It Only Takes an Instant

Posted December 1, 2014 by gensci
Categories: Teaching Mathematics

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Two images of a wave in Waimea Bay, Hawaii, taken 0.1 s apart, reveal how quickly things can change from beauty to chaos (or vice versa).

Two images of a wave in Waimea Bay, Hawaii, taken 0.1 s apart, reveal how quickly things can change from beauty to chaos (or vice versa). © 2014 David E. Shormann

I never cease to be amazed by the technology of today’s digital cameras. By pushing a few buttons, anyone crazy enough to bob around in a 6-8′ high shore break can capture some amazing beauty in God’s creation! But as the photos above reveal, the beauty only lasts for an instant.

It seems impossible that things can change from beauty to chaos, or vice-versa, in so short a time. But big changes can and do occur in even less time than the tenth of a second that elapsed between these two photos.

Strangely enough, instantaneous change, something us humans really can’t fully comprehend, is behind almost every major technological achievement of the past 300+ years!  How can that be? How can something we will never fully understand help us make all sorts of useful devices? Well, some things we just have to take on faith. Faith is at the heart of the branch of mathematics known as calculus. And calculus is all about the study of instantaneous change.

Subjects like calculus are easier to grasp when we consider the Author of every instant of time, and Creator of the biggest  and best instantaneous changes of all. Paul writes in 1 Corinthians 15:52 how we will be changed “in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised imperishable, and we shall be changed.”

Leonhard Euler (1707-1783), a devout Christian man considered the best mathematician ever, wrote that “it is God, therefore, who places men, every instant, in circumstances the most favourable, and from which, they may derive motives the most powerful, to produce their conversion; so that men are always indebted to God, for the means which promote their salvation.”

Euler understood God’s relationship with man and creation very well. He also understood mathematics really well, too! Much of the way we teach mathematics today comes from Euler’s textbooks on the subject.

In our new Shormann Mathematics curriculum, we believe that all 10 major topics covered, including and especially calculus, are best understood by connecting the study of mathematics to Jesus Christ, the founder of all knowledge, and the founder and perfecter of our faith (Hebrews 12:2).

Using Math to Unlock Mysteries, Reveal God’s Beauty, and Interact With Others

Posted September 8, 2014 by gensci
Categories: Teaching Mathematics

Tags: , , , ,

The following is the sixth in a series of posts covering Shormann Mathematics, Algebra 1, the newest product from DIVE Math and Science! Click here to read the complete document that covers Shormann Math core ideas, course description, and Algebra 1 table of contents.

Mathematics has so many uses, including modeling the beautiful arcing leap of a bottlenosed dolphin! Image ©2009 by David E. Shormann.

Mathematics has so many uses, including modeling the beautiful arcing leap of a bottlenosed dolphin! Image ©2009 by David E. Shormann.

Shormann Math teaches students that mathematics is the language of science, and therefore an important tool for unlocking mysteries and revealing the amazing beauty found in God’s creation. It is also an important tool for interacting with others, such as when buying and selling things. Shormann Math will train students to become skillful at using mathematics in a way that will help them become productive members of God’s world, using their talents to serve Him and serve others.

Shormann Math teaches students what mathematics is, and how to solve problems using mathematical concepts. Problem solving is simply the application of mathematical concepts in new situations. It is about building on foundations that have already been laid, using mathematical tools developed over the centuries and applying those in new situations to solve problems. This is the essence of deductive reasoning, which is simply about applying rules. Mathematics is primarily deductive in nature, while scientific investigations are inductive (about finding rules).

What follows is a partial list of areas that mathematics is used, and that you may see covered in a Shormann Math Practice Set. At least one problem in each Practice Set will be about one of these areas. If you don’t see an area you think we should cover, let us know. One thing is for certain, Shormann Math students will not be asking the “what am I ever gonna use this for” question regarding math!

Science: astronomy, chemistry, biology, physics, biochemistry, computer science, oceanography, meteorology, medicine

Farming: animals, plants, aquaculture

Natural history: geology, volcanism, genealogy

Business: marketing, accounting, finance, business startup, productivity, employment

Engineering: mechanical, electrical, petroleum, aerospace, civil, industrial, robotics

Architecture

Art

Sports: baseball, football, basketball, soccer, fishing, tennis, NASCAR, track & field, volleyball

Music

Use of Geothermal Energy Expected to Increase Worldwide

Posted September 2, 2014 by gensci
Categories: Teaching Science

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The number of geothermal powerplants worldwide is expected to increase by 73% between 2010 and 2015. Image source: geothermal-energy.org.

Electricity generated by geothermal powerplants is expected to increase by 73% between 2010 and 2015. Image source: geothermal-energy.org.

Most of the world’s electricity is generated using steam. Water is heated, generating high-pressure steam, which blasts out and spins a turbine. The turbine system creates motion of a magnet relative to wires, which in turn generates an electrical current, a phenomena Michael Faraday discovered in the 1830’s.

Powerplants mainly differ in the heat sources they use to generate steam. The most common heat sources are currently coal, natural gas, nuclear, and oil. A schematic of a steam turbine powerplant is shown below, courtesy of the South Texas Nuclear Project.

Schematic of a nuclear power plant. Notice how a closed loop of water is heated, passed over a turbine, cooled, and reheated. Image source: South Texas Nuclear Project.

Schematic of a nuclear power plant. Notice how a closed loop of water is heated, passed over a turbine, cooled, and reheated. Image source: South Texas Nuclear Project.

In the 21st Century, a growing trend is developing towards using geothermal heat sources. The amount of electricity generated by geothermal powerplants is expected to increase by 73% between 2010 and 2015. While geothermal powerplants are less efficient, they do have several advantages. The #1 advantage is they use the Earth’s heat. And beneath our feet lies an almost infinite supply of heat. 

Current geothermal powerplants are located where magma sources rise close to the surface. However, with improvements in technology, we should be able to access deeper and deeper heat sources. And, we can also vastly improve geothermal powerplant efficiency by using supercritical water(high pressure/high temperature) instead of steam. This was the goal of the Iceland Deep Drilling Project. In this first-of-its-kind system, they actually drilled into the magma, creating what is known as a magma-enhanced generating system. While the system is not currently operating, the project showed it is possible to use water near the supercritical phase, resulting in a much more efficient powerplant. 

The search for alternative energy sources continues as people become increasingly aware of the negative environmental impact of covering vast expanses of Earth’s surface with wind turbines and solar reflectors. God commanded us to be good stewards of His creation, and covering the land with windmills and solar reflectors is not a good management solution. Hopefully, cities and states will continue looking more and more at geothermal systems, with their small environmental footprint and low emissions. 

 

 

The 10 Major Concepts of Shormann Mathematics

Posted August 26, 2014 by gensci
Categories: Teaching Mathematics

The following is the fifth in a series of posts covering Shormann Mathematics, Algebra 1, the newest product from DIVE Math and Science! Click here to read the complete document that covers Shormann Math core ideas, course description, and Algebra 1 table of contents.Screen Shot 2014-08-26 at 7.57.55 AM

After years of teaching mathematics, researching math curricula and math history, and applying mathematics as a scientist and engineer, I concluded mathematics can be taught by covering 10 major concepts. The 10 major concepts are: number, ratio, algebra, geometry, analytical geometry, measurement, trigonometry, calculus, statistics, and computer math. While all 10 concepts can be taught in any K-12 course, specific concepts will be emphasized more or less at appropriate times. For example, number and ratio will be emphasized in younger grades, algebra in Algebra 1 and 2, etc.

I know what you are thinking right now, and that is “But CALCULUS is one of the 10 major concepts! How can you possibly teach calculus to an Algebra 1 student?!” Well, if you have even an 8th grade level of math proficiency, you know that if it took you exactly one hour to drive 60 miles, your average speed would be 60 mph. If you understand that, you already understand something about calculus, because calculus is really nothing more than studying rates of change. And yes, it gets more complicated than that example, but it also gets less complicated, too, so much so that there are things about calculus you could teach a kindergartner!

Most state mathematics standards do not include calculus, and none that I know of require calculus in high school. And the federal Common Core math standards include no calculus, and almost no precalculus either! However, the discovery of calculus is one of the greatest mathematical achievements ever! All the great technological achievements of the last 300+ years are in some way or another related to calculus! And proficiency in calculus opens the door for a student to choose any college major, while an inability to pass calculus limits a student to about 20% of college majors.

For high school mathematics, most home schools and private schools simply parrot whatever their state standards are, which means they complete Algebra 1, 2, and Geometry, and check off math on their transcript, not really knowing why they did math this way. With Shormann Math though, we want you to know why you are doing math differently. We are going to paint a broader brush than most math curricula, teaching math like a language, while at the same time helping you become proficient in standard Algebra 1, 2 and Geometry concepts. Along the way, rather than avoiding calculus because you heard it was scary, you are gently introduced to it. And, before you know it, you will be understanding more calculus than all your peers, and probably even your parents, ever did! Rather than an afterthought or a scary thought, Shormann Math makes calculus a normal, natural part of the curriculum, and culminates with a formal (and yes, it’s optional!) calculus course that will prepare students to receive college credit via CLEP or AP Calculus.

Done in a thoughtful and age-appropriate way, all 10 major concepts listed above can most definitely be represented in one way or another in a K-12 mathematics curriculum.

Mathematics History Matters

Posted August 18, 2014 by gensci
Categories: Teaching Mathematics

Tags: , , , , , , ,

The following is the fourth in a series of posts covering Shormann Mathematics, Algebra 1, the newest product from DIVE Math and Science! Click here to read the complete document that covers Shormann Math core ideas, course description, and Algebra 1 table of contents.

Screen Shot 2014-08-18 at 9.54.49 AM

What does Leonardo DaVinci’s famous painting, “The Last Supper”, have to do with geometry? Use Shormann Mathematics and find out!


History helps connect students to their world and their Creator.

Most modern mathematics curricula ignore math history. But core ideas have consequences, and studying history often reveals which ideas are worth repeating and which ones aren’t. Did you know that Isaac Newton, author of the most famous science book ever written (The Principia), based the format of his book off of Euclid’s Elements, the most famous math book ever written? Did you know Shormann Math bases its format off Euclid’s and Newton’s famous works, stating rules and definitions up front, and using these as the building blocks to learn new concepts? Did you know that modern mathematics has a rich Christian heritage? Well, if you use Shormann Math, you will learn all about these things and more! Whether or not you are using a classical, trivium/quadrivium approach to your child’s education, understanding mathematics within a biblical, historical framework will help students make more sense out of what they are learning and why they are learning it.

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Screenshot of a part of Lesson 9 from Shormann Mathematics, Algebra 1. Did you know there is a connection between the U.S. Declaration of Independence and Euclid’s famous geometry text, “The Elements?” Did you know Scripture predates Euclid’s main idea of “self-evident truths?” Shormann Mathematics uses history to connect students to their world and their Creator.

Shormann Math Core Ideas: Jesus Christ is the “Common Core.”

Posted August 5, 2014 by gensci
Categories: Teaching Mathematics

Tags: , , , , ,

The following is the third in a series of posts covering Shormann Mathematics, Algebra 1, the newest product from DIVE Math and Science! Click here to read the complete document that covers Shormann Math core ideas, course description, and Algebra 1 table of contents.

Jesus Christ is the “Common Core” of Shormann Math

Perhaps you have heard of the United States government’s “Common Core” curriculum. Perhaps you have also heard that a lot of people are concerned about it. Leading experts believe the Common Core’s mathematics standards will not prepare students to study science, technology, engineering, and math (STEM) in a selective four-year college. And a white paper by the Pioneer Institute concludes by saying

“At this time we can conclude only that a gigantic fraud has been perpetrated on this country, in particular on parents in this country, by those developing, promoting, or endorsing Common Core’s standards.”

Unfortunately, man and his ever-changing ideas are at the core of this curriculum. At DIVE, we strive to place Jesus Christ at the core of all our products, and we pray that this will result in students learning math and science for His glory and the service of others. So, even though our primary goal is NOT to prepare students for STEM, we believe by putting Christ at the foundation, just like the world’s original universities did, students will naturally learn to use mathematical tools that will connect them to their world and their Creator.

 

Should my Child Do Saxon Math 8/7 or Algebra 1/2?

Posted July 31, 2014 by gensci
Categories: Teaching Mathematics

Tags: , , , , , , , ,
Need more arithmetic review before Algebra 1? Then do Saxon Math 8/7. Fluent in arithmetic and ready for more algebra-type concepts? Then do Saxon Algebra 1/2.

For students ready for pre-algebra, we recommend Saxon Math 8/7 for most of them. 

A pre-algebra course is a normal and recommended part of a child’s math education as they make the journey from a world of mostly numbers (arithmetic) to a world of mostly letters (algebra). John Saxon, together with Stephen Hake, authored two pre-algebra textbooks: Saxon Math 8/7 and Saxon Algebra 1/2*. Both books prepare a student for either Saxon Algebra 1, or our new Shormann Math Algebra 1. So should a child do Math 8/7, Algebra 1/2, or both?

*Math 8/7 (pronounced “Math Eight Seven”), is designed for the accelerated 7th-grader or average 8th-grader. Algebra 1/2 (pronounced “Algebra Half”), is a pre-algebra course that has more algebra and less arithmetic review than 8/7. For home educators, our child’s skill level, not grade level, trumps textbook naming systems.

Math 8/7 is all most students need for pre-algebra

For most students, Saxon Math 8/7 is all they need to prepare them for Algebra 1. It has more arithmetic review than Algebra 1/2. Prior to Algebra 1, we’ve found that most students need more review of things like addition, subtraction, multiplication, and especially division. And division is related to fractions, and Math 8/7 has more review of things like simplifying fractions and converting between fractions, decimals, and percents. Fluency in these areas will almost guarantee a smooth transition to Algebra 1.

Prior to each lesson of the Math 8/7 homeschool edition pictured above, students complete a Facts Practice that will help them build fluency in the areas mentioned. Algebra 1/2 does not include this daily practice, although it does review arithmetic (including fraction/decimal/percent conversions) in the daily homework sets.

Math 8/7 provides daily Facts Practice that Algebra 1/2 does not. This daily practice provides an opportunity to build fluency in areas many students struggle with, like fraction/decimal/percent.

Math 8/7 includes daily Facts Practice that Algebra 1/2 does not. This daily practice provides an opportunity to build fluency in areas many students struggle with, such as fraction/decimal/percent.

Have your child do Saxon Math 8/7, Homeschool Edition, if you feel like they need to work on becoming more fluent in the arithmetic areas mentioned above. This is what we recommend for most students.

Have your child do Saxon Algebra 1/2, 3rd edition, if you feel like they are fluent in arithmetic and are ready for a little more algebra, which Algebra 1/2 provides.

Have your child do both books if they complete Math 8/7, but you don’t feel like they are proficient enough and maybe need to be eased more gently into Algebra 1.

The beginning of each Saxon course contains review from the previous course. Therefore, since Algebra 1/2 covers algebra in more detail than 8/7, there will be more in Algebra 1 that the Algebra 1/2 student has already seen. For students who are extremely fluent with their math, you may want to skip some of the first lessons to avoid unnecessary repetition.

If you have any other questions about pre-algebra, please leave a comment, or contact us at sales@diveintomath.com.


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