> The Basecamp office has a row of desks out in the open which we govern by Library Rules. We also have four private work rooms.
Well it is still a bad decision. Libraries are an open work space and still manage to provide a reasonable working environment. This is less about money but more about a wrong perception of work.
I’ve moved from YouCompleteMe to Deoplete. While YCM is good, the plugin support for deoplete seems a bit more extended and easier to set up (ymmv). Maybe I just messed something up every time I tried to rework my vimrc with YCM though, it’s probably likely :D
One awesome thing about math is it is self organizing. Many topics are composed of many smaller topics. As an anecdote, we used to say people “actually learned” (high school) algebra in calculus 1 and trigonometry in calc 2/3. In those cases it was more that to solve those problems required using algebra and trig coherently.
So to learn those 1000 pages is to, to some extent, learn a core set of techniques across many different contexts. It compresses the required mental load. For the exceptions to that rule, well, you can skim over them and know enough about them to recognize their applications later.
Sure it does. If you really wanted to read and understand this book and dedicated time each day to understanding 3 pages worth of content, you could read the book in a year. Remembering 3 pages of content per day (especially in math where concepts build on each other) is really not hard.
> Remembering 3 pages of content per day (especially in math where concepts build on each other) is really not hard.
I disagree. Or rather, I think that's unsustainable. Any given three consecutive pages from Spivak's Calculus are probably doable on a daily basis. But is would be legitimately hard for most people to go through three pages of Rudin's Principles of Mathematical Analysis each day and consistently retain that information. Axler's Linear Algebra Done Right is very readable, but Halmos' Finite-Dimensional Vector Spaces will start getting just as dense as Rudin. These are difficult textbooks even when students are well-prepared for them with prerequisite courses. Terence Tao wrote two books to cover (with better exposition) what Rudin did in one. I think it would be pretty hard to read consistently three pages of Tao's Analysis I each day, before he even gets to limits.
I think you're underestimating the intellectual effort here. In my opinion, even if you're reading a math book targeted to your level, committing to reading and understanding three days of material each day would become exhausting. A typical semester is 15-16 weeks, with lectures 1 - 3 times a week, and most undergraduate courses do not actually work through the entirety of a 300 page textbook. Even at that slower pace it's not typical for most people to ace the course. If you read three pages a day and had a solid understanding of it, you'd be absolutely breezing through math courses.
In my experience students need to really step away from the material and let it percolate for a bit every so often in order to solidify their understanding. I really don't think you can partition the material into equal, bite-sized amounts each day. The learning progression doesn't tend to be that consistent or predictable.
If you assume that "run 200 miles" doesn't refer to a single run but rather to the capability of running 200 miles, the analogy works much better. If you stop training the ability to run 200 miles vanishes even more quickly than an equivalent feat of learning.
Trying to figure out the proof of a² + b² = c² by myself, without looking up the solution, was somehow exiting. Being exposed to a riddle and trying to find the solution is kinda cool.
However, not solving it after 10 minutes left me feeling a bit dumb... :)
Real (non-trivial, non-obvious) problems that someone hasn’t seen before can take hours, days, weeks, years, whole careers, or sometimes centuries to solve. Some of them later turn out to be impossible (and for many we still just don’t know).
Real math education would have students grappling with relatively open-ended problems that take significant amounts of rumination and some cleverness to solve. It would explicitly encourage/reward close critical reading, creative brainstorming, planning, strategic thinking, generalization and specialization, executive control (e.g. time management), error checking, and clarity of exposition (including when asking for help after being stuck). There would be no shame in throwing out incorrect hypotheses, asking for clarification, getting stuck on a problem, making subtle mistakes which could serve as good examples for future improvement, etc. But skill and stamina at such work must be trained slowly, starting from an early age.
The problem is that current (US) math education instead pre-chews everything, assigns students lists of exercises almost identical to what they saw someone solve before, and mostly tests memorization/recall and willingness to do the same trivial task over and over for hours despite being terribly bored, under purely extrinsic motivation.
For people used to such math homework, the standard response to a single problem which takes >5 minutes to work through is to give up.
Depends on what the goals are. I don't think most counties want a large fraction of their population becoming mathematicians. They basically want most of their population able to use math not discover new areas.
What do you think it means to “use math”? If you want to train human spreadsheets you’re wasting everyone’s time: the spreadsheet already exists as a product for electronic computers, and is much faster and more reliable than a human will be at doing huge piles of arithmetic. (But anyway, even if you wanted to train expert performance at arithmetic, the way schools currently go about this is horribly inefficient and doesn’t take into consideration research on neuroscience / psychology.)
If you just want to train people to be unthinking drones who can follow narrowly specified procedural rules without understanding their context or meaning, then I guess the current system is a relative success.
In general, the point of mathematics education in primary/secondary school is not to train future mathematicians, but to teach people important problem-solving skills. The same skills are (to some extent anyway) useful in essentially any field you might name, from childcare to plumbing to legal analysis to fine art.
In particular: self-confidence that hard problems can be tackled and that anything one person can do another typically can also with training and effort, time management, lateral thinking, learning when to keep trying a strategy vs. when to switch and try something else, salvaging useful partial results from failed efforts, drawing diagrams, careful record-keeping of works in progress, more generally externalizing problem state so that it can be worked with outside your head, converting fuzzy problems into precise formal terms to they are amenable to careful logical analysis (including making explicit all of the assumptions involved in the model chosen), exploring the relationships between different problems, investigating specific concrete examples of general rules and generalizing from particular cases to abstract theorems, searching/skimming published literature for solutions to problems that are too much to handle or finding relative experts to ask for help and knowing how to do so productively, clearly explaining an original problem and its context and any simplifying assumptions and then clearly explaining a solution step by step, checking solutions by solving a problem multiple ways or doing quick sanity checks, .....
I could probably keep listing more here, but you get the idea. Anyone who plans to do any kind of real-world technical work will be at a huge advantage if they have significant amounts of problem-solving practice going back to childhood.
You might similarly protest that we should not bother reading and analyzing novels in school because few careers explicitly require reading/writing fiction, or that we should not bother with physical education courses because few careers require playing dodgeball, or that we should not bother with music courses because few careers require skill at playing the recorder, etc. etc.
There is a wold of difference between solving for X and proving P =/!= NP. Collage level math like DiffEq is still the kiddy pool with well known approaches that work.
Writing a new graphics engine, or baking a cake takes applying existing techniques, but not the fluid exploration of unsolved frontiers. So, when I say use math I mean take advantage of what exists not nessisarily add anything new.
So, yea we want problem solvers, but not thinkers.
Why do you think (paraphrased) “I think secondary school mathematics should focus on solving non-obvious problems” has anything to do with “proving P ≠ NP” per se? I’m obviously not suggesting that we should assign famous unsolved research problems to secondary students. I would instead hope we could assign students a variety of problems taking them between 10 minutes and a few weeks to solve (at their current level of skill), with emphasis placed on smart problem-solving efforts rather than on sorting students based on who gets the most right answers.
Most American undergraduate differential equations courses are taught as a list of recipes with little room for thought. Rather comparable to elementary school arithmetic drills frankly, though obviously involving more built up preparation. https://web.williams.edu/Mathematics/lg5/Rota.pdf
However, it is possible to assign difficult problems to students at any level from age 3 onward (see http://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf for an example of real mathematics instruction for preschool students; for primary students look up the work of Dienes; at the middle school level I think some Russian programs are pretty good https://bookstore.ams.org/MAWRLD-7/ etc.). It just takes more work for teachers to provide feedback about student solutions to such problems, it’s less amenable to grading by rote (and therefore not easy to check via standardized tests), and it takes more significant focus/attention/decisionmaking by teachers from moment to moment (and ideally more teacher background preparation). The students learn the subject more deeply, enjoy the process more, and learn significantly more transferrable skills.
Porting a graphics engine from one platform to another very similar platform (after having ported many other software projects between the same pair of platforms) or baking a cake just like all the others you have baked before might take nothing more than skillful application of well-established procedures, but making a new graphics engine in the first place (assuming it does something novel) or inventing a recipe for a new type of cake definitely takes problem solving skills.
I am saying there are different classes of what you are lumping under problem solving skills. Coming up with a new cake recipe does not involve building a new oven or measuring system. It's a well constrained problem. Use known techniques and apply some time and money gets a new recipe.
P vs NP on the other hand might not be possible to solve.
Yes, but inventing a new cake recipe (especially one fairly different than what you have baked before) uses a completely different set of skills than baking a cake, is the point.
You need to develop hypotheses about cake baking, test them empirically (e.g. by baking many cakes while varying one ingredient systematically), cross-apply knowledge from other cooking/baking experience, figure out workarounds to any problems that come up, at some point develop a high-level goal (e.g. mix a particular pair of flavors), and then check that your result matched your previsualized goal, tweaking the recipe in response to feedback until it comes out the way you want, keeping detailed notes matching recipes to results, etc. You need to have a much deeper understanding of cake ingredients and baking chemistry, and you need to work a lot harder at a higher level to invent recipes than to follow them.
If you are a cookbook author designing your recipe to be implemented by unskilled homemakers using unstandardized ingredients and equipment, or if you are a food chemist for an pre-packaged cake mix company, you might have an even larger set of concerns and required skills to invent a new cake recipe.
The kind of skills you learn while inventing new cake recipes might also be useful for solving other kinds of engineering problems. The kind of skills you need to follow someone else’s recipe to bake a cake are much more limited and domain-specific.
I must admit I still don’t understand why P vs. NP has anything to do with primary/secondary math education.
The problem is that current (US) math education...
Worth noting that the US tried the kind of math education you are suggesting (called the New Math initiative) and it failed miserably. The math education we are seeing today is largely born out of a counter reaction to that failure.
The New Math was something fairly different than what I am suggesting. It was an attempt at an alternate curriculum for primary/secondary school based on higher-level / more abstract mathematical topics, partially displacing study of arithmetic.
The New Math curriculum per se wasn’t so terrible, though it certainly had flaws (like anything invented from scratch out of context and not slowly developed and tweaked over time in response to feedback in a real-world setting). The bigger problem was that the proponents of the New Math didn’t have much buy-in from students, parents, teachers, school administrators, or the broader society, didn’t really do any outreach or teacher training, didn’t really produce enough supporting materials, and just dumped the curriculum on schools without support.
Parents and teachers didn’t know what to make of the curriculum (were unqualified to teach with or assess it), and didn’t feel involved in the process, and as a result there was a lot of opposition.
But what I’m talking about is not teaching different subjects per se, but teaching whatever subject in a different way, focused more on solving problems and thinking than on precisely mimicking teacher’s demonstrations or memorizing formulas. The current typical math pedagogy is patronizing, emphasizes memorization/recall and very careful attention to details (sometimes irrelevant details about formatting), teaches students that they shouldn’t try to think for themselves and teaches them to conflate getting the right answer with being “smart” or “good at math” and that anyone who makes a mistake or doesn’t know how to get the answer is “stupid” or inherently incapable.
I'm not in the US, but both of those seem to be greatly focused on the curriculum. OP is instead talking about the style of teaching and learning, which could be applied to practically any curriculum.
The intuitive geometric description I sometimes use with people is to have them imagine two duplicate rectangles and cut a diagonal in each to form four equivalent triangles. Mentally label the hypotenuse c, the long side b, and the short side a. Then arrange the four triangles into a square with the perimeter consisting of sides of length a+b. Now from the outside area is simply (a+b)^2. But computed from the inside area will consist of 4 * (1/2 a*b) + the area of the central tilted square with sides c or c^2. Simple algebra yields a^2 + 2ab + b^2 = 2ab + c^2 or a^2 + b^2 = c^2. A more fun problem is imagining why there are 180 degrees in any triangle. And for that matter where did that 360 degrees in a circle come from (last one hint: imagine 6 duplicate equilateral triangles and arrange them into a hexagon, assign the Babylonian magic number 60 to each vertex and sum the interior central angles, after using one side to draw a circle around the hexagon).
> I deliberately selected a college where I knew I would be pushed hard.
Learning a lot in a short period of time is strongly coupled with coping strategies and discipline due to time pressure and the high requirements. After this process you probably have gained some insights about yourself and further, confidence to have capabilities to read and learn through whatever comes your way. At least to a certain degree.
> It depends on what you want. I didn't want to waste time in college. I wanted to get all I could out of the 4 years.
However, I think there's still a lot of space for improvement. Especially towards technical or math expertise. A lot of students even lack of the bare basics just a few months after the exams are over.
And even more importantly, problem solving skills and critical thinking are, at least in my experience, neglected. One of the main reasons therefore is the lack of time. Thinking and solving a problem _yourself_ instead of looking up the solution or being able to reproduce a solution at your exams are two different things. With the former being the critical point at which universities currently fail the most.
Well it is still a bad decision. Libraries are an open work space and still manage to provide a reasonable working environment. This is less about money but more about a wrong perception of work.