I am not even going to attempt to summarize Mr. Jain’s discussion of the relationship between mathematics and the Trinity. Go. read. it. That is all I can say.

But we will get into the practical details, because the real purpose of this series is to, as you recall, compare the philosophy and practices of The Liberal Arts Tradition with those of Charlotte Mason — because I think they are overwhelmingly similar.

This book is not a practical book — it’s mostly a philosophy. But still, there are some nuggets we can work with:

- Practice and drill are necessary.
[A] rigorous foundation including extensive practice and drilling would have provided a foundation for the skill of arithmetic.

- Rote memory, however,
would not have qualified as an art, liberal or otherwise.

- Therefore, comma, any drill needs to be connected to understanding:
For Nichomachus, deeply understanding the necessary connections and relationships among the numbers would have been an essential element of the liberal art of arithmetic. Reason would have to be joined necessarily with imitation to produce art. Today the desire among math educators to cultivate “number sense” reflects this ancient desire to have deep reasoning in arithmetic.

While I often think that Miss Mason did not go far enough in regard to math — how I wish she had waxed eloquent about its wonders! — still, she understood these basics, and it’s reflected in her practices and also what she wrote. For example, Miss Mason believed that mathematics was badly taught mainly because teachers did not spend time teaching the “Captain ideas” — the very ideas which would inspire wonder.

Another idea of Miss Mason was to demonstrate everything demonstrable:

The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child. 2 + 2 = 4, is a self-evident fact, admitting of little demonstration; but 4 x 7 = 28 may be proved.

In fact the demonstration — the *understanding* — was supposed to precede the drill. This means that, technically, the memorization was *not *rote. An example comes from Mary Boole’s article from *Parents’ Review Magazine* {edited by Miss Mason} back in 1893:

No child should use a multiplication table until he has made one. Rule wide squares, and write the heading numbers. Give the child, as addition sums, two 2’s to add together, then two 3’s, two 4’s, &c. Let him write the result of each of these additions into its proper square of the table. He may take many weeks or months to complete it. He may begin to learn by heart the easy rows before he has filled in the more difficult ones. Let him learn from the copy which he has himself make.

In all, I would say that this is another place where The Liberal Arts Tradition and the philosophy of Charlotte Mason agree. In some places, I have thought that it was Charlotte Mason who added a lot to the conversation. This section is a place where I really think The Liberal Arts Tradition really shines, building upon Charlotte Mason’s work and bringing the philosophy of teaching math up to an even higher level.

In general, I think that perhaps the most important idea we can get into our heads is that *doing math well but without understanding is not*

*doing it as a liberal art*. It is the understanding that brings the freedom — and freedom is why the liberal arts are called “liberal.” This is why I think Tammy’s requirement that math be narrated — even though I so often fail to remember to do this! — is so important.

Amber Vanderpol says

I am really looking forward to reading this book after I finish the nature study book I’m reading for the conference. Your posts have really piqued my interest!

Kate says

I love the way you explain the distinction between “memorization” and “rote memorization” here, Brandy. This is so important–kids need to know the math facts absolutely solid, but it can’t be pure brute force memorization, because that doesn’t build a mathematical foundation for more difficult concepts.

Also, I wanted to thank you for the recommendation to read “Arithmetic for Parents.” What a gem! It’s the most helpful and accessible book I’ve ever seen for helping parents teach math, and he gave me a lot to think about. (The play-acting–>pictures–>stories progression for introducing the operations is worth the price of the book alone, I think. )