40 Hours to Learn Mental Math | The riveting conclusion?

last time I set out to learn mental math in about 40 hours the last video was the first 10 hours of that 40 hours this is the last roughly 30 hours or so I'm using Arthur Benjamin's Teaching Company course on Mental Math as the major Learning Resource anyways you can find all those details in the first video which was posted many many months ago on my channel now on to what happened next so I am nearing about 20 hours into this project and I'm about to take another diagnostic test but before I do that I wanted to give you some Reflections about how things were going at this point so I've still been tracking my hours as you can see here but as you incorporate something into your life these particular hours might become slightly inaccurate I find myself doing math problems if I'm waiting in line somewhere or if I'm lying in bed idly lying in bed and just I'm like yeah maybe I'll do a couple math problems s you know it doesn't have to be a sit down hardcore study session I'm just kind of doodling doodling with mental arithmetic the other thing is that this diagnostic test that I'm about to take and that I've been taking it's okay but as a measure of my skills it becomes less and less accurate over time and the reason is that the course is teaching more skills than the test is testing I'm still going to keep doing the test just to see how my results progress but to me it's become a little bit less important the other issue is that the test has a lot of easy questions and some really hard questions and what I get on any particular test just seems to just be kind of a mixture of those two so sometimes I'll get a bunch of easy questions and sometimes I'll get some hard questions and so in that sense the test is not as consistent as I would like it to be on to the important stuff about learning do I push myself to go faster to like solve these problems as fast as I can and like put time limits on myself and like feel really good if I solve a bunch of problems or do I push myself to not make any mistakes so in that case I'm spending more time double-checking that I got the problem right but I'm obviously not pushing myself for Speed generally speaking I think of speed as a happy result of growing expertise but not as a specific goal that we should be explicitly pursuing the learning benefits of trying to solve a difficult problem correctly without making any mistakes outweigh the learning benefits of trying to solve that really fast you can end up in situations where you develop bad habits because you've been pursuing speed and not pursuing correctness that said occasionally I do push myself on speed maybe 10% of the time or 5% of the time or something but generally speaking there's more value in learning the patterns well I think that holds true for any kind of problemsolving domain if I solve a problem in two ways and get the same answer not only do I know that I didn't make any mistakes but I also get the benefit of seeing which method was faster or easier and I get to see the relationships between the methods sometimes you see like oh that's why that pattern works sometimes two evidence-based learning principles conflict with each other for instance we have interleaf practice which says that I should be mixing up problem types A lot when I'm practicing and we have deliberate practice which says that we should be focusing our effort on the most challenging parts of the task for us inter league practice would say that I should do subtraction addition Division and multiplication problems all mixed up together deliberate practice would probably say that you should just do the hardest problems like the really large multi-digit multiplication problems and that is what is going to benefit you the best in the long run so what should I do in this this case I prioritize deliberate practice for a couple of reasons one reason is that I already have a base level of skill and I want to increase that skill to a much higher level so that sounds to me like a scenario where deliberate practice is more appropriate another reason comes from understanding the underlying mechanisms Behind These principles part of the value of interl practice is from distinguishing different categories of things from one another but I'm not having any trouble with that so it's not like I see a subtraction problem and mistakenly apply a technique that suits multiplication problems among the group of more challenging problems I do apply interleaf practice so I might solve like a 2digit square and then a 3x2 multiplication problem and then a division problem with a two-digit divisor finally if you can feel yourself learning it can be really motivational even though I've worked on this project over the course of several months I can still remember when solving a two-digit Square seemed hard to me whereas now it seems like an average difficulty problem I wouldn't expect to make any mistakes on it it's finding ways of seeing your own learning progress can be really important especially as a way of sustaining motivation over the long run so those are my updates last time I got 31 problems correct in 10 minutes and I attempted something like 38 to be honest I think I got a little lucky last time just from how easy some of the problems were so I'm not expecting to do that much better this round but hopefully I'll do some better let's see how I [Music] do so I did about as well this time as I did on the last diagnostic test there wasn't any real Improvement yeah is that a problem uh maybe now I might be just rationalizing my own poor performance but it's a good time now to talk about something that happens all the time when we learn a naive way to think about it is that the relationship between practicing or studying and performance should be linear so the more time that we put into practicing the better our performance will be in general that is true but when you look closely what you usually see is something a bit different so performance gains tend to happen in spurts so you'll you'll go for a long time in which you feel like man I don't even know am I improving am I not improving I don't know I keep practicing and then all of a sudden your performance might get better very very quickly and then you'll start to plateau again part of the reason for these plateaus is that performance indicators that's like the tests that I'm taking are only partial measures of the learning that is happening imagine you're learning to play tennis and your coach teaches you the forehand first and so you keep practicing the forehand and you get better at the forehand and then they teach you the backhand well you start learning the backand and it's not just adding a fact to your repository of fact in your brain right when you learn the backhand there's some similarities to the forehand like there's some muscle movement turning similarities but it might actually make your forehand a little bit worse at first as your body adjusts to making these two kinds of movements you need to learn how to do the backhand well while keeping up your performance of the forehand but then you also have to learn how to integrate these two together in order to hit balls that are coming at you in any kind of Direction so you might be hitting a backhand you might be hitting a forehand and so the integration in terms of how you move how you turn requires learning yet another kind of skill you're learning more all the time but the number of balls that you get in in these practice sessions is probably going to go down a little bit while you're learning the new thing now what I think is happening or perhaps what I hope is happening is that I am engaging in this integration process right now so I'm learning new techniques uh trying to get better at those techniques and trying to figure out how those techniques uh should be used with the existing techniques that I have so just to give you an example in the beginning I was trying to brute force my way through these 2x3 multiplication problems which are usually the hardest ones that I come across now that I'm making this video and I'm saying that it's hard they're not really that hard if you use the right technique but anyhow at first I was trying to Brute Force these problems but but then you can use little tricks depending on the nature of the problem to make it a little bit easier to do in your head and then I started to use a method called cross multiplication you can do this quickly on paper as well but you could just hold the numbers in your head as you go it takes practice to use these tricks it takes practice to recognize when to use certain kind of shortcuts and when to use cross multiplication and so I'm trying to improve multiple skills at the same time now another thing to consider in this long way of rationalizing my poor performance is what is improving the test is measuring essentially my speed and accuracy in solving the problem so speed and accuracy not improving but maybe the amount of effort that I have to exert to solve these problems is becoming less over time test is not measuring for that what about my stamina maybe at the the end of those 10 minutes when I cut myself off for these tests which is really kind of arbitrary now I feel like I could keep going at the same level and maybe in the beginning I could feel like oh man I'm really tired I really need a break something else to consider is whether you are making the same mistakes that you were before so one of the most common mistakes that I was making in the first 10 hours or so of doing this project is that I would literally do the wrong problem so I would add when I should multiply or I would multiply when I should divide I wasn't paying attention to the sign and so now I don't make those mistakes really anymore but those kinds of mistakes when you miss a sign for another potentially could mask a different kind of mistake so if I multiplied when I should have divided there was at least some possibility even if I would have done the right operation I would have still screwed up so we've peeled back one layer of mistakes and now we're making a different set of mistakes so that's okay that's another sign of improvement I think as I worked over the next few hours after this test I focused on a few different things 1 by3 and 1 by4 multiplication comes up very often in some of the harder multiplication problems and so you want to get really really really good like perfect at doing this at the moment I'm okay at doing it I also noticed I was having trouble dividing numbers by small two-digit numbers it should be easier for me to do and so what I focused on is just a little bit of time spent learning the multiples of say 13 14 I guess 15 16 are easy 17 18 19 just so I would be able to recognize more quickly when say you know what's like 9 time 19 or 9 time 17 when I see that result I recognize immediately that you know uh 17 goes into 153 nine times for instance I also spent a little bit of time working on What's called the major system which is a way of remember in large numbers through words so every number gets assigned a letter and then you create words based on the letters in the number you use it especially when you can't hold a large number in your head for very long so if you are solving a four-digit Square for instance the number is going to be pretty big and it involves several steps mentally and so it's helpful to have this major system to remember part of the answer while you work on another part and then you put put it together at the end after spending a couple of hours working on this I decided not to focus as much on this and the reason is that I wanted to become more automatic in being a able to calculate the smaller numbers and then maybe later I can do the major system if I wanted to you know looking back on it now you could say that those hours were wasted because I spent time learning something that wasn't really a part of what my ultimate goals are I think for this project but this is part of the learning process too figuring out what you want to focus on and what you don't want to focus on that's part of learning and so you can't really Escape that learning is at least partially about exploration having the mentality of like oh well if I only knew then what I know now well the knowing was part of the learning so it doesn't really make much sense to think that way because how could you have known unless you would have explored so that's my Spiel there so I'm GNA work some more on this over the next few weeks I'll see you in a few hours and I'll come back with another update so today is the first time I'm seeing real gains from a practice test I solved 100 problems in 25 minutes and I got seven wrong but I felt extremely confident especially over the first 50 problems or so and this was all done while my son was like distracting me asking me random questions and such I want to talk a little bit about the development of intuition and pattern recognition so just to give you an example of what the process of change is like uh earlier on when I came across a problem that asked me you know how many times does 17 go into 51 usually as part of a larger division problem I would take a guess and then I'd have to check right so I might guess it's not two but I might might guess three or four possibly and then I'd have to check oh oh like which one of those is correct then a little bit later because I got used to seeing the patterns and because I started directing my attention to the right qualities of the number it became a lot easier to guess correctly you have 17 you see 51 and you know that uh some multiple of seven has to end in one and and really there's only three that ends in one that it can possibly be correct so when you see that and you look at the one and you know okay it must be three and then you can double check if you want you say Okay three * 10 is 30 + 21 that's 51 great now when I look at something like this when I look at that 51 I'm not even really focusing on part of the number it's just that three pops into my head it just feels right three must be true now obviously I can go back and quickly check if I need to but I'm starting to feel more and more like the the the right number is just coming to me uh without a lot of explicit calculation you still I still have to do a fair amount of calculation but there are part parts of the calculating process that seem a lot more autom itic so I put a few more hours into this project and the last diagnostic test I did I got 91 problems correct out of 98 that I had attempted to do and this was in 21 minutes and 25 seconds this is okay but it's not great I really should not be missing any of these problems these are pretty easy problems at this point I'm wondering a bit how much of this is due to me just trying to run rush through the test rather than taking my time and making sure I get every problem correct now lately I've been focusing on three kinds of problems large division problems four-digit squares and two-digit cubes now all these problems are on the harder side of the mental math Spectrum at least in the context of this course you can get way way way harder problems if you get really good at this now I had done a lot of these problems during my practice before when I was first going through the course but it had been a while since I did them and I thought it would be a good idea to go back through them again and the first time I sat down to do a bunch of large division problems I made a ton of mistakes and so that made me want to focus on them more notice that all three of these kinds of problems incorporate smaller easier problems in them so a four-digit Square for instance requires that you do a 1 by4 multiplication problem and then keep track of how many zeros you have and then do a three-digit Square after that so they're good problems to do because if you make one little mistake at some point during the problem the whole problem is wrong right and so you can't get away with telling yourself oh hey I got 17 out of 18 problems right that's pretty good well if a division problem embeds 18 small problems together and is asking you to get every single one of those right before you get the whole problem right well then it's it's just a more challenging problem it gets you to focus on the mistakes that you're making more now I know I said earlier I was going to set the major system aside and not really focus on it as I did four-digit multiplication problems I just had to start using it again because it's really hard to remember the numbers if you don't the second time around I felt a lot more comfortable using using it and it seemed easy and helpful to me whereas the first time around it felt like I was almost faking it you know I was trying to use it but not really using it super effectively and with all that said I am going to stop the project here even though I have a few more hours left on my hours tracking sheet because I feel like I have gotten out of it what I wanted to get out of it and this part two video is going to be published many many months later than I wanted it to be I don't have any big desire to be a mental math competitor but there are two things I've gotten out of it one is a much better number sense a much much better sense of patterns within the natural numbers and I feel much more comfortable calculating things in everyday life and estimating things in everyday life if you can just do one thing it would be to pay attention to your mistakes understand the patterns in your mistakes and figure out how not to make those mistakes in the future this is really the essence of deliberate practice as you notice patterns in the mistakes that you make then you can figure out well what is the activity that I should do to make me better at this skill you can put a pause on one skill or one component of what you are learning and come back to it later there there's absolutely nothing wrong with that that actually is part of a healthy relationship with what you are learning right that is a form of managing your time and attention managing your cognitive load and often the second time that you come back to something you're going to be in a much better position to appreciate it or to understand how to do it or to be able to do it more effectively so the learning process that second time around is going to be easier it just is in all of these problem solving domains I really think that you should aim for correctness over speed I've never regretted prioritizing correctness over speed but there are times when I have regretted prioritizing speed over correctness you always need to be moving to more and more challenging problems more and more interesting problems more problems that embed other skills and other smaller tasks inside of them and this is a point that Arthur Benjamin makes in the mental math course is that it's only when you start doing some of the more advanced things that you understand those simple things you get really really good singled digigit multiplication when you are doing these larger multiplication problems but this also means that how you test yourself and how you assess your own skills have to change over time by the end of this project this diagnostic test that I started out out using and it started out thinking it was okay not great but okay really became more like these are a bunch of dinky problems to do doing the test wasn't interesting anymore because it just seems more like busy work and so you want to really avoid that feeling of busy work and move towards these more challenging more interesting problems another key point is to use the research on learning as a guide not as a shackle this means understanding some of the principles of effective learning but it also means having some sense of why they work the way they do and maybe even more importantly understanding the limits of these principles you know when does interleaving benefit you when does it not benefit you you're using these principles as tools to help you but it's not like you are bound to always use these principles at all times otherwise you're doing some kind of suboptimal learning experience no it's it's not about that at all it's about structuring your experience and learning from your own experience while you use these tools to help you the last thing that I want to say is that skill development and observation or noticing things go hand in hand it's not just that you're learning some arbitrary set of skills it's that the development of skills help you to see things differently and vice versa seeing things differently can help you to develop your skills as well all the stuff that Arthur Benjamin talks about in his course yes they are all kind of tricks right they're Mental Math shortcuts ways of making problems easier to do in your head but as you learn these tricks you're also seeing patterns that you weren't seeing before that for me is what was interesting to me that was what kept me motivated that if if that wasn't happening then it just seems like I'm turning the cogs in a machine and and it doesn't make it interesting thank you for joining me on this little journey if you liked this video I am slowly building a playlist of other learning challenges this is where I try to apply the research on learning to actually learn something you can check that out right there links to other resources are in the description thank you I'll see you next time

Channel: Benjamin Keep, PhD, JD

Views: 10,749

Keywords: science of learning, interleaved practice, deliberate practice, skill building, mental math, learning sciences, learning

Id: DtNGsdUifJI

Length: 24min 26sec (1466 seconds)

Published: Thu Oct 26 2023