tag:blogger.com,1999:blog-41893261411733128502024-03-14T12:04:16.550-04:00b's lawI write here to put down ideas. I don't try to edit. I don't try to be right.
I do try to disrupt.blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.comBlogger34125tag:blogger.com,1999:blog-4189326141173312850.post-16312817426708242642016-04-19T16:06:00.000-04:002016-04-19T16:06:03.241-04:00The Myth of Math AbilityMy colleague <a href="http://education.illinois.edu/faculty/rg1" target="_blank">Rochelle Gutiérrez</a> gave several wonderful talks at NCTM & NCSM related events this past week in San Francisco. In her talks, she proclaimed, "Mathematics 'ability' is not real, but the trauma associated with it is."<br />
<br />
Each claim resonates with me. We can recognize how many of our friends and loved ones readily proclaim they are not good in math. They are placing themselves in some way or another "less than" those who they view are "good in math." This may cause little trauma. But to quickly ratchet up the damage, when schooling and or testing teaches children what they should think of their intellectual ability, the trauma is clear. Children act out and/or reject school. They have been taught they are less than. I taught a Pre- Pre-Algebra class one semester. The school selected 18 ninth graders they deemed might need extra support for high school math. Within the first week, my students were fully aware that they were "the dumbest kids in school." "That's why we have a special class, just for us."<br />
<br />
I have also been in schools on the North side of Chicago, schools regularly praised for their success. The faculty are very aware of student scores on the Iowa Test of Basic Skills. This is because students are sorted by their percentiles. Kids in the 90th percentile take classes with other kids in the 90th percentile. 80th, 70th, and so on. In fact, faculty even refer this group of kids by name -- the 70s kids, etc. As ugly as this seems, the real trauma is evident in the kids know their own classification, and refer to themselves and others by this numbering / ranking system.<br />
<br />
The trauma is real.<br />
<br />
Is math "ability" real? Dylan Kane took this up recently in <a href="https://fivetwelvethirteen.wordpress.com/2016/04/16/math-ability/" target="_blank">response t</a>o reactions to his post on twitter. He offers some reflection on the idea, maybe willing to agree it isn't. He makes an important point about the futility of trying to convince students of this, or parents. I concur, but feel the most important audience for the idea is the teacher. I have three thoughts on math "ability," which I'll discuss from pragmatic to idealistic.<br />
<br />
1. The <i><a href="https://www.marxists.org/reference/subject/philosophy/works/fr/lyotard.htm" target="_blank">grand narrative</a></i> of math ability can effectively be disrupted in the mathematics classroom; sociologists Elizabeth Cohen and Rachel Lotan (<a href="http://www.uvm.edu/complexinstruction/about_ci.html" target="_blank">Complex Instruction</a>) have demonstrated this in their classroom research. Cohen and Lotan have studied student interactions in classrooms for 40 years. They first established that students learn when interacting. Applying expectation states theory, they recognized that what students expect of another with regards to competence influences that students opportunity to access or engage in the interactions. Less interaction means less learning. Complex Instruction (CI) is a instructional strategy that changes these expectations of competence, so that previously low status students have increased opportunity of learning.<br />
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What is important here with regards to ability is that CI begins with creating a multidimensional classroom (<a href="https://www.youcubed.org/wp-content/uploads/pdkarticlejuly2006.pdf" target="_blank">Jo Boaler</a>). Using language of "ability," it is about expanding students ideas about what counts as mathematical ability; or my words in the classroom are mathematically smart. The instructional treatment begins with the launch of a group-worthy task. The teacher identifies several mathematical abilities necessary for success with the task. Then the teacher states, "All of us have some of these abilities, but none of us are great at all of them. You will need your group members t=o be successful on this task. The teacher then follows this up with observing students demonstrate these various abilities. During or after the task, the teacher then affirms the different abilities seen expressed by class members at various times during the task. The teacher especially affirms the contributions of low status students. This change in the classroom changes students expectations for competence ("ability") they hold for one another.<br />
<br />
As Dr. Gutiérrez believes, this strategy also impacts the teacher's ways of seeing and judging children; the teacher acts his/her way into new beliefs.<br />
<br />
In this way of tacking the question, the response is less about a yes or no answer, but instead focuses on how to change how people perceive others and their own ability. By creating many ways to be mathematically smart, we can recognize mathematical abilities in all our students.<br />
<br />
2. And to come at the question, is math "ability" real? from one more perspective, I contend it is a social construct, a notion created only to organize the world as we experience it. In this way it is like <a href="http://www.nytimes.com/roomfordebate/2015/06/16/how-fluid-is-racial-identity/race-and-racial-identity-are-social-constructs" target="_blank">race</a>. And like race, how people are classified by ability most certainly results in consequence.<br />
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3. I agree with <a href="http://www.powercube.net/other-forms-of-power/foucault-power-is-everywhere/" target="_blank">Foucault</a> that power/truth/and knowledge are intermingled. And that it is power that produces truths. Thus, the discourse around mathematical ability has been shaped by the thought of the dominant culture, in this case Western thought--and more specifically <a href="http://imaginenoborders.org/pdf/zines/UnderstandingPatriarchy.pdf" target="_blank">imperialist white-supremacist, capitalist, patriarchal</a> thought. The idea of "ability" is a construct of this hierarchal worldview; it is a notion that in its existence affirms the power structure. In this way, I would argue that ability only is a "truth" because it serves power. Power works through its practices of surveillance, classification, exclusion, regulation, and normalization (<a href="http://www.amazon.com/Working-Ruins-Feminist-Poststructural-Education/dp/0415922763" target="_blank">Elizabeth St. Pierre</a>).<br />
<br />
In sum, our notion of math "ability" only exists because of the power structures that defined it. We as educators can disrupt this "truth" in our classrooms and in our interactions with family, friends, and elsewhere that we are turned to as an authority.<br />
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Not only can we disrupt, it is our moral obligation.blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com1tag:blogger.com,1999:blog-4189326141173312850.post-64821087957797867692015-07-30T12:52:00.000-04:002015-07-30T12:52:00.452-04:00The Ethics of a Relativist's Epistemology, and concerns of (Mathematics) Education<span style="font-family: inherit;">OK - another response provoked by a discussion with a colleague. Maybe I am creating the anti-blog; or simply the blog that is only extended comments (since I seem to really have trouble writing briefly).</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">A friend @doingmath asked (paraphrased some)</span><br />
<blockquote>
<span style="font-family: inherit;">I am curious is about the part of the discussion from 1:08:00-1:13:00 in the <a href="https://youtu.be/n5RjTIaJmBM?t=68m" target="_blank">video</a>, when the discussant sets up the speaker, Dr. Munir Fasesh, to elaborate about the connections between mathematics, violence, and freedom...as well as some connections to Freire's work. He says (roughly):</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="color: #666666; font-family: inherit;">"What I did not like about Freire's approach, is that he divided people into levels of consciousness. There are those that are more conscious and those who are less...and less...and less...</span><br />
<span style="color: #666666; font-family: inherit;"><br /></span>
<span style="color: #666666; font-family: inherit;">And the ones who are 'more' have to 'teach' the ones who are 'less.' Here, he is following the logic of education....</span><br />
<span style="color: #666666; font-family: inherit;"><br /></span>
<span style="color: #666666; font-family: inherit;">[Freire] changed not the logic, but the fact that he is now calling them more 'conscious' rather than more 'educated.' Dividing someone into expert or someone who is trained in the 'Freirean way' are allowed to really put people....to decide who is more conscious and who is not; Rather than to look at every person as uniquely complete. </span></blockquote>
<blockquote>
<span style="color: #666666; font-family: inherit;">Every person is like a seed. I cannot compare. Every seed has the potential to really grow and all we can do is to provide that seed with the right environment."</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">I take it you know my (post-)epistemological assumptions, but something about this perspective gives me pause any time I consider social perspectives. What are we to think when people hold worldviews that serve to perpetuate the marginalization of certain groups of people? Or that have implications for the sustainability of our planet? Should we really not acknowledge these perspectives as harmful? As "less conscious?" </span></blockquote>
<blockquote>
<span style="font-family: inherit;">What are your thoughts? What reactions did you have to Munir's plenary and ensuing line of questioning?</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">P.S. The only resolution I can come to right now is a type of "intersubjectivity of ethics" where people, collectively, must decide what is right and wrong...what is ethical. But what happens if people can't agree? Multiple realities? But what if one or more of those realities are harmful to others?</span></blockquote>
<span style="font-family: inherit;">My thoughts: The discussant, Dr. Julia Aguirre, asked roughly, "You make connections among mathematical knowledge, domination, freedom, and intelligence. Words that embody inequality are crucial [i.e. necessary] for domination, especially inequality in intelligence. Freire says knowledge is invented and reinvented through struggle. To prevent another from engaging in such inquiry is a form of violence; it is dehumanizing. Say a bit more about the interconnectedness of mathematics & these ideas."</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Munir seemed to respond: Freire is my buddy, but I disagreed with some of his ideas -- he divided people into levels of consciousness. Those who are more conscious must teach those who are less. He only replaced educated with conscious; he was still dividing people. This division allows people to judge others, and rank by consciousness, rather than to look at every person and </span>acknowledge that<span style="font-family: inherit;"> "every person is uniquely complete." Freire's language is </span>propagated<span style="font-family: inherit;"> by mathematics, that people can always be placed on a line. "<b>The worst inequality, that is the source of all inequalities, is believing that people are not equal in intelligence</b>. <b>And giving yourself the right to tell who is more intelligent than another: Arrogance."</b></span><br />
<br />
<span style="font-family: inherit;">The dilemma (paradox?) you seem to hit is certainly one I wrestle with. I wrestled more years ago, but am coming to decide there is no answer of the form with which I am used to finding answers.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">My first thought is that the basic notion *is* or *to be* is a sort of paradox of ethics, or ethical behavior. It seems to send or establish an ethical message throughout maybe every human culture, what is this notion of (related to) existence? </span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Being raised catholic we were taught “do unto others as you would have them do unto you.” I think if each action one takes is guided by that principal, we would come to the same dilemma you have pointed to, how I myself cannot judge the ethics of another if I would not allow them to judge mine. </span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">I don’t know much about ethics and morals, except that I suppose I have and practice my own. So, maybe the first part of my answer is that I am compelled to interact with people who hold worldviews that I consider dangerous / oppressive or simply disagree with, because I suspect that if enacted, such views would harm me and/or members of my family and community.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">I <a href="http://www.academia.edu/11978717/The_fabrication_of_knowledge_in_mathematics_education_A_postmodern_ethic_toward_social_justice_2012_" target="_blank">wrote previously</a> about another layer of ethics that I believe is baked into Radical Constructivism - that we need thriving (healthy, mindful, thoughtful) others different from ourselves in order for ourself to thrive.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Yet, I wouldn’t wish for that other person, with whom I disagree, to oppress me, shut down my opportunity to speak, thrive, live joyously, etc. -- So I cannot do the same to him.</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">NOW, a very different question; switch </span>from the<span style="font-family: inherit;"> ease of pontification to the pragmatism of daily living. Does the above change in the context of your employment as a math coach. I think it does. I think you have now become not simply a single entity, but a member of a group, with its own moral code. You are now compelled to follow the moral code of this group (or resign). So, what is the moral code of the educators in your school district?</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">There is likely a moral code for what is valued as education within your district, maybe even one specifically written for mathematics. I suspect it is stated in such unmeaningful words that everyone nods agreement at the collections of words, yet there is minimal shared interpretation. </span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">It would be nice to collectively identify/discuss this moral code, and from it define a vision for maths education in your district, and then a mission -- with specific, measurable statements. And assign a timeline. (i.e. a 5-year plan and how you will measure successes of it). </span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Now you will have a taken-as-shared (among district math teacher community) and public statement of morals, reflective of a <i>collective</i> ethics. Decisions should be judged against those morals. </span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">I don’t know what more to say; I am stuck because what I have said once again seems very idealistic. But I think you and your colleagues within district leadership must have careful talks such as the one suggested above, and set a 5-year agenda, because it seems to me that a school district MUST have a <i>collective</i> moral that <i>OPPOSES</i> that of the status quo, the community majority. You also must broadcast this district moral code, in a well-marketed way--because it will be counter-cultural, against the status quo, not what the proletariat has been programmed to expect. Your challenge will be for parents to embrace the Deweyan notion that <i>what is best for their kid is best for all kids:</i> </span><br />
<span style="font-family: inherit;"></span>
<blockquote>
<span style="font-family: inherit;">What the best and wisest parent wants for his child, that must we want for all the children of the community. Anything less is unlovely, and left unchecked, destroys our democracy.</span></blockquote>
<br />
<span style="font-family: inherit;">Any of those thoughts stick? sticky?</span><br />
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">And you have a P.S. People will never agree -- the idea of “agree” is a falsehood. Yes, of course multiple realities. BUT, as a member of a social group, your school district, -- your </span>administrator-level<span style="font-family: inherit;"> challenge is to monitor that level of agreement about the direction of the moral code. If the admin learn they have employees who bigots, they must act immediately to remove them from interacting with children. Same as if they learned they had an employee that was a sexual abuser. It is sad to me that that particular action has not been taken, and the bigotry is so normal that it is openly stated in emails and to my student teachers.</span><br />
blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com0tag:blogger.com,1999:blog-4189326141173312850.post-54265135998854001102015-06-13T21:41:00.000-04:002015-06-13T21:41:09.044-04:00How Children Construct Number - an effort to respond to Dr. Robert Craigen<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">Robert Craigen @rcraigen has confronted David
Coffey @deltadc and me @blaw0013 on Twitter, “Neither of you seems to know <b>how a child's first math concept arises</b>
- that's the Q on hand.” <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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.”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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 <i style="mso-bidi-font-style: normal;">confirmation bias.</i> But maybe we
all could be accused of that. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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 <i style="mso-bidi-font-style: normal;">math</i>, <i style="mso-bidi-font-style: normal;">concept</i>, and <i style="mso-bidi-font-style: normal;">arise</i>, 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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">My summary of his efforts to provide
definition to each term:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: .5in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">Math: the science of structure, form and
relation. <i style="mso-bidi-font-style: normal;">Math is a body of knowledge</i>.
Math is a precise language for technical and abstract subjects matter. Correct
usage and standardization are critical.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: .5in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">Concept: “happy with natural meaning of the
term in ordinary discourse.” He wrote a bit more, which I summarize as <i style="mso-bidi-font-style: normal;">a general rule or class</i>, and abstraction
from singular instances.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: .5in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">Arise: <i style="mso-bidi-font-style: normal;">to
come to exist, appear or manifest.</i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">And he asked for my definitions. Next, I will
give that a go.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">I am good with how @rcraigen defined <i style="mso-bidi-font-style: normal;">arise</i>, 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 <i style="mso-bidi-font-style: normal;">know</i> if that object <i style="mso-bidi-font-style: normal;">exists</i> in a Platonic sense.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">I am also comfortable with the definition
@rcraigen pointed toward for <i style="mso-bidi-font-style: normal;">concept</i>.
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. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">“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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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 <b style="mso-bidi-font-weight: normal;">MATH CONCEPTS ARISE</b> 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. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">And here is where a VERY careful definition
of <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">mathematics</i></b>
becomes important—is mathematics physical, social, or logico-mathematical
knowledge? Or maybe, <i style="mso-bidi-font-style: normal;">which</i>
mathematics is physical, social, or logico-mathematical knowledge?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">I draw upon a notion of science developed by
Imre Lakatos, a mathematician and scientist of the mid 20<sup>th</sup> 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. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 12.0pt;">
<span style="font-family: inherit;">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 <i style="mso-bidi-font-style: normal;">process) </i>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. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 12.0pt;">
<span style="font-family: inherit;">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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 12.0pt;">
<span style="font-family: inherit;">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. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 12.0pt;">
<span style="font-family: inherit;">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.”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 12.0pt;">
<span style="font-family: inherit;">Through
additional layers of reflective abstraction, the mind creates not just multiple
units bounded by their sensory material, but a <i style="mso-bidi-font-style: normal;">unity of units</i> – or in other words (whole) number.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 12.0pt;">
<span style="font-family: inherit;">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.</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_1MpI0r7MqZdDpLG8yk1FZpbwEy8Ekh9C19bmejxNflgEhHibtr6KDNgmRPEX9644USSiZg_LuMVt7iGqYOwMcSjGhQnkLJ2SoLThee67xkkdNnSpCevAGSHLXppzKfM0X4fLMA61SmkM/s1600/Screen+Shot+2015-06-13+at+9.02.14+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img alt="" border="0" height="317" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_1MpI0r7MqZdDpLG8yk1FZpbwEy8Ekh9C19bmejxNflgEhHibtr6KDNgmRPEX9644USSiZg_LuMVt7iGqYOwMcSjGhQnkLJ2SoLThee67xkkdNnSpCevAGSHLXppzKfM0X4fLMA61SmkM/s640/Screen+Shot+2015-06-13+at+9.02.14+PM.png" title="" width="640" /></span></a></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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.</span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">Vygotsky suggested that speech <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">can</i></b>
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.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit;">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 <b style="mso-bidi-font-weight: normal;">ARISE</b> in children.</span></div>
<div class="MsoNormal" style="margin-bottom: 6.0pt; margin-left: 0in; margin-right: 0in; margin-top: 6.0pt;">
<span style="font-family: inherit; font-size: x-small;"><i style="color: #666666;">To write the response above, I drew heavily
on </i><span style="color: #666666;">An Attentional Model for the
Conceptual Construction of Units and Number</span><i style="color: #666666;"> by Ernst Von Glasersfeld
(1981), in </i><span style="color: #666666;">JRME 12</span><i style="color: #666666;">(2) pp. 83-94.</i></span></div>
blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com2tag:blogger.com,1999:blog-4189326141173312850.post-78224201888314039832015-04-25T15:41:00.000-04:002015-04-25T15:41:01.745-04:00Mindset - useful or useless, or dangerous pop-psychologyIs math education having a mindset revolution or devolution?<br />
<br />
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.<br />
<br />
And so if we can teach children to believe that if they work hard, they can learn math.<br />
<br />
Now this pop-psychology takes a very ugly turn, in my opinion. A few comments:<br />
<br />
Serious Problem #1<br />
This whole message is endemic of the "white missionary paternalism" (<a href="http://muse.jhu.edu/journals/hsj/summary/v091/91.1martin.html" target="_blank">Danny Martin, 2007</a>), a revivalist & colonialist approach to rescuing those poor ______ (fill in the blank, underachievers, brown children, etc.).<br /><br />
Serious Problem #2<br />
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. "<a href="https://abagond.wordpress.com/2012/01/12/the-bootstrap-myth/" target="_blank">Pull yourself up by your bootstraps and you can achieve</a>."<br />
<br />
Serious Problem #3<br />
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.<br />
<br />
Serious Problem #4<br />
The mantra is just new words for teachers to continue to blame children for "not learning."<br />
<br />
Corollary<br />
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.<br />
<br />
Serious Problem #5<br />
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).<br />
<br />
Serious Problem #6<br />
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 <a href="http://www.alfiekohn.org/article/five-reasons-stop-saying-good-job/" target="_blank">dangerous technique</a>.<br />
<br />
Hopeful Consideration #1<br />
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.<br />
<br />
<br />
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.<br />
<br />blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com4tag:blogger.com,1999:blog-4189326141173312850.post-89994311674782462532015-04-25T14:11:00.004-04:002015-04-25T14:14:18.494-04:00A Sketch of why Math Education must go the way of Latin EducationI 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)<br />
<br />
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.<br />
<br />
Some words, phrases of Latin remain. In essence, new languages (plural with emphasis) have been built on top of and/or replace Latin.<br />
<br />
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.<br />
<br />
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.<br />
<br />
But that is only layer one of an argument,<br />
<br />
The real problem is (a.) epistemological, and (b.) ethical in nature.<br />
<br />
To TEACH math is to divorce children from their own reasoning, their own mathematics. We need to stop.<br />
<br />
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.<br />
<br />
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,...<br />
<br />
Will there be a math that only some people study? Yes--but a complicated answer (in my prediction).<br />
<ol>
<li>I think their will be those who study “Classical Math” -- as it will be labeled. Really, it is White Male Math.</li>
<li>Then there will be people who study Math of Societies. It will be more an Anthropological (or Historical or Cultural) approach.</li>
<li>There will be a (recognized) study of Math of Children, maybe a Psychological approach.</li>
<li>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.</li>
</ol>
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.<br />
<br />
it’s to rebirth, revalue, the people’s math<br />
<br />
<br />
Coda: It would be fun to rewrite <a href="http://donaldclarkplanb.blogspot.com/2011/02/10-reasons-not-to-learn-latin.html" target="_blank">this blog entry</a>, replacing "Latin" with "Math"blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com0tag:blogger.com,1999:blog-4189326141173312850.post-47211128688475176612015-03-29T14:05:00.000-04:002015-03-29T14:05:01.085-04:00Brief thoughts on purpose of HOMEWORK in high school maths<br /><br />Check out this report on homework: <a href="http://blogs.edweek.org/edweek/curriculum/2015/03/homework_math_science_study.html" target="_blank">Heavier Homework Load Linked to Lower Math, Science Performance, Study Says</a>. A better / more truthful headline would suggest that these researchers recommend one hour of homework per night.<br /><br />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 <a href="http://www.huffingtonpost.com/alfie-kohn/homework-research_b_2184918.html" target="_blank">Homework: New Research Suggests It May Be an Unnecessary Evil</a>. <br /><br />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. <br /><br />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”).<br /><br />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.<div>
<br /></div>
<div>
A few more essays / research summaries by Alfie Kohn can be found <a href="http://www.alfiekohn.org/homework-improve-learning/" target="_blank">here</a> and <a href="http://www.alfiekohn.org/article/rethinking-homework/" target="_blank">here</a>.</div>
blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com2tag:blogger.com,1999:blog-4189326141173312850.post-54560924909845062922015-03-21T20:45:00.001-04:002015-03-21T20:45:41.265-04:00More 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?<br />
<br />
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?<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
What shook some of this loose for me was a wonderful essay I read today by Vivian Paley entitled "<a href="https://www.dropbox.com/s/fu8o2iyxp46vdmq/On%20Listening%20To%20What%20The%20Children%20Say.pdf?dl=0" target="_blank">On Listening to What the Children Say</a>." 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:<br />
(1) craft mathematical questions (ones we as maths educators consider to be mathematical, and age appropriate)<br />
(2) be genuinely curious about what students say, how they respond; and<br />
(3) seek to build connections among children’s ideas and between children’s ideas and the “text” (i.e. discipline)<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
<br />
<br />
<br />blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com0tag:blogger.com,1999:blog-4189326141173312850.post-62928787966164252212014-09-28T16:19:00.000-04:002014-09-28T16:19:19.018-04:00A pre-service teacher's first read of Kamii's constructivism<span style="font-family: Arial, Helvetica, sans-serif;">I ask my elementary preservice teachers, all graduate students (5th year program in CA), to read Constance Kamii's "<a href="http://amzn.to/1rDaCEa" target="_blank">Young Children Reinvent Arithmetic</a>." (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.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">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.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Student:</span><br />
<blockquote class="tr_bq">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="vertical-align: baseline;">In the first Chapter of </span><span style="font-style: italic; vertical-align: baseline;">Young Children Reinvent Arithmetic, </span><span id="yui_3_13_0_2_1411932143429_560" style="vertical-align: baseline;">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.</span></span></blockquote>
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Me:</span><br />
<blockquote class="tr_bq">
<span style="font-family: Arial, Helvetica, sans-serif;">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. </span></blockquote>
<blockquote class="tr_bq">
<span style="font-family: Arial, Helvetica, sans-serif;">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.</span></blockquote>
<blockquote class="tr_bq">
<span style="font-family: Arial, Helvetica, sans-serif;"><span style="background-color: white; line-height: 19px;">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...</span> </span></blockquote>
blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com6tag:blogger.com,1999:blog-4189326141173312850.post-44908704476276307512014-03-14T23:51:00.003-04:002015-10-13T21:38:02.363-04:00On Constructivism and MotivationAs 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."<br />
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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.<br />
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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).<br />
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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 as changes to those modeled knowing structures. 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.<br />
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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.<br />
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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?<br />
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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.<br />
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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."<br />
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For now, an unresolved issue...<br />
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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.<br />
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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.<br />
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blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com1tag:blogger.com,1999:blog-4189326141173312850.post-36147252536562623392013-07-16T18:35:00.000-04:002013-07-16T18:35:34.754-04:00Just some thoughts regarding my research on Mathematical IdentityI 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.<br />
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These frustrations emerge from, in my opinion:<br />
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<li>researcher's lack of an explicit theoretical stance, and/or</li>
<li>an under-examined definition of identity, and/or</li>
<li>an under-examined definition of mathematics, and/or</li>
<li>an under-examined role of oneself as a researcher. </li>
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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. </div>
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I cannot understand efforts to define other's identity w/o foregrounding oneself as an observer, and as a researcher as a second-order observer ( doing science on what was observed). Said another way, some (social?) researchers seem to allow themselves an omniscient positioning.</div>
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Maybe that's my first concern. Second, is the major problem of a theoretical lens (e.g. Wenger, Vygotsky) that gives any a priori existence to knowledge. It allows the researcher to operate as though they were all-knowing, omniscient. The researchers' work is to examine others to try to understand how the other comes to know as one himself does.<br />
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In that context, it is sensical to make a comment such as learning occurs in social processes. (Or in this case, identity is not formed in the head, but in social processes.) the researcher sees evidence of their own ways of knowing in the activity (behavior) of another. And then attributes their own way of knowing to the observed.<br />
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A constructivist seeks to create models of the subject's identity, as that subject may "know" "it." But at all times, the researcher recognizes their role (as a knowledge constructor / constructor of knowing). So the researcher builds a model for the "identity of the child" -- known to not necessarily be the child's identity, but a viable model for it. One that seems to remain consistent across observed behaviors.<br />
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The researcher constructs this model in her head. It is a model for the identity of the subject, something that they have constructed for themselves ( in their 'head'?) I don't know how identity "lives" somewhere outside of the mind/body/whatever. A common statement among sociologists (cf Restivo).<br />
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Ok, 3rd, but ultimately the same concern, is that maths is so often I interrogated in the question of identity. If identity is constructed, it is constructed in relation to another knowledge/knowing that is taken to exist in advance; math is taken to have an a priori existence.<br />
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My identity work instead focuses on personal epistemologies and author-ity. I'm beginning a paper on this, to push out during the summer. I'm very curious to see where it will go.blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com4tag:blogger.com,1999:blog-4189326141173312850.post-1016104290531190492013-05-21T21:11:00.001-04:002013-05-21T21:40:14.395-04:00Planning PD for HS Math teachers -- Focus on Pedagogical Practices in PrBL<span style="font-family: inherit;">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 <a href="http://www.vanderbilt.edu/lsi/expert/documents/effectsofmathteaching.pdf" target="_blank">H&G's chapter</a>, as well as the <a href="http://www.nctm.org/uploadedFiles/Research_News_and_Advocacy/Research/Clips_and_Briefs/research%20brief%2019%20-%20benefit%20of%20discussion.pdf" target="_blank">NCTM Research Brief</a>.</span><br />
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<span style="font-family: inherit;">We plan an online conversation about the ideas in H&G's paper, especially the two principles of learning: (1) </span>being explicit about the key ideas, and (2) <span style="font-family: inherit;">supporting productive grappling are central to our thinking about the pedagogical work for teaching in a problem-based learning (PrBL) environment.</span><br />
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<span style="font-family: inherit;">Here I just write as a read, trying to capture some of my thoughts specifically about how to USE these two principles in a teacher professional learning community that is focused on the<b> belief structures and </b></span><b>pedagogical </b><b style="font-family: inherit;">practices</b><span style="font-family: inherit;"> necessary for </span><span style="font-family: inherit;">teaching</span><span style="font-family: inherit;"> more mathematics to more students. My main effort is to think about how our workshop participants (HS math teachers, instructional support providers, and administrators) will experience, interpret, think about... the ideas presented, and more importantly the formats in which they may be presented, read, experienced,...</span><br />
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<span style="font-family: inherit;">Onwards.</span><br />
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<span style="color: blue; font-family: inherit;">“When I use a word,” Humpty Dumpty said, in a rather scornful tone, “it means just what I choose it to mean—neither more nor less.”</span><br />
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<span style="color: blue; font-family: inherit;">“The question is,” said Alice, “whether you can make words mean so many different things.”</span></div>
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<span style="color: blue; font-family: inherit;">“The question is,” said Humpty Dumpty, “which is to be master—that’s all.”</span></div>
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<span style="font-family: inherit;">—Lewis Carroll, <i>Through the Looking Glass </i></span></div>
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<i><span style="color: purple;"><span style="font-family: inherit;">Alice wonders if words can mean more than one thing. Of course says Humpty, the meaning of a word is held by the user. The meaning does not get transmitted to the listener. The listener </span>constructs<span style="font-family: inherit;"> meaning for the word, and a string of words, by attending to the context and--most importantly--from her own experientially-reliant ways knowing.</span></span></i></div>
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<span style="font-family: inherit;">1. This NCTM version (written by Judith Reed) of Hiebert & Grouws' work presents itself as highly problematic for me. I believe it reads to math teachers as, teach with any style you want, do it well, and kids will learn. What Judith Reed (UGA friend and colleague) did not successfully emphasize is stated (in polite, and reflective of allowable conclusions among "scientists," is that:<br /><br />"In the other system, instruction is more slowly paced, teachers ask questions that require longer responses, and students complete relatively few problems in each lesson. At least under some conditions, this appears to yield skill efficiency coupled with conceptual understanding." (p. 2)</span></div>
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<span style="font-family: inherit;">Judith also wrote, "The features of teaching <span style="color: #cc0000;">that facilitate skill efficiency and conceptual understanding</span> do not fall neatly into categories frequently used to contrast methods of teaching, such as..." (p. 2).</span></div>
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<span style="font-family: inherit;">H&G wrote, "the features of teaching <span style="color: #cc0000;">we describe</span> do not fit easily into any of the categories frequently used to describe teaching..." They continue, "Although these categories and labels have been useful for some purposes because they capture constellations of features and treat teaching as systems of interacting components, they also can be misleading because they group together features in ill-defined ways and connote different kinds of teaching to different people.... Most of these categories, distinctions, and labels <span style="color: #cc0000;">are now more confusing than helpful</span>" (p. 380).</span></div>
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<span style="font-family: inherit;">Judith allowed the categories to remain on the table in this NCTM brief.<br /><br />2. Hiebert and Grouws write, "Different Teaching Methods Might Be Effective for Different Learning Goals" (p. 374). Precisely why school-based conversation about what type of learning goals the school would like to achieve, within maths classrooms, etc., should be settled AND decisions made about what evidence must be collected to evaluate success along these goals BEFORE determining best "ways to teach."</span></div>
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<span style="font-family: inherit;">3. H&G "present several patterns that connect mathematics teaching and learning. We begin by identifying one of the most firmly established but most general connections be- tween teaching and learning. Commonly referred to as “opportunity to learn,” this claim says that students learn best what they have the most opportunity to learn" (p. 378). Uri Treisman too spoke of this opportunity to learn in the Iris Carl Equity Address <http://vimeo.com/65731353>.</span></div>
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<span style="font-family: inherit;">4. Feature 1 of "Keys to Promote Conceptual Understanding" (p. 383): "A clear pattern across a range of empirical studies is that students can acquire conceptual understandings of mathematics <i>if</i> teaching <span style="color: #cc0000;">attends explicitly to concepts</span>—to connections among mathematical facts, procedures, and ideas. By <i>attending to concepts</i> we mean <span style="color: #cc0000;">treating mathematical connections in an explicit and public way</span>" (p. 383).</span></div>
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<span style="font-family: inherit;">A couple thoughts:</span></div>
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<span style="font-family: inherit;">• Attempting to understand their intended meaning of the word e<i>xplicit</i> is interesting to me. While the word is most commonly understood to mean something along the lines a "clearly explained/defined," I wonder if that is their intent here; why would they add the adjective "public" if they meant for explicit to mean "said out loud in a clear manner." I believe their meaning is to be something closer to "intentional." The continuation of H&G's discussion at top of 2nd column on p. 383 bears this out, "This could include...."</span></div>
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<span style="font-family: inherit;">• When explicit is meant to mean clearly explained, I think we sit once again squarely in the Direct Instruction camp, upon which student ideas/thinking are not foregrounded, and the teaching goal is to get all kids in the room to think like the teacher, rather than discuss/debate the ideas of the students. In this setting, the teacher works to make connections among student ideas, and maybe even to the discipline of Mathematics. Such connections I believe could be described with the adjective "mathematical," as it was used by H&G. My concern is that a majority of the teaching community interprets "mathematics" and "mathematical" as something, some body of knowledge, or some way of thinking that exists apriori to human thought, rather than mathematics being something of a human endeavor. </span></div>
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<span style="font-family: inherit;">• When Judith uses the adjective "important" when she writes "Making important mathematical relationships explicit" (p. 1), she further cements this a priori notion of these "Mathematical Relationships" that exist prior to the knower (the student).</span></div>
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<span style="font-family: inherit;">• I am concerned to by H&G's use of important, "a second feature of teaching that consistently facilitates students’ conceptual understanding: the engagement of stu- dents in struggling or wrestling with important mathematical ideas" (p. 387). Do the students get to deem what is important? Must the teacher deem what is important? Or is it the discipline, math educators, CCSS-M, the textbook? <i>Where does <a href="http://www.doingmathematics.com/2/category/authority/1.html" target="_blank">authority</a> lie?</i></span></div>
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<span style="font-family: inherit;">5. Judith incorrectly cites Boaler's doctoral work (and Fawcett): "Making important mathematical relationships <span style="color: #cc0000;">explicit</span> has been shown to support students’ understanding of the relationships in... secondary school geometry and algebra (Boaler 1998; Fawcett 1938)" (p. 1). In neither study was any effort to make mathematical relationships explicit. In fact, what H&G worked to point out from these two studies was that, "the findings from these studies typify those found in much of the literature that assesses the effects of instruction that <span style="color: #cc0000;">explicitly</span> attends to conceptual development of mathematics. Students receiving</span></div>
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<span style="font-family: inherit;">such instruction develop conceptual understandings, and do so to a greater extent than students receiving instruction with less conceptual content" (p. 387). Yes the same word is used, but to mean quite different things. I believe H&G's use here again shows they mean for it to be synonymous with "intentional."</span></div>
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<span style="font-family: inherit;"><br /></span></div>
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<span style="font-family: inherit;">6. I noticed that Jamila used the word "grapple," one not used by H&G. They use struggle, "to mean that students expend effort to make sense of mathematics, to fig- ure something out that is not immediately apparent" (p. 387). Here is where I get left behind by H&G: "The struggle we have in mind comes from solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed" (p. 387). [yes, they use grapple--the one time in the paper). Again, an adjective (key) on "mathematical ideas." Why do I belabor this point? Hiebert & Grouws rely on a learning theory, an idea of constructivism, that foregrounds a particular way of thinking and knowing that is to be known--in this case, mathematics ("important mathematics" more precisely). This mathematics-to-be-learned is defined in advance of the learner. This reflects the constructivism of Vygotsky (employed by & doing his research at the behest of the Russian Communist government--where education was for indoctrination), and the social learning theory of Lave & Wenger--that uses a metaphor for <i>cognitive learning</i> of a discipline to be like the process under which an apprentice begins to <i>behave</i> more like the master. </span></div>
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<span style="font-family: inherit;"><br /></span></div>
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<span style="font-family: inherit;">I am much more comfortable thinking about learning with Piaget in mind, who seeks simply to define what he sees in the child as mathematical, rather than look for mathematics in the child.</span></div>
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<span style="font-family: inherit;">In fact, it is quite interesting how H&G's 2007 publication fully ignores the constructivist "revolution" of the 1990s in math education. I suspected placing children's thinking at the foreground threatened the status of the discipline...</span></div>
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<span style="font-family: inherit;"><br /></span></div>
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<span style="font-family: inherit;">7. H&G conclude, based on their survey of many empirical studies, "</span>Apparently, it is not the case that only one set of teaching features facilitates skill learning and another set facilitates conceptual learning" (p. 390). But the <i>nature</i> of skill learning may be somewhat different. I don't think this come's out well in the NCTM paper. H&G seem to advocate for the attention to the foregrounding of conceptual development, because the nature of the skill development that emerges--a procedural fluency--seemingly is valued in the first two paragraphs of p. 391 (to the extent two people who value their identities as men of science are able to express bias/preference).</div>
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8. My conclusions: I think the two proposed pedagogical goals, (1) being intentional about foregrounding particular student ideas in classroom conversations, and (2) creating cognitive disequilibrium in students and then supporting the necessary struggle to allow students to find resolution, can be fantastic organizing features for HS math teacher's professional learning experiences. </div>
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I would suggest that we determine a way for the Workshop Participants (WPs) to build some personal, and taken-as-shared meaning for each of these ideas. The Instructor(s) will have to devise methods to create some disequilibrium about the ideas, create structurs for WPs to muck around in making meaning, and then be intentional about foregrounding certain ideas.</div>
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Next, conversations should emerge about how to put these two prinicples in action in the classroom. The mathematical content of the various workshops can allow for a context to think it. What pedagogical practices support? which work against?</div>
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As the week progresses, maybe a collective description of these practices could emerge. With weach (many) followed by specific, actionable practices a teacher can do.</div>
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Maybe the conclusion of the week could be school-based teams developing an action plan to Implement (Lesson Design) and Evaluate (collect data to assess that the implementation occurred--not to evaluate impact on student learning, we've already established that) the results of the Action Plan. </div>
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That final idea connects to Hiebert et al.'s (2012) recent paper in JTE, <i><a href="http://jte.sagepub.com/content/63/2/92.full.pdf" target="_blank">Teaching, Rather Than Teachers, As a Path Toward Improving Classroom Instruction</a>.</i></div>
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<i><span style="color: purple;">Coda: Humpty says much more. His final comment is about how one perceives their own ways of knowing, what role for authority when considering knowledge. Humpty speaks to an idea I am deeply curious about--personal epistemologies.</span></i></div>
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blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com4tag:blogger.com,1999:blog-4189326141173312850.post-7421398185316054492013-02-08T21:15:00.001-05:002013-05-21T21:15:25.001-04:00Mathematics is a SimulacraI write that <a href="http://www.esri.mmu.ac.uk/mect/papers_11/lawler.pdf" target="_blank">Mathematics is a Simulacra</a>, 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.<br />
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Andreas Quale wrote a nice <a href="http://www.univie.ac.at/constructivism/journal/7/2/104.quale" target="_blank">paper</a> that allowed me a fun curiosity for the evening.<br />
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Maybe it is productive and/or interesting to view this discipline of Mathematics as a human-invented <i>game</i>, 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.<br />
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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.blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com0tag:blogger.com,1999:blog-4189326141173312850.post-43867731676862749452013-02-07T11:26:00.001-05:002013-05-21T21:15:56.805-04:00what do I think of the CCSS?I bumped into a longtime colleague and fellow survivor of the <a href="http://www.math.cornell.edu/~henderson/courses/EdMath-F04/MathWars.pdf" target="_blank">CA Math Wars</a> at the <a href="http://creatingbalanceconference.org/?page_id=1305" target="_blank">Creating Balance conference</a> a couple weeks ago. She sent me this article and asked my opinion of the CCSS. I sat and wrote back, allowing some anger to emerge. Maybe you'll notice...<br />
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<a href="http://www.edweek.org/ew/articles/2013/02/06/20commoncore_ep.h32.html?tkn=XWZFeU8Olfm083V0xBy70dDbpDLjBvdWdUAJ&cmp=clp-edweek">http://www.edweek.org/ew/articles/2013/02/06/20commoncore_ep.h32.html?tkn=XWZFeU8Olfm083V0xBy70dDbpDLjBvdWdUAJ&cmp=clp-edweek</a><br />
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Thanks for sharing this. I have strong, but unreconciled opinions about the CCSS (and CCSS-M). Some unedited thoughts, maybe sequential, maybe nonsensical. [maybe I needed to attend school when there were standards for writing]<br />
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This sums up my general feeling about them, "Through the common core, public schools will be used to foster "economic fascism" in education, charged former U.S. Rep. Bob Schaffer, a Republican from Colorado." [dare I agree with a republican?]<br />
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The ideal of standards as they first emerged in math ed during the 90's reflected my values of what they might offer--a vision, an opportunity for professional discussion / learning, an opportunity for serious and focused debate over what to teach (e.g. NCTM's 1989 standards for curriculum & evaluation), how to teach (e.g. NCTM's 1991 professional standards), and possibly more.<br />
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But they turned away from professional guides to a hammer of accountability, at about the time of the 2000 NCTM Standards. By then, NCTM was viewing itself as a corporation that needed to give its customers what they wanted, rather than serve as a vision for the profession. They sidled up with corporations (the first I noticed, and called him on it, was Lee's connection with Duke Energy. Now, the drive for a "national curriculum" seems to only meet the needs of corporations (textbook publishers, test publishers, we'll come fix your school yokels, we'll run our own schools, etc.) who will be able to create a greater profit margin. This corporate agenda, rather than one about the health, happiness, and welfare of children, as well as our citizens. The hammer of accountability works efficiently to cleanly carve up a radically classist society.<br />
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What is more insipid is the propaganda behind efforts for standards, learning math, etc. We need math and science for our economy. That said right after saying our schools are failing in teaching math--while the UCA (United Corporations of America) is by far the wealthiest and most powerful nation in the world. The propaganda machine also sends the message, Math is Power (recall this NCTM campaign?). Well, logic might not say that implies no math means no power, but that is how humans, worse--children, interpret that message. A message that is if not directly stated, the undercurrent in virtually every school in this amerika. We have our success-stories in mathematics, often times a girl or a black boy, held up by white oligarchs' EXCEPTIONAL underclass child-the model to follow if blacks want to achieve this Amerikan Dream.<br />
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I guess I am losing focus on the CCSS. How might they be good (maybe)? By providing opportunity for professional discussion. At this time, I see teachers either being pressed to wonder, how will I teach kids "mathematical practices" - or thrilled to feel relieved of that crap they've felt they had to drive into the heads of kids for a decade, none of which was meaningful or fun to teach or learn. To quote a pair of Vista school district teachers who survived the Math Wars here in north county San Diego, "you mean, we are allowed to use these materials (IMP) again?" -- said to their district math specialist.<br />
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But, I hate the accountability and associated punishment and threat, as well as the carrot-dangling (the way money plays in this is sick). <br />
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I hate the standardization of thinking. The discipline of math does itself a disservice by trying to create a singular way of knowing, a narrow view of mathematics and of mathematical knowledge. It closes itself off to opportunity for knowledge creation. Further, it shuts out potentially more mathematicians by FAR than it invites in. (I suppose this serves mathematics and mathematicians well, however. The pinnacle of knowledge-power in amerikan society, if not $, its perceived intelligence, where mathematicians reside on the highest of thrones.)<br />
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I hate the soul-crushing message they send to children who can't/don't. <br />
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It is practically comical these are written seemingly without any awareness for who amerika is. Seriously, remove the schools from the nations wealthiest 25% counties/districts and re-consider what the goals of education ought to be. Take out the corporations need for workers. Take out the classist societies' need for a mechanism to delineate haves and have nots. Take out this seemingly human need [i don't believe this] to be better than another. Take out the governments' need for an emasculated citizen. Look at the child. <br />
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Every child is mathematical. No child NEEDS to learn mathematics--the discipline. Why are we as teachers so confused about what we should do?<br />
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Because we are each a success of a system that told us not to listen to our heart, not to listen to our body. <br />
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No grand conclusion for you. Just and end to my ramblings. <br />
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-brianblaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com2tag:blogger.com,1999:blog-4189326141173312850.post-66385600336654857052012-11-09T10:00:00.000-05:002013-05-21T21:34:26.642-04:00Math & Jazz: Learn by Doing or by Training?<span style="font-family: inherit;">Healthcare professional's first </span>pronouncement when beginning study in the field is to do no harm. Could we teach mathematics if we asked teaching students the same question?<br />
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<i>more to follow...</i><br />
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[Returning 5/21/13] Well, I began with some thoughts about math & jazz. probably decided to mentally improvise on this riff, and not try to scribble down the notes.<br />
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But as I came across this initiating strong today, I am reminded of some thoughts on modeling teacher education on healthcare education. The argument was that healthcare (and other professions) thrive because of the standardization of known "good" <=> "scientific" techniques allows for a well-agreed upon healthcare education structure and methodology.<br />
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Can this be compared to what it means to teach math well. Maybe yes, if "teaching" is thought of in a transmission model, and the outcomes for teaching are a particular standard way of knowing and thinking of Mathematics to be held by children.<br />
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Aaaargh! No, please no. I leave for another improvisation. Maybe I'll listen to Public Enemy's <a href="http://www.youtube.com/watch?v=8PaoLy7PHwk" target="_blank"><i>Fight the Power</i></a> as I drive home, sit down to read Priya Parmar's <a href="http://www.amazon.com/Knowledge-Reigns-Supreme-Transgressions-Education/dp/907787450X" style="font-style: italic;" target="_blank">Knowledge Reigns Supreme: The Critical Pedagogy of Hip-Hop Artist KRS-ONE</a>, or ...blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com3tag:blogger.com,1999:blog-4189326141173312850.post-20179472423781645872012-08-15T14:02:00.000-04:002013-05-21T21:35:11.973-04:00A Constructivist's View on Teaching Mathematics (for Social Justice)<br />
<i>This post is a long reply to Bryan Meyer's April 22, 2012 post at <a href="http://www.doingmathematics.com/2/post/2012/04/constructivism-vs-discovery-help-me-sort-this-out.html" target="_blank">Doing Math</a>. </i><br />
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Bryan, you won't be surprised in what I say. I think you've made a fundamental error of seeking to compare two very different issues, a theory for knowing (constructivism) with a theory for teaching (discovery learning). If you address constructivism as a method for teaching, there is no core base for which to make decisions on how to act. What seems to exist in the literature are no more than others' grievous misunderstandings / bastardization of constructivism.<br />
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If you treat discovery learning as a theory for knowing/learning, I would say you are much more strongly aligned to a segment of learning theorists who may not fully embrace the "radical" component of constructivism, that we have no access to reality and thus could not "know" in a manner that had been previously taken for granted. These sorts of more modern theory for knowing, that seem to be taken for granted in this "discovery learning" idea, do embrace a similar principal--that the learning mind is an active one--and thus are often labeled constructivist in some way. However, these constructivisms are "trivial constructivism."<br />
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Moving beyond the initial problem of your comparison--the titles of the two columns--I find your questions intriguing.<br />
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1. The point you attribute to Social Constructivists (SC) is what they believe to be what distinguishes them from (Radical) (Piagetian) Constructivists (RC). They are in error. In fact, Piaget gives great emphasis to the role of social interaction. SC's actual distinction from the RC is the manner in which they fail to deal with this "reality", what the RCist names "intersubjectivity" -- a taken as shared way of knowing. The response to this question, to me as a RCist, is that this is fundamentally not a question I would ask. As someone interested in SJ, I would argue that as soon as you ask the question, you have created an unjust space between people--there is no way that one could be deemed not correct, nor more correct. Each is assumed to be "correct," i.e. viable, for that other autonomous being. That to me is the healthy way of interacting with the other.<br />
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To pursue this question, briefly, in greater detail--I wish to deconstruct the portion of the question, "when students agree upon an ontological reality." I read this to mean that the students recognize themselves as several autonomous agents (cf. Maturana & Varela) have knowingly come to an intersubjective agreement about what they are intentionally thinking of as an "ontological reality." I take the students to be RCists in terms of their personal epistemologies; I assume they have agreed to intentionally give an "existence," albeit it unknowable or otherwise, to some mathematical construct. (I state all of this because it is how we most often operate in mathematics--we assume some entity to exist, and then operate on that notion as if it were an object.)<br />
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With this clarification, I would suggest that this opening condition you have created, when students agree on a reality and intentionally determine it to be different from that of the teacher or the mathematics community, I would suggest that is the pinnacle of educational success: when the child refuses the oppression of other's ways of knowing.<br />
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2. This question to me is fundamentally Dewey's in <a href="http://www.gutenberg.org/ebooks/29259" target="_blank">The Child and the Curriculum</a>. In considering your question, first I encourage both you and myself to recognize your role as an observer in considering the various portions of the inquiry, such as "more consistent with that of the mathematics community" -- that is a statement made by you, the observer. It says you see what you think to be the knowing you as teacher press towards to be that of the knowing you attribute to the mathematical community. Ultimately, what your question says, as a RCist, is that you are pressing the students to know as you to.<br />
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I simplify the dilemma posed here for myself in such a way that the dilemma simply no longer exists. Basically, there is no other way of interacting with the other. We always "draw" them toward our ways of knowing. It is our mind's way of seeking to maintain equilibrium.<br />
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Getting back to my basic read of your Question #2: Does the Constructivist learning theory imply a "Discovery Learning" pedagogical theory? I say no; it is dependant on the teacher's personal epistemology. If the teacher is a Constructivist, they would recognize the decision they make about accepting or rejecting the ways of knowing of others. A Constructivist who acted in socially just ways would value other's ways of knowing, and not seek to use to their advantage power relations to preserve the self-proclaimed superiority of their knowing.<br />
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Dewey concludes, "The case is of child." "The formulated wealth of knowledge that makes up the course of study...says to the teacher: Such and such are the capacities, the fulfilments, in truth and beauty and behavior, open to these children. Now see to it that day by day the conditions are such that their own activities move inevitably in this direction, toward such culmination of themselves. Let the child's nature fulfil its own destiny, revealed to you in whatever of science and art and industry the world now holds as its own."<br />
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3. I think my responses to 1. and 2. provide my answer to question 3. It is "agreed!" provided the teacher sees those content standards as anything other than what Dewey recommends. This allowance for "measuring against a greater authority" is some perverse myth, if not a blatant bigoted oppression. It is my believe that there are MANY powered structures, including those of racial status, economic status, and intellectual status, that all benefit from mathematics being the pinnacle of the truth regime.<br />
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What is the purpose of mathematics education, and how can it be socially just?<br />
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<ul>
<li>What <i>is</i> the purpose as math education is presently conceived? It is to replicate the racist society. </li>
<li>How <i>can</i> it be socially just? Adults can respect children's ways of knowing. A teacher can conclude their days by observing the child's ways of knowing, and deem them to be mathematics. In fact, it is certain that they are mathematical; and there is no way of knowing otherwise.</li>
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I do think it is quite neat to have come to a very similar questioning by Luis Radford in his paper "<a href="http://www.springerlink.com/content/t0r7648748616748/" target="_blank">Education and the Illusions of Emancipation</a>." He seems to have limited himself, however, to a world of math education as it seemingly operates now; rather than what I have offered above.blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com8tag:blogger.com,1999:blog-4189326141173312850.post-16745964557259644722012-07-17T14:46:00.000-04:002013-05-21T21:35:28.947-04:00My sister's can of worms<br />
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<span class="Apple-style-span" style="color: #333333; line-height: 18px;"><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">My sis, a Montessori teacher, asked Why do we need to invert the second fraction when we are dividing? I told her to go to the Kahn Academy!</span></span></div>
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<span class="Apple-style-span" style="color: #333333; line-height: 18px;"><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">Actually, here is her email. This has become an interesting curiosity for me. What can I "teach" "online" ? Why not jump in with the hairiest possible question, fraction division.</span></span></div>
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<span class="Apple-style-span" style="color: #0b5394; font-family: Arial, Helvetica, sans-serif;">On Jul 16, 2012, at 4:08 PM, Mary Halase wrote:</span></div>
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<span class="038395822-16072012"><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif;">So I have this question that I believe you could answer. Why do we need to invert the second fraction when we are dividing? so if I have 1/4 and you divide that by 1/3, why do we need to do the invert 1/3? I had an 8 year student ask me this question and for the life of me I could not answer that question, I just told him that is a rule that I have learned when I was young. He is the first student that has asked me that. I googled it and got some answers but the answers that I got seemed more confusing. So I would like you to help me with that question so that I understand it and that way I can go back and explain it to this young student. I am actually getting the answer to this question thru my montessori training. We are going into division with a fraction. Thanks for you help.</span></span></div>
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<span class="038395822-16072012"><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif;">Love,</span></span></div>
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<span class="038395822-16072012"><span style="color: #0b5394; font-family: Arial, Helvetica, sans-serif;">Mary</span></span></div>
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<span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">So, I made an attempt to respond to my sis (Mary). Oh, that is her real name. Please don't mis-treat her! She is a fabulous teacher--imagine, she actually asked the question "why is that?"!<br /><br />This is really meant as an experiment to me, to begin to consider Sal Kahn's challenge--to communicate/develop understanding(?) of mathematics "online." I totally know it isn't a greatly designed experiment, nor do I have any idea where it will go. I am just (quickly) giving something a try, with my patient sister.<br /><br />Here is what I wrote back:</span></div>
<span class="Apple-style-span" style="color: #6aa84f;"><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">My answer is that you have to make sense of this one for yourself. It is complex because a lot of operations (relationships) must be understood. Give this a try... (let me know!)</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">1. First, you must deeply understand division... both sharing and grouping ways of thinking. Do you? Play with that until each is second nature. Start with easy numbers, then use numbers that yield a fractional answer. Understnd what the fractional answer means.</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">2. Wait, why don't you go watch the Kahn Academy video <</span><a href="http://bit.ly/O5CWsa" style="font-family: Arial, Helvetica, sans-serif;">http://bit.ly/O5CWsa</a><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">>?</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">3. OK - totally kidding. Might be a joke you don't get, but in the land of math teachers and math ed researchers, this stuff is the bane of our existence. Back to really trying to help...</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">4.a. After really owning division, now its time to really know fractions. Here are a few ideas, then problems. Begin by drawing some line segment. Call that one unit of length. Now split it into three equal parts. Notice each is the exact same size, and that when all three are put together, they create the "whole"--the one unit of length. For naming purposes, call each of those pieces "one-third". (You know the fractional symbol.) Etc. for all other "unit fractions" (a fraction with a "1" in the numerator)</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">4.b. Take "one-third." Match that with one more one-third, so you have 2 one-thirds. OK? [with kids, a teacher should repeat this kind of naming over and over, i.e. 5/7 is five one-sevenths]. Imagine 3 one-thirds. The same as one whole, right? 5 one-thirds? OK?</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">4.c. Note that might be written 1/3 + 1/3 + 1/3 + 1/3 + 1/3 , which is equal to 5/3. What is 2/3 + 1/3 =? 2/5 + 4/5 = ?</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">4.d. Next equivalent fractions. Split each of your 1/3 (from 4.a.) into four pieces. How big is each piece? It is one-fourth of the one-third, yes--but what portion of the original whole is it? That is, how many do you need to make the whole? Your reasoning should go, I need 4 of these to make one-third. And 3 one-thirds make one whole, so I need 4 x 3 = 12 of these new pieces. [Aside: when a kid can reason in this way, they know multiplication. Until then, multiplication is simply a modified addition for children. This is a profound way of knowing multiplication, some children do not develop this as a result of school- (or life-) based activities until early teens, or later. It seems that children around 10-12 typically can.] So the new piece is called a one-twelfth. Given this, children can create and identify equivalent fractions. When teaching, they should be allowed to be extremely creative, such as arguing that "three and one-half one-sevenths is the same as one one-half." OK?</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">4.e. Addition next (which is the same as subtraction). How much is 1/2 + 1/3? Figure this out, and why--with no algorithm. Keep in mind that you should be able to justify any renaming of a fraction through the rules of equivalence developed in 4.d. Notice the answer is five one-sixths. 5/7 + 3/5 = ? Answer is 46 one-thirty fifths. Children do this stuff, with lots of time andp lay, they get quite fluent at it... Subtraction is just as easy/hard (because subtraction is the same operation as addition, for the child--maybe not for the mathematician.)</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">4.f. Back to multiplication, which is easier than addition actually. What does 3 one-sevenths look like? Write this either as 3 x 1/7 or 3/7. So it is true that 3 X 1/7 = 3/7. (again, equivalence being emphasized). </span><br /><br /><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;"> _______ _______ _______ </span>(I begin with 3, that is 3 one-wholes.)<br /><span class="Apple-style-span" style="font-size: x-small;">------- ------- ------- </span>(this part is two simply create/show the</span></span><br />
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> length </span></span><span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;">one-</span><span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;">seventh)</span><br />
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;"> ↓ ↓ ↓</span><br /><span class="Apple-style-span" style="font-size: x-small;"> - - - </span>(the problem 3 x 1/7 states, for each of the </span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;"> original </span><span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;">unit, I take </span><span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;">1 one-seventh of that</span></div>
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<span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;"> unit)</span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">After many whole number times fraction, where the first number means "for every unit in that first number, I must have the named number of units of the second number." Better said with the example. "For each of the three 'ones' I must have one 'one-seventh'." So 5 x 2/3 is "For each of 5 ones, I must have 2 one-thirds." So that is 10 total one-thirds, i.e. 5 x 2/3 = 10/3. </span><br /><br /><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;"> ___ ___ ___ ___ ___</span> (this sketch shows the 5 ones)<br /><span class="Apple-style-span" style="font-size: x-small;"> --- --- --- --- ---</span> (I redraw each of the ones with something</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> equivalent, 3 one-thirds for each one the ones</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> seems helpful here)<br /><span class="Apple-style-span" style="font-size: x-small;"> ↓ ↓ ↓ ↓ ↓</span></span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;">-- -- -- -- -- (for each of the 5 ones, I take 2 one-thirds,</span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> collecting 10 one-thirds in total)</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;"><br />What then, is 1/7 x 3? "For each of the one 'one-seventh' I must have three 'ones' (i.e. one wholes)</span><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace; font-size: x-small;">_______ </span><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">(this sketch for initial reference, my initial whole</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;"> </span>(unit), to build my 7 one-sevenths)</span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;"> ------- </span>(this is 7 one-sevenths)</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;">- </span>(this is my starting one-seventh. Next, for each of these</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> units, I take 3 (whole) of these units.)</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;"> ↓</span></span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;">- - - </span>(so, compared to the original unit, I have 3 one-sevenths.</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">1/3 x 1/5 = ?</span><br /><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><br /><span class="Apple-style-span" style="font-size: x-small;"> _____ _____ _____ </span> (this sketch for initial reference, to build my 3</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> one-</span></span><span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;">thirds)</span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;"> ----- ----- ----- </span>(this is 3 one-thirds</span></span><span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;">)</span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;">----- </span> (this is my starting one-third)</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;">Next, for each of these units, I take 1 one-fifth of these units. That is for the one one-third, I need one-fifth one-thirds. So split this into 5 parts.</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;">- - - - - </span></span><span class="Apple-style-span" style="color: #6aa84f; font-family: 'Courier New', Courier, monospace;"> (pretend that is the same length as the one-third</span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> above. I will take one of these one-fifth of one-</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> third.)</span><br /><span class="Apple-style-span" style="font-size: x-small;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> ↓</span></span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"><span class="Apple-style-span" style="font-size: x-small;">- </span> (compared to the original unit, what portion is</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> it? Well, I know I need 5 to make one one-third.</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> And I need 3 one thirds to make one unit, so I</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> need 15 of these pieces to make one whole; it is</span></span></div>
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<span class="Apple-style-span" style="color: #6aa84f;"><span class="Apple-style-span" style="font-family: 'Courier New', Courier, monospace;"> one-fifteenth)</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">Work that out for:</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">1/2 * 1/5</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">3/5 * 1/2</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">2/3 * 4/7</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">5/4 * 1/3</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">11/7 * 5/3</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">5. Write a division problem. Re-read it as a grouping or sharing task. Solve it--with the sketches. This will allow you to argue WHY for your answer when you divide to fractional numbers. A key: remember, fractional numbers are just another number... (instead of 3 one-wholes, it may be 2 one-fifths)</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">Some examples that may help:</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">9 ÷ 3</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">12 ÷ 7</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">3 ÷ 5</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">1/2 ÷ 3</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">2/5 ÷ 8</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">1 ÷ 1/3</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">6 ÷ 3/5</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">5 ÷ 2/3</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">1/4 ÷ 1/3</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">3/4 ÷ 2/3</span><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">3/4 ÷ 5/3</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">6. After lots of playing, practicing, doing... kids may be ready to analyze what happens for any a/b ÷ c/d and generalize a rule like what you know. </span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">7. Next problem for you is to write a contextualized (i.e. "real") problem for 3/4 ÷ 2/3.</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">I am sure you'll write me with a question, at some point...</span><br /><br /><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">-brian</span></span></div>
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blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com7tag:blogger.com,1999:blog-4189326141173312850.post-27294507841830340882012-05-27T12:20:00.000-04:002013-05-21T21:36:17.309-04:00Cogs in The Testing MachineI find the conversation below to get at the heart of a problem I face in my "work." My work, I suppose, is often under the guise of serving math teachers, schools and districts, to raise test scores. I accept this rue so that I can do what I define to be my work, work for a social justice. My goal is that children, all children, are schooled in ways that maintain their authority for knowing, acting, and being in this world.<br />
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The conversation was sparked by an @AlfieKohn twitter post, which I past to a group of colleagues working HARD to improve the math teaching and learning experiences for students and teachers at a large number of high schools. Names have been changed to protect the innocent (and the damned)--that is, except my own.<br />
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Alfie Kohn (@alfiekohn)<br />
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Video of kid being talked at about her scores (“data-based goal setting”): ow.ly/b7UKi. How *not* to do a tchr/stud conference<br />
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Hi all,<br />
I share this video because the data factory that education seems to being asked to act as seems to go unquestioned in the schools I visit. Mostly, it seems to make it very difficult to think about how to do work with admin. who see no problem with the activity in this sort of interaction. Maybe said slightly differently, I find myself challenged to work with admin. beliefs acknowledging that they may see no trouble with this interaction.<br />
-Brian<br />
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Brian- This video is ironic on so many levels. Watching it I am so appalled and yet I know for sure that in my district we pulled in all our students and did the exact same thing and were lauded for doing so! Watching the video makes me realize how meaningless and impersonal those conferences are. It seems ludicrous to imagine that this student will leave that conference with any concrete ideas for learning more math or with any sense that she is highly valued by the adults around her. I'm also wondering why, in this day and age, they would produce a video that is so blatantly non-diverse.<br />
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I think there are many fine administrators with idealistic goals for students and schools. They are being herded down a cattle chute in which the rhetoric about " what works" has centered on data. A few well -promoted successes in using data for school- wide reform resulted in large - scale, half-hearted, blindly executed imperatives that administrators have convinced themselves will help their teachers and their students. I think it only appears that they have lost sight of the real data source: students' thinking based on real work around learning math in class. I think we need to take back the agenda for school improvement and encourage the administrators who are looking for a way to re-gain a student- based approach to step up. We ned to promote and provide alternatives to balance this test score mania. Indeed early efforts in many districts centered on studying students' learning and understanding their thinking (for example re: learning). These innovative ways to look at students may have been ahead of their time but could be re- introduced to people looking for answers. Just thinking...<br />
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Happy Memorial Day. Heidi<br />
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Heidi, thanks for your thoughts. It is ironic (that may not capture my reactions totally, but at least in part) in many ways--diversity, total lack of relationship between teacher & student, I don't think the student voiced one thought of her own rather was given words and ideas and opinions by the adult, and more. It is also disturbing to me the way the student responds the whole time, as if this is OK, normal.<br />
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Yes, fine admin with idealistic goals. Parallel to most every teacher's classroom I experience with idealistic goals--yet their actions don't seem to match their beliefs. A challenge to me, someone who hopes to provoke possible realization of this incongruency, is to figure out how to create the space for the teacher or administrator to see differently their actions.<br />
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To paraphrase Neil Postman, children come with excellent crap detectors. How do I tap in to those still functioning mechanisms in adults, especially when the crap to be detected is tied up in their own identity?<br />
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-brian<br />
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<br />blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com1tag:blogger.com,1999:blog-4189326141173312850.post-43806795609894921182012-03-24T00:50:00.002-04:002012-03-24T00:59:57.375-04:00Knowledge as FabricationThe following is an excerpt from a soon-to-be-published book chapter I wrote a bit ago. It happened across my email just after I made some Twitter comments about constructivism and the fascism of teaching styles that are predicated on Discovery Learning. Many people equate the two--constructivism & DL--mistakenly.<br />
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The mistake begins with the idea that constructivism, a theory for knowing and learning, has anything to say about teaching. Or that any teaching theory could be based on constructivist principles. The problem is that no matter how or what is taught, learning occurs. Constructivism helps to account for what is learned, and how it is known. It says nothing about the effectiveness or quality of the teaching.</div>
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I am willing to say that if a teacher embraces the ontologically agnostic (i.e. we cannot know whether objective truths exist) status of knowledge that constructivism posits, that her teaching may be quite radically different. Maybe this passage hints something at that, but I have more thoughts on teaching--I may post on those later. I do think Dewey got it right in the <a href="http://ia700506.us.archive.org/1/items/childandcurricul00deweuoft/childandcurricul00deweuoft.pdf" target="_blank">Child and the Curriculum</a> (when read of course, as an ontological agnostic). </div>
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See more at <a href="http://www.oikos.org/constructivism.htm" target="_blank">Why Some Like it Radical</a>. </div>
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<b>Knowledge as Fabrication</b> (selection from <i>The Fabrication of Knowledge in Mathematics Education: A Postmodern Ethic toward Social Justice</i> -- in press)</div>
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This postmodern, post-epistemological, post-knowledge framework for a mathematics education draws upon the demand for attention to access, achievement, authority, and also action. The constructivist perspective redefines what access and achievement might be by posing the question: relative to whose mathematics? It repositions authority and authorship to take centre stage; and it binds in the responsibility for social action as inherent in each of these first three cornerstones. Acting upon our world is how we come to know it. Further, we need healthy cohabitants in this experiential world to feed back into our knowing of the world, which is populated by knowers other than ourselves. <br />
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Mathematical learners, like all learners, fabricate knowledge (where ‘fabrication’ is taken to mean ‘build’, ‘design’, ‘construct’). Although the field of mathematics education has seemingly embraced the constructivist notions of the active learner and the constructing mind, a ‘softer’ constructivism is most certainly enacted in schools (Larochelle and Bednarz 2000: 3), where the modernist truth agenda remains cemented in place. While the child’s active mind may be increasingly valued in policy documents and students’ points of view are elicited in the classroom, such elicitation only serves to determine what is ‘wrong’ about the students’ way of thinking or understanding. ‘Wrong’, used in this manner, comes from the perspective that there is a pre-existing knowledge – Mathematics, a truth-regime – that is to be taught. In the soft version of constructivism, the fabrication of knowledge takes on a different meaning; the knowledge fabricated by the learner is a concoction, an invention, a forgery. In essence, this soft constructivism encourages a view of the learner as one who constructs untruths, one who fabricates lies. Without question, this is an unjust and unethical perspective to have towards another – to an autonomous constructive knower other than oneself. <br />
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The political and social ramifications for a constructivist view on learning and the related constructed view of knowledge that I have been discussing have yet to be enacted in mathematics classrooms. They have yet to be taken seriously by schools and teachers when they conceive activities and goals for mathematics education. The current treatment of children as fabricators of knowledge, seeing them as little liars, may in fact be a greater injustice to these learners than teaching with the intent to deposit knowledge into their minds (to paraphrase Freire’s (2002/1970) banking model). In this insidious current model for teaching young mathematical fabricators, we engage them in activity, lending them a momentary belief that we are truly interested in what they are thinking about their world, and then we tell them how it ‘really’ is, how they should have reasoned, how they should think. We not only continue to act in accordance with a belief that language may somehow transmit knowledge – an illusory notion (Glasersfeld 1998) – but we enforce and enhance the modernist knowledge-as-truth agenda, forcing it onto the mathematical learner. <br />
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When unquestioningly engaged in an epistemology of soft constructivism, we treat the learning activity as a process of discovery while holding tight to a knowledge that is to be discovered, listening for (Davis 1997) cues to hear in the child our own ways of knowing this knowledge, this Mathematics. The pedagogical practices of the teacher devolve to a guess-what-I’m-thinking exercise; the pressure of time and the testing of this pre-existing knowledge drive the maddening process of an education that began with a hopeful premise – that children make meaning through active engagement with their experiential world, that children are knowledge constructors, genuine fabricators. If the radical epistemology of constructivism is embraced and the fabrication of mathematical knowledge is recognized, not as a construction of untruths, but as the development of other truths, as each learner’s ways of thinking and understanding, a different style of mathematics education must be conceived. Such a mathematics education would grow out of this postmodern constructivist epistemology, and its concordant poststructural concept of power/knowledge (Foucault 1982). Such a mathematics education would be in a position to embrace socially just calls for access, achievement, authority and action far more powerfully. Furthermore, this ethic for interaction, in particular an intentional teaching interaction, is not only more just but also helps bring about a relational equity among students. </div>blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com1tag:blogger.com,1999:blog-4189326141173312850.post-29064745725565406962012-02-22T03:16:00.000-05:002012-02-22T03:16:22.750-05:00must there be a curriculum?<span class="Apple-style-span" style="font-family: inherit;">A local HS teaching friend and I have been discussing the challenges of focusing mathematics instruction on respecting students' reasoning and embracing the need for children to invent mathematics (following on constructivist learning theory, especially that of Piaget; his students such as Constance Kamii, Seymour Papert, and Tom O'Brien; and Ernst von Glasersfeld). In other words, teaching with full recognition that children (all learners) are little scientists who are constantly creating and testing their own theories of the world.</span><br />
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This creates slippery ideas about what defines or constitutes "mathematics." I suppose there is a community of mathematicians that have a definition for what mathematics is. And there is a subset of that mathematics that might be described as School Math. This refers to a particular way of thinking (e.g. maybe the CCSS "standards for mathematical practice") and a particular way of knowing (maybe the CCSS content standards).<br />
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How does this mathematics reconcile itself with children's mathematics (cf. how Les Steffe distinguishes among three ways of considering mathematics)?<br />
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In discussion, my colleague wrote as he was reflecting on the quality of the projects in his school (a PBL school):<br />
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<span class="Apple-style-span" style="font-family: inherit;">I have been reading "Switch" for grad school which is all about creating change when change is hard. They talk about directing the 'rider' (analytical side), motivating the 'elephant' (emotional side), and shaping the 'path' (environmental side). I am considering trying to talk with my director about a plan in which we can help create change at our school by appealing to all three:</span><span class="Apple-style-span" style="font-family: inherit;"><b>rider</b> - define elements of good projects that serves as a checklist for design</span><span class="Apple-style-span" style="font-family: inherit;"><b>elephant</b> - operate under a slogan like "building better projects" (which would fuel our content meeting time); each teacher serves as an expert on one "bright spot" of project design</span><span class="Apple-style-span" style="font-family: inherit;"><b>path</b> - adopt integrated "strands" that have been created to create a connected 4 year program; eliminate use of ALEKS/Khan/etc.</span></blockquote>
I wrote in response (this is a portion):<br />
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<span class="Apple-style-span" style="font-family: inherit;">I wonder if there is a shared valuing [among your colleagues] of even having a path... I suspect some may give much more credence to the doing of projects and a "could care less" orientation to the content. (I am one who might think this is a healthy attitude..., but in opposition to the standardization of mathematics education that reflects the majority of educators' attitudes)</span></blockquote>
He responded with a couple questions. I restate them one at a time followed by by replies, his second question posted first. [please note, I am tracking this conversation here because the ideas are of interest to me to keep thinking forward on...]<br />
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<span class="Apple-style-span" style="font-family: inherit;">2. If they didn't exist AT ALL, what do you envision high school math ed to look like?</span></blockquote>
Needing to be brief:<br />
<span class="Apple-style-span" style="font-family: inherit;"><br />TOTAL IDEAL: I would like to teach in a school that was entirely directed by the students (read Summerhill). I really don't think kids would get screwed if they never studied school algebra or school geometry. I also think that if they would like to, they would learn much of it rather quickly.<br /><br />IDEAL: I wish that learning theorists could construct a learning trajectory that is observed in most children up through approximately age 18, or a variety of trajectories that would account for most (95%+) children. Yes, this learning trajectory would ultimately be based in an adult's way of knowing mathematics and his/her models of children's ways of knowing. It would NOT necessarily be exactly how any one child develops a mathematical mind.<br /><br />How this would be used: to guide the teacher in making decisions about potential developmentally-appropriate mathematical activities.<br /><br />If these trajectories were completed (currently, they are--basically--through approximately grade 7), curriculum designers could then develop example activities, or even possibly lessons (a chunk of activities) or units (a chunk of lessons) that mapped well onto that trajectory. [likely you see that what I am saying is not too radical, and that IMP might be an example of it]<br /><br />What would be most important though is that the curriculum is designed in response to the student(s) and their interests and their contexts, maybe in conjunction with the interests/needs of the local community. But I actually think the interests of the community would be reflected through the interests of the children.<br /><br />What I DO NOT WANT is an authoritarian, fascist structure as we have now. It is a curriculum backward-mapped from what someone (???) determined kids must know and then "backwards designed" into a mathematically logical sequence. Damned be the learner.<br /><br />OK - that is longer than brief.</span><br />
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1. I too wish we could de-emphasize/eliminate skills based standards. However, the reality of their existence and power over school future indicates we must respond in the best way possible? Doesn't this mean some sort of curriculum organization is necessary?</blockquote>
I think I didn't answer this.<br />
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<span class="Apple-style-span" style="font-family: inherit;">You may not be surprised, but my most honest answer to your first question is: ethically, we must respond in the best way possible. To me, that means FIRST developing autonomy and authority in children, a confidence in their own selves as reasoners. Second, (not as in second in case we are able, but second of only a slightly diminished priority) I think that children should be taught (or maybe offered the opportunity to study) the mathematics of the oppressor. [that was the language I decided to use tonight; you could call it School Math if you'd prefer] I think that children should have the opportunity to have a competence in this. More importantly, they should study School Math critically.</span></div>
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<span class="Apple-style-span" style="font-family: inherit;">I intentionally labeled the second goal as second because it should NOT take precedence over achieving the first.</span></div>
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<span class="Apple-style-span" style="font-family: inherit;">So to achieve the second, I agree "some sort of curriculum organization is necessary." I am actually forced to agree with that because that the teacher has some sort of influence within the classroom merely by the questions he/she asks, the coherence/organization is necessarily that which is enacted by the teacher. There is an unavoidable manipulation of the child's experiences that the teacher cannot avoid; hence an organization is present.</span></div>
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<span class="Apple-style-span" style="font-family: inherit;">A bit more solution oriented: A principal in the early 30's [google <i>Benezet Centre</i>] decided not to teach (formal) arithmetic to any students in his school through 4th grade; students were only to be engaged in rich quantitative activities & play. Then during the next year or so, the early grades content of formal arithmetic was taught. His children learned the formal arithmetic as well or better than other students - taught in the traditional manner. He extended the years of "free" learning eventually through 7th grade, and found that students could still pick up all the formal algorithms by the end of 8th grade.</span></div>
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<span class="Apple-style-span" style="font-family: inherit;">So, is a principal willing to commit similarly today? To take this leap of faith? risk? It happens in some places...</span></div>
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<br /></div>blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com0tag:blogger.com,1999:blog-4189326141173312850.post-48746982761341321652012-02-12T02:34:00.000-05:002012-02-12T02:34:26.679-05:00A Deconstruction of Learning -- Assessing -- TeachingA Twitter conversation with Chris Dell, a district math person in NorCal, pressed me to say, with some conviction, that assessment is not necessary for (a child's) learning. Assessment is only needed when an other (adult) wishes for a child to know or think a certain thing, or know or think in a certain way.<br />
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That assessment is not needed by a learner is clear to me--humans are learning machines. The comment I made pressed me (as did Chris' reactions) to think further on the role of assessment with regards to the teaching of mathematics.<br />
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Here is the start of the conversation, taking place in the twitterverse. At bottom I will make one last statement about assessment & teaching mathematics!<br />
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<span class="Apple-style-span" style="font-family: inherit;">CD: Students need frequent (formative) assessments for individual growth and performance, not only the one size fits all summative test.</span></blockquote>
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<span class="Apple-style-span" style="font-family: inherit;">BL: I call BS to the claim kids NEED any sort of assessment. Adults need assessment when they want a certain thing [e.g. for a kid to learn something in particular, or for a kid to think in a particular way]</span></blockquote>
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<span class="Apple-style-span" style="font-family: inherit;">CD: It is ludicrous to think that no assessment is the best plan. It's your definition of assessment that needs clarity.</span></blockquote>
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<span class="Apple-style-span" style="font-family: inherit;">BL: Not for me, it involves judgment. I know one can teach wo judgment.</span></blockquote>
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<span class="Apple-style-span" style="font-family: inherit;">BL: And I know people learn amazingly well w/o judgement. In fact, it keeps their self-concept in tact.</span></blockquote>
<span class="Apple-style-span" style="font-family: inherit;">So, again I say assessment is for the adult, not the child--this is my primary point. The second point is that judgment harms self-concept, either with "wrong" or with "praise" (read <a href="http://www.alfiekohn.org/f_news/fullnews.php?fn_id=5" target="_blank">Alfie Kohn on praise</a>). A person is internally programmed to learn, and to construct knowledge that is viable (i.e. true, consistent) in that knower's experiential reality. The child provided the space to determine for oneself what is truth (correct) maintains confidence, positive self concept, an sense of agency--the confidence to act in and on the world.
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<span class="Apple-style-span" style="font-family: inherit;">CD: If "judgement" is the goal of assessment than its not formative.</span></blockquote>
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<span class="Apple-style-span" style="font-family: inherit;">CD: @misscalcul8 I've heard and read some interesting ideas about feedback from Dylan William.</span></blockquote>
Judgement (defined): the cognitive <b>process of reaching a decision</b> or drawing conclusions<br />
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"Practice in a classroom is formative to the extent that evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers, <b>to make decisions</b> about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have taken in the absence of the evidence that was elicited" (<a href="http://www.springerlink.com/content/94753257u011w380/" target="_blank">Black & Wiliam, 2009, p. 9</a>)</blockquote>
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<span class="Apple-style-span" style="font-family: inherit;">"We use the general term <i>assessment</i> to refer to all those activities undertaken by teachers -- and by their students in assessing themselves -- that provide information to be used as feedback to modify teaching and learning activities. Such assessment becomes <i>formative assessment</i> when the evidence is actually used to adapt the teaching to meet student needs" (<a href="http://www.kappanmagazine.org/content/92/1/81.abstract" target="_blank">Black & Wiliam, 1998</a>).</span></blockquote>
I know Wiliam's work with formative assessment quite well. In a quick review of some of his work, I sense he feels that the goal of formative assessment is to draw upon evidence of student knowing/understanding <b>to make decisions</b> for teaching. Clearly, judgment refers to the process of making decisions.<br />
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Returning to the claim that "It is ludicrous to think that no assessment is the best plan" [my self concept remains in tact, despite your assessment/judgment of my claim] -- this is how I try to live with my partner, and with my students. I think my students and I learn quite a bit together--about mathematical things, and about teaching mathematics. And I think my partner and I learn much together as well.<br />
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I believe Brent Davis taught me a lot about this tension between assessment/judging and learning in a piece he wrote in JRME, 28(3) in 1997, titled "<a href="http://www.nctm.org/publications/article.aspx?id=17750" target="_blank">Listening for differences: An evolving conception of mathematics teaching</a>."<br />
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Chris - In the end, I seem to notice some internal contradictions in your thinking. I pointed out that you certainly wish that assessment were something other than judging. Also, on Dec 3 you recognized that focusing on students' weakness--which might be akin to "what they don't know/understand"-- impacts confidence [maybe identity, self-concept]. I don't think you would (formatively) assess only to focus on students strengths, would you?<br />
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<span class="Apple-style-span" style="font-family: inherit;">CD: If you focus on students' weaknesses, they lose confidence. [tweeted Dec 3]</span></blockquote>
<span class="Apple-style-span" style="font-family: inherit;">I don't think contradictions are bad, resolving them is interesting and leads to learning.
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<span class="Apple-style-span" style="font-family: inherit;">CD: @blaw0013 must have some deep scares with past assessment experiences.</span></blockquote>
<span class="Apple-style-span" style="font-family: inherit;">Lastly, I actually don't think I've been scared or scarred by assessments. Quite the contrary--they have boosted me to an undeserved social status.</span><br />
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<span class="Apple-style-span" style="font-family: inherit;">Concluding thoughts on assessment/judgment and teaching mathematics: Maybe I cannot get away from judgment when attempting to teach a particular mathematics, School Math as many call it, or the race-expression (so named by Dewey in <i>The Child and the Curriculum</i>, 1902) the Discipline of Mathematics.</span><br />
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<span class="Apple-style-span" style="font-family: inherit;">[T]he value of the formulated wealth of knowledge that makes up the course of study is that it may enable the educator <i>to determine the environment of the child</i>, and thus by indirection to direct. (p. 31)</span></blockquote>
<span class="Apple-style-span" style="font-family: inherit;">The teacher's knowing of mathematics allows the teacher to create a opportunities in which the child may learn. Dewey (1902) continues, </span><br />
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<span class="Apple-style-span" style="font-family: inherit;">Its [the teacher's mathematical knowledge] primary value... is for the teacher, not for the child. (p. 31)</span></blockquote>
Returning to my point above, the teacher draws upon their own mathematical ways of knowing to assess/judge that of the child to determine the environment of the child. Assessment is for the teacher.<br />
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However, <br />
<blockquote class="tr_bq">
<span class="Apple-style-span" style="font-family: inherit;">The case is of Child. It is his [sic] present powers which are to assert themselves; his present capacities which are to be exercised; his present attitudes which are to be realized. But save as the teacher knows...the race-expression which is embodied in that thing we call the Curriculum, the teacher knows neither what the present power, capacity, or attitude is, nor yet how it is to be asserted, exercised, or realized. (p. 31)</span></blockquote>
<span class="Apple-style-span" style="font-family: inherit;">It may be the teacher should work to assess the child's present ways of knowing in order to act. But, the teacher should be no means expect that his/her expectation for knowing/learning is where the child will arrive. Recognizing this and interacting in such a way may reduce the authoritarianism of judgment and create a freer space for both the child and the teacher.</span><br />
<span class="Apple-style-span" style="font-family: inherit;"><br /></span><br />
<br />
<span class="Apple-style-span" style="font-family: inherit;">So again, m</span>aybe I cannot get away from judgment when attempting to teach a particular mathematics. But I am quite certain I can develop/influence thinking/ideas within the child that are what might be captured more broadly by a term such as quantitative literacy, or numeracy--something I would notice after the fact as mathematical.<br />
<span class="Apple-style-span" style="font-family: inherit;"><br /></span><br />
<span class="Apple-style-span" style="font-family: inherit;"><br /></span>blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com7tag:blogger.com,1999:blog-4189326141173312850.post-69511455177317391382011-09-26T02:28:00.000-04:002011-09-26T02:28:21.075-04:00And a student wrote something that prompted me to feel like I've gained some insight into why educators (especially) incorrectly (?) perceive math to be such a linear march through topics that "all build on one another." She wrote:
<blockquote> While I have somewhat of a plan in order to work for social justice, I also have some questions that should be cleared up before entering a classroom. For instance, what will I do if only some students do not understand a topic while the rest of the class does?</blockquote>
My response went...
<blockquote>One of your first questions is important (in my mind) for all mathematics educators to resolve. In my opinion, a significant part of resolving this means coming to a much broader definition of what it means to be smart mathematically. It is most certainly not about racing forward through a linear track of math topics, as a textbook may have you believe.<br><br>
Challenge: Consider how to organize your instruction around key mathematical ideas--not marching through the pages of a book.</blockquote>
I think that there is something significant to the way math teacher's have become almost slavishly directed by the sequencing offered up in texts, that there is a loss of vision about the nature of the underlying mathematics. The text promotes a blindered view of mathematics; it makes it <i>seem</i> to be linear. It causes teachers to forget that mathematics could be more robustly thought of as underlying principles, big ideas to be explored, etc.blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com2tag:blogger.com,1999:blog-4189326141173312850.post-64074520783438610172011-09-26T02:18:00.000-04:002011-09-26T02:18:28.404-04:00A friend of mine wrote to me:
<blockquote>Hey, what are your thoughts on the Common Core of State Standards and the states using these to guide curriculum? Unfortunately where I was teaching they still use a traditional text to drive the classes and race through them chapter by chapter which is why we didn't get along. They told me to leave since I didn't fit in. I have a professor telling me that all high schools are on board and actually practicing authentic, problem solving, constructivist based learning. I was even told I was wrong about teachers in math still lecturing and using worksheets at the high school level. Unbelievable, I lost my job over it. Anyway, do you think NCTM's approach will make a difference in how states make any changes?
</blockquote>
To which I replied:<br><br>
No, NCTM's approach, nor the CCSS will change the rotten approach to teaching math. I think I would say there are three fundamental problems:<br><br>
(1) Americans think of teaching as delivery of information;<br><br>
(1b) American math teachers think the goal for teaching math is to be able to solve a given problem--whether that be the 15 second type, or even the larger, rich problem (as opposed to understanding the math behind the problem);<br><br>
(2) Americans have a shallow understanding of what math is, and what should encompass a quantitative literacy curriculum;<br><br>(2b) the math that we do teach is a eurocentric relic of the past--it does not recognize the present or future lives of any of today's students, except that it allows certain students to continue to have access to the privilege afforded those students for whom the education system is structured to make succeed; and<br><br>(3) the efforts of both CCSS & NCTM enhance problem 2b because of the strong support for national testing--naively demanding all kids across the whole country with immense diversity in life paths to think and be automatrons; (and)<br><br>(3b) I cannot imagine teachers, admins, schools, or districts reacting in any possible healthy manner to the oppressive accountability environment created by these efforts for standardization.<br><br>
Is there a hope? I would say that the 8 Mathematical Practices should be used to justify the use of rich tasks in the service of teaching the CCSS content standards. And to engage kids in rich tasks can change the emphasis from merely being able to do problems toward thinking and understanding.<br><br>
Potential for change? Yes. Will it? No.<br><br>
-Brianblaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com2tag:blogger.com,1999:blog-4189326141173312850.post-2152924269422284142011-06-09T13:55:00.000-04:002011-06-09T13:55:37.294-04:00Thud! - Comments on dy/dan re: Kahn went UnheardI posted this response to Dan Meyer's <i><a href="http://blog.mrmeyer.com/?p=10629">Salman Khan Isn’t A Fan Of One-Size-Fits-All Lectures</a></i> (post #46).<br />
<br />
I am just reading a paper by the math educator Paul Cobb, who states that an environment for learning mathematics should incorporate the following qualities:<br />
<br />
<blockquote>1.) Learning should be an interactive as well as a constructive activity – that is to say, there should always be ample opportunity for creative discussion, in which each learner has a genuine voice;<br />
<br />
2.) Presentation and discussion of conflicting points of view should be encouraged;<br />
<br />
3.) Reconstructions and verbalization of mathematical ideas and solutions should be commonplace;<br />
<br />
4.) Students and teachers should learn to distance themselves from ongoing activities in order to understand alternative interpretations or solutions;<br />
<br />
5.) The need to work towards consensus in which various mathematical ideas are coordinated is recognized.</blockquote><br />
I think what is of importance in conversation around teaching mathematics, lecture, text-based, video, etc., is the notion of authority: authority of <i>knowledge</i>, <i>author</i>-ity of knowledge, and authority for knowing. Each of these speak to the learner’s position of him/herself within the world, maybe called his/her (mathematical) identity.<br />
<br />
I suggest that when education gets treated as “reception of knowledge” all (?) that is important in the role of the teacher and of schools can be discounted. It is the present (and previous — see 1930′s for example) danger of the economizing of education.<br />
<br />
Cobb, P. (1990). Multiple Perspectives. In L. P. Steffe & T. Wood (Eds.), <i>Transforming children’s mathematics education: International Perspectives</i> (pp. 200-215). Hillsdale, NJ: Lawrence Erlbaum.blaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com4tag:blogger.com,1999:blog-4189326141173312850.post-70617062172855972392011-05-31T13:49:00.000-04:002011-05-31T13:50:36.185-04:00Conversation following "Waiting for Superman" & Ravitch Comments [Nov. '10]<blockquote>I believe the author, Diane, mentions to pay teachers well, lower class sizes, relieve the work load for teachers because otherwise teachers an average leave after 5 years, etc. It's a matter of convincing the masses to pay more for education.<br />
<br />
Also, though, the problem we have with some schools failing and others excelling is that they reflect the segregated nature of our society. If we could integrate our schools across ethnic lines and socio-economic lines, including admitting homeless kids and Learning and Behavior Disabled kids in selective schools, then we could work towards wider societal integration and move towards closing the Achievement Gap.<br />
<br />
The last school I taught at, Wendell Phillips High School, was a dumping ground. Students that had no where else to go would come to Phillips. We had a 20% homeless rate, and a Special Ed rate of 30%. For all the wonderfully motivated students that did attend Phillips, there was such a large group of troubled kids that they disrupted the learning process for all. We can't allow for such segregation to exist.</blockquote><br />
Agreed with Sean. It is not a systemic problem of education, "failed" schools are a batural outcome of a classist society, and corporate/capitalist mindset.<br />
<br />
The citizens of this country a sick with capitalism, that is sick with the disease called capitalism. There is no way of understanding community or common good because our the lenses we wear. We don't know how to interpret the "problem" of schools in any other way than to say, "failed."<br />
<br />
Another simple, absolute first step (along with Sean's/Diane Ravitch's) is to force the conversation to stop staring at deficits and relook to see positives and potentials. Every kid Sean listed, while possibly seen through deficit/difference eyes, sees a kid who is not Beaver Cleaver, or some cute quirky version of him. The measures we use to define good schools, good teachers, good education, and good kids must be rethought. That to me is absolutely more powerful and has way more potentials than the immediate fixes that Sean & Ravitch suggest. I think those suggestions are not bad or misguided, but tremendously inadequate.<br />
<br />
-brian<br />
<br />
<blockquote>It sounds like you're arguing against national or state standards of measurement to assess student achievement. It's good to have an awareness of what skills are generally expected from students entering college, for example. Good teachers will know this. Moreover, good teachers will know their students, their abilities, how to scaffold instruction to help their students learn as much as they can, and how to assess such learning. So, I argue that we have to keep working on getting good teachers in the schools, paying them well, supporting their craft, and provide job security so that they stay.</blockquote><br />
Standards and measurement are two distinct things. Standards in and of themselves are not bad. (Ask, and I can share some great book or article titles speaking directly to this point).<br />
<br />
What is your educational goal? …to educate for living or for future living? (Dewey asked this) …to serve the economy, i.e. the ruling class? (today's rhetoric) (a schooling Marx would vehemently reject). There are probably many additional possible responses, I know of view that fall far away from any of these versions.<br />
<br />
The trouble with today's image of education is its strong alignment with the American Jeremiad (the ever-present fear of our great society's eminent downfall, a christian belief structure set into the American mindset by the Puritans) & concomitantly, the "American Dream"--this myth that we live in a meritocracy <http://www.ncsociology.org/sociationtoday/v21/merit.htm> . What happens if ALL students met the Standards? What if ALL wanted to go to college? What if ALL believed they had the power to voice opposition to the government, or to refute the false power/status of the elite? The propaganda of the American Dream structures inequitable systems, not to be overcome, but to be extinguished.<br />
<br />
On a different note, I contend that if teachers found their work intellectually engaging and personally rewarding--due to that work, but also the joy of healthy & Happy communal relationships, pay & possibly even job security are much less the lever to keep teachers in place. BUt simply to attend only to increased pay and increased job security will do nothing to improve teaching, and I believe nothing to increase the attraction or retention of "good" teachers/people.<br />
<br />
<blockquote>To rebut my own argument, however, I heard a statistic stating that student achievement is only 20% connected to their teacher while some 60% is connected to family environment. I don't know, but a good teacher is vitally important. So is an engagement with the arts, health and fitness, and school cultural activities (school dances, etc.). Schools need money to have these programs (band, football team, pottery class…).</blockquote><br />
Your statistic is correct. But again, what was the measure of student achievement? There are other, much more important measures of student achievement entirely unaccounted for in that measure--actually, probably more than just unaccounted for, but masked. So agreed, good teachers are vitally important.<br />
<br />
And not to diminish our field, but anyone (maybe 97% of people) are good teachers. One context or another, we do teach,and we do teach well. It is a biological need of the species. What is it about education that makes only *some* good teachers?<br />
<br />
<br />
<blockquote>I really don't think you need to know rocket science in order to know how to help schools be effective places of learning.</blockquote><br />
TOTALLY agreed. I suggest that knowing "rocket science" is in fact harmful. Our formalized ways of knowing, of developing the "scientific mind", removes us from the powerful, and personal ways of knowing when we pause, erase our minds, and let our bodies know (do the knowing).<br />
<br />
-bblaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com0tag:blogger.com,1999:blog-4189326141173312850.post-73608832483569245762011-05-27T12:04:00.000-04:002011-05-27T12:04:05.472-04:00Radical Constructivism & Intellectual Fascism"As soon as… a trait is labelled as a good thing in itself, all non-possessors of that trait automatically are labeled as evil or worthless. Such an arbitrary labelling… is fascism" (Ellis, 1985, p. 9).<br />
If the notion of fascism is based upon some arbitrary belief that those who possess certain traits are intrinsically superior to others, and thus deserving of a higher status, or socio-political privilege, then those who subscribe to a humanist mathematics education are intellectual fascists. That is to say, those who subscribe to present day proposals for mathematics education are intellectual fascists.<br />
Paul Erdös spoke of the Supreme Fascist (SF), his agnostic way of referencing god, this person who he would accuse of “hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself” (Wikipedia, 27 May 2011).<br />
If I follow on this paper, it would be constructed along the lines of is has the RCist constructed a limited number of ways of knowing and then seeks to observe the child to “listen” (Brent Davis, 1997) for only these possible models. Or, maybe that the RCist has created learning trajectories that then encourages him/her to listen to children, create curricular opportunities, that assume these (and only these) possibilities.<br />
I don’t necessarily think so. But it is maybe the danger for the RC reader, to perceive this “limiting.” The RCist might say when the goal is to be the researcher in the teacher/researcher interaction with the child, the observing mind thinks with these models, but works to not limit ones hearing to only these; that is to especially be receptive to surprise. <br />
But when the teacher takes precedence in the teacher/researcher interaction with the child, especially in the context of the classroom, can the fascist be held at bay?<br />
<br />
Next: Critique Cobb’s work, for example, under this framework for teaching with a RC perspectiveblaw0013http://www.blogger.com/profile/13023564844812039091noreply@blogger.com0