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Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical,
subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or
physics (mathematicians conceived of black holes long before astronomers actually found any),
and allows more freedom of expression than poetry, art, or music (which depend heavily on
properties of the physical universe). Mathematics is the purest of the arts, as well as the most
misunderstood.
KestrelLowing wrote:Math is a difficult subject, but no more difficult than any other.
KestrelLowing wrote:However, it typically has been able to stay away from grade inflation due to the pretty much right/wrong nature of most math subjects. Is this the major reason math is seen as so difficult?
KestrelLowing wrote:One thought I've had is that from the very beginning, math is not always taught by the best people to teach math. Elementary school teacher are notorious for not being that great in math (although this is obviously not always the case). This causes math classes that are very formulaic and not terribly insightful as the teachers themselves learned math in a formulaic manner without much meaning behind the method. It could be that many teachers during those years when people decide what their favorite subject is are not particularly fond of math themselves and pass that onto their students.
doogly wrote:Ah, the solution to problems with math is to do less math. Wonderful, wonderful.
doogly wrote:I think if you offer math teachers a little more salary and freedom in the class room, you can attract more and better mathematicians. People who can impart some sense of the joy, can share it and for starters, experience it themselves!
markop2003 wrote:From what i've seen most complaints are not about it being hard but more about it being useless.
Ubik wrote:Hear the volume of my bladder through the volume of the splatter!
doogly wrote:Standardized testing exists, for better or worse, but I still wouldn't teach to it. It can just happen.
doogly wrote:Ah, the solution to problems with math is to do less math. Wonderful, wonderful.
Bakemaster wrote:As far as experiencing and imparting joy, it doesn't matter how excited or animated you are about math unless one of the things you're sharing with your students is the ability to perform highly on quantitative assessments.
Magnanimous wrote:(This starts at approximately algebra, I think.)
Bakemaster wrote:That attitude absolutely reeks of privilege.
Vangor wrote:
Calculation and abstraction needs to be dismissed. We have ubiquitous computers and a century of developmental study. What students need to understand is when we use a formula or process and how to gather the numbers, and this should be done by gathering numbers from the world. This is not to dismiss the usefulness of mental calculation for estimation purposes, essential to using computers properly, but something tells me professionals steeped in mathematics are not busy calculating by hand as we make students do, and those professionals have something real to work with besides.
KestrelLowing wrote:Ah, here's the video, it's a ted talk. It's really good if I remember correctly.
Vangor wrote:Bakemaster wrote:That attitude absolutely reeks of privilege.
He is not wrong in general. Some test strategies are useful, but modeling the curriculum around the assessment with the district or state or whatever provided summative assessments and worksheets and materials in general is ineffective in several ways for development and less effective than designing proper experiences.
Bakemaster wrote:He's not wrong in theory. In reality, the freedom to teach material in the best way for the deepest understanding is a privilege afforded only when teaching in certain environments. Your students have to have a certain level of preparation for the material, which is contingent on the quality of every teaching experience they have had before entering your classroom, which is contingent not only on the quality of their past teachers but also on resources, funding, curriculum, home environment, and so forth every year they've been in school.
As teachers encounter more and greater limitations on the time and resources available to them and on the aptitude or preparation of their students, teaching to the test becomes more and more attractive. What it's not is a good way to learn math; what it is is the fastest route to a student being able to demonstrate proficiency on the next test. It hurts the student's education in the long run, but the system is not designed in such a way that it effectively holds teachers accountable for their students' performance a few years down the road. It is designed to pressure teachers to produce quantifiable results for a group of students within the span of a single year or term. And it only takes a few weak links in the chain to undermine students' preparation to the point that their next instructor is going to have to cut a few more corners and gloss over a few more topics to continue to meet his or her standards for performance.
The value of being able demonstrate proficiency on the next test is further exaggerated when passing the test is a requirement for high school graduation (e.g. MCAS). If you have the luxury and privilege of teaching kids in honors and AP courses, who are expected to pass with ease, you can pooh-pooh test prep and turn up your nose at the thought of teaching to the test. Not so when you're working in an environment where your students are expected to fail. In such an environment, having the diploma is what's truly important to the student, not a rigorous, intuitive understanding of mathematical concepts. Having the diploma is what comes first because without it, no amount of excitement or authenticity is likely to break the cycle of poverty that prevents these children from realizing the successes and opportunities of the "good" students.
In all of this, I'm not trying to argue against making math exciting and authentic and favoring conceptual learning over rote learning. I'm simply suggesting that the benefits of this approach are disproportionately realized by the students with the least need, which is to say, those students who have yet to fall particularly far behind. There are few exciting, conceptual math topics to cover until after basic arithmetic, by which point the damage has already been done for the students with the most need.
KestrelLowing wrote:Actually, I think abstraction is absolutely the most useful portion of math - to take that abstraction, know how to use it, and realize that things can be used in more than one case. Humans are awesome in that regard. For example, when working with dogs, you have to 'proof' the behavior. Say you're teaching a dog to sit. If you teach that inside in your living room and nowhere else, chances are they won't sit anywhere else. You have to teach them in each place. While dogs will pick up on it quicker, they still need to be taught how to do it in multiple environments.
KestrelLowing wrote:Ah, here's the video, it's a ted talk. It's really good if I remember correctly.
Bakemaster wrote:In reality
Bakemaster wrote:what it is is the fastest route to a student being able to demonstrate proficiency on the next test.
Bakemaster wrote:I'm simply suggesting that the benefits of this approach are disproportionately realized by the students with the least need, which is to say, those students who have yet to fall particularly far behind.
Bakemaster wrote:There are few exciting, conceptual math topics to cover until after basic arithmetic, by which point the damage has already been done for the students with the most need.
Bakemaster wrote:(Who is Alia?)
Bakemaster wrote:For one group of students studying conceptual physics in particular, I was able to spend a good amount of time doing things in what I felt was the "right" way, but of course the entire nature of the course was conceptual.
Bakemaster wrote:But for many of the students with whom I worked on algebra and trig, it often seemed that an interesting presentation only inhibited their ability to solve the problems they were assigned.
Bakemaster wrote:So, if I had been their instructor instead of their tutor, what would have been different? This is not a rhetorical question. Instead of having an hour or two of one-on-one time with a student each week, I would spend three to five hours a week teaching several dozen students at once.
Bakemaster wrote:If this is a group of students who have never been on an honors track, most of whom are struggling both in-school and out, what's the best favor their teacher can do them? How many deficiencies in prerequisite material can any teacher possibly repair in the course of a term while also covering a full term's worth of new material
Vangor wrote:Should differ between "interesting" for the sake of generating interest and "interesting" by nature of developmentally appropriate and active. Teachers, sadly, incorporate a whole mess of fluff in order to generate interest in lessons which remain developmentally inappropriate and passive. This is the basis of the discussion, for me at least; math is scary due to being developmentally inappropriate and passive for the learner.
doogly wrote:You just called trig higher level math. I lolled.
But anyway, yes, that seems to make sense. But what kind of math class isn't full of examples from physics, biology, economics, and of course the joys of pure math? There ought to be mutual reinforcement, not subjugation of one of the fields.
doogly wrote:You just called trig higher level math. I lolled.
Zamfir wrote:doogly wrote:You just called trig higher level math. I lolled.
Then again, the far majority of people will never use trigonometry in their lives outside of high school tests. Even people who got a good grip on it in high school will mostly ignore it for the next years, and when they ever enocunter a situation where it could come in handy, they have forgotten too much of the details. So perhaps it's "lower maths", but there's not really a higher-than-trig maths for most people.
It's very tempting to see high school maths as an introduction, a preparation for the real stuff later on. Typical high school curricula are filled with that stuff: trigonometry, calculus, logarithms, they're taught because they are bread-and-butter for hard-science people. But not the obvious things to teach as last maths that someone will study.
And as far as I can tell, people catch on to that pretty quickly and accurately. If they have to study intro-math-for-mathy-people, it makes perfect sense to see the subject as pointless.
There might be a similar risk in using physics as guidance for maths. There's a lot of physics that's important for people to know about, but being able to calculate stuff is far less important. If you turn physics into a calculation class, it becomes less interesting for people who want to have a grip on how the world around them works conceptually, in favour of people who are preparing for a more in-depth schooling after high school.
Turiski wrote:So that's what we want of Computation; what do we want in Mathematics? Proofs, of course! And not Geometry-style proofs (Ah, hello there, Lockhart!); maybe later we can talk about that level of rigor, but not as a first introduction. For now, let's focus on games, let's talk about puzzles, let's learn some number tricks, then figure out why they work. We can teach proof techniques, maybe some logic. If I had (a lot) more time, I might want to think about curriculum, because it would obviously have to be done very carefully, to be developmentally appropriate - even if it's an elective, it still has to be effective.
** It occurs to me that it might be less arrogant to call it "Logic" instead of "Mathematics" but I definitely don't know enough of the history or psychology of the situation to say much.
KestrelLowing wrote:I guess I never really thought that people loose interest at algebra - that's when I finally started being interested. That's when it actually seemed real-world to me. Of course, I'm currently an engineering student, so that's obviously a huge bias here.
maxh wrote:Now we're in algebra. Some things make sense (I spend 5$ a month to get 100 texts and each text after that costs 5¢) but others... not so much.
Byrel wrote:You know, I hadn't really thought about the pure mathematics when I suggested totally eliminating it. You are right; there is a place for number theory, axiomatic geometry, game theory, etc. Perhaps some such elective should be available. But this math is completely distinct from what will be useful in most technical backgrounds. The real reason most people learn Algebra, Trig, Geometry (aside from being forced to), is not to then reason about the results. It is to use them. This math is the core of practical science and engineering.
Byrel wrote:So, I guess I agree that a Math (or Logic) course should be maintained, but I have to disagree at where to draw the line. Introductory Algebra? Algebra is at the core of any math-centric field. Algebra, with trig identities and calculus stacked on top. If you want to separate useful math from pure math, then the useful math has to include algebra and trig at a minimum.
Byrel wrote:On the other hand, this is still teaching people math that won't be necessary for their occupation. (How many lawyers need algebra? Burger-flippers? Truck drivers?) Maybe we should split math into two parts, Arithmetic (fractions, etc.), Logic (pure math, number theory, etc.) and absorb the classes that are direct STE prep into the science classes (particularly Physics).
Byrel wrote:On the other, other hand, maybe teaching people math they won't need will be helpful. As big boss pointed out, a lot of high school is not designed to prepare us specifically for our careers. My dad went so far as to say the goal of highschool was a trained mind; not really any of the knowledge in it.
Byrel wrote:Certainly most college students I knew that had trouble in math-based classes didn't struggle with the math; they struggled with problem solving skills, when applying it to math. They could recite all the formulas, properties, integrals. (As Zamfir noted, d(x^n)/dx is trivial.) But they still got Fs. Why? Because they couldn't solve the problems.
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