Posts Tagged ‘math’

What We Study and Why: Mathematics

Dear Reader,

This is part of an ongoing series in search of a reformed philosophy theology of education. You can find all the posts here.

Last time, we wrapped up the section of this series on practical details. You can find that summary post here. Today I’d like to begin a new sub-series on individual subjects. I have argued that the teacher’s attitude is paramount and so a large part of what we are doing here is just to frame each subject rightly. Whether you are a homeschooling parent or employed in a school setting, you may find yourself having to teach subjects that just don’t thrill you (what on earth does grammar have to do with the kingdom of God?). While we will touch on some practical details as well (why teach pagan myths? does everyone need calculus?), the main goal of this part of the series is just to show why we teach each subject.

There are a couple of big ideas behind what we are doing here, including: All truth is God’s truth; In education we lay before our students the things of God, primarily His general revelation which comes to us in many forms; and The purpose of education in the life of the believer is for the transforming of his (fallen) mind. (If you are just dropping in, I do recommend reading some of what has come before; see this summary post on the theory behind it all.)

With these goals and ideas in mind, we will ask for each of the subjects we address: Why do we study it? How does it point is to God? How does God reveal Himself or His truth through this subject? In answering these questions, we will look at Scripture whenever possible but we will also look at quotes from many other sources.

Finding God in Mathematics

Let’s jump right in then to mathematics. Most would agree that some level of math instruction is necessary. Beyond the basics, there tend to be two camps — those who see no need to go beyond the basics and those who find pleasure and meaning in higher mathematics. The problem is that there is a gap — we don’t convey the beauty of math when we are teaching the basics and so those who do not naturally enjoy it drop it as soon as possible and never get to the part where it seems to expand and take on a wider significance. The solution is to show that math is lovely even at the lower levels (that’s where the teacher’s attitude comes in again). So if you have lost the joy of math, or never had it, here are some quotes to inspire you:

The laws of mathematics point us to the Law of God:

“We take strong ground when we appeal to the beauty and truth of Mathematics; that, as Ruskin points out, two and two make four and cannot conceivably make five, is an inevitable law. It is a great thing to be brought into the presence of a law, of a whole system of laws, that exist without our concurrence,––that two straight lines cannot enclose a space is a fact which we can perceive, state, and act upon but cannot in any wise alter, should give to children the sense of limitation which is wholesome for all of us, and inspire that sursum corda which we should hear in all natural law.” (Charlotte Mason, Towards a Philosophy of Education, pp. 230-31)

Mathematics conveys eternity:

“But education should be a science of proportion, and any one subject that assumes undue importance does so at the expense of other subjects which a child’s mind should deal with. Arithmetic, Mathematics, are exceedingly easy to examine upon and so long as education is regulated by examinations so long shall we have teaching, directed not to awaken a sense of awe in contemplating a self-existing science, but rather to secure exactness and ingenuity in the treatment of problems.” (Ibid., p. 231; emphasis added)

Math underlies the universe. It may even be called the language of God:

“Mathematics is the language in which God has written the universe.”  —Galileo Galilei

Math is the foundation of many other fields, both sciences and arts. Its beauty can be seen even by non-Christian authors:

“Mathematical analysis and computer modeling are revealing to us that the shapes and processes we encounter in nature — the way that plants grow, the way that mountains erode or rivers flow, the way that snowflakes or islands achieve their shapes, the way that light plays on a surface, the way the milk folds and spins into your coffee as yo stir it, the way that laughter sweeps through a crowd of people — all these things in their seemingly magical complexity can be described by the interaction of mathematical processes that are, if anything, even more magical in their simplicity.

….

“The things by which our emotions can be moved — the shape of a flower or a Grecian urn, the way a baby grows, the way the wind brushes across your face, the way clouds move, their shapes, the way light dances on water, or daffodils flutter in the breeze, the way in which the person you love moves their head, the way their hair follows that movement, the curve described by the dying fall of the last chord of a piece of music — all these things can be described by the complex flow of numbers.

“That’s not a reduction of it, that’s the beauty of it.” [Douglas Adams, Dirk Gently’s Holistic Detective Agency (New York: Pocket Books, 1988) pp. 182, 184]

That’s all fine, you say, I am inspired but I am still teaching long division to cranky eight-year-olds. A couple of thoughts: I argued recently that when educating we must be careful not to provoke children. Math is a field in which it is very easy to provoke. It tends to come with a lot of repetition. I do think we should all learn to do long division without a calculator. But if I have ten such problems to do, I get my calculator. Why should we ask a second grader to do so many at once? Sometimes more is less (how’s that for a math concept?).

There is a certain progression to math; one can’t do algebra before learning to count. But that doesn’t mean the beauty of math needs to wait until high school or beyond. There are resources which are accessible at younger ages but which either introduce concepts usually reserved for later or give more of a big picture understanding of math, bringing out its complexity and elegance. (I will add a brief bibliography of some we have used at the end of this post.)

Lastly, there is the elephant in the room question: When will I ever use this? And its corollary (there’s a nice math word): Why do I need to learn calculus anyway? As for the first question, I reject the premise. Our approach to education is not utilitarian. Whether we will use upper level math has nothing to do with anything. The end we have in view is not the balancing of checkbooks or even being able to do advanced physics (for which I hear math is useful) but to bring glory to God which we do by learning about Him as He has revealed Himself through creation, and (as the quotes above are meant to show) mathematics is an integral part of that creation.

As for the second question, not everyone needs to learn calculus. We are finite people and time and energy spent on one subject come at the expense of another. So while I do think it is good to learn these things, beyond a certain point we must recognize that we are different — indeed unique, individual — people and that we don’t all have to learn the same things (see this post on core curriculum). So perhaps you don’t have to learn calculus.

I’d like to end with a plea — as I work on this section of the series, I am giving you my best ideas and resources but I could use some help. Please reply to this post or contact me if you can help with any of the following:

  • What questions do you have about teaching (insert subject here)?
  • Do you have good quotes about math, or any other subject, that you have run across, particularly about why we teach them and how they point us to God and/or teach us about Him and His creation?
  • Any favorite resources? Since math was our topic this week, feel free to add in the comments your favorite big-picture math resources.

Nebby

A Brief Math Bibliography

Base Five by David Adler. Picture book.

One Grain of Rice by Demi. Things expand exponentially in this picture book based on a Chinese folk tale.

Librarian who Measured the Earth by Kathryn Lasky. Biography of Archimedes. Middle years.

Mummy Math by Neuschwander.

History of Counting by Denise Schmandt-Besserat. Upper elementary.

Too Many Mittens by Louis Slobodkin. A wonderful author. Elementary.

Life of Fred Math by Stanley Schmidt (Polka Dot Publishing) — You may have heard of this alternative math curriculum. It takes a narrative approach and follows the life of 5-year-old math professor Fred. Though the author says the elementary books can be used as a stand-alone math curriculum, I was always hesitant to do so. They do, however, make a lovely supplement to whatever else you may be using. The stories and such may be overly silly for some but my kids always loved them. The elementary series is a collection of thin books with short chapters. It is easy to add in one chapter a week. Ages 10 and up could breeze through them pretty quickly. The upside of these books is that they introduce concepts that usually don’t come up until later such as set theory.

Here’s Looking at Euclid by Alex Bellos

The Number Mysteries by Marcus du Sautoy

Thinking in Numbers by Daniel Tammet

These three books are all of a type. They are roughly middle school level books (and up) that have relatively short chapters which disuss math concepts like pi, prime numbers, and how people in Iceland count.  I am sure there are many other such books out there; these are just a few we have used.

 

 

 

Guest Post: Joseph’s Life as a Mathematical Function

For a change of pace, I have a guest blogger this week. Elijah is 15, homeschooled, and apparently has way too much time on his hands.

Joseph’s Life as a Mathematical Function

by Elijah Van Vlack


So a few weeks ago Matt asked me the question of which mathematical function best fits Joseph’s life (Genesis 37-50). After a short discussion we came to the conclusion it was either the sine function or the cosine function. I was thinking later and I decided a polynomial function might actually fit better. After a little trial and error I came up with this function.

(X/10-1.6)^7 – 5(X/10-1.6)^5 + 7(X/10-1.6)^3 – 2.3(X/10-1.6) = Y

I know most if not all of you cannot imagine that so I made a graph for those of you who want the easy path.

graph1

Joseph’s life starts at 0 on the X axis, and the X axis continues to represent his age. (Kinda not really close to scale.)

The Y axis represents his favor among the people he regularly interacts with.

At his birth Joseph already had considerable favor with his father for just being the child of Rachel.  He rose after that to his first peak, but he was cast down by his brothers into a pit and sold into slavery. That is represented by the dip around 18 on the X axis.

His next high around 40 is in Potiphar’s house when he becomes next to Potiphar.  However his amazing good looks cause his next dip when Potiphar’s wife falsely accuses  him and he is thrown in jail. This dip is around 62 on the X axis.

Joseph then attains great favor when he interprets the baker’s and cup bearer’s dreams, along with being promoted to chief prisoner. But when his God-sent gift fails to get him out of prison for so long his faith in God (probably) begins to wane and his fellow prisoners doubt his God will ever save him.

However, in a dramatic turn of events God preserves Joseph and takes him out of prison and sets him as the next best man to Pharaoh. This final high starts around 108 on the X axis. He then goes into an unending high which ends with infinity in the Y axis representing his eternal favor with God when he joins his maker in Heaven.

To prove it really does go to infinity I made a second graph with a much larger scale.

graph2

As can be clearly seen from that graph, when Joseph goes to be with God it infinitely out-weighs all the troubles of this life. I haven’t come up with an explanation for why the line comes from negative infinity, so I cropped that part out.

It isn’t a perfect graph of his life, but I believe it proves that a polynomial function fits his life best.  It clearly fits better than our previous ideas of sin X = Y and cos X = Y, both of which are in the below graphs. I even included sin (X+0.5) = Y, which also had the possibility of being a better graph than sine and cosine.

graph3

In the graph above (sin X = Y) Joseph’s life starts at (0,0). However, in this graph he never reaches infinity with God. Also, every one of his highs is as tall as all the others.

graph4

The cosine curve (above) might fit a little better. It has Joseph’s life start at (0,1)  and he immediately goes down from there. It is probably a better fit than the sine in the fact that he is born with favor with his father, however it fails in the other things sine failed in.

graph5

I believe this graph, sin(X+0.5) = Y, fits best of the three trigonometric functions. In this graph Joseph starts with favor, but he gains even more favor before his brothers sell him. However, he still never reaches infinity and he would supposedly live forever. I think I have demonstrated that a polynomial function fits better than any of the trigonometric functions.

Thank you for bearing with me.

Why Study Math?

Dear Reader,

Do you ever notice that no one asks this? We might ask why about some of the higher maths like trig and calculus but we don’t ask why study math at all like we might for art or music or even history. It’s kind of a pet peeve of mine that the STEM subjects, as they call them, (STEM stands for Science, Technology, Engineering and Math) are so emphasized while others are neglected. But we never ask why we study math at all. It’s always good to consider these things though and in the section for this week’s Charlotte Mason Blog Carnival, Charlotte invites us to do just that.

Charlotte is arguing, as she often does, against certain ideas prevalent in her day. The big one here seems to be that studying math with train certain faculties in the child’s mind, will cause them to exist even. Charlotte, believing as she does in the personhood of each child, rejects the idea that we produce any such faculties in our students. She believes that the powers of logic and reasoning which they need are already in them. Nor does she seem to believe that these powers need to be trained particularly by us. She questions also whether the logical training of mathematics actually carries over to any other area of life (an idea which has also been floated in our own day).

So why study math? The answer, to Charlotte, is because of its own inherent beauty:

“We take strong ground when we appeal to the beauty and truth of Mathematics; that, as Ruskin points out, two and two make four and cannot conceivably make five, is an inevitable law. It is a great thing to be brought into the presence of a law, of a whole system of laws, that exist without our concurrence,––that two straight lines cannot enclose a space is a fact which we can perceive, state, and act upon but cannot in any wise alter, should give to children the sense of limitation which is wholesome for all of us, and inspire that sursum corda which we should hear in all natural law.” (pp. 230-31)

In our own day, I think we have lost this idea. We do really emphasize math (along with the other STEM subjects) but our goal is just to get ahead of other countries. We don’t study math for its own sake nor do we talk about its beauty or how it shows us the constancy and absoluteness of its laws. Charlotte discusses math under the heading “Knowledge of the Universe” which it is but knowledge of our universe has its greatest value in that it points us to the Creator of the Universe and tells us something of His character. This, then, is the true value of math and the best reason for studying it.

One final note, while math is important, Charlotte cautions us against letting it become too important. It is, she says, very easy to test and this fact tends to make it assume greater proportions than it should have:

“But education should be a science of proportion, and any one subject that assumes undue importance does so at the expense of other subjects which a child’s mind should deal with. Arithmetic, Mathematics, are exceedingly easy to examine upon and so long as education is regulated by examinations so long shall we have teaching, directed not to awaken a sense of awe in contemplating a self-existing science, but rather to secure exactness and ingenuity in the treatment of problems.” (p. 231)

This again is a warning that we need to hear.

Charlotte sums it all up very well:

“To sum up, Mathematics are a necessary part of every man’s education; they must be taught by those who know; but they may not engross the time and attention of the scholar in such wise as to shut out any of the score of ‘subjects,’ a knowledge of which is his natural right.” (p. 233)

Nebby