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Ep 21. The right to learn math with Daniel Ansari

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

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

Ep 21. The right to learn math with Daniel Ansari


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


You are listening to episode 21 of Chalk and Talk. My guest in this episode is Dr. Daniel Ansari, who is a Canada Research Chair in Developmental Cognitive Neuroscience. I first met Daniel over ten years ago, and I'm a huge fan of his work. I think you'll find, as I do, that he's very knowledgeable and sensible.


His passion for children is evident, and he stresses the fundamental right of every child to learn math. We discuss the importance of early numeracy skills and his research on symbolic representations. We talk about similarities and differences between math and reading. We also debunk the misconception of gender differences in math. Spoiler alert, there aren't any, and girls don't require different teaching approaches than boys.


Responding to a question from a teacher, Daniel discusses the role of manipulatives in teaching math. I also asked him to tell us about dyscalculia and to provide some tips for helping students who struggle with math. You'll want to stick around to the end because I asked Daniel to tackle some neuromyths. I know you're going to love this episode.


Now without further ado, let's get started.


[00:01:41] Anna Stokke: I am thrilled to introduce Dr. Daniel Ansari this morning, and he is joining me from London, Ontario. He is a professor and Canada Research Chair in Developmental Cognitive Neuroscience and Learning at Western University, and he heads the numerical cognition laboratory there. He has a Ph.D. in child health from the University College of London and an M.Sc. in neuroscience from the University of Oxford.


He has over 150 publications and book chapters. He is a frequent keynote speaker all over the world and to a wide range of audiences. He studies numerical and mathematical skills using both behavioural and brain imaging methods, and I'm really excited to talk to him about that today. Welcome Daniel. Welcome to my podcast.


[00:02:33] Daniel Ansari: Hi, Anna. Great to be here. Thanks for having me.


[00:02:36] Anna Stokke: I've had various psychologists on the podcast, so I've had cognitive psychologists and school psychologists. Can you explain what type of psychologist you are and what kind of research you do?


[00:02:50] Daniel Ansari: Happy to do that. So I think I would say that I study child development and that I use a variety of approaches. They range from, you know, traditional experimental psychology, cognitive psychology, but also cognitive neuroscience to get more at the level of the brain. What unites all these themes is a focus on child development.


[00:03:10] Anna Stokke: We hear a lot about reading these days, and I think that most people understand that, and we agree that learning to read is very important. What about math? So, how important are early numeracy skills? Do we know anything about the predictive nature of early numeracy skills? Do they predict later academic achievement?


[00:03:34] Daniel Ansari: Yeah, absolutely. So there's lots of good longitudinal evidence to suggest that early numerical skills as early as, you know, four years of age do predict later math outcomes. We have to be a little bit careful when we talk about this evidence because it is correlational and so we always need to always also look at the intervention effects, and we do have some evidence to suggest that if you intervene early, that is if you foster numerical skills early on, that has downstream cascading effects on the development of higher order mathematical skills.


But we also need to recognize that we can't just focus on the early skills. We have to have a model of sustained support for the development of mathematical skills throughout a child's educational trajectory.


[00:04:20] Anna Stokke: How can neuroscience play a role in studying how children learn math?


[00:04:26] Daniel Ansari: That's a great question. I think neuroscience is one of many different approaches we can take to understanding children's development of numerical skills. We shouldn't think about neuroscience as sort of sitting at the top of a knowledge generation hierarchy but as one tool that also has a lot of limitations.


But I think, if we really want to understand developmental processes and processes of learning holistically, anything we can look at - is it context? Is it the environment? Is it genetics? Is it the brain? How can we understand these things? But when we study neuroscience, one of the things we also, I think, really need to embrace is the concept that, of course, the brain and how it is processing information is dependent on the environment, which a child finds themselves.


So we need to sort of distinguish between cause and effect and understand that if we see a difference at the brain level, it doesn't mean that that's a biological difference. It could also be a difference in exposure difference in learning environments and so forth.


[00:05:31] Anna Stokke: So maybe we can talk a little bit about your specific research. I understand that you have done some research on mathematical symbols, in particular, our numeral system, our place value system, the Hindu system. So those are the numbers as we know them, like the number three, four, that sort of thing.


What have you done in that area? And how important is it for children to become fluent at recognizing and working with Hindu-Arabic numerals?


[00:06:04] Daniel Ansari: Symbolic representations are incredibly important for early math learning, and of course, children learn number words, which are also symbols, before they learn Hindu-Arabic numerals. And most of the evidence says that once children have an understanding of number words, it's very easy to transfer that understanding to Hindu-Arabic numerals.


And so, when we think about math education, I think one of the things, or math learning in general, one of the things that's really important to bear in mind is that children are always along a trajectory from concrete to representational to abstract. So, you know, when children start out learning about numbers, they will work with objects and later on they will work with hatch marks on a page, and then they will eventually transfer that to symbols.


And we can also move back and forth on that continuum, right? To, not a necessary, purely linear process, but a process where we go back and forth as well. So most of the evidence that we and others have gathered suggests that symbol learning, and especially Understanding the meaning of number words early in development, is really sort of a critical building block, if you like and that if you lack that understanding, that's going to have carry-forward effects.


So let's say you do have a rudimentary understanding of numerical symbols, but you're not really fluent in comparing symbols or ordering symbols. That's going to make it really difficult for you to do any kind of arithmetic, for example. So then it might look like you have a working memory difficulty, but really what you have is a difficulty in representing quantities and having a fluent understanding of the relationships between numerical quantities.


So I think it has important downstream effects and it's one of those milestones like maybe phonics and reading that, you know, children really need to be strengthened and given the opportunities to develop that fluency so that the downstream effects are positive rather than negative.


[00:07:55] Anna Stokke: This makes a lot of sense to me. I used to teach a history of math class. When I first started at the university, so I know a fair bit about the history of the development of the numeral system we use today, which really originated in India. It was kind of surprising to me when my kids were in school because there was this heavy focus on using manipulatives and even as a method actually for doing calculations, which seemed kind of absurd to me because the whole point of our numeral system is it's really easy to work with.


What I saw going on, it kind of almost reminded me of Roman numerals, which really held back the Europeans for many years in terms of complex mathematics and scientific discoveries because they're really clunky and difficult to work with. So what you're saying about symbols and moving on to symbols, not that we're saying that you shouldn't do the concrete materials.


It makes a lot of sense to me and, and it seems like a really important thing to me. But sometimes I feel like there's maybe a bit too much emphasis on the manipulative piece. And I'm wondering if you can say anything about that. Like where do manipulatives fit into math instruction?


[00:09:16] Daniel Ansari: That's a really, really important question. I think, for me often when we talk about educational topics, such as the use of manipulatives, we get too much in, in sort of a stance of either or. I think for me, it's important to think developmentally. And when you think developmentally, then you recognize that of course manipulatives are important early on, but you do want to get away from them.


Eventually you do want to move away from them. And then when we think about manipulatives, we also have to think carefully about what manipulative. So for example, if we take a stance where we say, let's make it fun for the children, you know, let's give them lots of colourful manipulatives, every one looks different.


Well, that's, that's going against all the research that we have that says you need to strip your materials down so that children can actually focus, begin to focus on the abstract properties of the sets rather than things that have to do with the appearance of the sets. So there's lots of research to show that, for example, young children are better at matching dots to dots than they are at matching, let's say, a pile of shells to dots because shells and dots look differently.


So we have to think about the nature of the manipulatives that we employ in classrooms. And we have to, when we think developmentally, always use manipulatives as the first stage towards an abstract mental representation of number.


[00:10:41] Anna Stokke: What you're saying then is if the manipulatives are too colourful and distracting, that actually maybe that could be detrimental. Is that correct?


[00:10:52] Daniel Ansari: Absolutely. We have the research to show that and it's, it's not just in, in math, it's in science education as well. There's lots of research, you know, our intuition is to want to make things colourful, busy, like almost like a graphic novel. We want the textbook to look like that. We want our materials to look like that.


The difference is children don't learn math for entertainment like they read maybe a Spiderman comic, but they actually have to build mental representations. So the quickest way of helping children build mental representations is to go against our intuitions and to strip the materials down and gradually increase them in complexity.


Because eventually, of course, we want children to be able to look at a plate of different kinds of fruit and be able to extract the overall quantity or the cardinality of that set, but initially we need to, in order to build that, we need to have very simplified materials.


[00:11:47] Anna Stokke: And I think you're right, it does go against our intuition. I mean, even say, if you're preparing a PowerPoint presentation, you really feel like you want to put lots of pictures and lots of colourful things on the PowerPoint presentation because you think it will make it more interesting for your audience. But in fact, it might be better to have less on the slides for the same reason that you're saying it's sort of distracting.


[00:12:10] Daniel Ansari: Exactly. You know, we are, all adults and children, we have a limited capacity of information processing, call that working memory, parts of executive function, but the point is that we cannot process tons of information in parallel. So when you prepare a PowerPoint presentation or any kind of instructional material you have to be very careful about not overloading because then your message won't be heard because our brains will be too distracted with all the other things that are going on.


[00:12:42] Anna Stokke: I'm just going to try to even get a little more specific on this. So let's say you're teaching a child to add and it makes sense to talk about actual things that you might add, you know, like if you have three apples and you have two apples, how many apples do you have in total?


So what sorts of things would you suggest as the manipulatives that teachers use in that situation?


[00:13:09] Daniel Ansari: That's difficult because I'm not an educator myself, but I would look for materials that are relatively homogeneous, that don't have a lot of distracting elements. And I would quickly also have children use you know, a variety of more homogeneous elements so that they can transfer, that they can sort of learn the concepts of abstraction.


And then I would also work with these manipulatives in different spatial layouts. So you know, if you have four objects, you can arrange them in a square, you can arrange them in a line. But the fullness still remains. And I think that's a concept that takes time to develop. And so giving children opportunity to work with these manipulators in various spatial layouts and transfer between different kinds of manipulators is important as well.


Again, it's, I think it's really important. And I emphasize this over and over again, because I'm a developmentalist, to always take a developmental lens for each individual child to try and assess the point at which they're at and how much abstraction they are capable of at that point and to use that as a starting point.


[00:14:15] Anna Stokke: I would like to ask you a question that I received from a teacher. So I have you know, I have a fair number of engaged listeners and sometimes they write to me and they ask me questions. And so this question came from a teacher who I think, I'm not sure which grade the teacher is teaching but I'm guessing grades five to eight based on the message so I'll read it out and hopefully you can help with this.


The teacher writes that there is a perception that struggling math students in grades five to eight will benefit from using math visualizations and manipulatives, number blocks and algebra tiles, et cetera. And the teacher's observation is that these attempts at visualizing math principles just confuse struggling students even more, so that it seems to be unhelpful. Trying to explain how to divide fractions using visual models tends to confuse the kids who are already confused.


The only students who seem to benefit are the children who are already really proficient at math and they're ready to see those visual abstractive models.They love the clever visual models, but they also don't need the help in the first place. And the teacher says that they would love to hear more about what the research says about the benefits of math visualizations for the struggling students in higher grades. So it's similar to what we've been discussing, but a more maybe specific question.


[00:15:37] Daniel Ansari: First of all, that's an that's an excellent question and it's an important one. However, I would say that it's difficult to answer that question directly because we don't know whether the children are having problems because of the visual models or because they lack certain skills that would help them engage with those models.


I think most of the research on visual models suggests that they are beneficial. For example, there's a lot of research on number lines and the use of number parts that allows teachers to essentially integrate, you know, whole numbers and fractions in the same representational frame. We also know that there is a very tight overlap between spatial processing and numerical processing in the brain.


There is some preliminary evidence to suggest that, in young children, if you engage in spatial training and spatial visualization, that has transfer effects to mathematics. So I think we should not give up on the connection between visual representation and mathematics, but the devil is always in the detail.


The question is about how do we provide visual models that help learners that are struggling in mathematics? And I think that's an important question. In fact, I've just submitted a Social Science and Humanities Research Council ground proposal on this exact question in the domain of fractions.


You know, do we teach fractions with - what visual models do we use? Pi models, do we use number lines? I don't think we have the full answers to this, Anna, but I think what we can say with some confidence is that there is a very strong link between spatial visualization and math learning. And I think we are well-advised not to give up on that.


[00:17:21] Anna Stokke: So some visual models make a lot of sense to me in the K to 6 sphere, that does make a lot of sense. You'd have to convince me about algebra tiles. I'm really not convinced that those are that useful. I mean, the entire point of algebra is to take an actual problem like a real world problem, turn it into something that symbolic that makes it easier to solve.


So to me, adding algebra tiles to this whole process, I have a hard time seeing the benefit of that. Someone could convince me otherwise, I do listen if confronted with evidence that goes against What my gut tells me.


So anyway, I think what I'm gathering from this conversation is don't give up on manipulatives and visuals that they are really good for at least starting out, right, and helping children to understand mathematical concepts.


The skills are important, but we do care about understanding and getting an idea of what's going on, but it's also really important to move past those and spend a fair bit of time on the symbolic piece. Is that right?


[00:18:38] Daniel Ansari: Exactly. And to, and to always think about it as not as it's not a binary thing, right? It's a, it's a developmental continuum where we can shift back and forth on it. And you know, if you look at, for example, how Singaporeans teach mathematics, they will use a model method as well as a symbol method.


So they have completely embraced this idea that in order to get mathematical concepts into the minds of children, they need to go through this transition. And they may need to shift back and forth in order to re-strengthen their abstract representations by interacting with with representations, be they on paper, be they on a computer screen, or be they manipulatives.


So, yes, I would say that we can embrace both concrete manipulatives and abstract representation, but that goal, of course, has to be that we get abstract representations into children's minds. And that's the beauty of mathematics and the beauty teaching, I think, as well as to achieve this higher-order level of thinking that involves symbols and relationships between symbols.


[00:19:44] Anna Stokke: I'm fairly familiar with the Singapore primary math series, which is a really neat program. And it's true, they use this, Bar Model method, and it's kind of a neat way to solve problems, but you really need to move on to the symbolic piece and we can't forget that Singapore curriculum actually is a lot more advanced than our curriculum.


It moves a lot more quickly. And there is also a huge focus on getting the basic facts down early, lots of practice, and that actually prepares students for problem-solving, in my view.


[00:20:23] Daniel Ansari: Yes, absolutely. And whenever we look at something like a Singaporean math curriculum, I think we need to look at the broader context as you've already outlined, that there's different, it's a different priority. There's different levels of investment in math education. There is, you know, the Singapore Academy of Teachers, which provides ongoing high level professional development.


There is a culture of avoiding bringing, you know, the most qualified teachers out of the classroom into administration. So there, we cannot fully replicate that, but what we can, we can certainly take a few principles and look with interest and with awe at it. And you're absolutely right.


There is a lot of emphasis on practice and we cannot get away from practice. Math is not something that is natural to the brain or to the mind. It is in that way, similar to reading, although it's very different in many ways, but it requires practice just like most things in life where we want to move from novice to expert, we do have to practice that.


The question is, how do we structure that practice? That's the interesting question. And I don't think we, we really fully know that yet. I think we still have a lot of work to do on that, but if we can all agree that practice is good, then we've already reached a very good level from which to begin a conversation about how do we then go.


[00:21:41] Anna Stokke: Maybe this kind of relates to what we've already started talking about, but I'll go there anyway. You wrote a note on LinkedIn about developmental trajectories. So you said something about foundational skills are the key to an unfolding developmental process that leads to cognitive flexibility and creativity and basic skills which require deliberate practice, and as you've already said, they're not a trivial side product of learning.


They are, this is kind of a nice phrase, “the undeniable cradle of such thinking.” I love that. In your opinion, to be most productive in this space, it's all about understanding human development and applying a developmental lens to our thinking.


Can you please expand on that?


[00:22:26] Daniel Ansari: Yeah, I guess that note sort of comes from years of being slightly frustrated with the state of debate in education and the way that debate intersects with political movements and where there is a painting of dichotomies that, in my view, are overexaggerated. And my resolution, at least in my own thinking about this, is to always think about the child and the child in a developmental context.


And when you look at that, you cannot deny the fact that two things can be true at the same time. We want children to reach a level at which they understand mathematics, but we also need to give them the skills. And those skills are the foundational competencies that I talk about in that post. It reminds me so much of the debate in reading, right?


We need to do away with this. I think everybody can agree that we want children to be able to comprehend text and to think about it creatively. We also can all agree that we want all members of society to have a high level of numeracy and being able to understand numbers and being able to use that to inform your behaviors and your decision making.


How we get there requires a developmental approach. It doesn't require us fighting about specific either this or that because I think we can all agree on the outcomes. It's the developmental process that we need to respect and where we need to recognize that we cannot put the horse before the card or I don't know if that's the right way of saying it, but we cannot demand, you know, children to understand something if they don't have the toolkits to build that understanding in the first place.


So that's what I mean by developmental processes. It helps me to reach some clarity on things that get very muddled and where people get very emotional. And in the end, that is not benefiting children at all.


[00:24:18] Anna Stokke: And certainly I've been one of the ones that's been arguing for, re-emphasis on, on skills. But I argue for what's missing. When I argue for skills, I'm not arguing against understanding. But I also think, with understanding, that's actually a fairly complicated piece of all of it.


Because some things are a lot easier to understand than others. So, for example, it's a lot easier to understand, say, the standard addition algorithm than it is to understand the long division algorithm. And I think some people think that students could understand everything all the time, but actually, some things in math are really difficult to understand, and sometimes the understanding actually comes after the skill, but it doesn't mean it's not important.


Of course, we can have both things is what I think you're saying.


[00:25:08] Daniel Ansari: Yeah, absolutely. And, I also don't just want to sort of leave on this sort of hand-wavy note and say, “both is good and we need to do both.” That's what I mean by developmental, right? In the early years, we do need skill building. It's undeniable. Look at how we teach reading or how we are supposed to teach reading, it's very systematic.


It involves a lot of direct instruction, a lot of repetitions. But that's, you know, nobody would tell us that we shouldn't do that in, when we send out kids to various sports activities, you know, that's also repetition practice. That's how you build a repertoire of skills. And then you have this beautiful brain that is in a position to combine those skills and to mix them up with one another to produce even higher order knowledge.

So that's, that's my perspective on, on all of these learning processes is that. Yeah, we need both, but we need to think exactly about how we do both, right? So we, we can't expect a young child at the very outset to be immediately understanding and to immediately be able to reflect and to use their metacognition when there are no pieces in place to apply the metacognition to.


[00:26:16] Anna Stokke: So do you sort of see a similarity between what we know about reading and phonics and breaking down words into sounds and how kids learn math?


[00:26:26] Daniel Ansari: I do and I don't. So I, I do in the sense that I think reading and math are actually quite similar in terms of, you know, both are not natural to the brain, at least in the symbolic form. We may have some intuitions around quantity and, things such as area and density and we can distinguish between sizes.


But the symbolic abstract representation is something that each and every child needs to learn. And we also know that there are many cultures around the world that don't have symbolic number systems because there hasn't been any kind of pressure for them to develop them. So we know it's a cultural learning process.


So that's, where the similarities lie. Differences are quite vast because reading is so beautifully linear. As a researcher, I'm almost jealous because it's like, you can say, you know, you need to be able to be sensitive to the sounds in language, then you need to map those sounds onto, onto symbols and that's how you build increasingly better and more automatic representations and eventually word forms.


In math, it's different, right? Because math is so many different things. We can talk about arithmetic. We've got a development trajectory for that, but then when we think about geometry and so forth. There are differences. That's why I like in British English, they call it “maths” with the plural because I think that's really instructive. So there are some unifying principles that transcend reading and math that I think we can appreciate, but there's also vast differences.


[00:27:55] Anna Stokke: I'm wondering if you can tell me a little bit about the research on gender differences in math. There's sometimes a misconception that boys outperform girls in math. I mean, I still hear people say things like this, which just, I find really surprising, but what does the evidence actually say about this?


[00:28:14] Daniel Ansari: That's a great question, Anna. And I share your frustration. I, it is so often that I encounter people who will have this stereotype that girls are innately less able to be competent in math than boys. And we know this has downstream effects. What the actual evidence tells us is that there is more evidence for gender similarities than there is for gender differences.


So, Janet Hyde has done a lot of this work and one of the pieces of research that she actually published in Science she used statewide assessment data to show that, you know, the effect size of millions of students were somewhere within 0.01, 0.02, so there wasn't much there. Now, many groups, including us, have also looked at this in younger children.


So we've, for example, looked in a sample of about 1,300 children from grades one to six at all kinds of basic number processing tasks, things like comparing two numbers, estimating a point on the number line. And we also find overwhelming evidence for gender similarities. So, and then you can look at the PISA study and one of the striking things if you opened any kind of summary of PISA math data and you look at gender differences, you can see, yes, in some countries, it tends that girls are better than boys and other countries it's maybe slightly different.


On an average, there's maybe a slight gender difference. Then look at reading. Reading is, across the entire world, all the countries that participate in PISA, girls outperform boys.


We don't tend to talk about that, yet we talk about something in math where there really isn't much of of an effect. Right? We, we see overwhelming evidence for gender similarities.


[00:29:59] Anna Stokke: That's really interesting. What is surprising to me is that girls consistently outperform boys in reading. Why would that be?


[00:30:08] Daniel Ansari: We don't know, there's probably a combination of biological and cultural factors that lead to that, but it is a, it is, and I'm not talking about a small effect size. This is a big effect. OECD countries are of course not representative of the entire world, but they cover several continents, and yet you see this consistent effect.


I think we can see this effect having downstream effects because you know, universities are, are admitting more female students than they are admitting male students, because more female students meet the admission criteria. And so this is real. And yet, we seem to be stuck in this unproductive and actually counterproductive debate around gender differences in math.


[00:30:47] Anna Stokke: I definitely share your frustration on this. And I think, there are some leading math educators who actually aren't helping with this. So the sort of thing that I hear that really upsets me is I'll hear things like, “Girls should be taught differently than boys.”


So, for example, girls are more visual, and there's, you know, in the US, there's this push towards these math-light courses. And one of the reasons that's being given is that girls are more visual and they'll become more engaged in math. And, steering them into these types of math-lite courses actually is not a good thing to be doing.


But I'm just wondering about. That claim, is there any evidence that girls should be taught differently than boys? That they react in a more positive way to visuals and that sort of thing?


[00:31:40] Daniel Ansari: Not that I know of, no. I'm not aware of that and I think that we really need to be thinking carefully about when we make these kinds of claims because that is a big claim to make. And if you don't have data to back it up, then you better not be making those claims and be treating all students as equal, and not to classify students into any kind of categories, and to use the best principles from the science of learning to structure your instruction, and don't put emphasis on these things.


Because if we start to put emphasis on these things, we do create, I mean, our students are aware, right? They are aware. They have, they are sentient human beings that pick up on climates and pick up on “What group do I belong to?” and “How does society see me?” and “How do my teachers see me?” “Oh, I'm a girl, therefore, they must be thinking this."


“I'm a boy, therefore, they must be thinking that.” That is the wrong approach. The more we can de-emphasize these these sort of theories of gender differences, the better off we are and the better off our students are as well. I think it is not a productive debate.We've got far bigger problems than to think about gender effects.


[00:32:54] Anna Stokke: So let's shift to dyscalculia, which I think you know a lot about and I really don't know anything about. So I'm interested in hearing about this. So what is dyscalculia and how common is it?


[00:33:08] Daniel Ansari: So developmental dyscalculia is defined as a specific learning difficulty in the domain of mathematics. Basically, when you have students who are non responsive to instruction and require additional approaches, different kinds of approaches and special educational approaches, those students could be defined as having developmental dyscalculia.


But we need to be very careful. Developmental dyscalculia is not something that you can go to your pharmacy and take a test. It is defined in a way that's slightly arbitrary. We set a certain cutoff point or we say, you know, this group of students is not responding to instruction. So it is not though it is as if developmental dyscalculia has a very solid definition.


But I think It helps us to allocate resources and give attention to students who are particularly resistant to instruction in mathematics. We also need to recognize that developmental dyscalculia is often co-occurring with other learning challenges and other developmental challenges such as ADD or dyslexia.


So it is not as though we can say dyscalculia is only a math learning difficulty. It often co-occurs with other learning difficulties.


[00:34:22] Anna Stokke: What sorts of things might you notice if you were working with a student and they had dyscalculia, like say a Grade 1 student or something like that?


[00:34:32] Daniel Ansari: The defining feature of developmental dyscalculia, this was really reported first by David Geary at the University of Missouri back in the 80s and then 90s where he did some pioneering research on trying to understand what's going on with students who are really struggling with math.


And one of the things that he discovered was that students who are really struggling with math learning have almost an inability to encode facts into their long-term memory. So, you know, students go through a process of learning, for example, addition, where they use their fingers and gradually they develop increasingly more sophisticated finger counting strategies.


So initially a student might do a problem like two plus five, go one, two, three, four, five - one, two, and then count the two hands up and then eventually they learn that, you know, “I can just hold up the five and go six, seven and I've got the solution” and then eventually it goes into their memories and they don't even need to use their fingers.


Children with developmental dyscalculia don't seem to make that transition. Even when you look at the strategies that they use, they seem to remain using these very inefficient strategies. So that's a hallmark of dyscalculia that, that it would be very quick to see. I mean, there's other things that are a bit more subtle, but the fact-retrieval difficulties are really striking.


Together with Stephanie Bugden, who's a colleague of yours at the University of Winnipeg, for her Ph.D. Stephanie followed a group of students longitudinally who had these persistent math difficulties, and it was really striking when we asked them how they solve problems. They would always report very basic finger-counting strategies and never would they report, “Oh, it's just in my head. I have these facts.” That's really what's striking about, about students with dyscalculia.


And I think this comes from and this is my hypothesis, and there's many hypotheses in this space, it comes from a difficulty in representing symbols in the first place. And then as the pace of education accelerates, these children are sort of left behind with these very immature representations that take a lot of their working memory.


And you know, just to hold them in mind and to work with them because they haven't developed that fluency. So those are some of the hallmarks that you would see if you have a student in your classroom that has a math learning difficulty or developmental dyscalculia.


[00:36:50] Anna Stokke: How frequently are students diagnosed with dyscalculia? Like I've heard about dyslexia a lot, but I'm not even really familiar with dyscalculia. So I think that says something, but I'm just curious about how frequently this is diagnosed and, are students getting the help they need?


[00:37:09] Daniel Ansari: Not very frequently is the answer. I mean, it's, it's interesting. I've been in this space, you know, for almost 20 years and the under recognition of math learning difficulties, whether we call them dyscalculia or specific learning difficulty in math, I don't care. But there are students out there who need a lot more attention in mathematics and there isn't you know, I often talk to school psychologists and they might not even know the term or they're very, they're very unfamiliar with it.


Dyslexia, they know everything and they know how to quickly diagnose it. They've got all the tools. So we've got lots of work to do in this space. I think we also have work to do in advocacy. So parent advocacy around reading is excellent in North America, right? We've got things like the International Dyslexia Association, we've got the Ontario Human Rights Commission Right to Read report. There's been a lot of advocacy when it comes to math.


There doesn't seem to be a sort of collective will to develop that advocacy. But I think it is incredibly important and there are many students I think even at university level who have dyscalculia, but don't know about it and they feel, they feel stupid, you know, and I get emails from grown adults adults in their 50s, 60s who write to me and say “Oh my god, I read your article about dyscalculia. I finally know what I had my whole life.”


And they tell me stories about how it's held them back in their lives and how knowing that they have this and knowing that it's legitimate, it's something that people, that they're not alone with this is tremendously empowering. So I think we have to do more work around dyscalculia and setting criterion and building advocacy and also within schools, ensuring that schools have tiered instruction where students can go to a special educational environment, but then also with the aim of always bringing them back to the main classroom and helping them to sort of catch up.


So, yeah, we have a lot to do, a lot of work to do, but I feel like I've been saying this for years. It hasn't really changed. And I think this is also, Anna, partly because maybe in Western cultures, we just don't care enough about math and this is one of the downstream effects of that.


[00:39:19] Anna Stokke: I also think that people are more likely to say, well, you're just, you know, you're not cut out to do math. You know, whereas in reading, if a child isn't learning to read, people will get upset. You know, this child has to learn to read. Everybody can learn to read.


 So I think that probably contributes to this issue too.


[00:39:38] Daniel Ansari: Yeah. No, I just want to say, because you said this, it is a very important thing. You said, “everybody can learn to read.” I think we need to have the same approach to mathematics. Everybody can learn math. And that means that everybody has the right to learn math, right?


And so if we start from that premise, if we start looking at, because I think sometimes people look at math, it's like “I didn't really like math. I'm not good at it. There's a couple of, there's some people who are good at math, and they're gonna do well in this sector of our economy, but I just don't belong there.” If we break that down a bit, and if we have the same approaches we have to reading, which, you know, everybody can learn to read.


Everybody can learn to be numerate. Of course, there's going to be some people, you know, some people who are going to be better readers always. You know, for example, I know that my wife is a faster reader than I am, but still, we're both readers, right? And we should have the same thing with mathematics. We can all reach a level of mathematics that allows us to be numerically literate.


[00:40:38] Anna Stokke: Absolutely. And if you have an article on dyscalculia I think people might be interested in reading about that. And so I will put it up on the resource page for this episode. If it's caught early, can a child with dyscalculia learn, say, can they memorize their times tables? Can they get to the point where they're using efficient strategies so that they can be on track with their peers?


[00:41:06] Daniel Ansari: I think so. It all depends on how we structure that remedial piece of instruction for them. And I think we do have some very good guidance. And one of the things that I'll send to you afterwards, I'm sure you probably have already seen is that the Institute of Education Sciences just, I think, two years ago published a very good practice guide on teaching students struggling with math in the elementary grades.


And it, it has a lot of detail and a lot of suggestions, concrete suggestions of how to structure instruction for students who are struggling with math. I believe that the principles in that book don't just apply to struggling students, they apply to everybody. But it's a really good starting point.


And yeah, I think we should never give up on any child, right? We should always hold out for hope that we can get to a point where they don't need tier two instruction anymore. And this is especially important in the early years. You know, we, children are so variable in their development trajectories in the early years.


That doesn't mean that they learn in different ways, by the way, I don't want to get to a learning style or differentiated instruction. That's not what I mean. I mean, the rate of change is very different in the early years. And so there's a lot of work that suggests that if we do too much diagnosing in the early years, we can also reach a point at which we have too many false positives.


And then of course it's negative as well. So yeah, let's not give up. There's good guidance. There's evidence on how we best instruct those students. Let's heed that advice and let's continue, you know, trying to get them back into mainstream education and to thriving in math.


[00:42:46] Anna Stokke: Let me guess. Explicit, systematic instruction, scaffolding, worked examples. That kind of thing. Am I right?


[00:42:53] Daniel Ansari: So they have six principles. I can't remember them all, but one of them is definitely systematic instruction, the use of word problems, but it suggests how to use use of number lines, the the careful choosing of concrete, what we talked about in the beginning, concrete representation, abstract trajectory, and also one of the principles is, as one part of math instruction, is to use timed activities.


And that's of course always going to be, as we both know, always going to be a sore point. But it is definitely in that report and, they do provide all the evidence in support of what they're claiming.


[00:43:28] Anna Stokke: This type of instruction likely works for all students. So why not use it for the whole class? I use explicit instruction for the most part with my university students who are very advanced, scaffolding, worked examples, these are all tried and true methods. You know, let's do it. We want people to learn, right? That's, at the end of the day, that's what it's about. It's not about, you know, ideology.


[00:43:51] Daniel Ansari: Couldn't agree more. We've got, we've got lots of good principles out there. How we get them, you know, I think to be fair to educators, I think educators are doing a tremendous amount of great work all around the world and in this country and in other places.


I think we do have to think about our teacher education models and because I, you know, I do a fair amount of professional development events, and I'm often shocked by the fact that there are seasoned educators in the audience who haven't heard of certain principles that have been known in cognitive science and the science of learning for decades. And that is an alarming state of affairs that I think we do need to address very quickly so that we get to a point. And I agree with you, Anna, we get to a point where we agree that there's instructional principles that work for all students, and that we implement those.


And then of course we still need to have tiered instruction because some students are not going to be responding, but eventually when they return to the base layer, the most inclusive layer of education, that they are, that they're able to pick up with those methods again because I think we do need to get to that point where we accept that there is, that there are some things that are definitely true about the way we learn. And they apply to, to all of us.


[00:45:11] Anna Stokke: So do you feel like doing some neuromyth busting?


[00:45:15] Daniel Ansari: Okay.

[00:45:15] Anna Stokke: I mean, what I think are neural myths, what do I know? I don't really know anything about neuroscience and I would never claim I did, but some people do claim things about neuroscience, which seem a bit sketchy to me. So I want to ask some questions about this. So, so here's the first one. I've heard things about brain plasticity. I've heard people claim that due to brain plasticity, the brain can be changed. That students with learning disabilities are just as able to be high math achievers as anyone else so is this correct?


Does this make sense?


[00:45:49] Daniel Ansari: Let's first look at what brain plasticity really is. Brain plasticity is a biological process by which the organism adapts to the environment. It has no value. It is neither positive nor negative. If tomorrow, for whatever reason, I had to amputate my arm, I would develop a phantom limb.


That's brain plasticity, but it's not pleasant. Brain plasticity also is the driver, of course, of all learning. So, in many ways, it doesn't really help us to think about learning because it is learning itself. If our brains weren't plastic, we wouldn't have schools. We wouldn't have universities. You know, we wouldn't be learning.


We wouldn't be, so I, I think people think that brain plasticity is some kind of magical panacea or it can help us to explain away differences in learning trajectories, but it's not. It's adaptation. That's all it is. So in many ways, I wouldn't even call it a myth. I would call it a misunderstanding of what brain plasticity is.


It doesn't mean that everybody is capable of everything. There are individual differences. Brain plasticity doesn't mean that we are a blank slate and that we can just write on it. No, it means that we have certain predispositions. There are individual differences, but we all have brains that, like, all of our whole organism is seeking to adapt itself to the environment.


[00:47:14] Anna Stokke: The next one, so working memory, and I think we've covered this a little bit and I've talked about it onprevious episodes and we know that the working memory, it can only hold so much at a time. Maybe four to nine items. We should always really be taking this into consideration as educators, working memory and its limitations. I was really surprised because I, I saw this person saying that they had this program, they had this way to increase working memory capacity, which would be great.


If we could increase working memory capacity, it would be so much easier to learn and we wouldn't have to worry about all these things like manipulatives being overstimulating or this type of thing. So is there any evidence that you can actually increase working memory capacity?


[00:48:00] Daniel Ansari: That's a great question. And the question that the whole field of psychology is sort of been grappling with for the last 20 years or so. So let's start at the beginning and working memory, as you say, is this limited capacity that we have in order to hold information online while we're doing something else, right?


That's, that's the working part of it. It's we are, we have this thing that we're holding onto. Thank you. While we're doing another subset of the problem, anybody who cooks will be very familiar with this. And so the, the issue here is that yes, we can train working memory to become better. So we can train somebody to become better at a working memory task.


But what we have unequivocally not ben able to establish, the thing that has been rejected, is the idea that you train working memory and then you get transfer because that's really what we want, right? We want better working memory for math and what we're talking about, or any other domain of learning.


But the problem is, if we train the working memory, it doesn't actually affect the math. So there's no far-transfer. So I would recommend that no school board, no school division, invest any resources into any programs that promise to train working memory. Train the knowledge. Also, working memory is tied to the knowledge that we have.


That's the point I was trying to make earlier about math. Math is always associated with working memory. And we ask ourselves why? The reason often why it is associated because the items that we need to hold in working memory are the mathematical objects, the symbols. And if those representations are not strong and not fluid, then of course they're going to take more of your working memory capacity rather than less.


So now people are starting to think about, you know, working memory is maybe not some isolated homunculus that sits independent of math and reading and science learning, but more that there is a working memory for math. There's a working memory for language. And I think that that is leading us into the right direction, but I think in education, we're really advised to focus on the content knowledge that students need to have rather than trying to indirectly get to the content knowledge by strengthening processes that, and, and mental mental capacities that students need to have.


[00:50:16] Anna Stokke: It's a very appealing idea, because actually math, it's kind of hard work. So like you, you talk about the developmental trajectory and I sort of think of it as it's just cumulative. To learn algebra, you need to know how to work with fractions like add and subtract, multiply, divide. And, to do that, you need to be able to actually do the addition, subtraction facts, et cetera.


And it would be great if we could find some way to skip all that stuff, but the reality is we can't. I like your advice. No district should spend any money on any program claiming that they can do this.


[00:50:54] Daniel Ansari: The evidence is so strong, you know, and I'm not saying this lightly to give such strong recommendations. It's because we have tons of metaanalyses, we have lots of intervention studies on working memory training, on brain training, they are resoundingly null results, like you cannot even find a hint that this is working.


So let's, let's focus on the things that work.


[00:51:21] Anna Stokke: You've heard of productive struggle. So the idea is that instead of teaching explicitly, you give kids problems, they'll get interested in the problem, you just sort of let them struggle, they persevere, and eventually they figure it out.


So this is something personally I would use for students that are at a higher level or, or who actually have a fair bit of knowledge myself. It's not that I'm against this sort of thing. My personal opinion is that it's likely a bad idea for a new learner. We'll hear things like you know, “making mistakes is a good thing.” And so does this make sense?


[00:51:56] Daniel Ansari: I would exactly repeat what you said, which is that it can work for students who are really experts, right? But novices, what are novices going to do? What are they going to struggle with? What is in their minds that they can use to struggle? Yes, they might, they will get frustrated. They will probably be put off math altogether.


But is that productive? So again, it's, it's really not, I don't want to sort of say, you know, productive struggle is a terrible concept. I don't want to say that. What I want to say is that you have to look at the where students are at and use it. If you've got a group of really exceptional math minds, Get them to work with problems.


That's what we do at university, right? With especially with our upper level students and then with our graduate students. But we also respect that we need to have introductory courses and we need to build knowledge. So I think it's an interesting concept, but it's not something that I would ever apply wholesale.


But always think about as you know, Paul Kirschner has always said always think about the trajectory from from novice to expert, I think, and then, you know, that productive struggle is not going to work for novices.


[00:53:07] Anna Stokke: No, it seems kind of obvious to me, but it actually is a fairly popular idea, which I find quite alarming, actually because I suspect a lot of kids are probably really, frustrated and feeling stupid.


[00:53:20] Daniel Ansari: I think we all forget as adults what it's really like to be a child and to be apprehending knowledge that you find completely inaccessible to begin with. So I think out of respect for children, we have to understand their position and their knowledge and not assume just because we as literate adults have thousands of hours of practice that we can now expect them to struggle productively at the outset.


[00:53:49] Anna Stokke: You mentioned something earlier and I heard the same thing when I had Dan Willingham on, you know, he said he, he gave this presentation to 500 teachers and he was very surprised that none of them knew anything about cognitive principles and how to apply those to education.


And you seem to echo that. I'm wondering, you know, there does seem to be this gap and it actually would be great if we could bridge that gap, so that more educators knew about some of these cognitive science principles that could really help us to teach children and adults for that matter.


So do you have any ideas for some ways to close that gap?


[00:54:32] Daniel Ansari: Yeah, I mean, this is part of the work that I'm trying to do in a very modest form with a center that I founded here at Western called the Center for the Science of Learning. And Dan was actually our speaker for our inaugural event. It's, you know, I see it this way, Anna. I see educators that I meet that are hungry for this knowledge.


Hungry. They want it. They go through all efforts to do it. They attend conferences and so forth. I see a culture around teacher training and teacher professional development that is maybe not as embracing of that as it could be. And the roots for that probably have deep historical cultural roots that are going to be very hard to untangle, but I think if we continue advocacy around this, we will succeed.


You know, I'm trying to do that in a small way here at my local institution by trying to find ways in which I can contribute to the teacher education program. So small steps, but we, we do need to get to a point where we, bring more of this information to those who can actually use it.


And there's some good models, you know, and around the world for how to do this. I think what's happening in reading instruction is really encouraging, really encouraging. You know, because it is, and you know, what's interesting is, is that in Ontario school boards started to immediately prepare for this. Faculties of Education, including my own, resisted it to begin with, resisted the results of the Ontario Human Rights Commission, and that's an example for me of like the gaps that we still need to fill in.


[00:56:04] Anna Stokke: I think that's fairly common and quite unfortunate I agree that what we're seeing in reading is really promising, but I think we have a long way to go in math and I, I really appreciate your work. And I appreciate your passion. And I really want to thank you for coming on my podcast and talking to me today. I've learned a lot and it's been an absolute pleasure.


[00:56:27] Daniel Ansari: Thank you so much, Anna. It's good to have this conversation. I appreciate you having me on your podcast. Thank you.


[00:56:32] Anna Stokke: More in just a moment. In the show notes, I've linked to a resource page where you'll find articles relevant to the conversation today, including recommendations for helping children who struggle with math. I hope you enjoyed today's episode. I've got another great episode coming out on January 12th. If you enjoy this podcast, please consider showing your support by leaving a five-star review on Spotify or Apple podcasts.

Chalk and Talk is produced by me, Anna Stokke, transcript and resource page by Jazmin Boisclair. Subscribe on your favourite podcast app to get new episodes delivered as they become available. You can follow me on X for notifications or check out my website,, for more information. This podcast received funding through a University of Winnipeg Knowledge Mobilization and Community Impact grant funded through the Anthony Swaity Knowledge Impact Fund.

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