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Transcending Traditional/Reform Dichotomies in Mathematics Education by Martina Metz, A. Paulino PreciadoBabb, Soroush Sabbaghan, Brent Davis, Geoffrey Pinchbeck, Ayman Aljarrah
Recent calls to find a balance between allegedly opposite instructional approaches–commonly dichotomized as "traditional" and "reform" approaches–have at times posited these extreme approaches as complementary, suggesting that the two may productively interact. For instance, Ansari (2015) claimed that "[I]t is time to heed the empirical evidence coming from multiple scientific disciplines that clearly shows that math instruction is effective when different approaches are combined in developmentally appropriate ways" (para. 14). In this paper, we argue that such seemingly contradictory approaches are in fact very similar in significant ways and propose a 'third way' that addresses important features not stressed by either approach.
On Practical and Theoretical Thinking and other False Dichotomies in Mathematics Education by Anna Sierpinska
1. INTRODUCTION
This paper, dedicated to Michael Otte, is about practical and theoretical thinking as complementary epistemological categories and the use of this distinction in mathematics education. The distinction is presented as one among many "false dichotomies" that are common in the domain. The dichotomies are first discussed in the light of Michael Otte’s papers on complementarity. An alternative view is then proposed in terms of couples of epistemological obstacles. A possible use of the practical/theoretical distinction in mathematics education is illustrated by means of a thought experiment about a teacher educator planning to discuss the use of manipulatives with his student teachers. The thought experiment points to the complexity of the system of objects of thought in mathematics education and the extreme fragility, in practice, of the distinction between theoretical and practical thinking. It also highlights the crucial role that epistemological analyses, such as those offered in Michael Otte’s papers, play for research in mathematics education. In his comments on one of my papers about epistemological obstacles (Sierpinska 1996), Michael Otte was saying that, where I saw a couple of obstacles, he could see only one, namely:
... the problem that in order to understand mathematics one has to take into account [the fact] that mathematics is simultaneously metamathematics … [T]he problem lies in an empiricist or concrete epistemology [that] does not think of mathematical objects in relational or structural terms. ... [M]athematics is difficult for the learner not because of the technical complications of its method, but because of the specificity of its objects. (Letter dated 24.1.1994).
He went on to say that he has been busy his entire didactical career with this one problem [the problem of sources of difficulties in mathematics learning] and with the question of the nature of mathematical objects and concepts. In my own research, I have tried to engage more directly with the practice of teaching, with designing and experimenting didactic sequences. Somehow, I always ended up discussing these same problems. They are very powerful attractors indeed in the dynamics of research in mathematics education.
2. EPISTEMOLOGICAL OPPOSITES
Theory of mathematics education is replete with pairs of opposite categories of knowing and thinking such as the empiriciststructural distinction mentioned above, intuition versus formal knowledge, instrumental versus relational, or operational versus structural understanding. In my research I first resisted using such global categories, explaining both the meaning of particular mathematical concepts and students’ difficulties by the existence of "epistemological obstacles" specific to concrete mathematical concepts. But, as I went on in my research, I realized (and thus agreed with Michael) that many obstacles were related not to specific concepts but to mathematics in general. And thus I ended up with, first, three categories of thinking in linear algebra: syntheticgeometric, analyticarithmetic and analytic structural, and then attributing students' difficulties in linear algebra to their tendency to practical as opposed to theoretical thinking (Sierpinska et al. 1997; Sierpinska 2000; Sierpinska & Nnadozie 2001).
It is tempting to think that these categories refer to some ontological reality; that there exists an identifiable brain activity such as, for example, theoretical thinking, with no trace whatsoever of its opposite, namely practical thinking. But, as Michael Otte has argued in (Otte 19902), these distinctions should be regarded as epistemological, not ontological distinctions. They are our simplified ways of knowing human cognitive activity in mathematics; they are not kinds of human cognitive activity.
I have argued that, whenever we see mathematical proof as involving only a mechanical aspect, we are driven to see that it involves, as well, an intuitive one. And whenever we are tempted to see mathematical proof as involving only a solitary aspect, we are driven to seeing that it is also a social matter. And whenever we are tempted to see a mathematical argument of the kind found in proof, namely a chain of tautologies or of equalities, as merely, or perhaps the ideal of, literal expression, we are forced to see that it is, in fact, essentially metaphorical. (Otte 19902)
This is why Otte preferred to speak of "complementarity" (p and not p) rather than of dichotomy (p or not p).
Play, Mathematics, and False Dichotomies by Douglas H. Clements & Julie Sarama
Let’s stop the cycle of "abuse"–or at least confusion–that stems from false dichotomies in early education. "Play vs. academics" is arguably the main one. Of course children should play. But this does not mean they should not learn, and even play, with mathematics.
Dichotomy method: A method for numerically solving equations in a single unknown.
 The essential dichotomy underlying mathematics  Data Structures Math Foundations 186What lies at the very core of mathematics? What is mathematics ultimately about, once we strip away all the hoopla and complexity? In this video I give you my answer to this intriguing question. Surprisingly, it is not really the natural numbers: they are fundamental, but not the most fundamental. And geometric objects, basic though they are, are also not at the core.
There is an essential dichotomy underlying mathematics: the "something/nothing" ideology... resulting in an intuitive natural progression that mirrors ordinary usage in arithmetic. "0" (zero) is considered the first natural number.
HOB comment: He speaks of the idea where there is something beneath the Natural Numbers, but does not also provide a third idea of something above the Natural Numbers, though such thinking is a commonality amongst different thinkers. The usage of zero as the first natural number is already used, but not taught as such as a public schooldriven given.
Exponential Dichotomy and Trichotomy for Difference Equations by A. I. Alonso, Jialin Hong, R. Obaya
AbstractIn this paper, a roughness theorem of exponential dichotomy and trichotomy of linear difference equations is proved. It is also shown that if an almost periodic difference equation has an exponential dichotomy on a sufficiently long finite interval, then it has one on (∞, +∞). ©1999 Elsevier Science Ltd. All rights reserved.
Discrete and Continuous: A Fundamental Dichotomy in Mathematics by James Franklin
The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics.
The False Dichotomy in Mathematics Education Between Conceptual Understanding and Procedural Skills: An Example from Algebra by Carolyn Kieran
Abstract—: The history of mathematics education provides ample evidence of the dichotomous distinction that has been made over the years between concepts and procedures, between concepts and skills, and between "knowing that" and "knowing how." In no field of school mathematics learning has this dichotomy been so damaging as in algebra. While reform efforts of the past decade have attempted to imbue algebra learning with meaning by focusing on "reallife" problems and their various representations, these efforts have missed the main point with respect to the literalsymbolic: that is, that conceptual aspects of algebra abound within the literalsymbolic and that these are integral to most of the socalled procedures of algebra. Both theoretical and empirical arguments will be used to make the point for adopting a different vision of the literalsymbolic domain, in which the procedural is so permeated with the conceptual as to render obsolete a primarily procedurebased view of algebra in school mathematics.
Admissibility and polynomial dichotomies for evolution families by Davor Dragiêevié
For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a (strong) nonuniform polynomial dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of a strong nonuniform polynomial dichotomy under sufficiently small linear perturbations.
Dichotomies or Binoculars: Reflections on the Papers by Steffe and Thompson and by Lerman by Thomas E. Kieren
Three Entangled Dichotomies in Mathematics: Inductive/Deductive, Defining/Proving and Arbitrary/Necessary. by Hals, Sigurd Johannes
Abstract: How can we understand the structure of mathematics in the context of school mathematics? It appears that there are challenges in teaching and learning mathematics that relate to its structure. This paper examines the structure of mathematics in the context of school mathematics and the need to explore and understand the structure of mathematics in relation to the school subject. The three dichotomies of inductive/deductive, defining/proving and arbitrary/necessary are used to elaborate on the structure of mathematics in the context of teaching and learning in school. Here I will explain these dichotomies while referring to the new school curriculum in mathematics in Norway and to the nature of mathematical argumentation in general. This will show that there is a bias towards the deductive side of mathematics, while the aspects that are inductive and arbitrary are overlooked. The analysis in this study points out the importance of the potential in developing mathematics teaching that nurtures the student's feeling for mathematics and mathematical reasoning, while taking into account the entanglement of these three dichotomies that we witness in school mathematics.
HOB note: Observe how the author is leaning towards a third option, thus illustrating that "beyond dichotomous thinking" can be expressed as a "Accordian Calculus" that has sprouted wings and begun to breathe beyond its current restraints.) In some examples, the author straddles the dichotomy line with three patterned examples being used in an attempt to describe what he perceives as dichotomies.
14 False Dichotomies in Education
 Learning as a skill, a concept, a process or a product: (One or another or a blend.)
 Learning as an observable behavior versus learning as a cognitive process.
 Teaching as an art and a science: (I’m starting to think this dichotomy is ridiculous, period. There is a real science to any art and there is an art to any science. The distinction is often a relic of an Enlightenment mindset.)
 Learning can always be quantified with a metric versus learning is seldom measurable and should be looked at through a qualitative and descriptive lens:
 Learning as linear or connective: I’ve heard the tree versus rhizome point way too often. Why not both? If you’re going to look to nature, can’t there be a place for both trees and rhizomes? Stories are powerful, profound and linear. Conceptual attainment is often webbased.
 Education as a system, a structure or a series of relationships
 Assessment as something descriptive, analytical or evaluative.
 Practical and theoretical: All theory is rooted in practical observations and has practical implications. All practices come from some theoretical mindset. Busting on one or the other seems kindof silly.
 Motivation being primarily extrinsic or intrinsic: I also see a dichotomy sometimes in negative or positive reinforcement. Spend some time and ask honestly about motivation and intentions and it almost always gets muddled. Our minds and motives are complex.
 Freedom and guidance.
 Classroom management and classroom leadership.
 Knowledge constructed internally versus externally.
 Universalism of theory versus theory as contextdependent: We have to recognize that there are some practices that work better than others while also being critical about the context and the role it plays in the construction of research and the implementation of theory.
 Transformation and restoration: We’ve lost some great vintage ideas and we have some great things going for us. At the same time, the system needs transformation. We need a dialogue that includes visions for the future while also bringing in great ideas from the past.
 ...So, it leaves me with a lingering reminder that truth is almost always paradoxical, solutions are almost always nuanced and that the best approach is to approach teaching with a healthy dose of humility … but still be bold. See that? There’s a paradox in that one, too.
Dichotomy meaning (A rating scale of different examples suggesting the meaning of the word dichotomy.)
Persistent Dichotomies (List of dichotomies occurring in Psychology)

Note: there are more examples on this same page.)
The persistence of dichotomies in the study of behavioral development by Timothy D. Johnston
Abstract: The inadequacies of dichotomous views of behavioral development that oppose learned and innate behavior, or genetic and environmental determinants of behavior, have long been recognized. However, they continue to exert a powerful influence on current thinking about development, often by way of metaphors that simply recast these old ideas in a more modern technical vocabulary. The idea that the information for behavior can be attributed to either genetic or environmental sources was originated by Lorenz and provides the basis for many current dichotomous accounts of behavioral development. Lorenz's "sources of information" metaphor for development is fundamentally flawed, however, as are those more recent accounts that are based on it. The alternative interactionist account of development, most clearly articulated by Lehrman, is a far more powerful and coherent theoretical framework for development, but it has not been broadly assimilated into psychology and continues to be widely misunderstood. In particular, the interactionist account does not involve a radical environmentalism, does not attribute all behavior to the effects of learning, and does not interpret development as a geneenvironment interaction. The attractive simplicity of dichotomous thinking encourages its continued application to the study of development despite the fact that it is clearly inadequate to the complexities of developmental analysis.
Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns by John R. Anderson, Shawn Betts, Jennifer L. Ferris & Jon M. Fincham
Abstract Students were taught an algorithm for solving a new class of mathematical problems. Occasionally in the sequence of problems, they encountered exception problems that required that they extend the algorithm. Regular and exception problems were associated with different patterns of brain activation. Some regions showed a Cognitive pattern of being active only until the problem was solved and no difference between regular or exception problems. Other regions showed a Metacognitive pattern of greater activity for exception problems and activity that extended into the postsolution period, particularly when an error was made. The Cognitive regions included some of parietal and prefrontal regions associated with the triplecode theory of (Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506) and associated with algebra equation solving in the ACTR theory (Anderson, J. R. (2005). Human symbol manipulation within an 911 integrated cognitive architecture. Cognitive science, 29, 313–342. Metacognitive regions included the superior prefrontal gyrus, the angular gyrus of the triplecode theory, and frontopolar regions.
H.O.B. note: Very often we encounter the usage of the term "Metacognition" that may or may not be paired with "Cognition". Much less is the case when a third possibility is encountered such as the tripartite lineup of Metacognition Cognition Subcognition... which undoubtedly has parallels in other subjects such as the three traditional social classes (upper middle lower), three stooges (Moe Larry Curly joe), three stop lights (red yellow green), clothing sizes (large medium small), speed dial selections (fast medium slow), three University degrees (PhD. Masters Bachelors), etc...
 Metacognition: the ability to think about thinking
 Cognition: ability to think (perhaps about others thinking but not one's own)
 Subcognition: precursor influences of thought
If you're afraid to ask what it is, it probably doesn't apply to you. The DunningKruger effect is a cognitive bias whereby low ability specifically leads subjects to overestimate their ability. Popularly: the tendency of stupid people to assume they're clever. Named after two social psychologists, the Americans David Dunning (°1950) and Justin Kruger.
The LaségueFalret syndrome (The nosological significance of Folie à Deux: a review of the literature)A psychiatric syndrome in which two persons share the symptoms of a delusional belief. Also known as folie à deux. Ironically, it took two scientists to describe it: Charles Lasègue (18161883) and JeanPierre Falret (17941870), both French psychiatrists.
The JarischHerxheimer reactionThe feeling of getting worse before you get better, sometimes called a 'healing crisis', as experienced by sufferers of syphilis, Lyme's disease or relapsing fever. Also known as a 'herx'. Named after Adolf Jarisch (18501902) and Karl Herxheimer (18611944), two dermatologists, from Austria and Germany respectively.
The LotkaVolterra modelA model describing the dynamics of biological systems in which two species, one predator and one prey, interact. Named after the US mathematician and chemist Alfred J. Lotka (18801949) and Vito Volterra (18601940), an Italian mathematician and physicist.
The MargolusLevitin theoremStates that the fundamental limit to quantum computation is 6×1,033 operations per second per joule of energy, thereby providing the theoretical horizon for Moore's Law. Named after Norman Margolus (°1955), a CanadianAmerican physicist and computer scientist, and Lev B. Levitin (°1935), a RussianAmerican mathematician and engineer.
The DansgaardOeschger eventsDescribes the rapid climate fluctuations that occurred 25 times during the last glacial period. Temperatures went up by around 8°C over 40 years. Named after Danish paleoclimatologist Willi Dansgaard (19222011) and Swiss geophysicist Hans Oeschger (19271998).
The KelvinHelmholtz instabilityOccurs when there is velocity shear in a single continuous fluid, or where there is a velocity difference across the interface of two fluids. Manifests itself as waves on a water surface, or wave clouds in the sky, for example. Named after the British engineer and physicist William Thompson, a.k.a. Lord Kelvin (18241907) and Hermann von Helmholtz (18211894), a German physician and physicist.
The HardyWeinberg lawPredicts that genetic variation within a population, in the absence of other evolutionary influences, will not change over time. Named after the English mathematician G.H. Hardy (18771947) and the German obstetrician Wilhelm Weinberg (18621937).
The MichelsonMorley experimentAn attempt to detect the existence of luminiferous aether, which was thought to permeate space and carry light waves. The attempt failed, because aether doesn't exist. Named after the American scientists Albert A. Michelson (18521931) and Edward W. Morley (18381923).
The SniderPellegrini Wegener mapUses continental drift to explain why similar dinosaur fossils can be found at different places on distant continents. The map takes doublebarrelledness to a new level, combining the name of the German scientist Alfred L. Wegener (19901930) with that of the French geographer Antonio SniderPellegrini (18021885), whose two surnames sound like a scientific theory all of their own.
Doublebarreled theoriesPage initially created: Wednesday, March 3rd, 2021... 1:19:25 PM
Page initially posted: Saturday, May 22nd, 2021... 7:17 AM
Herb O. Buckland
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