…If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education. Paul Lockhart, “A Mathematician’s Lament”
This weekend, I was privileged to gather at the Catholic University of America with fellow educators who feel the force of Lockhart’s now-famous Lament and who have taken steps over the years to do something about it. They are part of the classical liberal arts revival that is gaining momentum among Catholic schools, and recognize that many of these schools have found it difficult to integrate mathematics and the sciences into the liberal arts curriculum. These branches of knowledge, whose hold on the privileged place in education seems to grow with each passing year, pose particular difficulties for the classical liberal arts schools.
Based on the early modern developments of algebra and mathematical approaches to the study of the natural world by which men like Descartes, Galileo, and Newton transformed our understanding the world around us and the very way the human mind should hope to achieve knowledge of what is real, contemporary math and science arose outside of the traditional liberal arts curriculum, rejected its presuppositions, forced their way into education through battles in the nineteenth and early twentieth centuries, and always followed their own ways, even imposing those ways upon the properly humane studies. In these days of “teaching to the test”, even math and science have suffered a loss of coherence and meaning, leaving mathematicians and scientists like Lockhart, himself no classicist, mourning and weeping.
Today’s situation forces difficult questions on those trying to teach within the classical liberal arts tradition. Should the ancient Quadrivium – Euclidean geometry, the contemplative study of integral numbers, astronomy as a discipline distinguished from physics, and the numerical analysis of music — be revived in the upper school curriculum? To what extent? How are the modern developments in these areas to be resuscitated and integrated with both the ancient Quadrivium and the rest of the liberal arts curriculum, and, more importantly, to contribute to the formation of the young in the true, good, and beautiful?
All those gathered agreed that the study of mathematics and the natural world provides important opportunities for accomplishing central goals of a liberal arts education:
- the development of wonder
- the confidence that comes from arriving at knowledge of what is real through ordered investigation and thought
- the ability to judge well the character, extent, and limitations of that knowledge
- that students will be best prepared to take full advantage of collegiate courses in these areas if they have learned how to ask the right questions and measure the worth of the answers given
- That natural philosophy should play some important role in helping students integrate these branches of knowledge into a complete vision of the world that is, and in fortifying them against the reductionist mentality formed by cultural scientific indoctrination.
Michael Moynihan, Head of Upper School at The Heights in Washington, D.C., emphasized the threat posed by the philosophical presuppositions underlying contemporary relativism and explicitly contained in the framing sections of most high school science textbooks. Michael identified four profound errors that today’s young drink in with their mother’s milk [my metaphor] and find reinforced in ordinary education (which he is elaborating in his Bring Back Reason series).
- Everything is composed of little particles.
- There really are not such things as substances in the classical sense of something as a whole entity having a unified substantial existence.
- There are really no such things as natures.
- Living things are machines.
To combat these errors and prepare their students to be able to give reasonable witness to natural truth, The Heights is introducing a robust Philosophy program into its college-prep curriculum. For a number of years, Freshmen have spent a semester learning formal and (more recently) material logic (drawing on Martin Cothran’s Memoria Press texts). Beginning this year, Sophomores and Juniors are being introduced to concepts central to traditional Aristotelian and Thomistic natural philosophy and metaphysics. This sequence will help them see that the human mind has more, more fundamental, and more certain ways of knowing the real world than simply through contemporary science. Seniors will find their history class devoted to the intellectual revolutions that have characterized Western thought and provide an understanding of today’s cultural foundations which, together with an Apologetics course that culminates the theology sequence, will bring out the profound significance of the ideas studied in previous years.
Peter Orlowski, teacher at the recently formed Summit Academy in Fredericksburg, Virginia, expressed his conviction that teachers of math and science should first be teachers of human beings, seeing themselves as having the same fundamental goals as teachers involved in the humanities and arts. He witnessed to the creativity shown by teachers committed to these ends and given freedom to work. He tells students and parents that the best preparation for collegiate courses is often not found in learning parts of those courses ahead of time, but in learning what scientific thinking is about and how it proceeds.
Peter believes that the ancient quadrivium should continue to be a curricular foundation, with algebra, calculus and higher mathematics providing a higher level wisdom about these same subjects. Although as an individual teacher, he has not been able to incorporate much arithmetic or musical mathematics, Peter shared with us his approaches to astronomy, Euclidean geometry, and natural philosophy. In each of these areas, Peter consciously aims to foster student questioning, activity, satisfaction in knowing, and caution in judgment. Astronomy focuses students’ attention on celestial bodies, opening them to becoming fascinated about objects that can’t be produced by human beings. Peter’s students spend significant amounts of time observing and charting the movements of the sun, moon, and stars, learning to tell time by the positions of stars in the night sky, and experiencing trigonometric functions through carefully noting the annual changes in the shadows cast by gnomons. They think through the problems involved with determining the lengths of the month as determined by moon and stars, activities which culminate in a field trip to observe a lunar eclipse, which they can compare with their predictions.
Peter prepares students for the study of Euclid through weeks of discussions about where the study of shapes should begin. After debating the relative merits of the Pythagorean theorem, coordinate systems, what is most interesting, they inevitably move towards the simplest objects, culminating in the partless point. After this, they move carefully through demonstrations of Euclid’s first book, which culminates in the demonstration of the Pythagorean theorem. The training in demonstration has a profound impact on all of their studies, such as literature discussions, as students become accustomed to expecting that opinions will be held based on clear, evidenced-based reasoning.
Peter introduces students to natural philosophy through a sequenced reading of Greek philosophers, beginning with Thales and culminating in reading (with assisting commentary) the first two books of Aristotle’s work On Nature. For each thinker, students are required first to present the best arguments they can think of to support a philosopher’s view (such as Thales’ view that everything is made out of water), and then to argue against it. Often they anticipate the critiques and advances made by subsequent thinkers. This sequence includes a complete reading of Plato’s Timaeus, which they find challenging but deeply formative.
Edward Trudeau, one of the co-authors of the influential Educational Plan of St. Jerome Academy, is now deeply involved with forming the St. Jerome Institute high school, which will open its doors next fall. SJI intends to fully integrate the math and science sequences into its overall curriculum. By doing this, they hope to avoid forming in students the belief that those disciplines provide the only objectively right answers to questions, while giving them a deep understanding of how advances in these areas have profoundly affected human affairs. The carefully-sequenced modules will help students to live adult lives filled with the true, good, and beautiful, and also prepare them well for collegiate learning and fruitful careers.
Their four year program, which they call Natural Philosophy rather than Science, situates learning techniques within narratives that raise historical problems leading to the need to develop more sophisticated tools for problem-solving. Each year, problems are presented that relate to a common school-wide theme. The first year’s theme of “Exodus and Odyssey” invites such problems as those involved in discovering means of determining longitude, crucial for learning to navigate the globe, or the logistical failures in the provision system for Napoleon’s army as it moved through Russia. The second year turns to observing and understanding whole beings and systems, such as animal life and eco-systems, while the third year (“God and the Human Person”) follows the humanist-led turning inward through dissection and analytical reasoning. The fourth year draws upon all the previous years to address larger social problems.
The SJI approach always begins with what amount to elaborate, significant word problems which not only increase interest in learning techniques of solution but also provide iterative training in what is considered the hardest part of equation-based reasoning. Narrative context provides a motivating question, and an occasion to discuss preliminary ideas on how to approach it. Historical readings (such as Columbus’s logs) and activities deepen understanding of the narrative context. Students are then taught simpler and more complex tools (e.g. vector multiplication or matrix transformation) that they can use to address the problem. Higher level word problems train them in using the tools, which are finally applied back to arrive at more satisfactory solutions to the original narrative. Students are expected to engage in research and investigative thinking for homework.
Each of the presenters emphasized that they are only in the initial stages of enacting their ideas, and affirmed their intention to adapt in the light of experience. But their work promises to help students experience the power and beauty of math and science, intelligently and critically appropriate their riches, assess the evidence for questions of contemporary importance (e.g. biological and cosmological evotution, bioethical challenges, the value of sociological research), and be aware of the contingent character of scientific theory as well as its impact on human culture and history.