How Teaching Hampers Learning (Part 4)


“What is the use of this?” is probably the most asked question in school, whether spoken or unspoken. And it is the question most often ignored by teachers, giving such formulaic and even smart alecky answers as “you’ll find this useful for college,” or “you’ll need it to get a passing grade in my class.”

They do not understand that unless they answer this “why” with all sincerity, their teaching amounts to nothing, and their students are merely learning by rote, which is probably the most ineffective learning style there is because when summer vacation comes, you’ll be lucky if they remember even 10% of whatever they were able to memorize just to “get a passing grade.”

Indeed, students who understand why they are learning something, who display genuine interest in the subject, are those who do well in it — and even if they don’t understand it in the alloted time, they will persist until they get it.

In 1929, Louis Benezet, the superintendent of schools of Manchester, New Hampshire, wrote the following to a colleague:

“In the first place, it seems to me that we waste much time in the elementary schools, wrestling with stuff that ought to be omitted or postponed until the children are in need of studying it. If I had my way, I would omit arithmetic from the first six grades. I would allow the children to practise making change with imitation money, if you wish, but outside of making change, where does an eleven−year−old child ever have to use arithmetic?

I feel that it is all nonsense to take eight years to get children thru the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten−year−old child for a knowledge of long division? The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.”

He then proceeded to convince teachers in his school to conduct and experiment. They would eliminate teaching any form of arithmetic from the first to fifth grade. Instead, they would focus on allowing their students to express themselves, to learn to read and reason, to tell stories and give their own opinions. The result was astounding:

“The children in these rooms were encouraged to do a great deal of oral composition. They reported on books that they had read, on incidents which they had seen, on visits that they had made. They told the stories of movies that they had attended and they made up romances on the spur of the moment. It was refreshing to go into one of these rooms. A happy and joyous spirit pervaded them. The children were no longer under the restraint of learning multiplication tables or struggling with long division. They were thoroughly enjoying their hours in school.”

But even more astounding was when Benezet introduced arithmetic in the sixth grade level, these 12-year old kids who had no previous formal training arithmetic were very quickly able to catch up and even perform better than their peers in traditional schools. They did especially well in story problems that required a mix of general understanding, analysis and plain common sense. Benezet performed the same experiment in other schools in Indiana and Wisconsin, with the same results, showing that this was not merely due to chance or a fluke accident.

When we teach less, children learn more.

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

Originally published in Sunstar Davao.

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.