Episode 26 - The Language of Math with Paul Riccomini, PhD
Paul J. Riccomini, PhD, began his career as a dual certified general education mathematics teacher of students with learning disabilities, emotional and behavioral disabilities, and gifted and talented students in Grades 7–12 in inclusive classrooms. His teaching experiences required a strong content knowledge in mathematics and the development and maintenance of strong collaborative relationships with both general and special educators. He earned his doctorate in special education from The Pennsylvania State University and his master’s degree in education and Bachelor of Arts in mathematics at Edinboro University of Pennsylvania. His research focus is on effective instructional approaches, strategies, and assessments for students with dyscalculia and students who are low achievers and/or students with learning disabilities in mathematics. Additionally, he is a co-author of the bestselling Response to Intervention in Math (Corwin, 2010) and Developing Number Sense through the Common Core (Corwin, 2013). Additionally, Dr. Riccomini provides high quality professional development focused on effective mathematics instruction to school districts across the United States. As a former middle and high school general education and special education mathematics teacher, Dr. Riccomini knows firsthand the challenges and difficulties teachers experience every day when working with struggling students.
Danielle Scorrano: Welcome Paul Riccomini to the podcast. Thank you for joining me. And I have to say we at the Windward Institute have been delighted, grateful, inspired that you're a member of our faculty to share your expertise and application on math instruction. I have to say in my introduction, and I'll say to you, whenever I'm amongst the presence of math experts and educators, I feel this spark of energy and inspiration. I don't know. It's almost like this exclusive league of educators. So again, thank you for joining me. And as we start our interview, I've attended your classes at The Windward Institute, and I want to know more about your journey in teaching math. Like, were you always a math enthusiast growing up in school? And how did you transition from the K-12 space as a teacher to higher education?
Dr. Paul Riccomini: Well, that's such a great question. And, no, I was not actually a math enthusiast all this time. When I think back to my school days, math always came fairly easy to me until I hit algebra a little known fact, I almost failed ninth grade algebra and really struggled. I had to get tutored. My dad made me do extra math in the summer and it got me over the hump. But in reality, when I think back about that, I think that's maybe one of the reasons that makes me very focused and enthusiastic. Because I've been in that seat where I'm not understanding what's happening, I'm trying to get extra help and it's not resolving my issue.
So I really think that struggle that I had in that algebra class has put me in a really good position to do what I do now, which is really work with kids that struggle with math. So the kids that don't get it right away, they don't get it with the kind of the typical instructional approaches, the typical extra help. I think that has probably been the driving force with my work and why I do get enthusiastic about math because kids struggle in math. That's a fact, and it's not a new fact. It's always been this way. So anyway, that was my first reflection back to math. Then when I went into the university as a math major. I wasn't, I didn't necessarily have teaching on my mind per se, but along the way, you know, I really, I liked working with people, especially kids. So a lot of my part-time jobs, I was coaching, teaching, swimming, teaching skiing, and sort of the teaching part sort of evolved naturally. I added in the math education in college and my father was a school psychologist and he was very influential with me saying take a look at these kids that struggle. I got really interested in working with kids that struggle. It's frustrating, when they don't understand. But at the same time, it was very interesting to me is why a kid with learning disability is not getting it sitting right beside another student who is getting it kind of with the same exposure. So that's sort of what drove me. When I entered the classroom, I thought I was a really prepared teacher. And then I began to struggle like all teachers do and, you know, struggled with classroom management, curriculum and materials, you know, the whole nine yards. That drove me to the next step, which was, let's try to expand the research in this area. And that's what drove me into higher ed. As a faculty researcher, I pretty much focused exclusively on instructional techniques to support kids with disabilities or who are struggling.
DS: When you were talking about your story, a lot of your story mirrors mine. I loved learning in class. And I said in the introduction that math in some ways came really easy to me until I hit higher level statistics and geometry. I had been coaching children in soccer, similarly to you with swimming and then sort of fell into teaching as well. It's interesting that when you talk about supporting struggling mathematicians that you discussed the instructional techniques, what are those instructional techniques that you are focusing on your research and professional development? Basically, what is this area of expertise that you're bringing surrounding these techniques?
PR: Yeah. So if I were to sum up sort of in a phrase, what I focus on it, I would say it's specially designed instruction. Now within specially designed instruction, I'm focusing on the area of mathematics, and I call what I do high intensity instructional techniques. So it's taking techniques and intensifying them through very specific scaffolding through a very precise progression of representations through the abstract and that linkage. I do quite a bit of work on how you develop mathematical fluency, a topic that, you know, we've gone up and down rollercoasters around, around merry-go-round on this idea of fluency, but I do a lot of work with this idea of like, how do we facilitate fluency, especially with kids with processing disabilities that are just sort of inherently a little bit slower in terms of processing?
As of late, an area that I've really gotten excited about is this idea of retention. A lot of research focuses on that initial learning, like how do we get the kids to acquire concepts and skills? And in our field, special education and education in general has not really addressed retention. So, so once they learn it, how do we help kids retain that information across time? So that's been my shift and focus is this idea of retention.
DS: Hmm, that's interesting. And I do want to circle back with retention at the end of the episode. You talked about these representations, the modeling, especially for struggling readers. And to start, I want to focus on the history of math, not all history of math and humankind, but sometimes when I dream Paul, I depict a scene of Pythagoras, Euclid, Katherine Johnson, even my fifth-grade math teacher, Mr. Plato in this room across time, debating and ruminating the greatest mathematical questions of humanity. So right now, in 2021, are there certain research or instructional, I don't want to say trends, but current best practice in math that we should be focusing on?
PR: I think you're probably your first description of trends where a hot topic is, is probably a good descriptor here. You know, like education, math is not immune to what's the latest and greatest trend that's happening right now. And, we've seen, going back, there's always something there's calculators came onto the scene. It was the calculator. And then it was the graphing calculator technology. Most recently there was this idea of Singapore math was going to revolutionize the world because Singapore, the country had really high test scores. So we started hearing Singapore, math, Singapore, math, and that's kind of died away a little bit, but you know, sometimes you have to dig deeper than the catch phrase, trend name, and kind of dig into like, all right, well, why? Why is Singapore math getting all of a sudden, why are there their test scores? And generally it's not one variable that is doing that. It's a lot of different things. Some of it's the culture, simply the culture of the United States. I wouldn't describe it as pro math as much as it is pro reading so culture plays into it. But with Singapore math, it was a big on language. So their language is very closely aligned to kind of a mathematical language in terms of numbers and place value and things of that nature. But it was sort of an anchoring point of Singapore math is their use of representations modeling. Elementary teachers tend to do more modeling with things than middle-school concrete modeling, the concrete manipulatives kind of fade away as they go to upper elementary in part, because the concepts of math are getting a little bit more complex, but that was a big piece of Singapore math.
So there's that trend. Then we had the redoing of the standards. That that was a big trend that caught everybody's attention. I would say the standards were a lot less controversial as, as compared to the interpretation of how those standards should be delivered in curriculum and materials. That's where the people that's where a lot of folks sort of like, wait a second, what are we doing here? So, you know, the standards, the tests, I would say that, that right now, sort of a trend, a focus that I'm seeing emerge is retention that I mentioned earlier, but also language just in general, the language of mathematics and how that plays into a conceptual understanding, problem solving application and interpretation, which is really the ultimate goal of math is that they use it in a way that allows them to interpret the world.
DS: Interesting. I come from this as a perspective as a language teacher. One of the ways that I think developed probably the most agency when I taught math at Windward was my understanding of the language for students with language-based learning disabilities. And it was one of the reasons I think that I just loved it so much - was to capture numbers and language form and to help my students process and truthfully problem solve even the most complicated problems. I know you had written a number of articles on math, including one of the recent ones I read was the 2015 article you published with Smith Hughes and Fries. I'll put that on the website for those READers that are interested, but I want to know Paul, how do you and the research conceptualize the language of math?
PR: Yeah. So, you know, this is something that when I reflect back to my teaching, I realized, wow, I did not do a lot of language kinds of things. It wasn't a focus except when I taught geometry in school, everybody's like, well, you got to teach the vocabulary. Okay. But why not? Why won't we do that in algebra and everything else from elementary school up?
So, so this is something that really began to emerge as I was looking at the field and started in particular, that article sort of the impetus of that article was we know language is problematic in many other content areas like science in particular, as well as ELA. I mean, teachers in ELA spend a lot of time devoted to vocabulary. And it's just started getting me to think what about math and that's how I got into it. So how do I conceptualize language in math? So math is a unique, it's a unique content, different than most other areas because it's constantly expanding and progressing. And with that is this language piece. So we have formal content specific math language, but unfortunately we also have this shadow informal language that we often default to. The most classic example is how we read decimals. Most people will read it from a place value perspective. We will read it as 4.23 versus four and twenty-three hundredths. So how I conceptualize is language develops through a lot of opportunities to both hear and use the language in multiple formats, verbal, written, and so forth.
And what became clear to me was we really don't emphasize language. Now this is, this is 10 years ago. That's changed. I mean, I think language is definitely being emphasized much more now in math classes. Curricular materials certainly emphasize it because they usually have their vocabulary highlighted in their programs. They have a lot of language resources. So language really is what revolves around math. And a phrase that stuck with me was if kids don't understand the language of instruction, they're not going to be able to learn whatever we're trying to teach. So that's where I sort of really got interested in this language piece of mathematics.
So, you know, I'm a big advocate of teachers using math, correct language, not informal and it's tough. It's tough. You have to actually consciously make yourself aware to use the correct language or we default. The other piece of the language that I think sometimes gets overlooked is the symbol notation that comes into play. Now it's not as great at the elementary level, but the symbols or shorthand really explodes once you get into middle school and then certainly into the upper levels of algebra, geometry, and so forth. So language is really something that we need to focus on and we need to go a little bit above and beyond, how we approach it at math.
Now, that's that, that's where the catch 22 comes into play. There's too many vocabulary words to directly teach all of them. So you have to prioritize, but that's sort of how I conceptualize language. It's one thing to solve a problem and get the correct answer. It's a complete other task skillset to explain how you solve that problem and what the answer means, and that's where the language piece comes into play. And that is being captured. They're trying to capture that on the end of your assessments, that they have an open construction response where they have to explain it.
DS: Hmm, there were so many aha moments from when I was in the classroom. I don't know if you saw this or Harry you're on the call now that my eyes closed at certain times, because I was picturing myself in my math classroom. And when you talked about the decimals, I had so many times where students would say 4.3 instead of four and three tenths. And in that moment I had to be so cognizant of the usage of that and correct them using corrective feedback, saying it's four and three tenths, and then they would practice saying that. So I appreciate you talking about that, highlighting that. And the other piece was the use of the symbols. I was thinking when I was teaching algebra, when students were using X, the X to indicate like multiplication, like three times three, and then all of a sudden they're seeing three plus four in an algebraic expression. Then they would see it into an equation and how even just recognizing those symbols and the changes of symbols, what that meant, so it was important to explain that. When you talk about language too, I'm thinking of vocabulary and in that article, you cited something like thinking even just what product meant in multiplication. And as a teacher, it's like what, math is all about numbers, right? So what is this vocabulary thing you should be talking about in math? So I'm just curious too, if you can dive into more about what vocabulary means in the context of math.
Yeah. One of the exercises that I like to do with my students in my university classes is have them explain how you add fractions with unlike denominators and record themselves. Record the explanation, then transcribe that explanation and you will see how language-based math really is. Vocabulary runs through it. And vocabulary is the key. Now, language is both expressive and receptive using a word and understanding a word. So it's not just using, they got to understand it. So, when I focus on vocabulary words, a lot of times the vocabulary word is an access point. In other words, the kid can't access the content if they don't know the terminology. So we'll see that in word problems a lot. So the kids struggling with the word problem, it has nothing to do with the mathematics. It has to do simply with a vocabulary term that they didn't know what it meant in a math context. And so that's the other thing I thought you were kind of leading to is product in math class means something very different than a product in a social studies class or in ELA. It's a different context and as a different meaning. So we also have that element that is wreaking havoc with our kiddos too. Vocabulary is something that a teacher can address in a relatively short period of time. And we have lots of good things that teach vocabulary that are out there.
DS: Can you provide a couple of resources or maybe we can do it after you can give me a couple of websites or do you have just a couple of thoughts on resources for math teachers to access?
PR: Well, while I don't know if this is as much a resource, but various states will list like key vocabulary by grade level. Some do some states don't. So I would say the first place is check your state to see what information that you have. Like in Pennsylvania, they have a list of key vocabulary for math that's important for our end of year assessment. So check with your state. Second is use your curriculum. Everything should kind of flow from the curriculum materials. I guarantee if you have a core math program, you have a lot of vocabulary resources within that program. And then third, without getting too much into the weeds here, graphic organizers are a very well-established way to help kids learn a lot of different things, not just vocabulary, but a graphic organizer that I find particularly helpful in math is the Frayer model. Sometimes it's called the four corners.
And what I really like about that is it specifically forces the teachers, the kids to pay attention to both examples and non-examples. That's a big part of math is the non-example because it sets the parameter or the border of the concept or the skill that you're working on. So that those are a couple of resources that I would, uh, I have teachers encourage teachers to take a look at.
DS: I appreciate that. And as you're talking again, I was transformed back to my classroom when I was delivering instruction through direct instruction model and teaching concepts and vocabulary and context and having students practice it. And what I particularly loved, of course, was understanding the complexity of math, its language, its patterns, how it translates to students with their strengths and vast needs. But what I particularly enjoyed was engaging in a process of discernment in which I recognize what all my students brought to the table, both strengths and difficulties. And when I thought about my classroom of students, students that might even as presented as quote unquote stronger in math, had difficulties with a concept or procedure, and it was never the same. Some students had difficulties with their number sense. Others had difficulties across different types of topics. A student that had a difficulty with fractions may have been really strong in let's say decimals. So for me, it was just very interesting to observe and to think about how I supported those students. For you and for a question for you, how do we as math teachers support students when they do present this range of difficulties and strengths in the classroom?
PR: Yeah. So that's a great, great question. I think that's the million dollar question. How do we support kids when we have 25 kids in the classroom and all different variations? So, you know, one of the things again, and I kind of alluded this earlier, the content, the nature of our content, pretty much guarantees at some point, you're going to struggle in math. So my teaching experience, I taught middle school and high school learning support, which was kids with learning disabilities. But I also taught general ed math all the way through, you know, the upper levels of high school math. And I also taught gifted children and it doesn't really matter. At some point you're going to struggle in math. Now what tends to happen is the stronger kids struggle briefly and then get over the hump where the kids with learning disabilities tend to struggle more pronounced and long and so forth. So, you know, how do you go about supporting your students if you're a teacher and you've got 25 kids. We know they're going to struggle at various points. Some are going to consistently struggle. Some will barely struggle, how do we address it? I really think the first step in this point is having the data. So you have to have some data. Now, the data can be informal data like that you're picking up as you're having conversation with students. It also could be classroom data. So tests or quizzes that you might give, computer practice, as well as formal data that comes from state assessments, or maybe district benchmarks and so forth. But I really think that's the starting point is the data piece.
So, you know, what's happening now that is, can be a catch 22, a score does not really tell you what the kid is struggling with. The score tells you, Hey, there's a problem here, but you don't know what that is. That's where you have to get down to looking at the kids worked out problems. You have to look at their solution and you have to follow their thinking. One of the things that struck me early on as a teacher is when I would see kids making mistakes and I would, I would have the kid come up and I wanted to verify that was the mistake. And I asked the student to explain their work to me and they explained it perfectly. It was just wrong. So what that was telling me was that they're learning, they just learned it wrong. So I really think that data piece is the first stop. And then from there, you have to have a knowledge base of instructional scaffolding in how to do the scaffolding. And that's where it's going to vary. If you're trying to support someone with problem solving, well, that's going to be a different scaffold than if you're trying to support someone who is having issues with computation or a procedure. If it's a language problem, that's a different kind of scaffold than for fluency or for problem solving. It really depends then on where the child needs a support.
DS: Hm. So you talked about scaffolding and you talked about data or data. I say data or data anyways, I guess. I don't know why I'm tickled so much by that word. Anyways, so when you talk about data, I want to know more about errors and how you analyze errors when they do come in through a homework assignment or a test.
PR: Wow. That's great you asked. Do we have a couple of hours to talk?
DS: I mean, I have all day, Paul.
PR: This is something that I really got interested in early in my career and have continued to carry through it. I really like this idea of error analysis. Do kids make predictable error patterns essentially is the question. So back in the eighties, when artificial intelligence kind of exploded onto the tech scene, there were a lot of folks running students work through these AI programs to look at errors. And you know, what they found was in a basic subtraction problem, there can be four or 500 different kinds of errors that kids are making, which is not helpful, right. That's not helpful for a teacher, but what they found is the majority of kids are clustering their errors in about four or five errors. So essentially we have, we know there are predictable error patterns that occur. Now, some of it is probably, some of it is occurring naturally. Some of it is occurring because math evolves. So what you do for this problem, you don't do it for this problem, the kids over-generalize. But the error analysis is when you take a sample, several samples of students' work across a time period, never like for one day for an error analysis because maybe the kid just had a bad day that day. But you look across time and you get in and you study the kids solution. In other words, how they solve the problem. And I encourage you to study the correct, any problems that were correct as well as incorrect because that comparison will begin to zero.
It will allow you to zero in on the specific errors. Once you pinpoint an error, then that's going to help drive your instruction. Without that, reteaching remediation, acceleration, whatever word you want to use, is kind of like driving your car at night without headlights. You don't, you, you kind of know where you're going but you're not necessarily staying right on the road. So driving a car, that's obviously bad. If you drive off the road in math class, what's bad, is it leads to you using time inefficiently. So you're doing something that the kid doesn't really need help on. Therefore, you're not resolving the error and you're using up your valuable time.
So I really encouraged teachers to first select group of students. So what we're talking about here is intensive in nature. You're not going to do this on 30 kids. You're going to pick your five really struggling kids that kind of have you stumped and dig into their work. You do error analysis, whole class just in general, but what we're talking about now is sort of a really specialized, intensive dig into the kids' work. Now, once you look at the kids' work, you need to confirm it by having the kid come up and asking them some questions so that you're, you're making sure that you're not misinterpreting something that's on the student's paper.
DS: Hm, I like that follow-up too, and it reinforces the relationship with the child too, and helps them to know that you're supporting all the struggles that they have. Now, you talked about the group of struggling students in a math class and looking at using data to determine their errors. So now let's even think broader, what happens when a child is experiencing a persistent difficulty or even a disability? I'm thinking of even just the markers of dyscalculia. What happens when a child is experiencing this difficulty or a diagnosed disability, and when should teachers and educators be concerned?
PR: Yeah, so this is another really important question. And there's been a decent amount of research done on sort of the predictiveness of math performance at any given grade, and then what it means long term. So I don't know if there's a specific indicator per se. I will say that math struggles tend to go a little under noticed versus reading. When a kid can't read, then that's really clear. Math, we don't, it's not, it's not quite as clear, but there's plenty of research that show in kindergarten, we can begin to identify kids that are going to struggle in third grade because of some things that they have not developed.
As a parent or as a teacher, I think we need to be in tune to math as much as we are in tuned to reading. When kids struggle in math, it's often going to be a, a longer timeframe versus a short timeframe, but it starts really early in kindergarten, first, second grade, you can start to see the earmarks. The other thing that happens is we have kids that don't struggle at the elementary level, but then they'll struggle once they hit around fourth or fifth grade. And again, our content has a big role in this K to three, the primary focus is whole numbers. And then we hit third grade and then definitely into fourth grade, we take our whole number system and we expand it to rational numbers, fractions. So some kids don't struggle in the whole number system, but then fractions really kind of gives them struggles. And then we know if you have problems with fractions, then you're really going to have a problem with algebra later on. But something that I find very interesting little known fact here is that there's some research done, that looked at what is most predictive for dropping out or graduation, and interestingly enough, math washes out reading. So kids are going to drop out more so because of math. In math, what we've found is sixth grade math is a key predictor for graduation. So if a kid, if kids fail sixth grade math, only about 13% of them will actually graduate.
DS: Oh, wow.
PR: So that's something I think schools need to get a little bit more in tune to is sixth grade. You really know right there, who's at greatest risk to drop out and we should be really focusing some efforts on sixth grade. Now why sixth grade? Is there something in the content? Is there something with the kids? Is it the transition from elementary to middle school? Again, issues are complex. Therefore, there are complex explanations, but I think sixth grade is where you have a culmination of a lot of things in math coming together and having to be used in terms of the content.
DS: It's interesting that you talked about those key milestones from third to fourth grade to sixth grade. I didn't know that about the statistic for likelihood of dropout. I was even thinking when you talked about the transitions from fractions, from whole numbers to rational numbers, I even thought about the difference in integers, so positive and negative numbers. When I was teaching math from fifth through eighth grade, a fifth grader to an eighth grader had just this difficulty understanding the difference between a positive three and a negative three. So why would that be something that a child would have difficulty with?
PR: Yeah. So again, the unique nature of math is that you have a conceptual piece there, then you have the procedural piece, the computational piece, and eventually with integers, we get into the rules. So again, it's all about the combination or the simultaneous blending of those things together. It really makes mathematicians, mathematicians. It's not that you know all or you're conceptually strong and then procedurally weak. And it's not that you're procedurally strong and conceptually weak. It's really the blending of the two that is important. Why a student struggles with negative three and a positive three? You know, I think there could be a lot of reasons with that. Part of it is, we hold off on the negative integers to later in upper and middle school. So in elementary school we have a number line, but it's not really a number line. I said no way, because they're not allowed to go to the left of zero because that's the dark side in elementary school. There's all to the right.
DS: Like Star Wars, dark side, or what?
PR: Yea, right. Then, in middle school then we open it up to both sides. So I think it's just familiarity and exposure, how it makes sense. Some will say that kids in the north are better at handling negative numbers because we have winters that have negative temperatures. So kids see a vertical number line and the negative numbers on there, where down in the south or warmer parts of the state, they don't see that as much. I don't know. I think we don't see math outside of math class, as much as we should.
DS: I actually did a radical crowdsource of mathematicians, AKA students and school aged children. And that was one question, how does math apply to the real world? So we will get to those questions at the end. I appreciate you breaking down some of the reasons for students who struggle in math from diagnosed disabilities to just recognizing those patterns, and you have provided so many strategies when you have taught classes to us at the Windward Institute. I want to know for our READers, what are some best practices to supporting struggling mathematicians?
PR: Again, there's a nice wide continuum here. I like to look at instructional supports from a continuum where you have the most intensive supports, and then you have sort of the sliding scale where you have the least intensive supports. So I never liked to pigeon hole that a kid with a learning disability has to have "x" all the time. I'm not one that says this or that. And I think that makes sense in life. It's rarely this or that. It's kind of this continuum piece. Again, in math, you have to match the strategy or the scaffold or the practice to what you are trying to support. So problem solving, if they're reading a word problem and they're trying to problem solve, obviously there could be a vocabulary element or one technique that I like for scaffolding of word problems is to have kids interacting with word problems that don't have a question in it. So there is no question. All the information is embedded into the word problem and then lead a conversation, modeling a think aloud of what the numbers are representing.
So scaffold off the kids having to get an answer and you focus the kids on what does this number represent. And this discussion is not focused on the contents of the problems about baskets of cherries. That's not what the number represents. What we're looking for is the mathematical structure that it's representing- the beginning, the total, the ending, what I got, those kinds of phrases.
So that's one element. We don't always have to solve a problem to learn something. Second one, this is more at the upper grades is this idea of using worked out solutions as a learning tool. So a lot of times in math, the teacher models the problem and explains it explicitly to the students.
Then the students get an opportunity to try to solve it with guided practice. So there's this emerging body of research that is suggesting that giving kids fully solved problems can be an intermediate bridge from the teacher model to the students solving the problem on their own.
And again, the scaffolding element there is when they're studying a solution, they're not having to compute, they're not having to write or solve. They're focused on the process and it helps support their reasoning through the problem. I mentioned, the Frayer model, I think is excellent for vocabulary. For fluency, I'm a big advocate of more purposeful and planned practice through incremental rehearsal. That's where you shrink down the number of facts that you're targeting. They get multiple opportunities on each fact followed by some feedback. So those are just a couple of ones. If we get into the retention, the retention practices, space learning is excellent. Interleaving practice is another technique and then practice test retrieval. In this body of work for retention, they kind of found three key elements that are important to produce. Long-term learning, there's a timing issue. So it's when you revisit something and then interleaving is the purposeful mixing of skills. And then practice retrieval is this free recall exercise followed by feedback. Those three elements seem to be really important for attention.
DS: Hmm. You talked about fluency, and I come from fluency from a reading perspective. What do we know about fluency and math? How do we explain that?
Yeah. So, so fluency has kind of been this 900 pound gorilla that's been in the room for several years and decades, you know, memorize the facts, don't memorize the facts. At one point we all memorized the facts, almost too much memorization of the facts. Then we went to zero memorization of the facts to use the calculator. Again, when you're on the extreme ends of things, you're usually not going to address the situation. So when the standards were revisited, fluency became embedded in most of the standards, at least from that initial set of national standards that came out. It sort of worked its way back in. But you have different viewpoints on it. So sort of the way I approach it is more from a cognitive perspective versus a mathematical perspective.
Memorizing your math fact does not guarantee that you're a good problem solver, or you become an excellent mathematician, but if you don't know your math facts, then there's going to be a lot of problems or challenges that you're going to have to handle. From a cognitive perspective, when kids are as answering math facts, they're leveraging their working memory. So if they don't have the facts memorized or they're not automatic with it, then they're going to be using a strategy to find that answer. In reading, that would be decoding a word. In math, that could be counting on your fingers, making a 10 near neighbor, sing a song, whatever the strategy is, counting on the number line, whatever it is. What we know cognitively is when they when they're decoding a word at the letter level, or they're using a strategy to find a math fact, to do a computation, that's using their working memory. Working memory is limited. So the more that the kids have to use their working memory, the less capacity they have to do other things. In reading, decoding every word is going to impact your comprehension because you are focused on decoding and not comprehending. In math, when you're using fingers and calculating and so forth, you're focused on the computation. You're not focused on the reasoning, explaining or the interpretation. So having kids become fluent to automatic reduces the load on their working memory. It's simply a capacity use issue. Do you want your kids using all their cognitive capacity on a computation or do you want them using that capacity to reason, explain and understand?
DS: Yeah. I appreciate that cognitive look on it.
PR: Yeah, so that's the way I look at it. I also have a daughter who has a disability and mathematical fluency. So I've been, I've not just do research in this, but I've experienced firsthand how this issue will impact math in general. I also want to make a point here is that just having kids memorize their facts is not the solution, but it is part of the solution. And kids with learning disabilities, we have to be very aware that some kids, because of their deficits or their disability and their processing, they're not going to get automaticity on everything, but the more they have automaticity on, the better able they will be to handle the content as they move forward.
DS: Hm, you provided already in this interview so many beneficial strategies and resources and information so I appreciate that. Leading into my last question, you obviously have been a math educator and I'm glad that you brought up your role as a parent because in my last few questions of the podcast, I polled our READ listeners, parents, guardians, educators, as well as students actually nationwide because we had some entries, some participation from students around the country. I'm going to start with the first few questions from parents and guardians. And this was actually a common question that out of the parents and guardians, I polled kept coming up.
When I used to teach math or even talking with these parents that I polled with these school aged children, a common question that has arisen was, or is why do I have so much difficulty helping my child with homework? And one parent actually said to me, why is math always changing? When I look at my child's homework, it looks like a complete foreign language and even drawing on my own experience, it seems like when I was growing up, the way I learned math is perhaps somewhat similar, but very different. So what can you tell us about how parents can support their children at home right now with math? Is it changing? Has it stayed the same? What's going on?
PR: I get my ear bet on this a lot at my sporting events. I'm here to tell you, math has not changed. Place value is still place value. Math has not changed. So be rest assured the math is not different. What is different is the learning progression. How we go from point A to point B, that has what has changed.
When the standards got re-looked at, at the same time, we were also saying we need to do, we need to have more rigor. So part of what has changed and has led to parents' frustration, and by the way, I'm with you on this and I have a math degree and I taught math. Doing my kids homework is frustrating to me sometimes as well.
There's a progression that's different. So how we went from point a to point B, a good way a simple way to say this, and this is an oversimplification, is we went at lightspeed. The concept was introduced and we went right to the algorithm. There wasn't a lot of why. And there was just, you know the algorithm, can you solve the problem? Done.
So that is what has changed and no longer is it the expectation to know the algorithm to solve the problem. The expectation is to understand the algorithm so that you can then better utilize it. All right. Now this is all, you know, this is all yet to be determined, but that's the rationale. So the point of frustration with parents is they don't know where they are in the learning progression.
So you sort of to get this, "Well here. Why don't you just do it this way? This is how I did it." But the child is not there yet in their learning. So that is what really changed. What parents have the most challenge with is the curriculum writers looked at the standards and then they interpreted the standards in a learning progression. And so a lot of what is happening in between, the purpose of that is, that's not how they're going to solve the problem in the end. The purpose of that is to help them better understand how to solve the problem with the end. And that's where the parents get frustrated. Now, as far as how can parents help their kids.
I can always tell when school starts up, because my phone is usually blowing up with my friends, texting me about, Hey, can you show me how to solve this math problem? I know school starts when that happens. These are coming from educated parents. So it doesn't matter your level, math is frustrating to help your kids. But when I would help my daughter, when I think back, I always find it funny because I would say, so how did your teacher explain this to you so that I could help her? And I'm thinking now that I think back on that, well, if she knew how her teacher explained it to her, why would I be having to help her? Part of it is the disconnect. You're not there to help the child. I really think, what I encourage teachers to do is to make accessible the answers to all the homework problems at home. So parents can at least look at the answer so they can at least check. So one problem with homework is it's the delayed feedback.
They do it at home. They don't know whether it's right or wrong until they go in the next day. So I'm a firm believer that teachers should be providing all of the answers to the parents so they can check at home. This new body of research that I mentioned earlier about the solutions, what really is helpful for homework is that the teacher provides the solutions, not just the answer, but the solution to the problem.
What we're finding is studying a solution is as effective, if not more effective, than traditional practicing of solving problems. As a parent, ask the teacher for the answers for solutions, sometimes they'll do that. Sometimes they don't, some teachers already doing that. If that isn't there, then jump on your Google and Google the math problem. There's some apps out there that you can enter the problem in, and that will give you the step-by-step. Again, not for the student just to copy those down, but to allow you to see how to solve a problem. As a field, we have to stop hiding answers and solutions in math and start using them as part of the tool.
DS: Transparency. The second question I had from parents and guardians was why are we more likely to associate math with anxiety? And when I talked to some adults, some adults will get a complete look of joy on their face about math, but for the most part, I see these looks of fear like these eyes pop wide open. What is the research telling us about math anxiety and what does it also tell us about how the adult view towards math may influence a child's perspective towards math?
PR: So this is again going back to our culture in the United States about math and other countries are very different. You know, I really like to emphasize, just take a moment here. Kids are not born hating math or having anxiety towards math. Alright. It is learned now, where is it learned? Well, it's learned from a variety of areas. Well, one area that research is really showing that can be very profound is that parents pass their anxiety of math on to their children. So you got to remember as a parent you are the role model and kids are much more in tune to your behavior and your actions, then we realize as adults.
How you talk about math is picked up by the kids. So when you're helping with their homework and go, “Good Lord. I hate this. I hate this math, why this is why are you doing?” That is being picked up by the kids. When you say I'm not really good in math and you know, I hate math. You never use math. The kids are hearing that and they're learning and or you start talking about how, oh, math made me so anxious. Kids see that they hear that they learn it. Then when math starts to get a little challenging, then they saw their parents and then they default to, well, my mom, dad couldn't do it, so I can't do it. Parents are influential. The other piece is teachers, too. Teachers can create anxiety as well in math if they approach it in a less than positive fashion. And again, our culture in the United States is, you know, the way math, the way people that are good at math are portrayed is not always positive.
So I think your kids are hearing this, seeing this stuff and they develop anxiety. Then, then, then the way we do things in general with math and so forth can create it. But anxiety is learned through experiences. Now with that said, my daughter who struggles at math by fourth or fifth grade, she was, I hate math. Why do we have to do this? And she wasn't getting that from me at home because I'm presenting math very positively. So even if you are presenting it as positive parent, experience is going to eventually take over. So if you constantly experienced failure in any subject or any activity you will eventually learn to not like it and then avoid it and unfortunately that's what happens now.
DS: Yeah, I think those are really important last pieces to keep in mind for adults. Okay. So here are the school age, the questions from my school age children, not my school is children, children that we polled. These could be anywhere from kindergarten to, I think ninth grade was one of these. So the first one was, and I think a lot of kids have wondered about this. Why do I need to learn to solve word problems?
PR: Yeah. So that's the classic disconnect that everybody brings up. Word problems are not real life. Right. And, and, and, you know, you don't see a word problem when you're trying to figure out how much lumber you need your deck, how much carpet you need, or how much paint you need. It's not in word problems, you have to figure it out. There's certainly a disconnect there. Even if the word problem was talking about something in real life, that's not how we see math in word problems, but you know, my point to the student with the word problem is word problems are teaching us how to think through something logically and apply mathematics to solve it. Obviously, we need to do a better job as a field of showing how this word problem is going to be related to something we see in the real world and we just have not done a good job of that, me, including myself when I was a teacher.
DS: That's a good point. The second question is, I love this one. Why, if we have cursive letters, why don't we have cursive numbers? I never thought about that before.
PR: Great question from this little one here. Cursive letters, cursive numbers. I know different countries write their numbers, subtly different, but math is kind of the universal language with numbers. So numbers tend to be the same. That's a, no, he stumped me on this one. I'm going to have to say, I don't know.
I will say we don't have cursive numbers, but we got all kinds of symbols that you don't have in writing. We have lots of symbols, but he stumped me on that one. I don't have a good answer for that one.
DS: The last question is this is from the kindergartener. Can math be my job when I grow up?
PR: Wow. That's a, that's a kindergartner is asking that. Wow. Well, sure the job that comes to mind is an actuary. That's all about math. It's one of the top 10 paying jobs for as long as I can remember. You got to know a lot of math for that. So yeah, math could be your job, but you could be a math teacher. You could be an accountant. Although my accountant says he just uses Excel. Sometimes I wonder about the math skills, but either way, Yes, math could be a career or have a large element of what you do.
DS: Maybe we'll have the next Dr. Paul Riccomini in about 20 or 25 years.
PR: Never know!
DS: Thank you, Paul. I know we have spent an hour talking about the world of math, and I know that you have a lot going on and from your teaching to research, to being an instructor and a presenter and working in professional development. Was there anything that you wanted to add about what you're excited about in the future of research and instruction in math before we close?
PR: Yeah, well, at first off, I want to thank you for having this opportunity to talk about math. I think we need to do more of this. We need to promote math, promote what we're doing a bit more. As far as what I see happening in the field, I really hope that if we were to do this same conversation in five or 10 years, that I would be able to say that a lot of different things about what are the practices, what are the ways to support students in the field as we evolve. It's something that we are definitely expanding and growing, because math is critical and it's something that I think we'll continue to expand and evolve with.
DS: Thank you so much, Dr. Paul Riccomini. Thank you, READ listeners for listening to all about the world of math and instruction. And if you want to learn more about Dr. Riccomini, you, Paul are delivering a lot of professional developments for the Windward Institute so they can learn from you there. Is there anywhere else if people want to learn more from you?
PR: You can find me at Twitter at pjr146 is the best place to find me on Twitter. And then, at Penn State, I have a home page that you can check out too.
DS: Great. Thank you all. Thank you, READers. I should say now thank you mathematicians and thank you, Dr. Riccomini.
All READers are invited to take a journey in mathematics with Penn State professor and The WI faculty member, Paul Riccomini, PhD. Dr. Riccomini and Danielle discuss the language of math with the underlying principle – if kids don’t understand the language of instruction, they’re not going to be able to learn to their potential. Dr. Riccomini explains certain markers of why children may experience challenges in math, when difficulties persist, and scaffolds and instructional practices to support struggling mathematicians. At the end of the episode, Dr. Riccomini answers “crowdsource” questions from adults READers including how parents/ guardians can support kids at home, and he takes on the curiosity of school-aged students by answering their top questions. Dr. Riccomini invites all listeners into a math class to listen closely for the importance of its numbers AND its mathematical language.
Top READ Bookmarks
Each episode, host Danielle Scorrano identifies key takeaways or “READ bookmarks.”
1. The Language of Math
Math has its own markers of complex language such as vocabulary and symbol notations.
"If kids don't understand the language of instruction, they're not going to be able to learn whatever we're trying to teach. And that is true for the language of math."
Takeaways for instruction:
- Vocabulary plays a central role in math, especially when solving word problems.
- Prioritize and explicitly teach vocabulary terms in context
- Be mindful of multiple meaning terms, like “product” in multiplication. Read more about the complex nature of math vocabulary in a 2015 article here.
- Check state education websites and resources for curriculum related lists of key terms by grade.
"When I focus on vocabulary words, a lot of times the vocabulary word is an access point. In other words, the kid can't access the content if they don't know the terminology."
2. The Importance of Data and Error Analysis
"The nature of content in math pretty much guarantees at some point, children are going to struggle."
Takeaways for instruction:
- At some point, all students will struggle with math, but some children, especially those with learning disabilities, may struggle for a longer time based on their challenges.
- Math errors are predictable. Research shows that students tend to make math errors across four or five different clusters.
- Error analysis is more than examining student scores on assessments and consists of analyzing student processes and mistakes for a small cluster of students over a period of time.
3. When Children Persistently Struggle in Math
When children struggle in math, it tends to occur across a continuous time frame and can be identified across important developmental milestones.
"In kindergarten, we can begin to identify kids that are going to struggle in third grade because of some skills that they have not developed in math."
Some children may understand whole numbers but then struggle with rational numbers and fractions, typically around third or fourth grade. These challenges can later predict difficulty with more complex math like algebra.
"[Proficiency in] sixth grade math tends to be a key predictor for high school graduation."
4. Examples of Instructional Supports for Young Mathematicians
"In math, you have to match the strategy, scaffold, or practice to what you are trying to support."
- Graphic organizers, like the Frayer Model, support vocabulary instruction in math.
- Scaffolding problem solving to target student needs in vocabulary and language.
- Worked out solutions to show students the process to problem solving.
"There's this emerging body of research that is suggesting that giving kids fully solved problems can be an intermediate bridge from the teacher model to the students solving the problem on their own."
- Incremental practice to build fluency.
- Interleaving of skills and concepts, spaced practice, test retrieval, and free recall with feedback to promote retention
5. Supporting math students at home through
- Increased communication between teachers and parents/guardians on the learning progression in math (i.e. how students are expected to learn and show their mathematical thinking)
"Math has not changed. Place value is still place value. What is different is the learning progression."
- Provided worked out solutions and answers for parents/ guardians to review work and support students who may have immediate questions
- Encouragement for students to self-advocate to teachers
- Mindful awareness that adult views toward math influence a child’s perception. Research shows that a parent or teacher view and actions (whether explicit or implicit) toward math will shape how their child approaches the subject- especially during times of challenge!
"Kids are not born hating math or having anxiety towards math."
- Proactive, intentional supports and scaffolds early to mitigate challenges.
"If you constantly experienced failure in any subject or any activity you will eventually learn to not like it and then avoid it."
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About READ: READ, the Research Education ADvocacy Podcast connects you with prominent researchers, thought leaders, and educators who share their work, insights, and expertise about current research and best practices in fields of education and child development.
Note: All information and insights shared demonstrate the expertise and views of our guests.