ROMEO`O, let us hence; I stand on sudden haste.`

FRIAR LAURENCE`Wisely and slow; they stumble that run fast.`

Students in general grow impatient with the advice of Friar Laurence. I don’t blame them. I, too, sometimes find myself wanting answers more quickly than the nature of this or that question allows. Anyone who has gone skipping through the *Summa Theologica* attempting to get a fast answer to some abstruse Theological question knows what I mean.

As a matter of fact anyone who has ever picked up the *Summa Theologica* without having read the major works of Aristotle should know what I mean – although in this case it is more forgivable.

Picking up St. Thomas *at any time* is probably a good idea, just as long as the humble reader is willing to take many things on faith, admitting his own ignorance rather than having the temerity to find fault with St Thomas.

But strictly speaking one should not read the *Summa* without having mastered, to some acceptable degree, the works of Aristotle. (i.e. from the *Categories* straight through the *Metaphysics*!)

So I don’t blame any student for indulging in what St Augustine condemns as a sort of *curiositas* – or to put it more unpleasantly – a perverse desire to know.

What’s more, the fact that students suffer from a ‘disordered desire to know’ is not entirely a fault stemming from their youth. A fair share of the blame also lies squarely on the shoulders of the parents, teachers and the reigning educational establishment which all conspire with irresistible effectiveness in encouraging disorderly learning.

The ordinary parent is mostly (and understandably) concerned that his child be successful. A parent naturally wants to see his own child succeed in life.

Success is rarely measured in terms of wisdom.

The current prevailing fashion in education places a high value on productivity. The student is praised for his speed, accuracy and efficiency in “problem solving”, which as any Algebra teacher knows, does not require understanding.

As a matter of fact, the attempt to understand often gets in the way.

Why would anyone really want to know the meaning and significance of the terms “sine,” “cosine” and “tangent”? No Algebra book makes an attempt to explain what “tangent” has to do with a real geometrical tangent. The meaning of these terms simply does not matter if the end one is pursuing is not understanding but *productivity.*

These days, by the time students have completed middle school they all seem to have received a completely upside down intellectual formation. That is, students now appear to learn everything backwards and in the opposite order that any particular field of learning ought to be learned.

Without having learned Grammar, Rhetoric and Logic they have instead received a complete indoctrination in the atomic theory.

They know about DNA, Quarks, Plasma, Black Holes, Anti Matter and Negative Energy- and all these things before they can even write a complete sentence!

In Mathematics they are familiar with the Pythagorean Theorem well before taking a single Geometry class.

They can invert ratios, cross multiply and alternate proportions without even being able to say what a proportion is.

SCARY!

Without having taken a single class in what the Greeks used to call “Arithmetike,” they can talk about “numbers” positive and negative, “e”, “i” and “pi” and even the “square root of 2!”

They are blissfully unaware of the story of that unfortunate Pythagorean who was buried alive for his discovery of incommensurability.

But to get back to Friar Laurence, he says,

“Wisely and slow”

If we are to learn and obtain any wisdom ourselves, we ought to avoid the temptation to proceed swiftly.

The method of Catholic liberal education is absolutely contrary to the method of the world. The world would have children speed through text books and lessons and books in the futile attempt to become “current.” Students are supposed to “get up to speed” and gain skill in surfing the waves of data that sweep in from the four corners of the globe with inexhaustible fury. The crown of victory goes to the fastest.

But to run fast, that is the very characteristic of youth. And unfortunately like Romeo this sort of behavior can lead to very real peril in the physical life but even more disastrously in the life of the mind.

The method of Catholic liberal education is the method born from leisure.

It demands quiet. It demands slow reading, speaking and listening.

It demands lengthy discussion.

It demands orderly procedure.

Wisely and slow.

Would you happen to have a recommendation for how we should go about educating our children in arithmetic as as to properly prepare them for geometry and other studies? I see your objection but, with a lack of anything that seems to follow it, what are we supposed to do?

Hi, thanks for the response! Surprisingly quick considering the age of the original post! :]

Two other questions for you then — am I correct in assuming your kids are in or near high school? Mine are still small, the oldest and only actually involved really in school at this point is nine, and then the next is 4.5, so just barely starting things (not sure how some do this, but I haven’t tried to teach him things yet, we’re just starting to open Ray’s Arithmetic and work on establishing what 1 and 2 and 3, etc. are to him — and not the symbols, just the concepts) The youngest is 2 and we won’t talk about him, haha, way too young to do anything but show two fingers for his age! ;] Our oldest was in public school for kindergarten and half of first grade, then home and “taking” a public-school-like math “class”, which we soon ditched after its uselessness was discovered and I had recovered from the idea he needed to be taught like in school, but at home. Then we worked through the elementary Life of Fred series, if you’re familiar with it, and then Math-U-See for multiplication, division, and fractions. There was a smattering of percentages and decimals and very basic algebra in the last Math-U-See book. He went through this quite quickly, and since I’ve decided we should stop and review and we’ve gone through multiplication and division once more, with daily multiplication fact reviews. Far more details than necessary, I’m afraid, but it seemed the best way to explain what I’m thinking of when I think of arithmetic.. is this arithmetic? Is this all of arithmetic to which the student should be exposed in the earlier years? How might one go about exploring arithmetic and other math slowly and leisurely? Isn’t there some drill and memorization that is involved just to be able to call the student’s progression ‘mastery’? I’m still pretty new to schooling in the role I’m currently in and have taken quickness in understanding nearly anything I’m taught for granted my whole life so am woefully unprepared in the skills necessary to impart such knowledge and learning to others in the event they don’t just immediately “get it”. I’ve been reading lots about classical education and the proper way to do it, but math is where I hit a snag especially, I don’t see a way to make it virtuous, leisurely, and contemplative.. it seems that you watch it done until you see and get the steps and then replicate them. And I know that’s not the ideal, at least not from what I’ve been discovering lately, but no idea how to approach it differently!

Okay, end of that craziness, second question — are you thinking that a year of Euclidean math will plant the necessary seeds and that anything else is unrealistic in today’s educational atmosphere? Or have you just settled on that with your particular kids? If you were to do it all over again, would you stick with that plan, figuring it’s at least not harmful? Or would you do something differently?

Alrite, will wrap this up now! Please don’t feel obligated to hurry in answering, although I certainly appreciate any input, I’m just still mulling all this over trying to determine how to move forward from here. For now, we’re sticking with review, figuring that won’t harm anything! ;] Plus, it’s the end of the year, seems a bad point to start overhauling, might as well have slow repetition until we hammer things out than forcing stuff!

Thanks again,

Maria :]

Hi Maria,

Sounds like you are off to a great start! And your concern to teach mathematics in a way that is more than just something servile is fantastic.

My kids are at all levels and we are probably handling their mathematics education at the earliest levels in a way very similar to what you describe- learning numbers and the four operations of arithmetic, drilling, etc.

I suspect that Aristotle may have done the same with his kids. Learning to count and calculate, as well as shapes and solids etc… seems like it ought to dispose the mind for understanding Euclid’s contemplative and theoretical presentation. We do try to encourage our homeschooling kids to always stay on top of their math workbooks as a priority, especially since their solid math skills will enable them to focus more effectively on things like Latin and Greek as well as Algebra in High school without too much of a struggle.

But I am leaning on their Euclid class that they take in the tenth grade as an antidote to the servile presentation of mathematics that they encounter in most every other math class (including my own Algebra II class!)

We have a predilection for encouraging our children to think about going to a college like Thomas Aquinas College where the students take four year so math- and they study math in way that allows them to really think through the theory – they get to study and contemplate the central questions in mathematics and what the nature of numbers and magnitude is. And so all of the work they do in acquiring the speed and skill in calculating that they learn in grade school and high school becomes very useful for them then- because if they did not have that skill, this would be a stumbling block when they try to understand the theory.

For example- when they start discussing the origin and theory of calculus and whether the area underneath a curve really can be found through Integration- or something like this- if they do not understand the Algebra involved, then they will not be able to focus on the deeper philosophical question (i.e. is an area enclosed by a curved line able to be known through rectangles?)

I hope that makes sense.

But the danger is that – if students are exposed to nothing other than the conclusions of modern mathematics, it is difficult to see how they will escape a conception of mathematics as a purely human construct, as opposed to a science about God’s own ideas about quantity (to paraphrase Euclid:-)

Hello again! And thanks again for the response!

I suppose I should look into these Euclidean mathematics! I went to a run-of-the-mill Catholic school and then a public high school, so just went through the normal math up til about pre-calc and have since forgotten nearly all of it, so am in a weak position and need to do some shoring up myself!

Thanks for the info about Thomas Aquinas, btw, I hadn’t been aware of it until reading about it here.. We sort of have a lose plan that involves gaining a skill of some sort and then sending to college, but that all depends on whether our kids are willing to go that route of course.

Back to math, I probably will just stick to trying to cement in the basics for now, adding in more as we go along. For what it’s worth and in case it’s of any interest to you, as I’ve been searching the interwebs for possible strategies, I’ve come across math competitions like Math Prize and Olympiad and several others and have become curious about them as well — I’ll link to a presentation (you can watch the video or just open the transcription of it and read through it if you don’t care to sit through a video — that’s what I did!) by the man who wrote the Art of Problem Solving books, if you’ve encountered them, emphasizing the necessity to delve deep into math and learn skills that are widely applicable, not just skim the surface like is done in most schools. This seemed to align well and I’ve since given our oldest a few problems. I don’t know if it’s his stubbornness or his lack of ability (yet!) to think about things mathematically, but they’ve been challenging for him, even when I thought I was picking the very easiest ones. So thought they were worth a recommendation!

Again, thanks for the info and the opinion and advice!

Maria :]

(Here’s the presentation link, the link to the transcription’s in the intro to the video: http://mathprize.atfoundation.org/archive/2009/rusczyk)

(Here’s the Art of Problem Solving page link, if you care for that too: http://www.artofproblemsolving.com/)

(BTW, Fermi problems/questions are another example, though with no fixed answer, that make you think through them, hopefully thereby internalizing the processes.. I have been meaning to do some here and there in an effort to start thinking mathematically again, but they got lost in the jumble!)

(Alrite, enough talking from me!)