On the Stages of Learning and Doing Mathematics

I’m teaching Multivariable and Vector Calculus at Trinity College this fall semester and have students with all kinds of backgrounds. For some of them, this is their first semester of college and are used to year-long math classes rather than a whole course in one semester. The calculus they had was high-school AP Calculus which is at a different level than college-level calculus and can vary a lot depending on, say, whether it’s a public high school or a private science and technology magnet school. Some of them finished high school on Zoom because of COVID and for many, distance learning is not as effective as in-person learning. And then they took a break from math for a full year. Some of them don’t speak English as their first language. For others, they’ve taken the first two college-level calculus courses and have also taken Physics: Mechanics and Physics: Electromagnetism where vector calculus already appears. So this class is a breeze for them. One of my students went the first three months of his first year before losing a single point in any of his classes. English is not his first language and so he seems to appreciate that in a math course, the most important language to understand is mathematics (though of course that happens by means of English in my classroom).

There’s certainly ways to help students of different backgrounds and there are a lot of external resources online, in textbooks, in the form of tutors, study groups, flashcards, my office hours, etc. But I think that many of these external “fixes” don’t address the root issue. I think the root internal issue is that students don’t yet know what mathematics is really about. They don’t hold the mathematician’s view: that math is primarily about human understanding of structure and using powerful abstraction to get to the essence of an idea or to broaden the scope of what true statements can be said. Instead, the students’ views is, say, a view on what a high school math class is about: do homework, take exams, try to get a good grade. But these are almost two separate worlds. It’s similar to the difference between reading Hamlet and being able to name some characters and describe a basic plot versus being to write your own play that reaches the same acclaim as Hamlet (whether you like his plays, I think it’s undeniable that Shakespeare has made a lasting name for himself).

Well, of course it’s too much to ask for students to jump right into doing mathematics like a mathematician just as it would be unreasonable to expect someone who read Shakespeare to then be able to write like Shakespeare. And of course, most people won’t reach anything close to Shakespeare’s level and that is okay; they don’t need to. But one can’t always stay at a high school view of math and this class is supposed to be of intermediate level rather than introductory. So to illustrate various levels, here’s an extended culinary analogy.

Cooking

High school and intro college math: You are tasked with preparing a microwavable meal. You read the instructions, peel the plastic film off halfway and microwave the food for 2 minutes, let it rest for 1, and then microwave it again for 2 minutes. You take it out cautiously and put the food in a serving dish.

Intermediate to advanced college math: You are given ingredients and a concise, not-the-most-helpful recipe. The recipe tells you to wash, cut, peel, season. But do you use warm water and soap? Which knife should you use and how do you hold it safely? What does “season to taste” mean? You find yourself needing to make more personal decisions, unlike with a microwavable meal but also that sometimes, it doesn’t matter what knife you use so just use the one you like. Should you get the water boiling first for pasta or start the sauce first? You learn that you should add water into a pot first, then turn on the heat, then add pasta. If you do this in the reverse order, you’ll get burning pasta and then a burst of scalding steam. Overall, the recipe has given you the basic structure of preparing the meal but you realize that you need to have a more thorough understanding of the ingredients than before. For example, you may find that bell peppers cook at a different speed than onions; the recipe didn’t tell you that but you figured it out. You also found that if you cut the bell pepper into only four pieces, those are too big. Sometimes, you’re asked to prepare a meal without consulting the recipe book at all and you forget to add cilantro or overcook the rice. It happens. The intermediate college courses deal with simpler recipes, say, spaghetti and (premade) meatballs, while the more advanced courses ask you to make a whole meal where the main dish is beef wellington. At various steps, you need to make decisions such as what wine and dessert to pair with your meal.

PhD dissertation: You now need to find your own ingredients though often, you can just go to the grocery store. You realize that this means you’re responsible for selecting what is fresh and suitable. There is now no recipe book, only the instruction: “Make a seven course meal with seafood.” You decide to go with various dishes that have been tried before but you put your own spin on them.

Professional math research: You find that sometimes you need to grow or forage for your own ingredients. Who knows of a salt mine around here? Is this mushroom safe to eat? How do chicken lay eggs? And not only is there no recipe book, you’re tasked with creating something new. The most successful mathematicians are those that create something novel and delicious enough that others get excited about it and want to study their creations.

Learning and Doing Mathematics

I hope the culinary analogies are clear but it’s worth elaborating. High school and intro college math: a large part of math classes at this stage is about knowing very precise but also short procedures and then following them. Things are formulaic/mechanical and there’s generally one path forward (like microwaving). You may spend a whole year in a class called “Calculus” but if you really think about it, most of the class is about solving the problems using algebra tricks rather than pure calculus ideas because life is easier that way (students see limit definitions but then quickly, move onto power rule, chain rule, quotient rule, etc). Similarly, microwavable meals are easier to prepare than cooking. You don’t need to know how the meal was prepared, placed in a microwave-safe container, and sealed. You don’t need to know how a microwave works. You just operate the machine. Often in these math classes, the formulas are your machine or you even have an actual machine: a hand-held calculator to punch in numbers and get a result. And also a formula sheet so that you don’t need to memorize formulas, just like how a microwavable meal tells you the amount of time you should punch in to get the result. In high school, you might have math class 4-5 days a week, have work days where there’s no lecture and the teacher is there to answer questions. You also get to see old AP exams when studying because, let’s face it, the goal is to score well on the AP exam and that requires a different skill set than fundamentally understanding math. One can train for exams, like the SAT (the College Board companion to AP), learning words like “jejune” or “opprobrious” only to forget them soon after, without becoming a better writer or even having a better understanding of English.

Intermediate to advanced college math: Unlike before where there is really just one step to solving a problem (similar to how there’s really only one step to microwaving a frozen meal), the kinds of problems are more sophisticated and require multiple steps. You also find that when solving a problem, you’re not always given all the ingredients in a ready-to-use form and you no longer have calculators (not that they would help) and formula sheets to hold your hand. You have to do something to what you’re given so that they’re more useful. For example, if you’re given three points in $\mathbb{R}^3$ and asked to find an equation of the plane containing them, you need to first produce two vectors, then take a cross product to get an orthogonal direction. Then you need a second ingredient: a point on the plane and you realize you already have three to choose from and it doesn’t matter which you pick. This is like washing and cutting vegetables; they aren’t ready to go into the pot until you prepare them. Sometimes, you prepare minced garlic and then you need to set some aside for later and you understand it doesn’t matter which pieces you pick to set aside just as it doesn’t matter what point you pick for an equation of a plane. But this is something that students in introductory courses aren’t used to: having the freedom to make arbitrary choices. Students often get very confused when I solve a system of two equations with three unknowns and ask, “How did you get that $x=0$?” and I tell them, “I decided to set $x=0$. I didn’t go through any kind of logical deduction to conclude $x=0$. Rather, it’s my starting point.”

You also find that there are multiple ways to solve a problem, getting the same result by any of those means, just as you can use different knives, spoons, spatulas, etc to accomplish the same task. But you find that you also need to understand what it is you’re doing because some of these processes look similar; you can’t get by with only memorizing processes and formulas but need to know how and why they work. To find an equation that describes a line, you need a point and a vector. To find an equation that describes a plane in $\mathbb{R}^3$, you also need a point and an (orthogonal) vector. Students often confuse the two on the equation level though if you tell them to draw the objects, they could depict the difference. This is like confusing bell peppers and ghost peppers; abstractly, they’re both peppers and you can do similar things to them like wash and cut. But the flavor will be vastly different. Sometimes forgetting something in cooking leads to unimportant differences, other times, to enormous ones. That is the case in math as well. Having a single wrong sign can lead to entirely different results. And back to the line and planes situation. I assigned a two-part question: find an equation of a line through a point and then an equation of a plane containing the point which was orthogonal to the line. A student emailed me and told me they found an equation for the line but didn’t know what they should do to find a point and orthogonal vector in order to define the plane. I explained to them that they already had all the information they needed from part 1. There was nothing left to find; the two ingredients they needed were already staring them in the face but they didn’t recognize them as having more than one use. This is like cooking: the same ingredient can have multiple uses and you need to learn those uses.

But here’s something else that students often have a hard time with: not everything they’re asked to know is about solving or computing something in the traditional way of getting a number/quantity as the final answer. Sometimes, they’re instead asked to explain, identify, describe, prove, both quantitatively and qualitatively. For example, if the dot product of two vectors is $\vec{u} \cdot \vec{v} = -\sqrt{5}$, is the angle $\theta$ between the vectors bigger than $\pi/2$? Many students think that to answer this question, they need to compute $\theta$ but in fact, they do not; they only need to notice the negative sign.

Remark: By the way, the reason for why lines and planes are each determined by pairs of points and vectors in $\mathbb{R}^3$ is that $1+2=3$; i.e. the dimension of a line plus the dimension of a plane equals the dimension of the ambient space. In more direct terms, use the standard metric and Hodge star (this is just linear algebra).

You also find that the order in which you do things matters (like the burning pasta and steam example). If you want to compute arclength, the formula is $\int^b_a |\vec{r}’(t)|\, dt$. You need to first take a derivative, then take the norm of it, then integrate, and lastly evaluate at the bounds. If you first integrate, you’re integrating an entirely different class of object. The notation is sometimes very subtle. For example, $\vec{r}(t)$ compared to $\vec{r}’(t)$ is only different by a ‘ but the meaning is vastly different. We know that slight changes in symbols can lead to immense changes in meaning; e.g. “The teachers lead” and “The teacher’s lead.” In the first sentence, it makes sense to read it as the (multiple) teachers are leading. In the second, it’s one teacher who possess a lump of metal, say, a part of a lead pipe. Math tries to be less ambiguous with communicating ideas but to achieve this, one must respect what symbols and definitions mean precisely.

PhD dissertation: You have a thesis problem to solve and it’s not so clear what ingredients you need. But you can consult published papers and textbooks to see if those ingredients help. This is like going to the grocery store where there is an expected collection of items and you plan the specifics of your meal once you’re there and able to see what’s on the shelves. Sometimes, you need to be inventive. For example, the store doesn’t sell a sauce that you want to make so you need to figure out how to make it yourself. With math research in these early stages, you’re finding out that sometimes, you’re the first person to be trying something and there are no specific guideposts anymore though your advisor should, well, advise.

Professional math research: Sometimes, you need to develop your own theory (akin to growing your own ingredients). But once developed, you need to figure out what it’s good for just as once you have your ingredients, you need to figure out how to put it all together into a delicious meal that others haven’t made before.

Concluding Remarks

In all the above, I’ve described it as a very individual process but this is far from the truth. In reality, when cooking, you might cook with other people, you might get inspiration from a restaurant, you may talk to a friend who gives advice. So it is with learning and doing math. It involves other people, getting advice from them, being inspired by their clear way of thinking.

Also, I didn’t spell this out but hopefully it’s clear: learning math is like having recipe books to learn from and often, it’s about learning what’s been done before. Doing math research is more about being your own creative agent. You can still draw inspiration from other sources and you certainly don’t stop learning. But the way in which you learn may be very independent.

Lastly, I’ll reiterate what I said earlier. Many students may try to get through a course by using external means like tutors, study groups, online videos, etc. And they may do well enough to pass a course and I certainly support the use of such resources. But if they continue in math, there’s eventually a needed internal paradigm shift. They need to see that math isn’t about plug-and-chug formulas or merely getting correct answers without interpretation just like culinary arts isn’t about microwavable meals or merely filling your stomach, skipping over the labor and artistry. Just as a head chef expects more from a junior chef than frozen microwavable meals, higher level math classes ask students to go beyond the high school approach to math.

Mathematics is about human understanding, intuition, precision, and discovery of structures that emerge when we settle on a few rules that may be given by nature or by our own choice. As I wrote above, it’s about using powerful abstraction to get to the essence of ideas and to broaden the scope of what true statements can be said. Since it involves human understanding, there are human and personal elements to it even while trying to balance objectivity. I’ll end with a longer quotation from the first chapter of What is Mathematics? by Richard Courant and Herbert Robbins (1st ed. printed in 1941) along these lines. Note that it doesn’t mention numbers, shapes, functions, algebra, calculus, etc.

“Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science. Without doubt, all mathematical development has its psychological roots in more or less practical requirements. But once started under the pressure of necessary applications, it inevitably gains momentum in itself and transcends the confines of immediate utility.”