Ep 3. The Science of Math with Amanda VanDerHeyden

This transcript was created with speech-to-text software.  It was reviewed before posting, but may contain errors.

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You can listen to the episode here: Chalk & Talk Podcast.

 

Episode 3:  The Science of Math with Amanda VanDerHeyden

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Anna Stokke: [00:00:00] Welcome to Chalk and Talk, a podcast about education and math. I'm Anna Stokke, a math professor, and your host.

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You are listening to episode three of Chalk and Talk. My guest in this episode is Dr. Amanda VanDerHeyden. She is an expert in using screening measures and research-based methods to improve math instruction in schools. She is one of the founders of the Science of Math, which is a group of experts that are advocating for evidence-based math instruction.

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This episode will be useful for anyone interested in education, but especially for teachers and parents. We discuss the origins of the science of Math movement and their goals for improving math instruction. Along the way, we address some common misconceptions.  For example, do timed tests cause math [00:01:00] anxiety?Do standard algorithms hinder understanding? Is explicit instruction only helpful for struggling learners? We also talk about what happens when kids fall behind in math, and many other things. I will include links in the show notes to research studies that Dr. VanDerHeyden mentions in the interview. 

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Dr. VanDerHeyden is an engaging and passionate speaker with extensive knowledge and expertise. I really enjoyed our conversation, and I hope you do too.  So, without further ado, let's get started.

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Anna Stokke: I am excited to introduce my guest today. She is Dr. Amanda VanDerHeyden, and she is joining me from Alabama. Dr. VanDerHeyden has a PhD in school psychology. She's the founder of Spring Math through which she has developed academic screening measures and research-based methods for math instruction to be used in classrooms.

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For over 20 years, she has been working with schools to use [00:02:00] data to become effective. She has published more than 100 scholarly articles and seven books, and she's a frequent keynote speaker on effective math instruction. Dr. VanDerHeyden is one of the founders of the group the Science of Math. This is a group of psychologists, cognitive scientists, and math educators who are advocating for evidence-based math instruction, and we will hear a lot more about this movement shortly.

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Welcome to my podcast!

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Amanda VanDerHeyden: Thank you.

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Anna Stokke: So, let's start and talk a bit about your background and, and your career path. Can you tell us how you ended up working on improving math achievement in school systems?

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Amanda VanDerHeyden: Well, that was the problem I needed to solve, right? So, I mean, I've always worked in what we now call multi-tiered systems of support, and I got to do that in the late nineties in graduate school where Joe Witt was my mentor at LSU and Baton Rouge. And Joe had this wonderful idea [00:03:00] that maybe instead of spending all this investment and time and resource to meet and talk about why children were failing toward a path to make them eligible for special education services, which didn't always benefit them.

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That's what we were seeing data about. He had this wonderful idea that, well, what if we did sort of a functional analysis of the classroom environment and we could understand, does the child really stick out like a sore thumb from his or her classmates in terms of academic risk? And if not, perhaps maybe we should intervene at the class level.

 

So, we should be improving what we're doing every single day in terms of instruction. And then for children who don't thrive, when we do that, we could intensify the support that they get. And that ruined me really for the type of, you know, IES, you know, five-year research programs because I fell in love with this.

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Build it, test it in real schools. Break it, then [00:04:00] rebuild it the next day. So, it was this beautiful phase of training for me where I could do this work that truly caused children to learn to read and teachers to remain in the profession and get wind in their sails because suddenly they could teach children how to read.

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And so that was really how I cut my teeth. And I did, you know, my research was sort of disseminated broadly from my dissertation, which was a study of response, what we now call response to intervention. And from there I ended up, you know, in a district in Arizona where I was a leader and they hired me as a school psychologist.

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But I was really bringing this new process of RTI to their district. I was responsible for it across the entire district. And when we screened, we found a math problem. So, I go back to the leaders, this is 2002, and so they've just gotten their first year of accountability testing in hand, which is a very threatening place to be.

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This was brand new. It wasn't [00:05:00] even probably a very good test at the time. And I said, “well, there's a math problem.” And the leaders at the table with me, my colleagues said, “we kind of knew that - that validates what we were worried about. What are we gonna do about it?” And I said, “well, there is so much risk. I don't know who needs individual intervention.” And so that's really where I began to build class-wide intervention. 

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We rolled this out across this district. It became the first experimental study of RTI, which we, I published in 2007 in Journal of School Psychology.  It is still my most cited article. And in that article we reported findings like: The number of children made eligible under the category of learning disability went from 6% to three and a half percent in the first year; their achievement in math went from middle of the pack, 50th percentile, roughly in the state to rank order position one, and there they have pretty much remained ever since.

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And what was sort of stunning [00:06:00] about this to me, I, I'm still stunned by this when this happens, is it really wasn't that difficult. It was really a matter of doing the things that we've really known about since, you know, the 1980s, the 1990s, maybe even a little bit earlier. But doing them at scale, doing them thoughtfully and taking action based on, on what we learned and really letting the children's response, which we could characterize and measure in all kinds of ways, guide us as to what we needed to do next.

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So, our eye was always on the prize of being very student-centered, and we were able to move these big milestone indicators, like I gave you two examples. There were lots more too along the way.

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Anna Stokke: Well that's, that's really, really great. It sounds really promising. And, and let's come back to that sort of thing a bit later. I wanna ask you about the science of math. And first of all, what does that even mean and why did you start a science of math group? And, and what do you [00:07:00]think your group will accomplish?  What do you hope to accomplish?

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Amanda VanDerHeyden: Yeah. Okay. So, I mean, we understand that it's a little bit of a provocative name. We knew that, we totally co-opted what the science of reading kind of momentum that was happening. But here's the thing about that. We are all, it's a small circle. in the, in the, as you know, Anna, I mean in the world of like people who do scholarly work related to helping children learn better, helping systems instruct children in ways that will produce better outcomes for children.

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That's a pretty small community. And what I noticed is what began as this advocacy effort - the science of reading - very rapidly became bolstered and supported by my colleagues who have been doing research in that world. They just study reading instead of math, but they do the same type of work.  In fact, I published with lots of them many times.  And what we saw happen is those folks got on board and they were really [00:08:00] able to create this synergy between the research world and what parents were asking for and even what teachers - frustrated teachers - were asking for in systems.

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And we saw this potential to really be of benefit to children. And at the same time, you know, I've been doing this work -  I need to update my bio cuz it's actually closer to 30 years now. I mean, it started in 1996, so it makes me old. But, you know, I think what happened is myself and some Sarah Powell from Texas and Robin Codding from Massachusetts, we were talking to each other and I'm very close with Robin.

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I didn't know Sarah that well.  I, of course, I know her work and totally admire her professionally, but, but we're not friends or close, you know. But the three of us began to just sort of exchange emails. And it's funny because I was saying to Sarah, “Hey Sarah, you really should start this. Like, you gotta start this science and math thing.”

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And then Robin would email me and [00:09:00] go, “what are we gonna do?” We can't, like, and what we were experiencing collectively is going out into the world doing research, running system improvement efforts, which we have all done at scale many times. Even though, you know, I've certainly seen it said about me that I have zero instructional experience in classrooms.

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I mean, that's absolutely not true. I've only ever worked in schools and have been responsible for that at a district level and coached, you know, hundreds of teachers. But, but anyway, the point is we were looking at each other, having this collective frustration that we would show up to do our work, and we know that these tactics will benefit these children.

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And we could not get off of first base, we could not move forward because we had to answer the same questions. And the questions really came from these very common misunderstandings about how children learn and what effective instruction consists of. And that was just that was the impetus, was this collective [00:10:00] frustration.

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And Sarah Powell, I would say is really an, been an important leader for this group. I hope she doesn't mind me saying that because she really did organize us. She's the one who said, “all right, let's finally, let's do this.” And I hope we continue to do more, and that you see more work from us. I, I think some people are, pushing back a little bit and they're saying, “well, why do you get to pick what the science is?”

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And sort of the classic things that we also heard when the science of reading started. But you know what? Everyone is welcome, but everyone must, you know, be willing to sort of check your philosophical biases at the door and be open to what happens when you use an instructional tactic. And if you can use it, if you can, then you have to be able to operationalize it.

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You have to be able to evaluate it. And I don't wanna hear this nonsense about, “well, that's because your measure is not valid,” or “you're not measuring the right outcomes.” Well, then let's talk about that because [00:11:00] every outcome worth attaining can be measured.

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Anna Stokke: I think you're absolutely right about that. And I actually think it's really great that you, that you started this and, and reading does get a lot of attention as it, as it should, but math also needs attention and we do need to work on that and, and getting rid of some of the misconceptions out there because, we need children to learn math. It's just very important. 

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Amanda VanDerHeyden: It's pragmatic. I, I fell in love with your work, Anna, because we were on that panel together and I just remembered you talking about teaching at the college level and having these children land in your classroom who did not, never mastered the fundamentals of algebra.

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Anna Stokke: Oh, definitely. And it starts really far back. And you talked about people sometimes saying, “oh, you know, you don't have the right experience” and, and that sort of thing. And I get that too. And I'm a mathematician, I teach math all the time. And I teach at the university level, but a lot of [00:12:00] the problems start further back because math is just kind of like a, a ladder, right?

 

So, so that's why I got involved in, in advocacy as well. It's just, it's really sad to see. And then I also think about some of the students that didn't get proper math instruction, and they never even get to me. Like they, they would never even have any hope of going to university. 

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Amanda VanDerHeyden: It's un - it's unconscionable. It's, I mean, it's, it's educational malpractice every child. I say that all the time, like every child doesn't have to go to college necessarily. But every child should have the opportunity to do what they want to do in life and not have their lack of math proficiency be that barrier that keeps them from attaining their goals.

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Anna Stokke: Absolutely. So, let's dive into this and, you have a website, it's called www.thescienceofmath.com, and I would encourage people to visit it. And on that website there's a list of misconceptions. And let's talk a bit about those if you don't mind. So, and let's start, and I haven't talked about this one [00:13:00] yet, on, on the podcast and, and it's a big one, but a lot of people think that time tests cause math, anxiety.

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So, some, some people think we shouldn't give students tests at all actually. And then some people are okay with tests, but they don't think we should have timed tests. But what do you think about that? Do, do timed tests cause math anxiety? What does the evidence actually say about that?

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Amanda VanDerHeyden: Well, the answer based on the evidence, and you can always collect more evidence and we should have systematic replication and collect more. And a lot of people are doing this work now. But the short answer is, the answer is no. That is an oversimplified con conceptualization of, of anxiety with regard to math performance.

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Okay. Because what we know from the research is that there's a bidirectional relationship between skill proficiency in math and the experience of anxiety. And we can define that. We can measure that we, and there's great people doing this work. [00:14:00] There are only really a handful of studies that have, I mean, when I say handful, it's not even a full hand.

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It's like, you know, three or four studies that have been published so far that have actually thoroughly experimentally evaluated this question. And two of those studies found benefit to timing. So, it's not, it is not a foregone that timed assessment causes anxiety. But it's so interesting that it's not just a timed assessment, timed activities, period.

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It's so interesting to me that this resonates so heavily, heavily with people. It kind of reminds me of the old learning styles, right? Visual learners and auditory learners, which we knew were nonsense for years. We knew for decades, and you can still have somebody show up in your district and teach your teachers how to use this thing.

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That's not true. That never has been validated, and education is kind of ripe with those examples. Now, the consequence of taking this on face value because you read it in a popular book or [00:15:00] you heard it on the news or you talked about it at a cocktail party, the consequence is actually not benign. It's actually quite deadly to proficiency for children because what you do is you cause teachers - who actually love children - this is why they work in this space to avoid using timed activities because they don't wanna provoke this thing that they think is bad called anxiety for children. I totally respect that. 

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The problem is, as a psychologist, I will tell you that, you know, avoidance exacerbates anxiety. So, there are no effective anxiety treatments that do not involve exposure. So, the key for teachers to hear is that when you take time to practice off the table, you are in a land that is just not supported by policy or by evidence. And I will say, like, you know, I, I try to talk to folks who I would not say are on the same page typically with the work we are trying to do.

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And I [00:16:00] I will say, they often acknowledge, yes, you should do fluency building activities. And of course, N C T M in the adding it up report was very clear on that. Pitting fluency against conceptual understanding is, is not the way to go. It's not evidence-based, but the problem is you have these thought leaders who have, who do professional development at scale, who publish other thought pieces, not based on research.

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Or a misinterpretation sometimes of the research and say this to teachers. And what teachers take away is, “oh yeah, timed activities cause anxiety.” Well, if you do not conduct timed activities in your classroom as a math teacher for some portion of your instructional period, you are actually attenuating and worsening mathematical mastery for your students.

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And I use a graph that I show all the time to explain why we must have some timed [00:17:00]engagement with children. It doesn't mean we must always be timed, but when children reach a hundred percent accuracy, there is more proficiency to be gained. That's actually very meaningful, and that is the proficiency that is gained through fluency building activity, meaning it's high quality practice.

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It's not your mama's drill and kill. You know, it's not your old school drill and kill. It is thoughtful, engaged, high dosage of opportunities to respond where what you are actually doing as a teacher is you're making the work easier for the student to do. So, they're no longer having to really think about how to get to the correct answer.

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It becomes instant, easy, I'll say automatic, but I don't shy away from the word fluency either because that has a real behavioral definition since BF Skinner and that's what we use. So, but when you leave the fluency building on the table and you just get kids to a hundred percent accurate, what we will tell you is if you move the instruction forward into more challenging content, [00:18:00] you end up with these college kids that land on your doorstep who are not even, they can't remember how to solve the single-digit whole number operations.

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Anna Stokke: And I think that that kind of leads me into the piece about conceptual understanding, but I mean, it's great and everything if, if you can figure out what five times four is by adding up five, four times. But you do not want to be doing that when you're solving an algebra problem. And that's where the fluency piece comes in.

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And when we're talking about fluency building, like what kind of activities are you talking about? So, are you talking about giving students, timetable drills every morning and, and timing them? Or, or what, what exactly do you mean by that?

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Amanda VanDerHeyden: Well, okay. So, the way I've always taught it, and let me say too just before I move on, the person to look at for good data and I was just on a panel at National Association of School Psychologists. I was the discussant and every [00:19:00] panelist was presenting data on this question you just asked me. All right.

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And it was Robin Codding, and I'm gonna blank on Robin’s, student’s name, which - shame on me - but also another person, but Katie Maki - m, a, k, i, who is at University of Florida. She and Anne Zaslofsky, who is in Minnesota, they actually have an experimental, randomized control trial evaluating this question for which the answer is a resounding no - that timed activities do not cause anxiety. So, it's not out yet, but please pay attention to these data. 

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And, and Robin Codding, if you Google her and you Google math anxiety, she's written about this. She has also done, she did a wonderful webinar for us. It's a one-hour watch. I would show it to your faculty, so if you want, you can email me or I can send you the link and I'm happy to, to share that link so that you could, people could share that with their faculty if they wanted to.

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So, math anxiety now. So, what does high quality fluency building look like, right? If I said it's not drill and kill? Well, it really comes [00:20:00] down to, you know, teachers can aspire to arrange games and that is a wonderful way to conduct fluency building. But if you don't have time for that as a teacher, there's nothing wrong with simpler formats like response card interventions, even flashcard interventions, playing games, like we call it War.

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Some kids call it Top-It where you're identifying the greater quantity. Those are typically very easy to set up, but you build facility and fluency with understanding of quantity, which benefits the child when they work with within operations with those quantities and different formats. But yes, it could be that you're using a worksheet for that practice, but the key there is the following.

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So, it has to be a task that has been selected to reflect the, the children's rough for the grade or for the class, their instructional level of difficulty. I mean, this goes back to Gickling in Armstrong 1978, Herring and Eaton, 1978. These are [00:21:00] not new ideas. But the idea here, and this is what I think teachers profoundly lack this understanding, and I, I think it's the most important educational innovation in a hundred years.

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It's called the instructional hierarchy. So, when you understand that as a teacher, you know how to plan, you know, I would say one third, one third, one third of your math lesson, and one third should be devoted to high quality acquisition of new understanding. Okay. One third should be devoted to high quality fluency building instruction, and one third should be devoted to generalization and adaptation activities that you can create.

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Teachers love to work in the generalization and adaptation phase. They're willing to work in acquisition, but they don't always get every kid there and then they often skip the fluency building piece altogether. But the way this magic happens instructionally, and it's very powerful, like I really believe some of these colleagues who so vociferously disagree with me on Twitter when they could understand, I'm not [00:22:00] saying don't give kids challenging problem types they've never encountered and allow a process of inquiry or allow a process even of - shocker - productive struggle.

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That's actually okay with me. I think the mistake is when you use that in acquisition instruction. It's not appropriate in acquisition instruction. That causes anxiety. Robin and I have a paper on that. It's not out yet, but it's the, it's under review. But those, those particular activities, let's say productive struggle as a controversial exam example, I don't think it's that it works or not.

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I think it's, it works. When does it work? Under what conditions can it work? And my answer to that is when children are in the generalization and adaptation stage of learning, and this is what teachers don't have good systems to pay attention to. Or even the training to know, to pay attention to it, because it's not like it's that difficult.

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So, the idea is if you're introducing a new concept, children are not accurate. They're in an acquisition stage of learning. The [00:23:00] instructional tactics that you want to use are things like worked examples, a model, immediate corrective feedback, very well controlled task problem types, so that it's not, you know, you're, you're very specific about what you're trying to train, and you're not creating misunderstandings because of your sloppy presentation of the task.

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Okay. And then when children become very accurate and independent, they're ready for fluency building and fluency building is for an instructional level skill. That means children have acquired it, they are accurate, they can do it independently, they can explain to you how it works. Now they're ready for fluency building.

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And that could involve you know, a variety of things. It can be, this can be done with games, this can be done with, like I said, even like flashcards, response cards. Yes, even worksheets. As long as you've chosen the right one and you have enough problems that they're not gonna run out of problems to do before the time is up.

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And there should be a period of error correction and verbal rehearsal and think aloud, problem solving associated with that. Because [00:24:00] we're always building conceptual understanding. And then when children become accurate and speeded, which is the definition of fluency, then they're ready for generalization and adaptation instruction.

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So, as a teacher, what you have to take away from this is, in any given day, children cannot use, you cannot give a child acquisition, fluency building, and generalization instruction on the same skill. So, acquisition instruction is for new understanding, fluency building is for something they've learned how to do already, but it's, but it's hard.

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It is labored responding on their part. You can, you can know this based on well-published criteria. I mean, I'm happy to tell you what they are, but that's pretty detailed. And then generalization instruction has to be for a skill for which children have reached mastery. And I think what happens is many times it's not that teachers are using ineffective tactics, period. It's, they're using them at the wrong time.

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Anna Stokke: And about the fluency [00:25:00] building and, and the fact that maybe a lot of teachers don't spend enough time on that or skip that step, I think it’s because they're told not to do it. They, they're, or they're made to feel guilty. The principal walks past the classroom and what looks good to like good instruction to them is kids in groups talking loudly, playing with things, not kids sitting down and actually working out problems.

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But it, but it's a really important part of mathematics. Practice is what makes you good at it in the end, like in the beginning. Yeah. You have to know what things mean and you have to, you have to be taught properly, but you have to practice to get good at it or you'll forget it. 

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Amanda VanDerHeyden: Right. I mean, I think, I think it's fair to say, and I know this is provocative, and I know I, I say this in the most respectful way possible, but I think when you are part of a community, like for example, math education, that is disconnected from what mathematicians advise with regard to how to be [00:26:00] masterful in mathematics, I think you have a problem as a field, I think you should be coming to the table and having more conversations about, what that can look like, right.

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Or what that could, what that maybe should look like to bring greater benefit to children. But instead there's, there's an awful lot of influence from folks who are not really paying attention to the science.

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Anna Stokke: I think you've, you've gotten that exactly right. So, I wanted to talk a little bit about standard algorithms. So, a standard algorithm for the listeners, if you're not sure, cuz this might sound like jargon. Say if you're adding two three-digit numbers, you would line them up, and you would line up the ones column and the tens column and, and the hundreds column and, and you'd carry if necessary, et cetera.

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And it was interesting, and this is one of the reasons I actually got sort of involved in, in math education advocacy in the first place. My kids were in school and I was sort of shocked by what was going on. [00:27:00] Instead of teaching standard algorithms children were taught to calculate using a variety of strategies.

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Usually they were actually told not to use a standard algorithm. And so, for instance, using blocks to add numbers or separating the mode and, and doing these, sort of, what I thought were very convoluted methods for arithmetic. And I've even heard. people say that teaching standard algorithms is harmful to children, that it works against understanding, which I don't agree with, but I'm curious what you think about that.

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Do you think there's any evidence to support this idea that standard algorithms are harmful?

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Amanda VanDerHeyden: Oh, it's completely outrageous. There's no doubt. It's, it's a reaction to probably what many of us experienced. You know, I'm, I'm 50, so when I was in elementary school, and we are primary school, we were taught that you know, we were sort of taught the algorithm without any teaching around how it [00:28:00] worked or why it worked because there is beautiful, coherent logic that underlies the algorithms.

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And so to shy away from teaching them is like saying, “I'm gonna teach you to read without teaching you how to decode.” I mean, algorithms in math are decoding in reading, so it is just a profound, which we saw happen in the reading world. I mean, this is why the science of math is so important because the very same misconceptions are actually causing children harm.

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So, by far the, when you, when all fails, if you understood how that algorithm worked, then you could go back to that as somebody two or three years out from the initial instruction and you could get to a problem solution that works. And instead what we have done is exactly what you said. And it's, it's really, again, you know, when you're trying to establish acquisition and we're gonna - here's a jargon term - we're gonna say in, in the [00:29:00] behavioral world, that the goal then of that instructional episode is discrimination.

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But you fill it up with so many different ways to get to a solution, that actually worsens understanding.  It reduces efficiency of instruction and it can actually cause misunderstanding. So I think it's like a pendulum swing because moving away from sort of algorithm only instruction without any kind of explanation, you know, like the famous one to me is dividing with fractions and you'll hear teachers say things like, not much anymore, but you used to hear teachers say things like, “yours is not to reason why, just invert and multiply.”

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We don't want children to think of math as a bunch of tricks. There is a real reason that that works. So, we want them to understand that, but we want them to know how to use that algorithm because it is the most efficient, effective way to get to a correct solution.

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Anna Stokke: What you said really resonates with me [00:30:00] because you don't wanna just abandon the algorithm if, if it's understanding that you're worried about teach students why the algorithm works. That's not to say that they have to explain it every single time they use it. 

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Amanda VanDerHeyden: That's right.

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Anna Stokke: But, but, but we do need to tell them why things work. Everything in math actually does have a reason, and, and it can be explained and we need to explain those things when it, when it's something that a child can understand and, but they do need to work on the fluency building as well, and, and move past that stage where they're constantly thinking about understanding too.

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Amanda VanDerHeyden: Yeah, I mean, you know, in my work, my tool is called Spring Math and and I built it and it took me 20 years and I really built it because I was trying to train systems and help teachers do this work, and I felt they needed better access to materials to help them do it. I have 550 individual scripted interventions in math, covering about 145 skills from numeracy to algebra. And I wrote them all. [00:31:00] And so, you know, if, if you say, “oh, it's just procedural, it's just drill and kill,” then you haven't read these lesson plans because we always teach how and why an algorithm works.

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We are showing children, you know, how to, allowing them to practice solving on a number line, creating equivalent quantities, solving for an unknown. But every single day we also work on procedural skill and understanding. And the important piece of that, it's like you said, I saw this with my own kid too, you know, he would come home and what he had learned was a much more complicated path to a solution, which I could tell you right away - he's not gonna remember that in, you know, he is too much of a novice learner.

 

It looks cool to a teacher, but a teacher has 20 years of math, learning and experience and practice. So it, it's just a, it's just a misalignment of advice based on what makes sense maybe [00:32:00] to adults who have greater proficiency with regard to how children learn and move forward.

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And again, the, the thing about this is that misunderstanding is not benign. It really harms children going forward.

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Anna Stokke: So, another misconception you talk about on the website is, surrounding explicit instruction and by explicit instruction - some people may think of that as sort of more a traditional approach to instruction - where we, we actually explicitly teach people and you know, so the teacher, the teacher actually teaches the students the concept, gives feedback and, and works with the students with guided practice.

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So, there's a misconception out there sometimes that this only works for struggling students and that discovery-based learning or inquiry-based learning is actually better for most students. What does research say about that?

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Amanda VanDerHeyden: Yeah. That's just a profound mischaracterization of the wonderful [00:33:00] data that we have on this question. Okay. Like, I mean, so yeah, there's a misperception really, I think, when people say explicit instruction, which became a term because direct instruction was way too heavy. It sounded too heavy handed.

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So, the PR makeover was to call it explicit instruction. But then I think people think it's heavy handed. It is a teacher standing in front of the class lecturing the children. It's kind of like, you know, if you, if you ever watch the Christmas story at Christmas time on TV, I love that movie. It's sort of that what, like the way that teacher would teach.

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But what we know is actually, explicit instruction is about finding the right level of instructional difficulty, selecting the instructional tactics that are aligned with what the learner needs via the instructional hierarchy, and then delivering those things in a sufficient dosage so that children can actually progress.