# Mathematics Research at Secondary Level (High school Level)

I am a future teacher and am interested in incorporating a mathematics research unit where my students do their own research on an unsolved problem. I have a couple ideas on problems, but am difficulty finding scholarly results/benefits/examples of doing a research unit within the secondary setting. Would someone be able to point me to some scholarly articles and papers that could be used as evidence to convince administrators? Thank you!

• Having taught at a secondary level, and now nearing research, I strongly suggest that the "research" involved is something like doing a report on a modern or historical mathematician, and not actually conducting mathematics research. There are very few contemporary problems where the statement is understandable to highschoolers (much less undergraduates), and those that are understandable (e.g. Collatz conjecture) often lead to "crankery" -- students who believe that they have solved the problem because they do not understand the problem with their proof and cannot be convinced otherwise. – Opal E Jan 7 '19 at 17:21
• @OpalE "the statement is understandable to highschoolers (much less undergraduates)" -- Um, did you mean "the statement is understandable to undergraduates (much less highschoolers)"? – Rusty Core Jan 7 '19 at 17:56
• @RustyCore The question does not mention a country. Which one are you writing about? – Tommi Jan 8 '19 at 7:43
• This is a lovely idea, and great to do if carried out well. James Tanton has done great things with young people doing math that has not been done before. – Sue VanHattum Jan 8 '19 at 19:12
• @TommiBrander Indeed. I was writing about the U.S. – Rusty Core Jan 8 '19 at 20:32

This kind of echos Opal E's answer but with some specific problems I used for one of my "research projects" when I was teaching a class of standard level seniors (they were on the algebra II / Precal level). I also agree that modern day unsolved problems are extremely difficult to understand and even the easy sounding ones (Goldbach Conjecture, Collatz Conjecture, etc) are much deeper than they let on, making students believe they know much more about the problem than they really do. (EDIT: for a specific example of this, I showed a very bright student the " legendary question #6" and he was convinced he had found a simple solution after only 30 minutes. Most times, students think a simple question means a simple solution) Although, if you really do want to used unsolved problems, then this link could help get you started. They have unsolved problems for students to explore from every grade K-12. I haven't used their resources, but they do look like very interesting/engaging problems.

When I wanted to do a "research project," I wanted them to research the important landmark theorems of the past, the ones that marked major changes in the history or development of math. The students had to research what the problem meant why it was important in the history/development of math. I don't remember all of the problems I used but here's a short list:

1) Solving the Cubic Equation (eventually led to Galois Theory and Group Theory)

2) 4 Color Theorem (first rigorous proof completed via computer)

3) Bridges of Konisberg (Led to Graph Theory)

4) How easily can people factor a large number with the aid of a computer? (Leads to RSA encryption) Additionally, how can we tell if a large number is prime?

5) Is it true that for a given line and a point not on the line, there is only one line through the point parallel to the given line? (leads to the debate about the parallel postulate and the discovery of non-euclidean geometry).

6) How did the ancient mathematicians figure out the decimal representation of transcendental numbers? How do we know $$\pi=3.141592...$$? How do we find rational approximations, for example $$\pi \approx \frac{22}{7} \approx \frac{333}{106} \approx \frac{355}{113}$$

7) Which set has more numbers in it: a) The set of positive integers b) The set of all integers c) The set of numbers between 0 and 1? (Leads to the ideas of multiple infinities and how to show two sets have equal cardinality).

8) I think I also had a questions about what an irrational number is, how do we know $$\sqrt{2}$$ is irrational? Why was Pythagoras so upset at this. Are most numbers irrational or rational?

I can't remember if I had other problems for them too look up, those are the ones I can remember. Some groups did better than others, but the point wasn't for them to understand the proof of the Four Color Theorem, but just know how that one theorem represented a big change in rigorous mathematics.

I have taught high school for two years, and am now a Ph.D. student in mathematics. I continue to teach undergraduate courses.

Some high schoolers do participate in mathematics research. For example, the AMS put out a notice regarding the PRIMES program for advanced high schoolers -- who, first of all, are passionate about mathematics (such a thing is necessary to do mathematics research), and second of all, have at least a calculus background. The article then details the problems which students tackle, and the difficulty of devising problems to which a high schooler with adequate mathematical maturity can be built up to over a few months. Unless you are going to be teaching at a highly specialized school, I strongly advise against attempting to get high school students to do mathematical research.

However, if you intend for them to learn "research skills" in generality, I believe that a mathematical history project that aligns with your course goals would be an education-research-supported unit. I refer you to "Using History in Mathematics Education" (Fauvel, 1991), which discusses the notion of using history in mathematics education and summarizes some methods of doing so. A more recent discussion of this topic is found in "History of Mathematics in Mathematics Education" (2016) which discusses the following:

• Which history is suitable, pertinent, and relevant to Mathematics Education (ME)?

• Which role can History of Mathematics (HM) play in ME?

• To what extent has HM been integrated in ME (curricula, textbooks, educational aids/resource material, teacher education)?

• How can this role be evaluated and assessed and to what extent does it contribute to the teaching and learning of mathematics?

In my personal and professional opinion, tackling historical problems is much more accessible to high school students, and can give them the experience of devising their own question and seeking an answer in a real-world context.

Another possible way you could get high school students having a research experience without needing advanced knowledge is is via a multidisciplinary unit together with a science teacher. Mathematics, in particular statistics, is useful in answering many science problems. If your students have a statistics unit, you could have them be the "statistics team" and a science class be a "science team", and work together to produce a statistically and scientifically meaningful result for an end-of-year project. I did this together with an advanced science elective when I taught AP Statistics and it was very successful. I may add details later if requested.