Our high school principal explains the centrality of modeling to high school math:
55:23 This is a very important slide and one that you’ll hear me talk about a number of times.This strikes me as fundamentally wrong, but if you guys tell me it's sound, I'll have to revise my view.
Because modeling, we really look at modeling as the way to really permeate through all of the different levels of mathematics at the high school level. And we really look at it from a standpoint of pedagogy. When we talk about modeling, what we’re really talking about is the conceptual side of mathematics. Recently, and there’s been a shift to a very computational format for teaching mathematics, especially at the high school level. And that’s where we would start to break things down and scaffold them into very fine points. But what we have found is mathematics teachers over the last 10, 15, 20 years, when this pattern was happening, was that students were starting to learn their broader understandings of mathematics. 56:19 There was a big need to pull back and get back to the point of teaching to deeper understanding and to the conceptualization of math, not just about being able to compute the correct answer. So modeling we really look at as the link to be able to do that. It’s the opportunity to create real-life problem-solving situations where students need to understand the conceptualization of what’s going on in the math as well as how it relates to the real world.
Geometry, obviously, is when we start talking about shapes and sizes and the relative position of objects, and statistics and probability gives us the opportunity to start looking at mathematics and creating analysis and really looking into the chances of opportunity and things occurring.
57:07 So again, as I mentioned, modeling becomes a real important focal point for us. And the phrase that we’ve been talking about is it becomes this umbrella for us. It’s the umbrella that brings the whole mathematics curriculum at the high school level together, and a way for us to keep progressing through and thinking about how it matches up. So when we think about constantly naming and reinforcing the work that the students are doing we want to constantly bring them back as well to these broader-scale concepts. 57:39 So this slide and the next slide starts to talk about that even within those conceptual designs that I mentioned before, even within algebra, there’s an aspect of modeling that’s critical and important for them to understand in the algebra as well as the other mathematical concepts within.
Similarly you have functions here, and again, there are pieces of it that we pull out and we understand how do we create real-life conceptualization and contextualization for our students so that when they’re working through this, they understand again not just the specific calculation of an equation or formula but what it really relates to.
Similarly we do the same things in geometry and we do the same things in statistics and probability. Again, for me, this is about teaching to big ideas and perspective. We’ve been talking at the school about deep understanding and I said that would be one of those shifts that we keep coming back to, and I think that that’s really one of the most important messages that we can deliver about the mathematics instruction and how the Common Core starts to create a shift for us.
Irvington UFSD School Board Meeting - February 11th, 2014
My understanding of math, of what math is, is that …. mathematics is not essentially, or even first and foremost, a system for representing empirical reality. The fact that math so powerfully -- and so eternally -- does capture many aspects of empirical reality is, in my view, either a) beside the point, or b) creepy.
Math, as I think of math, has a mathiness that cannot be reduced to modeling; math is a thing unto itself and should be taught as a thing unto itself -- or, at least, students should be made aware of the fact that to a mathematician math is not just a code-writing tool.
(Again, setting aside the possibility that math is just a code-writing tool.)
I'm glad no one is making transcripts of my speech-tics...
ReplyDeleteFor me, it's difficult to say much of anything based on the things said in that speech. What you need are examples given by math teachers of lessons that they have (really) done. I think that's the only way to figure out what is meant by the gobbledygook. Could be that you (really) approve of what they're doing, or it could be that it seems like nonsense or no different from what has been happening already.
While I'm not too keen on real-life or context, I don't have any bias at all against modeling. But my working definition of modeling is any way of representing a problem/situation.
ummm - or make that any useful representation
ReplyDeleteI'm not at all opposed to modeling, but I **am** opposed to the disciplines being taught as purely the handmaiden of "real-world problem solving."
ReplyDeleteMath seems to have lost standing as a subject unto itself.
Of course, that can be said for all of the disciplines as taught by constructivists.
A transcript of me talking in my class would be FILLED with abrupt halts & reverse-courses, but it wouldn't be filled with gobbledygook. This is ridiculous, and this is the way our very well paid administrators talk to the board.
ReplyDeleteThe board NEVER HAS A CLUE what's going on.
I've got a couple of work samples -- I wish Blogger would let me post pdf files.
I'm definitely against teaching geometry as the study of 'shapes' and the relative position of objects.
And I would like, just once, to hear the word "proof" in a discussion of Common Core math.
I'm glad no one is making transcripts of my speech-tics…
ReplyDeleteI left out the speech-tics!
Is this a curriculum pedagogy statement from the math department? Who signed off on it?
ReplyDelete"We?"
Does the principal have a math degree?
This is just so much vague crap. Of course what they mean is that they want to approach mastery of skills from a top-down direction rather than trying to improve context and big picture ideas from the bottom-up. They think that skills are only rote.
Proper mathematical understanding needs to be built from a full understanding and mastery of skills. There are many layers of understanding and many of those layers are tied directly to mastery of skills. There is no such thing as being successful in math with rote skills unless the teachers are incompetent.
Since they are talking about high school, find out what textbooks they are using and which ones they are planning to change to. Are these changes for those on the AP calculus track or just for others? Is this being done for Common Core when Common Core specifically does NOT deal with STEM careers?
My position on this gives mathematicians the vapors.
ReplyDeleteMath has a privileged position in school curricula around the world. It's one of the "three Rs." No matter how tight the budget, math will not be eliminated.
How does it earn this privilege? By being beautiful? I see the beauty, but most educators see more beauty in art or music, and yet when unable to pay for more than two of math, art, and music, math will never be the one they eliminate.
Is it because it is fascinating? Again, it is to me, but less so than the deep natures of time and space, causality, and reality that I find in physics. But that's a matter of taste, and most educators are probably more interested in history than physics. Regardless, both science and history are eliminated before math. You'll close the school before you'll eliminate math.
Math is so privileged in the curriculum because it is so *useful* in so many fields *outside of math*. Even a drug-dealing dropout needs to learn enough math before dropping out to keep the accounting straight or there could be fatal misunderstandings.
And as you get higher and higher in math, it becomes both more beautiful and less generally useful, which makes it increasingly resemble other subjects at budgeting time. Calculus is more likely to be cut than music.
The more you make math the study of math rather than the development of useful tools for the rest of life, the less privilege math will deserve in the curriculum.
[Inevitably, this position will be misunderstood as advocating less math, when I advocate more; or advocating a mindless/procedural/no-understanding approach, when I say that tools you don't understand are tools you'll forget how to use; or advocating a teaching approach that ignores mathematical correctness, when I say that such approaches ultimately interfere with understanding, thus reducing usefulness.]
Catherine-
ReplyDeleteYou are describing what the Learning Technology Center at Vandy called anchored instruction when they developed it in the 80s. It's anchored in both the real world application and the desired conceptual understanding to be used for all problems that fit a given context.
It is also based on Piotr Galperin's work from the USSR. It's called theoretical-systemic instruction.
Under the Common Core, besides basic skills, math becomes simply a means of seeing desired relationships like some and few and more and less (part vocab lesson). You do the learning task as a group interaction or project. That becomes the assessment and then the cross-check is to see what math concepts the student applies to untaught or ambiguous problems.
I am laughing my head off. This is typical of a high school principal or other administrator trying to explain a deep concept that he or she knows nothing about. I teach computer science. You should hear it when our administrators try to explain what it is that I teach to others. Oy, oy,oy, Anyway, I wouldn't really try to parse this. What are the math teachers actually doing and saying?
ReplyDeleteRight, Froggiemama. This is like a tone-deaf principal trying to describe her high school's music program, or one who speaks only English trying to describe the foreign language program. Not to be taken seriously. The math program may indeed be this goofy, or it may be fine. There is no way to know, but we do know that the math department gets not much help from this principal.
ReplyDeleteIn my years as a math teacher, I have found that "modeling" is a code word that means "graphing calculators and word problems."
ReplyDeleteThe nice people at Texas Instruments will tell you how great the modeling approach is. They will come and teach a workshop on how great it is!
Bingo, Froggiemama!
ReplyDeleteAlso Anonymous!
Robyne - YET ANOTHER FORM OF BADNESS TO WORRY ABOUT. sigh.
Glen - I'm sure you're right, but I still don't want to hear that geometry is about shapes.
I want to hear the word "proof" AT LEAST ONCE in a pretentious, jargon-filled presentation on "deep" conceptual understanding.
The other issue re: fantastically pretentious verbiage no one can understand, at least here in my district, is that fantastically pretentious verbiage no one can understand has been used as a weapon against the board, against parents, and against taxpayers for years.
ReplyDeleteAs to whether the math program is or is not decent, here is my take -- and this is a 'take' as opposed to a fact.
ReplyDeleteI'm pretty sure the math department managed to fend off constructivist math for a good 10 years, and I don't think they're going to stop fending it off now that we have a new principal who thinks modeling is the essence of Common Core.
That said, I hear mixed reviews of the teaching in the math department, and of course we ended up leaving the district heavily because of the then-math chair....but the then-math chair wasn't a constructivist. Far from it (I think).
Irvington had for many years a very good reputation as a math school that I'm pretty sure was deserved. I'm also pretty sure that the math department had the wherewithal to say 'no' to the schemes of central administration.
Interestingly, when the high school presentation was introduced by our current (& outgoing curriculum director) her introduction basically said that the high school has been getting off scot free because kids in K-6 have to take state tests & high school kids only have to take Regents.
I found that **very** interesting; it seemed pretty clear to me that she was voicing a real sense of frustration that the high school never came around for any of the constructivist projects she helped bring into K-5.
Here you go: Tri-State Consortium 0, Math Department 1
ReplyDeleteDoes your high school have Algebra I, Geometry, Algebra II, Pre-Calculus, and (AP) Calculus? What textbook series do they use? Are the classes directly taught or do they use some specifically different approach? If they are currently using a direct teaching approach, is the administration pushing for a change in textbooks/pedagogy?
ReplyDeleteYou have to get past what comes out of their mouths and look at what actually is going on in the classroom. Ask the parents of students who know something about college STEM preparation to see what they have to do at home. Does your school have statistics on AP Calculus numbers and scores. How about SAT II Math 1 and 2? Of course, those numbers could be supported by parents at home, but it is a starting point of reference.
I am glad there are still places where teachers feel free to teach content without having their jobs threatened. Apart from tracking this by the declared intentions in the reports, I am in a suburban area where a huge % of principals have been replaced in 2 years. I call it the gypsy principal-gypsy super phenomenon where everyone knows their lucrative promotions are tied to pushing bad ideas.
ReplyDeleteThe suburban districts in many areas are pushing Ed Leader 21 or the new Consortium where they adopt the same humanistic psychology focus that destroyed the urban districts.
It is my fondest hope that as many parents and teachers as possible do not see the full force of what is going on for as long as possible. It means there are still islands where knowledge is the focus instead of changing behavior and beliefs.
Most of the high school math teaching that I've seen has not been constructivist, though the quality has been very mixed. Sometimes sounds like there is more constructivism on lower math tracks.
ReplyDeleteI came across a recent post on different instructional philosophies by a high school math teacherEducationrealist blog: math instruction philosophies and thought it might be of interest.
From "Cryptonomicon" by Neal Stephenson:
ReplyDelete----
[Alan {Turing} said,] “But Lawrence—when you really do math, in an abstract way, you’re not counting bottlecaps, are you?”
“I’m not counting anything.”
[…]
“There was an implicit belief, for a long time, that math was a sort of physics of bottlecaps. That any mathematical operation you could do on paper, not matter how complicated, could be reduced—in theory, anyway—to messing about with actual physical counters, such as bottlecaps, in the real world… It was believed that Euclid’s geometry was just a kind of physics, that his lines and so on represented properties in the physical world….
“It’s like this: when mathematicians began fooling around with things like the square root of negative one,…, then they were no longer dealing with things that you could translate into sticks and bottlecaps. And yet, they were still getting sound results.”
“Or at least internally consistent results,” Rudy said.
“Okay. Meaning that math was more than a physics of bottlecaps,” [said Lawrence]
Math (IMO) is best described dialectically: "pure" and "applied" math are both essential components of the discipline, pointing in opposite directions, and that opposition is one of the driving forces that pushes mathematics forward. Folks who think that only one or the other is important make my skin itch.
ReplyDeleteI don't think that dialectic is being illustrated here. The dialectic here is when one tribe, call them the "fakers", tries to talk about the work of another -- call them the "practitioners" -- work that they really don't understand. Yet they claim the authority to make decisions about the work of the practitioners. They talk about it in weird abstract ways that no one in the practitioners' tribe ever uses. They are like blind people discussing painting. They assume that it's a thing -- and they can natter on about helping art students to expand their conceptualizations beyond the choosing of "colors" to apply to spaces. You can almost forget that they don't really have any idea about what they are talking about. Then they reveal themselves when they talk about math involving "the calculation of a formula" or when they describe geometry as being about "the shapes and sizes and relative positions of objects." Can you imagine these phrases being spoken by anyone from the practitioner tribe?
ReplyDeleteI call it brain research misdirection. They do it on purpose. The "practitioners" are tricked into a discussion as equals. Actually, it's worse than that. Here they are talking about critical thinking and understanding, and here I am talking about mastery of the basics. You would think that educators would wonder why that's the case considering how well my son has done in math. They are disagreeing with so many experts in the field. It should give them pause. No, because that's all they have. They just find a few practitioners who seem to buy into their vague talk of understanding.
ReplyDeleteIt's even worse than that. Common Core specifically does NOT provide a curriculum path for STEM. Students can't get there without outside help in spite of all of their talk about critical thinking and understanding skills. Where do these STEM students come from? My hand is raised. Oh, that's right. I only want what I had when I was growing up. Blah, blah, woof, woof.
Thx for the post! Unfortunately, this sort of rhetoric is typical in PD sessions.
ReplyDeleteHere's my take - for what it's worth! :)
Mathematics exists far beyond our ability (or current need) to use it in modeling. This is what research mathematicians do - as far as I understand it. They discover mathematics. It exists prior to our having terms to describe it. Researchers define those words as they determine how their discoveries fit into our existing framework. (Sometimes realizing that those definitions were less than "ideal" as new discoveries are made.)
Anyway, back to schooling... and the beloved "modeling" - there's a nice write-up (imo), by a teacher who seems to understand the big picture, on mathematical modeling and how it relates to educating children over at Math is Fun! Examples are provided too.
http://www.mathsisfun.com/algebra/mathematical-models.html
That principal should go there!! Please! (yes, I'm begging!)
Of course, mathematics is useful! But its beauty does not depend on its usefulness. (i.e. "beauty" is the INDEPENDENT variable :)
Lisa