Definition of Spiraling

Multiplication, geometry, clock arithmetic, and art.

On the Same Page

Recently, I had the opportunity to work with a bright, creative student on improving his writing skills. We spent a great deal of time discussing how to assess the intended audience, and how diction, tone, and detail must be adjusted accordingly. In everyday conversation, a staggering amount of background information is assumed to be shared. When you write, you typically reach a larger audience, and you can assume far less.

Just for fun, I have selected a random reading comprehension sample from the net to see just how much core knowledge is required for real understanding. The test is aimed at students on the fourth grade level.

How many things can you see in the night sky? A lot! On a clear night you might see the Moon, some planets, and thousands of sparkling stars.

You can see even more with a telescope. You might see stars where before you only saw dark space. You might see that many stars look larger than others. You might see that some stars that look white are really red or blue. With bigger and bigger telescopes you can see more and more objects in the sky. And you can see those objects in more and more detail.

But scientists believe there are some things in the sky that we will never see. We won't see them with the biggest telescope in the world, on the clearest night of the year.

You might find it hard to imagine that stars die. After all, our Sun is a star. Year after year we see it up in the sky, burning brightly, giving us heat and light. The Sun certainly doesn't seem to be getting old or weak. But stars do burn out and die after billions of years.

As a star's gases burn, they give off light and heat. But when the gas runs out, the star stops burning and begins to die.

As the star cools, the outer layers of the star pull in toward the center. The star squashes into a smaller and smaller ball. If the star was very small, the star ends up as a cold, dark ball called a black dwarf. If the star was very big, it keeps squashing inward until it's packed together tighter than anything in the universe.

Imagine if the Earth were crushed until it was the size of a tiny marble. That's how tightly this dead star, a black hole, is packed. What pulls the star in toward its center with such power? It's the same force that pulls you down when you jump — the force called gravity. A black hole is so tightly packed that its gravity sucks in everything — even light. The light from a black hole can never come back to your eyes. That's why you see nothing but blackness.

So the next time you stare up at the night sky, remember: there's more in the sky than meets the eye! Scattered in the silent darkness are black holes — the great mystery of space that's because they're invisible. They're the mysterious dead stars called black holes.

• I can certainly see how this would be an intriguing passage for a fourth grader, but there is a lot that the author has assumed. For example, the reader must know what the difference is between the Moon, planets, and stars. Although context clues such as "the" instead of "a" and the capitalization of "Moon" might be enough to suggest that Earth has only one natural satellite, I suspect that this fact needs to be understood ahead of time.

• The fact that these things are visible each night requires an understanding of periodicity, if not necessarily rotation. The student must know that a telescope somehow magnifies images, and that the larger the telescope is, the more powerful its magnification.

• They must have an understanding of scale as it pertains to the decimal system of measurement. "Billions" is a lot of years, more than most adults can conceive of, let alone a child.

• There needs to be knowledge of the three common phases of matter, as well as chemical combustion. (Although it should be noted here that the flammability of certain gases in our atmosphere has nothing to do with the nuclear reactor of the sun. I don't know if the author is ignorant or finds it easier to massage certain key facts.)
  
• Students also need to understand gravity. They need to know that it is often related to the size of objects, which for the purposes of this paragraph, serves as an indirect measure of mass.

All this needs to be firmly embedded in the child's brain before any of this passage will really make sense. We can teach all the tricks and parsing techniques we want, but unless students have a lot of background knowledge, they are still going to have trouble comprehending what they read. I think this is why we are having so much trouble raising our reading scores on standardized tests. There has been too much focus on "context clues" and not enough on the shear quantities of information that must be shared to even get young readers on the same page.

Mathematics is Rated M for Mature

Pay close attention. I am about to suggest something that will make many of my math contemporaries put down their protractors and take up pitchforks against me. Before I get to the point, I'm going to attempt to outline the path that has led me to this heresy.

The following is a collection of ideas that have been espoused at one time or another on this blog.

1. The math education requirements set for aspiring and current elementary school teachers are far too relaxed.
   
2. Calculators are the latest in a series of tools, each adopted in turn for there superiority.
3. "It takes a certain maturity level to comprehend certain types of math." (Comment from Andy)
4. When engaged in the design process, sometimes weak links can simply be removed.

Now for some elaboration.

1. The Chinese say that in order to give each of your students a cupful of knowledge, you must have a pitcherful. Clearly, the people who determine the educational standards for elementary school teachers disagree. I recently had an opportunity to peruse a Praxis II practice test for elementary ed, and I was astounded at the difficulty level. The hardest question on the test involved little more than correct application, not derivation, but application of the Pythagorean Theorem. I think it is important to point out that the ancient Babylonians had already mastered this much. I appreciate the fact that there is much more that goes into teaching this just content knowledge. There is all the pedagogy and psychology, especially with the little ones. But the knowledge of how to teach becomes useless without mastery of what you are teaching, and in many cases, what they are teaching is how to hate math. And their students are learning it well.

2. Every time a new technology edges out an old one, traditionalists cry foul. What of the information that will be lost? What if this new technology is suddenly unavailable? This is the argument that naysayers employ against the use of digital calculators today. It is a valid argument, which is merely to say that it is not an outright lie. If a student is taught to perform arithmetic primarily by calculator, than that student forfeits the ability to use the "standard" pencil and paper algorithm, should the need arise.

As I type this, I am within sight of three calculators. The first is the built in application on the computer itself, the second is on my cellular phone, and the third is an actual hand-held with a total of 24 buttons. This machine, which probably retails for two dollars, has the ability to perform 5 arithmetic operations, can store values between steps, and can perform any calculation that would be required of the average person. Calculators are so ubiquitous that to suddenly be without them would mean one of two things, either society has collapsed or you are stranded on a desert island. In the first situation, I suspect there would be more to worry about than the ability to do long division, and in the second, simple finger calculation should suffice for survival.

Progress requires that we give up knowledge that our parents and grandparents depended upon. For example, can you start a fire without a match, can you even start a fire with a match, can you identify edible or poisonous plants, can you drive a stick shift, and the list goes on. When we trade that knowledge, it is with the understanding that we get something more from the deal. Maybe that is a dangerous assumption, but it has brought us safely down from the trees and into the modern era.

3. Before Andy made this comment, it had never occurred to me that the ability to understand math might depend on the maturity of the student. I have read so many stories on prodigies like Gauss, that I had assumed even the most advanced math could be grasped by a child, would that they had the right teacher. Now I am starting to see this may not be true. I have said many times that mathematics is the science of patterns. In order to see pattern, you have to be able to make connections between often disparate things, and that requires a healthy base of facts and experience from which to draw. Maybe children struggle with math simply because they do not have the mental and emotional background necessary to bring meaning to the algorithms.

4. The design process is just that, a process. Ideas seldom spring fully formed from the minds of their creators. Instead, there is a tedious and painstaking struggle to turn the initial concept into the finished product, and there are often heart-wrenching decisions to make along the way. As you watch the deleted scenes on any DVD, imagine how the director felt as the cut was made. You will notice that sometimes a different variation of the scene appears in the final cut, but often times it has simply been deemed unworthy and removed in it's entirety. It just wasn't working, and the faulty part had to be removed for the good of the whole.

Now for the synthesis.

Brace yourself. I propose that math education be delayed until the secondary level. I know that sounds crazy, but the more I think about it the more I love the idea. The two reasons we teach arithmetic are practical application for its own sake and as a precursor to later concepts. As I mentioned earlier, the practicality issue can be solved with a rudimentary explanation of the various operations followed by a brief tutorial on the use of a calculator. The issue of laying a foundation is much trickier. I can't even begin to argue that concepts touched on in arithmetic will not carry over to algebra and beyond. The latter is just a generalized version of the former. What I am suggesting is that school children lack the emotional maturity that makes that transition work. They have no concept of delayed gratification. They do not see that they are working toward something which may not become clear for several years. All they understand is that they are being forced to agonize over multiplication tables and long division and fractions, when they could just punch in the numbers on a calculator and be done with it. To them, it must seem like torture, and who's to say it isn't.

The other factor that conspires to defeat students from enjoying math is the poorly prepared elementary teachers. They often times don't understand themselves exactly how what they are teaching is laying the framework for what is to come, so all they can do is drill the lesson as it appears in their workbook. Reform math programs, which are well intentioned, often make the problem worse, because they require a greater mastery of subject matter from the instructor, not less.

Rather than spending those elementary years teaching students to despise math, we could devote that time to other ventures. Whether the extra space is filled up by music or reading or recess is a question for another day and another blogger. When the students reach the secondary level, then we can begin teaching real mathematics. It's true that they will lack the aforementioned foundation, but they will also lack the ingrained aversion to math. It should be a simple matter to teach long division algorithms along side polynomials or multiplying fractions with rational functions. Students will then be in a position to appreciate what the are learning and why they are learning it.

The math education system is broken. Certain links in the chain have rusted with time. Opinionated cognoscenti from all sides are locked in heated debate over how to repair it, but I think perhaps the solution may instead require total removal of faulty parts.

Or have I gone crazy?

Simple Arithmetic

Most of you probably think arithmetic is easy. All you computer programmers and tenured professors out there can add,subtract, multiply, and divide in your sleep, and you scoff at people who can't make change quickly in their head. Certainly, our nation's public schools think arithmetic is easy. They require elementary school teachers to have hardly any math background at all. And yet our elementary school students are using math technology 50,000 years in the making.

How Mathematics Happened by Peter Rudman explores in depth why and how civilized man came to depend on math. He focuses on the early years, beginning well before writing was invented, and giving an intriguing account of the birth of the science. I am in the process of reading it and it is truly fascinating. This isn't the first early math history I have read, but it is by far the most involved. Check it out. You'll never underestimate the power of the third r again.