Nothing Wrong with Rote

A popular view among educators is that rote learning is bad, bad, bad. Point out how well Asians do in international tests compared to Americans, and defenders will likely counter that maybe so, but Asian education depends on that bad rote learning, but we Americans, no matter how poorly we compare, are superior because we emphasize concepts and creativity.

To be honest, as a math teacher, for a long time I believed that rote learning, while undeniably effective, was merely second rate. Kids must need lots of memorization of mathematical recipes and homework practice to pound poorly understood mathematical procedures into their brains. As a math teacher, I believed that if skilled math teachers developed a strong conceptual foundation within children, the logic and elegance inherent in math would minimize the need for tedious, time-consuming homework.

As a junior high and high school math teacher, my ideas about foundation building were only theoretical. It was easy to look at my students, the products of elementary school instruction and conclude that elementary math teachers were doing a terrible job of building foundations. After all, there is plenty of documentation for the inadequate math teaching skills of elementary teachers. I could be charitable. I did not blame elementary teachers too harshly, because they themselves did not acquire mathematical foundations when they were elementary students. They cannot teach what they do not know.

However, until I came to China to work with elementary students, I had never had a chance to test my hypothesis that all students really need is a great foundation. Actually, for most students my hypothesis worked. They could demonstrate a deep and thorough understanding of the concepts I taught. I often assigned homework of only five to ten problems, and after this little bit of practice, they could reliably get the right answers.

However, there were a few students for whom concepts were not enough. One day they would demonstrate terrific understanding. The next day we had to start almost from scratch. I tried everything, every approach I could think of, including a lot more practice. What worked when nothing else did was lots of practice---yes, tedious, time-consuming practice designed to activate rote memory.

I finally concluded that if a child can successfully utilize the concept to solve problems, that's great. However, if the only way they can master the procedure is to repeatedly execute it until they reliably get right answers (my standard was 80% or more), then so be it. Better that then sending them on their way with nothing.

Sadly, as comparative studies show, too many American children lack essential conceptual foundations because the documented lack of teaching skill means teachers fail to actually effectively teach the concepts. Students also fail to do adequate procedural practice because of the American educational aversion to “boring” homework. So I say let's teach concepts and teach them well, AND let the students practice until they can get the right answers. Some students may need more practice than others. So be it.

Interestingly, recent research on children's and teen brains explains why children have such great memories, and how practice, even in an environment of complete concept understanding, is necessary to build brain pathways.

the whole process of learning and memory is thought to be a process of building stronger connections between your brain cells. Your brain cells create new networks when you learn new tasks and new skills and new memories. And where brain cells connect are called synapses. And the synapse actually gets strengthened the more you use it. And especially if you use it in a patterned way, like with practice, it gets even stronger, such that after the practice, you don't need much effort to remember something.

When we dismiss rote learning, we forfeit a valuable tool for building neural pathways in the brain.

See related posts:

Patient vs Impatient Problem Solving

Common Cart---Cart Before Horse

I Love Manipulatives...But

Cultural Sacred Cows of American Education

MacDuff: The New Math

Slaying the Calculus Dragon

No doubt about it. Many students consider calculus scary, right up there with monsters under the bed. Calculus is the Minotaur or St George's dragon of math at school. Sadly, schools have done little to undermine its almost mythological reputation, what with “derivatives” and “integrals” and those frightening numberless equations recognizable by the initial elongated “∫.” There is like, what? 100 equations, that, according to most teachers, need to be memorized.

It is a pity, because calculus is really the Wizard of Oz, terrifying to behold, but quite tame behind the curtain. Did you know that most people do calculus in their heads all the time? In fact, because the numbers associated with calculus are ever-changing, moment by moment, doing calculus with numbers is a bit pointless. A mother filling the bathtub very often does not want to stand around watching the water. She knows that the water is coming out of the faucet at a certain rate. She knows the bathtub is filling at a certain rate. Every moment the volume of water is changing. Yet, she reliably comes back to check the tub before it overflows.

The high school quarterback and his wide receiver communicate an even more difficult calculus on the field, seemingly by telepathy. The quarterback never aims the ball at the place where the receiver is standing. That mental math is too easy, more like algebra or even arithmetic. No, he aims the ball toward the place he hopes the receiver will be. In his head, he calculates the trajectory of the ball, the amount of force necessary (oh my gosh, not physics, too!), the speed of the receiver, and every other factor. And most of the time he gets it right, and the pass is completed. The fun thing about calculus is that the numbers are ever-changing. It is like hitting a moving target, whereas algebraic numbers thoughtfully stand still.

At its core, calculus is nothing but slope. Remember humble slope, change in y over change in x. Slope is an expression of rate, such as, change in miles over change in time, most commonly called “miles per hour” or mph. The graph is a straight line, so you can pick any two points to find the slope. But what if the graph is curved? If you were to magnify each point and extend its line, every point has a different slope. That is because straight lines illustrate rates like speed (velocity), while curves illustrate rates like acceleration (getting faster and faster each moment), just like the football getting slower and slower until it reaches the top of its path, comes to dead stop (but only for a moment) and then gets faster and faster.

A graph has three basic pieces of information, the x data set, the y data set, and slope. “Derivatives” are used to find the slope of a curve at any point when the x and y data are known. When you know one of the data sets and the slope, you can use “integrals” to find the other data set. Techniques like finding the area under the curve are used to get as close as possible to the exact answer. First you divide the area under the curve into rectangles all having the same “x” length. Then you add up all the area. Obviously, some of the rectangles are a little too small for the curve and some are too big, so your answer is only an approximation. If you shorten the “x” length to make narrower rectangles, your approximation will be closer. Integration allows you to find an exact answer instead of an approximation, however close the approximation may be. But there are “limits” (you have no doubt heard of limits). Your “x” length may be very small indeed, but it can never be zero, because then the sum of the rectangles would illogically be zero.

If you can find a skilled dragon slayer, that is, a teacher who can demystify it, studying calculus can be great fun.