Mathematics research is not something that most school teachers are able to experience at University. You can help these students effectively as they do their own research, even if you don’t understand their research topic or how to solve it.
The following is an edited extract of an article in Vinculum, a journal for secondary mathematics teachers published by the Mathematics Association of Victoria (MAV). We’d like to thank Vinculum and MAV for granting permission to publish part of the article here.
Mathematics research in schools
Dr Paul Fijn, School of Mathematics and Statistics, University of Melbourne
With the Maths and Stats Outreach Team at the University of Melbourne: Dr Susan James, Dr Cindy Huang, Dominic Maderazo and Cait Pryse.
Getting started
In many cases, starting with small examples will help with understanding the question/checking that you understand the “rules” defined in it. Some questions might be scaffolded to already do this, but it is always useful to start with some small examples. Students who enjoy using technology can also use this to work out some larger examples which, otherwise, might be too tedious to check by hand.
Guiding students
What distinguishes maths research from other fields is usually the abstract nature and the concept of “proof”. In other sciences, proof is typically an indication that it is the best theory currently fitting all known information. However, in mathematics it can be possible (but sometimes very difficult) to prove that a formula always works. While a proof might be the aim, it is usually better to think about justifying reasoning: progress is the aim, not simply finding the solution (if it exists!).
To help students learn to think in this way, here are some prompts:
- Does there seem to be any patterns in what you’ve found so far?
- Would presenting the information another way reveal more interesting aspects of the problem?
- Do you think that would work if the [problem] was [harder, larger, more complicated, different]?
- That’s a fantastic formula/theory you’ve come up with, is there a way to find out if it always works?
- Is there a formula you can find, rather than just calculating lots of possible answers?
It can also be difficult to help students find resources: some might be inaccessible to an audience with their background in mathematics (eg Khan Academy, depending on the topic), some are only known by those who use them frequently (eg the Online Encyclopedia of Integer Sequences, oeis.org), and others might be unreliable (eg any AI, particularly large language models, can give incorrect information).
However, this is the key to re-search: you need to actually look at a range of sources, and work out which ones might be helpful to you. Learning how to search: which words/phrases should I look up? Which results should I look at? Colleagues who trained as teachers in a field other than mathematics have an advantage here, where looking more broadly is encouraged. This can make it a collaborative supervision/guidance process, too, not just for students to collaborate!
Finally, it is important to think of research as a process, and trying to make progress: there will always be things they could have thought about, things they chose not to investigate, problems they didn’t solve. This is both normal and expected! You also do not need to know if everything is “correct”: the best research projects are those that explain the story of the research well.
Communicating mathematics
As a student who was good at maths in school, I undervalued the communication of maths: if you got the right answer, why were there such arcane rules about exactly which steps you needed to show? Fortunately, I learned the error of my ways and came to think of maths as not just the solution, but communicating the thought process and reasoning as well.
- Start with the story: explain the problem, and your approach to it.
- Using notation: mathematical notation can be very efficient, but a good guide is “use notation if it makes things easier to understand”.
- Visualising mathematics: a graph, number line or photograph of an experiment can be an excellent way to show a pattern or examples.
Critical evaluation
My favourite strategy when writing anything is to both start and finish by writing the introduction. Writing the introduction at first helps to clarify the problem, think about what story you want to tell, and explain how you chose what to do. (Re-)writing it last means you can make sure it matches the story you actually told, even if your plans changed from the beginning – this is a normal part of research; we don’t always end where we expected to!
Getting feedback (eg from family, other students) can also help: people who don’t know the problem or anything that they have done will quickly pick up when something is unclear or when something “obvious” has not been mentioned.
Celebrate!
Mathematics research (like all research) can be fun, interesting and you can learn a lot from it. It’s important to recognise that making progress (sometimes even just understanding the problem can take a lot of work!) is the goal, not “finishing” a project. Research is rarely “finished”: try to remind students that presenting progress is just as much of an accomplishment.