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 to 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 langauge 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

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

Our School Year: Math

Dear Reader,

As September nears, I would like to begin posting about our homeschool plans for the year. Math seems like a fairly easy place to start, and we actually did begin it this week.

We have in the past used both Math-U-See and LIfe of Fred (see this earlier post on LOF) and will continue to use both this year, though in a slightly different way. I have one child still in what might be considered the “elementary years”, at least as far as LOF is concerned. She is on MUS Delta which teaches division. Since we went through the LOF elementary books last year, the plan this year is for her to do the intermediate books. These will be done once a week or so as a read aloud with me. I expect the other children may try and listen in too. I know they say LOF can be a stand-alone curriculum for the early years, but I have never been quite confident enough to do this so MUS will still be her primary curriculum. A note on how we use MUS: I don’t tend to make my kids do every problem. If they get A right, they can usually skip B and C. The D page of a lesson usually introduces more review material so I will make them do that, and depending on how they do, parts of E and F too. Typically, toward the end of Delta, when one gets to real long division, I have children skip problems just because it is too many to do at once without one’s brain going googly. I’d rather have them do a couple of problems well than stumble through half a dozen.

My 11-year-old daughter is on algebra this year, I am also having her use MUS as her primary math. Though she is very verbal, the most of any of my kids, she doesn’t seem to get math concepts well from Fred’s narrative style so I would rather have her do MUS first. I am hoping there is time to then read Fred’s algebra book too, even if we don’t do all the problems.

My eldest, who is now 13, is doing LOF Advanced Algebra (note that LOF does Algebra 2 first then geometry, while MUS reverses this order). He has always taken naturally to math and finds it easy to learn from Fred. He can work through LOF on his own though I have told him if he misses problems and doesn’t know why, he should let us know. My plan is for him to use the MUS test booklet after he has finished Fred as a way to double-check that he has no gaps. The test booklet gives one test per chapter and if there are ones on which he seems to have missed something, we can then review that chapter in MUS without him having to do the whole curriculum.

For my almost 10-year-old, the plan is similar. We are starting with MUS and will use LOF to review and fill in any gaps. He is on fractions, which is the first level for which LOF has a thicker volume which seems like it could stand on its own as a curriculum (I know they say the elementary books can but I am just not there yet). I am not as confident in his abilities to read Fred and get the concepts but I figured I would give him a shot at it and see how he does. We can always switch back to MUS as his primary curriculum.

One more thing, I had been having ym oldest two read through First Lessons in Geometry and they will ocntinue with that, reading and answering questions orally about it.

And that is our math plan for this year.

Nebby

Is Algebra (or Grammar or Science or . . . ) Necessary?

Dear Reader,

Have you seen any of the many articles coming out lately on algebra and whether our kids really need to learn it? It started with a New York Times op-ed by Andrew Hacker saying that it was not necessary for most of to learn higher math (algebra and beyond). Of course this set off a storm of counter-articles and counter-counter-articles.

Hacker’s basic argument is two-fold: 1) algebra is too hard and leads otherwise decent students to drop out or underachieve and 2) most of us don’t need those kinds of math skills. The main counter argument (see this Washington Post article by Daniel Willingham) seems to be that even if the specific skills aren’t used, higher math is still valuable for how they teach people to reason and think. Willingham contends that “the mathematics learned in school, even if seldom applied directly, makes students better able to learn new quantitative skills.” He also implies that the problem is not unique to math:

“The difficulty students have in applying math to everyday problems they encounter is not particular to math. Transfer is hard. New learning tends to cling to the examples used to explain the concept. That’s as true of literary forms, scientific method, and techniques of historical analysis as it is of mathematical formulas.”

Then come the counter-counter-articles, like this one by Roger Schank. The essence of his argument is that the thinking involved in higher math just does not carry over into other areas of life. As the daughter and sister of mathematicians, I can fully relate to his observation that their lives do not demonstrate higher levels of reasoning than other peoples:

“Are mathematicians the best thinkers you know? I know plenty of them who can’t
handle their own lives very well.”

You’d be amazed by the soap-opera-esque scandals that go on in your local university math department.

Finally, we have this post, also by Roger Schank, arguing that almost all of what out students learn in high school is a waste of time.  Here perhaps there is some agreement with Willingham in that the problem is not math alone but rather it is an issue of whether any school learning is able to carry over into “real life.”

So what is the verdict? Is this all a waste of time? And, as homeschoolers, are there ways we can teach our kids that are not a waste of time?

Personally, I have used basic algebra (0ne or two variable equations, no exponents involved) when altering knitting patterns. But I don’t really see where my high school chemistry has benefited me at all. Schank debunks any foreign langauge learning done in schools though I did muddle through with my high school Spanish when I needed to get my husband medications in a pharmacy in Chile. So I guess from my own personal experience, I would say a lot of my high school time was a giant waste, but not all. Some bits here and there have been useful. Though my overall feeling is that I wish I had gotten a lot more education out of my high school. If my history studies were not beneficial, for instance, it is because they were so lacking. I do think it is valuable to know history but one has to know enough of it to see the patterns and the big flow of events. My education was too piecemeal to do any good.

Then to, especially when I think of educating my own children, I think it comes down to a matter of what one’s goals are also. If our goals are those of the public schools, to produce people who are able to go to college and/or get jobs, then a lot of what we do probably is wasted effort. It does not immediately advance the goal. But if our goals are broader, if we care more about who the student is, if we are trying to develop well-rounded, whole people, then a broader, more complete education is in order.

Both Willingham and Schank say that none of our school knowledge carries over well into the rest of life (the one thing they agree upon perhaps). But I think this is why Charlotte Mason (you didn’t think we’d get through a post without me bringing her up, did you?) approaches things differently. She sees the goal of education as allowing the child to form relations with the largest possible number of areas. “Education is the science of relations” is her motto, and her goal is to set students’ feet in a large room, to open up the world to them through these relations. If the problem is being able to apply knowledge from one context to another, then Charlotte begins work on this early on by encouraging children to make connections between diverse things.

So perhaps what we need is not to throw away all of high school but to change show we teach. We have become too utilitarian, and it is limiting us. We need to rethink our goals and to expand our vision.

Nebby

Life of Fred Math

Dear Reader,

I mentioned in my post on our homeschool plans that we have been using Life of Fred math and that I tend to get a lot of questions about it. So I thought I would do a post on it to share what our experience has been.

For those of you who haven’t yet heard, Life Of Fred (henceforth LOF) is a math curriculum that teaches through a narrative format. Basically, it  is the story of 5-year-old Fred who is a genius and is a math professor at KITTENS University in Kansas. You read a chapter about Fred’s life in which he uses math in (somewhat) real life situations and then at the end of each chapter is “Your turn to play,” section of questions and answers to be done by the student. There are two series. One is the more advanced series which begins with fractions, decimals and percents, and then pre-algebra and so on. This is the original series. Then more recently they have come out with the elementary series which is I believe 10 books titled in alphabetical order: Apples, Butterflies,  . . .up to Jellybeans. My oldest two children began in Fractions. The oldest is not in Algebra and the second is in pre-Algebra. My younger two are working through the elementary series and are in Goldfish now.

LOF says it can be a stand alone math curriculum though most people seem to use it as a supplement to their main curriculum. That is how we began. Now I would say we are using it alongside our other curriculum which is Math-U-See. The little two do both everyday. The older two can choose which one they do and usually alternate weeks.

The elementary series begins by teaching addition facts. My memory though is that it does not really cover the most basic ones like 1 and 2 pluses. So I would want a child to have some introduction to math before beginning it. They should know their numbers, including multiple digit ones and have some idea of adding and the simplest addition facts. I have been reading the series with my children and I would think this is the way to go with the elementary series. I have them write down the answers to the end of chapter questions and then we go over them. There are practice problems as one gets farther along but they are not many, maybe 5 or 6 at the end of a chapter and not in every chapter. So if your child needs more practice, you would have to supplement. When we started, my children knew how to add and subtract so we went through the first books quite quickly. We initially did 3 chapters in a sitting, then slowed down to 2 and now one. They love LOF. They always look forward to it, and there is no complaining. My 12-year-old read the elementary books on his own and still tries to finish his own work in time so he can come listen to me read it again to the little ones. LOF teaches many concepts which other math curricula leave out or save till much later such as sets. So I would always begin with Apples if you are doing the elementary series. If your child is up to fractions (meaning they know all the basic functions through division; this is the same order MUS takes the topics in though other curricula will differ), you could skip the elementary series all together.

The later series (which really came out first) has longer books which are designed to take a full year. They are include extra “bridge” problems every 5 chapters or so to make sure kids are getting th concepts. Pre-Algebra is actually two volumes, pre-algebra with biology and pre-algebra with economics. All Fred books dabble in other subjects from poetry to science, but these two are deliberate about it. One is supposed to do both. This would make pre-algebra two years long often since the books do get thicker and thicker. I have heard complaints that the economics book has a certain perspective which some parents find objectionable. So one might want to look into that before doing it, especially if you are on the more liberal end of the spectrum. I don’t think there is anything too controversial in the biology book but all the books do have  a very mild Christian bent. For example, there might be a passing mention of God as Creator.

When one gets to Algebra, there are other supplemental books with more problems and a plan for breaking the work down into chunks. Even though my son is in algebra, we have chosen so far to just stick with the main book. If he seems to need more, we may use the extras. But of course he is also doing MUS so he gets more problems there.

There is no question in our house that LOF is interesting and fun. The big question I think everyone has is will my kids learn math from it? I think that depends on your child. Many may need additional problems (though I think most curricula give far too many). My oldest has had no problems learning math concepts from Fred. He is a math guy naturally. My second is more verbal and artistic. You might think she would do well with this narrative form of math, but she finds it much harder to learn the concepts from LOF. For her, I think she needs to learn it and then use LOF for reinforcement and a different perspective. She likes me to read the chapters with her most of the time though sometimes she will do it on her own.

Those are my thought this far on LOF. Any questions? Anything I forgot to cover?

Nebby

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