The Dance of Number is the first volume in a math curriculum series whose vision and methodology is radically different from most. Its author, James D. Nickel, writes in the preface:
As a high school teacher of mathematics, I have observed far too many students who can’t do algebraic work because they have not mastered Arithmetic…I have also noted that very few students possess what I call the joy of number sense, i.e., a feel for numbers, the pulse of their patterns – how they work, how they interact, and how they reveal rational order.
This curriculum was written to address these important issues: a lack of mastery of Arithmetic, hampering continued progress, and a widespread lack of joy in studying mathematics.
I appreciate that The Dance of Number is divided into clearly delineated steps. Of course, lessons in typical math books attempt the same thing, building concepts upon one another from the simple to the complex. What I love is that this curriculum names it in terms of what is actually happening, right in the table of contents: the book is divided not into chapters or lessons, but into STEPS.
This an excellent vision, a fantastic way of instilling and nourishing a profoundly important understanding of how to learn math (and a good many other subjects, as well): seeing education as progression of ideas, knowledge, skills, and practice moving from the simple to the complex.
“Lessons” convey the idea of the passivity of the student, who receives the instruction. In contrast, the concept of steps evokes the active student, climbing the steps of learning with guidance from an instructor. This latter image captures a fundamental tenet of classical education: students as active, and critically important, participants in the process of their own educations.
John Milton Gregory writes in his classic, The Seven Laws of Teaching:
Knowledge cannot be passed like a material substance from one mind to another, for thoughts are not objects which may be held and handled. Ideas can be communicated only by inducing in the receiving mind processes corresponding to those by which these ideas were first conceived. Ideas must be rethought, experience must be re-experienced. It is obvious, therefore, that something more is required than a mere presentation: the pupil must think. He must work with a fixed aim and purpose – in other words, with attention. It is not enough to look and listen. If the mind is only half aroused, the conceptions gained will be faint and fragmentary – as inaccurate and useless as they are fleeting. Teacher and textbook may be full of information, but the learner will get from them only so much as his power of attention enables him to shape in his own mind (Chapter 3, The Law of the Learner).
This reflects what the ancient historian Plutarch once wrote, many centuries earlier:
For the mind does not require filling like a bottle, but rather, like wood, it…requires kindling to create in it an impulse to think independently. (On Listening to Lectures).
Accurately naming the progression of the parts of the curriculum in The Dance of Number shows Nickel’s clear comprehension of what Plutarch meant to convey: The student must be invited and inspired to take the steps, not simply be seen as a passive receiver of information out of context and unrelated to other aspects of life and learning. This curriculum also does a wonderful job of integrating mathematics with other areas of learning, such as history and languages. It’s sprinkled throughout with valuable and interesting definitions, quotations, diagrams, and pictures.
This idea of the dance isn’t simply the step-taking of accumulating knowledge and skills, however. The instructor and the student must also dance together, each with the necessary steps, to achieve the ultimate goal: an education. Eventually, of course, the goal is that the student will require no instructor, but will have learned all aspects of the dance and can carry it out effectively in self-learning. Furthermore, using the language of steps is a lovely reflection of the overall metaphor of dancing in this curriculum:
There is a dance, a harmony of order in creation revealed by law-like patterns, all reflecting God’s creational covenants…Mathematics, which is the search for patterns in the created order, or mind investigating matter, is reflective of God’s…order…Knowledge of the dance of number is an essential prerequisite for a proper understanding and use of God’s created order, which includes the study of higher branches of mathematics…A well-rounded study of mathematics should awaken you to detect these wonderful patterns (4).
I am reminded of something C.S. Lewis wrote in Letters to Malcolm. He is discussing church worship services, pointing out that:
Every service is a structure of acts and words through which we receive a sacrament, or repent, or supplicate, or adore. And it enables us to do these things best—if you like, it ‘works’ best—when, through long familiarity, we don’t have to think about it. As long as you notice, and have to count, the steps, you are not yet dancing but only learning to dance. A good shoe is a shoe you don’t notice. Good reading becomes possible when you need not consciously think about eyes, or light, or print, or spelling. The perfect church service would be one we were almost unaware of; our attention would have been on God.
This idea that as long as one is still learning the steps and noticing them one isn’t really dancing can, and should, be applied to mathematics just as Lewis applies it to reading: imagine being so familiar with the wonderful patterns, the dance, of mathematics, that you don’t even have to be aware of the steps – you would then have a joyful participation in its beauty the way an excellent reader can sail through a stunning sonnet and never stumble once.
That’s what I dream of for my own students – with respect to worship, with respect to poetry, and with respect to mathematics…
I see the promise of this curriculum already reflected in my student’s initial reactions to it when he said, with excitement in his eyes, “Wow! This book has already taught me some things about Latin, Greek, English vocabulary, Mathematics, and Theology!” As the poet William Butler Yeats is purported to have said (in what seems a clear case of borrowing from Plutarch), “Education is not the filling of a pail, but the lighting of a fire.”
My student’s mind has been kindled. He will take the steps.
For additional articles on the subject of teaching mathematics classically, see: