Children Dream

Both my parents were teachers at university. I saw them teach since I was very young, and I liked the performative aspect of teaching a lot. I also liked my teachers at school and was very impressed by some of my teachers. They were very impressive to me on the way they moved and spoke. Rumor has it that one day I arrived at my house, I was five years old approximately, and I out of nowhere started lecturing my teddy bears in full-on-force-of-nature fashion.

During a large portion of my childhood what I wanted to be was a teacher. I would tell this to my family and friends frequently. Unfortunately, the teaching job was (and continues to…) not be very well seen. Many times, I was told by people, sometimes closer to me and sometimes relative strangers, that to become a teacher would be to waste my life and opportunities. In truth, now at thirty-five, I wonder what prompts someone to say this to a six-year-old? In any case, I used to be frustrated about this frequently, but my parents would always be very supportive of this and say that it is the job of the teacher the most ungrateful but necessary job there is.

I only started to say I wanted to be something different from a teacher at thirteen, where I was considering maybe being a physicist (but, in truth, I did not mean it). Mathematics as a path was still not a thing for me and the most usual activity to perform in my free time was to give classes. Every school term I would decide on the topics I would study, create a schedule to follow on the evenings after returning to school, and then once I was back teaching on this. I had big boards in my rooms at the house, a nice desk, many teddy bears that I would put in front of me and deliver lectures on math, physics, chemistry, history, biology, literature, etc. Anything that seemed interesting to me, I would teach. Truth be told, many times I had no idea what I was saying completely, but many times I did.

Of course, as with any course, there would be some evaluation. I would design tests, usually quite challenging and complete, and then answer it for each one of the teddy bears. Then I would grade it. Of course, depending on the course, was the type of teacher I was. I had a complete cast of characters to perform. I could be a sweetie or devil itself.

One of the most vivid memories my father has of me asking as a gift for a birthday, or maybe Christmas, the total of 60 notebooks. I had 30 teddy-bears and each of them would get two notebooks: one for science classes and the other for literature and language. I did get the 30 notebooks, and I filled several of them. Not all of them, because some of those teddy-bears were very lazy.

Tutorials in Junior High School

I continued to engage in the above dynamics until I was fifteen years old. At that time, I was in junior high school, and I started to be more interested in spending time with my friends. However, the teaching vibe never really went away, it just mutated. I started to teach my friends on the topics of school because many of them used to struggle with studying or understanding certain topics.

For around two years, since the tutorials started, my friends and I would meet in the house of one of them and study. I would explain to them how things worked, we would go over examples, or do the homework again together, make notes. We would do practice tests. They started to improve their grades and their parents were fascinated, the teachers were fascinated, everyone was fascinated. It was quite peculiar. There was a version of truth in saying I was part of the popular kids because I was the teaching kid that also happened to be cool.

However, with time some different groups of people started to ask for my help. I had the conviction teaching should be for everyone, so I accepted to help most people that asked me. There are two experiences that I really remember from back then: one of them was when someone, who used to bully me in junior high school, asked me for help. I admit I was not amused but I accepted. It was the weirdest of experiences, because we didn’t know how to engage with each other…but he needed to approve, and I wanted him to stop bullying me. A lot of my friends told me I should have said no, but I disagreed: everyone deserves a chance. He approved the tests and at the end of junior high school one of my teachers told me she could not believe that change on him. I have never heard of this person again in my life after this.

Another interesting event was the following: it was late, around 9 pm, and I was called to help someone. I was tired and really did not want to tutor at that time. I put some impossible conditions, but they accepted. Out of options, I had to comply. I arrived at the house of these people and there she was, my student, and her mother next to her. First thing the mother tells me: good you are here, she is dumb, so we need this. I remember thinking: how do I follow up after that? I am sixteen years old at the time, so this kind of thing were very impressive to me, but I had no real way to react to them appropriately. I actually studied with her many times after that one, and she always did well. I have also not known of her since then.

At some point, teachers knew I liked teaching and that I was good at it. They were also aware I was helping the students at the school to study more and better. Thus, one day, the history professor asked me to teach his class, because he was not going to be able to be there. The direction approved of this, and I went to teach the class. At that time, I am in ninth grade, and I am going to teach the history class of another ninth-grade classroom. The lecture was one hour, the topic was The Aztecs way of life. I had prepared from the book we were following, but also, I had several books at my house on this topic I had read. So, I go into the lecture room, I pick up my chalk and I deliver. It was hysterical: it was a very good lecture, but at the same time I was sixteen, so I felt I was on top of the world. The lecture was not chaotic in a sense that the students were not paying attention, rather we were very engaged with it, and I was like possessed. A physics teacher, which was the responsible adult taking care of the situation in case it got out of hand, was so thrilled. She even hugged me after the lecture. They asked me to teach the same lecture to the other classrooms of ninth grade, and I did. And oh man did I deliver? Actually, the first one was the best, but the other ones were also very good.

However, I discovered something in that lecture about teaching: let yourself be possessed and when you know something, improvise with it. No lecture comes close to the ones that are improvised and successful. To this day, my best lectures are the ones that are improvised. Of course, this requires knowing the topic really well, and when they go wrong, well…it’s what we use to call an Epic Fail.

Olympiad Trainings

When I moved onto High School the situation also changed. Up to that point the schools I have been going to were private. At that stage I really wanted to go into public schools and I also wanted to participate more fully in the math Olympiads. This manifested in my teaching experience in two different ways: on the one hand, the public changed. The students I worked with in private schools were mostly middle-upper class teenagers. The ones I engaged with in public schools were not, their background was very different to what I was used to.

In high-school I went to the evening school (i.e. my classes started at 2:00 pm and finished at 9:00 pm somedays). Many of my friends had to work in the mornings, or were parents already, or had to be very active in the survival of their family. In truth, for many of them the school was a transactional period to achieve better jobs. It contrasted a lot with my experience of school, and university, with a place of knowledge for the sake of knowledge itself.

On the other hand, the type of training that was required from me was different. This time the challenge was different: firstly, I was trying to go into a competition that, as things improved, would require from me to confront people of higher and higher levels. It also required to study topics that were simply beyond our usual curriculum. Almost nothing from the Olympiads was covered in school.

These two things forced me to adapt myself to many new environments teaching related. In many occasions the students that wanted to participate in the Olympiad, and had good potential in it, were the same ones that required to work in the mornings in a supermarket to earn money for the family. The resources available to us were different in many occasions. I was used, in junior high school, to have access to basically anything in the school we had. At some points it might require some bureaucracy, but it was in the end achieved. In the public system, it sometimes was not there. I remember that in another of the public schools of the university there was no room for the math club to convene, so they decided to share the room with the cleaning department. So, in between mops and brooms, there was a board and chairs to study math for the Olympiad. This, for some readers might sound horrendous (and in some degree it is), however one did what was needed. The Olympiad world in a third world country.

Very soon I became one of the trainers of the mathematics club, because I started to improve in my development in the Olympiad. We used to have math club every day 12 – 1 for the interested members, and I was almost always there since 9 am. The official math club was only on Tuesdays (or Mondays). One time our performance on a competition was so dismal that I had a meltdown, and I imposed Math Club from 9 to 2 on Saturdays! We had a practice test every Saturday, and we kept scores, and see the rankings. I designed the tests and graded them. Surprisingly, people really wanted to participate and whenever they could, they would go (or they would face my wrath).

On one occasion, a Saturday coincided with May 10, which is mothers day in Mexico. That was no excuse! We had test, to the chagrin of many mothers. That earned me the nick name “Mamá Malors”, or in english “Mom Malors” and it has stuck with me up to this day. Sometimes, some of my ex-students back then and friends, still congratulate me on May 10.

What I developed, regarding teaching, in this period was the ability to teach technical stuff to people that, on the one hand, might not believe they can master it and also the development of plans and discipline to achieve a goal. I was aiming for the IMO, that was really a challenge. I had to be on the top six of the whole country and Mexico had been developing a very strong mathematical community through the years. Mexico has good mathematics. During those three years I really did not exist for anything else: I was a ruthless Olympic to the best of my abilities.

Through the Olympiad, both as a student and as a trainer, I realized the different ways to teach about mathematics. To teach students to engage in questioning and exploration. To have versality, but to do not be stubborn. To create new things but also to use what already exists. I learned to be humble, which is a hard lesson to take, because there will always be better people than you are at everything. The first time I experienced this in a way that was very difficult to overcome was in a state Olympiad training. In the state I participated we had, back in the day, one of the best math Olympics of the moment. He had already gone to two IMO’s at that time, won silver medal in one of them and bronze in the other. That day, our trainer was the first person to earn a silver medal for Mexico and we were only us three, because nobody else could make it. The trainer proceeded to write problem after problem in the board for us to work on, and the other student being more experienced than I was, just solved all of them basically after looking at them (of course, now I know that in great part that was because he had seen these problems before, but I did not know this back then). These were three hours of brutal humiliation; I was barely able to keep up writing the problems. I felt terrible, worthless, a mistake of nature.

If you cannot get over the shock, you cannot do it.

I think this is one of the most important lessons I have learned in mathematics and the way it is done. At some points it is unavoidably brutal, it is part of what mathematics is. This doesn’t mean we should make it more difficult and overwhelming than it is at times out of lack of opportunity and misguided narratives, but, when it comes to the real aspect of what mathematics can be, because of what it fundamentally is as opposed to what we make it be, then it can be shocking.

I try up to this day to be honest about this with my students that learn about math from me or that ask me about how it is to really do math. One should not engage with mathematics just because it can be beautiful, unless one is willing to get the tremendous shock of its difficulties as well. I do believe this point is very much hidden in mathematics education, olympiads and even undergrad education. Math is as beautiful as it is terrible.

Teachings in High School and Undergrad

When I moved onto undergrad, I was already used to teaching seminars by myself and to have a big habit of study. I organized with many friend’s seminar and we learned from each other. I also participated in the math Olympiad, now as an organizer, and taught a course in the High school of the state. In this sense, many of the things, teaching wise, that I was doing up that point kept on going on. With different people and with different topics but more less the same dynamics.

One of the first new experiences I had on teaching at that time was teaching mathematics to math teachers. I was asked to teach some courses to actual math professors from primary and high schools. This was completely different public: to begin with, I was a twenty-year-old teaching forty- or fifty-year-old people about how to teach math. A subject that they have probably been teaching for a long time, but also one that probably caused them certain apprehension. These were exercises on diplomacy.

The topic I choose was Diophantine Equations. I engaged with solving equations in integers using analytic geometry. I chose this topic because most of these teachers were professors of high school where analytic geometry is covered. I made extra effort on trying to use the most basic results and build from them. Designed activities they could themselves do with their students. And most importantly, to be super humble, as opposed to the arrogant teenager that thought he was superior because I was indeed studying mathematics. I had to avoid phrases like it’s trivial, follows easily, we all know, etc. We did everything carefully, I asked questions without singling people out, I wrote the arguments. I avoided showing off with spectacular examples that conveyed nothing except of how good I was. It was a very successful activity! I remember one of the teachers being very tense at the start and starting to relax as the course went by. I can almost say she enjoyed it at the end.

However, I promised myself never to do it again. It was very stressful.

Then I had some sort of hiatus from teaching and Olympiad endeavors because I needed to graduate and learn mathematics seriously. Thus, I did very few things in this direction for a while. In this time, my feelings and views toward the Olympiad changed and I never really return to it. Up to this point I have never engaged again in trainings or similar activities that go in the direction of competitions.

However, once I was in Toronto, I did decide to engage with the outreach department. In there I had the opportunity to teach advanced courses for high school students that were interested in mathematics. I also taught many times research mentorship programs for high school students. What moved me always in these activities was to teach material that was not childish, nor classical. What I mean with this is the following: in my experience, usually one teaches highschoolers that mathematics is fun by trying to surprise them with topics from probability, or topology or geometry that are very impressive. One insists on the ideas of questioning the stablished knowledge, on being creative, many times even saying that the point of math is not about repetition and hard work as in school(whatever that means). This indeed is true, but mathematics has another side that requires hard work, repetitive many times, discipline and that on many occasions is not fun. Thus, I try to convey this side of math as well when I teach mathematics.

I have taught courses to highschoolers on topics as: fractals and dynamics, or studying methods of classification that appear in mathematics. Another great one was on knot theory. We had homeworks that were intense. I made them read, write, engage. Many times I distributed papers and made them read proofs from them directly, so that they could see how math is really conveyed. We also saw online lectures on youtube. I made them present to grad students little presentations of five minutes. These presentations were very big obstacles for many of them: they wanted not to do it, and I always had to come up with the same fact: if you want to do math, you have to be able to get over the shock.

What has always moved me in my math teachings is honesty of what mathematics is. Not only what we would want it to be, but what it actually is. The terrible dimension that it possesses also needs to be shown, otherwise it is an incomplete aspect of it, and hiding it always comes back with a vengeance, because it doesn’t matter how the mathematics path is, eventually the terror of it comes out. It is better to know it is there, and you will find it, than to find out you were misled.

To my surprise, many times students engage very much with it in ways that surprise me. They have explained to me that they feel acknowledged. Many of them even decided to try for a path of mathematics once they take my courses, which sometimes makes me feel very nice but also uneasy. Mathematics is not an easy path; I do hope I have conveyed this appropriately.