I really had the problem of how to study mathematics without knowing anything about a topic and I would like to talk about it frankly because much of the information available on this topic is totally misplaced. The general advice is to do more practice problems or to watch more videos. And though neither of those things is wrong as such, they leave out the most crucial question, which is what makes you not understand the topic in the first place. It is there that the true answer is.
Let me set the scene. Second year of university. Real Analysis. I sat in lectures for three weeks straight and understood approximately nothing. Not a vague outline. Nothing. The symbols looked foreign. The proofs felt circular. I kept doing the problems at the end of each chapter and getting them wrong in ways I could not diagnose. I was putting in time but making no progress, and the exam was six weeks away.
What changed things for me was not a new textbook or a longer study session. It was stepping back and diagnosing the actual nature of my confusion rather than just pushing through it with more of the same. That shift in approach is what this article is about. Not inspiration. A practical framework for making sense of mathematics when it currently does not make sense to you.
Understanding Why You Don’t Understand Is the Starting Point for How to Study Mathematics When You Don’t Understand
You can only combat the problem before you can fix it. Widely speaking, there are three causes of a mathematics subject appearing impossible. The former is an incompatibility in prerequisite wisdom. Mathematics is a chronological subject that is not so much so in other subjects. You will never quite get some subjects of algebra unless you are thoroughly acquainted with fractions. Calculus will be a foreign language to you when you have weak algebra. Most students attempt to study a subject without realizing that they are not confused on the subject itself but, at the very least, a step or two further back behind the subject.
The second cause is the conceptual confusion, i.e. you know how the procedure works but not what it is. You can even go through the steps to derive a derivative because you do not know what a derivative is.
This kind of mechanical understanding breaks down the moment a question asks you to apply the concept in an unfamiliar context, which is exactly what harder exam questions tend to do.
The third reason is presentation mismatch. Some explanations of mathematical ideas just do not work for certain brains. The way your lecturer explains integration by parts might genuinely be excellent and still not be the explanation that connects for you. To know how to study mathematics when you do not understand is to find some other way of explaining the same fact than the one which has already left you puzzled.
The University of Queensland researcher discovered that students who could make the correct identification of the origin of their mathematical confusion (procedural, conceptual, or based on a prerequisite) did much better on follow-up tests when compared to students who approached all confusion as one undifferentiated problem fixed by more practice.
Going Back to Find the Gap in Your Foundation
When I was failing to understand Real Analysis, my tutor sat with me for twenty minutes and traced my confusion backwards. The proofs I could not follow relied on a strong intuitive understanding of limits. My understanding of limits was procedural rather than conceptual. I knew how to evaluate a limit but I did not really understand what a limit was representing. She took me back to that foundation, spent one session clarifying it deeply, and suddenly the analysis proofs started making sense.
This backwards diagnosis is one of the most powerful tools for how to study mathematics when you don’t understand a current topic. When you bump against a wall, ask yourself what has been just explained to you has already been assumed to be known. Then ask yourself, is this something you know deeply or shall I say on a surface level. List the prerequisite concepts and test yourself on all of these. When either of these of them bring uncertainty, that is where you begin.
Khan Academy, Pauls Online math notes and MIT OpenCourseWare have also free, excellently structured material, so you do not have to formally re-enroll in a course to step-back to a prerequisite. Two hours of solidification of a basic idea will frequently open the door to knowledge of the current idea sooner than ten hours of floundering with the current idea before that basic idea is established.
Changing the Explanation to Find the One That Connects
Mathematics has a long tradition of being taught in a specific formal way, and that way does not work equally well for everyone. If you have read the same explanation three times and it still does not connect, the problem is almost certainly the explanation, not your intelligence. Different representations of the same mathematical idea work for different people.
This is a core part of how to study mathematics when you don’t understand through conventional materials. You go looking for alternative explanations. On YouTube, 3Blue1Brown is an intuitively-focused, visual exposition of calculus, linear algebra, and other concepts that are often more engaging to many students than the textbook method. Brilliant.org develops mathematical knowledge by creating problem-solving scenarios that are interactive. NRICH and Mathigon offer instructional methods that focus more on exploration than process.
The idea is not to discover the simplest explanation. The aim is to discover the reason that renders the logic behind it to seem like a necessity and not a random choice. When you get mathematics at a conceptual level, then you no longer feel like you are taking steps. It is as though the steps are the most logical things that can be done as a continuance of what preceded.
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The Role of Worked Examples in How to Study Mathematics When You Don’t Understand
When a concept is genuinely new and confusing, worked examples are your most valuable resource. Not practice problems. Worked examples. The distinction matters. A practice problem asks you to produce a solution. A worked example shows you a solution being built step by step and invites you to understand why each step follows from the previous one.
The method that works best for me is what I call copy-pause-explain. I read through a worked example completely once. Then I copy it out by hand, step by step. At each step I pause and write in my own words why that step was taken. Not just what was done, but why it was the right thing to do at that point. This transforms a passive reading of a solution into an active engagement with the reasoning behind it.
Understanding how to study mathematics when you don’t understand requires engaging with the reasoning behind procedures rather than just the procedures themselves. The reasoning is what transfers to unfamiliar problems. The procedure alone only transfers to problems that look exactly like the example you practiced on.
Teaching the Concept as a Test of Real Understanding
The Feynman Technique is a well-known phenomenon in education research that specifically refers to the ability to describe complicated concepts in a simple way that is named after physicist Richard Feynman, known for such skills. The method is to explain something verbally using simple words as though you are explaining to a person who has no background knowledge of the concept. Where your explanation fails or becomes obscure are where you are weak in understanding.
I use this as a benchmark when I am working on how to study mathematics when you don’t understand a particular topic. As soon as I feel I know something, I put my notes aside and attempt to write out what it is. Not the procedure. The concept. What is integration really doing geometrically? What is known when an eigenvalue is found of a matrix? When I am unable to answer those questions in straightforward language, I really do not know the subject, no matter how many practice problems I have answered correctly.
This is not a pleasant test due to the exposure that it has. But these gaps are precisely what exam questions are aimed at. The questions which distinguish between high and average scores are nearly always those which involve the students in reasoning as opposed to merely following a memorised procedure.
Structured Practice After Understanding Is Established
After having a conceptual basis established, deliberate practice will really be productive. Practice at this level is not aimed at comprehending the information but at developing fluency and being exposed to the variety of possible presentations of the concept in questions. Solve problems of progressively greater difficulty. Keep a check on what kind of issues you misjudge. Go through the solved problems that you got wrong and implement a copy-pause-explain method to get to know the point in which your reasoning failed.
The difference between productive practice and unproductive struggle is whether you have the conceptual foundation in place before you start. How to study mathematics when you don’t understand shifts from being a question about practice methods to being a question about foundational comprehension first. Practice without comprehension builds speed on the wrong path. Comprehension first means the practice you do builds genuine, transferable ability.
Conclusion
When you learn to learn mathematics without understanding something, learning a subject in a different way is far better than learning it harder. It begins by finding out the actual reason behind your confusion instead of seeing all mathematical difficulty as one problem. It entails returning to prerequisites when the background is wobbling, getting alternative explanations where the normal presentation fails to resonate and working with worked examples in a manner that develops referring and not processing memory. When you come around to it in this way, the mathematics begins to feel different. Not simple, but necessary, and logical. And something you can work with is logical.