My niece reached out to me a few days back asking about tips for studying at school. She was specifically interested in any ideas I had about how to excel in her maths studies. I wrote up my thoughts for her and it occurred to me yesterday that there might be some benefit from putting these notes online as well.

Without further ado, therefore…

Study Skills Advice

As I understand it, your question takes two parts:

how do I learn / study better
how do I specifically learn / study for my upcoming exams (in particular for mathematics)

I've split my response up into two parts since they're somehow related but also different. Part one ('how do I learn new things') is sort of required for part two ('what do I have to do to get good grades on my exams'), since it's pretty hard to try to cram ideas and concepts into your head if you don't first understand them.

1. Learning to Understand

Something you just have to 'learn'

First off, you'll want to learn the small tiny 'fact-like' things first. Some of this you just have to put in some work and learn them. It'll help you a LOT when it comes to getting a grasp on the bigger-picture concepts. To give an example, if there are certain equations or definitions of concepts, it can help to take a bit of time and just learn those by heart (in whatever way usually works for you for that). Note that this is limited to just small fact-like things. For sciences, maybe there are some facts or properties of something derived from the periodic table, etc, that are important. In the specific case of quadratics that you asked about, it might be simple things like:

the usual form of a quadratic equation
the quadratic formula
'what is a discriminant'
what does the completed square form look like, and how do you solve it?
what does this or that quadratic look like when you graph it out?

The key thing is to isolated and master those tiny morsels of information / facts early on, since a) it needs to be done anyway and b) it'll help you with the big-picture stuff. Sometimes textbooks help out by highlighting or bolding those key concepts. Just make sure you're not just trying to learn EVERYTHING by rote, since that advice is as good as no advice in my opinion!

Learning for understanding: foundations

Something else I'll say early on is that there are some core foundations that will serve you well:

start with a / the problem — it will help to try to wrestle with the kinds of things you're expected to do with this particular topic you're studying. It can actually help you to do this at the very beginning of starting a new topic: before you even study things, take a look at the kinds of questions that you'll be tested on and expected to answer. Maybe even try to solve them. You'll learn a lot about the subject area just by rolling around in the mud and struggling a bit.
curiosity — this almost goes without saying, but you should try to find a way to be curious about whatever you're meant to be studying. It'll help a LOT with your motivation and interest, which will in turn help you keep moving forward and give you energy at moments when things get hard. Sometimes the trick to this is just trying to ask questions about whatever it is you're studying: why is this important? how does it connect to topic X or Y we already studied? what is most surprising about this topic? Sometimes it can also help to know a tiny bit about the history of a topic. Calculus is a bit down the road in terms of your studies, but back in the day (late 19th century, early 20th century) there were really epic debates between the various people who were developing this new area of maths. Lots of drama and falling out between people. So sometimes just knowing a bit about the personalities behind things can help.
growth mindset — I wrote a blog on this a while back but the core thing is just to believe that you have it in you to master this thing. Once you have that, and if you start from that place, then you'll have much of what you need to keep moving forward. And as one teacher once told me, as long as you never give up then you'll eventually master it. Sometimes you just have to be a bit patient. Most things come if you give them enough time :)

Retrieval Practice

This is sort of the core of most learning, in my opinion, and I have a lot of thoughts and practice around this. (Have read a LOT about what research recommends in this area, but I figure you're interested mostly in whatever is practical so I'll keep it mostly at that level.)

The core: if you want to learn something / master it, you have to retrieve it from your memory somehow. (Retrieve can mean lost of things: for German vocabulary it might mean knowing the gender or translation of a particular word. For Maths it might mean a particular formula, or even a high level understanding of how one concept relates to another.)

The important thing about this retrieval is that it will be and should be hard to do. (This is one reason why people don't necessarily enjoy doing this, and even fewer actually do it.) It's hard to struggle to put together a coherent explanation of topic x or y, but that struggle is what helps create neural pathways that cement the understanding going forward.

One possible way you could do this is to write short summaries of what you understood of a particular topic, then check your notes to see if you were correct. IMPORTANT: the key thing is to do this from memory / without notes. Otherwise you're not actually reinforcing materials in your head.

Another way of testing things (e.g. in your specific case of quadratics) is to do practice examples. Your school or textbook probably gives you some practice examples, but you shouldn't confine yourself just to doing those. Make up your own examples, or go online and find more examples (use Khan Academy or whatever).

It can also help to make distilled summary sheets at some point during your studies which gather together your understanding on a single piece of paper for the entire topic. Here actually you can see the one I made digitally for my own study of quadratics a few months ago:

The key thing with retrieval practice is to get a lot of it, and to try to make it the 'hard' / 'difficult' kind of retrieval. Mostly this starts with a blank sheet of paper and then you try to write down what you know about a topic, or a concept, or whatever specific thing you're trying to understand. Writing things down will help you realise (quickly! painfully!) which parts you don't actually understand. So it's as much to reveal to you which parts you need to work on as it is for anything else.

(There's a much-praised technique named after a well-known American scientist, Richard Feynman, and you can do something like this, too:

Write the title of a topic that you want to study / test yourself on
Write or map out an explanation of that subject intelligible / appropriate to a non-specialist. Do this from memory.
Identify any gaps in your explanation / understanding.
Relearn / restudy / interrogate to fill in the gaps.

You can use narrative / diagrams to condense and clarify your explanation. It's basically the same idea. And yes, bullet points or spider diagrams are all possible ways of doing this.)

Developing mental models

There's this idea that the whole thing you're doing when you learn something is developing 'mental models', which I personally find a bit hard to wrap my head around, but it is a thing… It's maybe the next layer up in what's happening when you try to learn something.

Mental models are, for me, about making a topic your own somehow. They're also about making the concepts of that topic manipulable somehow.

The 'making it your own' part has a lot to do with confidence, somehow, but it's also just feeling familiar and effective with the concepts that you feel comfortable solving problems in that area. If you see a problem, e.g., you know which techniques (or which subset of techniques) are needed to solve it. If there are multiple possible ways to solve something, you'll have a good feel for the tradeoffs: i.e. why this way is better than that way etc. In the case of quadratics, for example, we know that there is this amazing thing which is the quadratic formula, but you probably don't want to use that formula the whole time because it's easy to make a mistake with it and it's a bit cumbersome. Instead, we often use other simpler techniques that work for many (if not totally 100%) of the problems that you'll be exposed to.

One way to help develop mental models is to try to explain the topic to someone else. You already did a bit of this in the retrieval practice above: trying to explain it on paper is already some of this. But trying to explain a topic to people at different levels of understanding can be really clarifying. I.e. if you had to explain quadratics to a 5-year old it's probably different to how you'd explain it to a 40-year old. (Along with this, you can test out this approach by chatting with a chatbot about the topic. I'm sure you've heard of ChatGPT, but Claude is also another good option, esp for things like maths. The key thing is to start the conversation by saying something like "I would like to have a conversation about quadratics. I've been studying it and I'd like to test out my explanations of some core concepts with you. I would like you to tell me if things feel unclear about what I'm saying, or if you notice that there are some areas where I could improve my understanding.")

(While we're here, using things like ChatGPT to develop mental models can be useful. I will often have conversations that begin with something like "What is a good way to think about the discriminant in relation to quadratic equations? Please make your explanation simple to follow and use some concrete items in your reply, like only items that you'd find in a kitchen.")

Making mental models is hard! But the work you do to solidify things and make them your own is really worth it!

Anyway, the big point here is to reflect on what you're studying. Make sure to also give some time to connecting it to other things either in your maths studies or outside, or even life in general. It's not the best to just view everything completely isolated and disconnected from the other topics, so try to take a step back from time to time! (Unfortunately, most schools aren't built to encourage that process much, but it's important!)

Specific contexts

There are some other specific contexts that require different / more targeted advice, but you didn't mention them so I'll ignore them a bit. But language learning is one of them, and learning some kind of 'motor skill' is another (i.e. that requires coordination or physical movement like playing golf or the yoyo or whatever).

In Practice: Understanding Quadratics

To summarise the practical points listed above:

learn the small fact-sized pieces early on
get lots of retrieval practice (a mixture of examples of doing whatever the skill requires of you, and/or writing or explaining the topic at various levels)
develop mental models where you can.

2. Studying for Exams

I'll take it for granted that you agree that you can't study for something without properly understanding it, so somehow the things in part 1 are sort of a prerequisite for this section; you can't get ready for an exam if you don't understand what's going on.

That said, there are some tactical things you can do to help your chances of success once you do have an understanding of a particular topic. Note that for all of this, it's a bit of a question of picking which parts seem doable / manageable. It's probably unwise / counterproductive to necessarily try to do EVERYTHING :)

Ground rules

You should understand the requirements of the exam. Take a bit of time to read through some previous exam papers. I'm sure your teachers have also given you clear guidelines on what kinds of things to expect. That will give you a map for how to prepare, so be sure to do this.

Foundations: Exam Study

There are some basic foundations here which for various reasons get forgotten when you're under exam pressure, but it's good to remind yourself of these, since if you neglect these it'll negatively affect your ability to study etc.

sleep
eating things that nourish your body instead of just feeding cravings
taking breaks (every hour, ideally, get up and walk around for a minute or two)
minimise distractions (put your phone in airplane mode or in a lock box while studying)
movement in general / going out of the house for walks a few times a day is a good minimum.
'managing your energy' — this one's a bit hard to quantify / explain, but I'd say it's worth trying to embody the principle that you should only study as much today as allows you to keep studying tomorrow. I.e. if you overdo it and you study a lot today, but it's a bit too much and tomorrow then you can't do any study etc, then that was counterproductive. (Hope that was clear!)

Mnemonic / memory tricks

There are a TON of memory tricks out in the world. All of them are useful, but not all of them are equally useful for every situation :)

Things like the major system, the link system and the peg system are all useful, but they require a bit of time and probably also someone who knows how they work to explain them to you.

If you already have a bit of experience with these things, then I'd encourage you to use them in your studies, but if you don't have much experience then I'd say probably that it's not going to be the difference between an A and a B grade so probably it's a waste of your time to try to get into that in the run-up to exams.

That said, it would TOTALLY be a really useful investment to learn a few of these during summer holidays in a non-stress / fun way. You can play around with learning the order of decks of cards etc — I can explain all this if you're interested — and then you'll have that skill available to you if you need it next year or throughout your life.

There are some general memory principles that you can rely on in general terms:

when trying to remember something, make it memorable in your mind! so maybe try to imagine the concepts as characters in some kind of image in your mind, and use all your senses and bring in some shock or drama etc etc. (LMK if you want more of these kinds of advice. I have a lot, but not sure how useful it is for you right now.)

Spaced Repetition

This is a really useful tool, but it requires a bit of upfront (time) investment and unless you're feeling super comfortable / not stressed at all, I might suggest to add it to the list of 'things to learn about over summer / winter (?) holidays'.

Basically this means testing yourself with (digital) flashcards, but the twist is that you only get shown the flashcard at exactly the optimum time / day when you need to be tested on it. (There's a whole science to this which I won't go into, but there's a TON of backing to the fact that this is the way to make things get into your memory.)

The best option for this is a piece of software called Anki. It runs on your laptop and phone etc, but it has a bit of a steep learning curve mainly because the defaults it comes with aren't great. So if you were interested I could help you set that up, but the key thing to know is that using this requires a bit of extra work.

The main idea is that you create (digital) flashcards for all the things you need to know, and then every day you check in with Anki to review whatever cards it says you need to know. There's an algorithm that calculates which cards you should review. It should mesh well with your intuitive sense of how memory works: i.e. over time you slowly forget things, so Anki will prompt you to recall a particular concept just at the point before you forget it, since that exact moment is the best time to review it. When you review it at that moment and you get it right, it'll really strengthen your memory for that thing. If you don't remember it, then it'll reset the status of that card and it'll know to show it more often for a few days etc.

There are more manual ways to get the same effect, but (for a lot of reasons) they're not as effective since humans don't work / behave like computers so really using a digital tool is the only way to go.

For quadratics, to use my experience, I have a bunch of cards relating to that that I get tested on. Writing a 'good flashcard' is a bit of an art, and we can get into that if you're interested, but I'll just lay it here as an option for now.

Interleaving

This is a fancy word for saying: 'don't study just one topic on its own'. When you're testing yourself on things that you'll need to know for exams, make sure to switch things up a lot. This means doing one problem from quadratics, then another relating to trigonometry, and another relating to topic z etc etc.

There is again quite a bit of evidence that this makes you much stronger in your understanding / learning, even though (or maybe because!) it's a bit harder to do.

If at least part of your review of topics / facts are handled by Anki it'll take care of giving you random flashcards anyway, so this more relates to things like solving maths problems by hand.

So don't just do 50 iterations of the same maths problem, in other words. Make sure you're switching topics etc.

3. Next Steps

Check the practical suggestions above
let me know if anything's unclear / or you want to know more about how to do thing x or y
gather some problems to solve so you can make sure you're practicing the things you need to study
get into some good habits around retrieval practice (i.e. writing things down to test whether you know them or not)

All the things I wish I knew about studying at school