 Let's begin with counting. Now the type of students that I get when I'm trying to teach them counting is basically someone that's already learned 1, 2, 3 and they know at least all the way up to 10 or 11 or 12 or so, right? And what we have to appreciate when we're teaching someone counting is counting is just a language. Your natural language applied to mathematics, right? So depending on what natural language you're speaking, in English we go 1, 2, 3, 4, 5 all the way up to wherever we want to go, right? In Farsi it goes Yek Dosei Charpanj, it changes, right? In Armenian it goes Mek Ek Yerek Chorsing and it just continues on. In French you can do it, in Spanish you can do it, right? So every language has their own words they use for the specific numbers and that's really dependent on where you are, right? I'm more interested in when teaching someone how to count, I'm more interested in in terms of someone that's teaching mathematics to where focusing on where they have little hiccups, right? And this process is extremely personal, okay? There are kids in when they're learning how to count when it comes to English, okay? Because that's the language I teach in, right? So you have to sort of kick yourself into whatever language, natural language that you're using right now, right? But in English, in general, kids have a little bit of a harder time reaching, going from 10 to 20 because the number 11 and 12 are not derivatives of the number 10, right? So number 1 to 10 are basically just memorization processes, right? So we can write, oops, so we can write number 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Now, when you're learning how to count or you're teaching someone the counting process, what you have to keep in mind that in the English language, each one of these numbers has their own unique name, okay? That is not the case in other languages. In other languages, you count to 10 maybe and number 11 is, in general, if you do a direct translation, is 10 plus 1, 10 plus 2. Some of the languages, that's exactly what you end up saying, right? But in English, 11 and 12 have their unique name, okay? So some kids have a hard time getting 11 and 12 into their vocabulary. After that, we got 13, 14 and all the teens, right? So we have 13, 14, 15, 16, 17, 18 and 19. Now, in the English language, this is really easy to do, right? Because these are all just derivatives of teen, 13, 14, 15, 16, 17, 18 and 19, right? So there's a nice pattern there and a lot of people are just learning this. In general, they don't have a hard time with it. However, there are times where certain students, certain students that I've had, they have hiccups transitioning maybe from 15 to 16 or transitioning from 17, 17 to 18, okay? So if a student is having a hard time jumping from one number to the next number sequentially, let them go all the way through and then kick them back and test that transition, right? Sometimes you have to make the correction right away. Sometimes you have to help them along, right? Whatever you do, whenever you're trying to teach someone in general, it comes to mathematics, counting, addition, multiplication doesn't make a difference. Don't try to trick them in the learning process, right? Don't give them questions in general where you're trying to amplify their mistakes, amplify their hiccups, right? Make the hurdle, hurdle bigger, right? What you want to do is eliminate the hurdle that they have. Sometimes it requires you to let them make the mistake until you get to a certain number and then come back and correct it. Sometimes it requires you to stop them when they're trying, when they make the mistake and get them to repeat that. Sometimes you have to start back up here. So if they're having a hard time going from 15 to 16, right? Don't necessarily stop them there. Sometimes let them go all the way to 19 or 20 in general where I take them to. Sometimes kick them back to 13 and get them to start counting from 13 all the way up, right? So practice that, the location where they're having a hard time counting, right? From 19, what we end up having is a pattern that emerges when you're doing counting, which basically sticks all the way up to infinity if you want to go to, right, forever and ever and ever. And the pattern is basically from 19 you go to 20, right? And then you have 21, 22, 23, 24, 25, 26, 27, 28, 29, and then you're back to the next tens, right? The 30 and 30 starts off the same way, right? 30 goes 31, 32, 33, 34, 35, 36, 37, 38, 39, and then you're into the 40s. Okay. Sometimes this requires a certain amount of time. It takes time for a kid to be able to get their counting down right. The way I end up teaching this, if the student is just getting into counting and they've had some hiccups at the beginning stages, right? I get them to count from one to 10. Okay. It becomes easy to do because we have 10 fingers, right? So in general, when I'm getting them to count, what I end up doing is I tell them to hold up your fingers and to actually go through it. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, right? We do this a few times. We do it enough until they can do it without their fingers. Okay. And then we're going to the 11, 12. Okay. Get them to familiarize themselves with 11 and 12. Okay. So the next set of counting you're going to do, you're going to go 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So you're introducing them, letting them know to introduce another hand into their counting process, right? Once they can do 12 without their fingers, you kick them into the teens, right? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. Okay. And then we do 20. Okay. I never teach it all the way to 19 and get them to practice all the way to 19. I get them to 12, teach them the teens all the way to 20. So the next time that we're counting or the next step in the teaching process, right? I mean, it may be during the same session and maybe in a follow-up session or two sessions later, right? Depending on how fast the student is progressing, right? I get them to count to 20 and I do this often, right? Because if they can count to 20, they can count to whatever number they want as soon as they learn the hundreds and the thousands and stuff like this and we'll talk about that. Okay. So I get them to go 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. Okay. Without their fingers. Once they're comfortable with that, I teach them all the way from 20 to 100. Okay. So what we end up doing is we do 20 and then 21, 22, 23, 24, 25, 26, 27, 28, 29, and then you teach them the number 30. From there, and I usually try to do this in one session, maximum two sessions. Okay. Preferably, I try to teach them from 20 all the way to 100 in one session. Okay. When you're practicing a little bit of time all the way, counting all the way to 20 and then when they're really familiar with it, you lay it on hard, right? So you go from 20 to 29 and then you teach them 30. Okay. That's a new word, right? And then you say, hey, the process continues. 31, 32, 33, 34, 35, 36, 31, 32, 33, 34, 35, 36, 37, 38, 39, and then you go to 40. Okay. Once they're, you can see the lights shining in most students. Once you start teaching them this, right? Because they realize as soon as they know this, and they already know that, then they know how to count all the way to 100. Right. So you keep on doing this until you get to 100. Now, once they go through 100 once, okay, you've gone through the whole process. I get them to count from one to 100 every time I meet them until they're 100% comfortable counting from the number one to 100. Okay. Where most of my students that I've encountered have hiccups are, as I stated, in the teens. Okay. Because from one to 12 is just straight up remembering new words. Okay. From 13 to 19 is a pattern and they have to bring two words together. Right. The number they already know plus teen. Okay. So students in general have a hard time in the teens. Okay. Once they learn the tens, right, 2021, where they're connecting again, two words together, learning a new word, 20 and then connecting it up with one, right. So 21, 22, 23. And then you teach them another one, 31, 32, 33, and so on and so forth. Where students end up having hiccups when it comes to learning how to count are transitioning from 29 to 30, 39 to 40, 49 to 50. So what I end up doing when I'm teaching them how to count, I focus in on where they're having a problem. And in general, that's where the problem is. If the problem lays somewhere else, please focus on that as well. And the way you can focus again is what we talked about earlier, where you can let them either count all the way to a specific number you wanted to, and then go back and correct them. You can correct them right away and just ask them to repeat the hiccup, the problem spot, or get them to start off at a sort of a node, right. So if they're having a hard time with between 24 and 25, you can get them to count 24 to 25 multiple times, right. Or you can kick them back to a node where you've already moved on from the teens and you're hitting at about the 20, right, at an important point where the language changes, right. And then you go, okay, start from 20, you go 21, 20, 21, 22, 23, 24, 25. And if they have a problem, you say, okay, say that again, 24, 25, or let them count all the way here and do it again, right. When they're counting all the way to 100, the hiccups usually occur from the 9s to the 10s, right, 29 to 30, 39 to 40, 49 to 50. So I tend to focus on those. So keep your eye out, keep your, listen for the, where the student might be having a problem, right. And once they get to 100, okay, then they know how to count to 200, they know how to count to 300, right. It's just repetition of this, 101, 102, 103, 122, 135, right. So if they know 35, the 100 is just another word, right. So they're tagging on another word in front of words they already know, right. And most students catch up pretty fast with this. Where they end up having a hiccup is the transition from 199 to 200, 299 to 300, 399 to 400, right. And again, I focus on those things. I don't get students to count from 1 to 1000. That's a waste of time. It burns out the student. It emphasizes too much repetition, right. Because if they can construct the number, the word, you don't have to get them counting all the way up to the word. You can start giving it to them at random, right. So from 100, right, you're really teaching the pattern, right. From 1 to 100, you're teaching them how to count and become familiar with it so they can do it rapidly, right. From 100 to 1000, you're teaching the pattern. You're teaching them to add on the extra word in the front to make the word whatever it is that they want to say it is, right. So once they know how to count to 100, what I end up doing is I teach them the 100 counts, right. And then I go back, I start giving random numbers for them to say. Sometimes I introduce the random numbers when they reach 100, right. So when they count from 1 to 100 and it doesn't take it, it's very rapid, right. Start throwing down some numbers in front of them randomly. See if they can say it. If they can't say it, you know you still need to continue doing the count. You know you're finished teaching them or getting them to count from 1 to 100, right. Every time you meet them when they can read off 90% of the random numbers that you show them, right. So that's one way you can test to see if a if a kid, if a student is ready to move on to the next level. And once they reach that state where they can randomly pick off any numbers, I start giving them random numbers in the hundreds, right. 259, right, 345, 899, right. So I start dropping down random numbers. If they're having a hard time saying what those numbers are, go back and explain to them how the 100 counts go. Okay. Now once you they know how to count into the hundreds, the only thing left for you to do, if you want to teach them extremely large numbers is introduce the new names, the thousands, the tens of thousands, right, the hundred thousand, the millions, the billions, right. But you're basically going with the three digit markers, right. So let me show you how that works. So once a student knows how to count to a hundred and they're familiar with the hundreds, right. Then what I end up doing is I introduce them the thousands right off the bat. And then I show them the pattern, which is the same pattern for hundreds as it is for the thousands, right. So basically, if they know how to count to three digits, okay, I introduce them the thousands, which is the fourth digit. And I explain to them that counting or mathematics in general, and this is something that's extremely important to do, is constantly to remind the students that mathematics is extremely visual. So for them to use the pen and paper, use the symbols that we have in the language of mathematics and whatever natural language we have, to make the reading process easier, right, to let their minds subconsciously appreciate what's happening with the language of mathematics, right. So when we get into the thousands, almost right away, I introduce the comma, right, and I let them know to do a little separation so they're visually, automatically, their subconscious realizes that they're in four digits, right. And once they get into the thousands, you can, you can guess how this goes. You get them to randomly read some thousand numbers. So I don't necessarily get them to go thousand and one, thousand and two, even though I do sometimes, but in general, usually when a kid, when a student has learned where we've reached the level where they're into the thousands, then they understand that concept very well, right. So what I end up doing is I usually just kick it into five digits and six digits right away within, within minutes, really, right. So these numbers here, these are the thousands, okay. This is your hundred, that's your tens, and that, those are your single digits, right. In general, I don't really need to explain that, right. Once we reach this level of the thousands, right. If the students receptive to this, right away in the same session, I teach them the millions and the billions, okay. Because these are just two new words or three new words they're adding to their vocabulary. And in general, students that I've dealt with, they understand, they know the words a million or a thousand or a billion. They've heard it through their social networks, right. They've heard it in school, they've heard it in media, right. So the new word is not necessarily new to them. What I end up teaching them is, and what they really need to learn is, is how to put it all together to read off numbers, right. So after this, we got obviously the millions, right. And then you have your billions passed it. I don't have enough room right now to put it in. But basically, it's the same concept. You put three little dashes there, show them that. And what I end up usually doing is just placing random numbers on the sheet that we're working on. Let the pen and paper do the work for you as well. And that's not just for students, that's also for teachers, that's also for people, for parents, right. Or if you're helping a sibling learn mathematics, right. Or helping a friend learn mathematics, right. You teach them the millions, you teach them the billions. And then having this up, you start dropping numbers on the sheet, right. Start putting random numbers on the board. 25, 375, right. Start off with the low numbers that we know, right. And then put that number on. Don't put the commas on, right. Let the student do the work, okay. Let the student realize that they have to put a comma here. And if they're not putting the comma there, they're having a hard time reading it, then tell them to put the comma there. If they're able to read it right off the bat, without the help of the comma, without breaking it down, then don't emphasize this yet, okay. Only start emphasizing these breakers when you get to really large numbers, okay. Don't break the kid's pattern of thought. If they're having a hard time, introduce this, right. If they're not having a hard time, kick it up to higher numbers. Do that. See if they can read that. 375,679, right. 12,070, right. If they can't read that, tell them to look above and see where the comma is and to break it up, okay. And in general, they know how to do. Some students I've had, the comma when they're placing it, they make the mistake of going this way. This one's symmetrical. It works, right. But if you give them a number, let's say again, with five digits, right. Let's say we had, let's say we had 75,764, right. And your students having a hard time saying this, reading it. Tell them to put the comma. What I've noticed is some students start off putting the comma here. They think the three-way, the separation can occur from here, which it doesn't, which is something that you're going to have to end up correcting, right. So if they put the comma here, just tell them no. You're always going to count from this side, right. From your right side going this way, okay. That's one place that hiccups appear as well. But if they can't read it, put your commas in, right. Once they know how to read the tens, the hundreds, the thousands, the hundreds of thousands, kick them into the millions, kick them into the billions, right. And once you take them to this level, do not add the comma when you're asking the question. Get them to do it themselves, okay. So put that number down, see if they can read it, okay. Rare where I've had a student who's able to read this, or even an adult who's able to read this without putting the markers on, right. Let's put the markers on. Oops, not there. I almost put the marker in the wrong place, right. Watch your students, it happens. Now, this becomes easy to read, 76,077,074, right. Now, once they know how to read these numbers, and don't give them extremely hard numbers to read yet, right. What you need to do is start not necessarily trying to trick them, but start giving them numbers where they're similar, where they have to sort of pause and think about it for a second, right. So for example, you can give them these numbers here, 720, 702, 712, right. So you constantly need to push your students to challenge themselves, right. To put all this information into muscle memory, into their minds and make it automatic, okay. So this is the concept, this is how I go about teaching counting, okay. And it's very basic for me, like it doesn't take long. I don't want to say if it's basic, but in general, this process doesn't take long working with the new students. If they're, they already know the numbers from one to ten, and we're just adding on a little bit of information, right. And sometimes I, again, I get them to use their fingers. Sometimes I get them to use tick marks on a piece of paper, right. Maybe group them in fives, group them in tens, group them in fives is easy. You put four tick marks, right. One, two, three, four, you put a line over it, and that gives you a five, right. Sometimes I use that. In general, I don't have to go down to this state, but sometimes you do, sometimes you do. And again, this is extremely personal, teaching someone how to, teaching someone anyway in general is extremely personal, but teaching someone the language of mathematics becomes very intimate in general, because there's a lot of, when students that I've encountered, they hit an obstacle. Sometimes it's a little difficult getting past this obstacle, especially for mathematics. With physics, I haven't encountered that too much, because physics, it just uses the language of mathematics to teach them physical concepts, right. The mathematics applied to the physical world, right. In general, it's with mathematics. I haven't figured out why that is. I don't know if it's because of society or because of the new language they're trying to learn, because I've never tried teaching anyone a different language. So I don't know if this sort of struggle exists when people are trying to learn a natural language. I know, for me, it was a little frustrating learning English for the first time, but learning mathematics can really throw some people off, right. And that makes it extremely intimate, extremely personal. So you have to have patience when doing this type of thing, especially when you're teaching them counting, because that's their first exposure to mathematics. And what you want to do is eliminate the stress for someone when they're trying to learn math. So help them along. Don't give them any trick questions until you get to the end, where you're trying to test their ability to recognize the minor differences between the numbers, okay. And that's the process I use to teach someone how to count. I hope you find this useful. What we're going to do now is we're going to go into the process I go through to teaching someone how to add, okay. Let me take this slide down and we'll talk about that.