Information about How to Learn Mathematics - Part 1

1 00:00:00,586 –> 00:00:06,743 Hello and welcome back to a new video on the channel that is called “The Bright Side of Mathematics”.

2 00:00:06,943 –> 00:00:10,968 A channel that has now reached 100k subscribers

3 00:00:11,014 –> 00:00:14,505 and in order to celebrate that. I want to make some videos like that,

4 00:00:14,705 –> 00:00:20,336 that are not part of a huge video course, but explain a little bit how to learn mathematics

5 00:00:20,743 –> 00:00:25,514 and you see I want to start that by telling you how not to learn mathematics.

6 00:00:25,589 –> 00:00:29,350 So in other words I want to show you some common mistakes

7 00:00:29,550 –> 00:00:37,806 and I know a lot of mistakes simply because I made them myself and I teach mathematics for a very long time know.

8 00:00:38,371 –> 00:00:44,945 Indeed I checked that for this video and it turned out, I do this thing for the last 20 years now.

9 00:00:45,429 –> 00:00:54,600 In fact I started in high school. Explaining basic mathematics how I learned it. So basic calculations to other friends, to other pupils

10 00:00:54,671 –> 00:00:59,555 and then I continued that in the university with more advanced mathematics

11 00:01:00,000 –> 00:01:07,224 and then after teaching that for work at universities, you know see I continue that here on YouTube.

12 00:01:07,971 –> 00:01:14,424 However especially at the universities I met a lot of students, indeed motivated students,

13 00:01:14,500 –> 00:01:18,429 that still made a lot of mistakes while learning mathematics.

14 00:01:18,657 –> 00:01:24,567 Then let’s start with the first mistake here, which I call not having the correct prerequisites.

15 00:01:25,186 –> 00:01:30,087 In mathematics this is especially very common and it can feel very frustrating.

16 00:01:30,800 –> 00:01:35,468 It can happen to everyone and it also happened to me, when I studied.

17 00:01:35,668 –> 00:01:40,564 In my first year at the university I was sitting in a functional analysis class.

18 00:01:40,764 –> 00:01:45,100 Simply because I found it interesting and I wanted to learn that stuff.

19 00:01:45,229 –> 00:01:47,463 However I felt very stupid.

20 00:01:47,743 –> 00:01:55,441 The whole thing was very hard and I could not even ask the correct questions. I was not able to formulate what went wrong.

21 00:01:55,786 –> 00:02:05,650 The reason for that is easy to explain, because I didn’t even learn the basic real analysis parts that was needed for this course.

22 00:02:06,286 –> 00:02:11,819 So I was not stupid. I was simply not ready to understand functional analysis yet.

23 00:02:12,429 –> 00:02:19,506 In the end the solution was very simple. Very simple to understand. I simply changed the order of learning stuff.

24 00:02:19,900 –> 00:02:26,787 The key element I learned there, you could summarize as “mathematics is linearly to learn”. More or less.

25 00:02:26,987 –> 00:02:32,008 So the thing is you need to understand the foundations to move on.

26 00:02:32,208 –> 00:02:37,365 You need to learn the basic stuff first and then you can move to more advanced stuff.

27 00:02:38,029 –> 00:02:45,604 In other words if it’s really frustrating to learn a particular topic, you have to check what are the prerequisites you need

28 00:02:45,804 –> 00:02:48,357 and then you go back and learn them first.

29 00:02:48,743 –> 00:02:56,524 I can tell you another example, because once I explained the LU-decomposition for matrices to some students.

30 00:02:57,100 –> 00:03:01,300 The group was very motivated and there was no problem in the learning atmosphere.

31 00:03:01,557 –> 00:03:06,084 I could explain all the details step by step for this decomposition.

32 00:03:06,614 –> 00:03:12,790 However, during the whole explanation there were a lot of questions and I saw people were puzzled.

33 00:03:12,829 –> 00:03:16,549 It took me a long time to understand what went wrong.

34 00:03:16,749 –> 00:03:25,191 After some time I realized they weren’t struggling at all with this abstract concept or the abstract notions around it

35 00:03:25,391 –> 00:03:27,646 or even this algorithm I explained.

36 00:03:28,214 –> 00:03:33,311 The problem simply was that in my calculation I reduced a fraction.

37 00:03:33,843 –> 00:03:41,005 In the calculation there was something like 15/18 and after some steps I simply wrote 5/6.

38 00:03:41,843 –> 00:03:46,250 It took me some time to realise that for some people that was simply too much.

39 00:03:46,251 –> 00:03:50,964 It was too fast. It was too sudden. It was too hidden in the rest of the calculation.

40 00:03:50,965 –> 00:03:55,665 They couldn’t see that this fraction was indeed the same as this fraction there.

41 00:03:56,243 –> 00:04:02,053 So there was no problem at all in the linear algebra part. There was simply a problem in the foundation.

42 00:04:02,253 –> 00:04:05,695 In the foundation were you learn to calculate with fractions.

43 00:04:06,143 –> 00:04:11,717 Indeed, I would say this leads us immediately to the second mistake. The second common mistake.

44 00:04:11,718 –> 00:04:14,112 Which is that there is no practice involved.

45 00:04:14,900 –> 00:04:21,634 This is easily explained. If you just read a book about mathematics or about a particular topic in mathematics,

46 00:04:21,834 –> 00:04:23,483 you don’t really learn it.

47 00:04:23,486 –> 00:04:25,950 You only learn it if you practice it.

48 00:04:26,150 –> 00:04:34,829 Indeed, if you study mathematics at the university, you will notice that there is a big empathize on exercises, on problem solving

49 00:04:35,357 –> 00:04:39,940 and of course this is done, because that works. This is how the brain works.

50 00:04:40,140 –> 00:04:43,743 You have to train the brain such that you can learn something

51 00:04:44,200 –> 00:04:50,293 and we see, this immediately connects to our third mistake here. That there is no repetition at all.

52 00:04:50,871 –> 00:04:56,412 It’s a similar idea. You need some repetition such that you get used to new stuff.

53 00:04:56,612 –> 00:05:02,268 For example you need to reduce a lot of fractions such that the whole idea goes into your body.

54 00:05:03,000 –> 00:05:10,475 So you see, we have 2 important things here. First you need exercises. You need problem solving to understand something

55 00:05:10,675 –> 00:05:15,205 and then you need a lot of similar problems to solve such that you get used to that

56 00:05:15,757 –> 00:05:23,530 and I can understand this might be very hard for the learning process, because of our next mistake that happens a lot.

57 00:05:23,730 –> 00:05:30,234 This is what happens when you try to learn completely alone and then you don’t talk about what you learn.

58 00:05:30,434 –> 00:05:36,625 Indeed, there are some layers for this problem, for this mistake, because essentially you learn alone.

59 00:05:36,825 –> 00:05:44,654 You have to solve the problems alone. You have to understand alone. So learning mathematics is a lot of time spending alone, thinking.

60 00:05:44,854 –> 00:05:48,179 However, it’s clearly a mistake just doing that.

61 00:05:48,214 –> 00:05:52,229 Just learning for yourself and not talking about what you learn at all.

62 00:05:52,700 –> 00:05:58,341 The first thing that comes to my mind, what you miss by not talking about what you learn in mathematics,

63 00:05:58,429 –> 00:06:02,653 is that you don’t see that other people struggle the same as you.

64 00:06:02,853 –> 00:06:11,278 Of course also other people struggle to understand the concept, grasping everything that is needed, solving the problems and so on.

65 00:06:11,279 –> 00:06:15,812 So usually you are not alone if you are not understanding things

66 00:06:16,386 –> 00:06:24,029 and then talking about that, of course helps to find some missing links or some missing foundations you need to understand the topic.

67 00:06:24,714 –> 00:06:29,157 Indeed, there I would say this is a common problem with lectures in the university.

68 00:06:29,280 –> 00:06:37,036 Were there is just one teacher that explains everything and then you see only the people that really understood it, ask some questions

69 00:06:37,757 –> 00:06:41,889 and then most people in the audience feel intimidated or even stupid

70 00:06:42,089 –> 00:06:47,878 and don’t see that all the other ones or most of the other ones also have some problems there

71 00:06:48,386 –> 00:06:54,151 and that is a problem for the teacher as well, because some basic questions are not asked at all.

72 00:06:54,843 –> 00:06:58,507 So in summary I can tell you here, by talking to other people,

73 00:06:58,508 –> 00:07:03,669 by trying to explain a particular topic to other people, you understand more.

74 00:07:04,200 –> 00:07:12,517 Ok, then the last mistake I want to talk about today is that while you learn mathematics, you only learn isolated results

75 00:07:12,717 –> 00:07:18,557 and that’s a mistake, because different results, different concepts are always connected.

76 00:07:18,614 –> 00:07:21,747 Always related to some other parts of mathematics.

77 00:07:22,343 –> 00:07:28,279 Actually I would say ignoring such connections here is already a big mistake for your motivation,

78 00:07:28,479 –> 00:07:36,747 because how do you motivate yourself learning some isolated result if you don’t see how they are embedded in the whole structure,

79 00:07:37,257 –> 00:07:41,983 However, of course I know for a beginner it’s hard to see such connections,

80 00:07:41,984 –> 00:07:45,774 because everything feels isolated and not motivated at all

81 00:07:45,974 –> 00:07:51,060 and that’s the reason I put it here as a mistake. It’s also something the teacher can solve.

82 00:07:51,061 –> 00:07:57,174 So the teacher can make better for you by telling you, what are the prerequisites of a particular concept

83 00:07:57,814 –> 00:08:02,714 and then of course the other part is then telling you in which direction it leads to.

84 00:08:02,786 –> 00:08:05,650 So what can we do with our new results.

85 00:08:06,243 –> 00:08:13,357 Indeed, if you want to learn with such a network, you can do it with my real analysis course with the link in the description

86 00:08:13,358 –> 00:08:19,481 and I can tell you, it’s completely free. You can use it. You can watch my videos and you see the connections you need.

87 00:08:20,157 –> 00:08:27,859 Moreover there you also find exercises such that you don’t do the mistakes for learning mathematics I told you today.

88 00:08:28,059 –> 00:08:34,135 So I really hope that video helped you and then let’s meet in the next one. Have a nice day and bye.