To Drill It or To Explore It?

by Bill Lauritzen

There is heated controversy about the best way to teach math. I am probably somewhat qualified to comment on this. Of my 20 years involved in education, 11 of them have been as a mathematics substitute teacher, mostly in Los Angeles, and 3 as a math teacher. Thus, I have been in hundreds of mathematics classes, and have tried out most of the various methods of teaching math.

I have categorized these various methods into "drilling math" and "exploring math."

**To Drill Math**

Drilling math, proponents claim, is top-down, "back-to-basics," no-nonsense.

Drilling math might best be characterized by these words: drill, abstraction, memorize, practice, review, instruct, teach, indoctrinate, exercise, discipline, only-one-right-way, by rote, by lecture, read about, by procedure, listen.

In this math, in its extreme form, the students work individually, and are spoon-fed formulas, rules, theorems, procedures, etc., to memorize, drill, drill, memorize, drill, drill, drill (all completely disconnected from the real world), until they are stuffed completely full, so that they can then simply regurgitate this information on national tests such as the SATs, so, theoretically, that they can go to the college-of-their-choice, and will thus not end up as homeless bums or bag-ladies.

I have received this method and I have taught this method. I received it in elementary school, middle school, and high school, while the US was trying desperately to catch up with the "Sputnik science" of the Russians. I scored in the 700s on the math portion of the SATs, and went to the college-of-my-parents-choice.

I also received more no-nonsense math while I was a cadet at the US Air Force Academy, in Colorado Springs, where I graduated near the top 1% of my class. I went to Purdue, another engineering school, and took more math courses in the same way, getting straight "A"s for my master's degree.

At the end of all this drilling and memorizing, not only did I hate math, but I hated school, and never wanted to spend another day in a classroom.

Only when I turned 30, did I discover that I could explore math, that I could invent and discover things on my own in math, and this discovery delighted me and has stayed with me. (Recently, I was the annual guest speaker at the Dozenal Society in New York. Also recently, a school in Russia had a Versatile Number Day in which they studied my publication on these numbers.) So when I decided to study education, by working in it as a substitute teacher, as opposed to reading about it in a university, I chose to teach math.

I "drilled" the students on math when I first started substitute teaching. However, I knew this could not be the best method, and I was always trying to come up with better methods. (When I subbed at Garfield High for two months in 1982, I sometimes went by Jamie Escalante's class to overhear what he was saying.)

The drillers claim that the explorers are fuzzy thinkers (not to be confused with the book, Fuzzy Logic), that simply confuse kids, and lower test scores. Drillers also claim that explorers neglect to teach fundamentals, especially by allowing calculators into the classrooms.

Where I currently substitute here in Glendale, I sometimes see classes who have been over-drilled on math. I look into their glassy eyes, and there is no spark of curiosity. Sometimes there is a silent resentment. Oh, they are usually quite competent at solving the problems, as long as I don't change anything from the way it was presented to them in the text.

Drilling math might raise test scores artificially, like steroids can give strength, but what about the side effects? Like hating math. (Despite what they say to please their parents and teachers; I'm talking about what they tell each other.)

**To Explore Math**

The "explorers" claim that students can be taught to explore math in meaningful situations, so that they can then come up with the rules governing it on their own. The National Council of Mathematics Teachers (NCMT) endorses this approach. (Its standards are now available on-line.)

This method might best be characterized by these words: experiment, understand, develop, realism, guide, model, question, discover, focused play, many-right-ways, by grasping, by touch and manipulation, by invention, by guess and check, by experimentation, by application.

Explorers claim that drillers practice "drill and kill" math. Probably because it kills the spirit of exploration and creativity that is natural in every Homo sapian.

Let me give an example of how each side might teach one simple fact.

*Examples*

Fact: The three angles of a triangle add up to 180 degrees.

To Drill Math: the instructor might simply write this on the board, and then point out where in the book it also says this. Then he would do an example problem. He would sketch a triangle (with say, 90 degrees, 30 degrees, and x degrees) and show how you can get the missing angle, by adding 90 and 30, and subtracting the result from 180, to get 60 degrees for the unknown angle. Then he would have the students do about 20 or 30 example problems from the book like this. Later, they would be tested.

To Explore Math: the instructor might first introduce a broad problem from the real world in which angles and triangles are used, depending on the level of the class, such as finding the height of a tree. He would give every student a protractor (a device for measuring an angle; something that was never given to me throughout my entire education from Kindergarten through master's degree.)After the students knew how to comfortably use the protractors to measure and draw angles (something that could take several days), the instructor might ask them to draw three different triangles on construction paper. Then he would have them measure the angles, and add them for each triangle.

They might do all this in groups of four, though this is not necessary and, as in the real world, they might help each other out when they were having difficulty.

They might get 179 for one triangle and 181 for another and 180 for another.

The students would be allowed to use calculators in this approach, for this takes the drudgery out of the math in more lengthy computations, and allows more time for explorations such as this. Calculators can never replace understanding, however, calculators are what people use in the workplace (probably even by those who would tell us that they should be removed from the classroom).

The instructor would ask the students if there was a pattern? Yes, they might answer, all the angles of the triangles add up to around 180 degrees. They might be then asked to formally state this as a rule.

If I were teaching this class, I might ask the students to try to draw a triangle with 120 degrees or one with 200 degrees. Of course, this is impossible, but instructive. I might also have the students discuss whether it were ever possible to have a triangle in the real world of exactly 180 degrees. (The answer is no. A fact that drillers might never find out.) Then each group might present their results to the rest of the class. The work of each student might be kept in a portfolio for grading. All this would be with the real world view of eventually figuring out the height of the tree.

Sometimes the exploration might be "integrated," or combined, with another subject like science, English, or even history. This is how you find math in the real world. Mixed with other things.

These explorers here are learning about the world around them in a concrete manner. They look at it, measure it, look for patterns in it, describe those patterns, and apply those patterns in a new situation. How is that for fundamentals? You can't get more back-to-basic.

They are allowed to think and see for themselves.

There is nothing "fuzzy" about this approach. It is rock bottom, down-to-earth, rigorous, demanding, and challenging. Any discussion by the students should be strictly about math. Principles and fundamentals are uncovered, discussed, formalized, and applied.

**The Explorers' Failure**

I have seen several teachers apply this program well, and one teacher extremely well. When it's applied correctly, it can be a remarkable thing.

However, I have also seen this program totally misapplied by teachers.

For example, I've seen teachers read the solution to a problem in the teacher's guide (none of us knew how to do many of the problems as we were mostly trained by the "drill and kill" method), but somehow expect the student to figure it out, without the teacher's guide. These teachers should have modeled the correct solution for the students (given them the darn answers), even if it took several months of doing this before the students (and teachers) caught on.

Another thing these teachers didn't know was this: if you understand a problem, then you can usually solve a problem. I would sometimes spend a lot of time getting the students to understand the problem, and then they would usually come up with the correct answer on their own.

No matter which way you teach, you have to keep the class moving.

So exploring math has gotten a bad name in some places where the teachers were just not ready for it, or were not trained right. I can't say that every math "exploration program" or book is good. I have primarily used the Interactive Mathematics Program (IMP) that was developed by Berkeley and the National Science Foundation. Overall, it is a brilliant program, at times the work of genius, with only a few minor bugs.

**Summary**

In actual fact, there is a broad spectrum with abstract drilling on one side and realistic exploring on the other side. Many teachers lie somewhere in the middle. They drill their students, but also attempt to get them to understand why.

With abstract drilling, you can cover a lot of material. You can sometimes boost up test scores. With realistic exploring, you teach how to think, but you cover less material. "Drill and kill" math is better than no math education at all. At times, I lean toward drilling when I haven't completely thought through how to explore something.

Exploring math takes more money, more time, and more qualified teachers. It also may cause teacher burnout, as it sometimes requires more time and energy.

However, pure drilling math sort of reminds me of junk food. It fills you up, but you may feel sick afterwards, and you won't get the nutrients you need for the long haul.

In other words, to just drill math and only drill math is a sort of a imitation of a more natural, realistic, and very ancient education. A natural education using ruler, protractor, compass, and sextant. When geometry still meant geo-metry, or "earth-measuring" and not "theorem-memorizing." A realistic education that built the great pyramids, Stonehenge, the great cathedrals, and now the geodesic dome, the modern observatories, and the space telescope. An ancient education which nurtured the urge, designed through millions of years of trial and error evolution, to use the eyes, hands, and brain.

Can we afford to explore math? Or should we just spoon feed formulas and hope for the best? Or should we try for something in the middle, in which the teacher explains why? These are some of the questions that face parents, teachers, administrators, and legislators.

William Lauritzen has published several articles on science, math, and education.