Please sir make more videos continuing om how maths should learnt like this. Alot of info on learning mathematics online isnt actually helpful in learning mathematics properly in my opinion
In the video you say that the only way of writing the imaginary unit in terms of the real numbers is via a matrix. I dont think that is true since we can also write it as an ordered pair (0,1)
Thank you very much for your inspiration, as a high school student in Taiwan, I used to hate rote memorization the school is emphasizing, that led me approach to study only with the opportunity for intuitive learning. Now realizing that going to one side of extreme is wrong, I felt more pleasured in learning in both styles. I would say that you video has really changed my approach to learning. Thank you!
Do we have a mathematical model which extends this number line to 3D? What i mean is, the way i extended no. Line to 2D, is there something in math which extends it to 3D?
That is a great question! The answer is no, but a famous mathematician named William Hamilton tried looking for exactly that. He could not find it. Until one day he was walking with his wife along a canal in Ireland and he had a moment of genius: you can't create a 3-dimensional number system, but you can create a 4-dimensional number system! These are called the quaternions and they are endlessly fascinating. The pattern continues to produce an 8-dimensional number system called the octonions. But the pattern (R = 1-dimensional, C = 2-dimensional, H = 4 - dimensional and O = 8 -dimensional) stops there! This is also very interesting and mysterious
Going through primary and secondary school, and having no passion for maths pushed me towards rote learning in my first year of becoming aware of what I am learning in maths, as more and more concepts became exposed to me. My way of learning gradually evolved to become intuitive
I don't agree with this opinion at all, because to have good intuition you have had to do thousands of problems. On the other hand Rote learning is what would one call learning lots of concepts but few variations and not enough practice.
I agree that to /develop/ intuition you need to do problems. But here I am talking about intuitive learning as a modality of learning (not of understanding). Rote learning is simply getting the numerical answer or doing the computation without actually thinking deeply about what this calculation says about the objects involved. The point is that you can't develop intuition without rote, and rote learning is blind if you never ask intuitive questions to interpret the rote learning
Hi , i have a question and hope for an answer , there is some math rules that is usually hard to relate to first principles of the topic because mainly the proof behind them seems creative or the steps of the proof dosen't really tell you any relation to big picture and just seem just like symbol manipulation for example the sine rule proof , there is a geomtric proof wchich bassically tell you that the sine rule is equal to diameter of the circumcircle of the triangle but the proof seems creative so its kinda hard to stick because i can't logically relate the steps of the proof to big picture or the reason behind those steps so i ended up just saying what sine rule is telling from a geomtric point which is that the bigger the angle the bigger the opposite side , so really what im asking is there is some math things that seem kinda creative and its hard to find purpose maybe you encounter some moments like that , how should i tackle them? I don't like proofs that involve symbol manipulation and really when i go repeat the proof i forget steps because maybe the steps are not logically related . Maybe im wrong . Btw the video is really impressive what you are thinking when learning is the most important thing !!
You can “make” intuition by going over simple example many times. What we refer to as intuition is oftentimes only a result of repeated exposure to the phenomenon. We feel it is intuitive that a ball falls to the ground only becauee we’ve seen similar occurences many times in our life - there is nothing obvious about this phenomenon to a new born infant for example.
We best learn in layers. Some bottom layers will be easier to learn than some of the top layers. Ideally, we need to focus on those bottom layers first. With that said, many Anki beginners tend to create cards with too much stuff on them. So then, it can be worth breaking up those cards into multiple cards to make them more atomic. Atomic cards are usually easier to learn and test yourself on. But if those cards keep on coming back to you because you just can't learn them, at least, you know where your weakness is, and either you decide to rewrite the card, research the topic further, break it up further, or suspend that card from your collection if it doesn't seem to be worth the effort.
I personally used to spend a lot of time on intuitive learning, always trying to find why something was true, rather than going on and practicing problems. This rendered me with marks lower than what I would expect based on my understanding. Your video is the only source that addresses this issue. Thank you! for this wonderful video.
I had the same problem haha. What are some things you’ve done to improve your situation? I’m trying to deal with this by consciously spending more time doing problems.
@@ABC-jq7ve I have realised that there's a sweet spot, so when I start for the first time I still try to think why something was true, however now I don't spend say 1-2 hours trying to think how it maybe derived, I just move on.
Problem solving is great however I think what i really like about maths is how very simple ideas form complex results and how you can almost derive all maths (may be not possible) from bunch of intuitive ideas. So tbh I don't at all regret this habit.
Thanks for sharing your ideas! tl;dr: Introducing Anki (flashcard tool) into my learning process has allowed me to focus most of my attention on intuitive learning and problem solving, while spacing out the efforts needed for the memorization of facts over a sustained period of time. Up until after my 4 year undergraduate engineering program (2017), I almost entirely relied upon rote learning as my only objective was doing well in exams and getting a job. Thus, math courses were entirely a means to an end. Only after I started learning for the intention of developing a new skill (a foreign language like French), did I actually do some meta-thinking (how to learn stuff?). I now believe that my failure to have not acquired an intuitive knowledge of math and the other sciences was due to having over-emphasized the act of memorization when in fact human brains have evolved to dump facts/info that it deems not pertinent. But upon introducing Anki (flashcard tool) into my study workflow, my efforts for the act of memorization has been greatly spaced out over a sustained period of time, allowing me to place due emphasis on intuitive learning aka understanding as well as problem solving.
@@Podzhagitel The best cards use your own words, or your own diagrams. Ideally, you need to understand before you ankify things. I recommend you search for the "20 rules of formulation" by the creator of supermemo. Those are pretty good rules to follow. But keep in mind, initially your cards will be really bad, but over time, with the feedback you receive through Anki, you'll learn to create better and better cards (either that, or you'll cull the bad cards from your collection).
You can imagine what a number does (when multiplying) to the whole number line seen as a geometric object. For example, the number 2 can be seen as a dilation or 'zooming in' of the real number line. It moves 1 to 2, 2 to 4, -1 to -2, and so on. So the real number line is stretched out after multiplying by 2.
You know I think you should make a channel centred around explaining maths concepts like you have with some of your other videos, helping to develop mathematical intuition. I believe if you do that you could fill a RU-vid niche though to get more views and subscribers you may need to make the videos more eye catching and engaging. I personally don’t mind videos in this format but for a wider audience that may be a consideration. However overall it is a great video and this was positive criticism do understand that I am not trying to demotivate you and I totally support you in your RU-vid journey. Keep up the good work it will pay off.
Thanks, Haris - that is certainly my plan. I have many ideas for maths videos ranging from high-school level maths to research level and everything in between. Thanks for the advice and I agree with you!
@@zenmathmind It is very important to me these type of content, so thank you, society all over this planet needs better and easily accessible education, so what you do is very important. *Here i'm going to talk a lot about my views on learning math (i express myself better when i have the freedom to write without small limits). Ao feel free to read only if you want/have the time, i like to talk about my experiences in high detail. *Also this is the first video of this channel that i watch. RU-vid not always allows me to subscribe when i try (something about my number of subscriptions being bigger than my number of subscribers), but usually it only takes some time the platform to forget about it and allow me to subscribe properly (which is usually what happens) So, basically my experience on learning mathematics is the average, highly dependent on memorization (which i suck at), and overall awful. I love mathematics, even though i cannot understand it, which is like loving the flavor of something you cannot bear to eat. I'm austistic, so i don't know if this is something singular or simply not talked about more, but the way i'd like to learn mathematics is rather different from the experience one gets, at where i live. Mathematics when thought formally is usually put as a set of objectives to be achieved by memorizing patterns on a similar way monkeys are thought to do in experiments, but with more independency. This bothers me, how am i supposed to understand those patterns, if i don't know how they were discovered and why, which processes went through, in what contexts, from the earliest to the latest forms of a concept ? I would like to bother learning anything by studying it's purpose in a primary perspective, where i'm put on the place of each individual (s) and their historical and practical contexts, who were responsible for discovering/first documenting a subject of study, and then apply this same approach to learn the following breakthroughs of such subject. I'd like to be put under the skin of all these people who dealt with those issues by practical/theoretical causes, and why such causes are. I don't know if I could express myself better, but in any case I don't care to explain anything that is not clear. If all of mathematics is too long to be learned, then may i spend the rest of my life diving into it. Thank you for reading, God blessings be upon your and your family. Just felt like sharing those issues, because i would really like to experience mathematics this way ! I don't know any works that are similar to what I wrote or anything, but would love to be told of, or even if someone would start (or already has) something like what I expressed, be it in a book, blog, RU-vid channel etc, be it happening or yet to happen, i'd like to see and take part on such journey. Thanks a lot for your time !
