The following is the undergraduate thesis of Josh Lawman The argument Mr. Lawmen develops is that mathematics needs to be taught to children in a new way – a way that is effective, engaging, and connects up with broader learning about God, humanity, liberal arts, and sciences.
I am an admirer of Mr. Lawmen’s argument, so I post it here with his permission. – Keith
People of the Math
by Josha Lawmen
I. The Failure of the Utilitarian Approach to Mathematics
We have tried and failed to force feed the soul. Parading an atheistic mathematical pedagogy dressed up with Christianese, the modern Christian community has failed to train Christian mathematicians. As a community, we have lost sight of our vision of the Christian liberal arts education by incorporating a secular mathematical pedagogy which treats man as a machine which can be programmed. And so our schools fail to effectively serve either the city of God or the city of Man. The nature of man’s purpose on earth and the nature of mathematics should not have been forgotten.
As regards to the City of Man, our current use-centered education model does not help students in professional society – both those who heavily rely on mathematics in their profession and those who do not. There is a significant lack of met needs for those who use math in their profession. Every college math class whether it be BioStats, Stats for Psychology or Business Calculus is faced with students who have difficulty remembering even the most basic algebraic properties. High school juniors can manipulate giant fractions by applying multiple algebraic rules, while college juniors find themselves at a loss to simplify a2 * a5. Wait… do you add or multiply the exponents? Many of the business students I have tutored were embarrassed by what could not remember and did not have any ability to derive. They also felt that business calculus was a class to get through, not a class to learn from. They had a lack of meaningful carry-over understanding from high school and by their resistance to the subject commonly carried little from their business calculus class. They would not be comfortable applying a complicated math model to a business situation – before or after the class. The current utilitarian approach to mathematics is not equipping such students with the tools to continue to mature their mathematical ability in proportion with the demands of their profession.
In addition to these professional math users, we are harming non-professional math users. These students have gone through twelve years of math education and have little to show for it. They think somewhat more rigorously and they remember most of their times tables. But there is no substantial way in which one can assert that they possess the invaluable intellectual tool of mathematical thinking. In addition, they are jaded to a large area of human study and cannot profit from the deep joys and growth which could be had after high school. Math education has not increased their understanding of the world nor shaped their philosophy. Mathematical thinking does not deeply impact their life of the mind. It does not spur them to be more creative and more rigorous.
Both groups of students have failed to perceive math’s place in their universe. Math has become a specialized and annoying problem solving activity. Most have vague notions that math consists of “computation” to serve the economic and consumer market, and that it is a necessary element of intellectual growth for children. They do not perceive the great service math offers them as adult human beings.
Aside from not understanding how math fits into their life or the rest of their intellectual framework, they don’t understand math’s internal structure. They have no “road map” of the terrain of the mathematical universe, not even a bad one. Any road signs which they were given were forgotten. The most consistent and well-ordered system known to man seems structureless! They have notions of abstract symbols serving some final “result” with numbers seen as mere objects of computation. As a result of only receiving true opinions of the mathematical universe in school, they have no lasting knowledge of the mathematical universe.
Finally, math hasn’t been fun for these people. We have forced them to look at something we thought they should look at. Few feel grateful for being “forced” to do mathematics. And so, we graduate students with a bad taste for a subject that they cannot understand or properly place in their world.
It is acceptable, though lamentable, that a particular pedagogy is not serving the city of man. It is unacceptable and tragic that a particular pedagogy is not serving the city of God. In addition to these materialistic failings, our current model does not sanctify men. One of the most lamentable and truly tragic losses is that we are not properly shaping our students’ rational faculties. Math, one of the most noble subjects, is, as Plato points out through Socrates, an invaluable tool which turns the mind to contemplate that which is invisible. However we do not establish our students as students of the invisible.
As a result, our students are not being encouraged to explore and exercise their creativity. There is a distinct lack of mathematical creativity in our class rooms and homework. Tracks are laid and students are told to follow them. We provide an algebraic rule such as a^i * a^k = a^i+k and then we command that they apply this rule to 32 * 33, 68 * 62, … and so on. When we introduce a new question which may lead them to genuine exploration, we give them the answer immediately. In addition to this absence, we give no historical models of exploration. Thus the art of mathematics is lost with the blurring of the ancient vision of mathematics as mysterious, illuminating and transcendent. With a blurred vision and no tools to be creative, our students are handicapped in their hope to explore the mathematical universe.
The use of mathematics is thereby forgotten: the vast subject does not encourage an enjoyment of the world and God nor does it contribute to their understanding of the cosmos. They don’t understand the mathematical world – either on its own terms or how it fits in with the rest of the universe. We have provided no mathematical epistemology or ontology. Students have no such notions of what a number is and what they are doing when do mathematics. We bestowed a liberal arts vision and did not teach the students where to place mathematics within that vision.
There are a great many challenges to integrate math in a way that our students will see how it fits into a liberal arts education: there is no integration in the SATs, there are few places in secular society with healthy integration, and there is a lack of common math numeracy in our society. As Christians, one would expect that we would be able to help our students understand order, beauty and other attributes of God which are revealed in mathematics. Instead, the students graduate with bad notions of order (it feels cold and dead), and they are encouraged to take any shortcuts to avoid mathematics. In addition, these students are not mathematically eloquent. They have not been helped by their math classes to communicate ideas with others.
II. Mathematics: The Subject for Lovers
The Christian community appears to have a difficulty teaching mathematics. Instead of teaching mathematics in a fundamentally Christian manner, Christian mathematical pedagogical styles are fundamentally secular. We have dressed up the secular pedagogy with undefended claims that numbers reveal God’s glory. This secular model is very attractive because it is easy to implement: we already have a working model to imitate and our Christian examples need only to be as sophisticated as the common-denominator Christian child will accept. This system, however, is flawed. It is injuring our reputation as Christians and inhibiting our ability to share the gospel. We must renovate this system. We must teach our children mathematics. We must allow them to become adult souls.
In our math classes, it is evident that our students do not learn to be healthy human beings. The classes do not turn the students’ gaze upon that which is immortal and changeless, but that which is changing and corporeal. We are teaching them to live in a miniaturized world. In this reductionist world number means only quantity, order means only rigidity, beauty means only expression, and language means only words.
Yet, mathematics is one of the greatest subjects. Moderns were tempted to overthrow theology and establish mathematics as the queen of the sciences. Now, mathematics does not even assume her role as second only to the queen. Instead we have rejected her and placed a cheap impostor in her place. This impostor wears the mathematical garb but is hollow on the inside. Who is this impostor, but shadows of all that mathematics? Quantity and computation, mere shadows of the power of numbers, claim to be the whole story so-to-speak. This perspective glamorizes worldly success and mechanistic abilities. We must not depend on this feable, wooden view of nature to build our Christian mathematics. We must find a new foundation, God, and build from Him alone. Building from God will lead to teaching mathematics centered around love. If we teach the thing itself, what is valuable in its shadow will follow - but we must not teach our children to chase math’s shadows.
Instead, we will hope that our students will chase numbers driven by love for them. At present, it is not difficult to see that our students do not love numbers. We may wonder if it is not a bit unreasonable to ask them to love numbers, but we do not wonder whether they love them or not. We have such small hopes for our students in this regard. It may even be curious to imagine what signs or marks we would expect to see if our students did love numbers. Naturally, we would begin by looking for the common signs that are observable in any lover such as service, play, and self-perpetuating learning.
Service is the duty which the lover delights to fulfill and we should look to see students fulfilling this duty. As with all beings, we may best serve numbers by helping them fulfill their telos. Number’s telos will be explored in more detail in the next section on mathematical realism. Tentatively I suggest the telos of numbers is to manifest God’s Love. Numbers do this by reflecting many of His attributes and by drawing rational beings to Himself. Regardless of whether this particular model is true, it is evident that lovers help the beloved reach its telos. The lover, who plays, acts beyond the demands of service driven by what seems to be a divine madness. Everything that is in love, enjoys playing with its beloved. Man’s love of number should be no different. In any healthy lover, we expect hearty play. And, in addition to play, all men love knowledge of their beloved. Every husband, barring an impairment, knows the colour of the eyes of his wife. Love encourages a continal pursuit of such knowledge. Only the non-lover finishes learning about his “beloved”.
These marks of a lover are absent in our graduating children. They are absent in part because our curriculum and schools do not provide a framework for loving numbers. Our students do not understand what numbers are, and therefore cannot intentionally strive to help them fulfill their telos. Play is lost as we lose sight of serving and gazing up at numbers. And most obviously of all, our students do not not perpetuate their learning of numbers. Upon graduation, they do not build upon the foundations which their high schools laid for them. Many forever turn away from the sublime numbers believing them to be the shadows which they were shown in high school. They leave school and do not love numbers. If they catch glimpse of numbers later in life, they may fall in love, but no thanks to the schools.
Perhaps a comparison with loving dogs will help here. There are two types of ways to love dogs. One person loves dogs by playing with his/her dog. They love to feed him, pet him, train him, and walk him. They want to be with the dog, and they want their dog to be happy. A second person looks at a dog. They see how this dog desires to protect his master and how that echoes God’s desire for us to be protected. He learns from the dog about protection, mothering, loyalty, but he does not experience these attributes. In education today, we only encourage our students to love in the removed sense. They become observers of numbers, not participators in numbers. They love them from a distance. We, as teachers, have a responsibility to love numbers “up close.”
One possible objection is that we interact with numbers “up close” only when we interact with the physical world. This is not so, for in the physical world we only interact with reflections or echoes of each particular number. It is remarkable that we can understand so much of the cosmos through mathematical reasoning, but the numbers do not reside in the cosmos. Every circle which can be understood better by the number π, is understood by the same π each time. Thus we can learn to better see π the more we notice it in physical circumstances, but we should not presume to be studying many similar numbers all who are “π.” π is one.
As we try to understand how to love numbers it is also helpful to consider the way in which we love wisdom and beauty. Any man who loves wisdom would seek to better understand what it means to be wise, both for the application to his daily activities and for the good it would do his soul aside from any decisions he makes. We would also expect the man to enjoy seeing other men who were wise. Presumably he would delight in the fact that God is supremely wise. In a similar fashion, the lover of beauty would want to be beautiful. But it seems that they would be more interested in seeing and delighting in beauty than possessing beauty. They too would be delighted to know that God is supremely beautiful.
What of the lover of numbers? His search to understand what numbers are would not end. He would delight to posses numbers in whatever way he could, as well as delighting in whoever or whatever was most numbered. Emotions naturally follow love. Their presence is not necessary in order to love an object, but they naturally flow from the love. In this way we can hope to unite our whole person in a love for mathematics. It is not merely an intellectual ascent. Our appetite, our spirit, and our reason should incline itself to pursue numbers.
III. The Ontological Foundation for a Meaningful Love of Numbers
What are these numbers which we may fall in love with or reject? I will here propose a mathematical ontology which is consistent with our Christian educational vision and allows for a meaningful love of numbers. This model can be rejected or accepted - but we must have a mathematical ontology which is consistent with our pedagogy. If we are to center our pedagogy on love, we must be studying something – for love is of something surely.
In order to find out what sorts of things numbers are it is helpful to consider who uses them and in what way. Hopefully, we will catch a glimpse of the proper and most noble purpose of numbers as we survey their various roles and relationships.
We observe two users of numbers: God and men. Men use numbers for tasks such as engineering, IQ scores, economic regulation, counting, and taxes. God appears to govern nature with them. The workings of physics, biology, and chemistry are all bound by laws which rely heavily on the relationships between numbers. Observing this, Galileo reputedly said “Mathematics is the language with which God has written the universe.” Jesus also uses numbers in his parables, such as the parable of the lost sheep and the parable of the talents. There are many other passages in the Bible which include opportunities for men to learn from God through numbers: the dimensions of the tabernacle, the days of creation, the book of revelation, wisdom literature etc.
Aside from numbers’ use, they appear to have a strong relationship to many universal principles such as wisdom, transcendence, order, and beauty. I will briefly mention some of the most obvious connections.
- Omnipresence: on the surface, it may not be obvious to see where numbers are. But it is just as difficult, if not more so, to point to where they are not. Anywhere there is unity we have one, and anywhere there is distinction we have at least two.
- Beauty: we almost always see numbers in relation to beautiful things. Beautiful things in nature are quite evidently well proportioned. Aristotle claims number for the formal cause of the musical octave. In the visual arts we have the rule of thirds, to name just one aesthetic rule. Artists using this rule divide images into nine equal parts in order to have a beautiful composition.
- Eternity and objective truth: numbers don’t depend on time. So far as man call tell Math works yesterday, today, and forever. And they don’t depend on any individual person. In that way, they appear to be heralds of objective truth. Disagreement about the sum of two and two doesn’t change it from objectively being four. In addition to this, numbers prompt us to contemplate the infinite. Set theory has led us to believe that there are multiple infinite numbers (Cantor’s א0, א1, א2 …). As we contemplate these transfinite numbers we are encouraged to grapple more maturely with the way that God is infinite and eternal.
- Order: the set of counting numbers is the most perfectly ordered thing in existence. We notice that the universe itself seems to be kept orderly by the power of number.
Thirdly, it is valuable to consider our relationship with numbers. A robust mathematical epistemology is essential for understanding mathematics. Below are some cursory epistemic observations. For all intents and purposes, it appears as if we have been prepared to contemplate numbers. We are embodied rational souls, provided with many external “spring boards” to help us observe them; we are in a physical universe with counting, physics, Scriptures, and music. This material prompts us to contemplate numbers and guides us in our contemplation. But these external sources do not appear to be the only guide, since everything on earth is divisible and yet we still have a concept of one. Augustine argues that our concept of unity is internal for this reason (alternatively Aquinas argues that we learn of one from God’s divine simplicity).
Thus, we can contemplate number. And when we contemplate number, we contemplate some thing deeply related to beauty, omnipresence, order, etc. The presence of such divine attributes in number is incredible. No matter who you are, or where you are, this “face” of these attributes is present. Even if you lived your entire life in a dark room, the simple awareness of your ten fingers and ten toes could lead you to contemplate God’s immortal truth. In numbers, many of God’s attributes are always available for consideration. It appears that one possible purpose of number is to draw us to God. They present his majestic qualities and thereby naturally draw us to inquire, prompt us to explore, and encourage us to wonder. Possibly, angels and demons have a similar experience. As fellow rational beings, one would expect angels and demons to be aware of numbers, too. And through numbers they, too, are always faced with immortal attributes of God. Are numbers messengers to all rational beings?
Evidently the present ontological model implicitly present in our educational system (numbers are mental constructions of men) is found wanting. It is too “small” an existence for such entities. I would suggest we make the step from quantity to entity. This distinction is a crucial one: quantity helps us perceive entity, but quantity and entity are not con-substantial. If numbers really are related to the powerful attributes we mentioned earlier we would expect to be able to identify a more substantive construction than the mental model of quantities. Evidently, man is not the cause of such beings.
We find ourselves searching for an understanding of the existence of numbers. As Christians, we would expect God to have created numbers (if they exist). I will here present an adaptation of Augustine’s mathematical ontology in the framework of Aristotle’s four causes. This model is the “likeliest story” I have yet heard of numerical ontology. The reader may find this story unsatisfactory, but it is imperative that they only replace it with a more likely story which is consistent and beautiful.
As a brief review, Aristotle’s four causes are the material, formal, efficient and final causes. The material cause is the matter which “it” consists of (for Christians this “material” need not be physical, since non-physical things exist: angels, powers, dominions, etc). The formal cause is the arrangement of that material. The efficient cause, is the principle guiding the thing’s coming to being. And the final cause is the end for which it came into being, its purpose.
As was mentioned earlier, numbers extend God’s attributes into the presence of rational souls. Since this may not be all they do, we would expect that their final cause is either this or something greater. Once we have accepted this loving act (or something greater) as their final cause, it is evident that their efficient cause must be God. If God is extending Himself through numbers, He must be the one who guided their coming to being.
But what are they made of? What is their material cause? Unfortunately, we have few names for non-physical materials, which oftentimes people use as a reason to write off the existence of numbers in the first place. Augustine suggests that numbers are thoughts in the mind of God. They are objects of divine thought sustained by His Eternal Love. This model definitely accounts for the strong relationship numbers have with God’s attributes. It is attractive and holistic, and below is a brief reasoning through of his model.
- Divine revelation teaches us that God is one in essence, and three persons.
- Consider with me His Oneness. Surely God also considers His Oneness. He is always aware of his Oneness. He is always thinking ‘One’ and thereby continuously creating the number one.
- Now, consider his Threeness. God is three distinct persons by essence. The Threeness does depend on an inner Trinitarian addition operation: it lies in His essence. As with Oneness, God must be aware of His Threeness and contemplate it.
- It seems that God, in His infinite Wisdom, would also be eternally contemplating the relationships between these two numbers, and thereby construct the rest of the algebraic and transcendent numbers. (The sequential language is knowingly misleading here since God’s thoughts are not likely to progress in such a fashion).
- For example, from one and three we can derive all the other counting numbers from the simple property of addition. If God was thinking of three and one, would he not also have contemplated three plus one? If four, why not also four plus one? And so on.
Thus, the material cause of numbers would be the thoughts of God, and their formal cause would be the manner in which God arranges his thoughts. Arrangement by pure reason would certainly explain the attributes mentioned earlier (beauty, supreme order, etc.). Since we cannot imagine a time when God was not contemplating His nature, it appears that numbers exist in eternity.
One immediate objection may be that God has thought of other things such as pink unicorns, and so it would seem that our argument leads us to necessarily accept a ridiculous ontology for pink unicorns. I can think of at least three reason that the idea of a pink unicorn is different from one and three. First of all, God’s thoughts are not a suitable material cause to fulfill the telos of pink unicorns. The purpose of a pink unicorn is directly related to its material cause. And, as far as we are aware, an animal’s material cause is a collection of atoms. The purpose of a number does not require a physical material cause. Secondly, God appears to have chosen not to bring pink unicorns into existence. God thinking about elephants brought them into being on earth, but apparently He did not see fit to do the same with pink unicorns. We do not observe a physical manifestation of pink unicorns indicating the existence of this animal form. We have no other reason, except maybe desire, to believe they exist somewhere else. But we do see God assigning numbers to all things great and small, and infusing mathematical relationships into our lives. Finally, a pink unicorn is divisible. Oneness and Threeness, as parts of God’s nature are not divisible. As we already agreed, the Threeness of God does not depend on an addition operation. A pink unicorn is not a primary idea like this; it is an arrangement of other ideas.
Evidently, this argument is somewhat less successful to defend the other algebraic and transcendental numbers. In a sketch defense of the existence of these numbers I would suggest that three and four appear to be the same “kinds” of things. There is no difference in use and power between the two numbers. Secondly, I would point out that relationships between two numbers may be shown to have substance.
And what would this model mean for us? Well, when we contemplate the number three, we are thinking God’s thought after Him (albeit in an incomplete and imperfect manner) and contemplating His nature. We can really learn new things about God through numbers. And perhaps, some of His attributes are primarily taught to us by means of numbers. Their presence enables us, so long as we posses reason, to study God. If we are blind, or deaf or dumb, we may still wonder, we may still explore.
IV. Applying the New Values of Knowledge & Vision
We hope for knowledge and vision in our students. Implementing a realist’s mathematical ontology encourages hope for knowledge and vision, not simply development of analytical thinking. Math is, as Socrates points out in Plato’s Republic (Book 6), an excellent subject to help one turn one’s intellect to that which is. It helps students turn to contemplate substantial and invisible things. For numbers are, by their nature, deathless things - they cannot age and do not die. They should be studied and loved because of the type of their existence.
The first step to integrating a consistent philosophy into the class room is to reconsider our mathematical values. We have developed a K-12 culture which is in opposition to the mathematician’s perspective by claiming that only the answers are valuable. Thus our students hate to prove theorems - an activity which is a common favourite amongst mathematicians. We have valued only the destination, and consequently only valued the true opinion which will guide someone to that destination. Plato points out that true opinion will guide one as accurately to a destination as knowledge. However, opinion, is obviously inferior to knowledge.
True opinion isn’t bad. In fact, it can be very helpful in conducting a lot of the day-to-day mathematical decisions we make. It also helps develop thinking. But it is different from knowledge. Without knowledge, we cannot be lovers of something, we only love or appreciate things about that thing. And now this subject which could encourage students to love that which is, becomes an idiotic shopping assistant.
Our math text books have proofs for particular theorems, but we do not, at bottom, value knowledge. As a result the proofs, one way to come to know theorems, are perceived as useless by the students. We can and should help our students come to desire to know mathematical entities and to search them out.
One immediate way we can help our students search for knowledge of numbers is to give them a framework and experience at asking a “mathematical question”. A mathematical question is a question of mathematics driven by desire to know further the thing itself. With time constraints on over-ambitious computation goals we cannot hope to have all of their learning guided by these mathematical questions, but hopefully they will at least know what it is like to have a mathematical question. Likewise, we do not have time to prove all the theorems they learn, but we hope that they will experience the knowledge which comes with a proof.
Examples of some broad mathematical questions I would hope to provoke in my students follow in italics. Some questions are meta-mathematical questions - developing their mathematical framework, some are about individual mathematical phenomena themselves, and others are about living as a Christian. What is the relationship between addition and multiplication? What is a number? What is a function? What is a sine wave - does it exist? What is the most beautiful way to present the trigonometric functions? What does it mean to have an infinite number of primes? What is a prime number? What is the number 4? What is the difference between geometry and algebra? How does the Cartesian plane unite the two? What sorts of things might I learn about geometry from the Cartesian plane? What is a sequence? What sorts of things can a sequence show me? Could math ever change? Is proof by contradiction valid? What does it mean to serve God with numbers? Can I learn about God through math? Is there anything I can learn about God only through math?
My hope is that these questions will lead them to better see numbers and consider the invisible realm. If asked, I will offer my personal opinion to the students but I primarily desire the students to have questions like these within them. The key is that they are students’ questions. In doing so, the subject will harness their natural curiosity of the created world. The questions help them to search for meaning and find themselves delighted in the unexpected order found in mathematics.
V. Math Mythology: The Initiation Rites of the People of the Math
Implementing a mathematical pedagogy which is consistent with our liberal arts vision is all well and good if we have students who will receive it. Many students will resist the call to become lovers of mathematics and can be convinced by no reasonable argument. Their difficulty is not with logos, but mythos. They have a false and insubstantial math mythology.
We have culturally developed the myth for science as the means to conquer nature. This myth gained a great deal of success, especially as mathematics was infused into science during the scientific revolution. As science was mathematized, math was overlooked, considered subservient to this new goal of science. Our culture glimpsed the power of science and placed mathematics under its mast-head. But mathematics doesn’t properly fit under science - nor under sociology, business or psychology. It needs its own masthead – its own myth.
I propose we develop a “backbone” myth for mathematics. This myth points to the way in which math rules over all order and pattern. Numbers may not be the best way to represent some particular order or pattern (for example, IQ scores may inaccurately suggest an inappropriately strongly ordered model of intelligence), but the numbers rule over it.
This myth should embrace the heritage of great thinkers such as Ptolemy (whose geometric aesthetic sense drove the expectation that perfect circles ordered the cosmos), Pythagoras (for whom all is number) Plato (whose demi-urge fashions the world with forms and numbers in the Timaeus), Dante (who perceived spheres to represent divine, cosmic order), Copernicus (whose expectation of algebraic and computational elegance produced a revolutionary cosmology), and Galileo (with his vision that “Mathematics is the language with which God has written the universe”). The myth teaches the student of math to look from the particular to the general, from the physical to the invisible, and from order to divinity.
In schools with a great books program, we have the benefit of a myth network which willingly accepts this mathematical myth. If the students do not have a strong myth network, mathematics could be used as a way of building one. They should be able to give a good “meta-narrative” of mathematics - to tell a story of the universe according to math - not simply give a sentence answer on why Theorem 1.12 important to their lives.
In any group, there is a great, unifying power of the inside language. Math mythmakers need to harness this force. Infusing common words with meaning in order to create a new subject in their minds provides unique signs to point to their new mythology which distinguishes it from the cultural myth. Math is the subject they, as rational beings, always loved, but never knew. Our classes should point them beyond common mathematics to high reason and numbers themselves.
As they explore the mathematical backbone, the inside language helps them remember this exploration. Thus, they will develop within themselves a new robust math myth. With the myth comes a math ethic, which will encourage them to more deeply explore and face mathematical challenges they would not be able to face otherwise. For it helps them to internalize the desperate, visceral desire for pattern and mathematical beauty.
VI. It’s not Boring, You’re Bored
Humans naturally delight in being surprised by the naturally mythic nature of mathematics. Mathematicians find themselves repeatedly surprised when exploring the mathematical universe. This surprise reveals learning. It reveals that we have more fully grasped some beautiful thing. For example, consider a math student who affirms the belief that the mathematical universe is the most orderly network known to man. The student studies exponentiation and learns rules which tie together numbers in a curious fashion. As they become more proficient in mathematics they start noticing that there are always many roads to the same destination. Perhaps one day they learn a new road and are delighted by their surprise that it “worked”. The delight comes even without the hope of some later use. In this consistent path to the same destination they saw a little bit more of what it means for numbers to be well ordered. Thus as we have epiphanies, we learn more and more about the relationships between order, simplicity and beauty. Each time we are surprised we realize we hadn’t truly grasped what it would truly mean for something to be orderly.
There are many ways to encourage surprise and wonder. So many ways that we could be easily overwhelmed. It is only important to remember that numbers are intrinsically beautiful and wonderful – we do not need to project these qualities onto them. In fact our projections could prevent a glorious glimpse of the truth. We do not want to deprive our students of glimpsing the things themselves. Thus, our biggest step to encouraging surprise and wonder is to study mathematics ourselves and strive to grasp glimpses of them. There are multiple strategies that I would suggest to achieve the desired end in our students. The first tactic I would recommend is introducing the Rubik’s cube to your class. The Rubik’s cube is a wonderful example of many of the properties of numbers. It provides a small scale mathematical world where students are challenged to develop “theorems.” It becomes increasingly obvious to the student of the Rubik’s cube that many paths lead to the same road. The students are given opportunity to think creatively and critically about a system that is easier to grasp than the real number line. I would suggest buying multiple well-lubricated cubes and teaching the entire class the basics of the cube (solving has to do with moving blocks not stickers: there are 12 side, 8 corner, and 6 middle pieces, etc). Communicate that there are many different theories amongst speed-cubers on the best way to approach a Rubik’s cube and suggest they develop a strategy. I wouldn’t spend too much class time on the toy, but have them free to borrow and take home. You could also distribute them to those who have finished in class work early if they wanted to play with it instead of twiddle their thumbs. There are many other logic based games which are conducive to practicing developing theorems and learning to strategize abstractly. Two that immediately come to mind is Bono’s L game and the Nim game. The Nim game offers the possibility of working out a “perfect” strategy and Bono’s L game encourages students to develop “best practice” techniques. Challenge your students to develop a perfect noughts and crosses strategy. Ask them to think about whether there exists a perfect strategy for the L game, chess, and Connect 4. In addition to heavily logical games, there are many popular games which are governed by simple mathematics. Games such as backgammon, yahtzee, black jack, poker and craps are ample candidates for study. With these games, the teacher would ideally provide guidance in mathematical analysis, as it can be very difficult to step back for the first time and ask mathematical questions of games. Even without an in depth study, the teacher could, if effective, reveal the ways in which the students interact with mathematical models every day. With hope the students will wonder what mathematical rules guide their favourite game.
I would also recommend introducing students to mental math tricks. This is an excellent tactic as it encourages wonder each time the tricks “magically” work. Teach them a trick and ask them to discover the algebra behind it. Encourage them to develop their own tricks. In this way they will get experience asking questions of algebra in a fun, creative fashion. There is a dual motivation present to explore the numbers and to compete with their friends. Perhaps they could hold a mini math magic show for a younger class. This tactic capitalizes on the rule “success breeds success,” since mental math tricks give the students an opportunity to feel successful at mathematics, thereby gaining confidence to tackle more challenging problems.
There are two mental tricks which I have found particularly effective as introductory tricks: multiplying by elevens and squaring two digit numbers which end in five. To multiply a two digit number ab by eleven, simply place a+b in between a and b. For example: 6311 = 693 (6+3=9). If a+b is greater than nine, simply carry the one over to the hundreds place. For example: 8611=946 (8+6=14, carry the 1 over to the hundreds place and placed the 4 between 9 and 6). To square a two digit number ab where b=5, simply multiply a(a+1) and place 25 to the right of this number. For example: 252 = 625 (23=6). A second example: 752 = 5625 (78=56). There are many more tricks waiting to be deduced. Many popular and successful tricks are available in books such as Art Benjamin’s *Secrets of Mental Mathematics.
Another excellent way of encouraging wonder, love, a greater sense of familiarity with numbers is to teach students to memorize the first 40 digits of π. This, by sounding impossible can boost a sense of success and mastery. The exercise encourages a love for the number π. During this project I memorized the first 100 digits of π. Although the task was not particularly difficult, I have been overwhelmed by the “depth” of the number π. I have found myself surprised to grasp a bit further what it means for a number to be irrational. I am overwhelmed by the complexity and mere information in π. This exercise should not be graded so as to work purely on their excitement at being able to do the impossible (not all students will have this excitement and it should not be forced). Although it sounds impossible, teaching your students to memorize π to 40 digits is not particularly difficult using mnemonics. This system described below is used in Secrets of Mental Math mentioned earlier and The Memory Book by L. Harry and J. Lucas. First, introduce to the students the phonetic code that attaches the numbers 0-9 to a consonant sound.
The numbers are associated with the sound, not the letter itself. The system is as follows: 1 – t or d, 2 – n, 3 – m, 4 – r, 5 – l, 6 – j, ch, or sh, 7 – k or hard g, 8 – f or v, 9 – p or b, 0 – z or s.
After teaching the students this code, they must memorize: “My turtle Pancho will, my love, pick up my new mover, Ginger. My movie monkey plays in a favourite bucket.” They have now memorized the first 40 digits of pi! All they need to do is to simply “convert” the sentence in their mind (vowels can be ignored). The first few digits are as follows
- 1415 926 5 3 58…. My turtle Pancho will, my love,…
Using this technique it is possible to memorize first 100 digits of pi within an hour! Students will be delighted by this mastery of the number π. And, I would be very surprised if it does not increase their wonder at irrational numbers. It is mind-boggling to consider other irrational numbers after one has spent some time considering this particular number. They will find that there is so much more to the numbers they use than they previously thought. Finally, there are non-physical math games which can be introduced to students. My favourite to introduce to a “non-math” person is Four Fours. The goal of Four Fours is to create expressions for as many of the natural numbers as you possibly can using exactly Four Fours in each expression. No operator is off-limits (unless you want to make it more of a challenge). For example the first few expressions could be: 0=44-44, 1=4/4+4-4, 2=(4!/4)-sqrt(4*4). As the reader may have noticed, the last equation is not nearly as elegant or simple as it could be. A second challenge beyond creating expressions that work is creating the most simple and elegant expressions (I would replace the last expression with 2=4/4+4/4).
There are many mathematical games which are naturally fun and can captivate all. Discover what each student delights in most naturally and match the game that most visibly contains that thing. All men delight in fulfilling their nature as rational animals. Nobody hates all mathematical games, since mathematics is intrinsically interesting for rational beings. Each game which meets the students’ ability and maturity will lift them a little bit higher and open new doors for them. The pedagogical art is to give the right progression of steps to see more and more of mathematics. The teacher must know well the path which brought them where they are, and have a good idea of where they are going.
VII. The Expressive Mathematician
As the teacher must have an excellent ability to understand the terrain of mathematics in order to communicate to their students, the students must be able to articulate the terrain of mathematics in order to understand it. In all disciplines we perceive that it is invaluable for the student of something to be able to talk about the subject. We also perceive that students should be able to read further subject matter in any given field. We must be consistent with our math classes. Evidently, the first step to developing holistic, expressive math students is re-integrating mathematics into our liberal arts vision. Below are a few practical classroom steps to take after that integration.
In order to enable our students to express their mathematical knowledge in written English, I would recommend giving homework exercises beyond the half-hearted reflection questions at the end of a homework set. These are often graded as credit/no credit without a feasible way to meaningfully grade them. The state of the students’ answers and the teacher’s difficulty in grading is due to the structure of the homework set and the lack of value on mathematical literacy. Similarly with the true opinion/knowledge problem, we don’t really care about the reflection questions - we care if they could evaluate sin 2π. Aside from the re-assessment of values, we need to restructure math homework.
By the time students arrive at the reflection problems they have been using very computational functions of their brains. They have been retrieving formulas and applying rules often in a very non-creative way. The problems often leave students feeling mentally exhausted. It is then that we ask the students to tell us why they should care about the Pythagorean theorem. Not only are they too tired to properly consider the question, they are naturally inclined to treat it in the same manner as their other problems. What does the book say this theorem was important for? What did Mr. Lawman say was the impact of this theorem? They have not been prepared to step back and consider the marvels of mathematics and write about it. As a step towards restructuring homework it seems invaluable that the reflection problems are given as a distinct assignment (the English student needs to be able to distinguish between the memory words for the week and the creative writing assignment). I would also suggest placing value on proper grammar, punctuation, etc. in the math class so that students grasp a better understanding of the unity of the liberal arts vision. The students may be wrong in their responses and should be graded accordingly. They should also be graded for creativity and depth of mathematical searching. Finally, allow them to give some of their own questions. If the students have their own mathematical questions, use those.
Students of history, English, Bible, and sociology are enabled to read the relevant documents pertaining to further study in their field, math students are not. A portion of the students’ homework could be excerpts from famous mathematical essays. The more advanced students could make their way through entire papers which have impacted the mathematical community (Gödel’s dissertation for example). Evidently, many of these papers do not directly pertain to the techné which the students are to expect to graduate with. To fill this need I would suggest finding curriculum which communicates the idea of the math without simply using symbols and colorful boxes - the students should be have some hope to educate themselves further by reading math journals when they graduate.
Most, if not all, math students I have tutored have an internal ability to apply mathematical rules which far exceeds their ability to tell me what they have done. We should expect our students to be able to explain what they are doing. There are many ways to encourage students to be able to speak about what they are doing.
Assign a homework problem where half of the class researches one theorem and the other half researches a second theorem. For a fifteen minute portion during class require the students to “teach” their partner the theorem which they researched for homework. There are many other ways to encourage partner discussion of the mathematical ideas present in the classes. Teachers in schools with strong parental involvement could encourage their students to explain particular mathematical concepts to their parents.
Another way to encourage the students is to include a speaking component into the teacher’s evaluation of the students’ progress. A student could explain a problem on the board which the class knows how to do but which the teacher has not gone through step by step. In addition, the students could give very short presentations of a concept which has been studied earlier in the year. If class size permits, include a 15 minute interview portion on the students’ final exam. The goal of these exercises is to encourage original articulation of mathematical concepts.
With such excitement one wonders why we find numbers so boring. Numbers are intrinsically wonderful and awe-inspiring. They provide revelations about men, God and the universe. They shape our minds in wonderful ways. They are intriguing, mysterious, orderly, immortal, unchanging, and fundamentally other.
Evidently it is a challenge to help students access numbers to do their work. And evidently we have largely failed as a culture. By the time our students have reached high school they have a horribly inaccurate notion of mathematics. They reveal this when they ask “what do I do?” in math class not “I wonder if?” or “What is that?” As junior high and high school teachers, we must strive to break down their presumptions and rebuild the subject again from the ground up. The students think that math is about accumulating methods about “doing” – they do not get what math is.
It is interesting to compare mathematicians with math students in high school. Mathematicians don’t get tired of math – they love it the more that they study it. The student on the other hand “sees” that there is not enough in math to entertain them for the 50 minutes they have to be forced to think about it each day. It would appear, by the mathematician’s love, that there is something to love.
It is true that our students struggle with laziness and teachers face the challenge of immature minds wrestling with mature subject matter. But there is hope, for numbers are a thing which can be known and loved. And, as Aristotle rightly noted, “all men by nature desire to know.” In addition to such glimpses of invisible things which we naturally yearn for, mathematics allows man to be magician in a world that is in flux (see The Man Who was Thursday by G.K. Chesterton). It is one of the noblest of subjects if we but grasp it rules an otherwise chaotic universe, enabling man to live in this training ground for rational souls.
So let us help our students become People of the Math educating their whole person on the love, unity, myth, and knowledge of numbers. Let us welcome to the round table of high reason, where knights may quest to fight evil in the world with a sense of chivalry and number. Let them want to serve number, not simply use numbers as their slaves. Such a myth will help them to “go the other way” from applied to pure, from physical to abstraction, from shadow to substance. Then, they will not ask “how can we conquer the world?”, but “how can we find the Holy Grail?” - the ideal model for the pursuit of mathematical knowledge.
Appendix: People of the Math Blog Posts
Men not machines
“The difference between the pupil who works for himself and the one who works only when he is driven is too obvious to need explanation. The one is a free agent, the other is a machine.”
So says John Milton Gregory in “The Seven Laws of Teaching.”
His book seems quite excellent. It is, at heart, a very practical book. A philosophy of education book for the working teacher. It constantly encourages you to ask the question “how can I apply this to my class room?”
But I’ve been thinking about the role freedom and humanity has in learning. And this quote brought that thought to the surface again.
How can we be human and learn? Intuitively, I would think that learning is one of the most human activities (if not the most human).
If learning is the height of human life, how is it that we so tempted to mechanize it and use ‘learning’ to dehumanize people. This is one of my difficulties with modern mathematical education: it dehumanizes. Could we deduce from this piece of information alone that either we shouldn’t teach math or that we are teaching it wrong?
Mathematics appears to have the passive potential to fertilize our minds so that life within will flourish. But how can we do this? Gregory (along with Socrates) proposes that questions are the way to educate. Gregory states: “Questioning is not, therefore, merely one of the devices of teaching, it is really the whole of teaching.” He charges teaches to “Excite and direct the self-activities of the pupil, and as a rule tell him nothing that he can learn himself.”
But how can we do this with mathematics? What can we learn ourselves? What questions are appropriate. How can we not remember to apply the rule that “true teaching, then, is not that which gives knowledge, but that which stimulates pupils to gain it.”
The popular trend amongst high schools is to entirely reject that this model of learning applies to mathematics. But that is not how the ancients saw it. Most of the educators who have come before us have seen not drawn the stark line between mathematics and more soul focused learning. For Plato learning geometry was a prerequisite to enter his Academy and pursue wisdom.
How then can we properly ask questions of mathematics? How can we train our minds and our pupils’ minds for this?
Blind faith in the book
I’ve was reading another chapter of Gregory’s The Seven Laws of Teaching, when I came across this quotation: “Frequently no reason is asked for the statements in the lesson, and none is given. The pupil believes what the book says, because the book says it.”
Why, I wondered, is this a temptation? My best answer right now is that teachers lose confidence in their own ability and knowledge when faced with an authoritative sounding book. The educational system evidently perpetuates a reliance on some external agency which brings together our curriculum. This seems to reveal that teachers and students alike are not given books with obviously opposing ideas. Is the solution to read contradictory books and enter into the conversation they are having? Can we introduce this concept as early as, say, the 1st grade? Perhaps that is when we begin to teach students to believe everything they read.
The problem continues and is magnified with mathematical texts. Even the student who can begin to see that their history book is biased (as all must be - both good and bad histories), have difficulty noticing that their math book is as well.
Kurt Gödel, in his seminal dissertation: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (or On Formally Undecidable Propositions of Principia Mathematica and Related Systems for you non-German speakers out there), proved that we must find a foundation for mathematics outside of mathematics. Formally he proved that you can not build a formal system, which is provably complete and consistent, from a finite number of axioms. In short this means when trust math we are sort of “guessing” that it works. Pascal (who came before Gödel) provides a helpful model, the heart and reason, which explains our relationship to the first principles. He claims that we have faith (a kind of heart/intuitive knowledge) in the first principles and build upon our mathematical/reasonable knowledge.
In that we are called upon to have a faith in mathematics. But our students don’t believe this. They take the book’s word for it. They have blind faith in the book and presume that their faith is reasonable knowledge.
I would suggest that K-12 schools would introduce the freedom (oh scandal!) to disagree with the text book from an early age. They would the lead their students to truly inquire into the truth of all subjects, including mathematics.
Let them see the great mathematicians be disillusioned with Gödel’s paper.
Let them hear of the scandal of non-Euclidean geometry.
Let them feel the shock of the pythagoreans as they discovered (and rejected) irrational numbers.
A common text
In any discussion, all the participants must have a common text. Around the dinner table the ‘text’ could be common memories, in an academic discussion the common text could be Plato’s Republic. I am now wondering what is the best way of delivering a common text for a mathematical discussion?
For Euclid (and most of the math educators who use his Elements) it was proofs. Proofs served as the spring board for discovery or teaching.
For us, it is problems: computational exercises and questions asking you to mine paragraphs for a mathematical problem.
The common text has changed. How has this influenced what we think mathematical learning looks like? The conversations must different - but in what way?
Could a common text be intuitive?
Mathematical essays often have a backbone of very basic logic. The argument is fairly intuitive and could be understood by the non-mathematician. Mathematical symbols and reasoning allows us to understand an intuitive argument in a robust manner.
It seems like the process of approaching intuitive concepts with more rigor could be made more accessible. Most of my non-math friends (yes I have some, ummm… I do, honest), would be frightened by the math symbols in an essay such as Dedekind’s Continuity and the Irrational Numbers, but they could comprehend the argument with a moderate amount of effort. Should a paraphrase version of the essay be created to train students with a less extensive symbolic understanding? Could a loose rendition of the argument be presented to train mathematical thinking?
In this way we could encourage students to perceive the true work of mathematicians, and to practice it, instead of drowning in a sea of symbols.
Worker-bee motivation for project based learning
I recently heard a TED talk by Diana Laufenberg on learning. Diana shares that learning requires the freedom to make mistakes. Learning is not about finding the one right answer to fill in a bubble on a scantron sheet. Partly as a result, she has fashioned a project-based approach to learning.
Interestingly, she also pointed out that teachers do not have information in a sense which the student do not. This made me think that teachers now appear to be challenged to answer the question: “what are we are we offering to students which a computer can not offer?”.
She now works at the Science Leadership Academy, which successful implements a project centered learning community.
The following is found on their mission page:
“How do we learn?
What can we create?
What does it mean to lead?
These three essential questions form the basis of instruction at the Science Leadership Academy (SLA)
… SLA is built on the notion that inquiry is the very first step in the process of learning.
…At SLA, learning is not just something that happens from 8:30am to 3:00pm, but a continuous process that expands beyond the four walls of the classroom into every facet of our lives.”
Oh, did I mention that all of their students have laptops?
I found this school interesting. The rest of the site sells their school as a way of producing a highly successful worker. A curious blend of whole soul and worker bee goals. Interestingly their foundation for a project-based learning process appeared to revolve around business and work “success” instead a well articulated whole soul or religious goal.
How do we know math in a physical world?
I’m going to be using a quick and dirty definition of math to work with here: “the counting numbers and their relationships.”
So what does it mean to know the counting numbers and their relationships? Less, it seems that we can interact with numbers, in their purest sense, with our reason alone. Our bodies allow us to interact with things which participate in numbers (or are numbered?), but we can’t physically interact with numbers themselves. In this way it seems that the physical world can reveal properties about numbers and their relationships, but perhaps nothing which one couldn’t discover by pure reason.
Does the physical world help or hinder our pursuit of knowledge of numbers? The mini Plato in me is eagerly shouting “amen to that brother.” But I wonder. Could we understand the majestic order of Bessel functions if we never threw a rock into a pond and watched the ripples? Perhaps the physical world helps us to see the relationship of beauty (and order) with numbers. As I study abstract algebra I am awed with the intricate interconnectedness and find the complex structures beautiful. Put perhaps the whole of my person is not quite convinced by these classes alone. Perhaps the physical world helps me internalize what I receive on the surface level in a math class. Is the math teacher responsible to help me internalize? Or is a math class merely a sort of tilling of the field, a time of preparation of the soul. Perhaps it is only after the class that I can truly experience number, but the classes are necessary to get me to that point. Wordsworth provides a third alternative: reliving the ‘moment’ after the fact by perceiving an object with your inward eye. In his poem Daffodils, Wordsworth reflects upon the Daffodils he saw earlier that day with his inward eye. Can numbers, which are incorporeal, also flash upon that inward eye? His poem ends as follows: ”For oft, when on my couch I lie In vacant or in pensive mood, They flash upon that inward eye Which is the bliss of solitude; And then my heart with pleasure fills, And dances with the daffodils.”
Ten Thousand Simons
Understanding the best perspective to have on math and math education is not like Simon Says. That’s a shame. Unfortunately we have been playing Simon Says with conflicting Simon’s. We believe a little what Plato said and try to believe Machiavelli at the same time. How can we escape the many voices? Should we? If I step back from my math proofs (which I would hope, if nothing else would be unbiased) I see intrinsic metaphysical claims. In math notation for example we have the existential quantifier ∃ (there exists) which I use without a second thought. My symbols are deceiving me if I must say “there does not exist,” and only “there would exist in this system.” Gottlob Frege may have been a great mathematician, but how is it possible either to agree or disagree with him. I have accepted his symbols with a kind of blindness - not agreement or disagreement. I had not yet had the tools to question the implicit statements of symbols. How can we tread water in this sea of rhetoric? I don’t want to play Simon Says with ten thousand Simons. But how can I deny the ways in which my culture and experience has subconsciously molded my outlook on the mathematical universe? What is the best procedure to build my perspective above the mass of conflicting voices? Should I even want to?
Real Numbers = Infinitely Long Line?
I wonder in what way the idea of a line is different (or the same) as the set of all the real numbers. Richard Dedekind, in his Continuity and Irrational Numbers uses the similarity to begin the quest for a purely arithmetic foundation for the irrational numbers. 1. Let us assume the real numbers exist as a one dimensional domain. 2. Let us assume than an infinitely long line exists. Are 1. and 2. the same statement? If we assume the existence of 1. must we assume the existence of 2. (and vice versa)? Can you separate their existence? Mathematicians have historically thought of math problems as geometric problems which we now consider numerical problems (e.g. proof for an infinite number of primes: Euclid’s proof for the infinite number of primes in his Elements vs. modern algebraic proof). Does this essentially change our understanding of mathematics? Are we preforming the same reasoning functions as the ancients? Do we exercise more abstract reasoning and less spacial reasoning?
###Sonnets of pure reason
Paul Lockhart authors an excellent work on modern math education, A Mathematician’s Lament. He provides a vivid picture at the beginning of a musician waking up from a terrible nightmare. In the musician’s nightmare, the culture has made music education mandatory and soulless. Music class has become a study of music theory, and the students do not practice or listen to music themselves. They study the musician’s tools and shorthand without his art. Lockhart gives the “unifying aesthetic principle in mathematics… simple is beautiful.” He calls mathematics “the purest of the arts” where we are given free reign to play in an imaginary world. He reveals how mathematics depends upon natural curiosity and enjoyment of exploration. His description of the possible enjoyment of mathematics is excellent. His exposition of the flaws of route memorization and soul-crushing mechanistic exercises is superb. His argument seems water-tight. And then, he reveals his mathematical ontology. He believes we make up numbers in our head and that humans create the mathematical realm. As a result there is no objective truth and beauty to be appreciated by the mathematician. We will not grow our souls while doing math, but simply have fun (in a harmless way). His hedonistic approach to mathematics seemed weak. And I don’t think he really believes it himself (either that or his flowing praise of mathematics is simply self-referential). His view of human flourishing in mathematics (thriving as pattern making organic machines) is dismal and makes me not want to do mathematics. His voice is a welcome one against the stand for practical mathematics. He gives an excellent model for mathematics in which we explore ideas. But the lament falls (in my estimation) without a solid ontological foundation.