Math from a Christian Perspective

Could you take a moment to take a look at this new proposed book cover for Beyond Numbers? I would greatly value your opinion.

Here’s the current cover.

And here’s the redesign we’re considering (note that the watermark in the background is just temporary).

I’d really love to hear your thoughts about the proposed new cover. What does it make you think of/what does it say to you? What do you/don’t you like about it?

Note: If the covers are not showing up, please click here to view them.

Cornerstone Curriculum, publishers of the Making Math Meaningful curriculum series, is offering a rough draft of the first several hundred pages of their geometry course for free online. Based on a quick look at the course, it seemed to present geometry as a useful tool. I have looked at some of the author’s other resources and know he strives to help students really understand the concepts he presents and not merely memorize formulas.

Here’s a quote from one of the opening pages:

Geometry is all about measuring lines, angles, surfaces, solids, velocities and their interrelationships. In this study, you will act as a consultant, designer-planner, and builder. The projects will range from designing a tree fort in your back yard to planning the construction of a sidewalk and home on the hilly streets of San Francisco to charting the path of the earth around the sun. In the process you will learn the principles as well as the vast usage of geometry in everyday life. Geometry is used by graphic animators, artists, photographers, interior designers, engineers, architects, builders, construction teams, surveyors and doctors just to name a few.

The draft copy online does not contain answers to the problems, nor is it an entire course, but you could certainly use some of the application ideas or concept presentations from the free download. If you do, I’d love to hear how you liked it–as I’m sure would the author.

Greetings! I hope you all are having a wonderful and blessed summer!

I thought you might enjoy this interview I did a few weeks ago on HSLDA’s Homeschool Spotlight. It offers an overview of viewing and teaching math from a biblical worldview.

Spotlight on Math – Homeschool Spotlight with HSLDA from Paul Bass on Vimeo.

Spring has arrived! The azaleas are in full bloom, reminding me it’s time to head outside and do some planting.

While working in the garden, you have a wonderful opportunity to have your child apply those math concepts he’s been learning! Graphing, multiplication, addition, measurement, area, perimeter–these and other concepts prove helpful in the garden. Why? Because math is a way of describing God’s creation! We can use math in real life because God created man with the ability to “subdue the earth,” and because He faithfully holds all things together in a consistent fashion.

Kidsgardening.org offers an entire section filled with ideas for teaching math while gardening. While written for school teachers, most of the ideas can easily be adapted to a home setting. Be sure to take a look at page 2, which offers a variety of simple ideas on how different concepts apply (you may want to print this page to consult throughout the summer). The other pages offer more detailed ideas.

The suggestions offered, as well as others you might think of, could be used or adapted for a wide variety of ages–a young child can help you count the number of earthworms in a section of soil (or the number of seed packets you need/the number of seeds to plant in each hole, etc.), while an older one can start a plant business or calculate the surface area of leaves.

Don’t let the ideas limit you. Even if you aren’t planning on growing a vegetable garden, you can still have your child grow some plants indoors or on the patio and measure their growth–or design a pretend garden on paper.

Speaking of designing on paper, you may want to use graph paper, letting each square represent a foot. The graph paper should aid in visually seeing how much area each plant needs.

Lastly, here’s a page with handy math formulas. It explains how to find the area of your yard, the amount of mulch and fertilizer you’ll need to cover it, and more.

Hope you have fun using math in the garden! Please let me know how your gardening applications go.

Traveling…and math? Most of us don’t couple the two in our minds, yet a lot of math goes into getting from one place to another. Here are just a few examples, along with some links to resources you can use with your children.

Math is used to help design airplanes, to find a plane’s capacity, to schedule flights, and more. See www.planemath.com for some engaging, interactive activities illustrating some of math’s aeronautical uses. NASA also has some more activities and lessons on aerospace topics, many of which show some of the math involved.

Whether your travels involve air or auto travel, this page titled “Mathematics on the Go” offers some simple ideas of how you can illustrate math’s usefulness while traveling. Some of the ideas would make great family games on your next car trip!

Math’s applications in traveling is just one example of how math serves as a real-life tool–a tool we can use because of God’s faithfulness in consistently holding all things together.

While browsing the Internet today, I came across some fascinating videos that connected upper-level math concepts with real-life applications in an engaging, easy-to-understand way. The videos explore such varied examples as making fireworks and oil production–along with MUCH more!

The series are secular series, and I do not agree with some of the examples chosen and ideas presented in the videos, but they do contain very clear, helpful examples of math in action if you discern through some of the conclusions. It might be wise to discuss them together with your students afterward, and explore together what the Bible says about the various topics (the environment, health, etc.)

If you’re interested in the videos, I would suggest watching them soon, as the site dropped another video series on math they used to have, and purchasing the DVDs are VERY expensive. I think they rotate the content periodically.

Anyway, here are the links!

Algebra in Simplest Terms
Against All Odds: Inside Statistics

In my previous post on shapes, we briefly talked about how learning shapes doesn’t have to be confined to a textbook–how shapes help us understand and appreciate the shapes God placed around us.

Understanding how shapes respond to pressure–as well as lots of other math concepts–plays an important role in building bridges. Here are two bridge-building resources you could use with your children as a way to teach them to use math as a God-given, real-life tool.

Golden Gate Bridge – This section of the Golden Gate Bridge site offers lots of useful bridge-making links. There are links you could use with younger children, as well as ones for high schoolers.

Build a Bridge – While this interactive page doesn’t get into much of the math behind building bridges, it gives students an opportunity to explore the properties of different bridge designs and determine which design would be best suited for various situations.

Someone recently wrote and asked me if I had any information on pi from a Christian perspective I could share. So here are some thoughts on this mind-boggling–and incredibly useful–number.

What Is Pi?

Pi, symbolized π, is “A transcendental number, approximately 3.14159, representing the ratio of the circumference to the diameter of a circle and appearing as a constant in a wide range of mathematical problems.”[1]

As this definition explains, pi is a transcendental–a number that keeps going on and on. To get a better feel, take a quick look at the first 100,000 digits of pi–it’s a mind-boggling number!

Numbers such as pi defy our comprehension. As I mention in Revealing Arithmetic when looking at different types of numbers,

“The infinite nature of numbers reminds us of our limited knowledge. As James D. Nickel points out, ‘The infinite nature of the natural numbers has a way of telling man’s reason, ‘Under certain conditions, you can never know everything there is to know about me.’” Although our understanding is finite, God’s understanding is infinite. Psalm 147:5 tells us, ‘Great is our Lord, and of great power; his understanding is infinite.’ How foolish it would be not to trust Him!”[2]

When we look at pi, our minds should turn in awe and wonder at God’s greatness! Sadly, though, there “is almost a cultlike following that has arisen over the concept of π.”[3] It’s a reminder of Romans 1:20-23:

“For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse: Because that, when they knew God, they glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened. Professing themselves to be wise, they became fools, And changed the glory of the uncorruptible God into an image made like to corruptible man, and to birds, and fourfooted beasts, and creeping things.”

Has Pi Always Been Expressed the Way It Is Today?

Hardly! The symbol π is just a symbol man chose to help express that real-life ratio–the symbol has actually been used to mean other things! Like most math symbols, it has been adopted within the last several hundred years [4].

Throughout history, men have tried to more precisely define pi. I found it fascinating to read in π: A Biography of the World’s Most Mysterious Number that, when you really dig into the text, it appears the Bible accurately uses pi to the fourth decimal place (1 Kings 7:23)–the book concluded that ”such accuracy is quite astonishing for ancient times.” [5] Not surprising considering the Bible’s Author!

Where Do We Use Pi?

Pi has a way of showing up all over the place–a testimony to the same Creator holding all things together. The most obvious use is when dealing with circles (for example, pi is used to find the area and circumference of a circle), but pi also proves useful in less-obvious places, such as in sound waves, general relativity, movements of the heavens, and probability, to name a few.

Why Are We Able to Explore Pi?

Because God created man in His image and gave him the ability to explore His creation! We’re thus accountable to Him for how we use that ability.

Where Can I Learn More?

There are lots of materials online that share more about pi (you might start with Wikipedia’s overview or this historical overview of pi). Your library may also have some books that could prove helpful. One I particularly enjoyed is π: A Biography of the World’s Most Mysterious Number by Alfred S. Posamentier and Ingmar Lehmanne.

The thing to keep in mind is to turn the wonder at pi itself into wonder at the Creator of all things–the one who understands what only baffles our comprehension.

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[1] The American Heritage Dictionary of the English Language, New College Edition, 1980, s.v. “pi.”

[2] Loop, Katherine, Revealing Arithmetic: Math Concepts from a Biblical Worldview (Fairfax, VA: Christian Perspective, 2009), p. 125. Internal quote from James D. Nickel, Rudiments of Arithmetic: Foundational Principles in the Computation and Theory of Numbers, 1st ed. (preliminary draft) (U.S.: James D. Nickel, 2008), p. 294.

[3] Posamentier, Alfred S. and Ingmar Lehmann, π: A Biography of the World’s Most Mysterious Number (Amherst, New York: Prometheus Books, 2004), p. 13.

[4] It was “introduced in 1706.” Cajori, Florian, A History of Mathematical Notations: Two Volumes Bound As One (Mineola, NY: Dover Publications, 1993), vol. 2, p. 9.

[5] Posamentier, Alfred S. and Ingmar Lehmann, π: A Biography of the World’s Most Mysterious Number (Amherst, New York: Prometheus Books, 2004), p. 28.

Throughout history, men have used their knowledge of shapes to help them design buildings. Because of the consistent way God holds things together, we can predict how different shapes will hold up under pressure. One shape that supports weight well is the arch. You can easily see this by holding a piece of paper flat between your hands and having someone push down gently in the center of the paper. You should notice the paper bends easily under the pressure. But if you bend the paper to form an arch, you’ll notice the paper does not bend as easily. An arch shape holds up better under pressure than a flat shape does. [1]

Knowing this quality about arches helps us in designing buildings and bridges, many of which have an arch shape! It also gives us new appreciation for the design in our feet. If you run your finger along the bottom of your foot, you will feel multiple arches on your foot! God, the master engineer, designed the shape of our feet to support our body’s weight. Our feet are truly marvels of engineering!

If the foot were flat and rigid, fixed at right angles to the bone of the leg, walking would be difficult or impossible. The elastic arches also serve as shock absorbers to soften the jar resulting from walking on a hard surface.

The human foot is a miniature suspension bridge which is much more complicated than an ordinary bridge. Would anyone say that the Golden Gate suspension bridge just happened? Of course not, if he were truthful! But why do people assume that the even more intricate mechanism of the human foot could have just happened without intelligent cause or the workmanship of a master Engineer? [Allen L. Gillen, Body by Design (Green Forest, AZ: Master Books, 2001), pp. 43-44.]

The point? Learning about shapes doesn’t have to be confined to a textbook! As you teach your child shapes, you can be teaching him about the shapes all around us–and seeing the Creator’s wisdom and care in how He chose just the right shape for everything.

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[1] This experiment is based on one given in The Art of Construction. The book offers numerous experiments and information related to building. Mario Salvadori, The Art of Construction: Projects and Principles for Beginning Engineers and Architects, 3rd ed. (Chicago: Chicago Review Press, 1990). Originally published as Building: The Fight Against Gravity.

It’s resource week!

SCORE Mathematics Lessons – Check this site for some free, practical lesson ideas for grades K-12. Many of these lessons offer ideas on how to have children apply math for a specific real-life project, such as planning a trip or buying a car. Although written for a classroom setting, most of the lessons could easily be adapted to the home. It’s a site you may wish to bookmark and consult when you’re not sure how a specific concept serves as a useful tool or need an idea on how to actually have your child apply a concept. Please use your own discernment, as many of the lessons include Internet usage.

REMINDER! Next week is request week. Does anyone have a specific concept or question you’d like discussed?

Known as the discoverer of the laws of planetary motion, Johannes Kepler was the first to propose that the planets circle the sun in elliptical shapes rather than in circular shapes as previously thought. Although often thought of as a scientist, Kepler was also a mathematician. In his study of planetary motion, Kepler used extensive math, including definitions, geometry, trigonometry, algebra, and other math concepts.

This mathematician’s life both provides an example of math in action and is resplendent with lessons! Join me in taking a brief look.

God’s Plans Are Not Always Ours
Johannes Kepler’s life illustrates the important truth that God’s plans are much better than are own–and sometimes they surprise us! Johannes Kepler did not plan on becoming a mathematician–he set out to become a minister. But toward the end of his university studies, his professors recommended him for a math position.

The young minister-to-be didn’t like the idea of giving up his divinity studies. Although he eventually agreed to take the position, Kepler still planned on becoming a minister one day. But God had something very different in mind for Kepler, as Kepler himself later recognized. [1]

Kepler had always been interested in the movement of the heavens and had admired Copernicus and his sun-centered theory. As a professor, Kepler now had more time to investigate these matters. He spent years developing a theory to explain the movements of the heavens, only to later discover his theory was insufficient. Undaunted, Kepler kept trying. His belief in the universe as an orderly creation of God made him certain the movement of the heavens could be explained by geometry. [2]

In 1600, Kepler’s teaching career at the school came to an abrupt halt. Along with others who refused to convert to Catholicism, Kepler was told to leave the country! Yet although Kepler probably could not see it at the time, God had a plan to transform persecution and exile into a tremendous blessing.

Exiled from his own country, Kepler soon found himself assisting (and depending on the generosity of) the famous astronomer Tycho Brahe (pictured to the left)–that is, until Brahe died in 1601. After Brahe’s death, Kepler inherited Brahe’s position and records. Because he had been exiled from his own land and forced to take shelter under Tycho Brahe, Kepler now had the records he needed to discover the laws by which God caused the planets to orbit the sun. Who would have thought God would use a persecution and forced exile to help Kepler accomplish his life work?

Perseverance in the Face of Obstacles
One of the biggest lessons we can learn from Kepler is that of perseverance. Discovering the planetary laws did not prove an easy task. From a collection of numbers Brahe had made over a period of many years chronicling where in the sky Mars had appeared, Kepler tried to find some sort of orderly law that could express the way God caused Mars to orbit the sun.

Just how hard was this task? According to Robert Wilson, “It took Kepler eight years and nearly a thousand pages of closely written calculations before he cracked the problem and discovered his first two laws of planetary motion (the third was to wait another nine years).”[3] Can you imagine spending eight years on a geometry problem you are not even sure can be solved, then another nine years to finish the task?

Confidence God Created an Orderly Universe and Math Could Describe It
Kepler’s willingness to persevere came from his deep faith that God had created an orderly universe. Kepler longed to uncover that order so he might bring glory to His Creator and know Him better.

Kepler was unwilling to accept the “close” results obtained from the prevailing Greek cosmology of the universe, in which planets circled the sun in circles. Instead, he searched for a better model.

Questioning the Greeks was a huge step. For centuries, the Greek philosophers’ teachings had been taught as fact. To question them was equivalent to questioning proven fact. Kepler could only be so daring because he believed in God as the source of truth, not the Greeks’ human reasoning. [4]

Kepler’s Beliefs – The Good and Bad
No one reading through Kepler’s Harmonies of the World can doubt Kepler’s belief in God. He often paused in the middle of an explanation to mention his Creator, and sometimes even broke off into a hymn of praise. It seems almost as if Kepler still viewed himself as a minister, trying to uncover the glory of God throughout creation. His book on planetary motion ends with this tribute to God:

Crying out with the royal Psalmist: Great is our Lord and great His virtue and of His wisdom there is no number: praise Him, ye heavens, praise Him, ye sun, moon, and planets, use every sense for perceiving, every tongue for declaring your Creator…To Him be praise, honour, and glory, wourld without end. Amen. [5]

We can learn a lot from Kepler’s use of math as a tool to uncover God’s handiwork.

At the same time, though, Kepler’s theology and outlook on math were far from perfect. He carried over a lot of Greek mysticism into his beliefs about God and the universe. Forgetting that creation and our minds are both fallen, Kepler often drew unbiblical spiritual parallels and inferences about God. Kepler also dabbled in astrology (although he admitted it held no weight) and brought a good deal of mystical thinking into his astronomy.

Lesson? We need to be on guard against falsehoods and lies from our culture that try to creep into our hearts.

Conclusion
Within Kepler’s life, we see God’s sovereignty at work, using even an exile to accomplish His purposes. We also find a challenge to persevere–and to view the universe as God’s handiwork and worship Him while using math to explore it. At the same time, we find a warning to be careful about falsehoods that threaten to rob us of living in the completeness of God’s truth.

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[1] Max Casper, one of Kepler’s biographers, says, “Looking back later when, through the discovery of his planet laws, he had become aware of his ability, he recognized the voice of God in the call which had come to him. It is God who by a combination of circumstances secretly guides man to the various arts and sciences and endows him with the sure consciousness that he is not only a part of the creation but also partakes in the divine providence.” Max Caspar, Kepler, trans./ed. by C. Doris Hellman (New York: Dover Publications, 1993), p. 51.

[2] Kepler believed God was geometry’s creator. “For the Creator, who is the very source of geometry and, as Plato wrote, ‘practices eternal geometry,’ does not stray from his own archetype.” Kepler, Johannes, Harmonies of the World, in On the Shoulders of Giants: The Great Works of Physics and Astronomy ed. Stephen Hawking (Philadelphia: Running Press Book Publishers, 2002), p. 645. The Bible, however, never tells us God “practices eternal geometry.” We should not be surprised to find that parts of God’s creation are even more complex than geometry can describe. Nonetheless, Kepler was right in his general belief that God created an orderly universe, and that math records that order.

[3] Robert Wilson, Astronomy Through the Ages: The Story of the Human Attempt to Understand the Universe (Princeton, NJ: Princeton University Press,   1997), p. 69. For more details about the obstacles Kepler faced, see Max Caspar’s Kepler or Robert Wilson’s Astronomy Through the Ages: The Story of the Human Attempt to Understand the Universe.

[4] See Chapter 6 of Beyond Numbers for a basic overview of the switch from Greek thinking to biblical thinking that paved the way for the Scientific Reformation.

[5] Kepler, Johannes, Harmonies of the World, in On the Shoulders of Giants: The Great Works of Physics and Astronomy ed. Stephen Hawking (Philadelphia: Running Press Book Publishers, 2002), p. 723.

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Resources Consulted

Caspar, Max. Kepler. Trans./ed. by C. Doris Hellman. New York: Dover Publications, 1993.

Kepler, Johannes, Harmonies of the World, in On the Shoulders of Giants: The Great Works of Physics and Astronomy. Ed. Stephen Hawking. Philadelphia: Running Press Book Publishers, 2002.

Newman, James R., ed. The World of Mathematics. Vol. 1. New York: Simon and Schuster, 1956.

Nickel, James D. Mathematics: Is God Silent? Rev. ed. Vallecito, CA: Ross House Books, 2001.

Tiner, John Hudson. Champions of Mathematics. Green Forest, AR: Master Books, 2000.

Wilson, Robert. Astronomy Through the Ages: The Story of the Human Attempt to Understand the Universe. Princeton, NJ: Princeton University Press, 1997.

For our first specific concept post (see the schedule), I thought I’d offer some general thoughts on a part of math that confused me for years: algebra. Hopefully, this will help you see God’s handiwork amid the xs and ys.

What is algebra?
Algebra is the process of using letters and symbols to describe general quantities and relationships.

For example, say I head into a store with $20 and come out with $5. I would have spent $15.

$20 – $5 = $15.

My starting dollars minus my ending dollars showed me how much I spent in the store.

starting dollars – ending dollars = dollars spent

Let’s use letters to represent this relationship. We’ll use z to represent the starting dollars, y for our ending dollars, and x for the dollars spent.

z – y = x

We now have represented a general relationship that holds true for more than one situation! (We could use it to see how much we’d spent in any store.)

The above is one example of using letters to describe a relationship. By recording a relationship rather than a specific situation, we’re able to solve for unknowns and discover other relationships. This process proves useful in MANY areas (electricity, interest rates, gravity, laws of motion, etc.)

Algebra is based on the idea that certain relationships consistently hold true–that objects operate in a predictable fashion. Dollars do not mysteriously multiply in our wallets. Objects fall in a predictable way. The relationship between the power and voltage and current in an outlet remains the same. Why do things operate so predictably?

Because an unchanging God holds every aspect of this universe together! If God were not keeping everything together in an amazingly predictable manner, algebra would be completely useless. But because of God’s unchanging hand over creation, we can use letters and symbols to name and describe the predictable world around us.

What about all the rules?
A large portion of algebra textbooks focus on rules and conventions. Each “rule” is one standardized convention to represent a real-life consistency.

For years men did not use our current conventions at all! The graphic shows some different ways an algebraic equation has been expressed.

Why does algebra often seem so meaningless?
So often, algebra students completely miss seeing the amazing consistency algebra records because they get lost in the mechanics. As Morris Kline points out, “The usefulness of the techniques of algebra has caused many people to mistake the means for the end and to emphasize these menial techniques to the exclusion of the larger ideas and goals of mathematics. The students who are bored by the processes of algebra are more perceptive than those who have mistakenly identified algebraic processes with mathematics.” [Morris Kline, Mathematics and the Physical World (1959; repr. and slightly corrected, Mineola, NY: Dover Publications, 1981), p. 68.]

Conclusion
As you teach algebra, beware of emphasizing the means (i.e., the rules and conventions) to the point that your student loses site of algebra’s purpose–to record consistent relationships. Remember to let your mind pause and consider the greatness, power, and consistency of the God who, day in and day out, governs all things consistently enough for us to record general relationships and expect them to hold true in various situations. His power, might, and faithfulness truly know no bounds!

“I am the LORD: that is my name: and my glory will I not give to another, neither my praise to graven images.” Isaiah 42:8 (KJV)

Note: Watch for more on algebra soon!

The Golden Ratio – A Wonder of God’s Creation

Above is a link to a blog post one of the members of our church posted a few months back–I’m excited to be able to share it with you, as it does such a wonderful job sharing with pictures and text about this fascinating ratio. It ends with this fitting reminder:

Man is indeed without excuse, for God has put his signature on all of creation. “For since the creation of the world His invisible attributes, His eternal power and divine nature have been clearly seen, being understood through what has been made, so that they are without excuse” – Romans 1:20.

Several months ago, I felt like I’d hit a goldmine when I first discovered Google has digitized and made available for FREE numerous math books from the 1800s and 1900s, including some of my favorite ones I’d found in old collections. Written back before math was viewed as an intellectual pursuit, many of these old books team with practical word problems!

While they’re not for everyone, as the older English wording and time-period illustrations might confuse some children, these historic math books can be quite helpful! If you’re not sure how to show your child how a concept serves as a useful tool, try looking at some of the word problems offered in one of these books. I wouldn’t necessarily recommend reading through all the presentations, as they’re not easy reads, but many of the word problems are gems! A good place to start is the Table of Contents–the books are also searchable if you want to narrow in on a specific concept.

Below are links to a few of Google’s extensive collection. If you search for these books’ authors, you’ll find other books by them as well (you may want to limit your search to “Full view only” to see all those available on the site in their entirety).

Ray’s New Practical Arithmetic
Adam’s New Arithmetic
Stoddard’s Rudiments of Arithmetic
Practical Arithmetic (upper elementary/high school)

Reminder: Request Week – Next week is request week, so please leave a comment with what you’d like us to discuss!

It has been more than a year since I’ve posted anything–ouch! I think it’s high time for a long-overdo update : ).

First of all, Revealing Arithmetic: Math Concepts from a Biblical Worldview, is finished! The final book ended up being a more-than-200-full-sized page manual offering ideas and inspiration for teaching basic arithmetic (counting through exponents) from a biblical worldview. I can’t tell you how excited and grateful I felt when it went off to the printer…it’s been a long, but blessed journey. Thank you all for your prayers and encouragement along the way! If you’d like to learn more about the book or purchase a copy, it’s now in our online store!

Now that Revealing Arithmetic is finished, it’s my intent to focus more on this blog. I would really like to make it a place where we can explore God’s hand in math together. You may have noticed the blog has a new look and feel–and I’m hoping to add new content to it regularly.

My plan is to post at least once a week, according to the following “tentative” monthly schedule.

• Week 1: Specific Concept - Information, ideas, or musings about a specific math concept (concepts will vary each month).
• Week 2: A Concept’s History or Applications – An exploration of a portion of math’s history or one of its many applications.
• Week 3: Resources – A resource link or review.
• Week 4: Requests – Topic will vary based on requests and feedback received. (This is your chance to tell me what you’d like to see us discuss!)

As I try to begin/resume this blog, know that your comments (feedback, ideas, sharing, questions, etc.) will help keep me going : ) I look forward to learning and growing with you!

A few weeks ago, someone shared a link to www.mathworksheetscenter.com in a comment and I wanted to pass it along to you all, as some of the free articles there looked quite helpful. The articles are located at http://www.mathworksheetscenter.com/mathtips.

I especially liked many of the high school articles, as they gave some practical ideas on math’s applications. You could use each suggestion as a launching pad to have your student explore further.

The site also offers thousands of worksheets for a small membership fee. The lessons on the worksheets basically just explain the concept by offering an example problem worked out. A search online reveals other sites with free worksheets; however, if you’re looking for a wide variety of worksheets and don’t want to hunt all around to find them, the membership might be worth it for you. Be sure to view one of the worksheets first to see if the general quality/layout would work for you.

My prayers are with you all as you gear up for a new year of school. May each day be blessed with a deeper appreciation for God’s faithfulness and power.

In Him,

Katherine

I recently reviewed two math resources—Professor B’s Power Mathematics and Maximum Math, and reorganized my math review page to hopefully make it a bit more user friendly. I always hesitate when posting a review because not every resource or curricula works for every person. We all have different learning styles and specific needs. So please keep in mind that it’s not my intent to endorse any specific resource, but rather to describe some of the options so you can better decide what will work for your family.

My prayers are with you all as you make your plans for next year. It’s easy to get overwhelmed, taking our eyes off the Lord and putting them on the many decisions that need made. Remember to let God carry the burden—He delights in leading, guiding, and teaching you each step of the way.

By His Grace,

Katherine

I was recently reminded of the fact that many people use the term “biblical math” to mean something very different than I do when I use it. So I thought it might be beneficial to take a look at some different ways the term has been used.

Many use the term “biblical math” to refer to using math to unlock some sort of hidden code within the Bible. In fact, there have been numerous books written on this topic. Each one claims to have unraveled a mystery within the Bible’s pages.

If you’ve ever been intrigued by any sort of “biblical numerical code,” I’d encourage you to read the article linked to below by James Nickel. I found this article incredibly helpful in understanding why I felt such a caution and uneasiness in my spirit every time I read anything about these hidden codes. Mr. Nickel explains how these “hidden” codes really have Gnostic roots and are a distortion of the gospel of Christ and the Word of God.

http://www.biblicalchristianworldview.net/Mathematical-Circles/behindBibleCode.pdf

Although many people attempt to use these codes to defend the authority of the Bible or to proclaim some other truth, the whole idea of finding some sort of “hidden” revelation is not true and is actually very dangerous. God’s truth is available to all, not just those who can crack a special code. There’s a harmful elitism and mysticism in saying we need some sort of special knowledge to understand part of God’s message to us. God warns that He has hidden things from the wise and revealed them to babes (Luke 10:21). We’re to use the Bible to help us understand math, not use math to unlock the mysteries of the Bible. God has given us His Spirit, note a code, to help us understand His Word.

Other people use the term “biblical math” to refer to a curriculum that contains Bible verses and Christian morals. Yet although the student is being taught a biblical perspective on morality and learning biblical truth, he’s not necessarily being taught how to look at math itself from a biblical perspective.

I would define the term “biblical math” as an approach to math that leaves the student with an understanding of how math itself is viewed from a biblical worldview. The Bible gives us principles that impact the way we view and approach all aspects of life, including math. God is the creator and sustainer of all things and is a consistent, faithful God; hence, we can expect the universe to be consistent. Man is created in the image of God; hence, we’re able to observe the consistent way God governs all things and record that using math. Man is fallen; hence, we are prone to error and cannot place our faith in our own reason. God gave us the responsibility to work and “subdue” the earth; hence, God has given us math as a useful tool to help us with the tasks He has given us.

The above statements are just a few (and are simplified for the sake of space) of the ways the Bible’s principles should impact the way we approach math. When we take these and other principles and build our approach to math off of them, it completely transforms math from a meaningless exercise to something both meaningful and useful—and something that, above all, points us to the Creator (see Beyond Numbers).

I hope the above explorations of “biblical math” help clarify things a little for you. It’s amazing how differently a phrase can be applied, isn’t it? : )

A few weeks ago, I visited a local historic home with my aunt, uncle, and cousins. Something I saw there sparked a little research and has resulted in this post : ).

The chandelier hanging in the hall of the historic house was much too high to light easily using a ladder. But men had used the ingenuity God had given them to design a device whereby the chandelier’s candle COULD be easily lit—by just one person.

The chandelier was hung using a long chain, at one end of which was a cylinder that perfectly balanced out the weight of the chandelier. Since the weight balanced, the chandelier didn’t move unless a person pulled or pushed on the chain. By pulling or pushing on the chain, a single person could easily higher or lower the chandelier, light it, and return it to its former position.

As I looked at the chandelier, I realized this was math in action. Whether or not the original designers weighted the chandelier and the cylinder, I don’t know. But I do know that the weights equaled—and that math has been historically used to help design and use counterbalances for a variety of purposes.

[Photo taken at Sully Historic Site in Chantilly, Virginia.]

Merchants used to use a scale based on the counterbalance principle to weigh their products. On one side of the balance, they’d place the item to be weighed. They’d then add items of which the weight was known to the other side until the scale balanced. The whole process required quite a bit of math.

If you’re working on measurement with your child, consider having him or her build a scale! Easy instructions can be found at the website below. As you build it, thank God for creating us capable of designing devices—like balances—to help us.

http://www.campsite24.ca/balance_scale_eng.pdf

A recent trip reminded me of one practical way math serves as a useful tool–money conversion. On our trip, we saw prices listed in dollars, pounds, euros, forrents, and marks–and needed to use some quick math to figure out how much we were really spending : ). We employed addition, subtraction, division, and round/estimating at various times throughout the trip.

As you study different cultures in history, consider having your child learn about their money systems too. Simply search the Internet for “money” and the country’s name. You should be able to find some pictures and descriptions of the money in that country. Then search for exchange rates and see how the money compares to American money.

If your child is young and just getting used to American money and math facts, you could simply show your child the pictures of the money, explaining that different countries use different money systems just like they use different languages, and that math helps us compare prices. If your child is proficient with math, you could actually have your child pretend to go shopping in the foreign country and figure out how much something marked in that country’s currency would cost in American dollars. You could even set up a pretend shop!

Math is truly a useful tool!

Things have been piling up to post, so this post will contain quite a few different things.

Free Workshop
First off, I wanted to invite you all to attend a workshop this afternoon (Tuesday, March 18, 2008) at 1:00 p.m. CST I’ll be giving on teaching math biblically. You can attend right from your computers–and it’s free! The workshop is being done through Cindy Rushton’s Ultimate Homeschool Expo as one of their free, pre-conference preview chats. The chat will be archived and available after the conference begins to attendees, but it’s only open to everyone this afternoon. To join, follow the instructions at the bottom of this post. (I apologize for the late notice.)

Math Book Update
Next, after several drafts and a name change, my math book is actually nearing completion! The book is now titled Revealing Arithmetic, and serves as a guide for teaching arithmetic concepts from a biblical worldview. Revealing Arithmetic will walk through teaching basic arithmetic concepts from counting clear through long division, offering an overview, example presentation, reinforcement ideas, and much more. Thank you all for your continual prayers.

Math Information Online
We recently posted an article adapted from Beyond Numbers to our site and reorganized some other math information. I hope you enjoy!

I also recently found an article on biblical math online I think you some of you might find helpful. Vern S. Poythress helpful, although more intellectual, essay titled, “A Biblical View of Mathematics” is available online at www.frame-poythress.org/poythress_articles/1976Biblical.htm.

Please Note: I’ve not had a chance to look at any of the other essays on this site or by Poythress to know their merit or theological content. I mention this particular article because it looked helpful.

Instructions for Joining the Workshop (from Cindy Rushton)
We provide the software download-free. Here is the link:

http://www.cindyrushton.com/Conference.exe

Click to begin download or copy/paste into your internet browser and click ENTER. It will only take a few minutes to download into your computer, but you want to do it ahead of time so you don’t miss a minute in the Conference Room. After it is downloaded on your computer, it will set up an icon on your desktop that says CONFERENCE. Then, you will only have to click it to open up the room. Here are your instructions from there:

When you are ready to enter, you will enter your NAME or EMAIL ADDRESS in the first blank.

Add the following information when prompted:

Name of Room: Talk-a-Latte

Password: letschat (for Preview Chats)

Remember…your password is different for chats and other classes. It only works for the room when the room is open for the study.

Some time ago, someone wrote and suggested I write something on the golden ratio. I hope to finally post something on the topic today (this post has been in progress now for QUITE some time).

The “golden ratio” is a special name given to describe a ratio that seems to relate indirectly or directly to many aspects of God’s creation. The ratio is approximately 1.618 (see Wikipedia for a more exact definition).

To understand how we observe the ratio 1.618 in God’s creation, we need to take a look at a special sequence called the Fibonacci numbers. This special sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…and continues, with each new number formed by adding the previous two numbers together (1 + 2 = 3, 3 + 5 = 8, etc.). The ratio between most numbers in this sequence is very close to the golden ratio. This means that if you were to divide two neighboring numbers in the Fibonacci sequence, you’d get a number close to 1.618!

Since we find neighboring Fibonacci numbers all over creation, it follows that we also find the golden ratio all over too. For example, the seeds in any given sunflower are arranged in two patterns. If one of the patterns has 55 seeds; the other will have either 34 or 89—the number of seeds in each pattern are always neighboring numbers in the Fibonacci sequence! This also means that the ratio between the two patterns is always very close to the “golden” ratio. No matter how large or small the sunflower, one seed pattern always contains approximately 1.618 times the number of seeds as the other pattern. Guess what? This ratio allows for the most number of seeds to fit in any given sunflower! God sure thinks of all the details, doesn’t He?

If we were to look at plants, pinecones, or pineapples, we would again find Fibonacci numbers and the golden ratio and be awed again at the Creator (please see the websites listed at the end of this article for a more detailed explanation). Scientists have even found ways where things like the nautilus’ shell relates indirectly to the golden ratio. And artists and architects have discovered that rectangles based on the golden ratio are artistically pleasing (surprise!).

Many marvel over how the golden ratio (and numbers in the Fibonacci sequence) keeps popping up all over creation. As Christians, we know the ratio appears everywhere because God designed the golden ratio to have the properties it has, and then designed each part of His creation with infinite care and wisdom, using this ratio to give sunflowers, pinecones, and more just what they needed. He also created our minds to appreciate this same ratio as something “beautiful”—as testified to by the many buildings and paintings that incorporate this ratio.

Below are four sites that offer more details about the golden ratio. The first two approach the ratio from a biblical perspective; the last two do not have a biblical perspective, but contain some fascinating information and easy-to-understand explanations of the Fibonacci sequence and the Golden Ratio you might find helpful.

Note that one of the sites uses the name “golden mean” instead of “golden ratio” to refer to this ratio. The sites also refer to the “golden rectangle” or “golden section”—this is a rectangle whose sides follow the proportion of the golden ratio (one side is approximately 1.618 times the other side). Most of the sites also discus the Fibonacci sequence—remember that neighboring numbers in this sequence have a ratio that approaches the golden ratio.

I hope you enjoy!

http://www.answersingenesis.org/creation/v16/i4/golden.asp
http://www.biblicalchristianworldview.net/Mathematical-Circles/fibonacci.pdf
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html

“How many math concepts could you teach using just the objects on this table?”

The question came from my brother. We had just finished eating dinner, and I was brainstorming with him about some decisions I needed to make regarding the layout and content of Unveiling Math. 

I looked at the objects on the table—our leftover food and an assortment of dirty dishes. These were hardly objects one would ordinarily think to use to teach math concepts.

Yet as I began to answer the question, I realized that one could really teach nearly every math concept using just the items on that table. Addition, a method of recording the way God causes objects to add together can be demonstrated by “adding” the forks or cups on the table! Subtraction, multiplication, and division could be taught in a similar fashion by subtracting, multiplying (adding in sets), or dividing (splitting up) the silverware or plates. Fractions could be presented as one way of recording partial quantities by cutting up the left over food and demonstrating how each part could be represented. Decimals would follow in a similar line. The table itself presented the perfect springboard for presenting shapes and geometry. Since algebra is just a way of generalizing about quantities, we could really use an a or an x to represent the various objects on the table—or about the height of the table. Economics and statistics, as well as linear graphing and calculus, would come into play if we began to talk about the process of growing the food and selling it to the restaurant… 

I finally had to stop myself in amazement. Who would have thought that just a simple dinner table could prove the perfect classroom?

The other day, I had an opportunity to look at an Impressionist painting. Up close, the painting looked like globs of paint on a canvas. But from a distance, those very strokes of paint transformed into boats, shadows, and buildings—objects that simply would get mistaken for meaningless swirls of color unless the viewer took the time to step back and look at the big picture. Then, and only then, would he see a masterpiece.

As you teach math to your child, remember to step back and behold God’s masterpiece. Don’t get so lost in all the mechanics of writing and solving problems that you forget their purpose. Just like the brush strokes in a painting, they’re just the means by which we attempt to record on paper the beautiful and faithful way God holds this universe together. There is a beautiful masterpiece to behold in math.

Wanted to share with you all a math site I found the other day that offers a whole collection of free math worksheets on a variety of different math concepts. The worksheets are pretty much just paper drills, but those of you who are trying to assemble your own curriculum using an assortment of different resources may find this helpful. You could use a problem or two from here to provide the “drill” part of your curriculum, using practical math resources/real-life settings to do the majority of your teaching. http://www.math-drills.comAnd since writing the above, I found another site, http://score.kings.k12.ca.us/lessons.html, that offers a variety of mock situation suggestions you can use to help bring math into real-life. Don’t forget to point out to your child that math really is just a way of recording God’s creation, and as such is useful in helping us with the tasks He has given us to do.

Have you ever looked at your child’s math lesson and thought, I have no clue how to teach this. I’m clean out of ideas? If so, you’re not alone. We all face times when we’re simply not sure what to do or how to present something.

A few weeks ago, I knew I needed to write a blog entry on this blog, but I just couldn’t think of anything helpful to say. After a staring at my monitor blankly for a little while, I got up and headed out on a walk. While I walked, I prayed. What was something fall-related parents could do with their children? A leaf just starting to turn colors caught my eye. I wonder if there’s any way to integrate math with the falling leaves? I wondered.

Coming home, I decided to do a little search online on “changing leaves.” I later refined the search to “leaves math.” I found a wide variety of websites offering suggestions for ways to use leaves and math. Although I didn’t like many of the ideas as they were, they inspired other ideas.

For example, http://content.scholastic.com/browse/article.jsp?id=11901 had some interesting facts about leaves, such as that maple trees lose about 600,000 leaves each fall! I thought of how fun it would be to have a child try to count all the leaves that fall in your yard, pointing out that while this is impossible for man it’s not impossible for God. Surely the God who calls each star by name and knows the number of hairs on every person’s head also knows the number of leaves that are on (and falling off) each tree. He is truly MUCH greater than us!!!

In the end, other ideas came, and I didn’t even end up using my leaf idea for my blog. As I hit the post button on my finished blog entry, I thanked the Lord for giving me the ideas that I needed.

If you’re feeling out of ideas, I’d encourage you to take heart. God has the ideas you need. Seek Him. Ask Him. And watch Him give you the inspiration you need in His timing and way.

P.S. If you live in a part of the world where the leaves are changing colors, you may want to do your own search for “leaves math.” Then head out and enjoy the fall weather while you teach your children to use math to explore God’s creation!

I had read so many works on math’s history that I thought for sure I had learned all I needed to learn. But as I was trying to finalize a few concepts late last week, I realized that I needed to do more research. I needed to go back and read the original writings of antiquity so I could see for myself what I’d read so many summaries about.

Fortunately, we live ten minutes from a university, so I was able to pack up my computer and head to the university library. Still, finding original works wasn’t easy, but as I got my hands on a few mathematic source books I felt well rewarded. There was something special about seeing those actual writings. The Lord also used the experience to remind me of a spiritual principle.

Just like I had nearly settled for summaries, we’re often tempted to settle for intellectual knowledge or summaries others make of God’s truths. Yet we continually need to go back to the Source–God Himself. God is the true Source of knowledge in all areas. There’s no substitute for hearing directly from Him in His Word.

In my past couple of blogs, I’ve tried to offer some practical resources to help you teaching math biblically, and I’ll try to keep posting practical ideas as I can. But it’s my prayer that you will keep going back to the Source yourself so that God Himself can give you the ideas and inspiration you need. There’s such a joy in letting Him teach us–in letting Him step in where we thought all was hopeless. As the Creator and Sustainer of all the order around us, His ideas and wisdom are infinite! What a wonderful privilege we have to keep going back to the ultimate Source.

Rejoicing in His Care,

Katherine

P.S. As you may have picked up on in this post, I’ve been spending some time lately finishing up some research for Unveiling Numbers. I wanted to ask you all to pray. There is a LOT of finalization of concepts and decisions that still need made in order to get it off to the editor. Please pray that I’ll have the wisdom and discipline to both know what needs done and to get it done.

I’ve recently found out about a few practical math resources, and I thought I’d share them with you in case you might find them helpful.

1. http://www.livingmath.net/. This website looked like it had a lot of fun and helpful ideas on integrating math into everyday life.

2. Math on the Level. This new homeschool curriculum, although it doesn’t attempt to present a biblical world view towards math, it does a great job teaching math from real-life settings. It’s format was also very unique and more flexible/easy to modify than others I have seen. I posted my thoughts on
http://www.christianperspective.net/mathinfo/mreviews.htm#curriculum

3. Arithmetic for Parents. This book by Israeli math teacher Ron Aharoni came recommended to me, and I’ve been enjoying reading it. The book offers a lot of practical ideas about how to teach various concepts as useful tools. It’s available at http://www.sumizdat.org/.

A few weeks ago, I had the opportunity to attend a workshop on organization. Now, slowly but surely, I’ve been organizing my office. The first weekend after the workshop, I tackled all my math notes by organizing them into sets. This organization process reminded me of something I wanted to share here with you all.

As a student, I always used to struggle when my math book began talking about counting numbers, whole numbers, real numbers, irrational numbers, and other number groups. Who determined that these numbers were real numbers, while these other numbers were irrational ones? And what did it matter?

After I began to grasp the biblical worldview in math, I went back and looked at these different number groups and discovered that they, too, were really useful ways of recording God’s creation!

Whole numbers, real numbers, etc., are just fancy names we gives to numbers with certain characteristics. Just like Adam used names to describe the animals, we use these names to describe and sort the quantities God placed around us. Now, we could have used different names. We could have used whole numbers to refer to numbers with different characteristics. These names in themselves are not absolutes; they’re just names that we’ve adopted to help us easily refer to different types of numbers.

Referring to different types of numbers comes in handy. There’s no way any human being could ever remember every single number that can’t be divided by two. But if we refer to all these numbers as odd numbers and learn the characteristics of odd numbers, then we’ll be able to recognize whether or not a number can be divided by two fairly easily. We group numbers into sets because we can’t memorize or keep track of every single number and its properties–only God can do that!

One way you could help your child really see number sets as a way of sorting quantities is to have him organize his closet/bedroom. As you help your child organize his room you could teach math! Tell your child about how, when organizing, we sort objects based on their properties. Have him sort things in his room. For example, we might sort clothes by color (red shirts will go in this drawer, blue shirts in this one, etc.) or by type (long sleeve shirts go in this drawer, short sleeve shirts go in this one, etc.).

Explain to your child that, just like we put like things together to organize, mathematicians have organized numbers based on their properties. Whole numbers, real numbers, etc., all have properties that make them different and unique. We can therefore put them in different “sets”—the mathematical name used to refer to the different “piles” used to organize numbers. Having these “piles” or sets helps us easily refer to numbers with specific attributes, much like the name “long sleeve shirt” helps us easily refer to a certain type of shirt.

There are also a lot of household chores that you can use to teach the concept of sets to your children. Setting the table, folding the wash (by child or by item), or even, putting away the groceries (canned goods, refrigerator goods, freezer goods, etc). Just think–your math class can actually help you get caught up on your housework this week! More importantly, it can help your child see each aspect of math–even number groups/sets–as a way of sorting/recording God’s creation.

I imagine that many of you are pulling out the school books and getting ready to begin “school” again with your children (I put “school” in parenthesis because you’ve really been schooling your children all summer long too—we’re always learning). I thought it might be helpful for you to have a couple fall/back-to-school related ideas you could use to help bring math into everyday life, showing your child that math is a real-life tool we use to help us describe and quantify God’s creation.

–Have your child help you figure out how many pages they need to complete in their books each day.
My mom used to tell us at the beginning of the year how many pages/chapters we’d need to read each day/week. She figured out this number by dividing the total pages/chapters in the book by the number of days/weeks in a school year (minus any vacation days).

As you sit down to plan your child’s lessons, why not let him help you? Explain that we can use math to help us with the tasks God has given us to do. Talk to your child about how, by using math to plan the school year, you’re using math to help you with the task God has given you.

–Talk about Adam and Naming the Animals
You may want to open up Genesis 1 and 2 and discuss how right there in the Garden of Eden, God brought animals to Adam to name. The numbers 1, 2, 3 are just symbols we use, like names, to help us refer to quantities.

For some added fun, have your child try to come up with his own number system. Making up a number system is a great way to reinforce the idea that our number system is just one set of symbols that we use to help us describe God’s creation.

–How long until…?
Children are notorious for wanting to know how long it will be until something happens. At least I did! : ) How long until my birthday? Until Thanksgiving? Until dinner?

Take advantage of these every-day questions to show your child how math helps us keep track of time. Because God gave us the sun, moon, and stars to serve as signs and seasons (Genesis 1:14-16), and because He gave us the ability to develop calendars and time-keeping devices, we’re able to keep track of time. We can use numbers to represent the passing days and years.

Conclusion
I hope the above ideas will prove useful and inspire other ideas of your own!

May God bless you as you start the new year seeking Him in everything–even math!

It’s my prayer that the reflections and thoughts I share here will encourage you to keep seeking the Lord everywhere–including math. If you have a particular question or thought you’d like to see addressed here, please leave it in a comment. I am always appreciative of ideas .

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In Christ,

Katherine Loop