Synthetic Thinking and Math

by Karen Glass

Education is the science of relations. That’s the principle that underlies what I call synthetic thinking (explained more fully in Consider This). The principle applies even to arithmetic, which is, although we don’t usually fathom the reason, one of the liberal arts. An art is not made up merely of information to master--it is meant to used. Arts are “practiced.” We want to foster a relationship between students and the world of mathematics, and that is most readily accomplished when we treat mathematics as an art to be practiced.

There is going to come a time, in math, when a child is going to have to sit down and work through some complicated equations. That is either going to be a challenge met with confidence and a lift of the chin-- “I can do this!”--or with boredom and despair. We often speak of wanting children to love reading, love literature, love books, maybe even love history or science. We rarely speak of wanting to them to love numbers, and this is probably a reflection of the reality that few of us formed that relationship in our early years of education. Whether or not that relationship is formed will determine the response a child--and even an adult--brings to those complicated problems.

The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders. There is no one subject in which good teaching effects more, as there is none in which slovenly teaching has more mischievous results. Multiplication does not produce the 'right answer,' so the boy tries division; that again fails, but subtraction may get him out of the bog. There is no must be to him; he does not see that one process, and one process only, can give the required result. Now, a child who does not know what rule to apply to a simple problem within his grasp, has been ill taught from the first, although he may produce slatefuls of quite right sums in multiplication or long division. (Home Education, p. 254)

We often speak of a "Charlotte Mason education" being a paradigm shift, and nowhere is that shift greater than in the area of math. Charlotte Mason knew that math was about much more than getting the “right answer.” There is a relationship between math and the natural life of men, and it was this relationship that she wanted to foster first.

How is this insight, this exercise of the reasoning powers, to be secured? Engage the child upon little problems within his comprehension from the first, rather than upon set sums. (Home Education, p. 254)

In practice, this means that children should begin with what we call “word problems,” and those problems should be based upon real-life experiences that the child might expect to occur. For the smallest children, these math problems occur easily in course of living.

Home life is full of easy little arithmetic problems that bring the importance of numbers, as well as concepts such as quantity, equality, and one-to-one correspondence within the grasp of even quite young children. A family of four is joining us for dinner. How many chairs to do we need to add to the table? I can only find three clean spoons--how many will we need to wash so that we have enough? There are six cookies left in the box. How many can each child have?

Older children can figure how much five cans of corn will cost, or whether there is enough money for everyone to get double-scoop ice-cream cones, or if singles will have to do this time.

Older children may be given more complex, multi-step problems, such as “Joe gathered 87 walnuts and Tim gathered 28. They plan to share the nuts with three friends. How many will each of the five boys receive?” Charlotte Mason says that a child will perceive exactly what must be done in order to solve the problem, although “Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.” (Home Education, p. 255)

The more occasions a child has to use math in real life--and that might include playing games in which counting, adding (or subtracting) points, or other arithmetic plays a part--the more likely he is to develop an interest in and a relationship with math.

Math is needed for cooking, for science, for travel, for planning and purchasing, and the more integrated a child’s exposure to math is, the greater will be his appreciation for it. Once that appreciation is established, the extra effort needed to memorize math facts or unravel complex equations will be entered into more willingly. The child has no need to whine, “why do I have to learn this?” If he has developed a synthetic understanding of math, he already knows the answer to that question, and will likely also work out the answer to the arithmetic problem at hand.