The phrase “embodied design” was first coined by van Rompay and Hekkert (2001), industrial designers who used Lakoff and Johnson's cognitive semantics theory of conceptual metaphor to predict the emotional affect that humans would attribute to architectural structures, such as bus stops. There was also a fleeting unarchived use of the phrase by Thecla Schiphorst, circa 2007. Abrahamson recycled the phrase "embodied design" into the learning sciences to describe an approach to the construction of pedagogical materials and activities that enables learners to objectify their tacit knowledge in cultural forms relevant to disciplinary content (Abrahamson, ESM 2009). (An earlier paper appeared in the proceedings of PME-NA 2007.) Later Abrahamson elaborated thus: “Embodied design is a pedagogical framework that seeks to promote grounded learning by creating situations in which students can be guided to negotiate tacit and cultural perspectives on phenomena under inquiry; tacit and cultural ways of perceiving and acting” (IDC 2013, IJCCI 2014, Cambridge Handbook of the Learning Sciences, 2014).
Embodied Design Manifesto
We would like for learners to build on what they are able to do. And yet we recognize that tacit and cultural ways of looking at phenomena can be different. And so we ask how it is that learners can reconcile these, at times, incompatible views, and what designers and teachers can do to make this happen. That, in a nutshell, is what our lab has always focused on. We research that very gap between tacit and cultural views on phenomena pertaining to mathematical concepts, and we develop theoretical models to explain teachers' and students' cognitive and social actions that lead to reconciliation. The lab works both with sensory perception primitives, such as early sensitivity to color density in the visual field, and with human capacity to develop motor actions, such as learning new bimanual coordination.
This manifesto draws heavily, though at times selectively, on perspectives from genetic epistemology (aka constructivism, Piaget) and neo-Piagetian scholarship (instrumental genesis of Vérillon & Rabardel), cultural-historical psychology (Vygotsky; Cole; Saxe; Sfard), semiotic-cultural models (Radford; Bartolini Bussi & Mariotti), enactivism (Varela, Maturana, et al.) and radical enactivist cognition (Hutto & Myin; Sánchez-García), phenomenological and enactivist theories of mathematical teaching and learning (Nemirovsky; Kieren), philosophical underpinnings of embodiment theory (Johnson, Lakoff, Núñez), a push for dialectical conciliation of knowledge- and interaction-based conceptualizations of learning (diSessa), and a host of resonant views from ecological dynamics (ecological psychology of Gibson, dynamical systems theory of Thelen). The manifesto builds on a decade of research with insightful students, collaborators, and mentors. The manifesto essentializes relevant content that appears in our publications. We believe in these statements, because our empirical data suggest their viability.
- Mathematical reasoning is intrinsically sensorimotor and as such is necessarily spatial-temporal and object oriented. That's the brain we have -- mathematical reasoning co-opts the evolutionarily selected adaptive faculties of human being in this terrestrial and social environment. Yet the inherent sensorimotor quality of mathematical reasoning does not imply that it is always directly observable by another person: Mathematical reasoning may be manifest in overt external action, such as manipulation or gesture, obscured in covert mental simulation of imagined objects, or some functional combination of overt and covert actions. This is not to equate doing with knowing in some facile and dismissive anti-Behaviorist sense, because key to mathematical knowing is reflection, discourse, representation, and informed application. And yet mathematics learning is motivated by, and transpires within organized activity structures. Thus we are not shying away from a radical-constructivist view on the origins of knowledge, even as we profess, celebrate, and deliberately design socio-cultural enframings to stimulate learning: We build and facilitate fields of promoted action for the emergence of particular sensorimotor coordinations that empower individuals to construct and share disciplinary meanings.
- Procedural mathematical fluency is evident in successfully enacting the practice of performing rote algorithms -- rule-based processes of iteratively transforming propositions composed of symbolic and diagrammatic notation, with each set of operations generating a successive proposition. Procedural fluency is vital for engaging competently in activities that draw on mathematical tools to engage quantitative information, and this skill may facilitate learning new mathematical concepts and skills. And yet procedural fluency as a form of knowledge is not in and of itself necessarily indicative of conceptual understanding: procedural fluency is not directly indicative of analogous fluency in reasoning about meanings underlying algorithms. For example, knowing to transform a/b ÷ c/d into a/b × d/c does not imply having object-oriented sensorimotor analogs for visualizing this operation as the manipulation of quantities.
- Conditions can be created for students to develop sensorimotor schemes constitutive of new mathematical concepts. These conditions comprise dedicated environments -- fields of promoted action -- that motivate and frame sensorimotor learning. Essentially the activty architecture consists of a setting, resources, and a task. In the course of attempting to complete the task, a problem emerges whose solution constitutes the target learning. We are interested in solutions consisting of new, ecologically coupled sensorimotor coordination.
- Sensorimotor schemes develop dialectically with the coalescent constitution of concomitant phenomenal categories that the schemes engage. Thus while new goal-oriented operations are coordinated as means of acting more effectively in the environment, new aspects of the environment emerge from latency to saliency as the things we engage, that is, as the ontologies that the schemes operate on or through so as to effect the desired outcomes in the environment. Learning is discovering new structures in the dynamical perceptual manifold that come forth as mediating and enhancing our adaptive relations with the world. We call these attentional anchors. As we become conscious of these patterns as objects, we may articulate what they are and how we are wielding them. Thus new objects emerge from noticing consistent patterns in how perception guides action.
- Conditions can be created for students to encapsulate and articulate emergent phenomenal categories as proto-conceptual. These conditions comprise socio-epistemic framing as well as symbolic artifacts serving as potential frames of reference for the discursive elaboration of sensorimotor experience. Students use the provided forms to talk about what they are doing. Experienced teachers play rich and subtle roles in motivating, framing, and steering this learning process, using multimodal utterance to enact interactive and action-oriented tutorial tactics.
- The introduction of new tools not only enables students to talk about what they are doing -- it may change what they are doing. Thus instrumenting a field of promoted action modifies the action. Often, learning mathematical concepts demands of students to shift from these pre-mathematical categories, by way of quantification, to re-visualizing a situation -- how they perceive that situation, how they plan and enact purposeful action in that situation.
- Why would students shift from one form of engaging a situation to another? First, they need to recognize in the provided tools potential utilities for enhancing the enactment, evaluation, and/or explanation of an existing strategy. Yet in appropriating these affordances, they find themselves operating in a new way. This new way must be evidently effective. It may enable the student to arrive at the same inferences as did their previous strategy (what we call "inferential parity"), or it may enable the student to operate on the world with at least compatible results (what we call "functional parity"), though often adding greater control, prediction, and communication. These new ways of interacting with the world may offer students structured means of rationalizing their inferences and actions in a form apparently more resonant with mathematical practice.
- What happens once students shift from a pre-mathematical strategy to a mathematical strategy? Specifically, what becomes of the naive strategy that allegedly serves as the meaning that grounds the concepts? Moreover, what happens when students discover different mathematical strategies? We view phenomenal situations as affording different meanings, that is, different forms of goal-oriented contextualized engagement. Yet rather than call situations "ambiguous," we choose to think of them as "polysemous," because the different meanings are not incompatible (e.g., "duck" vs. "rabbit") but conceptually related (e.g., 2:3 vs. 2/3). Just how they are conceptually related is not always obvious, and yet we see advantages in students figuring out these relations.
- We believe that particularly powerful learning environments are those that expose for students the polysemy of situations, celebrating the various pre-mathematical and sophisticated orientations students bring to bear as valid and necessary, and creating structured opportunities for students to negotiate and reconcile these competing views. This reconciliation might come in the form of developing heuristic inferences -- inferences that may be naively articulated and only qualitative yet harbor frames for prospective quantitative work. As such, we favor opportunities for deep mathematical reasoning that is couched in sensorimotor engagement of simple generic objects, because we theorize such reasoning as constitutive of meanings to come.
- Ultimately, we hope for students to develop procedural fluency with symbolic notations. And yet students should be able to fold back onto the embodied dynamical meanings underlying their operations, so as to advance in their mathematical understanding.
Abrahamson, D. (Chair). (2018). Moving forward: In search of synergy across diverse views on the role of physical movement in design for STEM education [symposium]. In J. Kay & R. Luckin (Eds.), "Rethinking learning in the digital age: Making the Learning Sciences count," Proceedings of the 13th International Conference of the Learning Sciences (ICLS 2018) (Vol. 2, pp. 1243-1250). London: International Society of the Learning Sciences.
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.
ABSTRACT: Both ethnographic and neuroscientific research suggest that physicists solve problems by engaging in imaginary sensorimotor simulation of phenomena under inquiry. However, how this finding might inform high-school instructional practice is yet unknown. As educational researchers who are inspired by embodiment theory, we are investigating the potential roles that students’ choreographed physical movements in classroom space play in learning physics concepts related to motion. Here we focus on two case studies of high-school students whose summative projects for an instructional unit involving movement-based problem solving manifested deep conceptual and affective relations with the subject matter. Through qualitative analyses we attempt to build a coherent narrative of the subjective processes that led to these results.
Abrahamson, D. (2017). Embodiment and mathematical learning. In K. Peppler (Ed.), The SAGE encyclopedia of out-of-school learning (pp. 247-252). New York: SAGE.
ABSTRACT: Developed in intellectual disciplines as diverse as philosophy, linguistics, robotics, kinesiology, and cognitive psychology, embodiment is a relatively new paradigm for the field of learning sciences. This entry discusses the theory of embodiment, focusing on how the theory is informing new directions of research and pedagogy in the particular domain of mathematics education. More specifically, the entry addresses an enduring research problem in the learning sciences pertaining to the role of embodied action in the learning and teaching of mathematical concepts.
Abrahamson, D., Sánchez-García, R., & Smyth, C. (2016). Metaphors are projected constraints on action: An ecological dynamics view on learning across the disciplines. In C.-K. Looi, J. L. Polman, U. Cress, & P. Reimann (Eds.), "Transforming learning, empowering learners," Proceedings of the International Conference of the Learning Sciences (ICLS 2016) (Vol. 1, "Full Papers", pp. 314-321). Singapore: International Society of the Learning Sciences. ["Best Paper ICLS 2016" award]
ABSTRACT: Learning scientists have been considering the validity and relevance of arguments coming from philosophy and cognitive science for the embodied, enactive, embedded, and extended nature of individual learning, reasoning, and practice in sociocultural ecologies. Specifically, some design-based researchers of STEM cognition and instruction have been evaluating activities for grounding content knowledge in interactive sensorimotor problem solving. Yet in so doing, we submit, the field stands greatly to avail of theoretical models and pedagogical methodologies from disciplines oriented explicitly on understanding, fostering, and remediating motor action. This conceptual paper considers potential values of ecological dynamics, a perspective originating in kinesiology, as an explanatory resource for tackling enduring LS research problems. We support our position via an ecological-dynamics reexamination of the function of metaphor in the instruction of sports skills, somatic awareness, and mathematics. We propose a view of metaphors as productive constraints reconfiguring the dynamic system of learner, teacher, and environment.
Rosen, D. M., Palatnik, A., & Abrahamson, D. (2016). Tradeoffs of situatedness: Iconicity constrains the development of content-oriented sensorimotor schemes. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Sin fronteras: Questioning borders with(in) mathematics education - Proceedings of the 38th annual meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) (Vol. 12, "Technology", pp. 1509-1516). Tucson, AZ: University of Arizona.
ABSTRACT: Mathematics education practitioners and researchers have long debated best pedagogical practices for introducing new concepts. Our design-based research project evaluated a heuristic framework, whereby students first develop acontextual sensorimotor schemes and only then extend these schemes to incorporate both concrete narratives (grounding) and formal mathematical rules (generalizing). We compared student performance under conditions of working with stark (acontextual) vs. iconic (situated) manipulatives. We summarize findings from analyzing 20 individually administered task-based semi-structured clinical interviews with Grade 4 – 6 participant students. We found tradeoffs of situatedness: Whereas iconic objects elicit richer narratives than stark objects, these narratives may detrimentally constrain the scope of potential sensorimotor schemes students develop in attempt to solve manipulation problems.
Abrahamson, D., Sánchez-García, R., & Trninic, D. (2016). Praxes proxies: Revisiting educational manipulatives from an ecological dynamics perspective. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Sin fronteras: Questioning borders with(in) mathematics education - Proceedings of the 38th annual meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) (Vol. 13, "Theory and research methods", pp. 1565-1572). Tucson, AZ: University of Arizona.
ABSTRACT: The recent proliferation of technological devices with natural user interfaces (e.g., touchscreen tablets) is regenerating scholarship on the role of sensorimotor interaction in conceptual learning. Some researchers of mathematical education have adopted views from constructivism, phenomenology, enactivism, and ecological dynamics to interpret implicit sensorimotor schemes as both forming and manifesting disciplinary competence. Drawing on these views, this theoretical paper discusses what it means to develop a new skill by way of task-oriented interaction with objects. Beginning with sports then moving to mathematics, we focus on a subcategory of pedagogical artifacts that serve students as effective proxies for developing normative application of target schemes. We argue for the contribution of these views to designing artifacts for learning mathematical content.
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.
ABSTRACT: Design research is a broad, practice-based approach to investigating problems of education. This approach can catalyze the development of learning theory by fostering opportunities for transformational change in scholars’ interpretation of instructional interactions. Surveying a succession of design-research projects, I explain how challenges in understanding students’ behaviors promoted my own recapitulation of a historical evolution in educators’ conceptualizations of learning—Romantic, Progressivist, and Synthetic (Schön, Intuitive thinking? A metaphor underlying some ideas of educational reform (working paper 8). Division for Study and Research in Education, MIT, Cambridge, 1981)—and beyond to a proposed Systemic view. In reflection, I consider methodological adaptations to design-research practice that may enhance its contributions in accord with its objectives.
Abrahamson, D. (2015). The monster in the machine, or why educational technology needs embodied design. In V. R. Lee (Ed.), Learning technologies and the body: Integration and implementation (pp. 21-38). New York: Routledge.
OPENING PARAGRAPH: A while ago I consulted for a large-scale, federally funded effort to develop educational media for young children to learn mathematics. The project was based out of Hollywood, where the studio was abuzz with highly creative animators, scriptwriters, songwriters, and joke experts. The studio was still agog from a recent international award for their flagship product, and they could not wait to brainstorm the ‘merch’ that would surely emanate from the new project. It was all very flattering and alarmingly lucrative. As I was taxied and flown down and up California, wined and dined on cocktails and sushi, I began to affect a certain hero persona. Assistant professor in the Bay, big-time maven in LA. Abrahamson, Dor Abrahamson. And stir that latte, don’t shake it. My job title was ‘Education Expert.’ Yet gradually, I began to suspect this title was little more than a euphemism for ‘Imprimatur.’
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.
ABSTRACT: Design researchers should inform the commercial production of educational technology by explicating their tacit design practice in workable structures and language. Two activity genres for grounding mathematical concepts are explained: ‘‘perception-based design’’ builds on learners’ early mental capacity to draw logical inferences from perceptual judgment of intensive quantities in source phenomena, such as displays of color densities; ‘‘action-based design’’ builds on learners’ perceptuomotor capacity to develop new kinesthetic routines for strategic embodied interaction, such as moving the hands at different speeds to keep a screen green. In a primary problem, learners apply or develop non-symbolic perceptuomotor schemas to engage the task effectively; In a secondary problem, learners devise means of appropriating newly interpolated mathematical forms as enactive, semiotic, or epistemic means of enhancing, explaining, and evaluating their primary response. In so doing, learners heuristically determine either inferential parity (perception-based design) or functional parity (action-based design) as epistemic grounds for reconciling naïve and scientific perspectives. Ultimately embodied-learning activities may interleave and synthesize the genres’ elements. This taxonomy opens design practice into richer dialog with the learning sciences. An appendix lays out the embodied-design framework in a ‘‘how to’’ form amenable for replication both within the domain of mathematics and beyond.
Abrahamson, D. (2013). Toward a taxonomy of design genres: Fostering mathematical insight via perception-based and action-based experiences. In J. P. Hourcade, E. A. Miller, & A. Egeland (Eds.), Proceedings of the 12th Annual Interaction Design and Children Conference (IDC 2013) (Vol. "Full Papers", pp. 218-227). New York: The New School & Sesame Workshop.
ABSTRACT: In a retrospective analysis of my own pedagogical design projects over the past twenty years, I articulate and compare what I discern therein as two distinct activity genres for grounding mathematical concepts. One genre, “perceptionbased design,” builds on learners’ early mental capacity to draw logical inferences from perceptual judgment of intensive quantities in source phenomena, such as displays of color densities. The other genre, “action-based design,” builds on learners’ perceptuomotor capacity to develop new kinesthetic routines for strategic embodied interaction, such as moving the hands at different speeds to keep a screen green. Both capacities are effective evolutionary means of engaging the world, and both bear pedagogical potential as epistemic resources by which to build meaning for mathematical models of, and solution processes for situated problems. Empirical studies that investigated designs built in these genres suggest a two-step activity format by which instructors can guide learners to reinvent conceptual cores. In a primary problem, learners apply or develop non-symbolic perceptuomotor schemas to engage the task effectively. In a secondary problem, learners devise means of appropriating newly interpolated mathematical forms as enactive, semiotic, or epistemic means of enhancing, explaining, and evaluating their primary response. Whereas my analysis distills activities into two separate genres for rhetorical clarity, ultimately embodied interaction may interleave and synthesize the genres’ elements.
Abrahamson, D. (2012). Discovery reconceived: Product before process. For the Learning of Mathematics, 32(1), 8-15.
ABSTRACT: This article is motivated by a commitment to the ideas underlying discovery learning, namely the epistemological notion of grounded, meaningful, generative knowledge. It is also motivated by concern that these ideas have been implicitly misinterpreted in curriculum and instruction, ultimately to the detriment of students. Accordingly, I discuss an alternative, empirically based, theoretical articulation of discovery pedagogy that addresses the criticisms it has faced. The research question framing this alternative approach is, “What exactly about a mathematical concept should students discover via discovery learning?” I will pursue this question by reflecting on two case studies of children who participated in activities of my own design. Empirical data from these and other studies have served me over the past decade as contexts for inquiry into the cognition and instruction of mathematical concepts, an inquiry that, in turn, keeps feeding back into further design and articulation of design principles. In this essay, I will use these data to offer an empirically grounded “centrist” answer to the question of what students should discover, at least with respect to a particular class of mathematical concepts (intensive quantities) as embodied in a particular type of design (perception-based learning).
Abrahamson, D., & Trninic, D. (2011). Toward an embodied-interaction design framework for mathematical concepts. In P. Blikstein & P. Marshall (Eds.), Proceedings of the 10th Annual Interaction Design and Children Conference (IDC 2011) (Vol. "Full papers", pp. 1-10). Ann Arbor, MI: IDC.
ABSTRACT: Recent, empirically supported theories of cognition indicate that human reasoning, including mathematical problem solving, is based in tacit spatial-temporal simulated action. Implications of these findings for the philosophy and design of instruction may be momentous. Here, we build on design-based research efforts centered on exploring the potential of embodied interaction (EI) for mathematics learning. We sketch two emerging, reciprocal contributions: (1) a sociocognitive view on the role of automated feedback in building the perceptuomotor schemes that undergird conceptual development; and (2) a heuristic EI design framework. We ground these ideas in vignettes of children engaging an EI design for proportion. Increasing ubiquity and access to mobile devices geared to avail of EI principles suggests the feasibility of mass-disseminating materials evolving from this line of research.
Trninic, D., Reinholz, D., Howison, M., & Abrahamson, D. (2010). Design as an object-to-think-with: Semiotic potential emerges through collaborative reflective conversation with material. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the Thirty-Second Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 32) (Vol. VI, Ch. 18: Technology, pp. 1523 – 1530). Columbus, OH: PME-NA.
Abrahamson, D. (2009). Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics, 70(1), 27-47. [Electronic supplementary material].
Abrahamson, D., & Wilensky, U. (2007). Learning axes and bridging tools in a technology-based design for statistics. International Journal of Computers for Mathematical Learning, 12(1), 23-55.
ABSTRACT: We introduce a design-based research framework, learning axes and bridging tools, and demonstrate its application in the preparation and study of an implementation of a middle-school experimental computer-based unit on probability and statistics, ProbLab (Probability Laboratory, Abrahamson and Wilensky 2002 [Abrahamson, D., & Wilensky, U. (2002). ProbLab. Northwestern University, Evanston, IL: The Center for Connected Learning and Computer-Based Modeling, Northwestern University. http://www.ccl.northwestern.edu/curriculum/ProbLab/]). ProbLab is a mixed-media unit, which utilizes traditional tools as well as the NetLogo agent-based modeling-and-simulation environment (Wilensky 1999) [Wilensky, U. (1999). NetLogo. Northwestern University, Evanston, IL: The Center for Connected Learning and Computer-Based Modeling. http://www.ccl.northwestern.edu/netlogo/] and HubNet, its technological extension for facilitating participatory simulation activities in networked classrooms (Wilensky and Stroup 1999a) [Wilensky, U., & Stroup, W. (1999a). HubNet. Evanston, IL: The Center for Connected Learning and Computer-Based Modeling, Northwestern University]. We will focus on the statistics module of the unit, Statistics As Multi-Participant Learning-Environment Resource (S.A.M.P.L.E.R.). The framework shapes the design rationale toward creating and developing learning tools, activities, and facilitation guidelines. The framework then constitutes a data-analysis lens on implementation cases of student insight into the mathematical content. Working with this methodology, a designer begins by focusing on mathematical representations associated with a target concept—the designer problematizes and deconstructs each representation into a pair of historical/cognitive antecedents (idea elements), each lying at the poles of a learning axis. Next, the designer creates bridging tools, ambiguous artifacts bearing interaction properties of each of the idea elements, and develops activities with these learning tools that evoke cognitive conflict along the axis. Students reconcile the conflict by means of articulating strategies that embrace both idea elements, thus integrating them into the target concept.
Abrahamson, D. (2007). Both rhyme and reason: Toward design that goes beyond what meets the eye. In T. Lamberg & L. Wiest (Eds.), Proceedings of the Twenty Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 287 – 295). Stateline (Lake Tahoe), NV: University of Nevada, Reno.
ABSTRACT: Drawing on design-based studies where students worked with learning tools for proportionality, probability, and statistics, I appraise whether students had opportunities to construct conceptual understanding of the targeted mathematical content. I conclude that visualizations of perceptually privileged mathematical constructs support effective pedagogical activity only to the extent that they enable students to coordinate perceptual conviction with mathematical operations—intuiting that, and not how, two representations are related constitutes perceptually powerful yet conceptually weak situatedness. In constructivist learning, as in empirical research, regularity apprehended in observed phenomena is measured, expressed, and schematized. Students should articulate or corroborate visual thinking with step-by-step procedures, e.g., synoptic views of multiplicative constructs should include tools for distributed-addition handles.
Abrahamson, D. (2006). Mathematical representations as conceptual composites: Implications for design. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the Twenty Eighth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 464-466). Mérida, Yucatán, México: Universidad Pedagógica Nacional.
ABSTRACT: Positing that mathematical representations are covert conceptual composites, i.e., they implicitly enfold coordination of two or more ideas, I propose a design framework for fostering deep conceptual understanding of standard mathematical representations. Working with bridging tools, students engage in situated problem-solving activities to recruit and insightfully recompose familiar representations into the standard representation. I demonstrate this framework through designs created for studies in three mathematical domains.