About Me: A member of faculty at UC Berkeley’s Graduate School of Education, I am a Professor of Secondary Mathematics Education in the Area of Cognition and Development. I was born in Haifa, Israel, 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 conceptual 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.
Matific webinar series for the Indian Principals Network
September 30, 2020, India
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 ProLab, 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 has been investigating the sensorimotor roots of proportional reasoning. That’s where the Mathematics Imagery Trainer was born.
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. Also funded by NSF is the Debugging Failure that is evaluating a comprehensive pedagogical approach to fostering effective strategies for youth to cope with the struggles of computer programming. I joined Cal in August, 2005. On Dec. 15, 2011, at approximately 12:54:50 PM PST, I was promoted to Associate Professor with tenure. You can read the statement I submitted in Summer 2010, which elaborates on my work up to then. Then on Sep. 25, 2018, at approximately 15:42:23 PM PST, I was promoted to Full Professor. So now here is that statement, written in Spring 2017.
If you’re interested in joining the lab, please don’t be shy — just drop me an email.
Graduated Doctoral Students Whose Dissertations I Chaired:
- 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.
- Dragan Trninic: Body of Knowledge: Practicing Mathematics in Instrumented Fields of Promoted Action (2015). Then post-doctoral fellow at ETH, Zurich.
- José F. Gutiérrez: Signs of Power: A Critical Approach to the Study of Mathematics Cognition and Instruction (2015). Then post-doctoral fellow at the University of Wisconsin, Madison; now Assistant Professor at the University of Utah, Salt Lake City.
- Tim Charoenying: Fostering Embodied Coherence: A Study of the Relationship Between Learners’ Physical Actions and Mathematical Cognition (2015). Then software developer in Colorado.
- Yue-Ting Siu: A Virtual Water Cooler: The Ecology of an Online Community of Practice to Support Teachers’ Informal Learning (2015). Now tenure-track position at San Francisco State University.
- Kiera Naomi Phoebe Chase: Building Algebra One Giant Step at a Time: Toward a Reverse-Scaffolding Pedagogical Approach for Fostering Subjective Transparency Through Engineering Levels of Interaction With a Technological Learning Environment (2015). Now at Connect-Ed, Berkeley.
- Elisabeth K. Guneratne: From Instrumental Genesis to Digital Exodus: Supporting Urban Elementary Teachers Through Technology-Mediated Systemic Reform (2016). Now at Santa Clara University.
- Virginia J. Flood: Gesture as a dialogic resource in STEM instructional interaction (2020).
- Alyse D. Schneider: Roting Teachers: Constructivist math pedagogy and the development of non-teaching experts in teaching in the United States, 1820-1910 (2021). Now an algebra teacher at Oakland Unified School District.
- Leah F. Rosenbaum: Design and theory of diverse forms of participation in mathematics through Geometris, a collaborative, body-scale game (2021). Now a Post Doctoral fellow at Columbia University.
- Amelia (Milly) Farid: Undergraduate students’ definitional practices in mathematics (2022). Now a Post Doctoral fellow at MIT.
- Rachel S. Y. Chen: Being non-speaking in a speaking world: Surfacing the improvisations of autistic individuals (2022). Now faculty at Nanyang Technological University.
My Journal Articles (for the whole-ish list, go to our publications page)
- Abdu, R., Tancredi, S., Abrahamson, D., & Balasubramaniam, R. (in press). A complex-systems view on mathematical learning as hand–eye coordination. In M. Schindler, A. Shvarts, & A. Lilienthal (Eds.), Eye-tracking research in mathematics education [Special issue]. Educational Studies in Mathematics.
- Shvarts, A., & Abrahamson, D. (2023). Coordination dynamics of semiotic mediation: A functional dynamic systems perspective on mathematics teaching/learning. In T. Veloz, R. Videla, & A. Riegler (Eds.), Education in the 21st century [Special issue]
- Abrahamson, D. (2023). Almost in our grasp: The (slow) digital return of multimodal educational resources <Commentary> <Special issue>
- Benally, J., Palatnik, A., Ryokai, K., & Abrahamson, D. (2022). Learning through negotiating conceptually generative perspectival complementarities: The case of geometry. For the Learning of Mathematics, 42(3), 34–41.
- Abrahamson, D., & Mechsner, F. (2022). Toward synergizing educational research and movement sciences: A dialogue on learning as developing perception for action. Educational Psychology Review, 34(3), 1813-1842. https://doi.org/10.1007/s10648-022- 09668-3
- Botetano, C., & Abrahamson, D. (2022). The Botetano arithmetic method: Introduction and early evidence. International Journal of Mathematical Education in Science and Technology, 53(2), 516-534. https://doi.org/10.1080/0020739X.2020.1867916
- Lambert, S. G., Tancredi, S., Fiedler, B. L., Moore, E. B., Gorlewicz, J. L., Abrahamson, D., & Gomez Paloma, F. (2022). Getting a grip on geometry: Developing a tangible manipulative for inclusive quadrilateral learning. Italian Journal of Health Education, Sports and Inclusive Didactics, 6(1), 1–21. https://doi.org/10.32043/gsd.v6i1.604
- Lambert, S. G., Fiedler, B. L., Hershenow, C. S., Abrahamson, D., & Gorlewicz, J. L. (2022). A tangible manipulative for inclusive quadrilateral learning. The Journal on Technology and Persons with Disabilities, 10, 66–81. [Award: Best Submission—Dr. Arthur I. Karshmer Award for Assistive Technology Research]
- Pardos, Z., Rosenbaum, L., & Abrahamson, D. (2021). Characterizing learner behavior from touchscreen data. In D. Abrahamson, M. Worsley, Z. Pardos, & L. Ou (Eds.), Learning analytics of embodied design: Enhancing synergy <Special issue>. International Journal of Child–Computer Interaction, 100357. https://doi.org/10.1016/j.ijcci.2021.100357
- Tancredi, S., Abdu, R., Abrahamson, D., & Balasubramaniam, R. (2021). Modeling nonlinear dynamics of fluency development in an embodied-design mathematics learning environment with Recurrence Quantification Analysis. International Journal of Child–Computer Interaction.
- Abrahamson, D., Worsley, M., Pardos, Z., & Ou, L. (2021). Learning analytics of embodied design: Enhancing synergy. International Journal of Child-Computer Interaction, 100409. https://doi.org/10.1016/j.ijcci.2021.100409
- Abrahamson, D. (2021). Enactivist how? Rethinking metaphorizing as imaginary constraints projected on sensorimotor interaction dynamics. Constructivist Foundations, 16(3), 275–278. https://constructivist.info/16/3/275
Abrahamson, D. (2021). Grasp actually: An evolutionist argument for enactivist mathematics education. Human Development, 65(2).
- Tancredi, S., Chen, R. S. Y., Krause, C., Abrahamson, D., & Gomez Paloma, F. (2021). Getting up to SpEED: Special Education Embodied Design for sensorially equitable inclusion. Education Sciences and Society—Open Access, 12(1), 113–136. https://doi.org/10.3280/ess1-2021oa11818
- Ba, H., & Abrahamson, D. (2021). Taking design to task: A dialogue on task initiation in STEM activities. Educational Designer, 4(14), 1–21. http://www.educationaldesigner.org/ed/volume4/issue14/article54/
- Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020). The future of embodied design for mathematics teaching and learning <Original Research>. Frontiers in Education, 5(147). https://doi.org/10.3389/feduc.2020.00147
- Flood, V. J., Shvarts, A., & Abrahamson, D. (2020). Teaching with embodied learning technologies for mathematics: Responsive teaching for embodied learning. ZDM Mathematics Education, 52(7), 1307-1331. https://doi.org/10.1007/
- Abrahamson, D., & Abdu, R. (2020). Towards an ecological–dynamics design framework for embodied-interaction conceptual learning: The case of dynamic mathematics environments. In T. J. Kopcha, K. D. Valentine, & C. Ocak (Eds.), Embodied cognition and technology for learning <Special issue>. Educational Technology Research and Development.
- Rosenbaum, L. F., Kaur, J., & Abrahamson, D. (2020). Shaping perception: Designing for participatory facilitation of collaborative geometry. In R. Nemirovsky & S. Soury-Lavergne (Eds.), On the intertwined contributions of physical and digital tools for the teaching and learning of mathematics <Special issue>. Digital Experiences in Mathematics Education. 6(2), 213–232. https://doi.org/10.1007/
- Abrahamson, D. (2020). Strawberry feel forever: Understanding metaphor as sensorimotor dynamics. The Senses and Society, 15(2), 216-238. https://doi.org/10.1080/17458927.2020.1764742
- Shvarts, A., & Abrahamson, D. (2019). Dual-eye-tracking Vygotsky: A microgenetic account of a teaching/learning collaboration in an embodied-interaction technological tutorial for mathematics. Learning, Culture, and Social Interaction, 22, 100316. https://doi.org/10.1016/j.lcsi.2019.05.003
- Abrahamson, D., & Shulman, A. (2019). Co-constructing movement in mathematics and dance: An interdisciplinary pedagogical dialogue on subjectivity and awareness. Feldenkrais Research Journal, 6, 1–24.
- Abrahamson, D., Flood, V. J., Miele, J., & Siu, Y.-T. (2019). Enactivism and ethnomethodological conversation analysis as tools for expanding Universal Design for Learning: The case of visually impaired mathematics students. ZDM Mathematics Education, 19(2), 291-303.
- Morgan, P., & Abrahamson, D. (2018). Applying contemplative practices to the educational design of mathematics content: Report from a pioneering workshop. The Journal of Contemplative Inquiry, 5(1), 107-119.
- Green, C. A., Abrahamson, D., Chern, H., & O’Sullivan, P. S. (2018). Is robotic surgery highlighting critical gaps in resident training? Journal of Graduate Medical Education, 10(5), 491–493.
- Palatnik, A., & Abrahamson, D. (2018). Rhythmic movement as a tacit enactment goal mobilizes the emergence of mathematical structures. Educational Studies in Mathematics, 99(3), 293–309.
- Zohar, R., Bagno, E., Eylon, B., & Abrahamson, D. (2018). Motor skills, creativity, and cognition in learning physics concepts. Functional Neurology, Rehabilitation, and Ergonomics, 7(3), 67–76
- Barth–Cohen, L. A., Little, A., & Abrahamson, D. (2018). Building reflective practices in a pre-service math and science teacher education course that focuses on qualitative video analysis. Journal of Science Teacher Education, 29(2), 83–101.
- Chase, K., & Abrahamson, D. (2018). Searching for buried treasure: Uncovering discovery in discovery-based learning. In D. Abrahamson & M. Kapur (Eds.), Practicing discovery-based learning: Evaluating new horizons <Special issue>. Instructional Science, 46(1), 11–33.
- Abrahamson, D., & Kapur, M. (2018). Reinventing discovery learning: A field-wide research program. In D. Abrahamson & M. Kapur (Eds.), Practicing discovery-based learning: Evaluating new horizons <Special issue>. Instructional Science, 46(1), 1–10.
- Duijzer, A. C. G., Shayan, S., Van der Schaaf, M. F., Bakker, A., Abrahamson, D. (2017). Touchscreen tablets: Coordinating action and perception for mathematical cognition. Frontiers in Psychology, 8(144).
- Zohar, R., Bagno, E., Eylon, B., & Abrahamson, D. (2016). Creativity and cognition in embodied learning of physics concepts. Dance Now, 32, 24–31. (Original work published in Hebrew, in Israel
- Abrahamson, D., & Bakker, A. (2016). Making sense of movement in embodied design for mathematics learning. In N. Newcombe & S. Weisberg (Eds), Embodied cognition and STEM learning <Special issue>. Cognitive Research: Principles and Implications (CRPI), 1(1), Article #33
- Morgan, P., & Abrahamson, D. (2016). Cultivating the ineffable: The role of contemplative practice in enactivist learning. For the Learning of Mathematics, 36(3), 31–37.
- Abrahamson, D., & Sánchez–García, R. (2016).Learning is moving in new ways: The ecological dynamics of mathematics education. Journal of the Learning Sciences, 25(2), 203–239
- Abrahamson, D., Shayan, S., Bakker, A., & Van der Schaaf, M. F. (2016). Eye-tracking Piaget: Capturing the emergence of attentional anchors in the coordination of proportional motor action. Human Development, 58(4-5), 218–244.
- Chase, K., & Abrahamson, D. (2015). Reverse-scaffolding algebra: Empirical evaluation of design architecture. In J. Smit, A. Bakker, & R. Wegerif (Eds.), Scaffolding and dialogic teaching in mathematics education <Special issue>. ZDM Mathematics Education, 47(7), 1195–1209
- Abrahamson, D. (2015). Reinventing learning: A design-research odyssey. In S. Prediger, K. Gravemeijer, & J. Confrey (Eds.), Design research with a focus on learning processes <Special issue>. ZDM Mathematics Education, 47(6), 1013–1026.
- Hutto, D. D., Kirchhoff, M. D., & Abrahamson, D. (2015). The enactive roots of STEM: Rethinking educational design in mathematics. In P. Chandler & A. Tricot (Eds.), Human movement, physical and mental health, and learning <Special issue>. Educational Psychology Review, 27(3), 371–389.
- Abrahamson, D., & Chase, K. (2015). Interfacing practices: Domain theory emerges via collaborative reflection. Reflective Practice, 16(3), 372–389.
- Abrahamson, D., & Trninic, D. (2015). Bringing forth mathematical concepts: Signifying sensorimotor enactment in fields of promoted action. In D. Reid, L. Brown, A. Coles, & M.-D. Lozano (Eds.), Enactivist methodology in mathematics education research <Special issue>. ZDM Mathematics Education, 47(2), 295–306.
- Abrahamson, D. (2014). Building educational activities for understanding: An elaboration on the embodied-design framework and its epistemic grounds. International Journal of Child Computer Interaction, 2(1), 1–16.
- Abrahamson, D., Lee, R. G., Negrete, A. G., & Gutiérrez, J. F. (2014). Coordinating visualizations of polysemous action: Values added for grounding proportion. In F. Rivera, H. Steinbring, & A. Arcavi (Eds.), Visualization as an epistemological learning tool <Special issue>. ZDM Mathematics Education, 46(1), 79–93.
- Abrahamson, D. (2012). Seeing Chance: Perceptual reasoning as an epistemic resource for grounding compound event spaces. In R. Biehler & D. Pratt (Eds.), Probability in reasoning about data and risk <Special issue>. ZDM—The International Journal on Mathematics Education, 44(7), 869–881.
- Abrahamson, D., Gutiérrez, J. F., Charoenying, T., Negrete, A. G., & Bumbacher, E. (2012). Fostering hooks and shifts: Tutorial tactics for guided mathematical discovery. Technology, Knowledge, and Learning, 17(1-2), 61–86.
- Abrahamson, D. (2012). Rethinking intensive quantities via guided mediated abduction. The Journal of the Learning Sciences. 21(4), 626–649.
- Abrahamson, D. (2012). Discovery reconceived: Product before process. For the Learning of Mathematics, 32(1), 8–15.
- Abrahamson, D., Gutiérrez, J. F., & Baddorf, A. K. (2012). Try to see it my way: The discursive function of idiosyncratic mathematical metaphor. Mathematical Thinking and Learning, 14(1), 55–80.
- Abrahamson, D., Trninic, D., Gutiérrez, J. F., Huth, J., & Lee, R. G. (2011). Hooks and shifts: A dialectical study of mediated discovery. Technology, Knowledge, and Learning, 16(1), 55–85
- Veeragoudar Harrell, S., & Abrahamson, D. (2010). Second Life unplugged: A design for fostering at-risk students’ STEM agency. In H. Gazit, D. L. Garcia, G. LeMasers, & L. Morgado (Eds.), The metaverse assembled <Special Issue>. Journal of Virtual Worlds Research. Retrieved Aug. 12, 2010, at https://journals.tdl.org/jvwr/article/view/834/716
- Abrahamson, D. (2009). A student’s synthesis of tacit and mathematical knowledge as a researcher’s lens on bridging learning theory. In M. Borovcnik & R. Kapadia (Eds.), Research and developments in probability education <Special Issue>. International Electronic Journal of Mathematics Education, 4(3), 195–226. Retrieved Aug. 12, 2010 from International Electronic Journal of Mathematics Education
- Abrahamson, D. (2009). Orchestrating semiotic leaps from tacit to cultural quantitative reasoning—the case of anticipating experimental outcomes of a quasi-binomial random generator. Cognition and Instruction, 27(3), 175–224.
- Abrahamson, D. (2009). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47.
- Abrahamson, D., & Wilensky, U. (2007). Learning axes and bridging tools in a technology-based design for statistics. International Journal of Computers for Mathematics Learning, 12(1), 23–55.
- Abrahamson, D., Janusz, R., M., & Wilensky, U. (2006). There once was a 9-Block…—A middle-school design for probability and statistics. Journal of Statistics Education, 14(1). Retrieved August 12, 20010, at http://www.amstat.org/publications/jse/v14n1/abrahamson.html.
- Abrahamson, D., Berland, M. W., Shapiro, R. B., Unterman, J. W., & Wilensky, U. J. (2006). Leveraging epistemological diversity through computer-based argumentation in the domain of probability. For the Learning of Mathematics, 26(3), 39–45.
- Abrahamson, D. (2006). The shape of things to come: The computational pictograph as a bridge from combinatorial space to outcome distribution. International Journal of Computers for Mathematics Learning, 11(1), 137–146.
- Abrahamson, D., & Cigan, C. (2003). A design for ratio and proportion. Mathematics Teaching in the Middle School, 8(9), 493–501.
A Ramble: 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.
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:
Some Favorite Quotations:
- “But men may construe things after their fashion, clean from the purpose of the things themselves.” (William Shakespeare, Julius Caesar, 1623)
- “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).”
- “Careful 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)
- “To have only intelligence and talent is too little. One must also have energy, real interest, clarity of thought and a sense of obligation. …. Always write with attention and look on writing as a holiday.” (Daniil Kharms, 1937)
- “If I had asked people what they wanted, they would have said faster horses.” (attributed to Henry Ford)
- “Rules, like birds, must live before they can be stuffed.” (Gilbert Ryle, The Presidential Address, 1945)
- “There is nothing more practical than a good theory.” (Kurt Lewin, Selected Theoretical Papers, 1952)
- “Let be be the finale of seem.” (Wallace Stevens, in The Emperor of Ice-Cream, 1954)
- “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)
- “The 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)”
- “Mathematics, 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).
- “Insight is not necessarily gained by increasingly accurate quantitative descriptions of data, or by using increasingly complicated dynamical equations. Rather, we seek to account for a larger number of experimental features with a smaller number of theoretical concepts.” (Kelso, Schöner, Scholz, & Haken, Phase-Locked Modes, Phase Transitions and Component Oscillators in Coordinated Biological Motion, 1987, p. 86)
- “Space, for example, is not discretized. We’ve held onto a continuous intuition of it. In the same way, time appears continuous to us.” (René Thom, Prédire N’est Pas Expliquer — Interview with Emile Noël, 1991, p. 80)
- “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)
- “Which teaching methods, instead of transmitting contents, could elicit the gestures which allow access to the source experience that gives these contents coherence and meaning? Such a teaching approach, based more on initiation than transmission, by enabling children and students to come into contact with the depth of their experience, could re-enchant the classroom.” (Petitmengin, Towards the source of thoughts, 2007, p. 79)
- “When student is ready, teacher will appear.” (Bhante Sujatha, 2014)
- “We often confuse freedom with arbitrariness as though freedom were tantamount to doing something in a random way. But we are only really free or rather we savour our freedom when what we do is the necessary expression of what we are….This is what we mean by identity.” (Riccardo Manzotti, Dialogues on Consciousness, 2020, with Tim Parks)
- “Movere ergo cogito.” (I move, therefore I am)
Confession (well, you deserve one, gentle reader, having read all of this!): My nerdiest pastime is to find mean anagrams to the names of my numerous archenemies here. Oh, and another totally useless bit of information: My shortest publication ever is here, on p. 442.