5 Recent Mathematical Breakthroughs That Could Be Taught in Elementary School (but aren't)

In a previous blog post, I made the claim that much of the math curriculum is ordered based on historical precedent rather than conceptual dependencies. Some parts of the math curriculum we have in place is based on the order of discovery (not always, but mostly) and while other parts are taught out of pure habit: This is how I was taught, so this is how I'm going to teach. I don't think this needs to be the case. In fact, I think that this is actually a detriment to students. If we want to produce a generation of mathematicians and scientists who are going to solve the difficult problems of today, then we need to address some of the recent advances in those fields to prepare them. Students should not have to "wait until college" to hear about "Topology" or "Quantum Mechanics". We need to start developing the vocabulary for these subjects much earlier in the curriculum so that students are not intimidated by them in later years.

To this end, I'd like to propose 5 mathematical breakthroughs that are both relatively recent (compared to most of the K-12 curriculum) while also being accessible to elementary school students. Like any "Top 5", this list is highly subjective and I'm sure other educators might have differing opinions on what topics are suitable for elementary school, but my goal here is just to stimulate discussion on "what we could be teaching" in place of the present day curriculum.

#1. Graph Theory (c. 1736)

The roots of Graph Theory go back to Leonard Euler's Seven Bridges of Königsberg in 1736. The question was whether or not you could find a path that would take you over each of the bridges exactly once.

Bridges of Königsberg

Euler's key observation here was that the exact shapes and path didn't matter, but only how the different land masses were connected by the bridges. This problem could be simplified to a graph, where the land masses are the vertices and the bridges are the edges.

This a great example of the importance of abstraction in mathematics, and was the starting point for the field of Topology. The basic ideas and terminology of graph theory can be made easily accessible to younger students though construction sets like K'Nex or Tinkertoys. As students get older, these concepts can be connected to map coloring and students will be well on their way to some beautiful 20th century mathematics.

#2. Boolean Algebra (c. 1854)

The term "algebra" has developed a bad reputation in recent years. It is often referred to as a "gatekeeper" course, which determines which students go on to higher level mathematics courses and which ones do not. However, what we call "algebra" in middle/high school is actually just a subset of a much larger subject. "Algebra I" tends focuses on algebra as it appeared in al-Khwārizmī's Compendious Book on Calculation by Completion and Balancing (circa 820AD). Consequently, algebra doesn't show up in the math curriculum until students have learned how to add, subtract, multiply and divide. It doesn't need to be this way.

In 1854, George Boole published An Investigation of the Laws of Thought, creating the branch of mathematics that bears his name. Rather than performing algebra on numbers, Boole used the values "TRUE" and "FALSE", and the basic logical operators of "AND", "OR", and "NOT". These concepts provided the foundation for circuit design and eventually lead to the development of computers. These ideas can even be demonstrated with a variety of construction toys.

The vocabulary of Boolean Algebra can and should be developed early in elementary school. Kindergartners should be able to understand basic logic operations in the context of statements like "grab a stuffed animal or a coloring book and crayons". As students get older, they should practice representing these statements symbolically and eventually how to manipulate them according to a set of rules (axioms). If we develop the core ideas of algebra with Boolean values, than perhaps it won't be as difficult when these ideas are extended to real numbers.

#3. Set Theory (c. 1874)

Set Theory has its origins in the work of Georg Cantor in the 1870s. In 1874, Cantor published a ground breaking work in which he proved that there is more than one type of infinity -- the famous "diagonal proof". At the heart of this proof was the idea of thinking of all real numbers as a set and trying to create a one-to-one correspondence with real numbers. This idea of mathematicians working with sets (as opposed to just "numbers") developed momentum in the late 1800s and early 1900s. Through the work of a number of brilliant mathematicians and logicians (including Dedekind, Russell, Hilbert, Peano, Zermelo, and Fraenkel), Cantor's Set Theory was refined and expanded into what we know call ZFC or Zermelo-Fraenkel Set Theory with the Axiom of Choice. ZFC was a critical development because it formalized mathematics into an axiomatic system. This has some suprising consequences such as Gödel's Incompleteness Theorem.

Elementary students probably don't need to adhere to the level of rigor that ZFC was striving for, but what is important is that they learn the language associated with it. This includes words and phrases like "union" ("or"), "intersection" ("and"), "for every", "there exists", "is a member of", "complement" ("not"), and "cardinality" ("size" or "number"), which can be introduced informally at first then gradually formalized over the years. This should be a cooperative effort between Math and English teachers, developing student ability to understand logical statements about sets such as "All basset hounds are dogs. All dogs are mammals. Therefore, all basset hounds are mammals." Relationships can be demonstrated using visual aids such as Venn diagrams. Games such as Set! can further reinforce these concepts.

#4. Computation Theory (c. 1936)

Computation Theory developed from the work of Alan Turing in the mid 1930s. The invention of what we now call the Turing Machine, was another key step in the development of the computer. Around the same time, Alzono Church was developing a system of function definitions called lambda calculus while Stephen Kleene and J.B Rosser developed a similar formal system of functions based on recursion. These efforts culminated in the Church-Turing Thesis which states that "everything algorithmically computable is computable by a Turing machine." Computation Theory concerns itself with the study of what we can and cannot compute with an algorithm.

This idea of an algorithm, a series of steps to accomplish some task, can easily be adapted for elementary school instruction. Seymour Papert has been leading this field with technologies like LOGO, which aims to make computer programming accessible to children. Another creative way of approaching this is the daddy-bot. These algorithms don't need be done in any specific programming language. There's much to be learned from describing procedures in plain English. The important part is learning the core concepts of how computers work. In a society pervaded by computers, you can either choose to program or be programmed.

#5. Chaos Theory (c. 1977)

Last, but not least, is Chaos Theory -- a field of mathematics that developed independently in several disciplines over the 1900s. The phrase "Chaos Theory" didn't appear in the late 1970s, but a variety of phenomena displaying chaotic behavior were observed as early as the 1880s. The idea behind Chaos Theory is that certain dynamic systems are highly sensitive to initial conditions. Drop a shot of half-half into a cup of coffee and the resulting pattern is different every time. The mathematical definition is a little more technical than that, but the core idea is relatively accessible. Chaos has even found several notable references in pop culture.

The other core idea behind chaos theory is topological mixing. This could be easily demonstrated with some Play-Doh (or putty) of two or more colors. Start by combining them into a ball. Squash it flat then fold it over. Repeat it several times and observe the results.

The importance of Chaos Theory is that it demonstrates that even a completely deterministic procedure can produce results that appear random due to slight variations in the starting conditions. This can even be taken one step further by looking at procedures that generate seeming random behavior independently of the starting conditions. We live in an age where people need to work with massive amounts of data. The idea that a simple set of rules can produce extremely complex results provides us with tools for succinctly describing that data.

Conclusion

One of the trends in this list is that these results are easy to understand conceptually but difficult to prove formally. Modern mathematicians seem to have a tendency towards formalism, which is something of a "mixed blessing". On one hand, it has provided mathematics with a firm standard of rigor that has withstood the test of time. On the other hand, the language makes some relatively simple concepts difficult to communicate to younger students. I think part of the reason for this is that the present curriculum doesn't emphasize the rules of logic and set theory that provide the foundation for modern mathematics. In the past, mathematics was driven more by intuitionism, but the math curriculum doesn't seem provide adequate opportunities for students to develop this either! It might be argued things like "new math" or "Singapore math" are helping to develop intuitionism, but we're still not preparing students for the mathematical formalism that they'll be forced to deal with in "Algebra I" and beyond. Logic and set theory seem like a natural way to develop this familiarity with axiomatic systems.

Observers might also note that all five of these proposed topics are related in some form or another to computer science. Computers have been a real game-changer in the field of mathematics. Proofs that were computationally impossible 500 years ago can be derived a in minutes with the assistance of computers. It's also changed the role of humans in mathematics, from being the computer to solving problems using computers. We need to be preparing students for the jobs computers can't do, and my hope is that modernizing the mathematics curriculum can help accomplish this.

Do you have anything to add to this list? Have you tried any of these topics with elementary students? I'd love to hear about your experiences in the comments below.

Pre-Calc Post-Calc

Gary Davis (@republicofmath) wrote an article that caught my attention called What's up with pre-calculus?. In it, he presents a number of different perspectives on why Pre-Calc classes have low success rates and do not adequately prepare students for Calculus.

My perspective on pre-calculus is probably far from the typical student, but often times the study of "fringe cases" like myself can provide useful information on a problem. The reason why my experience with Pre-Calc was so atypical, is because I didn't take it. After taking Algebra I, I had started down a path towards game programming. By the end of the following year, where I had taken Geometry, this little hobby of mine hit a road block. I had come to the realization that in order to implement the kind of physics that I wanted in my game I would need to take Calculus. I petitioned my counselor to let me skip Algebra II and Pre-Calc to go straight into AP Calculus. They were skeptical at first, but eventually conceded to my determination and allowed me to follow the path I had chosen.

Skipping from Geometry to Calculus meant that there were a lot of things that I needed to learn that first month that many of my peers had already covered. I had never even heard the word "logarithm" before, had no idea what e was, and had only a cursory understanding of trigonometry. These were the topics I had missed by skipping Pre-Calc, and I was fully aware of that, so I "hit the books" and learned what I needed to know about them. By the end of that first month I had caught up to the rest of the class and by end of the semester I would be helping other students with those very same topics.

I think the most obvious difference between myself and the "typical Calculus student" was the level of motivation. Many of the students in Calculus were there because "it would look good on a college application". I was there because I wanted to be there. A common problem throughout math education is the "When am I ever going to use this?" attitude. I already knew where I was going to use the math I was learning. I had an unfinished game at home that needed a physics system, and every new piece of information I learned in Calculus made me one step closer to that goal. If you had ever wondered why a 4th order Runge-Kutta method is better than Euler's method, try writing a platformer.

The second difference was a little more subtle, but there were some conceptual differences in how I thought about exponential, logarithmic, and trigonometric functions. The constant "e" wasn't just some magic number that the textbook pulled out of thin air, it was the the unique number with the property that  \frac{de^x}{dx} = e^x and  \int e^x dx = e^x . When it came to sine and cosine, I would think of them like a circle while my other classmates would picture a right triangle. They would hear the word "tangent" and think "opposite over adjacent", but I thought of it more like a derivative. Sure, I had to learn the same "pre-calc" material as they did, but the context of this material was radically different.

A couple years ago I suggested that Pre-Calc should be abolished. The trouble with Pre-Calculus (at least in the U.S.) is that the course needs to cover a very diverse array of questions which includes exponential, logarithmic and trigonometric functions. I would argue that these concepts are not essential to understanding the basic foundations of Calculus. The math curriculum introduces the concept of "slope" in Algebra I, which is essentially the "derivative" of a line. There's no reason why we should be sheltering students from language of Calculus. The concepts of "rate of change" and "accumulation" can and should be connected with the words "derivative" and "integral", long before students set foot in the course we presently call Calculus. As students become more comfortable with these concepts as they relate to lines, parabolas and polynomials, then gradually step up the level of complexity. When students start to encounter things like surfaces of revolution, then they'll actually have a reason to learn trigonometry. Instead of trigonometry being the arbitrary set of identities and equations that it might appear to be in pre-calc, students might actually learn to appreciate it as a set of tools for solving problems.

I think this issue of Pre-Calc is really a symptom of a larger problem. The mathematics curriculum seems to be ordered historically rather than conceptually. I've heard Pre-Calc described as a bridge to Calculus. This makes sense when you consider the historical development of Calculus, but not when considering the best interest of students in today's society. Leibniz and Newton didn't have computers. Who needs bridges when you can fly?