Mathematical Knowledge and Problem Solving: Martha Alibali
1. How did you become interested in spontaneous gestures and children’s mathematical reasoning?
Well, I grew up as the second of 11 children and so I was surrounded by child development my whole growing up. I took a job in high school at a school for exceptional children. These were kids with severe and profound mental retardation, some with Down Syndrome, so kids with lots of different kinds of cognitive skills. And I found that extremely interesting, just to compare what different kids could do, what things were hard for kids, to see kids of different ages or different levels of abilities interacting together. I saw this in my own family, as well as at my job. And so I was very interested in those things. Then when I went to college at the University of Chicago, I had the good fortune to take a child development class as an undergraduate. And I hadn't actually realized until that point and time that there really was a scientific study of child development and that was something one could get involved in.
Then through a friend I managed to get a position in a laboratory. It was actually Janellen Huttenlocher's laboratory, were I was transcribing video data of kids interacting for one of her graduate students, Patricia Smiley, and found the work really interesting and wanted to stay in Chicago for the summer, so that I could keep working on her project. So she then introduced me to Susan Goldin-Meadow, who is another professor at the University who offered me a position in her lab to work on some projects about kids’ learning, about language and gesture. And that began the path to where I am today. I later became a graduate student with Susan Goldin-Meadow and went onto continue to study many of the same questions that I started studying in the lab as an undergraduate research assistant.
2. What is your current area of research?
> I have a couple of main thrusts in my research. One has to do with kids' mathematical thinking. I'm interested in kids' understanding, particularly of equations and other symbolic forms, like graphs and other representations that we use to express and convey mathematical information. I'm also interested in how kids and adults talk about their knowledge. And this has led me to an interest in spontaneous gestures. We often express things that we know in our gestures, so particularly mathematical and spatial ideas are often expressed in gesture. And I found in my work that these gestures often tell us things about what kids are thinking that are different from the things that they're saying in their words. And so I found this to be a fascinating window onto children's minds. And this has led to a more general interest in gestures as a form of representation that we use to support thinking and to communicate to other people. And so I study both mathematical thinking and reasoning, and language and gesture, how they fit together and what functions they serve in cognition.
3. Are spontaneous gestures international?
> The question of whether gestures are intentional is a really interesting one, and one that there's actually a lot of debate about. People think that they're unconscious. We do have some control about the way that we gesture, and I gesture differently because I'm talking to you face to face than I would, for example, if we were having this conversation over the telephone or over an intercom. My gestures would surely be different. So the gestures that I study are mostly either pointing gestures that point to objects or locations in the physical space where communication is happening, or representational gestures, which are gestures that represent the things that they mean. So if I were talking about a triangle and I made a gesture like this to convey to you what kind of triangle I was talking about, that would be considered a representational gesture; I’m representing what the gesture means. So pointing gestures are typical points; I could point at you or at the camera or at your piece of paper. So these are the kinds of gestures kids often use when they're talking about problems that they're trying to solve, when they're talking about ideas that they have. So we try to set up situations where kids have things to point to and talk about so that we can actually interpret the gestures that they produce.
4. What aspects of mathematical reasoning do you study, and why is this research important?
> One recent focus of my research has been on kids' understanding about the equals sign. The equals sign is a really fundamental symbol in mathematics. It's used, you know, from the earliest years through upper-level mathematics. But it's a symbol that kids often have difficulties with. And so I've been looking at kids' understanding of the equals sign in late elementary school when they're learning about equations, and through the middle-school years. One might think, surely by middle school students have a really good grasp of the equals sign, but in fact we've found that that's not really not the case. Kids tend to think that the equals sign means to put the answer rather than conveying a relation between quantities. And so you can imagine that, for kids who don't deeply understand the equals sign as a relational symbol, the transition into algebraic reasoning is actually much more difficult, because it's hard to understand, for example, why you would do the same thing to both sides of an equation if you don't realize that the equals sign conveys a relationship between the quantities on the two sides of an equation. So we've been tracking student understanding of the equals sign over the late elementary and middle-school years, and looking to see how that relates to kids' performance in solving equations and other sorts of mathematical tasks. One issue is where does this misconception come from? So why do students think the equals sign means "put the answer?" And one possible reason, one possible source of this misconception, is the kind of arithmetic practice that kids get in school. They see overwhelming numbers of examples of problems where there are operations, an equals sign, and then a place to put the answer. So they do lots of arithmetic practice. And so it's actually not surprising, when you think about it, that kids start to infer that the equals sign means "put the answer.” That's what they usually see it, the context in which they usually see it. The equals sign doesn't really mean, "put the answer," and so when kids move on to more complex problems that have operations on both sides, problems, you know, where there isn't necessarily a result, as a single object problems, you know, where they need to do more complex things than simply perform computations, then having a weak understanding of the equals sign can actually be a big stumbling block for kids.
5. What games or practice exercises help children improve their mathematical skills?
> Across studies we've done lots of different variations and things to get kids to think more deeply about what symbols mean. With respect to the equals sign, we've found that simply giving kids exposure to problem structures that don't have operations, and then the equals sign, and then the answer actually leads kids to start thinking about the equals sign in more sophisticated ways. This might be one of the reasons why kids start to understand it in middle school and beyond, because they're starting to see problems in other formats and other structures. And so one of the things that we do to help kids to understand is to actually talk to them about what it means but to give them exposure to problems like seven equals five plus two or six plus three equals five plus four; things they're not used to actually seeing, but that are actually meaningful statements that involve the equals sign that can get them to start potentially thinking about the equals sign in different ways. Another thing that we sometimes do with kids to help them understand these concepts is to make analogies to things that they understand in the real world. So symbols are abstract and the equals sign, it's like a foreign word, you know, a foreign symbol, a symbol in a foreign language, and by -- kids need to sort of ground that symbol to real meaning in the world. So linking the equals sign, for example, to a pan balance that, you know, that the equals sign can represent the situation when the pans and the balance, when the pan balance is level, you know, is a way, an analogy that sometimes is helpful for kids to start thinking about what the equals sign really means as a relation.
6. What surprised you most in your research?
I've consistently been surprised how difficult it is to get kids to let go of this idea that the equals sign means "put the answer." Some kids are ready to learn, and others are more entrenched in their idea that the equals sign means "put the answer," and understanding what makes one child more or less ready than another, you know, to take in new knowledge is I think one of the really interesting questions in cognitive development. Another thing that I've found surprising sort of over the years, I mean, from the early days, I was very surprised that kids' gestures could tell us things that they weren't saying. So a child might talk about a container of water and say, you know, that the container was tall, but they might gesture about the width of the container. You know, that, to me, that initial observation was made by Breckie Church and Susan Goldin-Meadow. That initial observation to me was very surprising, you know, that the gestures could actually say something different from the words? But in fact it's quite common and actually tells us that there's more going on than what kids are saying. And so, it's sometimes surprising, by looking at this other channel, how much one can start to see about what a child actually knows.
7. Describe the importance of gestures for children with language disabilities.
> I've done some work on looking at the gestures of kids who have a language learning disability called Specific Language Impairment. This is impairment that's thought to be specific to language. So these are kids who have typical IQ, good cognitive skills, but they have special difficulty with language. So their performance on standardized tests of language ability is at least a standard deviation below the age typical performance. And one thing that we found in a recent study was that these kids with Specific Language Impairment gestured at nearly twice the rate of their age peers in a narrative task. So, essentially, the task was one in which the kids watched a short cartoon and then needed to tell another person what happened in the cartoon. And the kids with SLI gestured at twice the rate of the typically developing kids, which we found very interesting. So, for these kids, their gestures might be expressing much that they're not saying in speech. And so, for these kids in particular who have strong knowledge but difficulty putting it into words, their gestures might be especially revealing window into their knowledge. And that's an idea that we're continuing to pursue. We also think for those kids and also for kids with other types of abilities, kids who have strengths individual and spatial skills that the gestures of the teachers might be particularly important. So teachers -- in some recent work, we're looking at gestures that teachers use in math classrooms when they're communicating about mathematical ideas to students. They use tons of pointing. They use representational gestures. The rate of gestures is actually pretty high in teaching. It seems higher than in ordinary conversation, although we haven't done a firm statistical comparison of that with the same speakers. But it seems high. And it isn't surprising that teachers would gesture a lot because they care a lot about what they're trying to communicate. So we're looking at what difference that makes for students in classroom settings. We have some early data suggesting that the teachers' gestures do make a difference for students, and we're now trying to look at what real teachers in real classrooms do with gestures so that we can evaluate better, more naturalistic, ecologically valid, you know, ways of looking at gesture, you know, that -- to see what difference they do make for students. And those might be particularly important for students with disabilities who might have difficulty. So kids with SLI might have difficulty taking in what the teacher is saying in the verbal channel. The teacher's gestures might be particularly important for kids with SLI, as well as for kids, you know, who have strong spatial skills but weaker verbal skills, even within the normal range. So we don't have great data to support that yet, but it's a hypothesis that we're interested in testing so.
8. Are there sex differences in children’s gesturing?
> We haven't really observed a lot of gender differences in kids' gesture. We see that both boys and girls produce lots of gestures when they talk. There's lots of individual variations, so some people gesture much more than other people. I'm someone who gestures a lot [chuckling]. And some people don't gesture so much. And so it seems not really to be linked to gender. There are a few studies in the literature, mostly with adults, not any that I know of in kids, that have suggested there may be a gender difference, that women may gesture more than men. But, really, the distributions overlap much more than are different. You know the differences are small when they're observed, and they're not observed in every study. So it seems -- my belief -- and actually I have some data to show that what matters more are things like verbal and spatial skills and how they're combined. And so, in a recent study with Autumn Hostetter, we found that -- this was with adults -- but we found that adults who had a big discrepancy between their verbal skill and their spatial skill, so, in particular, those who had spatial skill that was stronger than their verbal skill, those were the participants who were most likely to gesture, who gestured at the highest rate. So it seems that it isn't really linked to gender. It's more linked to particular patterns of cognitive skills. So it might also be linked to personality and culture and other sorts of factors that come into play. There are a lot of factors that influence the way that people gesture.
9. What is the paper-folding test, and what does it measure?
> We used a standardized test called the Paper Folding Test. It's a test where you see a picture of a piece of paper that's been folded up and a hole is punched in it and then there are, if I remember correctly, there are 5 choices that show how the piece of paper would look when unfolded. And so the paper's punched when it's folded, and so essentially what you have to figure out is where all the holes would be if it were unfolded. And so you get 5 choices and on each of those then people would choose which one; so it's basically a paper and pencil test, so involves sort of mentally imagining and manipulating this hypothetical piece of paper and figuring out which one would match how it looked if it were unfolded.
10. Where do you see your research heading in the future?
> One of the big questions that I'm interested in is how kids' knowledge changes. So I'm interested in mathematical knowledge and how it changes, so I'm interested in this transition from operational understanding of the equal sign to a more relational understanding of the equal sign. And I'm interested in transitions in kids' understanding of variables and transitions in strategy use in kids solving equations and different sorts of problems. And one of the things I'm particularly interested in these days is the role of problem encoding in some of these changes. So how -- what is it that kids notice when we present them with a math problem? What features of the problem do they notice? How does that relate to the knowledge that they bring to bear on the problem? And can we use that to predict either the strategies that kids will use at that point in time or the strategies that they'll use in the near future. So can we look at what they're noticing about the problem and somehow see sort of where they're going to go in development. So that's something I'm very interested in. That links back to the work on gesture, because I think kids' gestures both show us what kids are noticing, and I think that the teachers' gestures might guide children what to notice in the classroom. And so I think that’s actually -- that gesture actually -- particularly pointing gestures, plays an important role in guiding kids' encoding of what's out there in the classrooms. So I'm interested in those kinds of questions. Also interested in sort of the general question of how knowledge changes. So I focused on strategy change for most of what I've talked about, but I'm interested in change more generally. And so done come recent work on kids' understanding of variables, misconceptions that kids have there, and trying to sort of pin down what the different qualities of understanding of variables are. It's another symbol that kids have a lot of difficulty with. And so that's one that we're quite interested in right now. We're looking at math lessons to look at how teachers link pieces of mathematical knowledge together for students. So I've done a lot of work on the nature of mathematical knowledge, so there's conceptual knowledge and skills for solving problems. And we want kids to be able to solve problems in a conceptually guided way. We don't want them just to kind of do stuff. We want them to understand why they're doing it and to be know what -- to be able to use their knowledge to guide them to decide, you know, what approach to use. And so we're interested in how teachers connect concepts and procedures in the classroom, and what impact that has on students, so how developing concepts and procedures together can best be facilitated and what roll teachers' communication plays in that process.