Robert Craigen @rcraigen has confronted David Coffey @deltadc and me @blaw0013 on Twitter, “Neither of you seems to know how a child's first math concept arises - that's the Q on hand.”
And he restates, “The Q is: in what sequence does a child develop an association between names like "3" and numbers in the abstract? Here "in the abstract" is in the usual sense: Not "3 apples" or "3 fingers" but "3" as an isolated qualitative concept. "In what sequence": does the child start with abstract notion of "3" or learn it through repeated practice of counting? I say Sesame Street got this one right. So did your mother.”
Dr. Craigen is a practicing mathematician, leader in mathematics competition, stoker of the math war flames in the Manitoba region (quite late to the game from my California perspective), and evangelical Christian.
To unearth some work I haven’t tended to in over a decade, I decided to take up the question. Again, for myself. If you follow @rcraigen you will find that he uses social media to fan flames with illogical and/or irrational and/or unscientific rhetoric. He is supremely guilt of confirmation bias. But maybe we all could be accused of that.
Observing how @rcraigen operates in social media, and recognizing that as a mathematician, he likely as a deep value for the role of precise definitions in the axiomatic structure / logic of mathematical discourse and knowledge. So I asked him how he defined math, concept, and arise, so that I could stay as true to his definitions as possible toward answering his question. Or, if I veered, I could be clear in stating our differing meanings for these words.
My summary of his efforts to provide definition to each term:
Math: the science of structure, form and relation. Math is a body of knowledge. Math is a precise language for technical and abstract subjects matter. Correct usage and standardization are critical.
Concept: “happy with natural meaning of the term in ordinary discourse.” He wrote a bit more, which I summarize as a general rule or class, and abstraction from singular instances.
Arise: to come to exist, appear or manifest.
And he asked for my definitions. Next, I will give that a go.
I am good with how @rcraigen defined arise, except for a key point. I have a particular view of what it means to “exist” (and each other word he used). To exist for me does not mean that the object has an ontological existence independent of my knowing mind, but that in the organization of my experiential world, I now attribute an existence to the object. Because I am not omniscient, I could not know if that object exists in a Platonic sense.
I am also comfortable with the definition @rcraigen pointed toward for concept. I suggested the simple notion “general rule or class.” This is important to @ rcraigen’s question because he has suggested that he believes a child’s first math concept is number. I suggest this is arguable depending on how math gets defined, but I do agree to consider the concept of number for my response. And so number as concept suggests the child can discern if a particular object or collection of objects are examples of a particular number, i.e. belong to that specific class.
“Math” means different things in the different contexts in which I work. To be brief, with teachers math is rather static—it is a body of knowledge that kids must come to know. This body of knowledge is usually defined for them by an external source, state standards, etc.
For the mathematician, the “science of…” answer you provided is common. My sense is the mathematician has an appreciation for the growing body of knowledge that makes up mathematics, whereas the teacher is more focused on a static nature.
Related, for the mathematician, there is the invented vs. discovered sort of debate. Or it may be framed as math is the product of the human mind vs. math is uncovered by the human mind.
I hope it is already evident that one side or the other of the invent/discover notion of math changes greatly the effort to understand how one comes to learn or know mathematics. Is it something with an existence external or a priori to the knower, or is it something without this a priori existence, constructed by the knowing being.
Further, it seems both mathematicians and math teachers (and really everyone) recognize that math is not just a static thing, but it is both ways of understanding AND ways of thinking (e.g. Guershon Harel writes http://tinyurl.com/nva49yg). Hence my noun or verb question.
The math educator must have a well-defined theory for knowing and learning to do their work (I wish more of my colleagues valued this idea, and/or made the effort to be more explicit about their learning theory when writing). To become more specific to modern learning theory, the consideration of knowledge is less static, less noun-like. In the post-behaviorist, Piagetian tradition (which I firmly plant myself, having studied under Leslie Steffe, a collaborator with Ernst von Glasersfeld), there are three forms of knowledge worth considering when discussing how MATH CONCEPTS ARISE in children: physical, social, and logico-mathematical knowledge. Physical knowledge can be discerned through the senses. Social knowledge must be told to us by other people. Logico-mathematical knowledge is the knowledge of relationships, and relationships don’t exist until we make them. For Piaget, the schemas that constitute all three types of knowledge are constructed by the mind, and are not “known” and thus not knowledge to the knower until they are constructed. Until that time, the knowledge does not exist for that person.
And here is where a VERY careful definition of mathematics becomes important—is mathematics physical, social, or logico-mathematical knowledge? Or maybe, which mathematics is physical, social, or logico-mathematical knowledge?
With these definitions set, or at least made problematic in the case of mathematics, I next make a point about science and theory, as opposed to fact, before returning to Dr. Craigen’s question, how does number concept arise for the child.
I draw upon a notion of science developed by Imre Lakatos, a mathematician and scientist of the mid 20th century. For Lakatos, no theorem is final or perfect. It is incorrect to consider a theory to be the truth, rather that no counterexample have yet been found. Related, the theory for how children construct number can only be a model, since numerical concepts, as a process of the mind, are not accessible via observation. So in my efforts to report the scientific theory of how a child develops number, I recognize that it may not yet be final—a superseding theory may emerge, but to this time no contradictory evidence has violated the hard core of the theory I do my best to present next.
John Dewey wrote in 1895 that "Number is a rational process, not a sense fact," directing us to consider number not as a physical knowledge, but one constructed (a process) of the mind. Kant, nearly one hundred years earlier, named three categories of quantity: unity, plurality, and totality. Kant stated that each arises for the child from a synthesis of the a priori conditions of all experience, space, and time. Piaget does not take these conditions as a priori, rather they are constructions of the mind, a mind that organizes both itself and its world. While Dewey and Kant both pointed to the constructing mind as the source for the idea of number, that number is a rational process, and of the importance of the notion of unity, neither suggest how unity might be constructed by this active mind. Piaget also suggests the importance of one-ness in the construction of number (or number concept to better match @rcraigen), but is not careful to define what these concepts (unity, number) consist of. “Elements are stripped of their qualities and become arithmetic unities” (Piaget, 1970, p. 37). This negative description fails in that if a child has not yet constructed unity and strips an element of all other qualities, color, shape, texture, and all other sensory properties (physical knowledge, for those checking), that element is a nonentity for the child – not necessarily this unity.
Piaget defined number as an “operatory group structure, without which there cannot be conservation of numeric totalities independent of their figural disposition” (Piaget & Szeminska, 1964, p. 9). His work to understand children’s construction of number focused on number as a concept—attending to class inclusion and order—and took the construction of the unit for granted.
Neurophysiology has established that our cognitive structures differentiate or cut things out of a background and perceive each of them as a whole. That is to say, we do divide our various sensory fields of experience into separate parts, which then, in our cognitive organization, become individual things. A dog is not distinguished from the yard by merely the sensory differences alone, it is the work of the mind to organize the sensory input to recognize the individualness of the dog.
Ernst von Glasersfeld (1981) suggests a model for how the mind creates these conceptual units. This model is reliant on this work of the mind to use sensory input to recognize a “whole.” He draws on the pulse-like qualities of attention, allowing an organizing principle that operates independently of sensation. With this approach, he builds an attentional model for the conceptual construction that generates units, pluralities, and lots. The model is technical, but in sum it is first that the mind is able and does construct units. These unitary things are at first determined by their background, not yet abstracted. Once abstracted, this concept becomes a unit, which Glasersfeld argues aligns with Piaget’s “element stripped of its qualities.”
Through additional layers of reflective abstraction, the mind creates not just multiple units bounded by their sensory material, but a unity of units – or in other words (whole) number.
Glasersfeld concludes this careful description of a theory for children’s conceptions of unit and number: “when we speak of "things," "wholes," "units," and "singulars," on the one hand and of "plurals," "pluralities," "collections," and "lots," on the other, we refer to conceptual structures that are dependent on material supplied by sensory experience. Insofar as these concepts involve sensory-motor signals, they do not belong to the realm of number. They enter that rarified realm through the process of reflective abstraction, which extricates attentional patterns from instantiations in sensory-motor experience and thus produces numerical concepts that are stripped of all sensory properties.” This note is evident in the figure he provides that outlines his model for children’s construction of number.
Having concluded this expose on how the concept of number arises in children, I turn briefly to what I believe @rcraigen suggests when he mentions Sesame Street and my mother. It is a simplistic argument about whether or not children learn math (or in this case number) by listening to (and interacting with) others, I assume as they count.
Vygotsky suggested that speech can be used to create concepts. Piaget’s studies determine that concepts must be developed first, before speech can be mapped onto concepts to attain meaning—the words do not transmit meaning. Very simplistically stated, Piaget stresses concept development must occur before meaning is associated to what is intended through speech. Vygotsky argues for the use of speech to develop concepts.
First note is that the Piagetian perspective supersedes Vygotsky’s claim. I don’t think Piaget would eliminate the sensory input of speech as something that impacts the development of concepts. Second is that Vygotsky and Piaget’s epistemologies are much more aligned than divisions like this may suggest. In fact, because Vygotsky was a Jew working as a Developmental Psychologist in Stalinist Russia, much of his publications must take these contexts into consideration. Of course he lived through an era of Jewish persecution, followed by the socialist revolution and Stalin’s purges of people and ideas that he perceived dangerous to the soviet state. These threats were very real along with Stalin’s strict ideological control of Science. As a developmental psychologist, Vygotsky was likely expected to theorize how the state-sponsored knowledge is learned. For the constructivist, including Piaget and von Glasersfeld, the concern was not the construction of any specific knowledge, but just of knowledge itself.
In this sense, Vygotsky’s work might better help to understand how the discipline of Mathematics can be learned. While Piaget and Glasersfeld’s constructivism may help understand how knowledge that may be called mathematics or mathematical can ARISE in children.
To write the response above, I drew heavily on An Attentional Model for the Conceptual Construction of Units and Number by Ernst Von Glasersfeld (1981), in JRME 12(2) pp. 83-94.