Marvin Minsky: What makes mathematics hard to learn? (2008)
web.media.mit.edu
web.media.mit.edu
He said all knowledge has a half-life, the time it takes for half of what you know to be redundant or wrong. Math, he said, is the longest, measured in centuries or millennia. One should feel sorry for neuroscientists: they can go to the bathroom and half their knowledge will be out of date.
The only time I spent with him, but both the metaphor and his passion for math will stay with me.
Web development seems to be even worse than neuroscience in this regard. I feel like I have to perpetually study to be abreast of best practices.
The worst part is web development is a completely arbitrary made-up system without any abstractions you can learn and use in a different field of work. It does not even transfer to something as closely related as desktop/mobile GUI toolkits. I view "knowledge" like that as a waste of time and avoid it whenever possible. I wish there was a word for it, something to imply "garbage knowledge."
When you progress from arithmetic to abstract, it hardly gets easier.
Moreover, memorizing. You go from memorizing multiplication tables to other kinds of tables, like tables of equations giving various identities, rows of coefficients in series, and the like. Contents of various kinds of matrices.
The need to be precise and avoid mistakes never goes away. Manipulating a complex math equation is still a form of arithmetic. And it's harder because the underlying semantics means that something which is mechanically correct at the syntax level (easy to check) could actually be meaningless and wrong.
The simplistic notations used in math don't "keep up" with the increasing complexity of what is going on. They just get harder semantics. Notation which looks like multiplication or addition in such and such domain is just "sort of" like it, but, oh, here are the ways in which it isn't.
Minsky solidly makes the point that when math is taught as nothing more than an endless series of tedious drills with no purpose in sight, of course it demotivates kids. No one likes pointless Sisyphean tasks.
Many students, however, "rediscover" math when they get to their first class that involves theorems and proofs (often in high-school geometry) and when it starts being used in science coursework. It is then that they realize that math is a way of thinking and this can transform their opinion and motivation for math.
The best teachers find ways to relate mathematics to real life and real purpose. Yeah, it is always going to be challenging, but having a purpose creates motivation to get through the tedium.
High school geometry (at least in the US, and in my experience) is the worst of them all! Two-column proofs should be banned.
I had never heard of that monstrosity before and having googled it, it looks worse than programming in COBOL.
Also: it runs on machines that look like Cylons. You can't beat that. :p
To contrast, LOCAL-STORAGE seems like a regular global variable (.DATA/.BSS) segment. So is WORKING-STORAGE basically like a set of seeming global variables, that actually point into (the equivalent of) a shared mmap'ed file, such that other processes in the same session/pgroup end up mmap'ing the same file and thus sharing memory that way, without having to clone around the memory handle itself via forking/threading/IPC?
In this case, it sort of reminds me of the OS-level version of an anonymous Erlang ETS table held by a supervisor and passed to all the worker children to manipulate.
COBOL's type definitions are also patterns really - for instance you declare a field to be of type 9(7) which means a numeric field of length 7 and so on.
The whole setup gives you very fine control over the structure of your data. For example, you can describe lines of a document to be printed out with strongly typed fields to be filled in by your program, so a form basically.
It's also used as a template for documents like VSAM files. Those are flat-file databases and the COBOL program's working storage imposes structure on them.
Here's a small example:
01 FILE-RECORD.
__03 WS-ACCOUNT_____PIC 9(10).
__03 WS-SEPARATOR1__PIC X.
__03 WS-NAME________PIC S(25).
__03 WS-SEPARATOR2__PIC X.
[Underscores are formatting only, not COBOL syntax]
That declares a structure called FILE-RECORD with four named fields, one of numeric type with length 10, one of alphabetic type with length 25 and two of alphanumeric type with length 1. So that's one record in your file structure and you can fill it in with data from VSAM files and the like, then print it out, with the separator fields for formatting.
It's a strange thing, low-level and high-level at the same time, in a very nice way. It kind of grows on you. Well, it's grown on me anyway. Of course, YMMMV.
Btw, the numbers (01, 03) make the hierarchical relation between the root of the data structure and each of its fields explicit; I won't say I'm in love with that sort of syntax :)
Also, I'm a total no0b still, so I might have made some mistake above, apologies in advance if that is the case.
I had those in high school and I hardly see any problem with them. It's just writing the proof neatly organized into a table, with the reasons for each step given.
High school geometry introduces many students to the idea that math is a way of reasoning and this is _very_ different from the boring grind that they were used to.
My personal feeling (as a maths grad student) is that the utility of two column proofs depends on the field. For logic and algorithms sure but estimating integrals it becomes tedious quickly.
[1] http://research.microsoft.com/en-us/um/people/lamport/pubs/p...
Geometry is taught correctly when proofs are used as a tool to convey a deep intuition about a mathematical pattern. Check out Lockhart's book [1] if you want to see what it's like when done right, though there are many more examples. Two-column proofs are simply a tool for the lazy/unknowledgeable teachers to fill a geometry class.
I am aware of Lamport's work, and it's a specific tool for a specific subfield in which there is a plethora of false results. Ignoring the fact that most of theoretical computer science research and most of math research more does not fall into the category that Lamport is critical of (distributed computing), these temporary issues about rigor in academic publishing should have no effect on high school pedagogy. Instead, we should listen to the world's finest math teachers, who pretty much all agree that two-column proofs are awful. A quote from Lockhart [2]:
> Geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming. There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species.
[1]: http://www.amazon.com/Measurement-Paul-Lockhart/dp/067428438... [2]: https://www.maa.org/external_archive/devlin/LockhartsLament....
Unfortunately, most high school math teachers are undertrained, underpaid, overworked, and pressured to focus on exams. The two-column proof is a crutch. It's not the only awful part of high school math education, nor is it the sole bane of a student's math education (I never said it was). But it is the most egregious example of bad math education.
Just look at various cultural traditions, popular (and even serious) music, crafts, sports, ... sneeze ligion. Mindless repetition is everywhere.
The same kid that hates doing arithmetic will play the same video game for hours in which a small variation on the same set of events happens over and over again.
The kids you're describing are the smart minority, of whom a portion will go on into STEM type fields. They too will hit their mathematics ceilings. Most will go as far as a couple of third year undergrad math courses and that's where it ends.
Kids are naturally curious and fighting against their curiosity can easily lead to losing their interest.
That said, I started over in a community college, and finished up to trig. with ease. I found trigonometry, surprisingly straight forward. I went on to finish a year of physics. I didn't need calculus for my major, and heard it was really hard. Plus, I didn't want to ruin my grade point average with a class I didn't need. I now regret not taking calculus, but it's so much easier to learn something these days.
My struggle with math was two fold. I didn't care in high school, and I never had a firm grasp on basic math. I look back , and once I truly inderstood basic math, especially fractions, and percentages; it all became so easy.
I see a lot of kids still struggling with math. They get pushed along, and avoid any class that has math in it. I do blame our U.S. teaching system in this case. Keep teaching basic math until the kid can teach it to to their classmates.
Then, and only then move on to algebra, and trig? As to calculus, I didn't take it so I won't comment, but when I was applying to health professional schools, I didn't find one that required Calculus. Probally for good reason. My physics classmates biggest, silent stumbling block was they were horrid in math, and looking back--they just didn't know the basics, so every step up the ladder became more mysterious. Most of my classmates seemed to the solving problems in physics by memorizing the homework; not truly understanding the problems.
In fact, I've never learnt it. From time to time I still find myself multiplying on fingers; as a kid I used it a lot, now the table went into my head automatically simply from continued usage. The way to multiply two natural numbers from <6; 9> on fingers was one of the most important things my mom tought me :)
There's also a way as presented in the article (87 = 88-8), but I mostly use it when double checking, as I find it slightly more error-prone when doing it in head (and obviously very inefficient when done on paper). Works great for bigger numbers though.
Once in junior high school my math teacher caught me on multiplying on fingers and ordered me to learn the table for the next lession. Managed to pass it, but quickly forgot in following days. I never cared - I was always doing very fast and well on math classes anyway.
It's funny to hear you say that. I don't know how old you are, but lately I've been trying to teach myself some additional math that I never took before (like Linear Algebra) and refreshing on Calculus, and now - at the ripe old age of 42 - I swear I find it easier to learn this stuff than I did when I was in college 20 some odd years ago.
Maybe it's just a question of motivation, or maybe I have more context available now, or, hell, maybe I've gotten smarter... but whatever it is, I'm not going to complain, considering how you always hear about how our minds slow down as we age and things are supposed to get harder to learn.
Meanwhile, that dual sense- the pleasure the video game designer tries to create and the sense of importance the priest creates- these are fundamentally absent from our teaching methods in early arithmetic, which, at least in America, we call 'math,' not differentiating the symbolic thought processes from the repetitive arithmetic, incorporating little if any fun, and constantly failing to appropriately create a sense of importance (the most common teenage question in math classes is, "Why do I have to learn this? When am I ever going to use this?").
While there Do exist efforts to instill a video game's sense of fun to arithmetic exercises and a sense of near-religious importance to the symbolic thought of math, we are functionally applying a tourniquet to the mathematical understanding of the young by just force-feeding arithmetic drills and emphasizing mistake avoidance instead of proper thinking- and it's very little wonder that only some few students' mathematical passion survives the tourniquet.
Needless to say, my geometry grade was much better than my algebra grade.
Then, in college, taking pre-calc/Trig, it was back to a lot of boring repetition (except trig, that was actually interesting). But then in Calc I, it was fun again. The idea behind derivatives was actually interesting, and hooked me early on in Calc I.
Anyway, I always found that, other than tedium, the biggest challenge I had with math was being put off by cryptic notation. I tend to want to read things to myself in my head as I look at them on a page, and when I'm looking at (new/unfamiliar) math notation, and I don't know how to "pronounce" it to myself, my mind wants to just blank out and skip it.
Material that has an almost unnecessarily comprehensive vocabulary and notation practices guide at the beginning/end is the best kind.
I agree with a lot of what you're saying, but not this. The pattern is depressingly clear. People are taught math by teachers who don't really understand it and didn't like it themselves; most of the students wind up in the same place, and so each generation poisons the next. Elementary school teachers especially tend to be drawn from the subset of the population that doesn't like math.
And it starts with the idea that math is all about rote memorization. I would go farther than Marvin did with the flash cards. I would say, for each fact, let's work it out by the standard algorithm, then see if we can find two or three more ways to arrive at it: by distributing multiplication over addition (e.g. 7 × 6 = (6 + 1) × 6 = 6 × 6 + 6 = 36 + 6) or by factoring one multiplicand and reassociating (e.g. 7 × 6 = 7 × 2 × 3 = (7 × 3) × 2 = 21 × 2, which is easy because you can double both digits of 21 without generating a number bigger than 10).
So when I read the reply of the "traditional teacher" to the 6-year-old who had a novel way of computing 15 + 15 -- "Your answer is right but your method was wrong" -- I think that if this was a real anecdote, the teacher should have been fired on the spot, for not understanding the first thing about mathematics! But sadly, no one could probably have been found to replace this teacher who wouldn't have suffered from the same fundamental misconception. It's not about knowing the "right" way: it's about knowing many ways.
Luckily I didn't say that out loud, otherwise they might have made me hate mathematics and I wouldn't love science now (:
There's no correlation at all between being able to add up a grocery bill in your head and being good at abstraction.
My accountant is very good at mental arithmetic. I don't expect him to win the Fields Medal any time soon.
This is more relevant than it should be. Too many people leave school thinking math is mental arithmetic, and have no idea what symbolic manipulation is, or why it's useful.
Instead of memorizing multiplication tables, you instead have to memorize derivative formulas, or the formulas for the Fourier and Laplace transforms, or the difference between a bijection and an injection, or all the conditions on an elliptic curve that make it suitable for cryptography. I could go on and on and on.
The stupid notations that mathematicians use doesn't help either. All math should be done using S expressions (and I'm pretty sure Minsky would have agreed!)
R.I.P. Marvin.
And BTW, the notation actually sucks for paper-and-pencil too because of its ambiguity. See:
http://mitpress.mit.edu/sites/default/files/titles/content/s...
I'm sure that isn't in R5RS and I don't see it in R7RS. I found a description in the Racket manual.
In any case, I like how they are making the context explicit. In the regular math notation for the physics, you just say something like "the Lagrangian for the free particle" and then just spew out something with free variables sticking out of it, like velocity "v". The hacker in me of course asks, how the heck does that work if I have two particles? There is only one v? When we have two, we hand-wave in some subscripts: v1, v2, ... This system makes it explicit: it's the Lagrangian for a particle. So the particle appears as a parameter, and that parameter has accessors on it to retrieve its properties: velocity of the particle, etc.
The next step is algebra at 8th or 9th grade. Letters and numbers confuses people too.
In my case I succumbed at Analysis. I like applied math, but dislike constructing proofs.
And, on the converse, in previous centuries, you'd often see people only start learning math from the beginning in their late teens or early twenties—the time when they first attended "university" without previously attending a grammar or seminary school. These 20-year-olds would be able to pick math all up quite quickly, proceeding from addition and multiplication to calculus and beyond within a year or two.
This suggests to me that what we think of as "mathematical aptitude" is far more a measure of developmental age than anything else. I think a large part of why kids hate math is that we try to cram concepts into their brains when those brains aren't yet the right "shape" to take in those concepts. I expect that offsetting every part of the mathematical curriculum at least two years upward—if not far more—would do wonders for attrition rates.
My belief is that we could even fit all the same curriculum in by the end of public high school: although you'd be "squishing" more of the learning into the later years, the stronger minds kids would have at a higher age would allow them to acquire each successive concept both more quickly and more thoroughly, serving as a better base for the learning going forward, which would in turn be accelerated by that base.
[1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
If you've ever tried to tutor someone who "struggles with word-problems", the majority of the time it's because they don't actually understand the things they've been "learning" at all.
Similarly, this tends to be the difference between people who "get" programming and people who don't. If you ever talked to someone who failed a programming course, it's usually clear from their approach that every new language or API they learned became, to them, another set of rote facts to burn into their mind, never coming together to paint a picture of what the formal act of "programming" is generally.
In high school it's not so bad -- only 28% of high school math didn't major in math, and only 12% of our total math teachers in high school have no qualifications whatsoever. [1] In middle school 55% of math teachers didn't major in math, and about 30% didn't major, minor, or get a certificate in math! [2]
So -- you're right.
Effects Of Grade-Based Segregation http://web.media.mit.edu/~minsky/OLPC-2.html
Role Models, Mentors, and Imprimers and Thinking http://web.media.mit.edu/~minsky/OLPC-3.html
Questioning “General” Education http://web.media.mit.edu/~minsky/OLPC-4.html
Education and Psychology http://web.media.mit.edu/~minsky/OLPC-5.html
Wait, is this girl some kind of base-2 native?
What's 1 and 1? 2.
What's 2 and 2? 4.
What's 4 and 4? 8.
What s 8 and 8? 16.
What's 16 and 16? 32.
That's just five facts.
The game could occur socially between kids. It is natural to ask a question, then take the answer and "up the ante" by re-formulating the answer into a harder question, back into the other child's face!
Do you know 1 plus 1? Ha, two plus two? Oh yeah, how about four plus four, then?
The progression grows quickly, of course, and soon the interrogated subject breaks.
I get beaten at 65536.
Ah, if only "one twenty eight kay" were an accepted answer ... :)
If I agreed to pay you a dollar today, and double your pay
each day. How much would you get paid after a week? After
30 days?
Children raised on positional decimal arithmetic are "supposed" to figure that 10+10=20 or 20+20=40 and then add/subtract 5+5.
Of course it's easier to associate 15 with 16 than with 10 or 20, but the fact that she immediately knew 2·16 and was able to proceed further says something about either her experience with binary arithmetic or some tendency to spontaneously count in binary.
At least for me it was just silly shortcuts that stuck with me like 6+7=13 or 7*7=49. Maybe why I'm little fixated to 7 is that 5 and below is easy, but with 6 and up things get little more "math-y", so with some quick shortcuts like these it's easy to adjust to 7. Or most likely I'm just full of shit, that is usually the case.
Tricks only get you so far, you can't do 6+7 on fingers or 7·7 as (4+3)(4+3) (I had a huge problem with this one, too many numbers to track). So you fail. And when you fail, you memorize the problematic cases and go on with tricks based on those.
And yes, it's completely incomprehensible for people who learned the pencil-and-paper algorithms and think that those are the end of the world.
(2)(16) = (2)(15 + 1)
[0] http://slatestarcodex.com/2014/03/03/do-life-hacks-ever-reac...
5xN is easier to remember than 16xN on account of 5 being the smaller number.
2^n is logarithmic whereas 5*n is linear. Arguably, logarithms are not too complicated, even if linear seems to be a degree easier, seeing that the decimal system is also logarithmic as that's a denser representation.
edit: how to enter an aterisk as the multiplication operator sign
5 * N
5*N
5*N is easier to remember than 16*N
Versus
5 * N is easier to remember than 16 * N
And I think the word you want is exponential, not logarithmic. Related to each other, since exponential expressions become linear on a logarithmic scale, but exponential describes the growth of 2^n better.
By that logic, shouldn't it be easier to remember 7 * n than 10 * n, because 7 is the smaller number?
> 2^n is logarithmic
Also, as Jtsummers (https://news.ycombinator.com/item?id=10973381) points out, the function `n \mapsto 2^n` is exponential: its growth is significantly faster, not significantly slower, than exponential.
Two and two are four
Four and four are eight
Eight and eight are sixteen
Sixteen and sixteen are thirty-two
Inchworm, inchworm, measuring the marigolds
You and your arithmetic, you'll probably go far.
Inchworm, inchworm, measuring the marigolds
Seems to me you'd stop and see how beautiful they are.
(Kids Singing: 2 & 2 are 4, 4 & 4 are 8, 8 & 8 are 16, 16 & 16 are 32)
Even if she didn't know the song, she might have learned 16+16 from those.
2+2=4
4+2=6
6+2=8
...+8=16
...+8=24
...+8=32
Whole lyrics: http://www.teocio.es/portal/entretenimiento/canciones-danzas...
For example, rather than remembering 12 * 6 = 72, i recall that 10 * 6 = 60 and 2 * 6 = 12, hence 60 + 12 = 72. I don't need to remember as many things and for me i can do that much faster than trying to recall it.
My memory has never been very good, so perhaps this is why i take such an approach, as i find it overall easier.
Note: it works in pairs so the final 2*6 shows up.
Instead of memorizing algorithms, kids should be shown this. It's free gamification and, after some years, I believe it's a core idea of what is, and why math can be pleasing.
I actually was taught this and a number of other "tricks" in primary school. Estimation was a defined thing, and the idea of getting quick math results in your head was a real goal. Reasons given: job interview, pricing goods and services to understand value, fuel consumption, making things (that deck, how many boards, cost, etc...), navigation.
They took us through a lot of those cases. The first time I applied this was powers of 2 for computing. Learn the first 16 bits worth, and that helps with all sorts of things in computing, same as powers of ten and common easy to compute things do in most other areas of life.
So, what I did was make it a car game. Figure out how tall things might be, or prices, whatever comes up. Practice doing it helps to actualize the skill and once it's done, they will apply in in ways they find useful.
This style of learning works for everyone -- including children. There is a growing community of people sharing educational, animated gifs on Reddit that recognizes the value of visual learning.
Popular subreddits include:
- /r/educationalgifs - /r/mechanical_gifs/ - /r/GifRecipes/
While other kids complained about algebra, I recognized it as exactly what I had been doing with my little Apple II games, basically fumbling on my own. Now each math lesson was a chance to make my games even more interesting.
My first graphing calculator had a simple example of 3D point rotation. It was like learning a magical incantation! (This is pre-internet.) My friend and I were so excited at the possibilities and immediately wrote a primitive ray-tracer.
The point is: by connecting it to games, we were literally hungry to learn more math.
Cognitive maps (or mental models) are part of that.
I struggled a lot with this, and other meta-learning topics. Got it eventually, after much independent reading and testing, but why was this never a topic in school? I'd have gladly sacrificed a year of schooling just to learn about it, and I'd have made it up within another.
It's also the reason why I no longer believe that intelligence is the most significant factor in our capacity to learn. The same course done by the same person may be finished within a decade, a year or a week, with the only difference lying in approach.
The difference is so striking that calling kids that fail at it dumb or the more politically correct challenged is lazy pedagogy. It shifts the issue onto something outside of our control. But I don't think it is. We can do a lot better.
And it may be due time for a new approach. We don't need a large populace educated enough to man machinery or a filing system, as they're starting to man itself. We need youth that is confident in its ability to stride across vast depths of knowledge without getting lost. And current processes of education seem ill engineered to that goal.
Like I said though, pet peeve of mine. I'm sure we'll figure it out.
Math simply is hard. It goes against all our good-enough, subconscious brain functions we use every day to make basic "bigger or smaller" decisions. The problems mathematicians call "interesting" are counter-intuitive pretty much by definition and now imagine someone thinking like that writing a math book for children. The language used is vast and was built over literally millennia of mathematicians appending their own little modifiers above, below, and all around other mathematical symbols, resulting in monstrosities like the function for the Fourier Transformation which can actually be described quite intuitively geometrically.
Mathematicians want "simplicity" and "beauty" but their definition of either is absurdly different from any that would take into account basic human intuition. That's the problem. And there barely are any mathematicians that would admit that this could ever be a bad thing.
Then for people for which that's too abstract, I find this graphic helps a lot in explaining. http://static.nautil.us/1635_42a3964579017f3cb42b26605b9ae8e...
You picked a really bad example for the unintutiveness of mathematics.
Personally, my intuition gives up at spectral sequences.
Just a data point for you.
https://upload.wikimedia.org/wikipedia/commons/1/1a/Fourier_...
Thank you!
The problem is that an emphasis on avoiding mistakes leads to paralysis. Though he phrases it differently. Students don't want to be wrong so they just don't instead.
As he points out later in the piece, learning from our mistakes is key. If I never learned to walk because I fell the first time and failed walking class, I'd be crawling even today. An overly negative assessment of the students' works, an overemphasis on right and wrong, rather than correction and expansion to different methods when some method has failed a student, ends up producing many students who can't do math.
Substitute "Arithmetic" for "Interview Problems"
Honestly, for me, the issue is "one size fits all" that the vast majority of our students get.
I learned math just fine the way it was taught. Sure, it was boring at times, and I was mathematically inclined, but it worked for me.
For others, it clearly doesn't work.
We have to ask ourselves, what exactly do we want people to know who don't want to learn math. What is the "bottom line" that we need a citizen to know?
Algebra 2? Trig? Some combination of Algebra and basic math finance concepts?
That's the main question. And it's very difficult to answer for ALL students.
I don't know what the answer is, but adopting one method for all students isn't the answer.
Having read the article I find it well written but WHAT THE HELL?! Why doesn't he use pictures? simple diagrams? "sexy" stuff?
It seems that I'm having the same problem as the child is having: the form does suck and it discourages me from reading a boatload of articles like this one. If the presentation of the subject was better I could read 10 more and maybe, finally, take an interest in "teaching methods" that would lead me to being curious and accept any "bad form" of content :)
His target audience is composed of people who are interested in mathematics teaching (parents, teachers and such). He expects (correctly in my opinion) that these have handled and can handle long texts.
The subject matter is an opinion piece without much underlying data, so graphical representation is not required.
The writing style must be adapted to circumstances. Increases in picture density do not monotonically increase text appeal.
Now to nuance a bit this over-simplistic sentence I just written: do you think I am interested in mathematics teaching? I did read the article anyway so you might be right but I felt a bit overwhelmed by all this "jargon" when an example would have been so simple to understand.
After all, Mr Minsky was a teacher, certainly a mighty good one :)
A child was sent to me for tutoring because of failing a geometry class, and gave this excuse: " I must have been absent on the day when they explained how to prove a theorem."
No wonder this child was confused—and seemed both amazed and relieved when I explained that there was no standard way to make proofs—and that “you have to figure it out for yourself”. One could say that this child simply wasn’t told the rules of the game he was asked to play. However, this is a very peculiar case in which the ‘rule’ is that there are no rules! (In fact, automatic theorem-provers do exist, but I would not recommend their use.)
The game has rules, and you can learn them.
I can agree with that, but symbolic reasoning isn't everything. As shown by Gödel, you can't have symbolic system capable of proving every true fact about natural arithmetic without also proving some false "facts" about natural arithmetic.
The ultimate game doesn't have rules. On some level, math is just few pieces of meat speculating about abstract notions originating from and tied to their physical experience. Or something like that :)
Math isn't hard, it just doesn't work for those who don't know how to use it. And knowing how to use it means understanding what exactly you are doing. Many students don't bother about what does it mean to do when asked to solve an equation.
But at the same time, show the value of those answers, because on the face of it, knowing how much is 342x520+92 is not really valuable. Even knowing how much is any number plus any number is not that interesting for basically 90% of the population.
It's not clear to me that there's an obvious right answer here, but I do believe that most obvious answers are, to some degree, wrong. Namely, saying "just ask harder questions" doesn't take into account that then the student is trying to solve hard questions and learn a new method at the same time.
Teaching a more general method on easy problems gives time to become acclimatised to the method, while also offering sanity checks (by easier, faster, and / or approximate methods) to make sure that it's not going wrong. On the other hand, as you say, this approach seems unmotivated, and bores or frustrates students—there's no obvious right answer! I think that all that can be said is that one must find the appropriate balance for each individual student, or class.
In an ideal world maybe the teacher could tweak the problem each time they see a person using a different method, to make the problem one that forces a person to use the method they are trying to teach. In the real world you usually can't give each student that kind of attention. You just have to say, "Look, I know you could solve it that way, but I need you to try to solve it this way so we can learn this method."
https://www.maa.org/external_archive/devlin/LockhartsLament.... [PDF]
Well, I enjoy maths, but for me my entire mathematical education was always about memorizing tables. It was made worse by the fact that during Polish exams you cannot use advanced calculators(only ones that can do addition/multiplication) so for the final exams at the age of 19 I had to remember loads of different formulas, trigonometric values for common angles etc etc.
I'm not saying he was wrong, but in a lot of educational systems maths is 90% about remembering stuff.
Real math is about being able to derive this stuff when woken up at 5AM.
What you get in school is drilling kids to appear as if they understood math so that adults who don't remember this stuff anymore feel intimidated and leave with impression that their children are learning some rocket science and the "education system" deserves their money.
No teacher literally reasons like that and no one will tell you that, but that's exactly what they do when they are overwhelmed by a horde of barely interested kids who somehow have to pass the final exams. The other part of this insanity being politicians who set up those exams.
People who understand math can do X, Y, and Z.
Therefore we should make sure children can do X, Y, and Z.
Then they'll understand math too!
It is/was, but the Common Core Math standards are trying to change that. Children are now being taught how to look at addition, subtraction, multiplication, division, fractions, and other concepts from a variety of perspectives. Predictably, it's driving their parents insane because they were taught the "one true algorithm" for each of these tasks and can't help their kids when confronted with alien algorithms that change from semester-to-semester.
A majority of it is memorisation, which makes it both very easy in some respects, and hard in others. I still feel in the end it is a bit useless to learn it in the way we do now, seeing as computers can do it in a fraction of a second.
I am very grateful for all the help my parents provided for me without it i would not pass even the first math classes in highschool.
[1]: http://www.amazon.com/The-Math-Gene-Mathematical-Thinking/dp...
Math could be taught qualitatively.
First, one can give their students ten or so hairy problems where the goal is to write down what the problem is asking in the most explicit and gory details.
Then one can present another batch of hairy looking problems and ask the students what definitions/lemmas/theorems/corollaries look appropriate to apply to the problems and why?
Another batch of convoluted problems can be offered where the goal is to simply formulate one's questions regarding the problems in the most precise way one can. Asking the right, penetrating questions solves more than half the problem.
Ask the students to rewrite all the definitions/lemmas/theorems/corollaries presented in class in their own words.
Present students with a text full of unproven statements (most grad level math texts). The exercise here is to identify as many of these statements as possible. The bravest students are encouraged to supply proofs.
Give students a bunch of required problems peppered with generalizations and extensions of required concepts(typically reserved for big boy students) and ask them why they are required to solve these problems. What exactly to be gained out of solving every specific problem?
After all that students can tackle the required set of problems written up for them by the Department Of Serious Business. At this point math will come naked and explicit in front of one's eyes.
None of this is going to happen in classrooms, though. Students will have to learn these things on their own as per the traditions.
I hated math when I wasn't passionate about it. In school a lot of math classes were about route memorization, not going through or writing beautiful proofs.
Why don't they teach more abstract math in school? I love studying number theory and category theory.
One striking example: Training children to individually articulate their fingers makes them better at processing arabic numerals. [1]
[0] https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From [1] https://www.researchgate.net/profile/Marie-Pascale_Noel/publ...
Ah, the Henry Jones Sr. approach to parenting! I'm a fan.
Young Indy: I have to show you something!
Professor Henry Jones: It can wait. Count to twenty.
Young Indy: No, Dad, I-!
Professor Henry Jones: Junior!
Young Indy: One, two, three, f...
Professor Henry Jones: In Greek.
Young Indy: [rolls his eyes] Ana, theo, thea...
(from IMDB)
And, I basically don't have a clue about how they're being taught to do this stuff. Any sane person turns 2 x 15 into 2 x 16 - 2 or 10 + 10 + 5 + 5.
But this Common Core arithmetic is not quite sane. The intention is to teach reasoning skills, but the reality is that some curriculum-writer's idea of reasoning skills has become codified. Grumble.
You'll be missed, Dr. Minsky. So long, and thanks for all your teaching.
Everything is taught as a convention through brutally boring rote memorization and repetitive mechanical operation.
Unfortunately, the people creating the training materials are usually so far removed from a basic level of understanding, they're blind to their own expert bias.
What we need sre training materials that bridge basic to advanced understanding theough practical application. Except practical application is hard to measure so it'll never be used. In a traditional teaching environment.
Click on "Topic map" in the header — it is a real domain-expert-created topic map of many areas of math and science.
It's a free, interactive, crowd-sourced learning platform that does exactly this, especially for math — the founder is lead Coach of the USA International Mathematical Olympiad team, and a math professor at CMU.
f/d: I interned there last summer.
It's difficult to make initial progress and you see no path, with little rewards along the way, to get good at it. What's the reward at the end?
Math doesn't have a path and there's no "cool" prize at the end, for most people.
Computer programming has a path with lots of little rewards and "cool" prizes at the end. So, you get lots of self-taught programmers.
Create an interesting path for math with lots of "rewards" along the way and many more people would be interested.
I agree with the author in that a teacher's enthusiasm and perspective, and 'rigidity' in school can compound difficulties.
But as a physicist, the most difficult part of learning math, for me, is overcoming my own mental barriers. I need patience and focus to comprehend even the most simple sentences. If I don't want to learn/strain in the moment, then I won't.
On the other hand, in a subject like history a student can make good ground towards an answer by laying down fundamental facts and then arguing and reasoning back and forth. Even with minimal facts the student may use some approximate recollection of events and still be able to build an argument. The answer may not be complete, and it may not even be good, but the final outcome has the appearance of being complete, and is often somewhat far from the starting point of nothing.
Humanities subjects use assessments that provide intermediate rewards for a student. Mathematics offer no such respite. That is what I believe makes it intimidating. Even worse, some questions have intermediate solutions look extremely hairy, leading the student to believe that he has gone further away from the solution. In such cases, the intermediate step has punished rather than reward!
Very often suggestions for improving mathematics pedagogy is to make use of "open-ended" questions that admit multiple solutions (as Marvin did here). I believe most of us who make these suggestions understand the effect that traditional maths questions have on a learner. These suggested questions are just like those in history or literature that allow students to have a decent go at it and still make progress.
I don't know if I truly believe in the theory (hypothesis) that memorization makes it extremely boring. Until historical accounts are utilized in some argument (during a performance assessment), these are just empty facts to be memorized. And given the wide range of questions that can be asked in a history exam, the effort involved in memory work in history class seem to be more onerous and less rewarding. Yet I have seen students who are gifted in remembering in detail historical events, capable of building logical structures and narrative flow, and they struggle with maths. So clearly they must have committed a decent amount of effort to memorization - but not in maths but other subjects instead.
In any case, at the higher levels, rote work in mathematics is extremely important: it is very difficult to pursue an advanced class in probability theory without the basic tools picked up in real analysis, for example (that means all your theorems relating to sequences and continuity and functional analysis and so on). That usually involves some amount of "practice", thinking and solving problem sets before moving on to more advanced topics.
Algebra and Functional Programming taught together in Middle School is a huge win.
At this level, getting 3+2=5 was MUCH less important than being able to explain in writing that 3 apples and two bananas added up to a bunch of fruits in words. 4, 6, 8, 5, whatever. Alas, they never got 3+2=5 hardwired in this curriculum. But they were better artists for it!
And my kids did not learn math, because the curriculum was more about writing than math and explaining reasoning (I called it "math for people who don't like math and would rather write about it than do it"). So in elementary school, I taught my kids math using Singapore Math curriculum and let the school do its thing so it was balanced, and I volunteered weekly in math during the elementary school years so I was on top of what they were learning and how.
In addition, we always did math at home in informal ways without being explicit about it, from counting lug nuts on cars when they were toddlers through guessing the color of the next car to go by the stop sign on our walks to estimating which box of cereal was cheapest without a calculator to noting that the arrival rate at a traffic signal wasn't random in one direction because the only way cars could get to it was by going through another signal. Heck, even counting in binary using the 3 lights above an airplane row on visits to the grandparents - Look, with three lights I can only count to 8! And now with high schoolers and college kids, math jokes like log(fu) = log(f) + log(u) are the height of hilarity, or at least lighten the mood at exam time.
Point is - there is "studying math" (with whatever curriculum and rubrics) and there is "playing with math." As a parent, especially with younger kids, I found that it was great to establish both. Math isn't a "thing", it's part of life.
Ouch... I guess Minsky was a Mac guy.
I think some peoples brains are just wired to 'get' math just as some people 'get' programming. And some people just 'get' art. Certain people just 'see' certain things and you can't really teach that.
Crazy to read this sentence while I am halfway through Cat's Cradle.