Posted tagged ‘geometry’

How Shormann Math Teaches Proof

June 17, 2015
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Euclid’s Proposition 1 overlaying a pod of spinner dolphins swimming in a near-perfect equilateral triangle formation! The concept of proof applies to everything from building a rocket to the simple beauty of a pod of dolphins. Photo Credit

What is proof?

Proof is really nothing more than providing a reason for statements made or steps taken. In the standard American government school 3-year “layer cake” approach to high school math, the concept of proof is normally limited to some sections in the geometry layer. But proof is not a concept that is the exclusive domain of geometry. Shormann Math teaches proof in 3 main ways, by 1) studying Euclid’s foundational work on proof, 2) showing that proof is for all of math, not just a few weeks in geometry class, and 3) showing how proof applies in the real world.

Euclid and proof

Around 300 B.C., Euclid (330 – 275 B.C.) organized the previous 3 centuries of Greek mathematical work into a 13-volume thesis known today as The Elements or Euclid’s Elements. Scholars believe that only the Holy Bible has been more universally distributed, studied and translated. Starting with a foundation of 5 postulates, 5 axioms, and 23 definitions, Euclid proved 465 theorems, or propositions.While postulates are basically rules that are assumed to be true without proof, theorems are true statements requiring proof. Postulates are also referred to as self-evident truths.

Surprisingly, even though Euclid is considered the “Father of proof,” most American high school geometry textbooks mention little to nothing about Euclid. In Shormann Math though, students will learn who Euclid was, and the importance of his contribution to Western Civilization. Shormann Algebra 1 and 2 students will become very familiar with Euclid’s first 5 propositions, giving them a good understanding of proof technique. They will gain an appreciation for the deductive nature of geometry and geometric constructions, seeing how one proposition often requires the previous one. And they will also see the simple beauty and elegance of Euclid’s propositions.

Proof and mathematics

Perhaps one of the greatest flaws in the “layer cake” approach to high school math is that the concept of proof is almost always limited to a few weeks during the geometry year. In Shormann Math, we’ll do the standard triangle proofs and circle proofs, but we will also apply proof technique in other topics like algebra, trigonometry and calculus.

But how can proof be for more than just geometry? Well, proof is based on a type of reasoning called deductive reasoning (applying rules). Every single math concept begins with rules. And every single math problem can be solved by applying those rules. All of mathematics is deductive in nature, which means at any time, a student should be able to explain the rules (provide reasons for) they used to solve a problem.

Because Shormann Math is integrated, we’re able to help students make connections between the major concepts like algebra and geometry. This results in students getting a better feel for what mathematics is about, which will make it easier to learn. Instead of thinking that they are always learning something new and different, they will see how one lesson builds on previous ones, which makes it less intimidating.

Proof and the the real world

Sure, proof is important to mathematicians, but it’s also important in the real world. As we explain in Lesson 68 of Shormann Algebra 1,

“Supporting statements with reasons is a technique used by, and expected of, people that society refers to with words like professional, leader, wise, helpful, and trustworthy. People like Abraham Lincoln, the 16th President of the United States of America, known for his study of Euclid’s Elements and his application of the idea of proof to solving societal problems.”

We also helps students see the application of proof technique in the real world, as this table from Shormann Algebra 2 explains:

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Most importantly, proof is profoundly important in sharing the Gospel with unbelievers. God wants us to be ready to give reasons for the hope of salvation (I Peter 3:15). Our reasons should primarily be Scriptures we have memorized, or at least remember where to find them.


As you can see, proof is about a lot more than geometry! Shormann Math gives students a basic understanding of proof technique and it’s application to the real world. It’s a great tool to help them in their thinking, planning, designing and serving. If you think you would like your child to learn math in a more natural way that connects them to their world and their Creator, click here to learn more!

The New Shormann Math vs. Saxon Math and Common Core

April 21, 2015

Summary: If mathematics is the language of science, then fluency should be the goal, which means the traditional “layer cake” approach to 3 years of high school math (Algebra 1, Geometry, Algebra 2) is probably not the best approach. The shapers of Common Core’s math standards realized this, and now states that adopt their standards can choose between two high school “pathways”, either the layer cake or integrated approach, where students learn algebra and geometry together. John Saxon* actually pioneered the integrated approach in the United States back in the 1980’s, but his integrated approach was only one small part of his textbooks’ successes. His method of “incremental development with continual review,” combined with a constant encouragement for students to learn by doing, were the keys. Shormann Math builds on John Saxon’s efforts to really teach math like the language of science that it is, by not just connecting students to their world, but, more importantly, to their Creator. In doing so, students learn to wisely mingle concepts like science and Scripture, faith and reason. Doing so makes it easier to learn subjects like calculus, which really does require a faith commitment in order to make sense of it. Because of its obvious connections to God’s attributes, secular calculus courses steer clear of this, and in so doing make it much more difficult to learn. Shormann Math will change that.

*John Saxon passed away in 1996, and the company he founded, Saxon Publishers, is now owned by Houghton-Mifflin/Harcourt. They have since created some Saxon-in-name-only Algebra 1, 2, and Geometry textbooks. Click here to read our review of the new books and to learn why we don’t recommend them.

The 10 Major Topics of Shormann Math

The 10 major topics of Shormann Math, compared to John Saxon's books and Common Core standards.

The 10 major topics of Shormann Math, compared to John Saxon’s books and Common Core standards.

Measurement is a topic that is a natural part of any math course seeking to teach math as the language of science. That it’s missing from three years of Common Core high school math is a huge problem. As a science class and lab teaching assistant during graduate school, one of the biggest math-related struggles I remember was students’ inability to convert from one unit to another. And it’s not just Common Core, most government school standards are weak in teaching measurement-related topics.

All Shormann Math high school courses will keep students fresh with working with measurements. Computers are also a very real part of every students’ world, so knowing about some of the mathematics behind them should be a priority. And, as mentioned in the Summary above, calculus becomes a normal part of high school math when one of the priorities is to connect students to their Creator.

Foundations and Pedagogy

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As Euclid famously said centuries ago, there is no royal road to learning. However, some methods are definitely better than others, and the Common Core’s integrated pathway (click here to read an Education Week review) is definitely a step in the right direction. However, the integrated approach has it’s own challenges. To really teach math like the active, hands-on language of science that it is, you have to teach it like languages are taught, or sports, or instruments, etc. You teach students a little bit about something, give them time to practice it, and then build on it. John Saxon called this “incremental development with continual review,” which is missing from Common Core.

Also missing from Common Core is the importance of math history. Understanding why they are learning the different math topics makes math more relevant to students. Learning some things about the people behind the math concepts they are learning, as well as some of the great, and not-so-great things they did, makes math more meaningful. And the connection to history also shines a bright light on the rich Christian heritage of mathematics, especially regarding algebra and calculus. Showing students how God’s attributes are clearly revealed in mathematics can make a huge difference in their comprehension and success in the course.

Shormann Math’s emphasis on math history means that, in developing the course, I dove deep into the classic works of Euclid, Newton, Euler, etc. Rather than reinventing the wheel, this study of the classics allowed me to develop a curriculum that stands on the shoulders of giants (a phrase often attributed to Isaac Newton). It should be a huge confidence-booster to parent and student alike to know your course is built on time-tested and proven methods for learning math.

What 3 Years of Math Covers

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By focusing on what matters most, Shormann Math does more in 3 years than either Common Core or Saxon. In the first two years, Shormann Math covers all the concepts presented on the SAT (the new 2016 version), the ACT, and both the CLEP College Algebra and CLEP College Math exam. A full credit of geometry is integrated into the first two years as well. This is different from John Saxon-authored texts, which include the geometry credit in years 2 and 3 (Algebra 2 and the first 1/3 of Advanced Math). And before you think Shormann Math couldn’t possibly have enough geometry, consider that we will cover all the standards, like perimeter/area/volume, similarity and congruence, circle and triangle theorems, and proofs. In addition, we will show students how the proof technique is not some isolated subject you only learn in geometry class, which is what most students, and parents, think it is. We’ll introduce students to proofs by studying the master, Euclid, covering several of his propositions. We’ll do the standard triangle proofs and circle proofs, but will also apply proof technique in other topics like algebra. And students will learn how proof is used in the real world. They’ll even learn how geometry is used in art and architecture. And on top of all that, we’ll introduce non-Euclidean geometry in Algebra 2, diving deeper in Precalculus. We’ll also use CAD programs like Geometer’s Sketchpad to complete proofs and more.

Finally, Shormann Math will introduce calculus fundamentals. By year 3 (precalculus), Shormann Math students will be very comfortable finding limits, and will have a solid grasp of derivatives and integrals. We hope all students will continue on to Shormann Calculus, but if not, they will be more than ready for college-level calculus. Of all the courses in college, calculus is the subject that opens the door to virtually every college major, or if the student cannot pass the class, closes the door on about 80% of majors. The first three years of Shormann Math will give students the confidence they need to take college calculus, and be at a level to help their peers learn it, which can also open up opportunities to build relationships and share the gospel. And completing 4 years of Shormann Math will allow students to possibly prepare for and pass either the CLEP or AP Calculus exam, receiving college credit for their efforts.

But Saxon + DIVE Lectures do a lot of this already. Why make a new curriculum?

There are many reasons, here are a few:

  • We can build the curriculum on a Christian and historical foundation, rather than bringing these fundamentals in from the side, like we do with the DIVE Lectures that teach Saxon Math.
  • The one topic John Saxon didn’t integrate was calculus. We think it just might be the most important topic to integrate, and our current Shormann Math students are proving Algebra 1-level students can learn some calculus fundamentals!
  • We don’t know how long Houghton Mifflin/Harcourt will continue to sell John Saxon-authored textbooks.
  • We can take advantage of 21st Century technology and e-learning to provide more efficient and effective learning. Our self-paced e-learning format includes many powerful learning tools, including video lectures and video solutions to homework, all for about the same price as the Saxon home study kits. The following table lists some detailed differences between Shormann Math and Saxon Math.

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What is the prerequisite for Shormann Math Algebra 1?

Students who have successfully completed a standard pre-algebra course, including either Saxon 8/7 or Saxon Algebra Half, are ready for Shormann Math Algebra 1.


Do you have any questions? Feel free to leave a comment!

Click here for a detailed description of Shormann Math, including sample video lectures and pricing information.

The 10 Major Concepts of Shormann Mathematics

August 26, 2014

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

August 18, 2014

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.

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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.