Dr. Sharp: [00:00:00] Hello, everyone. Welcome to The Testing Psychologist podcast, the podcast where we talk all about the business and practice of psychological and neuropsychological assessment. I’m your host, Dr. Jeremy Sharp, licensed psychologist, group practice owner, and private practice coach.
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All right, welcome back y’all. Today’s podcast episode is on a topic that we have not talked about on the podcast yet, which is completely surprising given the number of episodes and the ubiquitousness of [00:01:00] this topic.
Today we’re talking all about math disorders, learning disorders in math. My guest, Dr. Paul Cirino is a true expert in this area. This was a very dynamic conversation. We cover a lot of ground and I really enjoyed it as well as taking away a lot of valuable information.
Dr. Paul Cirino is a developmental neuropsychologist whose interests, grants, and published works focus on academic achievement (reading and math), neurocognitive function (particularly executive function and attention), with both typical and neurodevelopmental populations. He has well over 100 Journal publications, and has been continuously funded by NIH, IES, and/or NSF for 20 years. Paul is the Editor-in-Chief of Learning and Individual Differences, and is active in many regional, national, and international neuropsychological societies, including 15 years on the Board of the Houston Neuropsychological Society.
We have a link to Paul’s published works in the show notes. He’s been cited about 10,000 times in Google Scholar as far as that says, and he’s done a lot of work in the field. This is a really fun conversation. We laughed a little bit. We talked a lot about math, which is fun for me, and I hope that you’ll walk away with some new knowledge in this area.
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Okay. I will not keep you in suspense any longer. Let’s get to my conversation with Dr. Paul Cirino.
Hey, Paul, welcome to the podcast.
Dr. Paul: Thanks for having me.
Dr. Sharp: I am so excited to be talking with you because I feel like I don’t understand how I’ve gone to 250 episodes on The Testing Psychologist podcast without talking about math disorder specifically. So I am very grateful that you’re here and excited for our conversation.
Dr. Paul: Excellent. I am too as well. And as we get talking, you’ll see that it’s not surprising math disorders get left behind quite a bit.
Dr. Sharp: Right. Yeah, it is like the redheaded stepchild. Is that an offensive term these days? I apologize to anyone if that was one of those terms I shouldn’t be saying anymore, but that’s the role of math disorders in the LD world. We know so much about [00:05:00] reading and spend a lot of attention on reading but not so much on math.
Dr. Paul: Yeah.
Dr. Sharp: Well, let’s get started with my typical opening question, which is, of everything that you could spend your time doing in this field, why this? Why math disorders?
Dr. Paul: Well, I would say two reasons. When I first became interested in neuropsychology, one of the first populations that I had interaction with was direct syndrome and I ended up doing my thesis and dissertation there. And another graduate student was also focused on that population and was focused on math. And I thought, oh, that’s interesting. I hadn’t thought about math in this context before.
And then throughout graduate school and after graduate school, I worked with several people who were interested in reading and reading disorders. And I started looking at the opposite. It’s like, how come people [00:06:00] don’t talk about math? And sometimes the conversation would veer over there.
Sometimes the occasional study would involve both reading and math. And I would think, huh, we know so much about reading, how come we can’t answer those analogous questions for math? And so it took off from there. And then since I’m a developmental person, I work with kids, all kids have the same job is to go to school. It’s one of the three RS. And so, because it’s so relevant to so many kids and it’s relevant to adults with our money, estimating time, distance, our 401ks, or 403B’s, tipping, our health, medical things, all involve numbers. So it was a lot of synergy for me.
Dr. Sharp: Well, I’m glad that you chose to go down that path because like we said, this is an area that needs a lot of attention. So let’s dig in. Let’s [00:07:00] do a little bit of definition here and terms. Can you briefly speak to just what we mean when we say math issues or a math disorder? What is encompassed in that?
Dr. Paul: Yeah. I would say that some people like to distinguish and have very clear and definitive terms when they say math problems versus low math achievement versus math disability versus dyscalculia versus math disorder. To me, I lump them all together. I would distinguish between an acquired problem. You have an event and now, all of the sudden, you cannot do math. But for the most part, I would consider all of those things to be the same thing.
And to me, whenever I hear any one of those terms, it typically implies that there is low achievement. [00:08:00] The score is lower than you would expect it to be. And that it is not due to poor instruction. It’s information that the person has been taught but is having difficulty actually learning. And that there is at least some modicum of cognitive skills and abilities that provide some base in order to do it. So that’s the way I would… I’m not so concerned with the words or the term that’s used, but to me, it implies you’re struggling to do something that you’ve been taught.
Dr. Sharp: I see. That makes sense. I’m going to out myself right off the bat and just go back to one thing you said which is, do you pronounce it dyscalculia? I always said dyscalculia.
Dr. Paul: Yes, I’ve [00:09:00] always said it that way. Although I can’t remember how many people I’ve heard say it. I always said dyscalculia.
Dr. Sharp: Oh my gosh, this is one of those moments I’m really having a reckoning here in my 40s. Like, oh my gosh. All this time and the wrong word. Okay, well, thanks for clarifying that.
So let’s talk about the different types of math disorders. The DSM, of course, lists different ways that we can be deficient or behind in math. Do those match what we’re seeing in the research or not so much?
Dr. Paul: Yes and no, I would say. In the reading world, a reading disability is most often, most researchers would define a reading problem as a word-level problem that it’s the inability to read individual words.
The [00:10:00] analogy and math is not exactly the same. I would say the closest analogy would be the inability to master your math facts. You and I, we hear 3+5 or 2*5, and we just say 8 or 10. We just do it automatically. Don’t have to think about it. And a lack of mastery of math facts means you might make an occasional error, but what it really means is I might be a little bit slower at doing it than you. So if I were to do 50 of them it might take me 80 seconds and it might take 60 seconds. So there might be a difference in terms of how readily that’s called to mind. And I think that that’s probably the closest equivalent to the basic difficulty associated [00:11:00] with math.
Now, our measures usually measure one of three things. So if you’re familiar with the WIAT, or Woodcock-Johnson, or the KTEA, there are generally three measures. There’s a computational measure. Here’s a bunch of problems. Do them. No words or anything.
And then the second kind is these math facts. Here’s a bunch of math facts. How many things can you do in a minute or three minutes? And then there’s this applied problem which I don’t really know what that is. It’s really everything and nothing. It’s like, here’s a graph. Here’s a picture. Here’s a lion and assign at a zoo. Here’s a time. Here’s a clock here. It’s really all over the place.
And if you think of that in terms of instruction to various degrees, [00:12:00] instruction-wise, kids are taught math facts. Kids are taught computational skills but they’re not really systematically taught this way of integrating this verbal component into it. There might be specific word problems but it’s less so a specific unit that is taught. It’s much more dispersed and distributed than is either the calculations or the math facts curriculum-wise.
Dr. Sharp: Sure. So that begs a question for me right off the bat of, would you consider that to be math skills or not, those applied problems, especially the ones you mentioned like the zoo sign and directions or the calendar. Does that still fall [00:13:00] under the realm of math per se?
Dr. Paul: I think it depends on what age you are referring to. In a young child, that’s all you have is the scaffolded types of math because 3, 4, 5. You’re not really doing too much in the way of written calculations and you’re not expected to have mastered your math facts. So all you have are these basic concepts.
And the way we learn math, like at a preschool level, we infer these informal principles. Each number has one and only one reference and the numbers go in order. We don’t say 1, 2, 5, we say 1 2 3, and kids learn this sequence before they even understand the concept of the number. Just like kids might spell ABCDEFG they might spell out 1, 2, 3, 4, [00:14:00] 5 without really understanding what three means and what three means relative to five.
And then at older ages, once you get into geometry and things like that, you can measure those things directly. And we know that reading and math are related to one another. So to the extent that those items get at the concepts underlying magnitudes and how they relate to one another, then yes, I would consider that math, but it’s a little bit more difficult to look at the pattern of Yeses and Nos, what you got correct or incorrect to try and understand what is the math problem that is happening.
Dr. Sharp: Right. I have a lot of questions that I think we’ll dig into here as we get into [00:15:00] some of the neuroanatomy and what’s actually happening in the brain, but just for my own sake to hopefully cue myself later on and not forget, I would love to talk about the role of language, certainly in math and retrieval of language but also what I think of as math adjacent skills come into play. So, the magnitude, the sense of direction, the visual-spatial skills for lack of a more specific term. So all those things. I just want to kind of bookmark that stuff as we get into some of the neuroanatomy here.
Dr. Paul: Yeah, sure. Of course.
Dr. Sharp: Let’s tap into that. Can you tell us what is happening in the brain when we say math disorder? I know people are like, Jeremy, come on. I know everything is dispersed in specific brain areas. It’s not so much [00:16:00] the right way to think of things. But as much as we can, can we make some generalizations and talk about what’s happening neuroanatomically with math issues?
Dr. Paul: Absolutely. And you’re right. Everything developmentally cognitive neuroscience-wise is all based on networks and interrelationship. And there was no specific math center just like there’s no specific attention center. There are several parts of the brain that are really quite specific and do some pretty specific things. But when you’re talking about a broad functional outcome like math, it’s going to be more distributed.
But having said that, there are certain nodes and certain ways that those nodes are interconnected to one another. The one caveat I would say is that for me even though I’m a neuropsychologist since I focus on kids, kids have [00:17:00] strokes, kids have neurological things that happen to them just not quite as commonly as adults would.
And so, I think there’s one ball game where you’re talking about, you have a stroke and now you’re no longer able to do math or you’ve had a head injury and now something has happened or you’ve had a tumor and it’s in this area. And now you have specific problems as a result of where that tumor is.
Outside of those cases, for me, understanding what goes into math whether we’re talking cognitively or neuroanatomically, I don’t draw all that much of a distinction between math and math disability because if something is important for math disability, it’s also important for math skills.
So [00:18:00] for example, working memory is related to math skills. Working memory is related to math skills when your math skills are weak and it’s related to math skills when your math skills are strong. So it’s not as though if there’s only relation in kids who have math disorders. So the same thing happens with the brain. The nodes are the same nodes that are active in somebody who’s good in math, somebody who’s bad in math, which you might find are different patterns of activation and different strategy usages that impact one or more of those brain areas.
And so having said that as a caveat, I can get back to your question which is, what are those nodes that you’re talking about? And I think that a good way to think about it is what happens [00:19:00] when we’re looking at saying, 52×24 or whatever.
First of all, we’re looking at it, right? So that means information is going to go to our occipital lobes. And it’s going to go from there to our ventral temporal lobes. So right in front of the occipital lobes, but lower in the brain. It’s going to go there because we have to recognize the symbols on the page. Just like we have to recognize words, we have to recognize those numbers, those Arabic numerals that we’re seeing 52. We have to recognize that as a 5 and as a 2 and as a unit 52 before we can do things with it and that’s the case for anything. So those areas are going to be active to the extent that it is something that we need to learn and [00:20:00] association for.
So for me, 5+3 is automatically 8. I don’t think about it. But if I’m in the 1st grade learning 5+3, then my hippocampus is going to be active in the middle part of my temporal lobe that’s important for learning and memory because I have to associate the stem 5+3 with the answer 8 because I don’t know that yet. And I have to repeatedly pair those things until it becomes automatic. Once it’s automatic, that hippocampus activity will go down because you don’t need to make those associations anymore.
Then the information has to get translated into a representation of the quantities, 52 and whatever I said, 38 or something like that. Understanding those magnitudes, areas of our parietal lobe are important for [00:21:00] understanding that specifically something called the intraparietal sulcus which is a pie in the parietal lobes. That’s an area that’s engaged when you’re talking about quantities.
I also need attention. I need to keep in mind which column am I on? Am I going here? Am I going there? What are the steps? And so attention, bilaterally, we need areas at the top of our parietal lobe, the superior parietal lobule to be specific. I also need to keep in mind all of these mental representations. I need to work on them. And so, my frontal lobe is going to be active in doing that as well. And so it’s all of these symptoms of systems or networks.
So we have the [00:22:00] occipital to the ventral temporal lobe that’s important for recognition and the establishment of identity. We have the hippocampus system that’s important for associational learning. We have our parietal lobes which are important for magnitude and for attention. And we have our frontal lobes that are important for bringing it all together and keeping track of where we are in our procedures.
And what we know is as we develop in age, parietal activity goes up, frontal activity goes down. In kids, the frontal activity is broader or larger, higher, whatever we want to use when the parietal activity is becoming engaged. There are two different studies that look at how these things are engaged to varying degrees but all of those systems [00:23:00] are happening at once. And so if I have a math problem, it could break down at any one of those nodes, or it could just be that all of the nodes are weaker or some combination. This node is not very active and then other connections are weaker than they might otherwise be. Does that sound… I’m trying to do it with words because I can’t show you a picture.
Dr. Sharp: I know. No, that was fantastic. I think that was a great verbal description of something that we typically think about happening visually. So thanks for going through that.
It just makes me think about the layers to the job that we do. And I know we’re going to get into the assessment component but the fact that at least for me, I went through grad school and [00:24:00] we were taught basically a discrepancy score model. And there was nothing to say about how a 6-year-old with low math might be different than a 12-year-old with low math or a 16-year-old with low math and the different types of math skills. And it just gives me an appreciation for these kinds of conversations digging into the nuances of the work that we do.
Dr. Paul: Yeah. I find that the more you know about something, you become a splitter rather than a lumper. If we were talking about anxiety and that was my specialty, I could slice and dice anxiety in all different ways. In a way that, well, anxiety is not my area. That’s when you worry, right?
Dr. Sharp: Exactly. Well, I like that we’re slicing and dicing math a little bit because it’s not just the monolith. You touched on certainly [00:25:00] working memory, attention, some other constructs, but I wonder if we could dig a little deeper into some of the neuro-psych constructs as we measure them, that get implicated in math along the way? Can you give any more detail there?
Dr. Paul: Yeah. I would parse the cognitive skills or subsystems or whatever you want to call them, constructs that are relevant to math into two categories. I would start with what I might call domains specific categories. So this is what I would call things that are relevant to math but they’re pretty much relevant mainly to math. They’re, for example, not very relevant for reading. They’re not very relevant for other functionals.
The most [00:26:00] basic domain-specific skills are what various people call magnitude estimation or number sense or counting skills or things along with foundational numerical systems, something like that. And the reason why math is interesting is, and that’s one of the ways in which it differs from reading because reading essentially is stolen from language, right? So we use language for reading.
250 years ago nobody knew how to read. The very small segments of the population but we all had language. And so we invented this method of learning and we co-opted [00:27:00] the systems for language that was already in place to do that. And so conversantly, what’s interesting about math is you have this sense of this is larger than that. That herd of bison is larger than this other herd of bison. We have a sense of that and animals have a sense of that.
And so math is in some ways more unique than reading and that we have some of these basic skills. And early on, that was super exciting to math researchers because we thought, wow, if we have this foundational system, and if we could just tap into that and measure it early, then I could go on to predict who’s going to have math problems later on.
The problem is that those skills do correlate with the math skills, [00:28:00] but the correlation is about 0.2. So we want it to be 0.6. We’d really like it to be 0.6 because then it’s a really tight connection. We can address it. We can do something about it. There’s a relationship but it’s a generally weak relationship. Weaker than we would like.
The other domain-specific skill would be any math skill that you learned up to that point. So for learning fractions, it might be whole number arithmetic. For algebra, it might be whole number arithmetic and fractions. For calculus, it might be whole number skills and fractions and algebra. So as you go up the math hierarchy, all the prior math skills are pretty specific to math as well.
Dr. Sharp: I see. That’s an interesting way to put it. I’m [00:29:00] just wrapping my mind around that. You have to have these prerequisite skills I suppose to continue to develop. Again, another difference from reading in the sense that, once we can read we can read., it’s not like you need to really read better to read more complex material.
Dr. Paul: Yeah. The words get more infrequent, they get longer, they have more syllables, the meaning is rarer but other than that, you’re still reading words.
Dr. Sharp: Sure. Okay. So we have these domain-specific skills that map on the math and there’s another…
Dr. Paul: Yes. The other area would be domain-general skills. And these are skills that are important to math, have been shown empirically in many studies. Hey, there’s a [00:30:00] relationship between this skill and math. Lots of empirical support and also theoretical support. Like it makes sense that this cognitive skill should relate to math.
The thing is, these domain-general skills relate to anything, not anything, but they relate to learning in general. So things like language, things like visual-spatial skill, things like attention, executive functioning, processing speed. You can just easily imagine how those might be important for math.
Like processing speed, okay, well, it’s math fluency. So it’s the ability to call these to mind quickly. Language where we talked about word problems. Well, it makes sense that language would be related. Executive function- working memory is the most common executive function. Well, that makes sense if you have a procedure on algorithms that takes [00:31:00] eight steps, you got to keep them. Which algorithm? What step am I on? What is the next step? It makes sense that it’s related to math but in analogy, as long as we’re talking about reading executive function is important for reading as well. Language obviously is important for reading. So that’s why I call them domain-general skills.
And there are lots of studies. All of those areas that I mentioned, language, visual-spatial skills, attention, working memory/executive function, processing speed, even things like fine motor skills because as kids learn that counting on their fingers are important for math. And I could give you five studies that are a meta-analysis that relate each of those domains to math. So there’s a lot of these domain-general skills that go with our feet forward into our ability to do math.[00:32:00] Dr. Sharp: Right. That does make sense. I had a supervisor a few years ago, I guess a consultant would be the right word that, and I’m paraphrasing here, described math fact retrieval as almost more of a language-based task than a math task. Is that something that you would agree with or not? And if so, why not?
Dr. Paul: Yes. So I would say that it is once it is done. So once it is done, it is not really math. For you and I, 3+5, 2×4 is not really math at that point. It was when we were learning it because we had to make those [00:33:00] associations and perhaps we did it on our fingers and perhaps we used those, I’m blanking on their name those units, those base 10 units, and the blocks, and you might have done that manually. And so at that time, it was maths. We were literally doing the adding, but once it’s done, it’s done. And it’s almost like another word. It’s like bacon and eggs or spaghetti and meatballs. 3+5=8. It’s just an automatic representation. And we’re generally not thinking of the quantity even because it’s so second nature.
Dr. Sharp: Sure. Okay, that’s good. I appreciate that clarification. That’s funny. It’s one of those things that I internalized a long time ago and have repeated many times. Now, this is a good time to probably check that out.
Dr. Paul: Yeah. [00:34:00] It’s also true that if you have math fact retrieval problems that you may be more likely to also have comorbid mood problems or the other way around. If you have a reading problem and your math is awesome, then if you’re going to have math problems, perhaps that’s the math problem that you might have.
Dr. Sharp: I see. That’s helpful. I know we’re going to talk about intervention a little later, so I’ll hold my questions around that. But I just so happened to have an 8-year-old right now who is struggling, my little girl is struggling to learn math facts. And so I’m very curious from a personal and professional standpoint what we might do to help them.
Let’s see. Is there more to say about math and just the development of these skills? I know we’ve talked about how [00:35:00] it changes as kids get older. It’s definitely hierarchical. Is there anything else to say along those lines before we move more into the assessment domain?
Dr. Paul: I would say that the demands of what we’re learning to do in school which is the way it is are going to change with development. And sometimes, that trajectory is linear. Like in build upon fashion, just like multiplication is repeated addition, right? It’s a fairly linear trajectory.
But the difference between whole number arithmetic and fractions, it’s almost like a cliff. They’re just different skills. And the rules of multiplication, you multiply two numbers, the product is bigger. Well, you multiply two fractions together, sometimes the product is bigger, sometimes it’s smaller and you’re like [00:36:00] I have to relearn new things. And sometimes that even gets in the way of what we’ve traditionally learned. And in fact, there’s a name for that. It’s called a whole number bias effect with one of the things that make fractions particularly stubborn to learn.
Dr. Sharp: Sure, it really does. I’ve never really thought of it that way. It does go against that major rule that we learned with multiplication. That’s very challenging.
Okay. Well, let’s talk about assessment a little bit. This is a big focus of our podcast, of course. So let’s start high-level and move down. So when we’re thinking about assessing math issues, at a broad level, what are the components that we need to be thinking of in terms of constructs, skills?
Dr. Paul: I think [00:37:00] it’s not unreasonable to start with where these big batteries begin which is, how good are your math facts? How well can you do computational things? And then how well can you apply those skills? So that is not at all a bad way to start thinking about what are common areas to assess. And like I said, I take that applied group with a bigger grain of salt than the other two domains. But when I’m assessing somebody, I assess those three areas. And there’s a reason why those are the three areas that are represented on most broadband academic achievement tasks because they’re useful and they’re relevant to a lot of things. So I would start there, definitely.
Dr. Sharp: Okay, that sounds good. This is [00:38:00] maybe opening a can of worms but that’s what we do, right? So why have discussions if they can’t be hard sometimes? So how about this whole concept? Do you feel like IQ testing or cognitive ability is important in the consideration of learning disorders?
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All right, let’s get back to the podcast.
Dr. Paul: I’ll give you a psychologist answer and say… and I will say two things that earlier I alluded to. You have some basic modicum of cognitive skills necessary to do the work.
As neuropsychologists, we’re typically focused on assessing specific domains at a level that’s a little finer grain than you find on IQ tests. However, IQ tests are very good tests. Psychometrically, they’re probably the best tests we have just in terms of their reliabilities and they predict a whole bunch of things, and [00:40:00] they do it really well.
So they’re very good tasks for a lot of things. It gives you a good sense of semantic language. It gives you a good sense of processing speed. It gives you a good sense of working memory. So do I need it? I could do a WASI, just a brief general thing, or I could just do a single matrix reasoning. There are some skills there. And I would personally be okay with that. But when you communicate a diagnosis, whether it’s to a school, to a parent, to a testing agency, whatever, they’re going to have their own requirements. And so you may need that information to serve those purposes.
To me though, I’m not really using an IQ test as an IQ test. I’m using it as a collection of cognitive skills that I’m interested in, some of which are more related [00:41:00] to math than others. But the underlying question is, do you believe in the discrepancy criteria, Dr. Cirino? And I would say, no, I don’t. I’m not as big a fan. And that comes actually not so much from the math literature, it comes more from the reading literature where it is very well established that if you are a poor reader, then you probably have poor phonological awareness skills. And whether your IQ is 80 or 100, you still have reading problems. And so you probably need some help with reading.
And I would say that in math, the same is true even if we can’t pinpoint the reason or as much as we would like to attribute it to something easy to compute like a discrepancy criterion, [00:42:00]it’s the way it is. So I’m not a big fan of the discrepancy criteria. And the DSM-V is really looking at underachievement, not just achievement relative to an IQ score. And that’s all the wording of the IDA, must not require, may use terms like that, that are a part of it. And that goes along with the science because just at the basic level, different scores are pretty unreliable scores when two tests are correlated, so they will definitely happen but there’s lots of inconsistency around their reliability.
Dr. Sharp: Sure. So I appreciate you diving into this. I have some follow-up questions naturally. I’m [00:43:00] always curious and I ask everyone who talks about learning issues on the podcast, I force them to talk about this question, which is, how do we identify learning disorders? This has been a fraught topic over the years. So I’m curious about the extremes of the range. So those at the lower end, those at the higher end.
So is it safe to say based on what you just shared that, let’s say we have a kid who has a fullscale IQ of 78 and their math scores are pretty commensurate with that maybe 74, 76, would you call that a math disorder or would you just attribute that to lower IQ, or is it worth distinguishing between those?
Dr. Paul: Yeah, I wouldn’t distinguish between them. I would say this is a child who’s really struggling in math and he’s probably going to continue to struggle with math unless we [00:44:00] do something. Now that child may also be struggling reading and they may also be struggling in writing, but in some ways, it’s neither here nor there. It is an issue.
And I think that the counterargument would be wouldn’t those resources be best used elsewhere for somebody who has a discrepancy? And that is the age-old criteria. But we know from the reading literature, if you teach reading, reading gets better and if you start at the same level, then reading gets better. And so that’s probably what’s going to happen. And we don’t have the resources.
You probably know this. Our schools in this country differ greatly from one another. So it is entirely possible for a school to have primarily high [00:45:00] achieving kids and for the kids who struggle to be the ones who are only a grade above their age level. Those may be the kids who are struggling. They may be failing those courses because the curriculum is so advanced.
On the other hand, you may be at a different school where everybody is struggling and you’re just trying to get by. And so the whole relative thing is relevant in those particular cases, but in one case, I think most people would agree that the latter children that I was talking about are more in need of help than the former.
Dr. Sharp: Sure. And that is a nice segue to the flip side. The other part of this question, kids at the upper end of the range who maybe have a let’s say a full-scale IQ of [00:46:00] 122 but then their math scores are 99, 101. Would you call that a math disorder?
Dr. Paul: I generally would not
Dr. Sharp: Okay. Go ahead.
Dr. Paul: because you can think about the flip side. So if you have two things, one is lower than the other, one is average, and one is superior let’s say, is the superior thing a strength? Because if I know nothing about you and I said, Dr. Sharp’s visual-spatial skills, right? Knowing nothing about you other than your name, I would say 100 because I would just ask the mean for what it is. I’m sure it’s 120, by the way.
Dr. Sharp: Thanks, Paul. 127 actually. Better than that.
Dr. Paul: So you would guess the mean, and maybe it just so happens that you have this really excellent skill in these [00:47:00] other domains. So we don’t say it’s a super strength that you have. We tend to be focused on, well, this skill is not as high as this other one therefore it’s a weakness. And it is a weakness. It is an episoden. It is a relative weakness. But when we say disability, just like when the ADA says disability, we mean a disability relative to the norm group. So if I have a disability of walking or eyesight, it’s not relative to people who have excellent red eagle vision, it is relative to the average person.
Dr. Sharp: That’s a nice parallel to illustrate this point. I appreciate you diving into that a little bit.
Dr. Paul: There is a lot of controversies, and there are a lot of people who would disagree with me on those things. I firmly acknowledge [00:48:00] that.
Dr. Sharp: Well, that’s fair. And that’s why I ask these questions. I’m always curious how people are thinking about this really steeped in it.
I do want to ask you while we’re in the assessment realm about one specific measure that you like, and then two, if you can speak to the, what’s the word, I don’t know if difference… that’s not the best word, but the way that a lot of districts and maybe even school psychologists and researchers are going toward more curriculum-based measurement versus the standardized batteries that we do.
Dr. Paul: To address that, what I would say is, I would start where everybody starts and everybody who’s out there who does testing probably. And to me, it doesn’t matter. Like if you use the Woodcock [00:49:00] Johnson and I used the WIAT or somebody else used the KTEA, I wouldn’t say, well, you’re only using the Woodcock-Johnson not WIAT. I won’t distinguish. I think that if you gave a student all three of those measures, you gave them all the computation skills, other than boredom, you would probably get a pretty similar idea of what their skills are. They’re all well-known. They’re all very well established. They have lots of evidence for all their principal validity going for them. So I would start with those measures.
The thing is, those measures aren’t particularly good at mapping onto the math curriculum. And they get worse as kids get past middle school or even in middle school. In high school, you’re almost talking apples and oranges in terms of what our tests are [00:50:00] measuring and what the student is getting. So I wouldn’t be surprised at all. Okay, your math computation score is 100, you’re failing algebra? I’m not at all surprised by that because they’re measuring different things. How many algebra problems did you attempt on math computations when you’re in 9th grade? One? It’s not surprising that there’s a difference.
For district assessments, I would say there are two kinds and they both have their place. In Texas, we have the STAR, it’s our state assessment. So do you know what the name of the Colorado State Assessment is?
Dr. Sharp: I want to say MAPS but I could be wrong.
Dr. Paul: Yeah. They’re different in every state but whatever it is, it’s these broadband skills that are tied to the [00:51:00] curriculum. And so it’s a measure at the end of 3rd grade or 4th grade, this is how kids do, and it’s used to grade the school relative to one another. And so, those district assessments, I think those are very good tests. They tend to be minimum standards tests. So they’re designed to say, this is what a 4th grader needs at whatever level. And so many of the items are clustered right around that area.
So these tests are really meant to form a perfectly normal distribution that has the tails where it’s designed to detect who’s really good and who’s really bad. It’s designed to detect, did you meet the minimum standards or not the minimum standards? Those tests are excellent across states. Psychometrically, they’re excellent.
I think where they get a bad wrap is the number of times and the extent [00:52:00] to which classes teach to the test and make the focus of the test the end for the curriculum. And then when you take the state test on April 1st, then you have two months of school where not much happens. And that’s the cynical view. And I know that that’s not the case for people who work in schools, that that’s not always the case, but it can be the case. And that’s the bad side. That’s the worst possible case.
The other assessments are what I would call progress monitoring assessments. So whether it’s Renaissance Star360 or aimsweb, DIBELS, or whatever it is, those are great measures too, because they’re designed to say, where’s Johnny now, is he getting better? I’m tracking progress over [00:53:00] time. So they have to be quick, right? So you can’t spend half an hour doing it. The teacher needs to take Johnny aside for two minutes and get an assessment. Is he learning his math facts? And so those definitely have their place as well. And they might map onto our math fact measures but they may not map onto like our applied problems kinds of measures.
Dr. Sharp: Right. So you discuss all of this. That example that you brought up at the beginning of how the curriculum diverges from our measures as time goes on, I think it creates an interesting conundrum for us where we maybe end up with a set of scores from our assessment that doesn’t quite match reality in either direction. It could be kids could do better on our measures. They could do worse than they’re doing in real life. And I’m curious if you have thoughts on how to reconcile those [00:54:00] differences, particularly when we’re trying to… well, actually it could go on in either direction if we’re trying to advocate for kids or if parents are saying, well, no, they’re doing fine.
Dr. Paul: Yeah. Well, the idea of something like the KeyMath which is a diagnostic test that has a lot more Southern areas to it, is geared more towards younger kids but I think it would not be a bad thing if there were more standardized measures of say, fractions or algebra or proportionate reasoning, things like that. I think they would be far more useful.
Sometimes what I might do is give an experimental measure, one from our studies because I understand how it correlates with math. And so I might do something like that in granted it has no norms or anything, but to [00:55:00] me, that’s how I help explain why that discrepancy might be occurring. And so, if a student is in high school and they’re taking algebra, if you’re not measuring fractions and you only have the whole number of arithmetic that they’re doing, maybe they’re failing algebra and they have 105 on the Woodcock-Johnson or whatever. That’s not entirely surprising because we’re not assessing what they’re actually doing in school.
Dr. Sharp: Right. So you would advocate if possible have a more specific measure on hand that might be able to dig deeper into some of these nuanced skills?
Dr. Paul: Yeah. I would almost rather have a word problem measure that uses just basic arithmetic. I might rather have a fractions measure. I [00:56:00] might rather have algebra measure. Things that correspond to what students are learning at various outboxes. And I think those are the main ones, right? Whole number arithmetic early, and their word problem analogs, then fractions and their word problem analogs which could be more proportionate reasoning and then algebra. And then in algebra, you could have graphs and tables versus computation versus a coordinate plain problem. Things like that. I think that we would find much more analogy to how students do on our tests relative to how they do in school if we had such measures.
Dr. Sharp: Right. You mentioned KeyMath a little bit ago. Are there other measures that come to mind that are widely available that might fit the bill?
Dr. Paul: No, there’s not. I mean, there is an algebra [00:57:00] test for college. I think that DTMS or something like that. The aimsweb, for example, has a bunch of different measures that are available. And I think you can apply for a license or so just like with the DIBELS or something like that to get at those things.
And then by any other means, it might be what are they learning? If Johny’s struggling in her 5th-grade class, what is the curriculum like in the 5th grade? Are they spending a lot of time on fractions? What are they doing in fractions? And then I can still look at applied problems or computations and see how did they do the few fractions problems or whatever problems and see if anything can green for that. And it’s less than [00:58:00] ideal, but at least knowing that there can be a distinction between them means you don’t really have to force the issue. I don’t know. It says 100 and so they’re fine. And we’ll just have to leave it at that, right?
Dr. Sharp: Right. When you mentioned just questions that you would ask a parent, that little one-off example made me think about a question that I should have asked a long time ago which is, when we’re interviewing or doing an intake for consideration of a learning disorder, math disorder. Are there any questions that you’re asking parents that fall outside the norm? Like how’s he doing in math or how are the math facts or whatever? I mean, are there any more layered questions that you might get at?
Dr. Paul: Well, I would say that a lot of that layered stuff is built [00:59:00] into whatever my developmental questionnaire looks like, but I think one of the things that’s relevant to me is, especially given states like Texas that aren’t part of the common core and how districts vary widely.
And so, if I’m in my area of Houston, if you tell me what school the child goes to, oh, is that a low achieving school? Is that are high-achieving school? Is that a private school? Is that a public school? Is that a charter school? And so, I’ll have some insight into what their level of the curriculum is and what it’s like. And I think that’s probably the most helpful information is who are you struggling relative to? Are you struggling relative to anybody or struggling relative to a super high achieving cohort?
Dr. Sharp: Right. That’s an important [01:00:00] distinction. In my interviews, I will sometimes ask those questions about math adjacent skills like the magnitude or time or estimating distance or whatever it might be, or maybe some sense of direction, that sort of stuff. Do you feel like that is worthwhile to know or is that extraneous info?
Dr. Paul: I would say it’s most relevant to know the younger the child.
Dr. Sharp: That’s helpful. Okay.
Dr. Paul: Otherwise it could be marker-based skills, but it might not be helpful diagnostically.
Dr. Sharp: Sure. So before our podcast, you were talking about some of the work and the thinking that you do around what do math disorders look like as kids get older, particularly in [01:01:00] college. And I wanted to make sure and touch on this because this was fascinating to me. So could you dive into that a little bit and maybe the trouble with sussing out math issues once kids get to college level.
Dr. Paul: So if you really think about it, all kids who go on to college and not everybody goes on to college, but about half of the people who go onto community college or two-year institutions and the rest go to four-year institutions. Two-year institutions, a lot of them are open enrollment. And so there is a placement test. And if you do poorly on the placement test and you have to take developmental coursework. The same thing at colleges, but relative to the level of the college, how selective the college is, they may not bother because you wouldn’t be getting into those more selective colleges but some of the broader-based public institutions, they may make you take a [01:02:00] developmental course.
And those developmental courses and those placement tests are essentially pre-algebra. And in most high schools in the country, you have to take algebra one and algebra two and pass them. And so you passed algebra one, Jeremy, you pass algebra two.
Now it’s two years later and you can’t do 8th-grade pre-algebra. It doesn’t make sense when you think. It’s this paradox, right? And there are lots of reasons for that paradox, but we’ll stipulate to the fact that our measures are not very good. We don’t have very good measures. The Woodcock-Johnson or the KTEA, or the WIAT, it’s not particular to any one of them are not going to be able to pick that up because you don’t have to answer a whole lot of those questions in order to get an average score. It doesn’t [01:03:00] mean you are going to struggle.
And when you think of what is normal or average for a 22-year-old, you have to include the other 40% of the population that isn’t attending college at. So you have to think of what is a functional life skill. And then we’re talking about everyday estimation, health literacy, health numeracy, those skills that are relatively important that are measured on some like worldwide. One of them is called a PIAAC, right? It’s worldwide and they do it in many countries that look at functional literacy, functional numeracy skills because that’s what you really need to get by. In your day today, you probably don’t need trigonometry for looking at your bank statement or figuring out a tip or something like that.
And so those [01:04:00] skills, it’s hard to say what’s average for those skills because what’s average for an average 22-year-old is different from an average 22-year-old attending a selective university, which is different from an average 22-year-old attending a community college. Those are different normative bases and that’s the basis on which you have to compare. And so the student who’s taking developmental courses and then fails the developmental courses, those are functional consequences. So I would call that a math disability despite a score of 100.
Dr. Sharp: Right. Well, the other piece of that, I suppose, is would you advocate doing away, I don’t know, adjusting the requirements? Why is pre-algebra a prerequisite for college when the majority of folks don’t use those skills ever? [01:05:00] Should we go more toward functional math instruction unless someone is specializing in engineering or physics or whatever?
Dr. Paul: That seems like a different podcast.
Dr. Sharp: That’s fair.
Dr. Paul: I would say that those kinds of algebraic skills, how one unit increases and a secondary unit increases as a proportion of that is relevant to actually a lot of day-to-day things. Gas or fuel consumption or medication usage and cumulative dosage over time or comparing cell phone plans or things like that use ins and outs and algebraic functions to get by.
Dr. Sharp: Okay. That’s fair. Maybe I was overcomplicated. I think about algebra as [01:06:00] abstract and complicated.
Dr. Paul: It is.
Dr. Sharp: Okay. This is good. I’m glad that you called that. Okay. I can’t pull my kids out of math quite yet.
Let’s move to intervention. I would love to talk about intervention. We can take this in any number of ways you would like. It’s a big topic. I might turn it over to you as far as how to organize this discussion around intervention and what works.
Dr. Paul: Okay. So what I would say is we can make some distinction between instruction and intervention. So instruction, being what every child gets, what curriculums should do for every child, like, good, bad, ugly math skills, whichever it is.
And so [01:07:00] on the instructional side, I think that curriculums in the US would probably benefit from more basic skills to mastery rather than my child’s in 2nd grade and I’m doing geometry and statistics and measurement and all of these things. Those aren’t bad things to learn and not bad to be introduced to, but I would say just like with word reading, we know that that’s the way most curricula for readings have gone. Let us teach the phonics-based word reading skills to every child and get them to some level. By the end of 1st-grade, you know the hundred, family words, and you can read 40 words per minute.
I think it’d be great to have similar goals [01:08:00] for maths. Regardless of whatever else you do in the curriculum, let’s make sure that every 2nd grader knows all of their addition and subtraction math facts because there are only 56 of them. It’s defined even more than words. Words are almost infinite in the number of words, but in math facts, there are only so many of them. So I would start there instruction wise and I would advocate for more continuity across states and districts and all of that to emphasize that.
In terms of intervention, and let’s assume that the school is fine, they do a decent job of teaching basic skills, and let’s say, we all agree that Susie has a problem in math. What are we going to do? And let’s say it’s a significant problem. And let’s say at 10th percentile of whatever measure, [01:09:00] computations, some kind of measure. What are we going to do?
Well, it turns out that a lot of the principles that we use for reading also work for math. So not like phonics but for example, a good reading intervention is systematic. It is explicit. It is scaffolded. It has motivational support. It has cumulative review. It has self-regulatory things to keep the child interested. It includes practice, direct instruction. That’s what I mean by systematic and explicit. It follows a chain and you measure progress on an ongoing basis.
If you just switch out the words, a good math intervention does the same thing but with different content. It is systematic, it is explicit, it is [01:10:00] scaffolded and in a way, scaffolding math is almost easier because again, there are only so many math facts and you don’t have to spend an hour. Okay. We’re just going to drill the 9s. 9*1, 9*2, 9*8, and we’re going to do that for an hour. That’s boring to even give. You can do the targeted practice. You can do five minutes of practice every day and things will happen. Things will improve. So you will gain the benefit from that practice.
With math, if you’re weak in math, you have to do math. If you’re weak in reading, you’ve got to read words. I forget who said it and I’m going to steal somebody’s saying, it was on some listserv and they said, there are no therapeutic bank shots. We’re just going to do this one [01:11:00] thing and then we hope it just explodes into everything else. We’re just going to train working memory and then it will have all these trickle-down effects on everything else that is related to working memory. And we know that that doesn’t work.
So I think those principles on the one hand and then cognitive science principles, which are things like the testing effect, scaffolding, interleaved practice works, things like those. Do you want me to say a little bit about those kinds of things?
Dr. Sharp: Yeah, definitely. If you could define each of those, that’d be great. Scaffolding, I think I get but the others, I’d be happy to hear about them.
Dr. Paul: So like a distributed practice would be that you have dedicated practice but it’s not all mass effect work. If you’re going to practice for an hour, it’s best to practice 12 [01:12:00] days for five minutes than it is one hour, one time.
Interleaving would be, I’m going to get you up to speed on your 9s multiplication facts, then I’m going to go to 8s but then I’m going to jump back to 9s. And just make sure that the 9s are still there. And so you do this in a cumulative way. You give feedback that’s incorrect, that’s correct, here’s why, here’s the correct answer. Things like that are all cognitive science principles that work for learning anything, learning a word list, learning words, learning math things.
And I would say for listeners if you go to like the What Works Clearinghouse, or if you go to the Institute of Educational Sciences, they have big broad practice guides that are called that you [01:13:00] can just look up those websites, and they’ll have all of these principles enumerated and there are Metta analytic views of what’s good instruction for math as well.
Dr. Sharp: That’s great. Was that the second resource you mentioned the Institute for Educational Science?
Dr. Paul: Yes, IES.
Dr. Sharp: IES, fantastic.
Dr. Paul: If listeners went to IES and just practice guides, you’d see a bunch of stuff come up.
Dr. Sharp: Sure. So just to be clear, I want to put a fine point on it, it sounds like the best math intervention is just targeted practice at the skill that needs…
Dr. Paul: But it is systematic and explicit and in a scaffold. So it’s not haphazard and it’s building towards a specific goal. So the goal is yes, to learn the 56 additional facts. [01:14:00] Then that’s a defined domain and you can reach that goal but the goal has to be more specific than to do math more. It’s too broad. Is it geometry or like with algebra, what is it? Is it to plot lines? Is it to solve for X? Is it to determine slope? You may have to work on those things separately. So you have to have defined goals and how those goals if you have sub-goals, how those sub-goals relate to the overall goals.
Dr. Sharp: You anticipated my question about those more advanced areas, how do you break down algebra or calculus when you’re trying to teach those skills.
Let’s see. Related to this, maybe one last question. We’ll [01:15:00] see where it goes. But one last question around, we get a lot of questions about where to take kids for math intervention. You’re nodding like you’ve thought about this before. I know it’s geographically dependent but are there any broad strokes to make here in terms of like, start with the school first or go to Mathnasium or how do you handle that?
Dr. Paul: I would say that any of those. Whether it’s KeyMaths on or a Mathnasium or whatever it is, is going to vary tremendously with the specific location that you’re at and who is there, and how tied in they are.
I would say that if anyone of any age needs a math tutor that starting with the school is probably your best bet, whether or not you’re looking for the school to provide intervention services. [01:16:00] If your child’s in 5th grade, sometimes the 5th-grade math teachers will do additional tutoring or a former teacher at that school, somebody who knows that curriculum because most parents will have fairly specific goals. And they’ll probably be happier with their child getting a B in their math class than moving a score from 98 to 105 on the Woodcock-Johnson.
And so because our tests diverge, especially in older students, Oh, how does this high school teach algebra? Oh, I know this former teacher. That’s the kind of tutor who might be needed. Or if it’s a peer tutor. It’s a senior who’s been through there. Depending on how it works, I would start there for people to do those interventions.
There are some [01:17:00] publishable math interventions like Pirate Math and the Phoenix Group. It’s great for like early on learning, 2nd and 3rd grade. And so some of those things are available to be used that are very well laid out. But again, if your specific goals match the goals of those specific programs. Other than that, I would say, start with the school and make sure whatever the tutoring is doing, that it is related to what, and it sounds like, well, duh, but it is true. Just because I’m a mathematician, doesn’t make me a good math tutor.
Dr. Sharp: Sure, that makes sense. Well, that’s solid advice. I will remember that and translate it to my parents here.
Well, I appreciate your time. This is super illuminating. It was nice to take the opportunity to shine the spotlight on math a [01:18:00] little bit. I wonder just to close, are there any resources you might recommend for psychologists out there who would like to get a better handle on math disorders? The neuroanatomy, the research, the assessment, anything is fair game here. And if nothing comes to mind, that’s totally okay too. I just want to give that opportunity just in case.
Dr. Paul: I would start with some of those things that I mentioned. Whether it’s IES or it’s What Works Clearinghouse or just like Meta-analyses and math, you’ll find a lot of good information. You can look me up. I’ve written a lot of stuff, and more is coming out. You can look up other very prominent math researchers. So for example, Dan Ansari, Western Ontario, Lynn Fuchs at Vanderbilt [01:19:00] daycare at Missouri, there’s a lot of people who studied Nancy Jordan in Delaware. There’s a lot of people who have devoted a lot of time and effort towards math and math learning. And a lot of their works are going to be representative of a state-of-the-art.
Dr. Sharp: Fantastic. Well, thanks once again. I really enjoyed it and I hope that our paths might cross again soon sometime.
Dr. Paul: Yes, I enjoy it very much. Thank you very much for having me.
Dr. Sharp: Okay, y’all, thank you so much for listening as always. Like I mentioned, I hope that you are taking away some new information or some interesting information that can impact your clinical work if you work with kids with learning disorders.
As I said at the beginning, if you’re a beginner practice owner or intermediate practice owner, and you would like a little support and accountability and coaching to help level you up in your practice, I would love to talk with you and see if The [01:20:00] Testing Psychologist’s mastermind groups could be a good fit. You can get more info and schedule a pre-group call at thetestingpsychologist.com/consulting.
All right. Thanks, y’all. It’s a pleasure as always. I will catch you next time with a business episode. Bye for now.
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