Math is not scary. Say it with me: “Math is not scary.” Perhaps you might tell me you grew up believing your sister to be a math oriented person and believing yourself to be a language oriented person (and this is the reason you think math is scary). This is not true. If you have a proclivity for one over the other, it is the fault of education and our culture—not the fault of your mind. We might have natural talents that make one area of study easier for us than another—although the argument could be made that those are still the result of cultural and familial influences—but your being pigeonholed into developing only one of those talents is the fault of education and culture, not your mind. God created us in His image and He is not a left brained or right brained God. You are not left brained or right brained either. You can image God—and by this I mean you can reflect the image of God to this world—with both sides of your brain; you just need to break free from the pigeonholing that education and culture have imposed upon you.
We also need to train our children not to think in this way. Every time we tell them we are not math people, that we are not good at math, or that we are scared of math, we are giving them the excuse to pigeonhole—and therefore limit—themselves too. We want to avoid this.
It is possible to teach math in a specific way that will help us to understand it the way we understand those subjects we enjoy. We were created to be discoverers. There are many ways we might enjoy learning, but independent discovery is one method which consistently brings joy to learning. The way the Saxon math curriculum is designed can help us to accomplish this. Some of you may not like Saxon math, but with a proper understanding of how it works and how it teaches, you will see how it instructs us in the very way that will allow us to enjoy math.
In Saxon Math 8/7, for instance, we see an example of how this works. Lesson 53 introduces the concept of “Multiplying Rates”:
First, Saxon introduces information about the new topic. In doing so, it presents all of the information students already understand—material they have already learned. For example, we can calculate how far a car traveling at a certain rate for a certain amount of time will go by multiplying, or how long it will take a car to travel a specific distance by dividing. This reminds the student of what he already knows (multiplying, dividing, fractions, and so on) so that the new material will be less scary; the student, therefore, begins with what he knows and can begin exploring the new concepts from a firm, familiar foundation. Saxon then introduces examples of the new concept; the examples always include the solution—the steps needed to solve the example problem.
Eight ounces of the solution costs 40 cents.
a. Write two forms of the rate given by this statement.
b. Find the cost of 32 ounces of the solution.
c. How many ounces can be purchased for $1.20
a. The two forms are
8 oz/40 cents and 40 cents/8 oz
b. To find the cost, we use the form that has money on top.
32 oz x 40 cents/8 oz (so that the ounces cancel and the cost may be calculated)
1280/8 cents (the numerators multiplied)
160 cents (simplified, or also expressed as $1.60)
c. To find the number of ounces, we use the form that has ounces on top.
120 cents x 8 oz/40 cents (so that cents cancel and the weight may be calculated)
960/40 oz (the numerators multiplied)
24 oz (simplified)
Jennifer’s speed was 60 miles per hour.
a. Write the two forms of the rate given by this statement.
b. How far did she drive in 5 hours?
c. How long would it take her to drive 300 miles?
a. The two forms of the rate are
60 mi/1 hr and 1 hr/60mi
b. To find how far, we use the form with time on top
5 hrs x 60mi/1 hr (hrs cancel)
300 miles (multiplied)
c. To find how much time, we use the form with time on top
300 mi x 1 hr/60 mi (miles cancel)
5 hours (divided)
Saxon goes on to provide a third example, but we can glean what we need to understand from these two examples. Let us take a look at what Saxon has done, analyzing it from a first person perspective by putting ourselves in the shoes of the student.
First, because Saxon has provided two examples, models, if you will, or types, of the concept, I (the student) am learning; I can compare the two examples to see how they are similar. In both of these examples the pattern of solving remained the same: I had to convert the given information to a form (cents per ounces and ounces per cents, as well as miles per hour and hour per miles). Then I had to determine what information I was trying to solve for and pick the appropriate form to use (the appropriate form was the one that would allow the unit of measurement we desired to remain while the other was cancelled). Then I solved the problem using multiplication or division, which I already understand. By encouraging me to engage in comparison, Saxon is pushing me toward independent discovery and towards learning how to learn.
Second, because Saxon presented me with what I already know before it presented the examples, the topic should be a little less scary to me. All of the math used in this lesson is not new, I just need to wrestle with one or two new steps. The examples will help me discover what those one or two steps are.
Third, because Saxon has given me examples to compare, I am discovering—through the proper use of questions and a developing spirit of inquiry—what the new steps in this lesson are. This is a key factor—perhaps the key—in this process: I am discovering it. Saxon is not telling me what I need to know; I myself am discovering what I need to know. Therefore, I am more likely to remember the discoveries and appropriate them as my own. I might even feel as though I have discovered something which Saxon had not been intending me to know! (For students in the dialectic stage, this is particularly exciting: they love it when they discover something on their own, especially something that their teachers or texts did not necessarily anticipate—or, so they think).
Last, Saxon provides mixed practice that allows me (or my child) to keep reviewing material learned in previous lessons. Just as I learned in Foundations, memorization consists of repetition, duration, and intensity. Saxon provides ample opportunity for all of these, even though I may not want to keep practicing old ideas. Boring as it may seem, the fact is, I need to go through these repetitive steps in order to truly master the material.
This process in Saxon math is very similar to what happens when I want to teach my son about a virtue such as bravery. I often find I cannot just give him a dictionary definition of the term. I need to provide him with types of bravery, models that he can examine, compare, and analyze for himself. For example, consider the following models from C.S. Lewis’ Narnia series: Peter fighting Miraz in Prince Caspian, Puddleglum stomping out the fire in The Silver Chair, and Reepicheep wanting to sail to the Dark Island in Voyage of the Dawn Treader. Having provided the types, I can ask my child to compare them to see what is similar between them. From this mental exercise, he can grasp a better understanding of heroism, and it is entirely his own understanding—not just a definition I imposed upon him and required him to ‘learn.’ Then, I encourage him to continue reading and, as a form of practice, find other examples of bravery and other virtues in the stories and events he comes across.
Saxon, then, is teaching us math the same way we learn the things we feel we enjoy more, in which we experience more pleasure. It teaches us what we know, it provides us types to compare, it asks us to draw out the lesson we are meant to learn, and it provides us with continued and ongoing practice. Essentially, it makes discoverers of us. For some, though, the difficulty may be in that they do not recognize that Saxon teaches this way. If that is the case, try to approach it this way for a while. Give it some time; practice. Your initial instincts will be to just tell your child what he should find when comparing the examples. Restrain yourself. Keep working through examples and asking comparison questions until he discovers the lesson for himself. Do this for several lessons. Eventually, you will find he is looking at the lesson this way himself—and he will be the better for it: he will have learned how to teach himself, which is nothing more nor less than learning how to learn.