[Numeracy 176] Re: Application vs. Theory
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Mon Feb 15 17:13:07 EST 2010
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Thanks Michael and all and thank you and Brooke for engaging us in this important discussion which is ever present in all applications of adult education.
First an aphorism: "There is nothing more practical than a good theory" (K. Lewin).
Second, "theory can have a wide array of applications from the recondite to the immensely practical. I've drawn on the classical pragmatic transition, especially John Dewey's theory of inquiry in which hypothesis construction is embedded within a critical phase of problem solving. http://www.youtube.com/watch?v=z3l46XhChxE. The underlying issue then becomes "what is the question or problem" for which an answer (however provisional) is needed in order to move forward.
Simple example. In a class used the negative/positive number line for illustration. I went to 6 on both sides of the line with ellipse before -6 and after +6. A student was quizzical. I inquired. She had thought the integers ended with 5 because that is the numerical example her regular teacher had used. Thus this student's implicit theory was that integers were somehow defined or bounded by the numbers (-5, +5). Obviously her teacher didn't tell her that but that was the implicit understanding she drew from the example. After I explained to her the definition of integers her understanding opened.
In Dewey's terms my use of (-6,+6) raised a problem with this student that required resolution for her to move forward in her understanding since (6) disrupted her previous construction of reality. This gave us an opportunity to discuss the definition of integers which aided in the broader discussion we were having on sets and subsets in which I was drawing on the set of integers as the more inclusive term to situate the more restricted sets of natural numbers, whole numbers and negative numbers. Thus these more restrictive sets were subsets of the more inclusive set of integers. Because my focus was that of teaching a basic concept I drew on the pedagogical principle that the concept could best be initially taught by making the initial example no more complex than absolutely critical to illustrating the concept in a manner that the students could firmly grasp. Then at that point if we were going to work further on set theory we could increase the complexity of both the theoretical presuppositions upon which set theory is based and the examples used, which would require much additional scaffolding.
Perhaps then what is required is some demystification of the definition of theory especially when we pit it against the "practical." The more fundamental issue, I believe is the relationship between conceptual understanding at whatever level or degree is needed as a tool in problem solving and concrete hands on application. Being both kinesthetic and conceptual I need to apply both in which I gain progressively clearer understanding of a given problem which then enables me to make additional progress in concretely working through a problem in which i both feel and think my way through it until I have actually mastered it, which may require some sleeping on it.
Thus, in the emerging field of of adult education mathematical pedagogy I posit as an operational hypothesis that conceptual understanding at whatever level needed to help grasp a problem is an essential counterpart in practically working through a problem step-by-step. More, sometimes a methodical step-by-step procedure can stimulate the core understanding needed to grasp a principle that will be needed to transfer it from one concrete context to another. Sometimes, a clear understanding of a core principle is needed to effectively engage in a step-by-step procedure. To the extent that our repertoire of instructional tools are extensive, all things else being equal, so will our capacity to effectively apply them in concrete teaching situations. A good theory described at a level that can be grasped is one such tool.
Date: Mon, 15 Feb 2010 10:23:02 -0800
From: tesolmichael at yahoo.com
To: numeracy at nifl.gov
CC: ooprc at comcast.net; HKerr at aol.com
Subject: [Numeracy 168] Re: Application vs. Theory
Greetings to all,
Will you agree that the goal of reading is to read with understanding? If we fail to read with understanding, even if whatever it is we understand will remain more or less subjective (because the meaningfulness minds derive from text will vary from person to person), we are in fact in the application phase of reading. We may have learned how to decode, including whatever rules we may be applying that assist us in decoding, but if that's all we do, we might as well not engage in reading at all.
I will agree that oftentimes we need the keys that open doors to application, perhaps more so in math than other disciplines. There are highly proficient users of calculators who have little notion of the mathematical processes calculators "engage" in, and yet, their ability to use calculators opens doors perhaps especially to gainful employment that would otherwise remain closed.
That said, as a teacher, my goal will always remain to place my students on a path of (self-) discovery, the very essence and meaning of education itself. Ultimately, it is the inability of so many to derive relevance and (personal) meaningfulness in what they are expected to learn that alienates them from what they are learning in the first place. This happens all too often in studies of math, and accounts for the self-proclaimed "hatred" of math so many of my students share with me.
The "golden rule" I referred to in post #150 (i.e., positive times positive and negative times negative both = positive, whereas negative times positive or positive times negative both = negative) is far more "practical," (vs. "theoretical") than might meet the eye. Once students delve into integers (undoubtedly a component of "functional" numeracy - just think of financial literacy), the rule provides one way of determining whether a sum, difference, product, or quotient is a positive or negative value (integer).
The rule, however, is more than an abstraction that lingers somewhere in ethereal space, and opens the door to the understanding of integers as they come to bear in maneuvering the demands of daily life.
I recommend checking the following three links, namely http://www.homeschoolmath.net/download/Add_Subtract_Integers_Fact_Sheet.pdf, http://www.homeschoolmath.net/download/Multiply_Divide_Integers_Fact_Sheet.pdf, and http://amby.com/educate/math/integer.html#mult-div. These sites might serve as good initial resources to "concretize" math, i.e. help our learners negotatiate meaning in it.
As for "because that's the way it is" pedagogical approaches, I'd contend that the purpose of education is not to instill that message, however "disabled" or "enabled" our students may be.
Michael A. Gyori
Maui International Language School
From: "Young, Krista" <krista.young at abileneisd.org>
To: The Math and Numeracy Discussion List <numeracy at nifl.gov>
Sent: Mon, February 15, 2010 4:49:17 AM
Subject: [Numeracy 163] Re: Application vs. Theory
I am not a mathematician, but have 16 years in adult education, and am one of the lead facilitators for the Texas ABE Math Initiative. I also teach an ASE/GED level math class. Personally, I can see that theory is important, and may lead to some deeper math understanding; however, in the real world, if I can get my students (and fellow teachers) to understand the application of math, I feel I have done my job. When teaching integers, I stress the addition rules using a number line, thermometer, check book - whatever works. When we hit subtraction - I use two strategies - one is that the subtraction is the "range" between the numbers using the number line, and two, this is the rule, and "Momma said so, so do it." (By this point in my class, I have created a level of trust and safety in the room, so they will take this direction very easily.) I understand that the mathematicians on this list are cringing right now, but this works for me and my students. They choose the strategy that works for them, and we move on. As long as they can continuously and consistently apply the rules, I am happy.
Also, in the real world of adult education, most teachers do not have the time or expertise to teach theory. Our math initiative has been created to work with teachers to give them strategies so they can work through their own fear of math. I cannot even see a practical way to introduce the theory that has been discussed here. My teachers would run out of the room, screaming.
Thank you for providing this forum.
Abilene, TX Adult Education
Texas ABE Math Initiative, Facilitator
From: numeracy-bounces at nifl.gov [numeracy-bounces at nifl.gov] On Behalf Of Denney, Brooke [denneyb at cowley.edu]
Sent: Sunday, February 14, 2010 11:05 PM
To: numeracy at nifl.gov
Subject: [Numeracy 160] Application vs. Theory
Your comments leave me feeling as a mathematician curious of how you came about your “logic”. To use the term absolute value to mean, “one that knows no positives or negatives”, is a paradox. That is, if you are talking about the mathematical operation known as absolute value (which does, have a positive connotation). Additionally, your comments about negative numbers astonish me and my fellow mathematician colleagues. Is it important to know the mathematical proof that states the logic of why two negatives when multiplied together yield a positive result? Or, is it okay for people to just “know the rule”? Several researchers have stated that adult numeracy learners need to be taught within realistic contexts; otherwise educators jeopardize de-motivating learners to learn. Do not misinterpret my response, I love to learn about math theory and logic but without application the concepts are often nonrepresentational.
I understand why you would choose to utilize the Cartesian Coordinate Plane to discuss all real numbers. Nevertheless, many students would consider that lesson mindless prattle if they did not have a prior frame of reference to build their cognitive skills from. Conversely, many students do know what it means to be below sea level or overdrawn in their bank account (and the like). Their frame of reference allows them to relate to the idea of negatives better. Albeit true, it is subjective in nature but isn’t all mathematics subjective?
I leave it to the discussion board participants, is it better for numeracy students or developmental math students to understand the application of mathematics or learn about the theory that lies underneath?
Math & Numeracy Moderator
National Institute for Literacy
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Numeracy at nifl.gov
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