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.

Student:
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.

Me:
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 http://www.vonglasersfeld.com/135, in particular beginning on p. 7.

Tuesday, July 16, 2013

Just some thoughts regarding my research on Mathematical Identity

I very often attend very research-y sessions on Mathematical Identity of students or some student group. Often, the student group in consideration is marginalized, in one way or another. I have a very difficult time during these presentations. I have named several reasons why it may be so (next bullets), but ultimately think it is simply my impatience/intolerance for people to see the world in my way. I'm such an adolescent.

These frustrations emerge from, in my opinion:
  • researcher's lack of an explicit theoretical stance, and/or
  • an under-examined definition of identity, and/or
  • an under-examined definition of mathematics, and/or
  • an under-examined role of oneself as a researcher. 
The thoughts that follow were written shortly after NCTM Research Presession 2013, during which I found myself rather underwhelmed by several sessions regarding children's mathematical identity. 

Tuesday, May 21, 2013

Planning PD for HS Math teachers -- Focus on Pedagogical Practices in PrBL

My teacher-leader colleagues in Delaware proposed focusing pedagogical conversations during summer professional learning activities for HS math teachers on 2 conclusions that emerged from Hiebert & Grouws (2007) book chapter in NCTM's Second Handbook of Research. We are reading H&G's chapter, as well as the NCTM Research Brief.

We plan an online conversation about the ideas in H&G's paper, especially the two principles of learning: (1) being explicit about the key ideas, and (2) supporting productive grappling are central to our thinking about the pedagogical work for teaching in a problem-based learning (PrBL) environment.

Friday, February 8, 2013

Mathematics is a Simulacra

I write that Mathematics is a Simulacra, what I consider to be a liberating recognition that maths is a human endeavor, it is created by people rather than discovered as an ontological reality.

Andreas Quale wrote a nice paper that allowed me a fun curiosity for the evening.

Maybe it is productive and/or interesting to view this discipline of Mathematics as a human-invented game, with a evolved set of rules that have created a wonderful space that is so at the limits of our minds that we cannot recognize it as such--a game, akin to chess, that is fixed, bounded, determined, one of many possible. Basically, this particular Mathematics is relative to its rules, rather than a real thing.

Some other species peers down upon us like rats in a maze. As rats, we see the walls constructed by our experiential knowing of our world/maze. We don't see our own constraints, as if from above--some omniscient perspective.