# Webinar Questions – Changes

**What are the fundamental things that have changed in mathematics during the last century and the beginning of the current century?**

Wow. Big question.

To really understand it you will benefit from an overview of mathematics as our ancestors developed it over time.

There are several things that are involved in the development and understanding of math.

- Math concepts, Numbers – Geometry – Algebra – Trig – Calculus – More
- Tools, to solve problems and understand concepts
- Experiments to understand the concepts and tools
- Heuristic Arguments Intuitive and Common Sense reasoning
- Rigorous Arguments Proofs
- Abstraction Creation of unifying abstract math concepts e.g. Group or Field

The ancients worked a lot with 1,2,3, and 4.

Euclid developed 5, in Euclid’s Elements for Geometry . This was challenging and somewhat limiting. For example, the Euclideans could prove that the definition of Pi as the ratio of C/D was independent of the size of a circle. But, they could not prove a formula for the area of a circle being CxD/4 or PiR^{2 }as we commonly represent it today.

Archimedes worked very hard on 4. He heuristically proved the area of a circle was CD/4. He also heuristically proved the formulas for the surface area and volume of a sphere. This is what he wanted inscribed on his Tombstone. It was an amazing feat, and presaged calculus that was to come almost two thousand years later.

Much of the technology used by the ancients was based on rules of thumb experimentally derived and perhaps heuristically proved.

In the 1500’s through the 1700’s, the concepts of calculus and differential equations were invented and the Fundamental Theorem of Calculus was heuristically proven. This was the foundation of modern science and technology. Newton and Leibniz both developed calculus heuristically and Newton used it in the creation of his theories of physics. However, when he published them in Principia Mathematica he utilized the formal Euclidean geometry and proof concepts to present his theories. Very hard to understand today.

Leibniz’s formulation of calculus was actually better than Newton’s and spawned a math revolution on the Continent that evolved into our modern math, starting with the Bernoulli family and then their genius disciple Leonhard Euler.

Euler in the 1700’s extended the work of Leibniz and Newton extensively utilizing brilliant heuristic arguments. Euler created the first modern math educational materials.

The problem with heuristic arguments is that they sometime lead to false results when false assumptions are made from weak intuition. When mathematicians began to deal with the concepts of infinity and infinite series, this was a problem.

So in the 1800’s mathematicians began to apply Euclidean rigor to the more advanced math concepts from calculus and infinite series and differential equations. A side effect of this was the banning of infinitesimals which mathematicians from Archimedes to Euler had used extensively in heuristic arguments. The 19^{th} century mathematicians couldn’t figure out any way to include them in their rigorous treatments. Subsequently they were abandoned in calculus textbooks. Engineers and scientists kept right on using them heuristically since they worked so well and were so intuitively understandable. There was quite a split between theoretical mathematicians and applied mathematicians and scientists.

Ironically, mathematicians were able to treat infinitesimals rigorously in the 1960’s and they are now back in rigorous math. Unfortunately, the calculus textbooks have not caught up.

The first half of the 20^{th} century saw a great profusion of abstract concepts and theories. This has a wonderful unifying effect on math. Unfortunately, rigorous proofs often are not very intuitive and heuristics were downgraded in favor of rigor in some of our textbooks.

Also, the tools that mathematicians had invented such as logarithms for arithmetical calculations were now superseded by modern technologies such as the computer. These new tools made experimentation and heuristic arguments much easier and more valuable in creating new concepts and understanding concepts in general.

These new tools made much of the math taught with the old tools obsolete. Many of the algorithms, a kind of tool, became obsolete.

For example, now one can learn all of the practical algebra and geometry one needs for practical industrial math very quickly by using the scientific calculator to perform all of the arithmetic calculations. Logarithms, and slide rules, the popular and best tools became obsolete in the 1970’s.

Then in the 1980’s computer tools made all of the manual algorithms of calculus and differential equations and linear algebra obsolete. Mathematica was introduced in 1988 and overnight revolutionized how STEM professionals solved STEM math problems.

These tools were expensive and, thus, did not make it into the SMC for high schools and elementary college. They stuck to the old obsolete tools.

However, in 2009, Wolfram Alpha, WA, a free version of Mathematica became available over the Internet. Now, many pre-calculus concepts and calculus and differential equations can be learned and mastered much quicker and easier. The SMC has not caught up with this at all.

The old algorithms taught in our SMC are like horse and buggy tools and the new tools like WA are like a modern automobile or jet plane.

Triad Math’s new five Tier math program is an example of a modern math curriculum that should supersede the old SMC. Just as the scientific calculator superseded the slide rule eventually something like the Triad Math program will supersede the SMC.

Until that happens our students are at a severe disadvantage and handicap compared to international students, or students lucky enough to take a modern program.

Home school students have the freedom to adopt the new modern program. Homeschoolers may lead the way to a reformation of the SMC for all students in our traditional schools.