For This Day
Volume 5 Issue 2
FOR THIS DAY A Vision for Math Education at CDA
by Jessica Seekamp, CDA Math Department Head
As classical educators, you are well aware of imaginary worlds, and I’d like to invite you to step into one with me today. Imagine yourself as Lucy or Edmond, about to enter Narnia. However, instead of a wardrobe, the method of transport is your mind, perhaps with the aid of a pencil and paper. Imaginary creatures play and lurk here! Not nymphs and fauns, mind you, but points, equilateral triangles, immeasurable numbers, like the diagonal of a square with a side length of one, and other, stranger numbers that exist only at a right angle to the real dimension, like the square root of negative one. There are many reasons you might enter this imaginary world. You may find yourself here to answer a question of your own curiosity. Other times, a question will arise in “The Land of
“Imaginary creatures play and lurk here! Not nymphs and fauns… but points, equilateral triangles…and other stranger numbers…”
Spare Oom and the bright city of War Drobe," but it cannot be answered in that world, because we must play with the imaginary creatures in our mathematical Narnia to answer it. British Mathematician Marcus du Sautoy said, “Mathematics is a place where you can do things which you can't do in the real world.”
My vision is for our young mathematicians to make wonderful use of this imaginary world to answer questions and solve problems; even for it to be become a comfortable and familiar place, where they have fun and play creatively with these imaginary creatures. Some of them are friendly and easy to understand, but many don’t give up their secrets easily. We must develop our students into patient, persistent problem solvers, with a myriad of tools available in their back pocket to help them coax the truth out of the difficult characters. Mathematics takes hard work. The more you do it, the better you get at it, but you must be willing to engage intellectually with those creatures that often appear unfriendly or unwieldy. Getting to know them is best done over hours of time and a cup of
Coram Deo Academy
www.coramdeoacademy.org
1
For This Day
Volume 5 Issue 2
tea, rather than hurriedly on the way to practice or in 30 minutes while distracted by texts and tweets. Isn’t that true of most complex individuals, though? Could you really expect a complicated personality to reveal their soul and deepest, darkest secrets to you through a few hurried encounters? Our imaginary and complex mathematical friends are no different.
The first goal of mathematics education at CDA, then, is to train our students to be patient problem
“We must develop our students into patient, persistent problem solvers…”
solvers who understand that failed attempts are part of the process, not a sign that you stink at math! They employ a variety of approaches and ways of looking at a problem – if one method doesn’t work, they try another. They draw diagrams, introduce variables as needed, and continue both attempting to and believing
that they can arrive at a solution until, in fact, they do.
A second goal of the math department is for our students to know the creators of this mathematical Narnia. This begs the question, is mathematics created or discovered? I believe the answer is… yes. These are the discussions we should have with our students! How is it that mathematical equations can be used to describe the motion of a falling body, the orbit of the planets, the propagation of waves, the engineering in a spider web, and a host of other phenomena in the physical world? Is this mathematical world a creation of our minds, or a place that exists independently of us? Nancy Pearcy in “The Soul of Science” answers it this way,
“the coincidence between the natural world and the mathematical world is not any more mysterious than the coincidences between the natural world and the auditory, tactile, and olfactory worlds… mathematical insight is analogous to sight, hearing, or touch. All are means for perceiving some particular aspect of a multifaceted reality…Mathematical knowledge is possible because of a corresponding capacity created in us by God. We appreciate the beauty in a work of art because we are created with the capacity to experience the aesthetic aspect of creation. We recognize contradictions in an argument because of a capacity for experiencing the logical aspect of creation. We perform mathematical operations because of a capacity to perceive the numerical and spatial aspects of creation…On the one hand, we root our work in the world God has created, and, on the other hand, we freely exercise the creativity and imagination He has given us.”
This, my friends, is why we must never say or accept statements like “We just aren’t math people.” We are created in His image and have the capacity to understand and reason mathematically! Very often, I believe, it is lack of opportunity, persistence, time or a growth mindset that lead to poor mathematical understanding, rather than lack of ability.
But back to my second point – our students should know the stories of the men and women who have helped to shape this mathematical world. Why were logarithms invented? Was it to torment
Coram Deo Academy
www.coramdeoacademy.org
2
For This Day
Volume 5 Issue 2
students with a bizarre new mathematical function, or in fact a sacrificial work done by John Napier (1550-1617) as he devoted 20 years of his life to creating mathematical tables which aided the work of scientists and astronomers? French mathematician Pierre-Simon Laplace later said that “Logarithms, by shortening the labors, doubled the life of the astronomer.” In fact, evidence points to the fact that the German mathematician, Johannes Kepler’s third law of planetary motion would never have been discovered without the aid of logarithms.
Our students will have far greater appreciation for mathematics if they see it as a growing body of knowledge, integral to the story of humanity. They need to be told that the scientists of the 17th century were motivated by a love for God and a desire to understand His creation! Many of these scientists and mathematicians believed that an ordered God created an ordered universe and gave us minds capable of understanding it! Rather than spirituality being at odds with math and science, it spurred scientific progress.
From the mouth of Isaac Newton,
“This most beautiful system of the sun, planets, and
“Mathematics takes hard work. The more you do it, the better you get at it…”
comets, could only proceed from the counsel and dominion of an intelligent Being...All variety of created objects which represent order and life in the universe could happen only by the willful reasoning of its original Creator, whom I call the Lord God.”
I could go on, and tell you of Kepler, whose discoveries often caused him to break into doxology and praise to God, or we could discuss the Greeks, including my favorite – Archimedes, arguably the most creative and genius mathematician of all time, a thinker thousands of years ahead of his day. The Greeks came “this close” to developing Calculus, but due to their superstition regarding things infinite, they just missed the boat. It would be almost 2,000 years before the torch would be passed to Newton and Leibniz. There are other fascinating questions of mathematical antiquity to consider. Where did irrational numbers come from? Those strange number-creatures, with decimals that never repeat or end? The Greeks, again, desiring for everything to be measurable, thought that surely they should be able to measure the exact length of the diagonal of a square with a side length one, but finally, begrudgingly, had to admit that this was impossible, excepts in the land of mathematical reality. Failures, iterations upon iterations, and the refining of ideas by many minds have brought the body of mathematical knowledge to where it is today. Our students need to understand that while they might be struggling to understand a concept, it took humanity a thousand years to accept it – so their struggle is ok! Theorems didn’t just fall out of the sky and land Coram Deo Academy
www.coramdeoacademy.org
3
For This Day
Volume 5 Issue 2
in math books. Mathematicians often struggle for weeks or years before finding the answers they seek, sometimes never finding success. Even Albert Einstein said, “Do not worry about your difficulties in Mathematics; I can assure you that mine are still
“A mathematician, like a painter or poet, is a maker of patterns.” -G.H. HARDY
greater.” Let us resolve to learn the history of our discipline! Only then will we be able to share the fascinating stories about the creation of this Mathematical world with our students, including the people, philosophies, and world views that shaped its formation.
Third, math education at Coram Deo should honor the discipline of
mathematics. The experience our students have in math class and in the tasks we have them do ought to resemble what mathematicians do: posing questions, finding patterns, making conjectures, defending their work, and collaborating with colleagues. G. H. Hardy in his book “A Mathematician’s Apology, says, 'A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.’” The art of connecting ideas and concepts is key. One of my favorite math authors, Paul Lockhart, says in “A Mathematician’s Lament”,
“Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity – to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs – you deny them mathematics itself.”
So how can we do better? We must stop concentrating on the what, while leaving out the why. Find ways to open up tasks to allow for creativity through multiple approaches or representations; include inquiry when possible; ask a question or propose a problem that your students don’t yet have the tools to answer, and see what questions they ask! Have them illustrate their ideas; require them to convince, critique, and reason. As our students propose questions and explain their reasoning to one another, they will begin to realize that not all solutions are created equal. Messy and meandering, efficient and getting the job done, or beautiful and elegant – your students will notice the differences and begin to engage in mathematical critique, an important element of any art, including mathematics. G. H. Hardy went on to say that:
“The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
The mathematical process is to notice patterns and pose questions, enter Mathematical Narnia through the wardrobe of the mind, play with these mathematical creations until they give up their secrets, and then come back to this world and answer the question originally posed.
Coram Deo Academy
www.coramdeoacademy.org
4
For This Day
Volume 5 Issue 2
I realize this may sound like a daunting task, but take it one small step at a time. First resolve to work on your own understanding of mathematics. Set high expectations for your students and require them to be patient problem solvers. Don’t bail them out at the first sign of struggle; help them exhaust their options as they tackle a problem. Study the stories and the history of mathematics, so our students will appreciate the beautiful, God-given human endeavor that mathematics is! Teach math in a way that honors the discipline. We are not just training students to be efficient calculators and swift algorithm executors, but ethical leaders and wise thinkers. Culture will be shaped for God’s glory as our students learn to “think God’s thoughts after Him,” including in the realm of Mathematics.
Mrs. Seekamp and family
_____________________________________
Be anxious for nothing, but in everything by prayer and supplication, with thanksgiving, let your requests be made known to God; and the peace of God, which surpasses all understanding, will guard your hearts and minds through Christ Jesus. Philippians 4:6-7
Coram Deo Academy
www.coramdeoacademy.org
5