About Me: A member of faculty at UC Berkeley's Graduate School of Education, I am an Associate Professor of Secondary Mathematics Education in the Area of Cognition and Development. I was born in Haifa, a gorgeous, diverse, and harmonious city straddling the ever-green Mt. Carmel on the Mediterranean littoral. I got an MA in cognitive psychology from Tel Aviv University and a Ph.D. in Learning Sciences at Northwestern University, then was a postdoctoral fellow at the Center for Connected Learning and Computer-Based Modeling. Also, I am a cellist. And a dad :)
Research Interests: I care about mathematics learning. I am equally passionate about building activities for learning and building theory of learning. Design-based research allows me to do both. I use the practice of designing activities as a context for conducting empirical research, and then findings from the research lead to new insights about the nature of learning, which in turns lead to better design, and so on. In a sense, I investigate processes of learning and teaching through attempting to enhance their potential: I study not "what is" but "what could be." The lab works with technologies ranging from paper and marbles to touchscreen tablets, agent-based simulations, motion capture, eye-tracking sensors, and artificial intelligence. We're trying to scale up our reach, so that any kid with access to a computer can work with our activities.
I publish about theory of learning and about methods for creating activities for learning. My focus has been on creating and modeling situations in which learners coordinate between informal and formal views of phenomena, because that is how I think about concpetual learning. I want to empower students to "get it right," and so I build situations where they can use their naive sensory perceptions, intuitive judgments, and physical coordination to interact productively with a set of materials, but then I introduce mathematical perspectives as complementary ways to "get it right." I study the micro-moments as students figure out what the mathematical views enable them, whether they should adopt these alternative ways of engaging the world. I am curious how teachers facilitate these fragile processes through speech and gesture, cues, clues, and metaphors. I use constructivism, social cultural theory, and ecological dynamics so as to make sense of these reconciliations.
|The Abduction of Peirce: The missing link between perceptual judgment and mathematical reasoning?
(Berkeley neuro-philosophy, 2008)
|Body of Knowledge: Grounding Mathematical Concepts in
(Learning & the Brain, 2015)
|Cultivating Mathematical Concepts: Insights From Ecological Dynamics (Utrecht symposium on
embodied design, 2015)
|The Eye Trick: grounding proportion in perceptual judgment||Project Proportion: learning proportion with the multiplication table||ProbLab: a suite of simulated probability experiments||Seeing Chance: concrete and virtual probability experiments||Kinemathics: fostering the sentorimotor roots of proportion||GEVAMT: building a virtual animated mathematics tutor|
I've researched a range of mathematical concepts. For my doctoral dissertation, I used the multiplication table as a source for learning about proportion. During my post-doc, I worked in a computational modeling environment to build ProbLab, a suite of interactive simulations for probability. At Berkeley, the Seeing Chance project further investigated whether students could coordinate informal and formal visualizations of randomness. Then Kinemathics investigated the sensorimotor roots of proportional reasoning.
Current Research Focus: Funded by the National Science Foundation, we are collaborating with UC Davis researchers to create Maria the humanoid virtual pedagogical agent. Maria lives in a large touchscreen. She teaches you how to move your hands in new ways that lead to learning new mathematical concepts.
If you're interested in joining the lab, please don't be shy -- just drop me an email.
Graduated Doctoral Students:
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:
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.