img.latex_eq { padding: 0; margin: 0; border: 0; }

Saturday, August 11, 2007

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?


Anonymous said...


I don't think that kids should be left in the dark until the secondary level. I do think, however, that they can start developing the right-brain visualization techniques and pattern identification that will serve them well later.
How? Imagine, if you will, a series of calculators that progress from K-6 that are highly graphical/visual in nature. Instead of drilling multiplication tables, when they punch in 3 x 5, the calculator displays:

1.) Three rows of 5 bananas
2.) Five rows of 3 bananas
3.) And then, the integer 15.

This way, visual representations of patterns are reinforced before they get the actual "answer."
Each calculator could be "flashed" year after year with different capabilities for the concepts associated with the particular grade level.
The technology is there - many learning games like Leapfrog already do similar things - it just needs to be brought into the schools.
What do you think?

Mr. Lucchese said...

What you are suggesting is only going to work for relatively small integers. Even if you have a screen big enough to show 215 bananas, having to count them would defeat the speed of the computation. The base-10 multiplication table requires the memorization of 45 entries, and any computation that would fit graphically on a screen could be done just as easily with a low-tech abacus of tasty M&M's.

Anonymous said...

While it may be true that maturity is required for math, I disagree that it should be delayed until high school. However, it might not be required until high school. I propose a system in which students are given the option of taking mathematics classes, but are not required. That way, nobody is forced to learn math (thus, no aversion developed) but those who are mature enough still have the opportunity to take classes. Since it requires a certain level of maturity to know taking an extra class now will help you later, a maturity test is already built into that system. Since fewer students would take these classes, a teacher with an actual degree/extensive experience in mathematics could be hired to teach these optional math classes. Students who chose to take these classes would of course get a boost when going into high school (such as being able to skip preliminary lessons, etc.).

When talking about maturity in my mathematics, be sure not to use a cookie cutter approach. Who are you to say that no elementary students are prepared for mathematical education? Instead, leave the door open for those who want to take it.

Mr. Lucchese said...


I spoke in sweeping generalities only because our elementary ed system is a one size fits all juggernaut. Of course, I realize that many young students will have the academic maturity required to understand and enjoy math. Having math served as an elective for any that chose to pursue it is certainly acceptable to me. I wouldn't want to deprive anyone from learning math at any age, I was only speaking of the masses who have math forced upon them before they may be ready.

Suma said...

Hi, as you said the latest calculators are adopted by everyone who is learning maths...The educational standards are to be determined clearly...

suma valluru

Anonymous said...

..but does this go against your original thoughts of skipping math exposure in their academic careers leading up to college? Also, if we are continually presented with math, from the amount of time we have to drink our coffee in the morning to the time it takes to fall asleep at night, then how can we expect our elementary students, or anyone for that matter, not be blessed with a math lesson during their elementary years? I do agree that some teachers do not have the ability to present math in a positive way but there are many wonderful teachers out there who do.

Anonymous said...

Andy, your example of bananas in a multiplication problem is how teachers are using the TERC program. Check out the unit of "Things that come in groups".

Mr. Lucchese said...

First, there is no way that math learning can or will be skipped altogether. It is far too ubiquitous. Students will discover number and shape through daily activities, and in their other classes. In early life, they need only be able to calculate, and for that they need very little math. Then at the secondary level, which begins around grades 6-7, we can start formally teaching them the underlying concepts.

Heather said...

I feel that in many subjects, children at an elementary level are not mature enough or cognitivly developped enough to really understand what we are trying to teach. When I was only in grade 11, I decided that all "real, important" learning happened in high school. I decided at that point, elementary school only prepares us for high school, exposes us to the basics of the subject matter, and I still feel that way. That being said, I think that exposure is crutial, and that children would really struggle with math in secondary if they were not able to so simple addition operations before the age of 12.

lost clown said...

Well as someone who hated, and i mean HATED math pre-trig, I am grateful that it was forced on me. I never, ever would have taken it and therefore I probably never would have become a math major. Opinions change over time. I, for one, am glad that math is compulsary.

lost clown said...

even though my algebra stinks, but I'm being forced to do it better so hopefully by the time I graduate I'll at least be average at it. If only life was all linear algebra. *sigh*