Math is not linear, so why do we teach it like that? | Hacker News Reader

Math is not linear, so why do we teach it like that?

prezi.com

96 pointssublemonic69 comments

Most of my school's funding is and was not dependent on their student's skills, schools get funding for attendance. When I went to school they checked attendance 6 times every day (once for every class), and gave you 3 bathroom breaks/year/class. Taking attendance took about 5 mins/ class, every time you went to the bathroom a teacher had to sign a form verifying you had permission. This comes out to about* 340,000 HOURS of wasted time in ONE YEAR FOR ONE HIGHSCHOOL. This isn't even considering many, many fundamentally wrong things with how grades and classes are structured and credit is awarded. For every competent teacher teaching a useful subject there are 3-4 incompetents wasting people's time with sophistry and selectively blind adherence to stated rules.

Public high school education in this country is a net negative. High school "education" has nothing to do with teaching students skills, its there first to benefit the people running the school, and second to make people obedient for factory jobs.

Math & science people tend to be humble & introverted because they're usually wrong about the solution to whatever problem they're trying to solve, and they spend all their time doing math/science and not talking to people. This is a bad thing. Every hour spent on spiffy presentations is an hour not spent on telling people public school's are doing it wrong. The author wasted his/her time on this* * , it's not going to change anything. Math/Science people who want to improve the state of math/science education should spend their time politicking, not science'ing.

* (5mins/attendance * 6 classes * 10 mins bathroom break form filling/a day * 2000 students * 34 weeks/year / 60mins/hour = 340K)

* * It is pretty cool, though.

It's worth sanity checking statistics - 340000 hours is equal to nearly 39 years, which seems a bit on the high side for a single school in a single year...

Exactly. Attendence checking doesn't take 5 minutes per student per class, yet that's what's being claimed by the math....

If we can measure time spent productively in man-months then I say we can measure time wasted by the same metric.

Actually if taking attendance takes 5 minutes of class time (which it easily can with a class size of 30), then all students are blocked from being taught while the teacher is busy so it does make sense to count that as (n+1) * 5 minutes where n is the number of students.

Fingerprint scanners for attendance would both speed up and parallelize this operation.

What could possibly go wrong with implementing a complicated biometric fingerprint scanner based system for our schools?

http://news.ycombinator.com/item?id=1221068

Well card swipes are obviously unworkable. You could maybe link it to student's mobile phones...

But the real solution is to have decent schools with classes small enough that the teacher doesn't have to take roll call because they know everyone.

The claim seems to be that if I waste an hour of my time and an hour of yours, then that's two hours down the drain. I agree with it. (Although it seems to be missing * 5 days in a week.)

I completely agree. And when you look at what is taught and how, you may be glad that instruction is not possible for at least part of the day.

I've listening to the mises.org podcast of Murray Rothbard's For a New Liberty and it deals with the subject extensively.

I've concluded that there really is no good reason to coerce people to use their brains. In fact, its impossible.

Maybe that's why Albert Einstein said “It is a miracle that curiosity survives formal education.”

Full Einstein quotation:

"It is, in fact, nothing short of a miracle that the modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry; for this delicate little plant, aside from stimulation, stands mainly in need of freedom; without this it goes to wreck and ruin without fail."

It's embedded in more context here:

http://learninfreedom.org/Nobel_hates_school.html

awesome.

maybe the best thing we can teach our kids is to not concern themselves with pleasing these people. that, and do our best to help them find answers to every question that they come up with.

* (5mins/attendance * 6 classes * 10 mins bathroom break form filling/a day * 2000 students * 34 weeks/year / 60mins/hour = 340K)

  Mistakes:
  Actual school days in 34 weeks = 5 * 34 days
  5min / day * 10 minutes bathroom break = 50minutes 
  Bathroom break form does not take 10 munutes per class.

More reasonable calculation 6 class * (( 5 min / day * 34weeks * 5days / week) + 3 * 10 min) * 2000 students / (60 min / hour) = 176K hours.

PS: Checking attendance takes less than 5 minutes, but adding in the time it takes them to fill out a form and I am willing to say 5 minutes lost to that type of activity.

An interesting quote I came across today from Edward Tufte:

One more example. If you are teaching math, hand out the proofs on paper at the beginning of class to all the students; then work through the written-out proofs aloud in class, following the proofs on paper.

That way your students aren't merely making notes and recording your words; instead they are thinking. I believe that students should THINK in class, not take notes. So give the students your lecture notes and go through them carefully in class, trying to insure understanding of each part as you go. Your voice in effect annotates and explains the material on paper.

(Of course, these ideas apply widely, not just to teaching math.)

http://www.edwardtufte.com/bboard/q-and-a-fetch-msg?msg_id=0.... -- It's the last paragraph under "E.T. ON TECHNOLOGIES FOR MAKING PRESENTATIONS".

We teach it like that because we learned it like that.

We also teach it like that because states have various standards with pretentious names and acronyms ("Essential Academic Learning Requirements" = EALRs [1]) that require students to have specific bits of knowledge at specific ages. This means that, instead of students exploring various mathematical subjects in parallel, they're stuck going through them in exactly the order presented.

I remember being in a room full of math profs and TAs discussing how to get more students interested in becoming math majors. I suggested a 100-level number theory course, with the rationale that it's simple, accessible, mathematically interesting, and makes it clear that there's more to mathematics than the "progressively-harder calculus-based classes" (up to DiffEq) most hard-science majors end up taking.

[1] 252 page, 8+ meg pdf: http://www.k12.wa.us/Mathematics/Standards/K-12MathematicsSt...

It's time for a favorite quotation about mathematics again:

"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.

http://www.amazon.com/gp/product/0387982892/

Wasn't there an attempt to do this in the 1960s? They tried to give students a better mathematical foundation for more advanced maths that ended up backfiring politically? Namely they were teaching rudimentary set and number theory to K-5 kids. This came at the expense of kids ability to multiply and divide, and when the press caught wind of this, the program was quickly shut down.

It seems like any attempt to restructure the math curriculum will be met with massive resistance from parents who were eminently satisfied with their own (likely poor quality) math education, and want their children to have the same.

You are referring to "new math," which is the kind of math instruction I had when I was a child.

I am reading just now a very interesting book The Mathematics Pre-Service Teachers Need to Know

ftp://math.stanford.edu/pub/papers/milgram/FIE-book.pdf

(posted on

http://hub.mspnet.org/index.cfm/13083

which warns that it may be a slow download)

which describes improved university courses for students who plan to become elementary teachers. That is a big emphasis in the United States now--international comparisons have shown that mathematics education of elementary pupils in the United States is lousy largely because the mathematical education of elementary teachers (at all levels) is lousy,

http://www.ams.org/notices/200502/fea-kenschaft.pdf

so United States mathematicians are trying to do something about that that is more effective than the 1960s attempt at "new math."

Yes, Stillwell's book, mostly aimed at mathematics students who will go on to be mathematicians rather than schoolteachers, is also an outcome of thinking about curriculum reform. He describes his motivation for writing his excellent book as attempting to understanding concepts of mathematics he still didn't understand after he earned his Ph.D. at MIT.

"mathematics education of elementary pupils in the United States is lousy largely because the mathematical education of elementary teachers (at all levels) is lousy"

This is something I've noticed in the time I spent working with teachers. It amazes me how many 4th-6th grade teachers don't understand fractions, but are trying to teach them to kids!

What do you mean when you say they don't understand fractions?

I mean, I have watched groups of elementary school teachers work on the same sort of problems they assign (fractions being one example) and struggle mightily. They understand the basic concept of what a fraction is, but many of them get bogged down in the algorithms because they don't really understand what the algorithms represent.

What is a "common denominator" beyond "the thing you put fractions over to be able to add them"? Many of the teachers I've worked with would struggle to explain this to students.

Is the US high school mathematics curriculum really as linear as described here? Do you really work through, say, "geometry" in one go, and never revisit it?

My mathematical education was a bit more like advancing on several fronts at once. A chapter on geometry, then a chapter on basic algebra, then some more advanced geometry, then some more advanced algebra, then some basic trigonometry, which enabled you to understand some more advanced geometry, then...

It's certaintly very linear, but I wouldn't say you don't go back to it.

Assuming you don't take any "advanced" math in college, yes, it is in fact that linear.

It's complicated by the fact that often you spend 2 weeks learning a concept and then have the teacher say "It's going to be on the test but you're not going to use it until halfway through next year so try and remember it!" Most everyone forgets it, so we waste another 2 weeks re-learning it next year.

Sorry, I couldn't finish the presentation because of that stupid animation.

really? I thought it was beautifully done...

Probably. But I cannot tell, because in opera mini this whole "essay" is just an empty white rectangle.

The animation is awesome, it makes the logic connection much more clear and easier.

Prezi is really revolutionize traditional reading habits. It transforms chapter-by-chapter/slide-by-slide, liner reading, to a more hierarchical and intuitive way. It takes less effort to make sense, so you can focus more on digesting and re-thinking, not only comprehending.

Prezi is very nice and I've seen a couple of presentations which have benefited from the hierarchy and demonstration of relationships. However, this presentation seemed to be animation simply for the sake of animation. The spacial relationships seem to have almost no bearing on the logical relationships. I too was unable to finish the presentation.

I totally agree with you! This is a great example of why more features hurt and less features are more useful. I know that the design is great and the transition or animation is nice looking. But the main goal of the users is to read the content not to enjoy the beautiful animations. The animations become a hindrance for the readers. It distracts the readers from the content. It slows down the reading. When adding features, don't just be impressed by the technology. It is crucial to think about what is the goal of the users and how this feature can help achieving this goal. Never add a feature because it is fancy or just because you can do it.

The most useful part of this link was finding prezi, which is awesome, and I hadn't heard of it before.

Yeah that was pretty snazzy, and yet somehow seems so obvious!

Yes; I'd never seen an essay arranged visually like that. In parts it was not better, just different (but not worse), but there were several points where the format considerably added to the essay. Now I have to go check out the rest of that site.

For me, it was much worse than a normal essay, because it was too slow. I dislike having to wait between chunks of text like that. It was cute and gimmicky at the beginning, but I quickly lost patience.

The website is certainly cool for presentations, but it's a terrible format for an essay.

I assume you found the "autoplay" button? I just clicked through the whole thing, and the short animation-induced delays were actually welcome. It would have been unbearable if I couldn't control the speed, though.

If you were racing through the presentation, could you have seriously been reflecting on its contents? Much of the visual movement was designed to accompany a process of thought. In short, you might consider chewing your food instead of swallowing it whole.

I am a very fast reader.

Every time I say this, someone accuses me of not thoroughly understanding what I read. But I've excelled at pretty much every reading comprehension test I've ever taken, so it's getting kind of old.

For some reason, over the past year or so it's become almost standard in the humanities (but not in science/engineering). I'm in CS but occasionally go to interdisciplinary conferences, where every CS person will have a PowerPoint, while every humanist who has a digital presentation at all will have a Prezi (to a close approximation).

How did LaTeX-presentations fare?

Ah, those happen too, though not as much at the type of interdisciplinary conferences where you'd also have a lot of humanities people--- seems to be mainly a thing for people who need equations in their presentation, esp. in cs-theory. I sort of prefer latex presentations myself, but just because I like editing source-code-like stuff rather than editing a presentation in a GUI tool.

> [...] seems to be mainly a thing for people who need equations in their presentation, esp. in cs-theory.

Indeed. At the International Conference on Functional Programming last year LaTeX was the norm. I enjoyed the hand-written slides somebody had produced on their tablet pc.

A delightful misuse of Prezi for zooming around with needless effects from place to place rather than moving within content.

I thought it was a little disorienting too, maybe just not having experienced it -- but after a bit I saw that it perfectly matched the actual content, and spirit, of her argument.

Alison Blank (the author) should write a math book instead of making this presentation. I think that would be a great way to prove her point (which I agree with).

I also agree, but how would you write a nonlinear book? "Choose your own adventure - Math" would be a tricky textbook to produce.

Well, I don't think any book/teaching method would entirely dismiss linearity, it would just make things less linear (as compared to the current way of teaching).

As a start, I suppose you could have a topic in algebra... then use that to discuss probability. From there, you can view something as geometry and teach some of that, etc...

Maybe that doesn't work. So maybe, you teach math by asking a question and exploring to find an answer. As you explore, you inspect different approaches to a problem. Later questions can use earlier points as analogies.

To be honest, I don't know quite enough math to even give a list of questions/topics. But, I imagine a good math teacher could do something of this sort.

Its probably best to start at the University level and move your way down as you learn lessons about teaching these things.

As for your idea, CYA-Math, I like it in some ways. "To see how this connects to this, go to page 34. To see an analogy to something else, go to page 54." I agree though very tough to build.

though very tough to build

A robust cross-reference system could keep the page numbers automatically up to date. I'd edit the textbook in a wiki style interface, then use some automated software to commit it to a specific order with page numbers.

This is one area where a digital textbook would be incredibly useful.

If you want a good "non-linear" math book, try the Art Of Problem Solving volumes.

I hope my (little) daughter gets a math teacher with A. Blank's sense.

A similar point is true of all parts of knowledge, of course: I was convinced to send my daughter to a local hippy dippy lab school when I saw the teachers facing the "But what if my child is Gifted?" question from yuppie parents. The teachers vehemently rejecting separate 'tracks' for the 'gifted' in favor of the Blank plan: if some students master something early, you needn't move them on to the 'next' thing, since this is in some respects an illusion. Rather you get them working on something cool that is 'off to the side' from the point of view of the curriculum sequence, whereby inter alia they learn the unlimited character of possible knowledge and might strike something that would really get them absorbed.

--This horrified the linearize-their-way-to-Yale yuppies in the audience, but it seemed like genuine wisdom to me. (That they were really losing people with this also attracted me.)

We'll see how it works in practice. I have a feeling Blank's utopia would be too expensive given the significance attached to education in the present age.

I wouldn't count on your daughter having a wonderful math teacher. I suggest teaching her yourself in addition to what she learns in school. My daughter isn't old enough yet (7 months), but I plan on teaching her daily.

Because high school curriculums are much easier to plan when they are linear. At my high school the curriculums for English and Social Studies were also linear. Only in science did we get to choose which order we took Chemistry, Biology and Physics in.

Curricula, not curriculums...

Apparently Latin wasn't on the curriculum at that school. :(

Latin was part of the curriculum, an optional one that I did take. Luckily the lingua franca of this site is English.

Both are correct.

Well, when I learned Latin, the plural of -um was -a. Just like the plural of -us was -i. People say campuses, but they also say alumni (not alumnuses). One should be consistent, imho...

We speak English, not Latin. Latin may be the origin, but the context of this conversation is in the English language and English has adopted the word Curriculums, which is consistent with other English rules.

"." not "..."

Artistic freedom. I am the lizard king, I can do anything.

We teach math linearly, because we follow the child's brain development.

numbers are quite an abstract things, and proportion is even more so. the next step, the percentage, add relative point of views too. you just can't teach that to a child that literally can't tell his right from his left.

Well, you can, but he won't really understand it and it is effort down the drain.

So I know that here we all are CS/EE/whatever, and probably knew more math then our elementary school teachers, but there is a reason to these steps, even if the said teacher never know about it.

Doesn't just apply to math. Applies to education in general. It's like someone one day decided that it's ludicrous someone might actually want to learn something to do something cool and decided to take the safer systematic approach to education. Only problem is that this systematic approach actually disadvantages those who want to learn, especially the ones with ADHD ('cause it's harder for them to shift focus to something they aren't immediately interested in).

Because time is linear?

Or is it? http://en.wikipedia.org/wiki/Imaginary_time

Imaginary time is a trick in quantum field theory; roughly: it goes away at the end of the calculation.

But humans can understand and think in parallel. The teaching process is serialized. Concurrent pedagogy FTW!

Her rectangle problem is wrong. Here are four rectangles with the same area and perimeter:

x = 3, y = 6; x = 4, y = 4; x = 5, y = 10/3; x = 7, y = 14/5

The formula is fix x > 2 (one side), let y = 2x / (x-2) (other side). Derived from xy = 2x + 2y.

As I recall the problem was not "there aren't any" but "there aren't any more"

I got dizzy reading that and gave up half way through. Stick to math and let someone else design your presentations :)

the root of the problems in the education system lie in its prussian derived methodology.

http://www.johntaylorgatto.com/underground/

compulsory schooling is stupid and wasteful, just like compulsory anything.

I love everything about it. The message, the presentation. Ugh if only I was taught this way, makes me jealous. This is how my calculus teacher in college taught. It was incredible compared to anything before that.

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