The Life of the Math Lesson

As a girl growing up on the plains of the Midwest, most of my ideas about the mountains were informed by the children’s classic, Heidi, the story of a little girl raised by her grandfather in the Swiss Alps. To me, the mountains were the place where heaven and earth met—filled with birdsong, happy friendship, and flowers nodding gently in the breeze.

Years later, I would again marvel at the magnificent heights with Heidi, this time reading her adventures aloud to my own children. As they breathed in the mountain air and drank in the golden sunlight of the imaginary mountains, I recalled that Charlotte Mason described mathematics as a mountainous land—one that rewarded the climber. Using her approach, we can make the math lesson every bit as vigorous, delightful, and health-giving to our children as the Swiss Alps were to Heidi. We do so by focusing on ideas.

Today we are closing out a series on the Atmosphere, Discipline, and Life of math lessons. We’ve already talked about how we can create a healthy atmosphere and how we can encourage good habits during our lessons. Today let’s turn our attention to the third part of this series: Life in our math lessons.

A living teaching of mathematics is direct and humble, opening a realm of beauty and truth for a child to explore. By avoiding anything that clouds the simple underlying ideas found in the science of Numbers, a mountain view can unfold in our very own homes, beginning with the very first lesson and the formal introduction to the number one.

We start by asking a child to point out one of something in the room, such as one dog, one book, or one cup; having her continue to point out a great many things that might exist singly in the room. Now the symbol for one is written down neatly for the child to see and is told that whenever she sees the stroke 1 she knows it stands for one of something. Next, number cards or loose numbers are spread out on the table in front of her and she is asked to pick out all the 1s from the group of figures. Finally, the child learns to write the number 1 herself, as neatly as she is able. Do you recognize how closely this resembles beginning reading lesson?

In this way the written symbol becomes an idea full of interest—whether it’s one kitten, one doll, or a single pencil—it’s now a friend that is pleasant in her eyes.

While notation is taught with intention in the math lessons, work at this point remains largely oral instead of workbook or worksheet driven. The physical acts of writing and reading are often labor-intensive for a young child and we don’t want the mechanical processes to overshadow the vital ideas of the lesson—even while working with facts. Writing in early math lessons is usually reserved for days when work is going exceptionally well and is considered a real treat.

As a child progresses, she’ll count upward and back, gaining the idea that there’s an order of things expressed by numbers. She’ll also get ideas of magnitude as she becomes aware that the symbols signify quantities that continue to grow greater. A sense of wonder occurs when a child realizes she could count day in and out and never reach the end, yet all her counting is done by the simple arrangement of only 9 figures and a 0. When a child finds that 2 + 2 = 4 and can never equal anything else she’s brought before an absolute truth. As she progresses up the mountain, she’ll learn a hundred other truths revealing math is not only of practical help in our daily life but it’s also a realm of beauty and wonder waiting to be explored, with exciting things to discover that aren’t of our own invention.

Practically speaking, a living teaching also means there’s no need to spend a small fortune on elaborate or contrived manipulatives. In fact, they often require over-teaching with too much importance placed on the things rather than the ideas they’re meant to convey. Simple everyday objects serve as handy tools for the investigation of ideas and are a stepping stone to more abstract thinking. Using a variety of objects—such as coins, buttons, and beads—gives the idea that math is everywhere and belongs to everything, rather than a subject relegated to a certain time of day or bound to one particular manipulative.

These everyday objects are so helpful in the presentation of ideas. For example, a foundational idea to math is that of place value. Putting objects, such as sticks or beads, into ten-bundles that are broken apart and put back together makes exchanging tangible rather than an abstract process. When we want to do some other exciting things with numbers, but it would be unwieldy to bring chickens, lambs, and kittens into the room, beans or buttons and the imagination serve the purpose. Once an idea is grasped, facts proven, or the child is working easily with objects, they’re put away and stimulating mental work is begun.

Whether beginning work with numbers or introducing new concepts, engaging questions of an interesting or practical nature also spark the imagination and aid in logical thinking. For example, a question such as “Bethany needs 9 peaches to make a cobbler. She’s picked 5, how many more does she need?” not only requires mental effort and fixes attention, it also gives the idea of subtraction as the complement of addition as the child discovers the answer can be obtained using either operation.

Charlotte Mason recognized that mathematics, like music, is a living language that rings clearly with undeniable logic, able to feed a child’s mind without the literary presentation imperative in other subjects. Though storybooks aren’t used to teach concepts, when introducing a new branch of mathematics or theorem, a few sentences of the history of discovery or captain thinkers can ignite the imagination and excite interest in the subject. Any biographies of famous mathematicians or histories of math read outside of the math lesson should adhere to the same standards of other living books by drawing the reader in and putting her in touch with vital ideas.

Let’s return to our little Swiss heroine as she’s taken from her life of glorious freedom in the Alps to one of confinement in the city, where she’s sent to be companion to the frail homebound Clara. As Heidi prepares to begin lessons alongside Clara, the little girl describes her kind tutor in this way

But mind, when he explains anything to you, you won’t be able to understand; but don’t ask any questions, or else he will go on explaining and you will understand less than ever.

I know I’ve been guilty of this. Charlotte tells us that giving rule after rule or lengthy explanations can cause a child to lose her way. Instead, she could be led to discover rules for herself through investigation.

Most of us grew up with the classic formula of “Memorize the algorithm and apply the algorithm to a number of practice problems” without ever understanding the ideas behind the process.

For example, when learning to add fractions, a child is given a dry rule: “To reduce an improper fraction to a whole number or a mixed number, divide its numerator by its denominator,” then she memorizes this rule before working a number of problems with its use.

Imagine, though, what happens when a child experiences the thrill of arriving at rules herself through investigation.

For example, we ask a child, “Can we can add kittens and puppies together?” She responds, “no.” Then ask “How could we?” We could if we gave them the same name, such as “baby animals.”

Then, “Can we add a 1 half dollar and 1 quarter together? First we must give them the same name—we must call the half-dollar 2 quarters.”

“Can we add books and rabbits together?” “No.”

“Why?” “Only things of the same name can be added together.”

Now, the child has stated a rule in her own words. Now, this rule can be applied when we ask, “How much is 1/3 of a day and 1/8 of a day?” What name can we give it? “Hours, or 24.”

In this way,  the child understands the rationale behind finding the common denominator when working with fractions. They must have the same name before adding them together…in this case, 8/24 + 3/24.

This living teaching that focuses on ideas remains key to the Charlotte Mason math lesson whether working with elementary arithmetic or more advanced math concepts.

Our students perform sharing exercises before they ever learn the symbol for division or the notation of fractions. Before ever hearing the terms “mixed numbers” and “reduction of improper fractions,” our child has been been receiving the ideas in a hands-on way through money problems, as well as her work in weights and measures while using things like a ruler, a scale, and measuring cups.

And, we know from the use of living books in a variety of other subjects, such as history and literature, that ideas cannot be forced into a child. Just as we need to digest food ourselves for our bodies to receive the nutrients, our students must work to grasp the ideas in math.

This isn’t accomplished passively by hearing a virtuoso lecture or watching a video. It happens when we allow the child time to wonder, grapple, and even touch the idea with everyday objects or through relevant interesting examples. In this way, the “why” behind the process unfolds in a most magical way. The ideas are then worked on by the child through the components of “New, Review, and Mental Math, too” that we discussed in previous posts.

Just as the power of ideas found in Heidi touches our emotions and draws us to ever-greater heights, the unfolding of ideas through the living teaching of math allows a child to form a happy relationship with numbers. While it takes effort and some places may require slower going, this humble and direct teaching leads a child upward into the exhilarating world of mathematics, where absolute truth is revealed and every step is taken on solid ground.

For more guidance in a Charlotte Mason math education, take a look at the Charlotte Mason Elementary Arithmetic Series.