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 Susan Milner.
Susan Milner taught post-secondary mathematics in British Columbia for 29 years. For 11 years she organised UFV’s secondary math contest, where her favourite part was coming up with post-contest activities for the participants. In 2009 she started Math Mania evenings for local youngsters, parents and teachers. This was so much fun that she devoted her sabbatical year to adapting math/logic puzzles and taking them into K-12 classrooms. Now retired and living in Nelson, BC, she is still busy travelling to classrooms and giving professional development workshops. In 2014 she was awarded the Pacific Institute for the Mathematical Sciences (PIMS) Education Prize.
First things first, thank you for taking the time for this interview!
Thank you so much for inviting me to participate—I love talking shop!
One of your passions in life has been to enhance public awareness and appreciation of mathematics—a passion that has led you to develop and become involved in a wide variety of outreach activities, including workshops, classroom visits, and public events, for which you received the Pacific Institute for the Mathematical Sciences (PIMS) Education Prize in 2014. As Alejandro Adem, a former PIMS Director, remarked, “Susan Milner is an outstanding educator, who has worked tirelessly to share the joy of mathematics with countless students and teachers in BC” (PIMS, 2014).
What drew you to study, and then to teach, mathematics in your younger years? What fuels your outreach work today?
All the way through school, math always seemed like fun, especially when I learned something new. Those aha! moments are really quite addictive. I remember loving to figure out patterns for myself—staring with the times table in Grade 3, re-inventing parts of modular arithmetic so I would know how at what time I had to go to bed in order to get the right amount of sleep, and teaching myself function notation in order to write math contests.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]To me, the most appealing aspect of mathematics is that people can reason with each other, without resorting to emotional brow-beating, and without appealing to authority, power, or wealth.[/perfectpullquote]To me, the most appealing aspect of mathematics is that people can reason with each other, without resorting to emotional brow-beating, and without appealing to authority, power, or wealth. From the time I was 10 years old, Mr. Spock has been my hero, not because of how much he knows, but because of how clearly he lays out arguments.
At university, I first studied philosophy and classics, taking calculus courses just for fun during my first degree. I thought about doing graduate work in philosophy, logic in particular, but that seemed impractical, so after I finished a degree in mathematics, off I went to do more math. Of course mathematics is more practical than philosophy, right? I ended up doing logic in topos theory, an abstract branch of category theory, which is in turn a highly abstract branch of mathematics. So much for practicality!
I then fell into teaching. When I took what was supposed to be a year off after my Master’s degree to teach at Okanagan College, I found that I thoroughly enjoyed working with adults. It was exciting to try to help them make sense of complicated ideas, to watch light bulbs going on, and to watch students get excited about what they were learning. That was always the best part of teaching, at any level, in any course.
Now that I have retired from teaching, I get to focus exclusively on having fun with students of all ages. I love watching them get caught up in a new game, figuring things out, and developing their abilities to think logically. I also really enjoy the excitement many teachers express as they watch their students get highly engaged and demonstrate abilities that they didn’t know they had.
One of the main ways you have sought to share the joy of mathematics is by introducing children and adults of all ages to a variety of mathematical puzzles and games [see www.susansmathgames.ca, as well as Issues 1(3), 1(5), and 1(8) of The Variable]. You have also suggested that, perhaps, “puzzles and games can play a small part in changing the Canadian pattern of having the math-averse and math-fearful pass on their issues to the next generation” (Milner, 2013, p. 12).
The question begs to be asked: What do puzzles and games have to do with mathematics?
The games I like involve logic, not chance and not knowledge. The types of reasoning we use in these games are exactly the types of reasoning that mathematicians use all the time to explore ideas and prove results. For example: “If A is true, then B must follow,” “If C were true, then D couldn’t be true, but D must be the case, so C cannot hold,” “If E were true, then there would be no way for us to achieve F, so E can’t be true,” “G is true if and only if H is true”… Don’t those abstract statements make your head spin? Yet I have heard many, many children say things exactly like that in their own words, while they were trying to work out some part of a puzzle.
Some games involve a lot of spatial reasoning, both two-dimensional and, occasionally, three-dimensional. Shape (geometry) is often sadly neglected in the school curriculum, particularly in the middle and high school years, yet it is a huge aspect of the advanced mathematics we need to model our world. It’s also highly appealing to most people, being tied so closely to the physical world. If we can explore reasoning using that appealing physicality, we are likely to reach far more students than we would otherwise.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]It is essential to be aware of the difference between what we know for sure and what we hope, guess, or want to be true. I think this is essential for sensible discussion in all aspects of life, not only in mathematics.[/perfectpullquote]Beyond that, playing with these puzzles encourages what have come to be called “mathematical habits of mind,” which include, among other attributes, persistence, attention to detail, and willingness to start over.
One of the most necessary aspects of all of the games and puzzles I take into classrooms is that it is essential to be aware of the difference between what we know for sure and what we hope, guess, or want to be true. I think this is essential for sensible discussion in all aspects of life, not only in mathematics.
And what do puzzles and games have to do with “school” mathematics—that is, the mathematics that students learn in elementary and secondary schools?
Most of the games I first introduce to classes use nothing beyond counting, two-digit arithmetic, and awareness of shape and colour. The reasoning, however, can be surprisingly sophisticated.
There are some puzzles you can use if you want students to practice some basic skill. For example, the Rectangles puzzle relates shape to the factoring small numbers, Kakurasu is terrific for adding and subtracting, and Mathdoku (KenKen) is good for arithmetic of small numbers. As far as I am concerned, though, in each case, the puzzle has to be interesting enough that the “good for you” aspect is all but invisible.
I don’t know about Saskatchewan, but in British Columbia the curriculum has recently started to include more of an emphasis on pattern recognition, reasoning, and puzzle-solving. I’ve been delighted with that, as it makes it easier for teachers to point to the learning outcomes they are meeting when they use my materials.
As you share on your website and in Milner (2013), even students who have had unpleasant experiences in “formal” mathematics courses typically enjoy mathematical puzzles. In some cases, you have found that opportunities to work on puzzles increased both confidence and skills in the subject among students who didn’t typically see themselves as “math brains.” Why might this be so?
All of us have had experiences where we’ve failed at something. If we’ve failed at it several times, many of us tend to be very wary of putting ourselves in the position to fail again, and even if we do try, maybe with encouragement from someone else such as a teacher, we get so anxious that we can’t function properly. There seems to be no way to break out of that downward spiral.
It’s difficult to change one’s picture of oneself, especially if negative aspects have been reinforced over the years; however, I think that “something completely different” can be a huge help. When a game or puzzle catches the attention of people who’ve had a rough time with math, they often seem able to break the cycle of negative self-talk because this doesn’t feel like the math that has caused them so much anxiety. Nearly all of my most math-anxious pre-teaching students found a type of puzzle or two that they could solve easily, after which they became hooked and sought out harder puzzles. Once they succeeded, they were of course eager to keep succeeding.
Also: We tend to be better at tasks we see as interesting, or at least we are more likely to focus on them. It seems that playing with patterns and solving puzzles is a very human activity—we just like playing games. The stakes are lower, so there is less to be anxious about. For the math-anxious, games that involve manipulatives are even better than those involving only pencil and paper, because people can just sweep away all evidence of their wrong answer.
On your website, you remark that “mathematics is important, hard work,” but that “most mathematicians will admit that they think of their work as play” (Milner, n.d.). And in Milner (2013, p. 10): “Yes, mathematicians play all day!” Could you elaborate?
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]At some point during my years of teaching, it became very clear to me that the way mathematicians talk about problems is very different from the way many people imagine.[/perfectpullquote]At some point during my years of teaching, it became very clear to me that the way mathematicians talk about problems is very different from the way many people imagine: “Let’s play with it” comes up remarkably often, as does “that’s an elegant solution.” At the start of term, my students laughed at me when I talked like that, but if I’d done a good job in the classroom, they would eventually start using that language, too. I am not alone in thinking that having a playful spirit and appreciating beauty in what we do lightens everything up and frees us to be more creative.
One of the things I love about abstract mathematics is that we can take a (small) set of definitions and maybe an axiom or two, and build an incredibly far-reaching system; think of classical geometry. So what else is a game, but trying to accomplish something within an agreed-upon a set of constraints? The bonus is that mathematics is remarkably good at modelling the world—it is not uncommon for apparently outrageously abstract mathematics to turn out to be essential at some time in the future. Number theory is a good example: According to the mathematician G. H. Hardy in the early 1900s, it was the purest form of mathematics because it had no use whatsoever; in particular, it could not be used for war or commerce. Now we cannot live without number theory in our internet security, which is essential to both war and commerce.
Is it a mistake, then, that mathematics curricula have largely ignored the playful nature of mathematics, and the role of play in learning? (On your website, quoting Plato, you write: “Do not keep children to their studies by compulsion but by play” [Milner, n.d.].) If so, how might play be integrated into the teaching and learning of mathematics, beyond the primary grades?
Most definitely! It seems to me that most people work much harder at their games, hobbies, and sports than they do at what they consider their “real” work. Motivation makes a huge difference, as any teacher knows.
To start, our own fear and/or anxiety has to go out the door, as students pick up on that immediately. We need to model what we believe—if we are going to see where a train of reasoning leads us, let’s actually follow it, not try to force it into a particular direction. (Think about how that connects to science, politics, everyday life!) If we end up in a mess, let’s go back and figure out what happened.
I startled several of my colleagues by setting project problems in calculus that I had no idea of the answer to—as a class, we would agree on the problem and the ground rules, then I would set the students loose to figure it out and write up their best answers. They couldn’t figure out what answer I wanted, because I didn’t have one, so the game became more like actual applied mathematics: They had to present their arguments in a way that would convince me that they had a good solution to a messy problem.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]Genuine enthusiasm is impossible to fake and impossible to resist, so I think the trick may be to figure out what really excites you about a particular topic or problem, and then… to play with it.[/perfectpullquote]And just to be clear, when I talk about playing, I’m not talking just about the gamification of learning, if that means turning it into a sequence of little steps with small rewards for succeeding in each one. That can be effective if it’s not over-used, but playfulness comes in many forms and we are not all suited for all forms. You can probably come up with several widely-differing examples of playful teachers, conference presenters, and mentors in your life. Genuine enthusiasm is impossible to fake and impossible to resist, so I think the trick may be to figure out what really excites you about a particular topic or problem, and then… to play with it.
Within two minutes of walking into a new classroom, I can sense if the teacher has a playful attitude towards mathematics. Sometimes the students will already have played some math/logic puzzles, but sometimes the playfulness has revolved around solving difficult or open-ended problems together. Students fortunate enough to have a playful teacher tend to focus quickly when faced with a new game: “What’s the goal? What do we know?” The whole class is much more likely to participate in the doing-an-example-together stage of my introduction, being willing to take turns supplying answers and listening to each other. Not that they are all perfect angels, but I’d say that students who are used to playing math games of one sort or another together are more likely to respect each others’ contributions than are students in a class that has focused solely on getting the right answer and moving on to the next question.
Teachers in the latter type of class are often very surprised by which of their students turn out to be good at spotting patterns or at thinking ahead several steps in a logical chain. I love it when that awful stereotypical distinction of “math-mind” versus “not a math-mind” gets broken, both in the mind of the teacher and in the minds of the students themselves. I was delighted and touched when a teenaged First Nations student said to me in the hallway an hour after her class, “Thank you for bringing your games to our class. I really liked Towers. It’s a smart person’s game. It made me feel smart.” I’d like to think that she learned something about her abilities and gained some confidence that day.
You share a wide variety of mathematical puzzles and games on your website, www.susansmathgames.ca, many of which you have shared over the years with K-12 students in classrooms or during workshops across British Columbia. But surely, there are puzzles that don’t make your list.
I definitely try out more puzzles with classes than appear on the site. I put something up on the website only once I have enough classroom experience to be able to describe a reliable way of introducing the game and once I am sure that my introductory puzzles are graduated appropriately for students to move smoothly from one level to the next.
In your view, what makes a good mathematical puzzle or game? Do you have a personal favorite?
For classroom purposes, a “good” puzzle has simple rules, is visually appealing, starts out fairly easily, and progresses through several different levels. Students should be able to see themselves solving harder puzzles at the end of 20-25 minutes. I hope that they can experience a couple of “aha!” moments, even minor ones, because that leaves them excited and wanting to do more.
The best puzzles to start with are the ones I’ve shared in various editions of The Variable [see Issues 1(3), 1(5), and 1(8) of The Variable –Ed.]. Hidato is very easy to explain and everyone seems to enjoy it, at any age. Rectangles is great for any age from about mid-Grade 3 to adult. Towers is terrific from about Grade 5 or 6 and up, but you should start with manipulatives so that everyone can understand the rules. Kakurasu takes a bit of effort to figure out, but students from about Grade 7 and up have found it exciting. Set is wonderful for the many types of games you can play once you know “the rule.”
While I find it exciting to learn a new game, I tend to use familiar games to clear my mind and help me relax. There are a few games I find absorbing enough to play regularly online: Kakurasu, Calcudoku (Kenken), Three-in-a-Row (Unruly), Magnets, Neighbours (Adjacent), and Kakuro. Lately, I’ve gotten into Nurikabe again, and I’ve really enjoyed Slant and Futoshiki (Unequal). All of these appear on the BrainBashers site (www.brainbashers.com) and/or Simon Tatham’s Portable Puzzle Collection (www.chiark.greenend.org.uk/~sgtatham/puzzles/). Set can also be played online (e.g., at www.setgame.com/set/puzzle) and clears the mind remarkably well.
Thank you, Susan, for taking the time for this conversation, and for sharing some of your favorite games and puzzles with our readers!
Ilona Vashchyshyn
References
Milner, S. (n.d.). Why puzzles? Retrieved from http://susansmathgamesca.ipage.com/why-puzzles/
Milner, S. (2013). Puzzles in my life. CMS Notes, 45(5).
Pacific Institute for the Mathematical Sciences. (2014, March 27). Susan Milner is awarded the PIMS 2014 Education Prize. Retrieved from http://www.pims.math.ca/news/susan-milner-awarded-pims-2014-education-prize
