When chefs write recipes, they put those instructions on paper not by looking at the words they've written and wondering which other words would be a good fit. Rather, the chefs imagine and experience sensations of flavor. You know, "Aha! If I only added some lime to that salsa, it would be fantastic on the tilapia!" When composers write music, they don't ask themselves which are the most logical notations to put in the musical score. Rather, the composers imagine and experience sensations of sound. That's actually called to "audiate." You know, "Oh, if only a horn played the second theme now in a lower register, that would be mesmerizing!" So they make their decisions based on inspiration, creativity, and sense of structural constraints. Only then they put that down on paper in the musical score. And when mathematicians write proofs or solve problems, they don't expect the meanings to be in the symbols themselves. Rather, meanings are experienced as sensations. Now, these are not sensations of flavor or sound but of action in space, action on imagined objects that are experienced as real or realistic, even if they are not substantive in the sense of concrete things in the actual world, and even if they are not truly palpable -- so I am speaking about visual, kinesthetic, and somatic experiences. It's only after they think that way that mathematicians articulate their thoughts on paper or the blackboard. All this has been known for a century now, through ethnography and philosophy, and recently through empirical research. This is not at all to mitigate the challenge of articulating sensations in formal signs. Rather, it is to say that there is no point in education programs that ignore the formative role of embodied reasoning in mathematics learning. Embodied design is an approach to instruction that recognizes the formative role of sensations and motor-action in generating new meaning and then negotiating these new meanings with the formal mathematical structures that children are to adopt, understand, and apply as complementary to, and empowering of their sensations. We view this approach as vital, and we view technology as offering new opportunities for implementing these transformational experiences in classrooms. Our research base offers examples for the effectiveness of this approach. In all this, we recognize that the educational transformations we envisage are systemic: they demand of all stakeholders -- policy makers, teachers, students, and perhaps parents -- to think in new ways of what it even means to understand mathematical concepts.
About Me: I am an Associate Professor of Secondary Mathematics Education in the Area of Cognition and Development at UC Berkeley's Graduate School of Education. I was born in Haifa, a gorgeous, diverse, and harmonious city straddling the ever-green Mt. Carmel on the Mediterranean littoral. My background is in cognitive psychology, which I studied at Tel Aviv University. Then I achieved a Ph.D. in Learning Sciences at Northwestern University and continued as a postdoctoral fellow at the Center for Connected Learning and Computer-Based Modeling. I am a cellist.
Research Interests: I'm a mathematics-education researcher. Specifically, I am a design-based researcher of mathematics education, which means that I use the practice of pedagogical design as a context for my empirical research. That is, I investigate processes of learning and teaching through attempting to enhance their potential. These investigations are informed by, and lead to the development of theoretical models for understanding and critiquing these phenomena. In turn, these studies inform the development of frameworks for effective design. In March 2014 I gave a ISLS NAPLeS webinar in which I explain my work.
Ultimately, my research is about meaningful learning of mathematical concepts, and so I examine how students build and keep meaning for the cultural-historical forms that have evolved in the service of the mathematical discipline. In my attempts to promote and understand meaningful learning, I have been greatly inspired by Luis Radford's semiotic-cultural theory of objectification, because the theory has enabled me to conceptualize meaningful learning the way I had come to perceive it through my studies, that is, as the intentional, guided, mediated enmeshing of subjective sensations and received structures. Moreover, the theory has enabled me to hone my research on micro-events where the clash of subjective and received resources culminates. I am particularly interested in how students and instructors' work together to reconcile these oppositional resources. Specifically, if naive and mathematical visualizations of phenomena differ, how do learners adopt the mathematical view while keeping the naive meaning? For example, how do students adopt a rise-over-run analysis of a sloped line whose steepness they sense as a holistic, presymbolic, embodied sensation of intensity or effort?
As a learning scientist practicing the design-based research approach, my empirical work explores the viability of integrating cognitive, sociocultural, and semiotics theory as a means of better understanding and supporting effective artifact-mediated interactions between experts and novices. As such, my research spirals through engineering, implementing, and evaluating technological systems supporting deep conceptual learning. Typically, these empirical studies and data analyses bear threefold systemic impact in the form of actual prototypes, theoretical models of cognition and instruction, and heuristic design frameworks for utilizing emerging technology.
I am currently exploring embodied-interaction learning environments as multimodal inquiry spaces for teachers and students to negotiate the conceptual core of fundamental mathematical notions, such as proportionality. In this project, students remote-control virtual objects as a means of effecting a targeted goal state of a technological system. Via guided, recursive problem-solving that includes trial, error, and reflection, students discover and rehearse a new spatial-dynamical bi-manual action sequence -- a "math kata" -- and then signify this embodied skill using mathematical means of enactment and representation that we introduce into the space. We have documented cases of students who bootstrapped curricular concepts by spontaneously engaging available mathematical forms to enhance and explain their embodied performances. By investigating informal-formal interfaces such as these, we attempt to theorize how culture re-analyzes and signifies tacit perceptuomotor activity and how individuals come to appropriate these significations. For example, students' operational strategies focused on gradually increasing the spatial interval between their two moving hands are recruited and re-described as proportional growth of two individual hands moving at constant yet different rates. I am interested in how individual learners wrestle with these imposed coordinations. By understanding these processes, I hope to inform the practice of instructors charged with facilitating these processes.
For the main, our research laboratoy creates situations for which students can use their tacit knowledge to arrive at mathematically correct intuitive inferences. Next, we have the students construct mathematical models of these situations, only that we have modified these canonical models -- still keeping their mathematical integrity -- so that the students can bring to bear similar tacit knowledge in making sense of these tools. For example, we had students build the sample space of a binomial experiment and guided them to arrange it in a form resembling a histogram of the anticipated outcome distribution of the experiment. Typically, students negotiated between their intuitive ways of looking at the world and the formal ways that they are to adopt. The cognitive process of reconciling these two views is the very stuff that conceptual learning is made of. I also explore the impact of Complexity Studies perspectives and methodologies on education research and have been arguing for the use of agent-based modeling to advance theory of individual learning in social context. During my tenure as a Spencer Postdoctoral Fellow, I developed computer-based modules for learning probability. I have published in the Handbook of Mathematical Cognition
, Cognition and Instruction
, Educational Studies in Mathematics
, Technology, Knowledge, and Learning
(member of the Editorial Board), For the Learning of Mathematics
, The Journal of the Learning Sciences
(member of the Editorial Board), Mathematics Teaching in the Middle School
, Mathematical Thinking and Learning,
the Journal of Statistics Education
, and ZDM: The International Journal on Mathematics Education
, and I contribute regularly to major national and international conferences. An informal introduction to my work can be seen here
Current Research Focus
: In our current project, "Kinemathics: Kinesthetically Induced Mathematical Imagery
," we are studying the emergence of mathematical concepts from embodied-interaction inquiry. To do this, we hacked Wii remote-action technology. Students problem-solve a mystery device by manipulating virtual objects on a large computer display. We then layer mathematical objects onto the display, such as a Cartesian grid, and observe how, if at all, students adopt these objects and, so doing, adapt their reasoning. This is very exciting work, and we have been inspired in particular by our insights on the nature of artifact-mediated mathematics instruction. You can link here
to a couple of video demos, a few conference papers, and a journal article. These days, we're converting the design to Xbox Kinect.
As a development of the Kinemathics project, we recently were awarded a grant from the National Science Foundation, along with our UC Davis collaborators: Gesture Enhancement of Virtual Agent Mathematic Tutors
. The idea is to create an avatar tutor who faces you on the other side of a large touchscreen and teaches you how to move your hands in new ways that later become signified mathematically.
Also, I have become increasingly interested in identifying and articulating the tacit, informal pedagogical practice of parents in families with academic traditions. As a mother interacts with her toddler who is problem-solving a mechanical toy, how might she best balance his challenge and frustration, so that the child develops a sense of self-efficacy and the epistemic disposition that he can "solve the world?" How might we leverage insight into these implicit mediation strategies so as to document efficacious heuristics and, moreover, embed them in widely disseminated artifacts that will contribute to a broader inclusion of underprivileged populaces into intellectual trajectories?
On Dec. 15, 2011, at approximately 3:54:50 PM EST, I was promoted to Associsate Professor with tenure. You can read the statement
I submitted in Summer 2010, which elaborates on my work up to then.
Graduated Doctoral Students:
- Sneha Veeragoudar Harrell: Second Chance in First Life: Fostering Mathematical and Computational Agency Among At-Risk Youth (2009; then two-year post-doctoral fellow at TERC)
Message to the World
When you think of math, what do you see?
I bet you saw all kinds of equations and stuff, right? Ok, that makes sense, because that's what we see in textbooks, that's what the teachers write on the board, that's what we write on the test, and that's, well, what we see of math most of our lives. I mean, what else could you see?
Well, it turns out from research in mathematics education that being able to solve mathematical problems by following some sequence of rules -- an algorithm or solution procedure for writing and manipulating symbols -- is perhaps important for passing a test that assesses that kind of knowledge, but that the rules are soon forgotten and no sense of the content remains.
Kids study, yes, but they don't learn.
Actually, it turns out, thinking mathematically -- I mean really thinking, not just plug-and-chugging numbers -- is more about bringing to mind all kinds of ideas and manipulating these ideas creatively.
Hmm... the idea of an "idea" is a bit vague, isn't it? I'll get back to that in a moment. So bear with me.
But what this all means is that what many mathematics teachers are teaching and many tests are testing are actually just the tip of the iceberg in terms of what kids need in order to think mathematically; in order to really understand.
So, sure, I share with all educators the desire that children will learn and become fluent in mathematics. I mean, I'm a dad to kids in the educational system, and I want them to do well. Yet I'm concerned whether kids are actually making any sense of what they are learning rather than just following rote procedures, in the good case.
In fact, my life work is to create and evaluate new types of activities for kids to develop deep understandings of mathematical concepts. So I'm certainly not some kind of mad scientist up in the ivory tower inventing some weird ideas that are detached from the terrestrial need for core fluency with symbols. I've tutored thousands of hours, and I've been thinking about mathematics education for 20 years now.
Still, my worry is that basic sense-making is the most important yet most neglected aspect of mathematics education, to a great deal because of the accountability pressures that teachers experience. Because we teach for tests, we throw out the sense-making baby with the... wait, that metaphor didn't work, so let me try another one. Oh yes, the assessment tail is waging the sense-making dog. Hmm... not quite. Well, you know what I mean -- kids don't get math.
It starts from about fractions, right? The big crisis. Why there? I could tell you about the difficult transition from counting numbers to rational numbers. There's lots of research on that. But what is at the heart of this crisis? I mean, sure, so we can write papers about this until the cows come home. Or jump over the moooooon. But what can we do about it?
Let me tell you a story.
Just the other day, during my office hours -- I'm a professor you see -- I was chatting about this to an eager undergraduate student who is planning to become a teacher (yoohoo!). So I asked her what ‘addition’ is -- what is the meaning of the arithmetic operation of “addition”. She answered, “It’s just, you know” [gesture 1 + 1 = 2 using “magic” finger trick ]. And I said, yes, that’s certainly one way of thinking of addition. Now what is multiplication? And she said, “Well, you keep adding the same amount over an over, like in 5 times 3.” [gestures gyrating hand with 3 fingers splayed, 5 wrist rotations] Again I said, yes, that’s it. So I asked, “What is proportion?”, and she said, well… you have two numbers like in a ratio, and then you multiply them both by the same number and you get another ratio that is equivalent.” But this time she only gestured the numerical procedure - like, what you do to the digits themselves -- she didn’t seem to have any image underlying this rote procedure, she didn't handle the quantities themselves. That is, sure, a proportion is the set of two ratios that reduce to the same quotient. Ok, does that help you, uhhhhm, 'feel' what a proportion is? I mean, right, you told me how to make a proportion but what *is* a proportion -- what does it look like?
That's what I care for -- helping kids build useful mathematical pictures in the head. Well, not exactly pictures and not exactly in the head -- more like whole-body dynamic schematic images, sort of physically moving pictures... very simple pictures, like the "addition" picture we did with the fingers before [gestures 1+1 like before], so that we can all have a familiar gesture for proportion just like we have for addition. Let me show you what I mean. [enter demo]
Some Favorite Quotations:
- "What is the use of all these symbols; why not begin by showing him the real thing so that he may at least know what you are talking about? .... As a general rule—never substitute the symbol for the thing signified, unless it is impossible to show the thing itself; for the child’s attention is so taken up with the symbol that he will forget what it signifies." (Jean-Jacques Rousseau, Emile, Book III, 1762)
- “[If someone says to you] ‘I struggled but still did not discover,’ do not believe him” [Talmud, Megila 6b], because the struggle in and of itself is a great discovery, a great find indeed.” (Rabbi Menachem Mendel of Kotzk, 1787 – 1859)
- "Let us fix our attention not on the line as line, but on the action which traces it." (Henri Bergson, The Creative Mind: An Introduction to Metaphysics, 1903)
- "It is by logic we prove, it is by intuition that we invent." (Jules Henri Poincaré, Mathematical Definitions in Education, 1904)
- "[C]areful inspection of methods which are permanently successful in formal education….in arithmetic….will reveal that they depend for their efficiency upon the fact that they go back to the type of the situation which causes reflection out of school in ordinary life. They give the pupils something to do, not something to learn; and the doing is of such a nature as to demand thinking, or the intentional noting of connections; learning naturally results." (John Dewey, Democracy and Education, 1916)
- "The tasks which face the human apparatus of perception at the turning points of history cannot be solved by optical means, that is, by contemplation, alone. They are mastered gradually by habit, under the guidance of tactile appropriation." (Walter Benjamin, The Work of Art in the Age of Mechanical Reproduction, 1935)
- "You can kill the King without a sword and you can light a fire without matches. What needs to burn is your imagination." (Constantin Stanislavski, An Actor Prepares, 1936)
- "Telling a kid a secret he can find out himself is not only bad teaching, it is a crime. Have you ever observed how keen six year olds are to discover and reinvent things and how you can disappoint them if you betray some secret too early? Twelve year olds are different; they got used to imposed solutions, they ask for solutions without trying" (Hans Freudenthal, Geometry between the Devil and the Deep Sea, 1971, p. 424)
- “[T]he ultimate truth of the puzzle: in spite of appearances, it is not a solitary play: each move made by the puzzle solver, the puzzle maker made before him; each piece that he takes and takes again, that he examines, that he caresses, each combination that he tries and tries again, each blind groping, each intuition, each hope, each discouragement, were decided, calculated, studied by the other.” (George Perec, Life: A User's Manual, 1978)
- "Of course naturalists seek to recast their tacit knowledge into propositional form as soon as possible, since, without so doing, they cannot communicate with others, and probably not even with themselves, about their findings. Yet to confine the inquiry itself only to those things that can be stated propositionally before the fact is unduly and insensibly limiting from the naturalist's viewpoint, since it eliminates to a large extent the predominant characteristic warranting the use of the human-as-instrument." (Guba & Lincoln, Educational Technology Research and Development, 1982, p. 245)
- "Most human beings do not see themselves, or their minds, as serving the process of evolution. Nevertheless, it would represent a major phase change in the evolution of human consciousness for such a realization to occur and to be acted upon. (Jonas Salk, Anatomy of Reality: Merging of Intuition and Reason, 1983, p. 80)"
- "[M]athematics, like music, needs to be expressed in physical actions and human interactions before its symbols can evoke the silent patterns of mathematical ideas (like musical notes), simultaneous relationships (like harmonies) and expositions or proofs (like meoldies)." (Richard Skemp, The Silent Music of Mathematics, 1983, p. 288)
- "The computer will have a profound impact on our educational system, but whether it will enrich the lives of students will depend upon our insight and our imagination." (Abelson & diSessa, Turtle Geometry, 1986, p. xv)
- "In ontological designing, we are doing more than asking what can be built. We are engaging in a philosophical discourse about the self—about what we can do and what we can be. Tools are fundamental to action, and through our actions we generate the world. The transformation we are concerned with is not a technical one, but a continuing evolution of how we understand our surroundings and ourselves—of how we continue becoming the beings that we are." (Winograd & Flores, Understanding Computers and Cognition: A New Foundation for Design, 1986, p. 179).
- "People have very powerful facilities for taking in information visually or kinesthetically, and thinking with their spatial sense. On the other hand, they do not have a very good built-in facility for inverse vision, that is, turning an internal spatial understanding back into a two-dimensional image. Consequently, mathematicians usually have fewer and poorer ﬁgures in their papers and books than in their heads. (Thurston, On proof and progress in mathematics, 1994, p. 164)"
- "Until cognitive scientists recognize this essential role of the body, their work will remain a mixed bag of ad hoc successes and, to them, incomprehensible failures." (Dreyfus & Dreyfus, in Perspectives on Embodiment: The Intersections of Nature and Culture, 1999)
- "To nooger; namely, to ootz a finger into a sticky place and wiggle it, pinch it, insinuate it until you find a way through without poking a hole into something important. .... I can't think, exactly, how I learned it; or how properly to teach it. But I did, both." (Sid Schwab, M.D., Surgeon)
- "If you know something, then you know it -- it's the answer. But if you understand something, then you understand the question to get you to the answer, kind of." (DB, Grade 5 student, Seeing Chance project, California, Dec. 2005)
- "Cogito ergo soma." (me, now)
Confession (well, you deserve one, gentle reader, having read all of this!): My nerdiest passtime is to find mean anagrams to the names of my numerous archenemies here.