Hi Bell Ringers! I freed one of my most popular posts this holiday weekend, so all of you free subscribers can get an idea of what’s going on behind the paywall.
This interview is with behavioral psychologist and SpringMath founder Amanda VanDerHeyden on the reasons so many students aren’t proficient in math. In this video interview (+ transcript below), we talk in-depth about the instructional hierarchy—the four stages of learning that every student needs to move through to get to mastery. She also gives us a primer on the research base, and why teachers should pay attention to it.
And BONUS: at the end, VanDerHeyden takes reader questions.
If you like this kind of thing, consider becoming a paying subscriber!
Today’s letter includes:
What is the instructional hierarchy? An explanation of the four stages of learning that all students move through in all subjects, and how educators can use the instructional hierarchy framework to improve student learning
Why educators’ instructional tactics should depend on how proficient/fluent students are at the task—varying student needs determine which tactic to use
Why helping students achieve mastery on discrete skills is so important to future learning, more complex tasks and being able to generalize and apply learning to problem solving
Key research references pertaining to the instructional hierarchy—go to the very bottom of this newsletter, below the transcript, to find Amana’s recommended list of research for further study
Full Transcript:
Holly Korbey: Hey there, Bell Ringers. You are watching Interview with an Expert, and I'm journalist Holly Korby. We are here today with an expert, Amanda Vander Heyden. She is a behavioral psychologist and researcher with a very long list of accomplishments, and I'm going to read a couple of them to you right now.
She has been a panel member for the NIH, a standing panel member for IES at the U.S. Department of Education, and many others. She has served on a number of boards, including the RTI Advisory Board for the National Center for Learning Disabilities. She's also authored a number of policy guides and position statements and delivered testimony on the use of multi-tiered systems of support, otherwise known as MTSS, and response to intervention. She is president.
of Education Research and Consulting in Daphne, Alabama. And she has a faculty affiliation with the Wheelock College of Education at Boston University. she, Amanda knows a lot. And the reason I know Amanda is because she's the creator of Spring Math, a whole class research-based intervention that's focused on learning math facts. And I got to know Amanda because I reported on Spring Math twice because of the intervention's impressive results. So welcome Amanda and thanks for being here.
Amanda VanDerHeyden: Thank you, I'm happy to be here.
Holly Korbey: Yeah. So today we're going to talk about the instructional hierarchy. And when I was doing my reporting on spring math and on early foundational math and how students learn it, that was the very first time that I'd heard that term and I'd heard it from Amanda. And the same goes for a couple of my editors as well who had never heard of the instructional hierarchy. So I thought today what we could do is we could talk about what it is, how teachers can use it, why it's important to know about it. And I'm gonna now just kind of turn this over to Amanda with my very first question, which is, what is the instructional hierarchy?
Amanda VanDerHeyden: This is so exciting. It's just, it's endemic to the way we practice school psychology, behavioral psychology, intervention work in schools. It's not a new idea. I'm gonna give you a little bit of the history, but the basic idea is the consideration of task difficulty as an important dimension of your measurement relative to what kids can do.
when you decide which instructional tactics they need. So for example, if a task is brand new and children don't know how to do it, then the goal of the instruction is really about establishing correct and complete understanding. The behavioral jargon for that is to establish stimulus discrimination. And the specific ways we do that are things like modeling and guided practice and worked examples and immediate corrective feedback.
And very importantly, if we do the wrong things, we will actually worsen learning. So for example, if an understanding is brand new and you say, okay, wrestle around with that in small groups and see if you can figure out how to do that. That is 100 % contraindicated according to the science of how humans learn from behavioral psychology. Now, once children get into the fluency building stage of learning, the tactics have to change. So the things that worked in acquisition will no longer work. And now your instruction really is about dosage of opportunities to respond. I know we're going to talk about this in more detail, but the takeaway here is that the idea is that all learning progresses through these predictable stages of development and it's really important to know where you're situating your learning strategies, your tactics, your instructional tactics relative to measured learner proficiency in terms of the task. Do you need to make the task a little easier because you're giving practice opportunities so you need them to be in the fluency building stage of learning or do you need to stop and do an acquisition lesson before you allow independent practice? It's that dynamic back and forth flow of instruction that's actually measurable, knowable, and very, very powerful.
Holly Korbey: Great. And I think the reason this came up when I was visiting these schools in Philadelphia, Amanda and I went together to Philly to look at some of the schools that were implementing spring math. And so the instructional hierarchy comes up because of how a lot of
early math gets taught. So I'm wondering if you can just, because what happens is they, kids are in the acquisition stage, which is like the beginning stage where they need a lot of concrete, direct information, but a lot of math curricula are set up almost in the opposite way.
Amanda VanDerHeyden: Yeah, I have this slide that I show all the time and I think there's so much like we'll call that math reform. Okay, this is like the math education perspective, which is not well aligned with the cognitive and behavioral psychology perspective on learning, which we could call the science of learning. Okay, but the math ed folks, I mean, I think the big mistake that they make their intentions, their aspirations are very good. They want children to not just be rote, superficial robot like learners, kind of like reciting information that they have not, they don't really understand. They have not contextualized in a way that will actually be useful for them after they master how to do it. They want equity. They care very much about equity, for example, but I think then the tactics that they go on to recommend based on those very solid aspirations are completely, unintended, in an unintended way, they're ineffective and actually harmful. And it all comes back to, you know, if they understood and considered the way children learn, or even animals learn, we will talk about this today, if they did understand that science, I believe they would have an easier time adopting more effective and promoting more effective techniques for kids.
Holly Korbey: And so what this ends up looking like in math class is that many kids don't learn their basic math facts because they just aren't given enough practice or time or whatever, so they can't do the cool, imaginative, creative problem solving that we all want them to be able to do, right?
Amanda VanDerHeyden: Absolutely, and I mean, I just think that's exactly how it plays out. The teacher has learned either through a preparation program or professional development delivered at the school or the very curriculum materials that are available to that teacher, the teacher has learned that tactic Y is an evidence-based tactic or a tactic that works or a tactic you should use. But there is no such thing as a tactic that works all of the time for all learning. That's the whole key of the instructional hierarchy. So I say to teachers all the time, it's not that I don't like productive struggle. I mean, the data are abysmal on productive struggle, which is a very common technique in math. But what I will say all the time is I think the biggest problem with productive struggle is most teachers are trying to use it in the acquisition stage of learning when it is absolutely contraindicated according to the science of how people learn.
Holly Korbey: Okay, so now, okay, you brought, this is a great segue. Let's talk a little bit about the research base behind it. You know, all of our listeners are really interested in understanding—What's the research that shows that this instructional hierarchy works?
Amanda VanDerHeyden: Yeah, so it's kind of funny. I just started jotting a list of all the references that I want to give you so you can have an annotated reference list maybe to pair with this episode. And many of these...sources or things that are in the public space you can access but it's so funny to me in my career I've run into this a few times like I remember one time in probably the 1990s I had somebody reach out to me and say well can you send me the research evidence that modeling is an evidence-based tactic? I'm like my gosh that's like that's not like a couple of papers that's like a body of work could you just not be lazy and go read the literature you know there's that but but it made me sort of reflect about what what would this be like for like a brand new teacher or I mean even a veteran teacher who wants to consider this new idea and I kind of have to be a little critical of the scholarly world that we have there's not a good paper that really connects the dots well for people so as I started
Holly Korbey: Could you write it? You get on that, please?
Amanda VanDerHeyden: I mean, I was thinking today I should write this paper. okay, so I'll tell you this, I'll show you this. So the critique would be, oh, it's from 1978. Well, this is where it started, in my mind, all right? But I'll situate this for you and tell you what this is about. So basically, the instructional hierarchy comes from behavioral psychology and very famous and well known founder of that field is BF Skinner. This is the person who discovered the laws and principles of human behavior. and the way that he did that it and he tells a story that Owen White recorded that he believed his greatest discovery was rate of learning and he used rate of learning in animal models and then later people used it in human models, of course pretty quickly as a way to characterize performance improvements Skinner would not say learning but performance improvements and it was sensitive. It was something you could manipulate environmental conditions and you could say, okay, these are the things we can do to reduce
the occurrence of a behavior and these are the things we can do to increase in occurrence of behavior. Well, in learning, we want to increase academic performance. We want to decrease things like errors, right? So, he really discovered rate of responding and then the subfield came along called precision teaching and they added the dimension of accuracy to rate of responding working entirely in academics, okay?
And in this field, there were two researchers, Clay Starlin, actually his wife too, Anne Starlin, so Clay and Anne Starlin in Minnesota, and there was a guy named Eric Houghton who began to publish all kinds of work in the 1970s, studying and recording words read correctly per minute, digits correctly completed per two minutes or one minute, to characterize improvements that resulted as a result of giving instruction, to characterize reduction of errors over time. So that was a whole field called precision teaching and as early as the 1970s those guys especially Clay Starlin had written an article about you could characterize student proficiency and in 1982 I believe it was he got very clear about you would characterize that to drive what they needed instructionally.
And at the same time, it was like this hotbed of discovery up in at Brown University of Washington. That's where these guys were. So this is the original chapter for me. I always cite this as the starting place in 1978. And this is important because Herring and Eaton wrote the second chapter in this book. And they laid out this whole notion in a very clear way that you could take this new information that had come into the world via perspective teaching and then was picked up and became curriculum-based measurement, which is in every school in the country now. This is the original book, 1977, Dino and Merkin. They never use the words curriculum-based measurement in the book.
This is going to make this very confusing for the graduate students and everybody else who wants to understand this. But in effect, this is the first CBM book that was ever written. This is also the first RTI or response to intervention book that was ever written. So for me, response to intervention, later MTSS came out of this work. So let me pull this together in a coherent way if I can. I love to talk about this stuff.
Holly Korbey: I know, that's why I asked you!
Amanda VanDerHeyden: So it was born in behavioral psychology. You have these Precision Teaching folks who came along and that's Starlin and Houghton and they start playing with this metric, okay? They're the first to define fluency. So words right correctly per minute, correctly completed per two minutes. They're beginning to characterize like Eric Houghton was identifying.
RAPS criteria, so what is the level of proficiency or fluency that is associated with retention, that's the R, application and performance standards, that's the RAPs, that's been expanded over time. And then, Stan Dino says, I think that's a good idea. And he situated it always in a system of program improvement, which really became RTI and MTSS.
And what he did that was so beautiful for the world, for my money, the most game-changing discovery ever, is he took what was a behavioral observation—we'll just stick with words read correctly per minute—and he turned it into a score. So, Dino at the University of Minnesota was using that beautiful little metric to graph performance improvements to begin to study what are the conditions under which these can be a universe of scores. So, we can now characterize whole school distributions of performance and so, every screening system on the planet is using this now to characterize for universal screening, to identify who needs intensified instruction, and all of that was born in those books in the 1970s.
But the instructional hierarchy was so important because that is the first scenario, the first case, the first framework that said, hmm, okay, measurement's really important. And we have figured that out. We have a very sensitive way to know, are you improving? Are the problems declining? But what we didn't have was a way to have that very clearly drive instruction because weighing a cow does not make it fatter. Does that, right? So their idea spelled out very clearly, acquisition—
Holly Korbey: Yes, absolutely.
Amanda VanDerHeyden: These stages of learning: acquisition instruction, fluency building stage of learning or instruction, and they had two more stages that they called generalization and adaptation. But in effect, I talk about those together because I think they're both forms of generalization and it simplifies it for teachers and we just, don't think the added complexity is worth it. So that's how I write about it. Matt Burns and I wrote a book. This one is, I think we did this in 2010. It's with Guilford and chapter lays out a little bit of that. I have a paper that I wrote with Ben Solomon and published in the last couple of years for School Psychology. I can send it to you because you can get it on ResearchGate and it tells a little bit of that history too that I just went through pretty quickly. Okay, but it's not just one book in the 1970s. So what happened after the 1970s is a...We used to call them the Eds. I don't even know if Ed Lentz is still around. I know Ed Shapiro is a dear friend and he's not alive anymore. But Ed Lentz and Ed Shapiro wrote a prescient article in 1986 in School Psychology. And it was about, and I think the title was Functional Assessment of Academic Problems or maybe Academic Skill Problems. This was a radical idea that you could go into.
A classroom and a teacher says I have a problem, this child is not learning and instead of taking that child out into a little office one-on-one and doing a battery of assessments and reaching conclusions about what may or may not work for the child, you could actually assess the classroom environment in tandem with the instruction that the child is receiving. Okay what are the classroom expectations for learning in math today? Okay does the child, is the child in the frustration level of performance for that sample? Using this one curriculum-based measurement took a behavioral observation score. Let's sample back a prerequisite skills are lacking. Do they need fluency building, or does the child need acquisition are different instructional tactics.
Eds wrote that seminal paper. By the way, we're all sort of related, the people who worked on this stuff. So lineage-wise, it gets real interesting real fast. Ed Daly, then for me, he's the guy in the 1990s. He was writing about this a lot and he was working with Brian Martens. Brian was the first doc student of my major professor. Tanya Eckert, I think, I can't remember who her major professor was, but she's worked with Brian all his career, so she's on one of those papers too. And there were just a number of important papers that Ed Daly specifically published, at least, gosh, six, seven, eight empirical, well-designed studies. Now, he was calling this work brief experimental analysis. This is why it gets confusing. So you've got B.F. Skinner, then you've got…the precision teaching people who were funny about publishing their work. Then you've got Dino and Merkin and then the subsequent CBM crowd. Then you've got Shapiro over here, Shapiro and Lentz also in school psychology. Now you have Ed Daly and he's publishing a lot and publishing in the behavioral journals, but he's calling it brief experimental analysis. Every once in a while he would call it the instructional hierarchy, but not in every article. And then my most favorite paper, bar none, just experimentally, which was published in the Journal of Applied Behavior Analysis in 1998, was published by George Knoll, who was one of my major professors at LSU.
And he, I can't even think of the title of his paper, but he also did not put instructional hierarchy in the title. He probably referred to it as brief experimental analysis or functional analysis, which is another way to look it up. So you can see it gets very confusing. Daly and Scott Ardwin, Scott Ardwin was Brian Martin's student and also worked with Joe Witt. Lots of connections. They edited a special series for Journal of Behavioral Education in 2007. Matt Burns and I published a book on this concept in 2010 and there have been a number of empirical pieces. So one of my most favorite articles from my entire career I did with Matt Burns. We published it in 2006. I think we published it in School Psychology Review. That was one of my favorite journals back then. And what we did is we looked at the level of proficiency that forecasted not just retention of math skills over time, but faster learning of more complex related content. So we could derive specifically what's the level of fluency that you need to demonstrate on single kind of single digit multiplication facts that forecasted when we gave you multi-digit multiplication facts with and without regrouping, you could master it faster than kids that had a lower level of proficiency on the component skill. We knew what we would find because the science is the science and it's no different than what Skinner was demonstrating with, you know, rats and pigeons in his lab and later the PT people began. I'm serious, you know, there's an expression in that world, the rat is always right. I say the learner is never wrong. Right?
Holly Korbey: Right. And so this is kind of why I wanted you to explain this to people. You know, maybe a lot of us don't know, aren't familiar with all the names and everything. But what I wanted listeners to understand is that this research is actually very old and it's well documented and it's empirical. So the next question is, I think that what we all want to know now is why don't more educators and schools and curriculum developers know about it.
Amanda VanDerHeyden: Well, isn't this interesting? I mean, my question for you is why do the math curriculum developers ignore the science of how humans learn? Because, you know, there's also a beautiful body of literature, Asha Jatendra, Diana Bryant, more recently, Ben Clark and Chris Dobler has probably the most recent, my favorite study from 2012, but Doug Carnine published a paper in 1997. Scott Baker published a paper, I think in 2002. And those studies are independent research teams evaluating math curricula for the principles of evidence-based instructional design. And those are gonna include things that I'm talking about.
Measurements, sensitive measurement of performance, measurement of the environment, task difficulty, the instructional hierarchy, guiding what you do instructionally. That's all considered part of evidence-based practice now. And so they evaluated curricula going back to the, you know, like 2002, the first paper I'm aware of is 1997. As the Carnine paper, and it's a ringing indictment of math curricula. And then if you look at a summary of effects of various math curricula on student math achievement, it's basically a zero effect size. So nobody is surprised curriculum developers in effect do not attend to the science of how humans learn yet. But then there's the second problem in education. There's way too long of a delay between discovery and revision and substitution of less effective practices for more effective practices.
Holly Korbey: Great. So. I think that one thing that is important for everyone to understand, and I want you to explain a little bit about this, is that this isn't just a math issue. The instructional hierarchy, the stages of learning don't apply just to math learning. Am I right about that?
Amanda VanDerHeyden: 100 % applies to everything. Like the paper that I did with Solomon, we presented reading data, we presented math data. You know, it's the basis for our theory of action in Spring Mouth. It's all I know how to do. Research goes on all the time where people, I mean, I published a paper with Robin Cotting in the last year or two that was a single subjects experimental design. it is, I don't know if we used instructional hierarchy in the title, we probably did not, but because I can't remember, but it's in the, certainly it's the basis for what we did and that's true for certain you Katie Mackey has a pretty recent paper and Journal of Behavioral Education Looking at multiplication performance, but even if like in my world if you look at dosage research in the space of MTSS.
Much of that work has been done in math, which I love, but even that is very much situated in the instructional hierarchy. So for example, a reviewer would not think well of the experimental rigor of such a paper if they did not verify before they study dosage that you're working on fluency building and you're looking at opportunities to respond in particular. Dosage would look very different in an acquisition stage of learning. It's not about practice and opportunities to respond. There it's more about the discriminability of the conditions surrounding your response, okay? And anyway, you know what? Succinctness is not my gift.
Holly Korbey: It's okay. No, this is really fascinating. So like, how does this translate for teachers? Like, I'm so interested in the where the rubber meets the road moment. How can teachers use that, this knowledge in class? How do they change what they're doing to make it more aligned with the instructional hierarchy and how you know the brain learns?
Amanda VanDerHeyden: Yeah, okay, let me talk to the teachers. Where's the pitfall that usually happens? You turn this light on for teachers and suddenly they wanna measure every kid and get very specific, you know? It's a natural desire because when you see it work and it always works, okay? It's addictive. mean, show a teacher who doesn't wanna help a kid learn, that's what they show up for, okay?
So what I want to say to teachers is, in your core instruction, you do not have to be so very, very precise about it. Let me tell you why. You can think of your core lesson, this is what Matt Burns and I have been telling teachers to do this since, my goodness, 2002, easy. Take your lesson and chunk it into acquisition, fluency building and generalization. Okay, so one third, one third, one third of whatever block of time you have, okay? So the one third of it is devoted to acquisition.
Okay, one third is devoted to fluency, one third is devoted to generalization. Now if I had just done a training with your people and showed them the data and how systematic it is and it really is that systematic, which is why we can write software to do it and we can produce very reliable results because it really is that formulaic, but by definition, the acquisition section of your instruction is going to be targeting a task that is a relatively new understanding. Okay, by definition your fluency buildings part of your lesson is going to target a task that students have already acquired the understanding for. They know how to respond correctly and they understand the conditions under which a response would be incorrect. They are ready to build fluency which is about making giving the correct response easier. Okay, that's the whole characteristic of fluency which the cognitive psychology folks will call automaticity.
If you want to say those aren't exactly the same, they are 99 % overlapping constructs in my mind, okay? And then the generalization stage of your lesson is where you get to do the cool stuff that teachers want to do that they've been trained to do. That's your small group. That's your productive struggle if you want. I'm gonna call it a generalization opportunity, but here the key is whatever applied work or challenge opportunity you are giving children to do, they are at
mastery level of performance on that particular task. So think about it this way. Acquisition, this is brand new understanding, it's what we're learning this week. Fluency building is what we learned a week or two ago. And generalization is a review skill. Maybe we learned it last month, maybe we learned it last year.
Holly Korbey: Great. Wonderful. Okay, so now this brings me to my final question.
Why is it so important that we all understand this when we're designing our lessons, when we're thinking about how to build, especially math skill, over time? That math is something that you have to build over time.
Amanda VanDerHeyden: Yeah.Right. And so is reading. mean, all learning can be situated in a skill progression, a logical skill sequence. You know, back in the probably 1990s, 1980s, teachers, special ed teachers called it a task hierarchy or a skill hierarchy. Later, you know, probably because of Clements and Sarama people began to call these learning progressions, right? And there's nothing fancy about that except to say, you know, I can't really teach you a walk probably until you can stand. So there are prerequisite milestones you need to hit. And this is true for any behavior that we want to teach. I taught my cat to ring a bell for food. We started tapping anywhere around the food bowl. If the cat tapped the food bowl, he got the food. And then we shifted it and made it a little harder for him. That's what we do. It's behavioral shaping. This is what Skinner discovered. This is all we do. But I will say, for classroom teachers, you can be a little loose about it. You don't need to measure every kid every second, okay?
But once you get into in the MTSS world, giving intensified instruction. So for kids who are not thriving in core instruction, you have to get very precise about it. And then you must have technically valid skill assessments for all of the skills in your skill progression. This is exactly how spring math is designed, for example, and it's very specific. There are specific frustrational, instructional, and mastery rules associated with specific assessments for 144 skills from numerous.
Holly Korbey: Right, and so you basically do these little assessments and you figure out exactly which skill.
Amanda VanDerHeyden: There's the protocol. Not just the skill, but then do you need an acquisition lesson because you have not acquired it? Or do you need fluency building and it needs to be adjusted as your performance improves, at least weekly.
Holly Korbey: Right.
Amanda VanDerHeyden: Okay. So to me, when people talk about, can remember when personalized learning, we all became kind of the buzzword and it was just like something that happened on the computer. But people like me are like, well, what does that mean? What's the decision rule? What's the theory of change? And like there were vendors that were like, well, that's proprietary. Like, well, then that's also not researchable. Like we can't, you have to be able to define your independent variable. Right. And so it turns out the independent variable of giving the child the right instruction at the right moment in time, which means you adjust it week to week. We use the instructional hierarchy. I mean, every great researcher I know does in my world. That's the magic. That's the secret sauce. But teachers don't, I think the sort of, you know, novice mistake, you you get excited about it and then you think you really have to be that precise in core instruction, you can be a little softer in your approach because it's more efficient. So you can say, here's the goal this week, we're gonna do acquisition instruction. We're gonna assume nobody knows how to do it. If one or two kids came in and they could already do it, it's not gonna kill them, but I'm gonna be, it's like a 90 % solution. You know what I mean?
Holly Korbey: Right. Yeah, absolutely. Okay, I hope that everyone now has a better understanding, I do, about the instructional hierarchy and how you can apply it and where you can apply it. And Amanda is gonna share some references with us that will be included in this episode. So you can look it up and get more information if you wanna find out more. I thought it would be fun. I know you don't have very much time. If we could take some questions from readers about math specifically, are you up for this?
Amanda VanDerHeyden: My gosh, this is the most fun part. I wish people would invite me to come do keynotes and the whole thing could be Q & A.
Holly Korbey: Yeah, well, these are really good questions. My readers are really smart. So, OK, first question is from Jerome. He wants to know what are effective strategies to develop fluency and math facts that should have been mastered in elementary school when the student is already in algebra?
Amanda VanDerHeyden: Okay, this is the great answer for that. Like you don't have to think about it in such a complicated way because learning is learning is learning. So the things that work for high schoolers also work for sixth graders also work for cats, okay? I should put that up on the web. My cat ringing the bell for food. It's the same strategy. if you are in, so first of all, you have to, we can probably assume that they understand conceptually the process of addition, right?
But if not, we would want to reteach that and we could do that very efficiently. That is a single protocol that you could easily give and it would involve some modeling, some guided practice, some correction, maybe of incorrect responding and explaining how they made it correct. Those are ways you can really make sure they understand how to give the correct response mapping it on a number line. But most likely these kids are have did get that instruction.
And what they're really missing is they never got to mastery. And this is very anticipated. Like it's a predictable result. If you did not get kids to mastery, they're not going to retain that skill over time. is not my idea. That is, you know,
Houghton and Starlin and Dino and Merkin from the 1970s demonstrating this in large data sets. So what you want to do is you want to give those children, those high school kids, do we call them children? I don't know.
Young adults, I have young adults myself, they're into college, but you want to give them a high dosage of opportunities to respond at the right level of task difficulty. So, you know, the mistake is to act like you can just march forward with the more complex skill training and ignore the fact that they have not mastered the prerequisites.
Holly Korbey: Right.
Amanda VanDerHeyden: You know, Anna Stokke says math is relentlessly hierarchical and people refer to it as cumulative and all that is absolutely true. You know, imagine just you're trying to teach kids how to factor and they can't multiply. They can't even follow your acquisition lesson on factoring because they can't do the computations.
This is, by the way, part of the reason just giving up on kids and giving them a calculator is so deadly. And by the way, you're going to hear me get on a soapbox about that. I'm writing a piece right now for our next newsletter. My kid just took the MCAT over the holiday. He did pretty well. I was really excited. But you're not allowed to use a calculator on the MCAT. I know if you know that. So he came downstairs after his practice, told me he would have done better if he had a calculator. I said, can I have your scratch paper? His scratch paper was multi-digit multiplication, long division using the standard algorithm. By the way, all under timed conditions because you have to finish a section in a certain amount of time. And it gave me sort of this renewed perspective of, you know, as an eighth grade teacher, a seventh grade teacher, a sixth grade teacher, you do not have a right to make a decision for a kid that an entire field of study is closed to them because you decide to give them a calculator in sixth, seventh or eighth grade. You undermine their mathematical development and skill mastery and you really hold them back from future opportunities and that's just not your role as a teacher. So I know I got a little off track there, but the key is don't give up on your high schoolers. If they master those fundamentals, what you're really doing is giving them a vaccination against failure. You are enabling faster mastery of complex related skills like Matt Burns and I demonstrated in 2006. So practice, delayed corrective feedback, goals for faster improvement, adjusting the task difficulty based on mastery. if they're working on single digit multiplication facts and they get to the equivalent of 80 digits correct in two minutes or 40 digits correct in one minute, then you increase the difficulty a little bit. So maybe now you introduce one by two to three digit multiplication without regrouping because in effect it's the same skill. And then you introduce the regrouping and if they remain at a baseline level of fluency, which would be about 40 digits correct in two minutes or 20 digits correct in one minute, then you can continue with fluency building. If they don't, then you back up and do an acquisition lesson.
Holly Korbey: And I mean, this is exactly, and I've talked about this before and I've talked about it with you, my son ended up at Mathnasium in the early years of elementary school because of this exact thing. And that's exactly what they do, but it's very expensive. So it makes a lot of sense for schools to get it right early on because it saves you in multiple ways.
Amanda VanDerHeyden: Well, and by the way, think about the way that opportunity gaps are opened up. So it's only the kids with resources and also it's not fair to your kid to have to go for extra training when what happened in the 60 minutes of math instruction was not adequate. And by the way, all that body of work I talked about, Asha Jatendra, Diana Bryant, Chris Dobler,
Doug Carnine, Scott Baker, that is a ringing indictment of math curricula. All of those studies found that math, and this will resonate with your readers, they already know this, and your followers, that math curricula do not provide a sufficient number of well-controlled practice opportunities. That's a major limitation of all math curricula.
Holly Korbey: Right, okay. Okay, so let's go on to the next question. Okay, this is from Jake. What does a good math intervention look like in elementary and what's different from literacy interventions?
Amanda VanDerHeyden: Well, I do it exactly the same way myself. I mean, that's the wonderful thing, I think, about being trained in behavioral psychology. To me, behavior is behavior is behavior. It doesn't matter if I'm teaching a child how to read or if I'm teaching a child how to do math or I'm teaching a child how to walk properly in line and follow a transition routine. Behavior is behavior is behavior. There are antecedent things that will work, behavior management of the child, the responding, and then consequences that are especially important depending on the stage of learning that you are.
So it works the same in reading. We do it exactly the same. identify the skill progression. assess whether the child is in, you know, is it for the expected performance right now, are they frustrational, instructional or mastery? If they're mastery, you don't have a problem. If they're instructional, build fluency. If they're frustrational, give acquisition support. If they are frustrational, you may do a process where you sample back through successively easier prerequisite understandings, which these examples
I could tell you them. I did this on a podcast for Kate Winn in Canada because she's a literacy person. So she said, do this for reading. I'm like, Kate, I'm a little rusty, but I can do it because we did it for years. And you find the sweet spot to intervene. Now the key to that, okay, by the way, that sampling back process is called survey level assessment. Again, that's part of our problem is there's lots of names for stuff, but it's called survey level assessment in curriculum based measurement, CBM.
And there's a reason you start at the goal skill and take little slices back. And I have heard a little bit of the hullabaloo in reading, the science of reading about, you know, targeting phonemic awareness and, you know, some of that. And what I will say is it's really important to keep in mind that there is a reason we start at the grade level goal skill and go back in little slices and we find the first place we can intervene because we want to intervene on the skill that is most proximal to the expected grade level performance skill. Because then our probability of transfer is what the cog size will call it, generalizations, what the behavioral psychs will call it, is much higher. Do you see what I'm And you know, when it turns out, you don't have to fix every little tiny baby skill that you've ever missed. You don't.
Holly Korbey: Right, right. You don't have to go back to the beginning. You just figure out where the break happened, where the, yeah, okay. So interesting.
Amanda VanDerHeyden: That's right.
Holly Korbey: Okay, so this is the last one and this is my favorite question. It's a little long. This comes from Jonathan, but I'm really interested in hearing your answer on this. So he says, our current standards and curricula often feel overloaded with content, strategies, and areas of focus, which can leave little time for ensuring mastery and providing any real-time intervention. So if we were to streamline instruction and focus only on the most essential concepts needed for success in Algebra 1, we might be able to achieve deeper understanding. But he talks about number sense. if you add 99 plus 98, what a student should be able to do is look at it as 200 minus three, that shows number sense. He says, however, our students can recite all these strategies they've learned. To do this kind of thing, they often struggle to apply them flexibly.
So they might be familiar with multiple strategies, but they don't recognize the easiest way to solve that problem. So he is asking, do you think there is optimal timing for introducing these multiple strategies? And how do we balance developing number sense and then overwhelming students with too many strategies and problem solving techniques?
Amanda VanDerHeyden: Yeah! This is predictable by science. Like I love the question actually. And I mean, it made me have like 72 thoughts while you were talking, right? Of things to say, and I didn't preview these questions. So, yes, there is a science to this. The science of learning would say that when in an acquisition stage of learning, you use something like multiple strategies, then what you are doing is you're teaching and you're doing a, using a level of stimulus control or presentation of stimulus that's much broader, okay? Well then as a result, your rate of skill mastery will be slower.
Theoretically, however, you will develop a more robust understanding on the other side. The
alternative is more tightly controlled stimulus presentation, faster skill mastery, but it might be a little less robust. So this is actually knowable according to science, and I think it's really important because the key takeaway here for teachers to understand is that there is not a tactic specifically that you should be married to. It should be the tactic that will work given the
of the children in front of you. One of my favorite people on the planet, Kristen Ring, says, teach the students in front of you. It's a pretty simple kind of concept, right? So sometimes you'll hear teachers say, should I just teach, or this is the way I like to do it. I teach the plus ones, then I teach the plus twos. Well, that can actually be very inefficient because most kids can tolerate the acquisition of sums to six, for example, without having to slice it that way. And you don't give up mastery on the other side, like something we can answer that question. Don't make that bad decision. It's going to take you three weeks to get them there instead of one. So use the more efficient method. So his specific question about the use of sort of like, I'm going to put that in the category of creating equivalent quantities to make difficult math easier to solve, which is, very important.
Holly Korbey: Doubles and plus ones and all those things. Yes.
Amanda VanDerHeyden: Many of those tactics will naturally emerge for students as they have exposure to the procedural problem solving. The problem is we have teachers hammering kids in the acquisition stage of learning and I'm sure the cognitive psych people would have theories about interference that go, you know, multiple skill interference maybe that goes on in that phase of learning that's deadly. You know, something else, but it's the same idea that when you're trying to focus on teaching a kid five different ways to get to a solution, you are actually confusing the child.
And if you just give the child procedural skill practice, many children will naturally discover add the double plus the one or if you suggest it, once they have a level of facility with it or fluency with it, okay, it's not to me, it doesn't live in acquisition instruction, it lives in fluency building, because it is a way to make it even easier to get the correct answer.
Holly Korbey: Ooh, that's really good.
Amanda VanDerHeyden: For kids who don't naturally do it, you can back up and teach it using acquisition, fluency building, generalization support. But that's gonna be a tiny percentage of kids. So it's kind of like, again, it's another example of teachers are hearing, okay, this is how we need to do it, totally divorced from whether or not it works, whether or not it's well aligned with the science of learning.
Holly Korbey: Wonderful. This has been awesome. So thank you so much. thanks listeners, you're here. This is the bell ringer. This is interview with an expert. Amanda is going to provide us with a list of resources. I'm going to include it at the bottom of this episode. And we'll continue this conversation on the instructional hierarchy. Thank you so much for being here.
Amanda VanDerHeyden: Thank you.
Learn more: VanDerHeyden’s research citations for the instructional hierarchy
Burns, M. K. (2007). Reading at the instructional level with children identified as learning disabled: Potential implications for response-to-intervention. School Psychology Review, 22, 297-313.
Burns, M. K., Codding, R. S., Boice, C. H., & Lukito, G. (2010). Meta-analysis of acquisition and fluency math interventions with instructional and frustration level skills: Evidence for a skill by treatment interaction. School Psychology Review, 39, 69–83. https://doi.org/10.1080/02796015.2010.12087791
Burns, M. K., & Parker, D. C. (2014). Curriculum-based assessment for instructional design: Using data to individualize instruction. New York: Guilford.
Burns, M. K., VanDerHeyden, A. M., & Jiban, C. (2006). Assessing the instructional level for mathematics: A comparison of methods. School Psychology Review, 35, 401-418. https://doi.org/10.1080/02796015.2006.12087975
Burns, M. K., Riley-Tilman, T. C., & VanDerHeyden, A. M. (2012). RTI Applications, Volume 1. Academic and Behavioral Interventions. New York: Guilford. (226 pp.)
Codding, R. S., Shiyko, M., Russo, M., Birch, S., Fanning, E., & Jaspen, D. (2007). Comparing mathematics interventions: Does fluency predict intervention effectiveness? Journal of School Psychology, 45, 603-617.
Daly, E. J., Martens, B. K., Hamler, K., Dool, E. J., & Eckert, T. L. (1999). A brief experimental analysis for identifying instructional components needed to improve oral reading fluency. Journal of Applied Behavior Analysis, 32, 83–94.
Daly, E. J., Martens, B. K., Kilmer, A., & Massie, D. (1996). The effects of instructional match and content overlap on generalized reading performance. Journal of Applied Behavior Analysis, 29, 507–518.
Daly, E. J. III, Witt, J. C., Martens, B. K., & Dool, E. J. (1997). A model for conducting a functional analysis of academic performance problems. School Psychology Review, 26, 554-574.
Deno, S. L. (1985). Curriculum-based measurement: the emerging alternative. Exceptional Children, 52, 219-232.
Deno, S. L., & Mirkin, P. K. (1977). Data-based program modification: A manual. Council for Exceptional Children.
Eckert, T.L., Ardoin, S.P., Daly, E.J., III and Martens, B.K. (2002). Improving oral reading fluency: A brief experimental analysis of combining an antecedent intervention with consequences. Journal of Applied Behavior Analysis, 35, 271-281. https://doi.org/10.1901/jaba.2002.35-271
Gickling, E. E., & Armstrong, D. L. (1978). Levels of instructional difficulty as related to on-task behavior, task completion, and comprehension. Journal of Learning Disabilities, 11, 559–566.
Haring, N. G., & Eaton, M. D. (1978). Systematic instructional procedures: An instructional hierarchy. In N. G. Haring, T. C. Lovitt, M. D. Eaton, & C. L. Hansen (Eds.), The fourth R: Research in the classroom (pp. 23-40). Columbus, OH: Merrill.
Haughton, E. C. (1971). Aims- growing and sharing. In J. B. Jordan & S. Robbins (Eds.) Let’s try doing something else kind of thing: Behavioral principles and the exceptional child. Arlington, VA: Council for Exceptional Children.
Haughton, E. C. (1980). Practicing practices: Learning by activity. Journal of Precision Teaching, 1, 3-20.
Lentz, F. E., & Shapiro, E. S. (1986). Functional assessment of the academic environment. School Psychology Review, 15 (3), 346–357.
Maki, K. E., Zaslofsky, A. F., Knight, S., Ebbesmeyer, A. M., & Chelmo-Boatman, A. (2020). Intervening with multiplication fact difficulties: Examining the utility of the instructional hierarchy to target interventions. Journal of Behavioral Education, 30 (1), 1-25
McComas, J. J., Wacker, D. P., Cooper, L. J., Asmus, J. M., Richman, D., & Stoner, B. (1996). Brief experimental analysis of stimulus prompts for accurate responding on academic tasks in an outpatient clinic. Journal of Applied Behavior Analysis, 29 (3), 397-401.
Noell, G. H., Gansle, K. A., Witt, J. C., Whitmarsh, E. L., Freeland, J. T., LaFleur, L. H., Gilbertson, D. N., & Northup, J. (1998). Effects of contingent reward and instruction on oral reading performance at differing levels of passage difficulty. Journal of Applied Behavior Analysis, 31, 659–663.
Shapiro, E. S., & Lentz, F. E. (1985). Assessing Academic Behavior: A Behavioral Approach. School Psychology Review, 14 (3), 325–338. https://doi.org/10.1080/02796015.1985.12085178
Skinner, B. F. (1953). Some contributions of an experimental analysis of behavior to psychology as a whole. American Psychologist, 8, 69-78.
Starlin, C. M. (1971). Evaluating progress towards reading proficiency. In B. Bateman (Ed.), Learning Disorders, Volume IV. Pp. 390-465. Seattle, WA: Special Child Publications.
Starlin, C. M. (1982). Iowa Monograph: On Reading and Writing. State of Iowa, Department of Public Instruction. Des Moines, Iowa.
Starlin, C. & Starlin, A. (1974). Guides for continuous decision making. Bemidji, MN: Unique Curriculums Unlimited.
VanDerHeyden, A. M., Solomon, B. G. (2023). Valid outcomes for screening and progress monitoring: Fluency is superior to accuracy in curriculum-based measurement. School Psychology, 38 (3), 160-172. doi: 10.1037/spq0000528. PMID: 37184958.
White, O. R. (1986). Precision Teaching — Precision Learning. Exceptional Children, 52, 522-534.
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