“Heavenly King, Comforter, Spirit of Truth, You who are present everywhere filling all things, Treasury of good things and Giver of Life, come and dwell in us.” – An Orthodox Prayer
I love this prayer. Especially the last part, “Come and dwell in us.” It reminds me that everything I know comes to me because it has been embodied in something. For me to get that 2+2=4, I have to “see” it embodied in a picture, some examples, or a few manipulatives. For me to apprehend that whatever I sow, I will reap, I need to see it embodied in a little seed that grows into a plant. For me to see that God loves me, I need to see it embodied in the very life of my Savior.
“All things that can be known have number; for it is not possible that without number anything can either be conceived or known.” -Philolaus the Pythagorean, fifth century
I am most familiar with the ability of a story and myth to embody a truth or idea and present it to the reader in a beautiful and accessible way. I have experienced this reality over and over again, in my life and the lives of my students, and it is truly stunning. However, as I research the quadrivium, I am beginning to see the powers of mathematics accomplish this same thing, but in a much more precise, yet mystical way. The Mathematical arts show us embodied reality in a way that nothing else can.
I have, to be honest, I feel ill-equipped to be writing these posts on the Quadrivium. I am just beginning to touch on the mathematical arts and only receiving vague glimpses of its beauty. Nevertheless, I know that if I do not try I may never apprehend more, and that would be a tragedy to remain where I am. Therefore, this article, and the next represents some of what I have discovered about the mathematical arts of the Quadrivium: Arithmetic, Harmonics, Geometry, and Astronomy. Today we will look at Arithmetic a little closer.
In the Liberal Arts tradition by Clark & Jain, they point out that arithmetic is the art of discrete mathematics. According to tradition, Harmonics is an extension of arithmetic, in that it is discrete mathematics in time. Discrete mathematics has to do with individual quantities and their relationships to each other. For example, a discrete way of looking at a line on a graph would be to notice all the individual points on the line, as opposed to the range of points or just the line as a whole.
This definition of arithmetic came from Nicomachus’s De Arithmetica, which is one of the most imitated math books in the history of mathematics.
“For Nicomachus, deeply understanding the necessary connection and relationships among numbers would have been an essential element of the liberal art of arithmetic.” – Clark & Jain, The Liberal Arts Tradition
Developing number sense within ourselves and our students is something many of us desire. We all intuitively know that those who have a sense of numbers “see” more and can do more within the world of numbers. After reading Clark & Jain’s words about arithmetic, I think I am beginning to see why that is. They point out that mathematics if studied classically, ought to lead us to work, wisdom, worship, and wonder. There is something contained within the practice and study of math that transforms, which makes perfect sense. Otherwise, it would not be one of the liberal arts. It is just so astonishing to me that MATH has such power. I grew up crying over math and learning to despise it. I have had to come to terms with the fact that just because I do not see the beauty in something, does not mean it is not there. Therefore, I have committed myself to studying mathematics until I see its beauty. I refuse to give up. I expect however that my awakening to the beauty of mathematics will not come by driving my math facts, but rather from allowing myself to wonder about it.
“Students who encounter mathematics in wonder are far more likely to commit to the rigors of its work.” -Clark & Jain, The Liberal Arts Tradition
Clark and Jain describe the difference in the tenor of the ancient math books from that of today. They say there was an air of magic to the way the ancients talked about numbers. I took a peek at Nicomachus’s De Arithmetica, and it is true. At first I was not sure that I was reading a math book at all.
I feel certain that for us to recover arithmetic in all its glory we need to regain some of that magic, and we need to play with numbers.
To regain the magic we need to read books like:
Beauty for Truth’s Sake by Stratford Caldecott
The Joy of Mathematics by Theoni Pappas
We also need to have conversations like the one I had recently with Daniel Maycock of Polymath Classical Tutorials and we need to attend things like the CiRCE Conference, where we can hear speakers talk about beauty we have yet to apprehend.
To play with numbers we need to do things like:
Finding as many ways as possible to name a quantity
Of course, we can simply do these things no matter what math curriculum we use. Some math curriculums have these elements present in them more than others. My personal favorites are Ray’s Arithmetic and Rod & Staff. Ray’s especially has a way of getting to the business of playing with numbers right away. I use this curriculum with my youngest daughter, and I already see the fruit. She sees numbers everywhere, and there is such delight involved. I use Rod & Staff with my older two. I needed a little more support for them, and Rod & Staff provided it. However, I cannot tell you how many times I have thought “I wonder if the Rod & Staff people used Ray’s as direct inspiration for their text?”
The truth is, no matter what curriculum you use it will be your attitude that mostly directs the experience your children have with math. So, maybe you are already there, Great! But, maybe, the first thing some of us need to do is have a real conversation with the Lord about it. I know, we are talking about math, but if everything in this world is a way for me to know my Lord, don’t you think He would want to help me seek Him? Therefore, when I am struggling with some academic area, I treat it as any other area of struggle. I repent and I ask for guidance. I seek mentors, whether they are in person, online, or in books. I give focused attention to doing better and I trust the Holy Spirit to bring Pentecost for myself and my students.
What are you doing to bring math alive in your homeschool? Have you seen the magic in math? I would love to hear your stories. Join the conversation here.
Expanding wisdom, extending grace,
Resources used in Research
The Liberal Arts Tradition: A Philosophy of Christian Classical Education by Kevin Clark & Ravi Jain
Beauty for Truth’s Sake: On the Re-Enchantment of Education by Stratford Caldecott
Khan Academy Video: Abstract-ness: The general idea behind the word ‘abstract’
Khan Academy Video: The beauty of algebra: Why the abstraction of mathematics is so fundamental