I could follow the lectures and examples in my mind.

Then came our first exam and I stared at the paper and wondered where the problems came from because they looked alien. I struggled to solve them but they seemed ten times as difficult as the lectures and examples. I barely passed that exam with a grade of D which in my mind stood for “deflated” as in it deflated my ego and I went back to diligently taking notes every class.

I guess it’s different when you have Ph.D. level professors. I went through calculus, linear algebra, graph theory and so on. But I couldn’t find my old groove. I didn’t have classmates or peers that I could have that sort of friendly competition and  camaraderie I had with Anthony and Eric in high school.

So I went back to being an average math student.

Originally published in Sunstar Davao.

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

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My Love-Hate Relationship with Math (Part 1)

I wasn’t a bad math student. I had a pretty good understanding of it in my elementary years, but I wasn’t that fast in operations. I had trouble memorizing the multiplication table at the higher digits. If you asked me to do 8×7 for example, I would still have to tick off the multiples of 8 on my fingers: 8, 16, 24…and so on until I got to my seventh finger, then write down the answer — I would do that all the way till high school.

I remember that at the start of every year since around grade 4 or 5, I would have trouble remembering how to perform certain operations like adding, subtracting, multiplying and dividing fractions and decimals, or how to get the least common denominator (LCD) and how that was different from getting the greatest common factor (GCF).

Despite that, I was chosen to be one of a handful of “math-gifted” students in sixth grade though I didn’t really feel all that gifted. Looking back now, they must have been pretty desperate. We were asked to cut short our lunch break and come in 30 minutes earlier for special lectures on advanced topics. Again, looking back, was that really a reward or punishment for being “gifted”?

I don’t remember much from those classes except that I struggled to keep up and that it highlighted what I hated most about math — word problems. In all my elementary years, I never understood how to solve a word problem. There was no step 1, 2, 3 to it. At least when multiplying fractions, as long as you memorize the steps, you had a pretty good chance of getting the right answer.

But word problems frustrated me.

Oh I would occasionally get them right but I never felt confident with them and I dreaded seeing them in quizzes or exams. No matter how I studied, I couldn’t prepare for a word problem. Thankfully, teachers didn’t make exams full of word problems, and so I survived elementary mathematics.

Then came high school and the start of algebra. Oh my, here I was trying to remember how to properly multiply decimals and subtract fractions, and now I have to deal with x, y and z, and sometimes a, b and c as well? But given some time, I was able to make some sense of the algebraic rules although factoring left me confused for a good long while, especially quadratic square trinomials.

Somehow, I survived freshman algebra. Now on to my sophomore year.

We began with a review of the basics, how to perform operations with variables, the laws of exponents, and so on. Shortly after that, we were introduced to equations, and then our teacher handed us a page of homework to do over the weekend.

When I got the page, I was terrified. It was full of word problems.

Originally published in Sunstar Davao.

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

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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.

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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.