The Two Faces of Math

This is short article on a different perspective on learning math. Steiner school approach math in quite a different way and yet covers all the main concerns we, as parents, would normally have. Think of this as an overview of what may be possible if only one class, like math, were percieved in a different way...

The Two Faces of Math
One of the greatest anxieties of parents, particularly in the current times is how well will our kids will do in math. Of course we worry about the other subjects too, but math in particular seems to attract special attention. It is not unusual that many an entrance exams or preparatory exams to universities only tests two subjects: language and math.

With language we don’t seem as concerned. We figure, I suppose, that eventually and because we use it daily, our children will develop a proficiency in language that will allow them to survive. Some may develop an extraordinary proficiency that will turn them into authors. Unfortunately, we don’t view math in the same light. Math seems to have something “extra” that requires special effort to prepare for in order to effectively overcome. And, there are realms of math that are totally unapproachable, as far as the average person is concerned.

So why is it that we have such a relationship with math? Most of us, I am sure, have some horror story from math class. And, rightfully perhaps, we all want our children to do better than we did. All this, however, does not reveal to us what is this hidden “something” in math that makes us have such a relationship with it.
I think it is important, first of all, to understand that math has two different qualities. The first, the one we are most conscious about and concerned about is what you may call a mechanical quality. The mechanical quality is the reason why we drill in math; for example, 2+2=4 all the time. There is no shortcut. Effectively we have to remember this. The most common mechanical drill in math is the multiplication table. We need to remember this table in order to do multiplication. There is no shortcut. Hours of math exercises we traveled through the various grades in school made this quite clear to us. Either we learned and memorized or we didn’t. The fact that this is a mechanical skill does not mean we have to spend hours hunkered down in a chair memorizing. It can also be achieved through wonderful games with younger children where they do certain movements or activities in relation to learning these drills. Eventually, it forms part of their memory and they simply have to recall it to use. Whatever the case may be, this is the quality of math that we are most familiar with and have, perhaps, the greatest expectations.

There is, however, another quality of math that we are less conscious about although we encounter it just as frequently as the mechanical quality. This is what I call the spiritual quality of math. Math was the quality to become. What could this possibly mean? If I asked anyone to show me the number 1 in the world, they will find that this is impossible. Try it. Find the number 1 in the world used as a noun and not an adjective and you will find that this is impossible to do (or at least my friends and I have not been successful). The reason is more obvious that it looks… the number 1 does not exist in the world. It only exists in our minds, as a concept streaming from a time before our birth. We cannot effectively show what 1 looks like on earth. However, we all understand what it is. As in the equation above, 2+2=4, effectively does not exist in the physical world; however, it is true nonetheless as we experience it in our minds or hearts or wherever. It has a potential that can be unlocked. It is spiritual in the sense that it is not physically manifest in the world.

Herein lies the challenge of appreciating and understanding math. In our anxiety to ensure that our children are “good in math” we tend to confuse the two qualities of math. In other words, the rules that apply in the mechanical side of math do not apply to the spiritual side of math and vice versa. This is a tough one to grasp. A child who drills and drills in math improves their memory but not necessarily their understanding; whereas a child who works purely on understanding math does not necessarily develop the skill to do equations quickly. The two are separate. One exists as a skill, the other exists as a picture in our heads. As parents, when we complain that “our child can’t do math” what are we really complaining about? Are we complaining that the child’s mechanical skill in math needs improvement or the other way around? Could it not also be possible that the real challenge in developing any proficiency in math is linking the two, the mechanical with the spiritual while each retains their rules? Drilling does not build understanding. It builds habits and mechanical skill. Conceptualizing does not build a mechanical skill. Math requires both.

The burden on the teachers is tremendous. They must guide the children through an experience that will allow them to learn to bridge the two qualities of math. How can this be done? There is a simple way to link the two: a word problem.
Word problems are stories loaded with numbers. All word problems could easily begin with “Once upon a time…”and end with “…and they lived happily ever after.” In fact fairy tales, like those compiled by the brothers Grimm are loaded with numbers. A great way to appreciate a word problem is literally to treat it like a story, which it is. The quest in the story is how to make everyone live happily ever after. People live happily ever after because there is a balance, a harmony in the kingdom, the land, between the prince and the princess. That harmony is what math is all about. It even has its own symbol “=” to indicate this harmony. The final line in any word problem includes the “=” sign. Word problems are very useful because they help us build a picture in our heads where the numbers live and yet it has a practical application because in the end, there must be an answer. When looked at this way, we come to realize that making a good word problem maybe a bit of a challenge. Nonsensical word problems are just that much more difficult to resolve. In stories, the good prince always defeats the dragon to win the princess. Thus word problems should have a similar quality.

The objective of the word problem is for the child to develop an understanding of the relationship between the spiritual quality of math and the mechanical quality. In this way, the child begins to develop the bridge between the two. It is a great pity that when we grew up, solving word problems remained mechanical, i.e. we extracted the numbers and fit them into equations, totally ignoring the story. Who knows how many great stories we might have missed. As a result, word problems have degraded into statements so that the numbers can be easily extracted. They ceased being stories. They became mechanical exercises.

If we look at math in this way, we also begin to understand that there is a time and place for everything. I will use a 12-year system as an example (I grew up in one so I am most familiar with it). In the first four years (grades 1-4) math is in its formative state. This means that during these years, it is good for the child to learn the qualities of numbers (this includes the four basic operations and fractions). The four basic operations and fractions work because of the quality of numbers: they can be broken apart and put together again. The first four years in grade school is all about learning these qualities. If we don’t come to grips with the qualities of numbers, we will not be able to do any more math.

The middle years, grades 5-8, are about building bridges. This is where children encounter genuine word problems for the first time. They begin to see that numbers and math have intrinsic qualities that produce wonderful combinations. However, there is also more. To be able to bridge to the more spiritual quality of math, they will have to learn to build bridges. Thus word problems become a great learning step during the middle years. Geometry is introduced and geometry is great material for word problem or stories. All of geometry, which means to measure the earth, is spiritual in the sense that it is not visible in the physical world. We see expressions of geometric shapes; but as we learn about geometry, we also learn that these geometric shapes are pure imagination.

These middle years are then followed by upper school (grades 9-12). Here math enters a highly spiritual realm, algebra, trigonometry, calculus among others. Understanding these subjects requires a good foundation in the mechanical qualities of math and a good imagination, hence the spiritual qualities of math. Otherwise, there is no possible way to actually grasp the significance of a=bc or even the idea of an irrational number. To do higher math, an imagination cultivated on stories such as fairy tales and word problems, is essential.

As we look back on how we learned math, and try to reflect on the less memorable moments, perhaps we can see how our teachers then muddled up the various qualities of math and numbers. We hope, as we raise our children now, that a different way of looking a math may make it enjoyable and wonderful. Just like a good story.