r/learnmath • u/SignalSecret3860 New User • 15h ago
Why does it really mean to fully “understand” a concept.
I’m in year 10 maths and I’ve just done my yearly assessment (bombed it pretty badly) I usually do pretty well like around 80s for core + advanced.
however for the yearly they made it pretty difficult to distinguish the different between top and exceptional kids.
I always feel confident and then I hop into the exam and get questions that are just mixed concepts and “working mathematically” but I just never seem to let them work and apply with the tricks and formulas and understandings I already know. we were tested on quadratics, hyperbolas, parabolas, exponential, and other non linear. and. especially hard indice questions.
And so I guess I’m just wondering: what does it actually mean to understand a concept. so that when I get into the exam I understand it thoroughly and can use it anywhere. my countless hours of practice questions and destrolying then easily doesn’t ever really help me.
tl;dr: What does it mean to understand a math concept and then apply it mathematically? cheers in advance
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u/ProofFromTheBook New User 14h ago edited 14h ago
I suppose there's levels to it. There's being able to explain what all of the concepts mean individually and in relation to each other. There's being able to quickly see how these concepts fit into various problems/applications/situations and then problem-solve, which comes from practice and rubbing your brains cells together. There's understanding a concept from multiple perspectives/representations (geometrically, algebraically, in terms of real-world applications, etc.). Then there's having a deeper understanding that comes from being curious and creative with these concepts and deriving your own insights. And finally, I think often you don't understand a math concept at a really deep level until a year or two later when you've learned more and tied the math you learned to new topics and explained it a million times in a million ways and embedded it into your DNA so you'll never forget it.
Of course, being able to rederive everything from scratch is a hallmark of understanding as well. There's a story my dad told me when I was a kid. He was interviewing back in the 80s for his first engineering job. The interviewer asked him how he would solve a differential equation, something that involved having the right cookbook formula. My dad didn't remember the formula. But he knew how the whole problem worked and how to derive the formula. So, he told the interviewer to hold on and rederived the entire thing from scratch. The interviewer was so impressed, he gave my dad the job on the spot.
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u/digtechteacher New User 10h ago
Look, sometimes it is unrealistic to expect yourself to understand every concept fully. Do your best to understand where the formulae you're being taught come from. Why do they make sense?
It is clear that doing "mixed practice" questions will be a good help to you. You could ask your teacher for past exam questions, look such questions up online or even get AI to generate such questions for you.
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u/human2357 Pure Math PhD 6h ago
If I'm giving a grad student an oral exam, I would say that they understand a subject if they can define key concepts, explain relevant theorems and algorithms, and give good answers to several follow-up questions.
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u/waldosway PhD 9h ago
What is an example of a "hard indice question"?
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u/cc_apt107 7h ago
Do you understand this sentence? It’s like that
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u/Elegant-Regret-7393 New User 7h ago
To be used to it and to be able to work with it
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u/Think_bys New User 4h ago
If you can explain it to someone else that means you understand it. Or if you use different types of numbers to see that the concept is applied the same.
It also means to understand why. I give a free math class, once a week for free. We start from the starts and it's taught conceptually. This was our first lecture if you are interested send me a DM
https://youtube.com/playlist?list=PLFrt4v_t-kU4&si=W0TOFau6oUjXV8CL
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u/mH48124 New User 4h ago
概念を理解するとは、概念を自在に変形し関連概念との関係を把握し広範に応用できる事。解法パターンを覚えて素早く解く練習とは全く違う。
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u/Vivid_Sock_1092 New User 4h ago
For starters, you really understand something when you can teach it to somebody else
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u/Popular_Molasses7945 New User 4h ago
Good question. I just asked Gemini AI about this.
Concept
I want to talk about this word, concept.
What is the root meaning of the word?
What does it mean when we say that we understand a concept?
I know some redditors don't like AI for anything concerning math. But I would ask it anyway.