Why Standard Teaching Fails Our Students - Part 4

T1I Worksop: Candy Problem segment 4
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Speaker 4: [00:00:00] Hey everybody, it's Cheri Dotterer, your classroom coach.

Speaker 4: We are here at Tier 1 Interventions. I am with my co host, Jonily, and we're going to be talking today about the candy problem.

Speaker 3: Counting by ones. Ones. So my skip counting number is one. So if I go from three to six in three counts, you see this goes back to earlier today when we talked about those bubbles and the, you see how abstract that is? But do you see how necessary it is? Okay, so if I go from three to six and it's going to take me three counts, what is my skip counting number?

Speaker 3: Everything comes back to what is my skip counting number. My skip counting number is one, but now I have to go from three to five. So all I care about right now is my skip counting number more than one or [00:01:00] less than one? That's all I care about with kids. If I go from three to six My skip counting number is one.

Speaker 3: 4, 5, 6. If I go from three to five with three counts, I'm still doing three counts. I need three counts. If I go from three to five with three counts, the question is, what is my skip counting number? Now notice I am not going to answer this, and I don't do that with students because I don't. I will come back to, they need a second interaction if I push this too far and solidify the answer.

Speaker 3: Now, hear me on this. There are certain places where I do not confirm or deny the answers. The reason is the kids have to come to it themselves. When I figure it out for them and tell them, they witness it, they hear it. [00:02:00] But when they figure it out for themselves eventually. They know it forever. It's ingrained in them.

Speaker 3: That's the difference. So I'm not going to solidify this. I'm just going to say to them one last time, I'm going from three to five in three counts. It's three o'clock. I'm going to stop at five o'clock and there's going to be three segments of stuff that I'm going to do.

Speaker 3: What is my skip count? Now, and I'll say this to them again, going from three to six. With three counts, my skip counting number is one. Going from three to five. Now what happens in the classroom is, I keep repeating these things over and over. The minute I put answers here, all [00:03:00] thinking stops. This is the art of a chef.

Speaker 3: Sarah, I love this. We should just call it like chef math. This is the art of a chef. Chef Math. Chef Math. And I am Gordon Ramsay.

Kirk: Oh, I can't really drop some curse words. What? You know who else follows recipes but sometimes does their own thing? Who? Witches.

Speaker 3: Oh, I would surely be one of those. This is witchcraft.

Kirk: That's what you're doing. You're teaching us witchcraft. Yeah.

Speaker 3: You know what? I, if I was, it's a good thing I was born in this time period because if I was born in that time period.

Speaker 3: I would have done been burned at the stake by now. Like I am way too progressive for my shirt today, though.

Kirk: For a lot of now too, by the way.

Speaker 3: I know. Please be patient with [00:04:00] me. I'm from the 1900s. Hey, I'm a 75 model. Y'all listen to this. Okay. I'm a 75 model, which means if you're a 75 model, you're going to be 50 this year in 2025.

Speaker 3: Okay. So it's the countdown to 50 for me. Alright? I am in my midlife celebration, so it's all coming out. Anywho, I keep repeating a lot of these same things because what it does is it continues to trigger thinking for students. And I will have students in my class, they'll be like, 3 2 3rds! 3 2 3rds!

Speaker 3: And I'll be like, oh that's interesting, okay then what's this one? 4 2 3rds! Like they'll get real irate, but I don't confirm or deny, I might acknowledge them because I'll write [00:05:00] all the answers that kids are saying, but I'm not going to resolve this. Now hear this, if you truly want to teach with the brain in mind, and you want to have a benefit on neuroscience this is the science of math.

Speaker 3: This is the science of math. If you want to truly engage the brain, you will leave some questions unanswered and some problems unsolved, and you will know, you will have an art. And a gut that tells you which ones. Because what happens in the student brain, and what's going to happen in your brain, when you leave here today, if I don't resolve this, your subconscious is going to continue to think.

Speaker 3: Actually, for some of you, your conscious mind is going to keep thinking about it because that's just what you do. I am wicked. Green. Oh, Amy. Oh, [00:06:00] girl. Okay. But, with students, they're not going to want to admit they're thinking about math outside of math. But the only way I can get kids to think about math outside of math, and to continue to have them think, reason, and sense make, is to leave some things unresolved.

Speaker 3: What that does to our brain is it works our subconscious mind. So kids are thinking about this without even purposefully thinking about this. When we can actually do that in mathematics, that is, it was written down here, but that is the number one way that we're going to increase memory and retention of content.

Speaker 3: Is when we can engage their subconscious mind outside of the classroom. And the way that we do that is to let math linger. We leave some Questions answered and some problems not solved. This is an important one to not come to a resolution today. But my point is, look at how in Algebra [00:07:00] 1, I can still go back to this same basic cycle.

Speaker 3: Now, we can trial and error things. Okay, let's count by halves. Let's just road count by halves, but starting at three. Okay, so we're at three. Three and a half. Four. Oh, that would be four and a half. So is my skip counting number going to be bigger or smaller than a half? Okay, bigger. Let's say my skip counting number is one.

Speaker 3: Because some of my Algebra 1 students don't know that there are numbers between a half and one. Let's just be real here. We have to fix these deficits, and I'm telling you how to do it. So then I'm like, oh, so if we skip count by ones, then we start at three, four, five, ooh, then that's six. So is our skip counting number going to be more than one or less than one?

Speaker 3: Less than one, but more than a half. Now, Amy, do you want to talk to us? No? Okay. Now, we know that our skip counting number needs to [00:08:00] be more than a half. But less than one. If kids even know that there are numbers more than a half and less than one, the most popular one they tell me is three fourths. So then what I do is I'm like, Okay, we're gonna go on a whole lesson next week, guys.

Speaker 3: We're gonna forget about this today. But next week, and I might get chart paper, because I want to remember to do this. Guys, next week, we're not going to do it for a week, because we need time for this to marinate. We gotta marinate the chicken, okay? Gotta marinate. We know that the food is better after we marinate.

Speaker 3: In a couple weeks, guys, here's what we're going to do. We are going to skip count by three fourths, okay? I'm going to write this on a chart paper. Couple weeks later, I'm gonna come back and I'm gonna be like, Okay, let's skip count by three fourths. Three fourths. Six fourths. Oh, they're gonna want to say six eighths.

Speaker 3: See, we're gonna do a whole [00:09:00] probably 30 minute lesson just skip counting by three fourths. Now, some of you are gonna be like, Jonalee, I teach Algebra 1 and I don't have time for your witchcraft. Okay, I don't have time for your witchcraft, because I actually have real content and standards to teach, okay?

Speaker 3: And I'm gonna be like, okay here's what you give kids on that day, okay? We give them this equation as our stimulus. Two thirds x plus twelve equals negative six plus twelve. Three fourths X. Okay? Is that real standards enough for you? Okay, now, my third graders are still working on solving this. Because this [00:10:00] is Jesse and this is Kay.

Speaker 3: Jesse starts with 12 and gets two thirds of a dollar a day. Kay starts owing 6. And gets three fourths of a dollar a day. This is exactly what kids need to improve number sense. Because we can develop this entire counting cycle. All the way up to rate and function. That's the progression. With my third graders, I never want them to solve for X.

Speaker 3: This is not about solving for X. This is about improving number sense through counting. It's about improving operation through counting. That's it. If we can do that, we can leverage higher level math [00:11:00] achievement. So with my 8th graders or my 6th graders, okay, over here this isn't too bad. We can be at 12 and we're going to count by two thirds.

Speaker 3: So the next number is going to be 12 and two thirds. Now, I could really give this counting exercise to high schoolers and you would be shocked at the number of high schoolers that cannot give us the next number. That's a problem. So what they're going to tell me is 13 and two thirds. Now, if I'm doing this as a lesson, I am going to correct them now.

Speaker 3: I'm going to be like, it's not 13 and two thirds. I'm going to put 13 and two thirds because what is, what did I count by to get from 12 and two thirds to 13 and two thirds? What's the difference there? How far apart are those numbers on a number line? What's the distance? The distance is one. So you've just [00:12:00] switched from counting by two thirds to counting by one.

Speaker 3: So it can't be thirteen and two thirds. So let's try something else. What if we say it's twelve and four sixths? Then I need to get my paper strip out and I need to be like, okay, let's show two thirds. Remember, thirds are yellow. Y'all, I know I'm jumping around today, but this is a combination of all of these tasks.

Speaker 3: You can't really just teach one task at a time. You can if you're just getting started. All of this comes together. Color is essential because it will help kids associate. My kids know that the yellow, they have to overlap when they paper fold and get three layers. My kids know that the yellow paper strip is thirds and three parts.

Speaker 3: We might need lots of yellow paper strips. We might have, every kid might have three yellow paper strips. Ooh, if I have three yellow paper strips, [00:13:00] How many thirds is that? One, two, three, four, five, six, seven, nine thirds. If we're counting by two thirds, guess what? I take my paper strip and I mark, and I'll show you in a second, I mark two thirds

Speaker 3: I marked two thirds of two different paper strips, okay? Two thirds, that's one count. Here's another two thirds. What do I have? This is not an easy question. Kids still want you to have four sixths. But I actually, I have four. I have four shaded in. But what size are those parts? What do we call those parts?

Speaker 3: They're thirds. Two thirds, four thirds. So this is going to be, this is going to be

Speaker 3: I just lost it. 12 and four [00:14:00] thirds. Okay. Now some of you are going to freak out cause you're going to be like. You can't do that. That's 13 and a third. Okay, fine. We'll worry about that later, but my kids understand it right now as four thirds. And then we'll get another paper strip and let's do another two thirds.

Speaker 3: Okay, and then we have two thirds, four thirds, six thirds. Okay, now some kids will be like, oh, wait a minute though. We have these extra two thirds. That could be our six thirds. Some kids won't see that. They're going to have to do a new paper strip for every two thirds. And that's okay. See, that's how we give them accessibility.

Speaker 3: I'm going to release kids to just play and skip count. And the reason we did that is my skip counting number here is two thirds. But it's [00:15:00] not until kids are going to have lots of counting experiences to be like, wait a minute, if I have to go up two numbers with three counts. That is the phenomenon of the fraction two thirds.

Speaker 3: It's just if I have four candy bars. Oh, but there's five of us. How much candy bar is everybody going to get? The numbers already tell us. If I have four candy bars. Now, this is good because I'm going to the blue paper strip. The blue paper strip is how we do fifths. To do fifths, we bring those edges to the center.

Speaker 3: We don't overlap. We pull them apart. The reason is, with fifths, we create a space in the middle with the two flaps. But the space in the middle has to be the same size as each flat. I need three equal parts [00:16:00] here. Once I have three equal parts, I can crease. And then where those edges are, that defines another crease.

Speaker 3: And when I do that with the sharing the middle strategy with five, because it's an odd number, I get fifths.

Speaker 3: So when I am dealing with fifths, Or other fractions. This color will help associate this. Let me pause for a moment. Thoughts, comments, questions.

Kirk: Just want to say that what you have said, I have experienced over and over again. And everything you've said is right. I don't know that I'm adding to the conversation, but you're right. And I have seen that. [00:17:00]

Speaker 3: And we see it. We keep seeing it over and over again. And we keep going back and doing the same things slower and louder.

Speaker 3: Other thoughts?

Speaker 11: Just that I I'm excited because we're getting ready to teach constant proportionality. And so my kids have a pretty good understanding of the candy problem. So I'm excited to see what they're going to do. If I just give them those ordered pairs, what they're going to do.

Speaker 3: Love it.

Speaker 3: Love it. Love it. Love it. Other thoughts?

Speaker 12: I like just coming back to paper folding a lot. When you threw out that, the slope and I don't think I'd really, I've never thought of slope that way. How you looked at the count for x and then you said if I want to do three counts for three to five, And so I drew that like on my [00:18:00] own version of a paper.

Speaker 12: I was like, okay, so three is at the left and five is at the right. Then I want two lines and then my five. But then by drawing that for myself, I was like I know four is exactly in the middle. So I like. Reference to my four there. And then I was like, Oh I like just split that middle section and a half so I could do that on the other side.

Speaker 12: I just thought through what you were asking me to do. So I can see how if you are creating these experiences for students, like over time and continually coming back to that, that you're you are giving them tools that then they can use for problem solving in a new scenario.

Speaker 3: Let me show, what Sarah is saying is brilliant and you're missing it.

Speaker 3: You're missing it. Okay. So let's, okay. So I just grabbed a white paper strip because I don't know what color this is going to be really. I should because it's [00:19:00] three jumps. It should be the yellow and I'll show you the yellow in a minute. But what Sarah is saying is, okay, I put three because remember the look, look at you girl.

Speaker 3: See, I drew it. Sorry. She put three on this is repeated reasoning. This is iterating. Because the way we use paper strips, we put a certain number on one edge, a certain number on the other edge, and then we have a certain number of spaces. So I have to go from three to five, and I have to go there in three counts.

Speaker 3: So I'm like, oh shoot, that's a yellow paper strip because that's our thirds. Then I did a dotted line like she said, and she's I know four is there. I know four is there. But then if I actually do that on the yellow paper strip to associate the color, look at what Sarah has here. In her drawing, and I might say to students, tell me about Sarah's drawing now just because Sarah explained it and Sarah just because Sarah explained it and Sarah made a drawing of it [00:20:00] doesn't mean that the kids are going to miraculously understand it and that's what we do as teachers, we explain it and then we make it a drawing of it and we explain it again and we get really excited, but 70 percent of our kids still don't get it.

Speaker 3: And that's okay. That's what we should expect. That's what we should expect. So then I'm going to say, I'm going to split you into groups and you guys are going to talk about Sarah's picture and tell me about what she did. And that's all they're going to do is talk about it. Pure brilliance, Sarah. But Sarah.

Speaker 3: Without the reference task exposures, without the multiple, and how many times would you say you've interacted with me on paper strips now? Just a guess.

Speaker 12: Oh my gosh. Oh, a lot of times and yet not enough. At least. Yeah, at least probably. Okay. Yeah.

Speaker 3: It takes that many interactions to apply it like this [00:21:00] at this deep level.

Speaker 3: And it's the same with our kids. And we still don't know the capacity that paper folding can take us to. This is the beauty of reference tasks. Through intervention, through tier one interventions course, this year is actually tier one interventions level one. You need this. Before you can get to the deep mathematics next year, we're going to do tier one interventions level two.

Speaker 3: So anybody that comes to level two. We are assuming has already had and experienced all of the modules of level one, whether you did that live or whether you did it self study because there's so much more to these reference tasks. But I can't just give you the final outcome without all of this foundation.

Speaker 3: You this is how you [00:22:00] actually experience it as teachers. That is a direct model of how I need kids to experience it as well. Other thoughts, comments, questions.

Speaker 3: The final thing that I am going to do is, you know that the ultimate counting result is rate. Rate and function. What needs to happen is, kids need Some kind of template so that they can actually start to organize. Okay, after round one, boy one gets, has two pieces, boy two [00:23:00] has three pieces.

Speaker 3: Round 2, and I can have all of my sensory stuff, kids can use tallies, they can use numbers, whatever. Okay just like Jesse and Kay, just like Jesse and Kay, where Jesse starts with 50 on day 0, and Kay starts with 0 after day 1, 55, and 7, day 2, 60, and 14. This is what I mean by counting and our ultimate goal is rate and function.

Speaker 3: Our ultimate goal is rate and function. So kids are going to act out candy problem with pieces, whether it's real candy or blocks or sponges or tallies or whatever it is. We give them as much sensory experience as possible. In round two, boy one has four pieces and boy two has six pieces. Round three, [00:24:00] six, 9.

Speaker 3: Round 4. 8. If I'm doing this right, if not, yell at me. Round 5. Round 6. Round 7. Round 8. And then I, so kids are just practicing counting. They're practicing counting.

Speaker 3: How do I know when to stop? How do I know when to stop counting? Let me give you another example of what I just did with my third graders last week. We looked at this problem 8. 5 divided by 0. 5. My third graders know, and I have audio tape of this, my third graders know division is skip counting. The second number is my skip counting number.

Speaker 3: They also know that decimal 5 is 50 cents and 50 cents is half a dollar. So I'm going to skip count by decimal 5 to [00:25:00] land on 8. 5. Now again, we can use Decimal 5, we can use half, decimal 5, 1, 1 decimal 5, 2. So I have to count by decimal 5. That's my skip counting number. I can write it out, I can use paper strips, I can use pieces, whatever.

Speaker 3: And I stop when I land on 8 decimal 5. That is the phenomenon of division. We have proceduralized division in schools. But that's the, that's not the natural phenomenon that division was meant to be. In the world around us, the natural, innate, intuitive, ingrained, natural phenomenon that division is skip counting to land on.

Speaker 3: That's what it is. You can't deny that. I didn't make that up. I didn't, [00:26:00] like, Why are these teachers teaching something I never learned it that way? It's not about learning it. I didn't make this up. This is how division naturally occurs in nature. I'm not, so we skip count by decimal fives to land on eight decimal five.

Speaker 3: My third graders are like, okay, two of those is one, four is two. Oh, eight. Okay. Eight would be double the amount. It would be 16 of those. And another one is 17. So they know it takes 17 skip count. Do you see the power of counting? If all we ever teach is counting, we can teach all of our standards. That's why when I was showing you the candy problem standards, that's why there are so many of them.

Speaker 3: Because we are just teaching counting. How do I know when to stop? Let's get a different color here, and then I'll be done. I'll wrap up. I could go on for another couple hours, but you guys have things to do. Okay? [00:27:00] In this column, How many candies have been passed out in round one? Boy, boy one got two, boy two got three.

Speaker 3: So five candies. Now, Sarah, you just matter of factly, like an hour ago, said, and I know about two fifths and three fifths. And I bet half the people on here and most of the people listening are gonna be like, where's she a fifth? Because every round, five candies have been passed out. So after round two, ten candies.

Speaker 3: After round 3, 15, and after round 4, 20. So I'll ask my question again. When do we stop passing out candies? Kids are pretty intuitive at this point after going through the experience. When we get to 90. That is why earlier I said it's going to be glitchy if you have two boys 90 [00:28:00] candies.

Speaker 3: And a, and boy one gets one for every three pieces boy two gets. Okay, it's gonna be glitchy. I don't say don't do it. That's how we move into division and remainder and fraction division. The ultimate goal, though, the ultimate goal of counting is rate and function. So there are three things. That we need to teach weekly to every student in all the grade levels.

Speaker 3: Rate, function, and iteration. This is iteration. This is repeated reasoning. That is the essence of repeated reasoning, mathematical practice number eight, and iteration. Now, what I can do with this is, on my coordinate plane, Someone said a little bit ago, I'm getting ready to teach constant and proportionality.

Speaker 3: Okay, constant and proportionality means, [00:29:00] If I extend my table in round zero, we're saying that it's zero. So this is an exact proportional, a direct proportional relationship. Grade seven don't miss this. So there is a zero, zero point for boy one and boy two. Both of their candy gettings. is directly proportionate because there are two factors for directly proportional, and that is it must go through the origin.

Speaker 3: It must have a starting number of zero. It must have a 00 ordered pair, and it must be linear. It must be skip counting by the same thing over and over. Boy 1 is skip counting by twos. I'm not going to go through the graphing part, but I, these are my ordered pairs. 1, 2, 2, 4. Okay, so boy 1 has this linear [00:30:00] graph.

Speaker 3: Now boy 2, is that graph going to be steeper or less steep than boy 1? Now if kids don't know, we have to write the ordered pairs for this. Which is round 1, boy 2 has 3 pieces. So 1, 3, 2, 6. So we may need to just write out the ordered pairs. And I go 1, 3, 2, 6. And what I start to see is That boy 2's line is steeper.

Speaker 3: What's the slope of boy 2's line? 3. And what's the slope of boy 1's line? 2. So I can actually write boy 1 as a function, y equals 2x. and boy two as y equals [00:31:00] three x and this is a very simplistic version of a function and then i could say jesse and k when does boy x boy one's candies equal boy two candies meaning let's solve this system substitution wise And what you're asking is, where do these lines intersect?

Speaker 3: Is there no solution because they're never going to intersect? No, they actually do intersect at day zero, at the starting point, because it's a constant of proportionality. It's a direct proportional situation. Take this equation, Which we got the equation from the situation. Now, we're going to take this equation and take it back to the situation [00:32:00] and eventually graph this.

Speaker 3: You see how we can do this with the same strategies. That we did this.

Speaker 4: Hey, everybody. This has been Tier 1 Interventions. We'll see you next week. Bye, everybody.