May 27, 2020
High school math alone does not adequately prepare students for the intensive curriculum of university calculus. In university, your professor expects you to have already acquired the confidence and substantial knowledge on the basics of math, allowing you to learn more advanced concepts towards forming a deeper understanding. In addition, many college majors and medical schools expect prospective students to take a university calculus course. Students, without the necessary preparation and proficiency in math basics, risk failing the needed calculus course as they struggle to keep up with the demands of a university curriculum and expectations of math skill. A University Calculus Preparation Course (UCPC) is designed to ensure your confidence in developing concrete understanding of fundamental math concepts, giving you a head start in achieving a successful post-secondary career.

To provide further insight on the importance of a UCPC, we interviewed Dev Gokal – a University of Toronto math specialist student. Along with his experience as a teaching assistant, he has seen at first hand the struggles of being ill-prepared for university calculus:

Q1: What is the difference between high school math and university math?
D. Gokal: High school calculus only scratches the surface of the subject; MCV4U0 only covered basic notions of limits and derivatives, before moving on to linear algebra during the second half of the term. In university, first year calculus is generally a full year course or offered as a split between two courses. All of high school calculus is covered within the quarter of the time in a university calculus course before moving on to new concepts.

Another main difference between high school versus university would be that students are expected to study the material in advance before attending the lecture. This expectation allows the lecture to be dedicated to not only teaching the material, but also for students to ask questions to get a deeper understanding of some parts that they may not have easily understood.

An important caveat to note is that, due to the large class size, it will be difficult for the professor to attend to the needs of every individual – hence why professors encourage students to read ahead so that they are not overwhelmed during lecture. The pace of lectures tends to be quicker than in high school, as almost all of the time spent during lecture is dedicated towards instructing and lecturing as opposed to splitting a period in half; one being teaching and the other being time to work on exercises.

In high school, calculus is based on formulas that you are expected to memorize and use to solve problems. In university, you are introduced to the deeper underlying theory which involves mathematical definitions, theorems and lemmas.

You will see the tools that you have been using in high school developed in a new light and rigorous manner that will help you form a deeper understanding of those tools. Elementary functions such as sine, cosine, the exponential, the logarithm will all be defined rigorously using their Taylor series expansion.

Testing will also differ, as they will require you to not only show your understanding of the material by making you compute integrals and derivatives, but rather prove certain statements by using definitions and theorems. A high school test question may look like this:

Calculate the following derivative:

Whereas in a first year calc course you may see something like this:

(Chapter 9, Q15 from Spivak)
One can easily see that a certain level of mathematical maturity and understanding of the theory is required to be able to answer such questions, compared to the relatively simple computational ones often seen in high school.

Q2: Why is this relevant for computer science and engineering students?
D. Gokal: Engineers and computer science students are natural problem solvers.

In engineering, it is often the case where you are tasked with finding a novel and optimal solution to fit the needs of your clients; one that may not exist yet. While it is easy to take an already existing solution/standard approach to a problem, this may not be always viable to do, and as an engineering student it is your responsibility to come up with new and optimal ways to solve problems. This can only be done with a strong understanding of not only HOW things work, but WHY they do the way they do. Having the ability to think logically with abstraction is a very important and powerful skill to have to be able to create these new ideas and solutions to problems. This is also extremely valuable when one is trying to troubleshoot a problem they come across; by looking at the problem abstractly, it is possible to see how each part functions individually and is connected in a given process/object allowing one to quickly find the root of the issue.

As for computer science students specifically, most people that are not within the field are under the impression that doing a degree in computer science just means that you learn how to code and design software with a bit of theory on the side here and there. This could not be farther from the truth. In first year, basic coding skills are taught and cultivated while the rest of undergrad is spent developing the theory behind essentially why computers and programs work and run the way they do. It is expected that, in your second year of study, you will be taking an introductory course on the theory of computation (such as CSC236 at UofT) which introduces things such as inductive reasoning, runtime analysis, program correctness and the study of computability as a whole. These things directly require proof techniques that you will have learned in your first year mathematics courses which is why getting a head start is so important. If you take upper year computer science courses, most of the textbooks that you will see for those courses will be focused on the theory and the underlying mathematics of the theory. Courses in hot topics, such as machine learning and artificial intelligence, assume that you have a very strong background in math and statistics and that you have the tools and abilities to think abstractly and logically; the tools learned in your first year math courses.

Q3: What benefit will I get from a University Calculus Preparation Course?
D. Gokal: By doing this UCPC, it will allow you to sharpen your critical thinking skills, as it forces you to understand the basics. You will see how the basic definitions are used as building blocks to understand behaviour of the objects that you are studying.

The use of mathematical logic and reasoning in the program is easily transferable to other subjects; the more exercises and proofs you write, the better you will get at explaining your ideas in a logical and rigorous way.

In addition, learning to be precise and rigorous when expressing your ideas is a necessity in any STEM-related field. This is why studying calculus rigorously at the university level will not only give you the tools you need to start thinking analytically (and you may think that you already know how to think analytically, but this is often the exception than the norm), but also transforms the way you think about problems and how you learn material.

In high school, students tend to heavily rely on examples to reinforce their understanding of material. However, those who have built these skills will have a habit of diving down to the most fundamental level to understand those building blocks from which the big picture is made up of. This ability of working with these “fundamental building blocks” is extremely important if one is interested in pursuing research in any area. I am currently working with a professor and a colleague in order to attempt to classify different kinds of real division algebras. In doing so, I’ve had to use the skills in logic and reasoning acquired from my first and second year classes to be able to take this seemingly huge problem, break it down to the basics and work my way up from there.

There are a lot of problems that you will encounter in STEM (math and comp sci in particular) that are easy to conceptually understand from a high-level and to break down. To be able to take those ideas and convey them rigorously in a way that can be understood clearly by anyone is an essential skill that many should have but, ultimately, do not.

Q4: Why does it matter now?
D. Gokal: This is probably the best time to learn and to do a UCPC. Given the current COVID19 situation restricting what we can and can not do, there is no time like the present to learn these valuable skills to get ahead of the curve. Building strong foundations is easier now when you have time to spare in learning this material that will help you become a better problem solver and will allow you to hone your critical thinking skills. But most importantly, it will be much less stressful later on when you are taking this course either in high school or in university, as you will have gained a much deeper understanding of the material than your peers. You will be able to understand WHY things are the way they are and be able to show them in a rigorous manner as opposed to blindly accepting propositions and theorems.

At Math Project , we have our University Calculus Preparation Course to help students achieve calculus proficiency before university. Within two months of our university calculus preparation course, our students gain the confidence and skills needed to excel in university calculus and in any STEM field. Enroll now by visiting our contact page or calling +1 844 628 4243.