3 concepts you need to succeed in calculus: S2 E9

Speaker 1 0:00
I had an anthropology class in college. Okay, I love the class. The teacher was amazing. I know nothing about anthropology, but I got an A on all my tests because I crammed for six hours the night before, and I just rewrote the notes, and then I studied them, and I set them aloud and rewrote the notes and studied them and set them aloud and rewrote the notes and studied them and set them a lot I went in and regurgitated all of that five months later, I couldn't have told you anything. If I were to take that same anthropology test five months later, I would get a 20% the way that we're teaching in typical, traditional classrooms is exactly what I've just described. That is not going to increase retention of content. It's not going to increase memory. The way that we have to ingest content is through interactions over time with space in between. Hey everybody, it's

Cheri Dotterer 0:54
Cheri Dotterer. You are here at tier one interventions, where we are looking at activities that you can do for the entire classroom and teach all kids at the same time, but we can also expand these activities that we're teaching to small groups and one on one sessions. So for you occupational therapists that might be listening to this podcast, you might hear a lot of math, but you're also going to hear some of the non academic connections to what is going on with math.

Speaker 1 1:29
Jay Z here. Jay Z in the audience. Joni zupanzik Here, I'm going to do is make that transition into our rigorous, complex math content that is fully accessible for all we've said a lot of things today, and I want to reflect on some of those things. Daniel Pink, in his book drive, talks about human beings and human natural behavior and what drives us, what motivates us. One of the number one things that drives us is autonomy and choice. Not too many choices frees us. We don't know what to do with too many choices, but no choice creates negative emotions and feelings, so we have to have just right number of choices. Reference tasks will do that to us, and I'll talk about that in just a moment. Another thing that motivates and drives us and gets us to persevere and do things independently and makes us feel good, is a sense of belonging. And we talked about this a little bit ago, where if I feel like I matter, and if I feel like the teacher thinks I matter, and if the teacher has indicated that they like my thinking and I'm impressive and I've added to the conversation, then I feel like I belong in Daniel Pink's book. Drive two of the biggest indicators of human motivation and initiative and drive. The book is called Drive is autonomy and choice. So autonomy, having choice, and, number two, a sense of belonging. I matter. Now we're not going to talk to any teacher that's going to be like I would never. Teachers want their kids to know they matter to them, that we do what we do that is what motivates everything a teacher does, because all we want is for our kids to know that they matter in this world. However, do we have the correct facilitation and instruction that sends that to kids, or are we doing the opposite and getting an A an opposite negative result from them? We're going to talk in a moment about how do we create that instruction and facilitation so that we get these positive responses, positive emotions, positive feelings. And all of the strategies that I'm going to teach you are called responsive teaching. They are responding to the student perspective. Now why do we want to do this? Not just because it's all about this, feel good, whatever we want to do this, because when kids have core positive emotions and feelings, when they have good experiences in our classroom, it is going to, in turn, increase the memories of that content. It's going to increase the understanding of the content, and it's going to increase the retention of that content. Months later and years later, think about cramming for a test in college, okay? I had an anthropology class in college. Okay? I love the class. The teacher was amazing. I know nothing about anthropology. But I got an A on all my tests because I crammed for six hours the night before, and I just rewrote the notes, and then I studied them and I set them aloud and rewrote the notes and studied them and set them aloud and rewrote the notes and studied them and set them a lot, I went in and regurgitated all of that. Five months later, I couldn't have told you anything if I were to take that same anthropology test five months later, I would get a 20% the way that we're teaching in typical, traditional classrooms is exactly what I've just described. That is not going to increase retention of content. It's not going to increase memory. The way that we have to ingest content is through interactions over time with space in between. The way that we do that is with what we call reference tasks. We teach all content through what we call the Dirty Dozen. There are 12 reference tasks that teach all content from preschool through high school. These are the 12 reference tasks that teach all content from preschool through high school. These allow for responsiveness in teaching. It allows for students to engage in a positive, deep experience. And on these tasks, there are also sub tasks. Today, we're going to focus on what the outcomes of the standards are like. What standards do we want to teach first? Then how do I get kids to master those standards through this task? That's the whole premise of it. These tasks are the vehicle to the standards. This page here gives some of the standards for each of the tasks and what standards these tasks teach two not only do we need to know what the tasks are, but we also need to know what the content is for each of the tasks. These are the kindergarten standards that I can teach through this one task. These are the first grade and I'm going to scroll quickly because I just want you to get the shock perspective here. These are the second grade standards, the third and so on. Just with this one task in high school, the standards are rate and function in high school. There's really only a few foundational concepts that we need kids to know. I did a research study a number of years ago. I'm from Ohio, the Educational Service Center in central Ohio. I was a part of a three year group that we were studying High School in higher ed. The big question was, why is it that our kids that are going to college are placing into remedial math classes? If those kids are replacing into remedial college math classes, think about the kids that aren't going to college, where would they place? It's a problem that kids are coming out of high school and placing into remedial math classes in college, classes that you don't even get a college credit for. This is an issue. We did a research study, and in Ohio, all of us participants spanned out, and we surveyed algebra two teachers. And our question to algebra two teachers was, what is it that kids need to be successful on Algebra Two is there like 35 things like, what? And I'm telling you, it boiled down to five things after we did the survey, five things for Algebra Two, and they were solving equations, solving systems. So the relationship between two equations, radicals, square roots, radicals, or square roots of a numbers perfect squares, factoring and graphing, graphing, meaning on a coordinate plane. But the foundational skill of that is understanding number line, because the coordinate plane is just a double number line, a vertical and a horizontal number line that we overlay numbers on to look at the relationship between two variables. It was amazing that all of mathematics boils down and just to just a few things, my question was, shoot, if there's really five things, then what can't we accomplish this? Here's why my question end up being, then, what do those five things look like in kindergarten? What do they look like in third grade? What do they look like in sixth grade? See, we're missing the boat here, and we're missing opportunities. I did a lot of my own study. I took those five things I've boiled down all. Mathematics into three things, and I'm going to share that with you right now, because what you're going to see is how the reference tasks fall into each of these three categories, and how all the standards also fall into that. Then I'm going to be like, Okay, here's monkeys, Penny jar, Jesse and Kay. Then I'm going to show you how it does fall in. But I feel like it's important today, in this course, to back up a minute and be like, Okay, what's important? Here's another thing I want to share, because this isn't just me and my opinion the College Board years and years ago, because when I was in my school district that I taught for Gahanna, which is in central Ohio,

Speaker 1 10:48
years ago, I'm going to say 20 years ago from now, we actually created what we called a math vertical team in our district. Fifth through 12th grade teachers in math got together and talked about the verticalness of mathematics. Now things went sideways, because some of the conversation was, how do we have kids solve equations? Do we have them solve them vertically so they write the next step underneath, or do they write the next step beside? And I'm like, holy freak, we're asking the wrong questions. I'm not going to sit in these meetings if this is the question we're asking. And then the other question was, Oh, do we allow kids to use calculators or not? That's the wrong question. The right question is when, or actually, what is the goal for when I have kids using calculators, and what is the goal for when I don't want them to use a calculator? And then we ebb and flow between the two. I wasn't real happy with how we implemented it, but we then structured the norms there, and we followed a book by the College Board called math vertical teams. You can't even buy the book anymore unless you buy it used I have one, and I guard it with my life. It's called the math vertical team toolkit. After we had done our research and I had done my own research. I then went back to the math vertical team toolkit. And I remembered, in the math vertical team toolkit, it said for kids to be successful in AP Calc, for students to be successful in AP Calculus, there are three concepts that they need deep awareness of. There are three concepts in all in mathematics, there's only three that they need lots of interactions with, experiential, contextual, conceptual learning and starting any many years before calculus and what the college board said was rate function and accumulation rate, in its simplest terms, for first grade is just skip counting. Is next level, skip counting. What function means is, if I skip count by sixes, what is the 12th number when I skip count by sixes. Function is efficient rate. I don't want kids to always have to skip count 12 times to get the 12th number when I skip count by six. I want kids to have a different type of thinking, to be like, Okay, I'm going to plug in 612 times, and I'm going to get this result. Function is about automaticity and fluency. Rate is just the skip counting. Rate is skip count. Now it's a whole lot more. Don't just rate skip counting. Function is operational. Skip counting, efficient skip counting. What's the 100 and 52nd number? When I skip count by six, it's being able to do that very efficiently, accurately and fluently. And there's lots of ways to do that. If that doesn't screen fourth grade standards, I don't know what does okay? Do you see what I'm saying? And then accumulation is like snow accumulation. Oh, the accumulate. We got four inches of accumulation. Accumulation is about, let me transition, because what I brought to teach you today, before I show you my three things and my curriculum guide for grades one through nine are the two big ideas, and it's going to describe accumulation. So I want to do this in context. If we are going to continue to study reference tasks, and if we're going to continue to teach on these targeted standards. There are two things that we're missing. There are two things that are in the standards as early as primary grades. They are literally in the standards with these words, and I guarantee and Amy, you might be the exception to the rule for. Guarantee first grade teachers definitely don't know one of these words. But I also guarantee a lot of my sixth, seventh and eighth grade teachers don't know one of these words. There are two things. If we taught on these two things, we would be able to not only teach the majority of our standards, but improve number sense memory and retention at the same time, and they are knowns bones is a word in the standards at many grade levels, and we'll define this in a minute. But here's the one that connects to accumulation, because, remember, the college board said there are three concepts that are going to guarantee success in AP, Calc, rate, function and accumulation. The other word that is in the standards, and it's verbatim there in first grade and in other grade levels, is the word iteration. Kids need lots and lots of experiences with unknowns and with iteration, however, and you're like, I don't know iteration accumulation. Help me understand it. Don't worry about it yet. I will as we go through our task. But when I say unknowns, here's our misconception. As adults, we think, Oh, solving equations, inverse operation, solving equations and using inverse operation and using the procedure does not increase an understanding of the concept of unknowns. Unknowns is about equivalence and balance. So as many equations as you have your kids procedurally solved, it's doing nothing for the conceptual understanding of unknowns, which means it's doing nothing for the conceptual understanding of rate and function. When I see unknowns, what I'm looking at are those two words that the College Board says, Look, if we want ultimate success in AP, Calc, it's got to be grounded in these concepts. Iteration is a mathematical practice standard. The mathematical practice standard that I connect it to, mostly is repeated reasoning. We do not do repeated reasoning well because we think repeated reasoning is solving 10 of the same problem. We think repeated reasoning is repetition and those are not the same repeated reasoning is, and I'll give you a specific example, skip counting by a number on a 100 or 120 chart, and finding out if that number lands on 100 or not, I'm going to skip count by fours, four, 812, 16, blah, blah, blah. Does four land on 100 Oh, four does how many skip counts? Does it take?

Speaker 1 18:07
That is a process that I can iterate. Now, this is a little different. This is more of a repeated reasoning rather than an iteration, because iteration is repeating the same action over and over. I'm going to do the repeated reasoning part of this. I'm going to say, hey, remember last time we skip counted by fours to see if we were going to land on 100 today, we're going to skip count by eight. Eight is double four. Let's see four lands on 100 let's see if eight does. See that's a repeated reasoning the reference tasks are the ultimate repeated reasoning experiences, because we're going to introduce all of our reference tasks in the first 15 days of school, and then every month, we're going to give another interaction with the same reference tasks we introduced at the beginning. That's how we get repeated reasoning. The 120 chart is one of our reference tasks. Then three months later, I could say to kids, Hey, get out your 120 chart. We haven't done 120 chart for three months. That would be bad. I like kids doing 120 chart more than that, but let's say we don't do 120 chart for three months. Get out your 120 chart. I ask I say my favorite three words, because I want kids to tell me when I tell them, they hear it but don't learn it. When they tell me they learn it and create good core memories, I say, take out your 120 chart. Tell me about what we do with the 120 chart. We skip count by numbers and try to land on 100 that's what repeated reasoning does. Engaging kids in repeated reasoning experiences allows me, as the teacher, to give kids a task and have them work independently on it, so that I can pull small groups and do small group intervention within my classroom during and while the rest of the kids are independently persevering because they've done this task multiple times so they know how to do it, I might. Say to my kids, you know what our number today is, 12. What do you think we're going to do with the 120 chart? We're going to skip it up by twelves, and then we're going to figure out if it's going to land on 100 or not. Now there is a next step to that. The next step is make 12 the denominator of a fraction 112, how much money is a 12th of $1 now let's paper fold. Now see that there's all these other repeated steps that come from reference tasks. What I'm describing is similar to iterating, but iterating, in and of itself mathematically, is repeating the same exact thing over and over again. Paper folding is another one of our tasks, and iteration is an action. Actually, let me get a white paper strip, because I'm going to fold it in half. There are three different color paper strips, white, yellow and blue. When we do paper folding, or when we do fractions, my kids have to decide based on the denominator of the fraction whether we use a white, yellow or blue paper strip. The white paper strip. The iterations are halves. So my first iteration, my first action is to fold in half. If I do a second iteration, I do the same exact iteration, the same exact action. I folded in half twice. A third iteration, I fold in half three times. I fold in half four times. An iteration is the same action repeated. A big idea in mathematics that is, iteration is fractals. So you can look that up. I'm not going to describe it, okay, the iteration is the action, and then what is the result after repeated actions. Now, let me just stop for a moment. If you're listening to the recording, I want you to reflect on this. How often do you teach you can't explicitly teach it. So that's the bad that's a bad question. How often in your classroom do kids experience repeated reasoning activities, not repetition of solving. See, repeated reasoning is not about solving. Repeated reasoning is about an iterative process. By engaging in that iterative process, we start to create core, foundational memories.

Cheri Dotterer 22:19
You guys have a great month and look forward to seeing you at Saturday math and impact Wednesday before next month, talk to later. Bye.

Transcribed by https://otter.ai