On Fri, Aug 20, 2010 at 11:34 PM, Robert Hansen <email@example.com> wrote: > Anne asked... > > "learning - my own and my students. This leads me to question, "What is abstract thinking, exactly, what are its components and their relationships." > > What is Abstract Thinking? > > A triangle is abstract along with its constituent elements, segments and points. Another term you could use for this (triangle) example is "imaginary" because a triangle cannot actually exist physically because in the physical world "things" always have thickness or they wouldn't exist at all. >
In another namespace we might say triangles, as abstractions, are "pre-frequency" i.e. independent of any time dimension, which is also the size dimension. Shape (angles) but not size, other than relative size among shapes of no energy content, as is characteristic of this "math world" (or "Platonic world" people sometimes call it -- Sir Roger Penrose has a "three worlds" trichotomy in his namespace, not saying I subscribe to that).
> Numbers are also abstract. We map them to physical properties (height, length, weight, count, etc.) as fits our needs at the moment but the numbers themselves have no physical existence, they are imaginary abstractions, yet they have a strict and undeniable order to them. I am not just speaking of their cardinal ordering, I mean all of the mathematics that supports their existence, albeit a non physical (abstract) existence. >
As a matter of logic, one might wonder, after reading the above, if it were possible to create a language in which numbers *do* have physical existence, as if they're prevented from having it by definition, then this detracts from the intrinsic sense of the claim. At least a few physical numbers, for contrast, would sweeten the pot, keep things less empty and tautological.
Immediately, I think of computerized numbers, which disappear when you pull the plug, i.e. they require energy to "stay alive" in the computations. These could be our physical numbers then -- a first move on the chess board at least.
> Technically speaking, "reasoning" is always abstract because it is in our mind. The "things" that might have inspired the thoughts may be physical and real (external to our mind) but how we model and compare them is internal and imaginary. We build models and abstractions of what we see and decorate these models with increasing detail and relationships. However, they are never exactly the same as what we see but over time become more and more similar to what we see. So I guess you could say that when our thinking involves modeling then it is abstract (which I find to be synonymous with reasoning). Math is unique in that it is entirely an abstraction and we further refine it based on what we see in the abstraction, rather than based on something we see in the physical world. >
This use of "internal" is somewhat imaginary and metaphorical i.e. to localize mental imagery as being "in the head" promotes all kinds of confusions about the physicality of location, when it comes to so-called "mental events". Funny how Robert has gotten all philosophical on us. He'll be quoting Wittgenstein next thing you know, or maybe Kant.
> But we must be careful on what we mean when we say "reasoning". Many attribute "reasoning" to be the single most important quality that separates us from other species. However, the gap between us and other species is so large that according to that comparison "reasoning" could be anything from shopping and driving cars to proving mathematical theorems. Shopping and driving cars can be and often are nothing more than conditioned behavior, albeit a behavior much more advanced than found in any other species. Reasoning, however, operates at a level beyond conditioned behavior. >
Have you heard of the Greenpeace plan to grant a PhD to all dolphins, honorary degrees through cooperating universities? There's some hope the fisher-folk will be less brutal if they realize they're murdering college professors. Whales are another story -- the plan there might be to give them high rank in some Navy (some non-humans already have rank, I kid you not).
> I don't think we have to define and sub-define what "thinking" is to have some understanding of what goes on when one student gets a subject and another does not. When students get a subject they get it pretty much the same way and when they don't get it they don't get it pretty much the same way, excepting of course special situations like Pam's students who have conditions that might mask their true ability to "get" or "not get" the subject. AI is inching up on what "thinking" and "getting it" is, although I am not suggesting that this knowledge will explain what having a successful and happy life is. But the difference between getting it and not getting is always either a lack of effort or a lack of understanding and the latter is more often than not a case of the student not having or at least not yet acquiring the ability to reason. >
I think AI is 98% bunk and 2% clever marketing.
> And that is the reason we are having a conversation on what "thinking" is in the first place. It is because all students are not successful at it. We don't have these heated discussions about walking and talking or chewing food. And even when we have a solid theory of "thinking" we still will not know why many students are not successful at it. That understanding will not come until we know a hell of a lot more than we currently do about the brain and genetics and even then we will probably find some sort of chaos theory at work that quasi-randomly predisposes each of us to different interests and paths. >
There are different philosophies about what the teacher, trainer or coach is for, in any given situation.
If your model is the teacher is responsible for students "getting it", and the snake oil teachers claim to have some secret for making them "get it", then you will think differently about the situation than if we're thinking of the teacher as role model student, obviously excited about the knowledge domain, engaged, likes to muse about it during free time, contributes research etc.
The latter is more the standard to which college professors are held, but there's no reason all grades might not adopt these same standards.
In that case, the teachers we need are simply those avid scholars with sufficient interpersonal skills to keep the school a congenial and collegial environment, safe enough to promote and nurture learning among those drawn into being studious.
> In fact, I am going to make that a law. When the educationalist states that they can make all students learn because they have unlocked the secrets of thinking then they are lying because knowing what thinking is has nothing to do with knowing why some students can't do it, anymore than saying I know what music is thus I can make all students musicians. Some might counter that law by saying that they have uncovered the secret to "learning" not "thinking" and I will say that is immaterial because they are both the same thing. You can't have one without the other. >
Even in an ideal world wherein all students are like that slave boy Socrates put through his paces, i.e. quite able to access the Platonic realm and grok generalities animal, vegetable and mineral, we still have plenty of room to debate sequence, content, the best routes to water, when to plant pumpkins. In other words, not every "education reformer" is interested in reforming human nature or revealing some new clever theory of some "inner child". Some just want to stay up to date, in a world that's anything but static (reform is therefore ongoing, is not about finally "getting there").
Take me for example. Our local paper, the Willamette Week, just ran this front page story going through the litany of what makes Portland, Oregon an open source capital of the world, and how we could potentially merge this well deserved reputation with our stellar and cosmopolitan health care services to pull off some big economic coup... and yet how many public schools around the state include anything about SQL on their math tracks?
Q: "What?" you say "SQL, on a *math* track? But that's just computer science!"
A: "So is computer science offered as a for-credit subject, a direct step towards getting a high school diploma?"
Q: "No, it was cut from the budget, was an elective in any case, but that doesn't change the facts about SQL".
A: "Can you imagine a discrete math course that tied set theory ala Venn Diagrams and boolean filters into practice creating and selecting from tables?"
A: "Then lets make sure every NCLB school has something of that when doors open in September, or call ourselves lazy good-for-nothings like WW hinted we might be."
Q: "That's pretty harsh, teachers won't have time to bone up before then."
A: "No problemo, we've got Youtube and ways to bus around Intel engineers as teacher trainers and substitutes, here to help with the upgrades. Silicon Forest, at your service."
Q: "But aren't Intel engineers all from foreign countries where they speak Asian and Arabic and stuff like that?"
A: "Well, yeah, so what's it to ya? We're cosmopolitan. Also, lots of witches and gypsies in open source, but that's nothing new in our pioneer state. We're also home to many Russians, our first pharmacy was Chinese."
> So, we can sit here and discuss "thinking" all day long and define abstract thinking, higher order thinking, lower order thinking etc. Let's just say that most of us here know it when we see it and move on to why some students can't or don't think. >
I used to attend the Florida school system and was amazed by all the racism and sexism going on. My international school circuit was more enlightened, and Portland felt more like home when I finally returned here in the 1980s, and started a new career after high school math and history teaching, also working in publishing (McGraw-Hill) and political maneuvering (Project VOTE!).
> Awareness > > You are playing basketball with another player and you are not that good. Your shots, while being in the general direction of the basket, have a random nature to them and seldom go in the basket. But the other player seems able to make his shots with almost no effort. You ask him how to make shots and he gives some pointers and then proceeds to show you actual examples of making shots. You attempt to mimic his movements and style yet your inability to make shots remains. He shows you again and you swear to yourself that you are doing what he is doing but obviously you are not because you are not achieving the same result. The other player is more aware of basketballs, baskets and shooting than you are. In fact, basketballs even feel differently in his hands than in yours. >
One secret to getting good at basketball is having a private hoop somewhere and just shooting over and over until the cows come home. Skateboarding, learning an instrument (e.g. keyboard) -- takes a lot of investment of solo time, which means a belief in and vision of yourself.
One of our current work / study students (global U) was an engineer for a computer science corporation but took it into her head to become more of a musician and subversive (somewhat in that old student organizer sense, familiar from the Kennedy-Johnson-Nixon era). This required some years of self discipline, learning music theory, chords, and the process for tapping one's deeper levels to come up with whatever haunting lyrics and eerie tunes (she's a bit on the gothic side, some call it "death metal"). She would never have gotten this far by aping some teacher, with her Perl or SVG skills either (she was lucky to have a polymath teacher in high school from the sound of it).
> My point here is that while we share many autonomous abilities, like the ability to throw a basketball in the general direction of a basket, we are, for whatever reasons, not equally and personally aware of these processes in such a manner as to take them to the next level. Reasoning and abstract thinking require one to have that very same form of personal awareness. If you want students to learn beyond that which is autonomous (rote) then you have to get them connected and aware of what reasoning feels like and this has proven as hard to do as making every student a basketball player or a musician. >
There's not just one IQ or EQ. Over-simplified models lead to pigeon-holing in ways that fail to optimize, squander "human resources".
If you don't have ways of opening doors, even as others are closing, then you simply create a discouraging environment, what I specified above as the anti-thesis of an educational one.
If your students are always struggling with the idea they might be failures, all washed up, no where to go, because they don't know what reasoning "feels like", then maybe you live in the wrong state where the NCLB schools don't even have hi-def TV in the geometry classrooms, unlike in Korea.
> Obviously, my description of "ability to reason" sounds very much like a talent and acts very much like a talent and that is certainly not my fault. We know of the numerous instances where students go though 12 years of school only to have rote skills that are not unlike the conditioned behaviors that make up the majority of all of our lives. We know that even after showing them the pointers, showing them the examples and having them drill, they still don't get it, very much like the analogy to shooting baskets. >
Which means they were probably never turned on enough to wanna practice alone, even during free time.
Where the teacher is role modeling, enthusiasm may prove infectious (may not), and Johnny goes off to study in earnest, because the payoff seems there.
The teacher is not trying to "force" Johnny to get it. In fact, said role modeling teacher may be on Youtube and Johnny has as much chance of meeting her personally as he has of meeting Britney Spears.
There's a popular spelling teacher on Youtube, among 15 year old boys especially, whom you might have learned "antidisestablishmentarian" from (a word I used recently in one of my blog posts).
> But let's assume that reasoning is a learnable ability. I think I would be safe in saying that it (reasoning) would be one of those abilities that once learned are unforgettable (like riding a bike). I point that out because since we have so many examples of students in high school mathematics that seem to lack the ability to reason (their work is rote at best) then they must have lacked the ability all along or at least not learned it at any point prior.
Or they may be "playing dumb" in some cases, because they're unimpressed by the management and don't want to be seen, by other students, as having sold out to some idiocracy that has no business being in authority.
Back to this visiting student, the corporate refugee, there's no question but she's a genius. Yet she didn't want to enslave herself to some airplane company that flies the likes of Rush Limbaugh around, if ya catch my drift.
Some kids withhold their participation because the school seems corrupt, is abusive, isn't really about teaching useful skills. It's about social control and sending the message they're failures, that they just "don't get it".
I've seen schools in Florida where I'd have been sympathetic if the kids just didn't show up. What's the point? No SQL, no hi-def TV. It's just not funny anymore, to live in such squalid poverty, within eye sight of Cape Canaveral and EPCOT. Here's the America I'm talking about (a Danish production):
Note: I'm not saying Oregon is poverty free. On the contrary. For one thing, we don't get all the defense money that Florida does, nor all the retirees (our climate is not known for its perpetual sunniness).
> This is the motive behind "concept" driven curriculums that profess to teach students to think rather than do. But those have been around for quite some time, certainly long enough to see an effect and if there has been any effect it has been negative (as shown by declining test scores). And the "concept" curriculums have the same problem any other curriculum has, they operate on the principal of "fire and forget". They assume that if you teach in a certain way then the results are a given. Obviously, the results are never !
I don't look at test scores so much as the quality of the professors (of whatever discipline) as a measure of a curriculum's worthiness. Are they up to date in their field?
In the case of math professors, you wanna know what they know about space-filling tessellations, especially the tetrahedral ones. That's a hot topic and if all the professor knows is calculus, well that's a problem.
If the end result is all hyper-specialists, no polymaths, no generalists, no liberal arts types, then your pipeline is broken. Time to agitate and subvert, and to suggest full tuition refunds for any put through it so far, as clearly they were ripped off. Debates welcome, pro and con.
"Unemployment benefits" should be called "scholarships" and the work / study programs they get you access to should be more rewarding then just mailing your resume to private security firms that'd send you to Afghanistan. Infrastructural improvements right here at home could be a part of the curriculum. Learn how to make cooking shows using produce from urban farms? That's closer to what our BFI might fund, why not Uncle Sam (or is the defense lobby against civilians having food security? -- which side are they on?).
> a "given" regardless of the curriculum and this is the basis of all this testing. Since we cannot assume results just because we used the "right" curriculum, we must authenticate the results with tests. But look what happened. The (NCLB) tests and the cutoff scores are almost all frauds. The only widespread tests that have authentically measured reasoning ability are the SAT and ACT but they are given at the end of school. >
The tests themselves need to be authenticated, compared with standards. It's chicken-egg as if your standards are bad (e.g. no SQL, no space-filling tessellations) then your tests will simply validate (but not authenticate) these dubious design decisions. It's like quality control at a meat packing plant, where all you look for is whether the price is high enough. Whatever happened to pride in one's work?
One of the positive impacts of sputnik was it reminded Americans of competition on the block and almost overnight, many former goof-offs had sufficient reason to hit the books, start boning up.
Everyone wanted to be a nuclear physicist all of a sudden, the next Einstein perhaps.
New Math pandered to these desires, helped many become more like Bertrand Russell (a little propositional calculus, a little Boolean logic... nothing the "one room school house" had featured when agriculture was the only game).
Now though, nationalist reactions don't work. Japan's Venus probe is due to enter orbit in December, but science has internationalized. The USA no longer has the capability to launch humans into space, but then Energia could well open offices in Portland and employ faculty at OGI, just as NASA could have facilities in Siberia or the Stans.
So what do we have as levers anymore, if not nationalist fervor and a fear of communism? Is the terrorism thing working out? All I see is decaying infrastructure and schools falling further and further behind (universities included).
> If you are really sincere about getting students to "think" and thereby really get the material then you will have to test for this reasoning ability (not the skills) early on and be very honest with the results. The nice thing about reasoning tests is that they don't require many questions. And when you find students that are not reasoning (and this is very obvious even without a test) you will probably have to work with them individually and intensely to get them to experience the feeling of reasoning, to become personally aware of it. >
Might require Ritalin or Prozac in some cases, judging from how Americans do it today. Or do you expect a revolution?
> This is basically what I do with my son. We have played around with fractions but he didn't really understand them (nor did I expect him to) but the other night we were working with multiplication and some simple division (actually un-multiplication) and I was asking him what half of 16 is, and half of 10 and so forth. Then, just for the hell of it, drew a number line on the whiteboard and asked him what half of 6 is and had him show me on the number line. Then half of 4. And then... I asked him what is half of 3? His first response after some confusion was that there is no number (there) but of course I kept pressing him to think about it. After a minute or so he blurted out "one and a half!". I really didn't expect him to get it that soon and he caught me off guard. Naturally, you want to extend this new notion of "halves" and ask what is half of 5 or 7, but i also realized that this was more than just getting the concept of "halves", this was actually one of his first tru!
I want high school students to *at least have the opportunity* to write a Rational Number type using operator overloading in Python (for credit, as a part of their accredited work).
That Litvins text starts down that path in one of the exercises but Euclid's algorithm for the GCD hasn't yet been introduced (comes later in the RSA chapter), plus Gary and Maria aren't that interested in my "math objects" approach (object-oriented).
> e "reasoning" experiences and I decided to focus more on developing his awareness of how he did what he did rather than what he did. You need the skills and concepts because you need material to reason with (it's the chicken and egg paradox) but the critical piece, what I call the "catalyst" in the other post, you have to tease out of the student and enable the student to own it and become aware of it. >
When you have 40 students in each class, some with parents at home, others in a boarding situation, parents back in Japan, you won't be able to take this individualized approach except maybe in helping with the exercises.
That's why your role as a role model is important. You're more like a stage performer or on-camera studio talent in that case, or more like a behind-the-scenes editor in my case (I'm not a camera hog, just a few Youtubes show my mug, mostly in costume so I might be mistaken for Barney or Big Bird? -- neither wears a hat (so Evil Bert?)).
> In any event, I don't think you can "teach" reasoning, at least not in any direct sense. If there is success to be had you must tease it out of the student. Some (like Bill) seem to equate reasoning with axioms and logic but I don't think that is the neurological basis of reasoning. I think reasoning is a very personal experience and you must become aware of what it feels like and master that feeling. Like any other highly developed skill. You can't just tell a student how to do it or give them pointers. Doing so reminds me of all the times a coach told me to keep my eye on the ball. I would think to myself "What the heck do you think I am doing? Closing my eyes? Of course I am keeping my eye on the ball!" That was until I saw videos like this one ... >
I think your notion that "reasoning is a feeling" is rather quirky.
You never mention the "left brain" versus "right brain" stuff, which even if somewhat metaphorical at times, is rich in supplying an opposition. What your theory lacks is a strong plot line, as what's the opposite of "reasoning" in your mythos?
I think you'll cast something insipid as a dramatic foil, like a sense of failure, whereas you should have this arty intuitive side that's capable of earning big bucks in some circles.
Students get talked out of being hyper rational and geeky by the prospect of being a rock star for example. Yet really being a musician takes lots of talent and practice, and actually some kind of reasoning, as writing chords, crafting lyrics, is a rhythmical computation. There's also the business of amplification, MP3s, podcasts, iTunes, publicity and labeling (branding)... before ya know it, you're really having to think!
My point: we shouldn't put "reasoning" on a pedestal as the only object to idolize. Other virtues matter, and traditional schooling has been about those as well.
I can't tell you how much of my 3rd grade was about teachers haranguing us on what it means to be rude, such as not giving your seat to an older person on the bus, or to a woman with a child. Being selfish and thoughtless was what needed to be whipped out of us (sometimes literally).
But this was a school for future English (as in citizens of the UK), not really the same model (we also studied the Bible a lot).
In the USA, the schools are not supposed to teach ethics of any kind apparently, and it shows. Hard to find a ruder, more backward nation, when it comes to manners, though I expect one could find some. Probably varies by state quite a bit too.
> http://www.youtube.com/watch?v=QvZ7prb43Lk&feature=youtube_gdata_player > > My first thought was "Ok, so I wasn't actually keeping my eye on the ball" and my second thought was "How the heck does Federer do that!" I have good reflexes but I never had the feeling of what he is experiencing in that video. In fact, If you asked me before seeing videos like that, I would have said that coaches just mean "pay attention" when they say "keep your eye on the ball" because no one can literally keep their eye on the ball at that speed. Well, obviously I was wrong since Federer is keeping his eyes on 100 mph balls as if they were going 10 mph. In fact, you need slow motion video just to watch his head, let alone the ball. And most of the reason that I didn't realize what "keeping your eye on the ball" really meant was that I hadn't experienced anything like it. Reasoning is the same way, it must be experienced in order to even be considered. Unfortunately, we don't have slow motion videos of reasoning. >
I think Einstein's world view pertains.
Some people who consider themselves lightning fast thinkers, actually come across as slow-as-molasses pendants in other frames of reference. A "speed o light" whiz kid in his local community, goes to the big city and finds there are as many ways to be slow as to be fast.
Likewise "reasoning" as in "thinking" means different things to different people.
Some would say it's "unthinking" to quit one's job in a successful engineering company in order to fight against some dominant (yet idiotic) paradigm. Others, with more sense of a time line, would say we're forever grateful to role models who'd do this in the past, as sometimes the status quo is just corrupt and doesn't deserve the talents of we the people. Says so right in the Constitution.
Back to my local scene, I'm still persuaded that we won't have our bonanza economy (the one Willamette Week hopes we'll have) if we kowtow to ETS and wait for some computerized testing service in Princeton, New Jersey to tell us what our math and science standards should be.
That's like going to a horse and buggy factory for a new transmission for one's hybrid car.
Oregon has to pioneer, per our destiny, and if there's a role model in the picture it's more likely Asian than anything east coast, i.e. I see no examples worth imitating anywhere east of the Mississippi.
> My point to this long post, aside from the beginning talking about "what is abstract", is that reasoning (the human form, not the Bill form) is, like every other talent-like skill, not a given and will not spontaneously appear as the result of any curriculum, traditional or not. Some students obviously bring it with them from the get go but for those that do not, if it is possible, then you will have to tease it out of them and make them aware of what it feels like and coach them to master it. Is this doable? The data suggest to me that it isn't doable to the extent that every student will master algebra 2. I don't know why it works out like this, I just know that the distribution is what it is. That doesn't mean that I don't think students' ability to reason cannot be improved, but if that is to happen then you will really have to look at it closely in the primary grades, stop lying about it, and create authentic tests that prove you are achieving it. >
And I'm thinking algebra 2, as mostly spoon fed around the nation, is another "mystery meat" long overdue for a redesign.
Rather than challenge the existing math track however, which I consider broken and moribund, I've agitated for a bright shiny new *parallel track* that starts to get beyond calculators and beyond flatland (they go together).
I've not been working in a vacuum and many share my vision. If I received an Academy Award, I'd have that proverbial long litany of co-workers and supporters to recite, with appreciation.