Dear Student:

Congratulations on accepting our offer! Becoming a university student is an important and exciting milestone in your life, as well as a watershed moment in the lives of your family and all of the people who have played and will play a role in your journey.  That includes me, your first-year mathematics instructor.

Thank you for your email, and for sharing your thoughts and concerns. You concluded your email with the question: “What can I do to better prepare myself for studying math at university?”

The purpose of this letter is to address your question and put it in a broader context, so that you better understand some of the challenges that both of us will face in the coming months.  Think of it as my way of welcoming you to our university and to my classroom.

I must warn you that nothing you have experienced in your schooling will compare to the change you will encounter. After spending several hours in front of your computer screen for a virtual lecture, you will be expected to do assigned readings and homework; you will feel exhausted, completely drained of all your motivation and energy. You will be lonely because you are unable to meet your peers—many do not turn their cameras on during lectures, nor join the class chat.

Or, disoriented and confused, instead of small and cozy classrooms, you will find yourself sitting in a large auditorium. You will look around but not recognize a single face. You will see my tiny figure in the distance, standing by the lecture hall podium, surrounded by computer screens. My voice will reach you through loudspeakers mounted on the ceiling high above your head.

Transitioning to the new online, in-person, or hybrid environments, while taking on demanding and complex mathematical content, may feel like a perfect storm.

It is well known that the transition to university is a significant “rite of passage” for many (likely, all) of your peers. Although researchers in mathematics education have studied the transition from secondary to tertiary mathematics (that’s what it’s formally called) for many years, it does not seem that today’s high school graduates have an easier time adjusting to the demands of university mathematics than their predecessors did 25 years ago. On the contrary, I believe that this transition has become increasingly challenging over the years.

It may comfort you to know that the challenges you will be facing in the coming weeks are common for anyone going through a major change in their life.

Researchers in education have conceptualized the transition from high school to university by applying the anthropological notion of the rite of passage.  A rite of passage consists of a sequence of events that enables an individual to deal with, and overcome, a “life crisis.” Identified across all cultures and societies, and going back thousands of years, rites of passage are present throughout an individual’s life, and include socialization, the transition to adulthood, beginning and ending spousal relationships, and retirement.

Important events take place as an individual undergoes a life crisis—in your case, of course, the “life crisis” is caused by your transition from high school to university. Through the anticipation of changes near the end of secondary schooling, and then by formally graduating, the routines in your life have been disrupted; in anthropology, this is called a separation phase.

Once you complete the transition and assume the full identity of a university student, you will be accepted into the new community (the incorporation phase). The purpose of any rite of passage is to make sure that any individual undergoing it successfully completes the transition and evolves a new identity.

What happens between the phases of separation and incorporation, namely the moment where you are right now? This so-called liminal phase is of particular interest to me, as it is a crucial phase during which various activities aimed at making the transition to university smoother should take place. What does the research tell us about transition and the liminal phase?

First, a transition cannot be accelerated. You will need time and patience to address your problems, stress, and fears; to digest large amounts of information and reflect upon it; and to start acting according to the norms of the new environment. Needless to say, you will need (a lot of) time and patience to succeed in your math courses.

Very likely, you have been invited to join a weeklong orientation program for incoming students. Please plan to be there, as you will hear important things about attending classes, note-taking, seeking help, time management, and academic integrity—to mention just a few important aspects of university life. Orientation activities are helpful, but they will not answer all your questions, nor will they meet all your needs.

The most important benefit of orientation events is that you will meet people with whom you will share your life for the next four years: your peers, instructors, and university staff. According to British cultural anthropologist W.V. Turner, “Among themselves, neophytes [i.e., participants in a rite of passage] tend to develop an intense comradeship and egalitarianism.” In other words, what universities aim for in their activities—stimulating social interactions between incoming students—is, as confirmed by anthropologists, not just useful but also necessary.

A rite of passage is a continuous process that follows its own dynamics. Rarely does a person undergoing transition ask themselves: “What do I do now?” Continuous learning, working toward accepting the rules and norms of the new community, and interacting with its members every day provides necessary guidance and support.

However, once a student graduates from high school, they often find themselves in a vacuum. Their high school teachers are no longer there, and with university instructors and university support months away, who is going to help them answer: “What do I do now?”

Orientation activities help, of course, but there is more to it. Perhaps the strongest lesson from anthropology is that, ultimately, it is the individual who must navigate their rite of passage. Thus, the most relevant question in this liminal phase of the transition is exactly what you asked in your email: “What can I do to better prepare myself for studying math in university?”

You used the pronoun “I” in your question. By doing so and by acting accordingly, you are making an important step—deciding not to wait to see what your future university will do for you but, instead, taking initiative into your own hands. For me, this is a sure sign of your maturity, something that is very much needed in this stage of your academic development. I wish I would receive emails such as yours more often.

There are many situations and challenges in life for which we cannot fully prepare. Reading books or watching videos about swimming or playing a piano are very different from jumping into water and swimming or playing a simple sonata. Nothing can fully prepare you for that rush of blood when you realize you will not have enough time to complete two important assignments due the same day; nor for the experience of trying to remain focused on your work while listening to your roommate complain; nor for the frustration you will feel when (you think) you studied hard, but still fail a test.

There are things that you can do to better prepare yourself.

First, there is the internet. A quick search will lead you to a web page of your new academic “home”—say, the mathematics department where I work. There you will find a link or a tab named “Undergraduate students” or “Prospective students” or something similar. You will soon find all kinds of useful information: for instance, outlines of math courses with the names of instructors. Are you not sure how the linear algebra course will be run, or what is assumed to be known on the first day of classes? Find the course instructor’s email and ask them! Will there be a review of functions in the calculus course? Reach out to your older peers, for instance, through a society of Math and Stats students, and ask for their experiences. What is that “Mathematical Reasoning” course about? Post your question on our university’s Reddit or Twitter!

This may still feel like reading about swimming—but it is a good start!

There is one thing that you can do, which will indeed feel like jumping into water and swimming: you should warm up and prepare for your math courses before you come to university. Mathematics builds on itself, and I expect that my students will be able to recall facts, formulas, and principles they learned in high school. Evidence suggests that students who struggle with first year mathematics courses often have gaps in their prerequisites (fractions, inequalities, graphs, and so on). Trying to absorb new material while having to review prerequisite material is demanding, frustrating, and time consuming.

What mathematics should you review? A simple Google search “review math before university” might lead you (depending on your location, previous searches, etc.), for instance, to the University of Waterloo page: “What’s the best way to prepare for first-year math?” The next link is even better – you will find a free copy of the Mathematics Review Manual, published by McMaster University. The Manual identifies exactly what high school material incoming students need to review, gives a brief overview of relevant theory and formulas, and provides examples and exercises that will help you verify your understanding of the basics. The next link, the University of Toronto page “Entry Level Math Preparation – Department of Mathematics” offers a self-test, fully solved examples, and practice problems for all math topics that you need to review.

So, to answer the question from your email: act now! Warm up and review high school mathematics before the first day of classes! Anecdotal evidence that I have collected over many years suggests that spending a bit of time reviewing mathematics before classes start (say, one hour a day for three weeks in August) is extremely helpful. If you do this, you will see that the perfect storm of transition will feel lot less challenging. And that tolerating your whiny roommate might not be as bad.

Good luck!

Looking forward to meeting you in September,

Your Math Instructor

Andie Burazin is an assistant professor, teaching stream, in the department of mathematical and computational sciences at the University of Toronto Mississauga; Veselin Jungić is a teaching professor in the department of mathematics at Simon Fraser University; Miroslav Lovrić is a professor in the department of mathematics and statistics at McMaster University.