After some years working as teaching assistant, professor and researcher I realized that expressing my ideas in the paper is something that requires a lot of practice. Students/mathematicians are used to have a voice at the back of their heads saying out and loud: “I know how to solve it, but I don’t know how to write the solution”. The world makes more sense in our heads, doesn’t it?
Well, one could argue that you truly know the answer when you know how to write it clearly. For the sake of the argument, let’s assume that there could be a situation in which a person knows how to solve a math problem but doesn’t know how to write the solution in a way that make sense to others. Notice that, this kind of situation is specially problematic in education, because the teacher has to grade students’ knowledge based mostly on their writings.
So, in this post I’d like to give four tips for the students who want to improve their writing-math skills. These tips are far from being a recipe. They may be seen as lighthouse to those who are lost in this endless sea of arguments.
But before we dive into the tips, let me say briefly how I, as a professor, see a math test:
A test is a moment when the students have to convince their teacher that (1) they know the main definitions of the subject; (2) they know how to use these definitions to derive other definitions (in theoretical courses) and/or to apply them in concrete or numerical questions (in practical courses); (3) they know the main methods of computations.
Ok, now let’s see some tips that can help you to convince your teacher that you truly know what you know:
1. Be boring! Tell the end of your novel!
Maybe the crucial difference from writing math to writing a novel is that in math you gain points by being boring, in a novel, you don’t. What I mean is that in math is desirable to be very clear from start to finish. Whereas in literature that’s the opposite. Could you imagine a version of Lord of the Rings whose first line is: “A society is formed to destroy a magic ring and one of its member, Frodo Baggins, managed to find his way to destroy it.”?
Well, this is more or less what you should do when you are writing mathematical reasoning.
Start by explaining to the reader (professor) how you will solve the problem. Cite the main theorems you will use and/or the techniques involved.
Example (linear algebra): “We will solve this linear system by applying the Row Reduction Algorithm to its augmented matrix in order to get its reduced echelon form. Then we will solve the equivalent linear system.”
Example (Calculus): “We will solve this integral by using the integration by parts process twice.”
Example (Optimization problems): “We will write the volume of water inside the cube as a function of time and then we will derive it in order to find its critical points and finally its maximum volume.”
Explaining your plan it is a good idea for three reasons:
- It shows you are aware of the main techniques and/or theorems;
- Sometimes, it saves you.
(Sometimes, it is possible to judge whether or not the student understood some concept by their plan to solve the problem. This helps you because even if you make some mistake in the process, the professor knew from the beginning what you were trying to achieve and then she or he can ignore some of those mistakes.)
- It gives you a chance to review your solution
2. Call the characters by their names
In mathematics we define things and give a name to them (although we always try to prove that different things should have the same name). Important theorems have names, the mathematical objects a particular field investigates have names, some rules have name. So, call things by their names. There is a reason why scientific community has its own jargon (and it’s not to feel smarter than others). We use specific terminology because each symbol/notation can encode a large amount of information, so we can say deep things with just a feel names. In a nutshell: jargon and specific notation let the scientific papers short!
Example (Linear Algebra): “The reduced echelon form of this matrix induces an equivalent linear system, so it is enough to solve this new system”
Examples (Probability): “We are under the hypothesis of the Central Limit Theorem, so we can apply it to the sequence of random variables to get the convergence in distribution”
Example (Calculus): “By the Fundamental Theorem of Calculus the aforementioned function is continuous.”
Calling things by their names show you are aware of the “great players” of the subject. Moreover, it shows you have some erudition/maturity and have devoted some time to learn the concepts.
3. Don’t hide anything, you aren’t writing a Crime Novel
Tell the reader (professor) the reason why you did things the way you did or at least let clear what you are doing.
Example (Linear Algebra): “We will replace the second row of our matrix by two times the first row plus the second row” (or use some notation that say the same thing in a shorter way)
Example (Calculus): “Denote by u = sin(x) and by dv=cos(x). Then, integrating by parts we have the following…”
It is important to have in mind the following: it is your duty to convince who is reading your test that your solution is right and that you know what you are doing.
Every time you make a mistake in a step which is not clear you bring some shadow to your answer. The point is that your teacher won’t be able to distinguish if you just made a slip or if there is something deeper involved. And I think when professors see themselves in this kind of situation they may tend to assume you don’t know what you are doing.
4. Let clear your final answer
You are not Kafka, so not finishing your answer won’t make you famous (actually it didn’t make Kafka famous either).
In many subjects there is a proper way to express your final answer. In linear algebra, for instance, you have the parametric description at your disposal to describe the solution set of a consistent linear system.
Frequently, the final answer in mathematics is a set of objects that satisfy a given property. So you can use set notation to describe your answer.
Example: “The solution set for the inequality is
The above tips can make your answer clearer, more elegant and can help you to show to the world the beautiful things you have learned. Expressing your ideas in the paper is something that requires hard work as the most things that matter in life. So, I suggest you practice by doing exercises and writing your solutions as clear as possible. Then you can share them with your classmates and/or professor.