On Tue, Dec 24, 2013 at 3:30 PM, Robert Hansen <email@example.com> wrote:
> > You seem to always shoulder students with the burden of suffering a math > deficit, whereas it may be the school's curriculum and/or teachers and/or > the whole culture of a high school that starts closing these doors. > > That?s not true. Most of my years here were spent studying curriculums, in > detail. I blame the schools and curriculums for not nurturing the students > who are good in math. The only students that actually continue on after > school and use and apply this gift. The little bit of math that the other > students pickup is soon forgotten. >
I don't think what we call "math" in K-12 should at all try to style itself as some miniaturized version of what a college math major might study. It's its own animal.
Numeric literacy includes using measurements and currency, telling time, arithmetic, appreciation for data visualization techniques and conventions, familiarity with rounding / rounding error / significant digits, an understanding of statistical arguments and concepts such as mean, mode and median, standard deviation (bell curve), interest, probabilities, the wave and particle nature of energy...
Then I'd add some broad brush stroke overview perspective as a navigation aid, tell some history.
This history and context part is a part normally left out I realize, but is what you'd find in a Time-Life trade book such as 'Mathematics', influential on my younger self:
I think many of those who survive the intensely dull presentation of grade schools are saved by trade books (not textbooks) such as the above, and today by stuff in the Internet, such as Mandelbrots and Mandelbulbs 'n stuff.
>> "Math dependent degree" includes a lot of them, including just about >> anything healthcare related. Maybe 3% want to be math majors, whereas 40% >> seek a math-dependent major. > > Arithmetic dependent, probably, but not math dependent. And certainly not > pure math dependent. >
Arithmetic for sure, but also using measurements and currency, telling time, appreciation for data visualization... is SQL arithmetic? I'd say not. Microsoft Access did a lot to expose cube farmers to the power of SQL in terms of pay scale. Get the DBA certification from Oracle or whatever and watch the income bump up.
When I started my training as a high school math teacher (Jersey City, 1980s), just about every math teacher I met was training to go in the other direction: towards private sector IT. We lost a huge army of math teachers to careers writing Visual Basic. It's not too late (in theory). Many of those VBers would like to jump back into teaching math, but we won't let 'em (barriers to entry are high).
Here's the paradox: computer science is a fancy extra most run-o-the-mill schools can't afford, but the high property tax areas can, so the only kids who know about XML / HTTP / CSS in any detail are privileged, get the good summer jobs and internships, join the nonprofits. People talking about the achievement gap never mention our solution / proposal in Oregon: let math teachers extend their curriculum such that all this vocational / applicable stuff stops being so elective / dispensable. Let them learn programming *for math credit* for a change, rather than making 'em burn out on calculus. I was sorry the politicians failed the IQ test and let it languish, but not surprised. Idiocracy is widespread.
> >> Some students have just never had the benefit of an optimized learning >> environment. They have every potential to catch up and surpass, and yes, >> that potential is as yet unfulfilled. Not unusual. > > I am not saying that is not true. But this is not the course I would point > that ?some? to. This course is not designed for the student that found > their calling, even if they just found their calling. Can this course help > a late student find their calling? Maybe, but I would design it quite > differently. Similar topics but very different treatment. This course lacks > a crucial element. It doesn?t prove to the student that they are really > good at this, and that is what a calling is. >
You should, design something quite differently and put your name on it. You evaluate the work of others but do you ever compete? I'm out there with my alternatives at least, however incomplete or whatever. But then I ramble through STEM, don't confine myself to the M part. For a long time I've marketed as ~M (not-mathematics -- just something quite like it).
> >> That doesn't mean it's wrong to ill advised to steer them to courses >> like Devlin?s. > > My point was that Devlin?s course wasn?t actually a ?course?. The topics > are good for the stated purpose, but the treatment is off. You?d have to > read the textbook I guess. And I?ll wait for the course in February to make > a final judgement. >
I'm looking forward to when Robert Hansen's course is one of the offerings. Why not put your insights to good use?
> >> Making sharp distinctions between pure and applied mathematics is not >> essential. > > Making sharp distinctions is always essential. And this one isn?t > complicated. I like pure math, but my colleagues and I didn?t find our > calling in solving everyday technical problems from pure math. I am talking > about engineers, software engineers, financial analysts, the list goes on. > That is why they call it ?applied? math. Can you also study pure math? Yes, > but it obviously isn?t a requirement for everyday problem solving because > the vast majority of people solving everyday problems don?t have a lick of > pure math in them. To say it is a different path is an understatement. When > Lou makes comments about engineers not knowing ?math? my first thought is > ?Who doesn?t know that?" >
Making sharp distinctions is often a complete waste of time unless you really love nit-picky religious wars that even further degenerate.
What stands out about you is you seem quite conservative in having swallowed whole many of the very bad design decisions made by your ancestors. You seem to really like the status quo. That's unusual in my neck of the woods where everyone is raging against the machine. We almost forget your kind exists.
In my view, STEM is STEM and attempts to atomize it further are usually quite misguided. What the letters stand for hints at the general domain, but if you admire polymaths like Hofstadter you have likely absorbed their oft-cited aversion to academic compartmentalization. Call it the revenge of the natural philosophers; they're snobbish against "over-specialists" though respectful of competence, in whatever form.
> >> Textbooks are very often formulaic, having you find roots of this and >> that polynomial, slopes of this and that line. What may not fall out of >> such formula-based learning is any confidence of being able to solve >> "generic problems" that life my pose. > > That depends on the textbook. Unfortunately, the textbooks of today are so > remedial in design that they never get to the enjoyable part of applying > all these new skills and insights to a compendium of problems like our > textbooks did. They are so focused on the majority of students that are not > good at math that they never get to the pace that occurs with students who > are good at math. Because of the algebra mandate, the purpose of math > education in secondary school has changed from choosing math as one of your > life's pursuits to just learning some math. And this has had a large impact > on qualified students finding their muse. It was easy in our day, you could > be good at math, like you could be good at music, you could be good at > sports. Now you can only be good at music or sports, but not math. > >
I wouldn't go back though. Don't take away my Google, my Youtube.
> >> At some point in a student's career we let them know about "numerical >> recipes" and the fact that whole books of them are out there for >> consultation. > > Let?s not paint a rosy picture where there isn?t one. Numerical recipes is > not a best seller. And schools are not turning out as many math artists as > they once were that could appreciate ?Numerical Recipes? or even understand > it. >
Knowing what algorithms apply: that was useful to me for example. I knew how to use the Internet to find Qhull, free software for finding a maximum convex hull given a smattering of points in XYZ. Just what I needed. And free. I incorporated Qhull into my studies of Waterman Polyhedrons without knowing in detail how it worked. But yet I could use it well enough to get sensible results. Quite a bit of math savvy resides in such "fitting the pieces together" skills, is what I'm saying.
> >> Those who use mathematical techniques to grapple with problems are less >> in need of great memorization skills than great research skills. > > The evidence I see is that you can?t have one without the other. There is > no shortcut to being smart and good at what you do. You need the reasoning > and you need the memorization. I have never seen someone make it with > "research skills". Those candidates stand out right away. >
I'm thinking a strong mathematician often has great research skills, including the ability to find the five or six other people in billions right now focusing on similar problems. I have this friend that calls me to talk about phi-scaled tetrahedrons, assemblies thereof, variations on that theme. It's not a conversation he can have with just anyone. People glaze over when you start talking polyhedrons at 'em. His name is David and he's the guy who invited me to Minnesota so we could go visit Magnus Wenninger, which we did. Magnus, age 91 (92?) is a big fish in that tiny world of polyhedron fanatics.
> >> Where are the cookbooks? How do you match a problem to an algorithm? >> Those are skills too, beyond proving. > > Now you are agreeing with me on the difference between applied and pure > math. But there are plenty of cookbooks out there, in everything. And I use > them all the time, but you have to be smart to know where to look and how > to apply them. Again, there is no shortcut to being smart and good at what > you do. The lack of google was never a barrier to being smart and good at > what you do before the internet, so how can it be an enabler of the same? > It isn?t. >
Just what I was saying: "smarts" includes knowing where and how.
> >> Finding the theorem and realizing it's relevance, locating the right >> recipe in the library, is all part of what one needs. Good road maps. >> Overviews. Most K-12 textbook math is fairly bereft of such concept maps. > > No. If this were true then over the years I would have seen a steady surge > in qualified candidates for the technical positions we post for. I have > seen a study surge in the breadth of expertise in the qualified candidates > but no surge at all in the number of qualified candidates. A sharp decline > in fact. >
Why? Do you think they're teaching research skills any better? The tools have improved, definitely, but some of the high school libraries I've visited actively discouraged using the Web. They call it the Free Web and point out that the most trustworthy and best information still costs money, so if you want good grades, look through stuff you have to pay for. You may think I'm kidding, but I have photographs of the posters in the school library, spinning just such falsehoods. Poor USAers, so oppressed by mis- and dis-info (and yet they consider themselves "well off"? "Developed"?).
> Think about what you are saying. > > Has not the internet made research incredibly more easy than our day? > Certainly you remember the before time. Before the internet. >
Easier yes, but are research skills being taught? My teachers (an unusual crew, mostly expats) consciously taught them and told us that's what they were doing. Concept maps, card catalog, note cards, thesis, outline, footnotes -- all stuff mathematicians need to know about, and then some. Bibliographies of bibliographies... Princeton had lots of those.
> > If road maps and research had anything to do with conquering these > technical fields, wouldn?t we have seen a surge rather than a decline? >
Not if no one is learning to drive.
> Being smart and good at what you do is the key. That is what needs to be > nurtured. Not excuses. >
Yeah, I know. I find approximately no one is as smart as they could be, but we live in a culture that expects and rewards dummies. Being smart means being lonely at parties.
But that's cultural and we have subcultures. I put a lot of hope in those. The speech and debate world, for example. Go Cleveland Cannibals!