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.

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.

Awesome post! I always tell teachers that children are capable of soooo much more than the present educational system requires of them. Thank you for this.

Absolutely! And I appreciate your suggestion for collaboration between natural language teachers and math teachers.

I agree. I think students can develop an understanding of the concepts behind these areas of mathematics and learn how to use the computational tools that are useful for working with these concepts. I think that areas which are interesting to explore lead to students being interested in learning more about these areas, which leads them to wonder how the ideas work, and this leads them to being driven to understand the concepts behind the phenomena.

different bases for number systems - binary, octal, hexadecimal but allow all students to design/choose their own base - this will show that diff base numeral systems are really just diff "languages" of quantity - helps students "see" that there is more than one way to skin a cat

Part of the reason those aren't taught in elementary or even high school is that most math teachers know nothing about them. How many math teachers are actually mathematicians?

Here in the UK, graph theory is taught at school to some extent, at A-level.

I'm not sure I agree that teaching chaos theory is totally a good idea... it's certainly a fun thing to have a one-off lesson on, since the ideas are fun and graspable, and there are lots of exciting pictures... but to actually solve any problems in it, and understand what you're doing, is pretty hard.

I nominate Pick's theorem as a great topic for schools. It's genuinely useful, especially in the computer age, is easy to apply with an unexpected but still fairly simple statement, and its a great example of a mathematical proof. I have some slide on it here: http://richardelwes.co.uk/wp-content/uploads/2012/02/PickEhrhart.pdf

I recall getting taught set theory as part of the new math curriculum. It worked for me but for many people it was just confusing. Possibly because most teachers weren't familiar enough with it and didn't get proper training about how to present it.

Interesting, but I find the post title to be largely false. Many of these things I was taught at some level as a student (and I was in elementary school in the 90s). Now I just completed my elementary education bachelors and again, much of this was represented in curicula we looked at, either in math, or in science. The rest (particularly much of the fields of #3 and #4) is avoided at an elementary level as it is simply not developmentally appropriate. Much of the subject matter of all five of these deals heavily in the abstract, and the brain doesn't deal well with that until around the middle school years or later.

I was exposed to set theory early in elementary school (2nd or 3rd grade), graph theory in 4th grade, and boolean algebra around 7th grade (all in the early 1990s).

However, I don't remember really loving set theory or boolean algebra.

There were many things that I really enjoyed in math class, such as working with different bases (1st or 2nd grade), algebra (7th grade), geometry proofs and trig (8th grade), statistics and the first few units on differentiation and integration (high school), and linear algebra and its uses in image processing and structural analysis (college).

I think the difference between topics that were enjoyable and felt mind-expanding and made me want to be a mathematician, versus topics that were intimidating or just tedious and made me wish I were doing something else... was the teaching. A good math teacher can take almost any topic in math and make it about DISCOVERY. Before even revealing what is being taught, the teacher states a problem that can be solved using the upcoming material. You FEEL that there is a unique answer, but you don't know how to get to it, and now you WANT to learn how. If a specific and simpler form of the problem is presented, maybe the students can solve it using common sense, and then the teacher can ask "How do things change when we add this complexity? How could we deal with it?". A technique is proposed either by the teacher or by students, and is used successfully. The teacher then asks "Will this work every time? Can anyone think of a time when this would not work?". Someone - teacher or student - comes up with an example where that technique wouldn't work. Then they either try to patch the problem-solving methods further, or say "Solving that kind of problem would require much more complicated techniques. They're called such-and-such if anyone wants to look them up later, but we'll only teach this in the classroom about a year from now". Everything from division and fractions (2nd grade) to multivariable calculus can, and should, be taught this way. THAT is what thinking like a mathematician is like.

I didn't enjoy boolean algebra or set theory because they were taught by bad teachers who went "Now we are going to learn X. Here are some definitions. Here are some properties between these kinds of things. So for example, when a problem looks like this, you get to the solution by doing this. Got it? Now do five examples by yourself and raise your hand if you have a question". Bo-ring!

I loved graph theory, though, which I did in 4th grade. The fact that all polyhedra we could think of (except ones with holes) have an Euler Characteristic of 2 seemed like MAGIC to me. Why was this true? The proof involved flattening the polyhedra onto a plane. The equivalency between solids and these 2D diagrams was super interesting. And then the proof (removing either one edge, or X edges and X-1 vertices, in either case combining two faces into one, until you have two faces - an "outside" one and an "inside" one bounded by X edges and X vertices) was AMAZING. It was the first time I had been exposed to a proof, and to the idea that you could pick apart WHY something was true in math. You could OWN the truth of the mathematical process in a way I had never experienced. Because it was a process of discovery, not just a simple fact.

So, yeah, as long as things are taught that way, I think kids will appreciate "mathematician thinking" even without learning topological or computational topics like these. (And by the way, everyone here has read Paul Lockhart's "A Mathematician's lament", right? If not, Google it! It makes the same point buch much more richly).

What you are saying about learning things in historical order agrees with something I have noticed in economics. More recent developments in the field, like game theory and agency theory, are considered advanced, when in fact they are accessible and interesting to new students.

I had set theory in third grade (involving cows, pigs,etc.) and I loved it, but it never got integrated into the rest of the curriculum and I can't say I got a lot of use out of it.