In this monthly column, we speak with a notable member of the mathematics education community about their work and their perspectives on the teaching and learning of mathematics. This month, we had the pleasure of speaking with Patrick Maidorn.
Patrick Maidorn has been a mathematics and statistics instructor at the University of Regina for the past twenty-one years. Apart from teaching undergraduate classes, Patrick has also been involved in the development of several mathematical outreach programs for students in Grades 1-12, including the University of Regina Math Camp and Math Circles, as well as the Canadian Math Kangaroo Contest.
Patrick grew up in Luxembourg, where he attended the European School. He holds a Bachelor of Science from the University of Guelph, a Masters of Mathematics from the University of Waterloo, and a Bachelor of Education from the University of Western Ontario. Realizing that each move took him further west on the map, he made one more westward leap to settle in Regina. He hopes to eventually get used to the cold winters of Saskatchewan. After two decades, he is still waiting.
First things first, thank you for taking the time to have this conversation!
You have been recognized for your hard work in mathematics education and outreach, including by the Pacific Institute for the Mathematical Sciences (PIMS), who awarded you the PIMS Education Prize in 2016. What kinds of mathematics camps, competitions, or other outreach activities are you involved in today?
In the past year, I’ve focused on expanding the size and scope of the outreach activities offered through the University of Regina. For example, the new Math Circle program is an extension of our evening problem solving sessions. The original sessions were mostly directed toward Grades 7-10, whereas the new format includes four concurrent classes that are available to all students from Grade 1 to high school. Their focus has also changed a little. While problem solving is still a large component, there is now also a bigger emphasis on open exploration of mathematical topics, partially in hopes of developing students’ sense of academic curiosity.
The decision, in particular, to include very young children in the Math Circle has led to a lot of interesting experiences. My favourite, by far, is witnessing a young student have a mathematical insight or make a mathematical connection far beyond their current curriculum level. The students’ energy and enthusiasm is also wonderfully infectious (if a little exhausting).
The University of Regina is are also continuing to run an annual one day Math Camp, as well as hosting the Canadian Math Kangaroo Contest on campus. Both of these events take place in March of each year.
What do these activities have to offer elementary and high-school students beyond their in-school experiences with mathematics?
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]There are many options for pursuing sports and creative arts outside of school. The options for students with academic interests are more limited, which is why these events are so important.[/perfectpullquote]Having such events offers those students with an affinity for mathematics an opportunity to explore the subject further. It also brings students into contact with other peers who are enthusiastic about mathematics. If a student has an interest in sports or creative arts, there are many options available to pursue these things outside of school. However, the options for students with academic interests can be more limited, which is why I feel that having these and similar events is so important.
Throughout the years, you have been instrumental in organizing several math competitions in the province, and were a lead writer for the Saskatchewan Math Challenge, a provincial math challenge co-sponsored by the Saskatchewan Math Teachers’ Society and the University of Saskatchewan for students from grades 7 to 10. From where do you draw your inspiration for new puzzles and problems?
First and foremost, I love the playful aspect of mathematics. When I see, read, or experience things throughout the day, my first instinct is often to play around with the ideas, look for patterns, and make mathematical connections. For example, when I first moved to Saskatchewan, I didn’t know much about the rules of football. When I was watching a game, a friend explained the strategy behind trying for a touchdown or a field goal (i.e., balancing the probability of success and the possible pay-off). My first thought was “This would make a great ‘expected value’ problem for my stats class!” Now, I still don’t know much about football, but I’ll always remember the touchdown and field goal point values.
Of course, many new problems are variations of what has come before, and I have to give credit to the mathematical community for being so generous with its resources. There are many authors who will invite others to try out, share, and modify their own problems and puzzles. So, when I come across an interesting problem, I often play around with it and see if it can be expanded or adapted for other purposes.
In your view, what makes a “good” problem?
There are so many ways to answer this question. It really depends on what the purpose of the problem is. Am I posing the problem in a contest to test or challenge the students? Am I using the problem in a classroom to introduce a new concept or to have the problem’s solution open up a new avenue to further ideas?
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]A good problem has mathematical value. Its solution should expand our existing knowledge, create connections between previously unrelated ideas, or lead to an entirely new problem.[/perfectpullquote]Ideally, a problem should be both accessible and interesting. This means that were we to encounter the problem, we would want to explore it further—even if it wasn’t a question on an exam or a problem posed in a class. Something about the problem triggers us to think “Hmm, I wonder how this works?” If there is an internal motivation to start manipulating the pieces of the puzzle before us, we’re much more open to discovering pathways to a solution. Also, a good problem has mathematical value. Its solution should expand our existing knowledge, create connections between previously unrelated ideas, or lead to an entirely new problem.
Take, for example, the classic problem “How many squares can you find on a chessboard?” This question is immediately approachable at any level. It invites us to ask questions to refine the problem, such as “What exactly counts as a square?” It also leads us to start looking for patterns, as soon as we realize that the final answer will be larger than we care to count to using our fingers and toes. Further, working through the solution teaches us the value of being systematic. Finally, having solved the problem, a natural follow-up question might be “Is there a quicker way to do this for larger grids?” This could connect to the idea behind summation formulas, and then you’re on your way exploring Gauss’ addition algorithm or developing the sum of squares formula.
What advice do you have to offer to students who would like to enhance their problem-solving skills, beyond exposure to a variety of problems and practice? What are some of the skills, habits of mind, or strategies that effective problem solvers rely on?
Even though you have asked me to look beyond “practice,” I think this is such an important aspect of learning mathematics that I want to highlight it again. When we see the same kind of problem a number of times, we learn strategies that can become part of our permanent mathematical toolbox. Then, when we encounter a truly new problem, we are much more likely to recognize a familiar element in the problem that might at least suggest a possible solution path.
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]To really benefit from practice, a good guide is important—be it a teacher, a textbook, or another resource.[/perfectpullquote]To really benefit from this type of practice though, a good guide is important—be it a teacher, a textbook, or another resource. A guide can help us select problems that are appropriate to the stage that we’re at. Just jumping between random problems can lead to a lot of frustration, as we have to start at “square one” each time.
What drew you, personally, to study—and then to teach—mathematics in your younger years? What fuels your teaching and outreach work today?
I enjoyed many subjects in school—science, computing, economics—but I quickly realized that it was the underlying mathematical aspects in each that I enjoyed the most. This led me to the decision to study mathematics in university. It also is a big part of my motivation to teach. I want to help inspire and foster in others the same appreciation I feel for mathematics—whether it is math’s many connections to the world around us or its intrinsic beauty. My favourite reaction is when a student tells me that they started a class with trepidation and ended up surprised that they actually enjoyed it. It might not happen often, but, in that moment, I feel that I made a difference.
Interviewed by Ilona Vashchyshyn
