Saturday, June 13, 2015

How Children Construct Number - an effort to respond to Dr. Robert Craigen

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 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.

Saturday, April 25, 2015

Mindset - useful or useless, or dangerous pop-psychology

Is math education having a mindset revolution or devolution?

My simple understanding is that it has been empirically confirmed that if you don't believe you can do it, you won't; and if you do believe you can do it, you will. Of course not completely in either case, but hopefully this simple summary is not too incorrect.

And so if we can teach children to believe that if they work hard, they can learn math.

Now this pop-psychology takes a very ugly turn, in my opinion. A few comments:

Serious Problem #1
This whole message is endemic of the "white missionary paternalism" (Danny Martin, 2007), a revivalist & colonialist approach to rescuing those poor ______ (fill in the blank, underachievers, brown children, etc.).

Serious Problem #2
It also contributes to the reification of the particular Mathematics that serves as a racist institution, ensuring the maintenance of power and privilege. It is simply the false mantra of the wishful American meritocratic belief system. "Pull yourself up by your bootstraps and you can achieve."

Serious Problem #3
Turning toward epistemological and ontological faults: the singular "Math" of the mantra is most certainly "School Math" -- what you are told you must learn in school. This is (a certain) other people's mathematics, it is history, it is the knowledge of the elite, privileged, powered.

Serious Problem #4
The mantra is just new words for teachers to continue to blame children for "not learning."

It continues the deficit approach to perceiving/knowing/teaching children; to focus on what they lack, what they need INSTEAD of what they bring, the brilliance of their ideas, etc.

Serious Problem #5
The mantra gets applied to teachers who think that some of their children cannot learn math. Again, a deficit orientation to teachers as learners (for those of us thinking about teacher professional learning).

Serious Problem #6
The empirical observation prescribes no solution to how to "change" someone's mindset toward mathematics. The best we seem to have are a new set of glossy posters, some rah rah enthusiasm, and new words through which to praise effort, a dangerous technique.

Hopeful Consideration #1
This mindset revolution may in fact be best for the attendees of the revivals, those who's mindsets may now be at least partially changed or more open to believing that all kids do learn.

The *real* mindset revolution would be to escape the shackles of mathematics and see oneself and every other human as authors and generators of mathematics, as brilliant, and insightful, creative, and mathematical. And to stop measuring any of these, especially mathematics, against some foolish declarations of or standards for what counts. To agree to be measured by the powered regime keeps that regime in power.

A Sketch of why Math Education must go the way of Latin Education

I wish to post an incomplete train of thought about the impending extinction, or need for extinction, of Math Education. (at least how we know it now, at least in secondary grades)

Some core principles of communication and literacy that Latin was an early reflection of still remain. How our literacy works is clearly a function of Latin. The development of many “rules of English” follow upon natural structures of Latin; yet are non-sensical in the structures of English--sort of why modern English is so weird, hard to learn.

Some words, phrases of Latin remain. In essence, new languages (plural with emphasis) have been built on top of and/or replace Latin.

That is what will happen (has already, has always already happened) with Math. The mathematicians will continue a study of (some particular) Math, and will develop more Math following that particular logic chain, way of thinking.

But so much other “logico-mathematical” reasoning, knowledge, etc. (in the Piagetian sense) has emerged in the 2000 years since Euclidean Geometry, the 1000 years since Algebra, and the 300 years since Calculus, that we eventually have to look backward and wonder why we’re teaching dead content.

But that is only layer one of an argument,

The real problem is (a.) epistemological, and (b.) ethical in nature.

To TEACH math is to divorce children from their own reasoning, their own mathematics. We need to stop.

And secondly, Math as an adjudicator of who is elite and who should lick my shoes can no longer be allowed to thrive. It presently operates as our primary way to define our social hierarchy in the Western World.

Further on this ethical (moral?) side, math is used to cheat, steal, rob, and kill. Period. Of course it is also used for parallel wonderful and beautiful reasons. BOTH OF THOSE QUALITIES OF MATH MUST BE STUDIED as an integral part of studying mathematics as a human endeavor, as multiple knowledges/ways of knowing,...

Will there be a math that only some people study? Yes--but a complicated answer (in my prediction).
  1. I think their will be those who study “Classical Math” -- as it will be labeled. Really, it is White Male Math.
  2. Then there will be people who study Math of Societies. It will be more an Anthropological (or Historical or Cultural) approach.
  3. There will be a (recognized) study of Math of Children, maybe a Psychological approach.
  4. Related, there will be some way of studying the Mathematical Activity of human beings, in there daily life, work, and play. This might be a Social approach.
I wish not to make this edict, "Math Education must go the way of Latin" as grandiose as it sounds. It’s about breaking the stranglehold White Male Math have on the mathematical minds and souls of our children, and our society.

it’s to rebirth, revalue, the people’s math

Coda: It would be fun to rewrite this blog entry, replacing "Latin" with "Math"

Sunday, March 29, 2015

Brief thoughts on purpose of HOMEWORK in high school maths

Check out this report on homework: Heavier Homework Load Linked to Lower Math, Science Performance, Study Says.  A better / more truthful headline would suggest that these researchers recommend one hour of homework per night.

HOWEVER, this sort of research concerns me because it suggests a causal effect from # hours of homework and test scores. Better stated, my concern is that the public will read into the science that a causal effect is the TRUTH. Clearly this study could not imply a causal relationship, and in fact many other studies negate that assumption. Alfie Kohn presents a discussion of a few key studies about the effects of homework, and in particular Math & Science homework at Homework: New Research Suggests It May Be an Unnecessary Evil.

Although it is reasonable to argue that Alfie Kohn may have bias, it seems to me his “bias” is to question or deconstruct the invisible assumptions that schools and U.S. Education writ large seem to operate under.

Me personally, I cannot bring myself to fully accept what Kohn suggests and advocate for NO HOMEWORK at high school level. But my take is different than the norm for school homework routines, but I find it is not usually argued with by math teachers. I think kids should have 20-ish minutes of math homework 4-5 nights per week, plus once every 2 weeks or so a bit larger of a project to complete, such as a portfolio demonstrating learning or a more formal paper about a mathematical investigation undertaken (i.e. a “POW write-up”).

The nightly homework should balance practice of recently developed procedural strategies with some challenging applications, sharing/talking about learning with others (family), and brand new ideas to consider in preparation for the next class. Of course, not all each night. Certainly the HW is not to be scored right/wrong but rather “credit” be given for doing some mathematical thinking between class meetings.

A few more essays / research summaries by Alfie Kohn can be found here and here.

Saturday, March 21, 2015

More Interesting: Elementary or Secondary Maths?

Dan Meyer recently posted on twitter, "I think elementary math pedagogy is more interesting than secondary but I don't know if I can get excited about the math." I found this statement to stick with me, I couldn't shake it free. I think maybe because it had truths in it for me personally, but maybe it didn't capture my feelings exactly?

To begin, I am personally strongly drawn to secondary maths pedagogy. My reasoning is because of all we don't know about teaching HS students, but more strongly because of all we do know, but (culturally) (politically) refuse (?) to implement. Why the refusal to ask children to interact in our classrooms, why the fears of wrong answers, why the avoidance of children's own algorithms, why the persistence of discipline separation, of labeling by math ability, the maintenance of the belief of a necessary linearity to maths learning, and why does tracking maintain?

Gosh, and I said nothing about math. I am aware of how purposeful that was. I actually really love engaging in mathematical activity, especially with others and *especially* with children--because they are less like-minded than me, and most colleagues who have grown up as star mathletes.

So this speaks somewhat to why I've wrestled with Dan's comment. When I hear "math pedagogy" my mind goes to the attending to and making sense of the thinking of children. That thinking *is* maths (children's mathematics, as Les Steffe defines it). In fact any mathematics is the thinking/knowing of oneself, one attributes to himself, or one attributes to others (which is still ones own mathematics); or what we refer to as that which is unknowable but we interpret to be mathematical--the (maths) knowing of others. I.e. that children's maths. So for Dan, what math isn't he all that excited about? Without the opportunity to be more precise constrained by the Twitter character-limiting structure, I would guess it is his own mathematics that he has either attributed to elementary or hs.

This seems very fair--it is not a just critique to gauge what another person should and shouldn't like. My interest was what about the sentence did I agree with, but maybe still didn't sit well.

What shook some of this loose for me was a wonderful essay I read today by Vivian Paley entitled "On Listening to What the Children Say." Of course it reminds me very strongly of Dewey's "Child and the Curriculum," yet Vivian laid out a bit of a pedagogical structure in the paper--one that I think could help the field of mathematics education reconstruct our profession. Here is my summary:
(1) craft mathematical questions (ones we as maths educators consider to be mathematical, and age appropriate)
(2) be genuinely curious about what students say, how they respond; and
(3) seek to build connections among children’s ideas and between children’s ideas and the “text” (i.e. discipline)

Further along the essay is what most stood out for me: Maybe HS students are so much more similar to our own ways of knowing mathematics that it is harder to see the differences and harder to be wrested away from our knowing is the knowing. But even more, Vivian suggests "older children have already learned to fear exposing their uncommon ideas." We don't get to hear what maybe brilliant and very non-standard mathematical wonderings, insights etc. in high school. I wonder if this is why many are less drawn to HS pedagogy.

I have recognized for a long time that "trajectories for student mathematical learning" seemed highly underdeveloped in HS. I thought that was mostly because the mathematical terrain (for me, this means the mathematical schemes constructed in young minds) became far too complex, interwoven, and of course non-linear. But the more I think, maybe it is so much more because we cannot get over ourselves and our own ways of knowing when studying older children in the position of an observer. The knowledge we see is far too close to our own knowing. Furthermore, it is far less likely for HS children to be open with their uncommon ideas.

This may be what Dan alluded to in some manner... I think I can now sleep better and let this comment go. Whether it is reconciled for me or not, is a different matter.

Sunday, September 28, 2014

A pre-service teacher's first read of Kamii's constructivism

I ask my elementary preservice teachers, all graduate students (5th year program in CA), to read Constance Kamii's "Young Children Reinvent Arithmetic." (if you haven't read it yet, you must) Of course I don't expect all my students to suddenly recognize and fully understand constructivism as a theory / model for how people learn, and especially not apply this model to one's own knowing of the world. But the text seems to really push students to recognize that kids need space to experience and interact, rather than be told and drilled through worksheets. So I will take that success any day.

I thought I would share one student's response to chapter's 1-4, and my response to her. I am curious about comments from anyone who happens to read this. Simply, I wonder what impact her present take-aways from the book may be, and to what effect my response may elicit further consideration.

In the first Chapter of Young Children Reinvent Arithmetic, a key concept that stood out for me were three kinds of knowledge and how each of them can be expanded through math, just through different means. The physical knowledge is expanded just by observing objects and the social knowledge is expanded by showing students what has been constructed for them to know. The most important and effective knowledge for math,of course, is logico-mathematical knowledge allows for students to be the constructors of their own understanding. This is achievable by the teacher creating an experience for students so that they can be the authority in their mathematical thinking as we talked about in Friday's class. I appreciated how in class we were able to both be a part of and observe this type of set-up when we constructed our own conventions about consecutive numbers. We were provided with the launch to think about at home and try to create expressions for the 1-2-3-4 puzzle. In class, we were then able to discuss our expressions with peers, either agreeing with or disagreeing with one another. Then we were prompted to explore some patterns we saw that connected these problems together and express a convention to share. I can definitely see how this style (launch-explore) is much more beneficial than merely showing expressions.The second chapter talked about representation and a major take-away for me here was the use of manipulatives as being a symbol for numbers. Kamii talked about how kids prefer to  use pictures to represent numbers when counting on. Chapter 3 was all about how social interaction is vital to understanding math and logic. Cooperation is necessary because it both mutually benefits the learners in the group when they can decenter and constructively criticize one anothers work and explain the "why" to one another. I appreciate how Kamii relates this to being  beneficial to moral development as well. This was demonstrated in our class session, like I stated before when we compared our expressions and either agreed/disagreed with each other.  Finally, chapter 4 was about allowing fro autonomy rather than promoting heteronomy. A key takeaway was allowing for children to make decisions for themselves by giving them choices not just in the intellectual realm but also in the moral. The title for the Kamii book is appropriate because it is all about creating opportunities that allow students to invent ways to connect math to the realities they experience. This is the only way they will take ownership and have a positive experience while learning math.

You have made an important mistake in trying to understand how we know people learn. The purpose for naming three types of knowledge is more about what the knowledge is of -- the physical world or the social world, or the knowledge that is built upon previous knowledge through the brain's inventions (constructions) of relationships between those other forms of knowledge. One human cannot transfer that knowledge (mathematical or otherwise) to anyone else--ALL knowledge is constructed. So the work of the teacher is to teach (math) "indirectly" -- because there is no other way. Even when "telling" - the student's mind must take in the perceptions (teacher's voice, images, etc.) and construct knowledge from that.  
When children think (all people), we work with images in our mind. When we put those images on paper (for example) -- within that representation of our mental are all the ideas and connections we associate with that. Then we can operate on the picture, reducing the taxation on our mind to hold all of those thoughts while operating on them. As a learner, we invent our own manipulatives -- only then can a manipulative be that representation of our mental images. And thus, usually the (forced) use of manipulatives in a classroom just become another physical or social knowledge for children to take in, rather than to invent the relationships among children's current knowledge that the manipulative is intended to invoke.
Your summary is quite excellent. Your response reflects a *core* misunderstanding of learning theory, one that plays itself out in classrooms across the country, leading to the poor learning opportunities you have observed. Children will not "invent ways to connect math to the realities they experience" first because math is not something external to them, ever. Math is external to no one--everyone of us has our own mathematics. And the "realities they experience" also may not be the most precise use of words for the modern learning theorist -- again, reality (while it may exist out there) is not so much experienced as it is invented by us as a biological organism living in this "world." And since no one can actually know a "true" reality (assuming there is one), we have no way of ever knowing if our knowledge is correct or not. We only have viability in the world as we experience it--that we remain alive, functioning, happy, and we have other humans who seem to confirm our knowing of this reality... 

Friday, March 14, 2014

On Constructivism and Motivation

As a constructivist, I'm promoted to wonder why math teachers obsess about children's motivation. Mostly, I experience this obsession in a negative manner -- "my kids aren't motivated." Or more productively, in a way that turns the responsibility inward, "I'm trying to figure out how to motivate these kids."

I can't help but interpret that thinking about trying to convince the learner to swallow some nasty medicine. But arguing that School Math is a nasty medicine we're trying to force down kids throat is a different post.

For me, ideas about motivation, especially those of intrinsic and extrinsic motivation, live in the world of behaviorist learning theory. A learning theory that western culture knows so well we have a hard time knowing/thinking outside of of it (like fish & water).

The present constructivist theory of knowing and learning, superseding behaviorism, really messes up the idea of motivation. At first, it makes a different definition for learning. It is not a definition that relies on a "what" is to be learned (what overwhelms us as math teachers), but instead focuses on hypothetical models for knowing and defines learning changes to those knowing structures modeled. So "what" is to be learned is recognized as an idea of the teacher, and something they want to "see" replicated in the learner. Now motivation has become more of a problem OF the teacher, not a lacking in the learner.

What I mean by a problem OF the teacher--as opposed to FOR the teacher. Lack of motivation of students to learn math, to do homework, etc. is the "standard" way of thinking about motivation as a  problem of the teacher. The teacher wishes to see behaviors replicated in their students. So totally behaviorist, I can't help but wish to suggest a whistle and a bucket of fish strapped to the teachers hip for rewards to the trained dolphins, er, students.

The idea of motivation as a problem OF the teacher is that when a person's experiential reality is not sitting properly, as it should, that person wishes for it to change -- so as the person's knowing of her experiential reality doesn't have to be shifted to account for something that just doesn't fit. The nature of a knowing, autopoetic organism (i.e. a human) is to maintain its own inner nature, its equilibrium. Herein lies its "motivation." This is the case of a teacher, and of the student. Each's motivation is at odds in their structural coupling. The motivation question shifts to first wonder why would the teacher wish to see her own way of knowing replicated in the knower external to herself, a knower she attributes as functioning in ways alike to herself but is a separate entity. It is the case that she wishes to coerce the other to behave a certain way. The motivation to "teach" is hers, and hence her problem. The question that should be asked is why ought this be the motivation OF a teacher?

A more ethical interaction between two cognizing subjects, even when one may have a socially defined role as teacher and student. The teacher ought not seek a change in behavior in the student, rather strive to invoke an inquiry process that she suspects may lead to a way of knowing that creates for the learner a greater viability in the learner's known world.

For me, I consider the individual knower / learner to be fundamentally goal-directed, i.e. motivated. This is a core principle to their remaining viable in the way of experiencing their world. Any nudge I can give this learner toward experiencing some joy by taking up some logic-mathematical disturbance I can create, might be the extent to which I can concern myself with motivation of a learner. As you can see, I am beginning to find myself thinking in circles about this idea -- when it comes to imaging the work of "teaching."

For now, an unresolved issue...

P.S. Lets not lose sight that Cognitive Psychology remains mired in the tar pool remnants of Behaviorist ideas, especially definitions for learning. The theory gives a reality to knowledge, and fails to problematize the observer stance.

Coda: It is likely I ought to have left this discussion of learning and motivation to von Glasersfled himself. One place he writes directly on motivation is here, in particular beginning on p. 7.