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Who or What Broke My Kids? (powersfulmath.wordpress.com)
177 points by ColinWright on May 31, 2014 | hide | past | web | favorite | 198 comments



Nobody broke the kids, they were doing exactly what they were taught to do.

I'm not a fan of the education system in the least bit, but something about this rubbed me the wrong way. She's probably exaggerating when she said she had a "meltdown", but all the same, what are you supposed to think as a kid when your teacher has a tantrum in front of you because you're doing what you were trained to do? To the kids she's just part of the institution, and in this particular case the institution is sending them very conflicted messages. "Think for yourself! As long as you're thinking what you're supposed to be thinking."

In any case, I think the real problem with the education system is it's constantly trying to bullshit kids into "learning" things they really don't need to learn. (And they don't really learn it. Quick sanity test: what do you remember about high school trigonometry? But you "learned" it right? Or did you, really?)

Most kids aren't stupid. They're naive, but they're not stupid. They can tell when their time is being wasted. They're out to give the right answer and not think about it because when you're literally trapped in a system like that, the most logical thing to do is minimize effort and not make waves until you can make your way out. So that's what kids do, and then pompous teachers are surprised that kids aren't "passionate" about these things. Well no kidding they're not! Why should they be?


> Nobody broke the kids, they were doing exactly what they were taught to do.

I would say that the fact that that's what they were taught to do is what broke them. In other words, trapping them in the system (and I agree with you that they are trapped--I certainly felt that way when I was in school) is breaking them.

> what do you remember about high school trigonometry?

Everything. But that doesn't really contradict your point, because (a) I have actually used trigonometry many times since high school, both for work and for fun, and (b) I didn't actually learn trigonometry from my high school trig class; I already knew it (or at least most of it), because my Dad had given me a book on trig some time before and I had read through it on my own, because I was interested in the subject matter.


In the same line of thinking, I can remember most of my classes from high school. But that doesn't mean just being able to recall something made it a good use of my time.

For example, Spanish class. Now some people are fine with the language classes effectively just being a deep social studies curriculum around one culture where most of the time is spent on rote memorization for arbitrary reasons, but while I can remember my vocabulary, I can't speak real Spanish at all. But then again, I never did in class either. 5 years of Spanish, including two required semesters in college, and if I hear a Spanish conversation I'm completely lost. If they slow down I can translate the words in my brain bank and figure it out, but I never spoke the language.

In the same line of thinking, I read Shakespeare, Lord of the Flies, The Great Gatsby, Uncle Toms Cabin, etc and never actually got anything more than factual recall of the story plots to show for it.

Of course I remember the math, because I liked the math and specialized in the math and went into software in part due to the math. But the entirety of HN is deeply biased because almost anyone here is an intellectual who likes knowledge in general, so they didn't mind the English and Reading and Social Studies classes even if they didn't directly contribute to their future employability or character.

But most kids are not like that. It requires both luck and intellect to realize that you should make the most of the prison you are stuck in for your childhood. Many don't get lucky, and the mental synapses that would lead them to embrace the education system to extract value from it have a hostile reaction and abandon it.

And that doesn't mean they are dumb, it means they didn't get the right combination of random teachers to inspire them. If they could be inspired at all - it requires a specific brand of personality to be compelled into a repetitive environment and find value in it. Some just won't be able to no matter what.

None of it changes the fundamental flaw in modern education, which is the need to inspire children to learn the things that interest them most and learn about that. And to try to inspire them to find interest in all the things we routinely parrot from their textbooks and lectures onto tests for them to have a "placement" on. And if they don't? The parroting is rarely doing any good, and is only doing long term damage to the childs ability to reason and have a passion for learning. I don't think knowing the plot to Uncle Toms Cabin is worth that.


>But the entirety of HN is deeply biased because almost anyone here is an intellectual who likes knowledge in general, so they didn't mind the English and Reading and Social Studies classes even if they didn't directly contribute to their future employability or character.

Out of curiosity, was this what it was like for most people here?

by contrast, I virulently hated any course that didn't pertain to computer science. I had my speciality picked out at a young age, and my frustrations with the education system subsist almost entirely on it being unwilling to accommodate that choice.

I assumed that was a common sentiment here. Maybe not?


I remember everything about highschool trig. And chemistry. And physics. And certainly the rest of math. We might waste the time of the kids that are going to end up flipping burgers, but we don't waste the time of the brighter kids. If anything we move too slow.


So I own a burger restaurant in the bay area.................

You would be surprised by both the number of people that can't do basic math (i.e. figure out what combination of coins leads to $8.72) and the number of people that want a minimum wage job while they already have a college degree (worst being one in CS...in the bay area? really? followed up by a foreign MBA...)

Also, one day I was standing at a bus stop near the restaurant waiting to go back home and I listened to 3 ~18 year olds talking about their college classes. This is roughly how the conversation went: "that fucking bitch (at the local college) won't let me take statistics. just because I failed algebra 2 doesn't mean I can't do math. I'm really good at math! I passed geometry and shit! why can't I just take that statistics class?" "yeah algebra is really hard. I took regular algebra in high school and i had a lot of problems...I think I have to do it again" "I'm really bad at math. I have to take a couple math classes to graduate, so I might just drop out of school. I can't do that shit, I don't know how you can go on about wanting to take statistics". Then they went on about how science, english, and other general education requirements are bullshit. So I'm standing there - went to college at 16, forgot more math than all of them combined will ever learn, read biochem texts for fun on holiday and I _want_ to have the luxury to study hard and go to med school. My parents sacrificed a lot to do everything they could for me too - their parents couldn't. Life is unfair.

I had flashbacks to high school where everyone in my program within an inner city school were in honors precalc or AP calc AB/BC and AP statistics classes, but we shared PE classes with the "regular" school and we helped a lot of our fellow students learn basic algebra to pass high school exit exams and tests in between activities. Makes me want to figure out how to overhaul education in the US - it's garbage and it's failing everyone - my program full of "geniuses" and the single parents working a job struggling to graduate high school alike.


> I remember everything about highschool trig. And chemistry. And physics. And certainly the rest of math.

Report back when you're above 22.


I'm 30. I don't remember everything, but I do know the basics, and most importantly I remember what it is and what problems it can solve.


Couple of decades.

Anyhow, the important part about math isn't the memorization. It's the process.

Geometry and trigonometry aren't that useful by themselves. Even Algebra is marginal. Probability and statistics are way more useful. However, all of it comes together when you hit calculus which is VERY useful.

Can I integrate from memory anymore? No. Can I solve a differential equation from memory? No.

I'm going to use Mathematica if it's a one off. I will bone up on the process if I'm hitting it too often.

However, I'm always doing Bayesian probability estimates for all manner of things in programming, business, manufacturing, etc.


I'm turning 29 in a couple weeks.


Just because someone isn't good at mathematics (or whatever subject) doesn't lead them to a life of flipping burgers, nor does it mean they're less bright.


Yes, schools train and select for obedience. Certainly not intellectual excellence. (Such as contradicting what the fool in front of the class says; or deciding for yourself whether attending one day is an effective use for your time, compared to reading an engrossing book.) If intellectual excellence were the goal, the institution would look very different.

Those who succeeded in such a system tend to enjoy thinking it's a meritocracy, and implicitly consider themselves bright. Others obviously have a different perspective.


Well put.


And then they go to lives of selecting against people with more gumption. It may be good for institutional stability, but not so great for organizational innovation and adaptability.


It certainly can mean that they are less bright.


It could mean they're more intelligent and more creative too, just perhaps not having the patience or mental structure for learning and applying mathematics, chemistry, whatever. So that makes comparisons a bit of a moot point, and so the focus should be a) getting people into activities where they're learning things they're motivated and interested in learning about, and b) treating everyone kindly, which means having compassion which means having understanding about how intelligences work, how people work, and not using language that tells a person what they're be or not be if they do one thing versus another. If you tell someone that they'll do well at something, they'll do better at it than if you tell them they'll do badly.


> We might waste the time of the kids that are going to end up flipping burgers, but we don't waste the time of the brighter kids.

I would have said we really waste the time of the brighter kids. (Though we waste their motivation more.) We just waste it in a different way. Bright kids could pick up the math they need in a very short period of time on their own, but we present the material only in drips and drabs at a glacially slow pace with lots of repetition, working on the assumption that they're not really paying attention and probably forget everything during breaks.


Ask a mathematics professor to remember all the random formulas that kids are force fed in highschool trig. You are as likely to get a blank stare as an answer. Are they the "slow kids"?


No you're not. You're just making up tore own statistics. If mathematician can't recite it quickly, she can certainly develop the formula. Really, trigonometry? This is pretty simple stuff. Is it that the surprising to some of you that there are "programmers" on "HackerNews" that might, at whatever age, still know such elementary mathematics?


>I remember everything about highschool trig. And chemistry. And physics. And certainly the rest of math.

Who cares? Honestly? Do you think Jeff Bezos busts out his impressive high school trig in the middle of the day? Do you think Larry Page even remembers the formula for transfer of moment?

If all you learned for HS was the formula for conversion from moles to molecules and not how to apply logical thinking and abstract processes into practical applications you suffer from the same problem that the OP was complaining about.


I think your comments are misplaced and I think using billionaires as examples is a poor way to try to win an argument. Larry Ellison is probably not a very good person either, does that mean all the learning about treating people well in elementary school is a waste of time?


As a parent of three bright children with excellent development reports your comment makes me want to do you considerable violence. Regardless of whether my children were skilled in mathematics or not, if I overhead your derisory attitude to kids learning abilities I would probably punch your teeth out.

And no, I wouldn't be sorry. Sometimes violence is the best teacher and learning not to refer to other peoples children that way is probably far more valuable a lesson in life than your highschool trig.

Just remember, for every child you refer to as a burger flipper - X probability will have fathers standing 6' 2", weighing 240 and a former linebacker.


Possibly excessive snark but I'm having a bad day so screw it.

I find it ironic that ZenPro is so quick to violence.

That said insulting people's children is never nice, but everyone is someone's child so sometimes it's a necessary evil.


I never said I was quick to violence. I am normally a very calm person.

That comment was ridiculous. In this regard we define "someone's child" as a non-adult. Technically an adult is someone's child but it is unlikely they require their parents protection in some way so in the spirit of the comment it is fallacious to include adults in the reasoning.

Simply insulting kids under the age of 10 and pigeon-holing what they will do in life based on a narrow prejudice deserves a hiding frankly. It also neatly sidesteps the phenomenal salaries that people in the fast food industry can earn if they want to progress.

In addition, Zen is not the opposite of violence. Actually, it is one of the aims prior to entering into martial combat.

http://en.wikipedia.org/wiki/Mushin


Your propensity for violence is likely the least impressive thing about you.


You are right. It is.

Thank you for the compliment.


Like 3pt141597, I remember everything about high school trig. Not only do I remember it, but as it turned out, 10 years after leaving school, I actually started having to use it regularly (graphics programming). I also remember high school level calculus. Sadly I can't say the same for university-level maths, which I regularly have to break out the text books for when I need them (stats and probability, complex numbers being my two regular bugbears).

When I think about it, I struggle to find a single useless class from high school. If anything, I regret mostly that I didn't make enough of the opportunities that my school provided me - I really regret not having put in the effort in art classes for example.


I thought lightly of all the arts classes I had to take in school and that my parents wanted me to take for "a well rounded education". Drawing, painting, ceramics, piano... It took me almost 10 years after high school to realize how useful it's been for me in so many random little ways. Everything from being able to craft some of the things I needed for my wedding (designing my ring to sewing a ring pillow and painting table numbers) to just everyday mundaneness when I'm meeting with a client and I can just draw out what they're thinking and get the point across easily. The piano especially useful because it's so relaxing and good for taking my mind off stress/pain better than painkillers.

Never in a million years as a kid would I have thought sitting in an art history class or 10+ years of playing a piano would come in as handy as all the obsessive studying I did of science, english, math and other classes. I'm so disappointed courses like that are disappearing (along with home ec - never had that, but my husband can't mend clothes or cook...) but so many people are struggling with basic math/language skills that I can understand why.


Kids do what they're told. Problem is there are several types of learning. Among them are skill learning and another is exploratory learning.

For much of education's history - we teach skills. That we can test easily. That's fine - kids need skills. We also need to teach creative and exploratory thinking. But teachers must tell the kids to be mindful when they are working exploration mode.

Perhaps mornings are skill learning and afternoons are creative/exploratory. Perhaps you can test exploratory skills but don't ever give an evaluation to it back to the child - as they probably won't get evaluation for it in the real world.

In exploratory learning - you will need to test or check your own work. These kids need to learn that. There may not be one right answer - but a person should be able to evaluate their own hypothesis and not get upset if they're wrong. That is a life skill we could all use. Why do we have grade it anyway - a teacher can see when a kid is slacking off and grade on that.


Exactly.

When I follow these thoughts to their logical conclusion, I always reach the Sudbury model: let the kids figure out what they want to do and learn. Humans (and kids are human) are smart and savvy. Give them a system and they will learn to navigate it to their advantage.

But give them no system, let them find their own way, and they can achieve anything.

Teachers can see the symptoms that the system is letting down the students. But it is extremely hard for them to accept that it is the whole structure that is problematic requiring a massive redesign.


That completely ignores the fact that people don't know what they don't know, or tend to erroneously assume that something they don't know about isn't relevant.


There should be some allowance for self-direction, but humans are only "smart and savvy" over a period of several hundred generations and not any specific individual. By the time a kid figures out that he should have learned something, he'll have great-grandchildren (assuming he's still alive at all).


I remember trigonometry, I think must of us here do as well. You're talking about something. You're promoting this notion that kids should be uneducated because knowledge is useless and it's better to simply learn the lowest of material possible to help attain low-skilled jobs. This teacher is bemoaning the thinking patterns of kids, I think it is a very different issue.


You seem to have made quite a few assumptions about what I'm supposedly advocating based solely on the fact that I'm critical of the current system. (You might notice I've made no prescriptions here, only commentary on the status quo). But please, by all means continue to argue against this strawman you've constructed, don't let me ruin your fun.


Oh, sorry. But, it was a straw argument. You stated no one really remembers trig anyway. Just stating than any criticism is attacking a strawman does not imply the argument was not imply the argument was not made of straw to begin with. :)


They rarely think about what they are doing as long as at the end of the day their answer is “correct”.

Agree. These kids are just playin "THE GAME".

I don't know why this is bad. It shows the kids are smart.

At the end of the day, the only use of education is getting stuff.

The kids are focused on the "getting stuff", and see the eductation as hoop to jump through.

The truth is they are right.


> what do you remember about high school trigonometry? But you "learned" it right? Or did you, really?)

<aside>

Soh Cah To-ah

sin over hypotenuse

cos over hypotenuse

tan over adjacent.

I'm REALLY glad I learned that, That's helped me a lot. Though, it doesn't really take away from your argument.

</aside>


I'm 36, and math in my youth was always about getting the right answer, and proving how I got that answer, even up through college calculus and discrete mathematics. Heck, that's one of the things I really enjoyed about math, when compared to, say, English.

So, if math isn't about getting the right answer, it's been broken a long time.

Even today, math as I've explored it (which admittedly hasn't been much) has right and wrong answers. Sure, there's a bit of fuzz in the answers now, but a probabilistic model which can't determine ham from spam some majority of the time isn't kinda right, or even on the spectrum of right. It's just wrong, and needs to be fixed.


When I was a kid I used to love solving the same problem in multiple ways. I remember during a written test in high school, I solved some pre-calculus problems in 4 or 5 ways within the allotted time. The coolest part for me was seeing that, no matter the approach, they all led to the same answer. I didn't need to wait for my grades back from the teacher to know if my calculations were right. I got confirmation by simply comparing the answers that each method I used provided. I admit that at times periods in Trigonometry provided a two second scare when the results were equivalent but appeared different on paper. Once I had to get my teacher to change my grade precisely for this reason.

Anyway, getting the answer right and seeing which different angles I could take to get to those results were entirely part of the reason why I loved math. The answer wasn't per se the number at the end. The answer was, for each method I employed, the right sequence of logical and mathematically sound steps to that final value.

I would have never really been so fascinated by mathematics and related disciplines if it weren't for that sense of absolute, meritocratic, objective wrong or right.


Weird, I'm 26 and in college math was all about 'how you got there'. Our tests were like essays, a single problem in either paragraph form or formula form, and several pages to write down how you achieved your goal. Following proper procedure was rewarded, even if the answer is wrong.

In high school, however, my school implemented a 'pass the test first' policy. We learned what the Regents (spelling on this) exam would have and learned only that. Still, most of my teachers always graded the procedure and not the answers (in fact, my teachers suggested we look at the answers afterwards to compare if our answer is right).


Even if you have to show your work, a lot of students will just blindly try a few canned approaches until they get something that looks about right. Actually understanding what you're doing is definitely more efficient, but a lot of people muddle through without really understanding the reasoning.

One of the worst examples of that mentality I saw in a first year university course was a multi-part question where one part had the student calculate something and the next part had the student do something to sanity check the answer. One student got the first part wrong and then fudged the working in the second part to get it to say that the first answer was correct. Of course, this completely misses the point of checking the answer. Worse, they came up to me after class and argued that they deserved full points because the second error was "a consequence of" the first one. They seemed totally perplexed that I didn't agree.


> So, if math isn't about getting the right answer, it's been broken a long time.

There is a problem where children just learn algorithms. But they don't really understand numbers or how this algorithm works. And the bang numbers in and hope for the best.

Show them "speed = distance / time", and give them distances and times and they're fine. Now given them some speeds and times and ask them to find distance and they're stumped.

Math should be teaching children how to think, not just how to put numbers into a calculator.


Math has right answers, but there isn't anyone to tell you what they are. Most of the time you're doing math for real, when you get an answer the next step can't be to check the authoritative answer. The next step has to be something like checking with additional [mathematical] reasoning, running an experiment, (if you're doing math about the world), or consulting with your peers.


"...and proving how I got that answer..."

That's the part that counts, and is largely missing today.


I am not so sure you are talking about about the same thing as the article. Usually, when "proving your answer," I found there was still an expected procedure for you to follow, and you would lose points for deviating from that procedure even if what you did was mathematically correct. What the article talks about is learning how to reason about a problem even if your reasoning is questionable and even if you don't arrive at an answer.


That is unbelievably terrible. You would lose points for a mathematically sound justification?!?!

For example, when solving quadratic equations, I often used trial substitutions and the fact quadratic equations have at most two solutions to justify answers, instead of using quadratic formula. Why wouldn't this get the full mark? (I would actually argue this is a better procedure if you are looking for integer-only solutions, for example.)


What do you mean by "trial solutions"? Were you just substituting numbers for x and seeing if they worked out? If that is what you mean I can't see why it would be a better procedure than using the quadratic formula.


Yes, I meant substituting numbers for x and seeing if they worked out. To get integer solutions, it's enough to try divisors of the constant term. Try it yourself. It's faster.


Yes, it is faster for small integer values. Most numbers aren't small integer values. Test questions are usually designed to give nice clean answers but the real world isn't like that. They were trying to test your ability to solve quadratic equations in the general case. Your approach doesn't demonstrate that ability.


I'm not at all convinced that has changed.

I've read the Common Core (and the Australian equivalent) and they certainly require proving how answers were obtained.


When I was a kid, each subject had its own attractions. The notion of a "right" answer was one of the things that I enjoyed about math. It didn't make me any less inquisitive. I often tried to come up with unorthodox ways of arriving at those answers.

Naturally, there's a side of math involving getting an approximate answer that's good enough to be useful. One thing I've noticed as my kids progress through school is that most science being taught is qualitative or descriptive.

Adding some quantitative science to the K-12 curriculum would be a way to teach a more fuzzy approach to math. Naturally, science has its own notion of "right" that could be learned through the formation and testing of quantitative hypotheses.

There were many courses where I could get an "A" by writing a mountain of drivel that anticipated the teacher's social biases. I was thankful for the boost to my GPA, but the "no right answer" aspects of those subjects frustrated me. Oh well, different strokes.

One thing the standardized tests might be doing is to focus too much on memorized algorithms. And it may not be necessary. My son had a teacher in grade school who didn't even teach the standard algorithms for things like two column addition. Instead, the kids spend time trying to come up with their own methods. Yet his students get excellent scores on the state tests.


> math as I've explored it (which admittedly hasn't been much) has right and wrong answers

Try spending some time digging into the foundations of set theory. A good starting point is looking into the Axiom of Choice and the Continuum Hypothesis: are they true or false? Is there a "right" answer to those questions?


Proofs don't have a single correct answer they have a range of acceptable solutions. Any time your told to show your work there tends to be a multitude of acceptable answers.


I think he likes math because it's consistent. E.g. if I prove that some natural number "x" is prime in decimal, then "x" is also prime in binary, ternary, octal, etc. Counting "x" in a different base doesn't change the symmetry. Likewise, arriving at a theorem via different proofs just describe the same mathematical object in different ways.


That's not the point: they have a correct sequence of steps.


Hardly any math has a correct sequence of steps. (Unless you meant a sequence of correct steps?)


But you are aware that tests on math, particularly for school kids, generally do have a correct sequence of steps, right? The point isn't the answer, but rather to demonstrate mastery of a particular technique. In this case, the test should present problems where one particular set of steps is arguably the best or most obvious one. If the student uses a different set of steps but gets the correct answer, then perhaps the question needs to be asked in a different way (e.g. "Use Euler's Method to determine...").


Not really. Consider the following question, for example (taking from an actual eighth grade math test):

"Combine the equation 8x+y/2 = 1/24 with a second equation so that the resulting system of linear equations has (a) exactly one (b) zero solutions. Justify your answer."

That's leaving aside the fact that there are multiple ways to solve a system of linear equations with two variables to begin with (since you don't even need to solve such a system to answer the question).


But if you're teaching a particular technique, then the test would not be whether you can answer the problem but whether you understand how to use the technique.


And that, in a nutshell, is what's wrong with American math education. Too much focus on rote learning rather than actual understanding. What you're describing is an accounting course, not mathematics. Mathematics is about problem solving, not plugging numbers into a black box and blindly following a prescribed sequence of algorithmic steps. You do not want to just test whether students can follow a procedure; you want to test whether they actually know what they're doing when they're using it (even if you specify the procedure, e.g. Gaussian elimination).

This even applies to simple problems. See the comment by Aur Saraf responding to the original article: http://powersfulmath.wordpress.com/2014/04/30/who-or-what-br...


How is a student going to do any math if you never teach techniques? And how are you going to know if the student actually learned it if you don't test them? I never said they should be the focus, but you need to test them sometimes.

Also, algorithmic thinking is one of the more powerful ideas we have right now. You can trust that a Turing machine has certain properties without being a genius yourself, for example.


Take something real simple "Solve X + Y = 2; X + 2Y = 4 using substitution."

At the basic level that has 4 choices, you can substitute X or Y in the top or bottom equation.


WTF? A proof certainly does have a correct sequence of steps. You can't just reorder the steps in a derivation for example.


What kind of proof? Here are 102 proofs of the Pythagorean Theorem. http://www.cut-the-knot.org/pythagoras/index.shtml


Pretty much all of them?

Take the Euclidian proof (#7). The order of the reasoning is important, because each step builds upon what was proved in the previous step.


You can think of proofs as chains of logic. Assuming A, B, and C.

  (A+B) -> X, (A+C)-> Y
  (X+Y) -> Z
Now you need to prove both X and Y before Z. But, you could prove Y before X so It's not a strict question of order but one of dependence.

Going back to the proof 7 the line For the same reason also means the order is not strict and you could swap those lines without changing the logic.


Till my senior year of HS, I thought math was easy, but not very interesting. This was basically for the reason you describe -- there's one obvious right answer, and a set of steps you can mechanically apply to reliably get there. When I finally reached the creative math classes[0], I was surprised to find that math could be so engaging (and difficult relative to all the previous courses). All math may ultimately have a right answer, but there's much more flexibility and even subjectivity as you progress

That's also a big part of why I got into Computer Science. Computer Science[1] courses are basically a flavor of the interesting kind of math. I really enjoyed working through algorithmic and computability problems because it wasn't always obvious how I should approach them or if I'd even be able to figure them out.

I think it's sad the vast majority of students will never even reach a math course where they aren't performing calculations that could trivially programmed into a computer. It seems like there should be a way to give students a taste of that earlier on, but I'm not sure what the best approach would be.

0: This kinda started with integrals in Calc II, but didn't get really interesting till the my first 3000-level University course on proof by induction and the like. 1: Not programming or engineering, although those can be fun in their own ways.


Math is getting the right answer. Statistics and probability is not exactly math. They deal with chances. So she is wording it in wrong way.

Also, I should note that at METU no mathematician major student was able to graduate from statistics department as minor. They all gave up. That's what our professors told us. That is why math and statistics are very different subjects. It is like math and chemistry or math and computer engineering. Statistics do not deals with math, it has its own separate topic.


Practical applications of math are about getting answers, and unless you do math at university, that's probably all you'll ever do. But the problem is that if you don't learn to understand how the math works and why it gets you correct answers, you're unlikely to spot mistakes and likely to accept misleading reasoning and calculations at face value.


For me, practical applications of math are about understanding how a system works. More often than not, it really doesn't matter what the exact number is that comes out the end (the precision is often beyond my ability to act), but rather the understanding that well, if I take this action, these other things are going to happen at about this magnitude, so I need to offset them by doing something else, etc. etc.


From what I can tell, the author of the article wanted her students to engage in philosophy rather than mathematics, but maybe didn't make this clear to them.

Mathematics is indeed all about rigor, proof, and finding "correct answers". Philosophy, on the other hand, takes a more lenient approach, with emphasis on the process of thinking, rather than the final result.

Perhaps the children in this class merely thought that a more mathematics-oriented approach was what the teacher desired. That's reasonable, if that's how their previous classes had been done.

Apparently the students themselves adjusted quickly enough, once the teacher made her expectations clear. As the article states, "Things went so much better the second time around with not one student asking me if their answer was “right”".

So this could very well come down to a simple misunderstanding between the teacher and her students, in this rather isolated case.


I disagree with your claim that what the teacher wanted the students to do was not really math. Instead she was claiming that they (and apparently you) have a narrow conception of what doing math involves.

The "right answer" consisting of a valid proof is very different from the "right answer" to a numerical computation. The thinking and communication process is surely more relevant to the former than the latter. And if anything, as you say, the former is what math is really about.


Yes, "mathematics" is a relatively narrow concept, even if it embodies a great deal of knowledge.

When it comes to mathematics, the exact form of the outcome isn't very important. Maybe it's a valid proof in one case, maybe it's a specific number in another case.

It's the degree of rigor that's important. Rigor is what separates mathematics from other fields of study.

Thinking and reasoning about abstract concepts aren't without value, of course. But if this is done without an extreme degree of rigor, it probably should not be considered to be mathematics.


Something tells me you've never sat in on an algebraic topology class where the teacher draws a donut and a pair of pants on the board, along with a line or two, and calls it a proof. A key characteristic of mathematics is that it can be made rigorous, but we certainly don't work in full rigor day to day. A look at Principia Mathematica should demonstrate simply why that would be infeasible.


You just blew my mind. I now want to learn everything I can about algebraic topology.


Q: How can you tell there is an topologist in the cafeteria?

A: Because they can't tell their coffee cup and donut apart.


I up voted this because it made me laugh, even though I don't understand.


Fundamentally they both have only one hole through them, so under the right sort of deformations you can turn one shape into the other: They have the same topology.


As a practicing mathematician, I disagree completely. Much mathematics is done with the understanding that the ideas can be made rigorous, and the point of educating people in mathematics is to allow them to craft arguments and revise when they find holes. One can and should introduce rigor gradually over time, and get the benefits of learning to reason and doing mathematics.

When it comes to mathematics, the exact form of the outcome is extremely important. Proofs that are aesthetically pleasing are better than proofs that are not. Proofs that glean insight are better than proofs that do not. In fact, most people don't care about whether a theorem is true unless the proof provides sufficient insights.


If you want mathematics to be it's own little walled garden free from ambiguity, that's fine. But don't you think it's kind of important that people be able to translate between problems in the messy non-mathematical universe and mathematics.


I think the actual issue is their approach to learning, rather than their approach to math. The exercise with probability was meant to help them understand what probabilities are, so the process was more important than the result in that case. When learning math, it's often more important to make sure you're taking the right path toward the answer. And there's also a place for exploratory activities like this, where you figure out what your assumptions are before worrying about whether they're right.


You are right.


This is just another Common Core hit piece. More subtle than most I've seen, but the story only has one point: Common Core is bad, because standardized testing of students promotes rote learning in schools. I don't agree with this, but I hope it helps you understand the article.


It is absolutely not a Common Core hit piece! The teacher talks about how providing context (teaching!) allowed Common Core material to provide a very worthwhile lesson.

"If we are to truly make progress in getting our students to understand the concepts presented in the Common Core to the depth intended we must help them learn to stop looking for a right answer and start looking for a right reason."

Talking to a lot of teachers it is often not the provided Core material that is the issue, but not having enough time to actually provide the structure and context to help the kids internalize why any of it is important. It's also not a surprise that is the case, given that at least where I live (California) the school year is measurably shorter and class sizes are measurably bigger than they were when I was in the same grades.


Oops, I missed the paragraph under the last picture. I know a few educators at a private school, and they said the hardest part is proving that the classes they already teach cover the Common Core requirements. Admittedly, they have much smaller classes than public schools.


I firmly believe there should be two math subjects taught at school. One being "Calculating" where students do the boring calculations and all the stuff, calculus, algebra, etc. and another one where the interesting and challening part of math is taught in an accessible and fun way.

There are a lot of topics in math and related to math that could be explored in that second, new subject, probabilities, paradoxa, symmetries, basic set theory, even concepts of linear algebra/groups, etc. that are normally taught in university can be broken down to really great middle-school or high-school lessons. The only problem with that in normal math classes is, that the angry parent will quickly complain when that is talked about in a normal math class because they think that their kids should be bored to death by calculations that their smartphone does better already.

I was recently talking to a retired high school teacher who had taught language classes (he was really not a maths guy). We started talking about password security and piece by piece we discussed how easy those passwords could be guessed and in the end he had a good understanding of the concept of probability/information/entropy. This really made me think, if a really non-maths person can enjoy and understand a topic that a lot of students who have gotten STEM degrees directly refuse (something like "I do not understand what entropy is, that is something about the my room being in order or not but I have given up .....")

To repeat myself: There are great concepts and modern curricula available, sometimes for decades. However mathematicians, math teachers and parents are just not supportive in chaning anything substantially.


> One being "Calculating" where students do the boring calculations and all the stuff, calculus, algebra, etc. and another one where the interesting and challening part of math is taught in an accessible and fun way.

Wow. That can start a flame war. I am a huge fan of set theory and probability but since when calculus and algebra are boring? Sure if all you do is evaluate bunch of integrals, it is but what if you are learning about Fourier Series/Transform? It requires vector algebra heavily as well as calculus and I think it is just as stimulating as anything else. Sure it involves "calculating" but it is not a bad thing, calculating is at the heart of the subject.


My point - and I could have expressed this more clearly, granted - was that you have one class for doing the numbers (the boring stuff), and one for doing more theory and explorative stuff. I did not want to imply that calculus and linear algebra are boring by themselves. In practice, linear algebra is not at all taught in school, people are just forced to train a few landmark calculations (gauss algorithm, etc.) as fast as they can.

What I would very much like to see is that people get a good unterstanding of essential concepts, kernel, image, projections, etc.

As you mention, calculating is important, and thats why I propose two classes, one being about calculating, one being about the theory, in practice with the focus on _results_ in school, the conceptual part is left behind.


In that case fair enough. The problem would be that understanding higher level concepts one way or another requires the underlying mathematical rigour and mental agility to do the "boring" stuff. So it would not be dividing existing curriculum into to but doubling the amount of lessons. I would have loved that when I was in school but I know that that course despite all of its beauty would be extremely boring to most other people.


I think there are many topics in math that are left out because students have to train a few numerical algorithms. For example I was not at all exposed to logic in my math classes (if I was, it was very informal and only on a side note). However people would greatly benefit from this, probably even more than sucessfully calculating the shortest distanc between a line and a given point.

The additional class would have room for some algorithms as well, some graph algorithms maybe (DFS, Djikstra), maybe also Binary Search, etc. A lot of those concepts do not require extreme amounts of calculation skills but they train the mathematical mind.

We expose students in calsses on literature, geography, biology with various concepts, yet we limit contact to math on a narrow subset (roundabout requirements of an engineering college class for college-preparing courses). If you do not end up in STEM or CS, you will probably only use the rule-of-three after having left high school for 5 years.

Just discussing questions like: "You have a scale and 12 billiard balls. One is lighter than the others. How many times do you have to weigh balls until you find the lighter ball?" would be inherently beneficial to students. Students afraid of equations can work on this as well and even have fun.


Thanks for this! I wrote but didn't post a comment about how I hate that I'm starting to feel guilty for my lifelong interest in and enjoyment of a bunch of stuff that now seems to be scoffed at as "boring".


> Since when calculus and algebra are boring?

After reading your post, it sounds like it's pretty boring until you start hitting differential equations. That's a long time for most people to be bored with calculating before they get to the good stuff.

> Calculating is at the heart of the subject

As long the emphasis is leaving students with an understanding why it works and giving them the tools to create new methods that they can prove work.


I always found the symbolic manipulation to be somehow enjoyable; combining different elements, working up equations that were forgotten and tying it all together to answer a problem (abstract or otherwise). But then in day-to-day life I tend to find calm in mental arithmetic too.

It's perhaps a little like practising piano - the syncopation and variance of a piece that is an accomplishment offers rewards in itself. But, nonetheless playing simple scales or arpeggios can be an almost meditative exercise.


My son's elementary school homework this week included, after a series of arithmetic exercises involving adding pairs of even numbers, this sequence of questions:

What happens when you add together two even numbers? Is that true for all even numbers? Why?

Part of me was overjoyed to see an actual mathematical question being posed, but... what exactly was my second grader son expected to write under the 'why' part of the worksheet? There was a couple of inches of blank space left for him to fill in his answer. Is the teacher going to read his answer there and mark is right or wrong and hand back the homework? I don't know if I have much hope that this will be used by the teacher to gauge each child's understanding in order to facilitate further classroom discussion. Have the kids been taught a particular explanation in class which the homework is expecting them to reproduce? What is the teacher's goal in sending home this question as an exercise?

The fact that two even numbers always add up to an even number, as do two odd numbers, is an interesting insight into the shape and flavor of numbers. It gives you a practical tool - you can use this knowledge to parity check mental calculations and intuitively reject wrong answers - as well as an insight into the idea that esoteric mathematical properties numbers have can have interesting consequences when they interact, which is kind of the essence of a lot of mathematical thinking.

Is that always true? Why?

are probably the most important questions in mathematics. But... I have no idea how you teach that, except by having a one one one conversation with each student about it. I don't think you can get much value out of asking them on a single page worksheet and sending it home as homework, though.


I really hope your child's teacher doesn't just mark it right or wrong, but provides a counterargument for a flawed answer.

The question of "how do you teach one to answer 'why?'" is a difficult one, but here's what I understand is the established way of doing it. You give the student a problem, they think about it for a long time, and give their answers as to why. You discuss their answers with them, and then after they revise their answers, you present an elegant and correct answer. Then you repeat.

It's interesting, for example, that nobody asks how to teach one to answer "why" when you're, say, analyzing the motives of a character in some piece of literature. What do they do? They discuss, then they revise, and at some point the teacher presents either a good piece of student work or some other established ideas. Is mathematics really that different just because in the end you can tell with certainty that an answer is correct or incorrect? English teachers have one on one conversations with their students about their work, and they make students have discussions with each other about their thoughts on a daily basis. Why can't math teachers do the same?

I think part of the reason math is plagued by such questions is that most people don't realize that a proof (aka the answer to "why?") is not just correct or incorrect, but has aesthetic properties. Nevertheless, it's clear that mathematics is set apart from this on day one. Just compare any high school math syllabus to a syllabus in another subject [1]

[1]" http://j2kun.svbtle.com/what-would-math-class-look-like-if-i...


> I really hope your child's teacher doesn't just mark it right or wrong, but provides a counterargument for a flawed answer.

His son's teacher isn't going to do this. First, she didn't design the question... even if some public school teachers have the capacity to do this, they don't have the time or resources to design curriculums. Second, if she didn't design it, she likely does not have the wit to be able to argue about it without doing research. Third, public schools aren't places where they make time for each student trying to inspire genius, she really is just checking boxes down a list all semester. Fourth, doing so won't improve his scores on a standardized test. Finally, there was only two inches of blank space for the child to write an answer, she definitely isn't going to write him a thesis on the margins of the paper.

Public education is blind idiot god, handing down edicts from on high, some dumb and some smart, some sane and others insane, some incomprensible and some not. And often enough, all of these at the same time. The bell will ring, it will be time for a new lesson, and in that one he will be taught to never leave any multiple choice question unanswered because "they unanswered questions are scored just the same as wrong ones, you might as well guess".


I gave a two-inch "proof" elsewhere in this comment thread. You don't need a thesis to reason about even numbers.


You're not a school teacher, are you?


Your comments reek of pessimism and resignation. How can you hope to educate anyone with that attitude?


I have two young children, they won't attend a public school, or a private one for that matter.

I'm very optimistic about their education. However, there is no hope of educating masses of children in large assembly-line public school systems, because those systems aren't designed to educate.


>math is careening towards you and you should brace for impact

Hilarious, another great article from Mr. Kun.


The "Why?" question might not be so crazy if the kids had just learned about base 10 addition and if the book defined "even" as "numbers ending in {0,2,4,6,8}". Intuitively, the kids would need to figure out that the digits in the tens, hundreds, etc places don't matter (this is implied by information-flow through the "addition algorithm"). Then it's a simple matter of proving the 5 cases given.

If that's actually what they intended, I'm impressed with the material. It plausibly teaches proof-by-cases (real math!) at a very young age, but more importantly it teaches the mathematical skill of using what you know to prove what you want to know. I felt mildly clever after cracking the puzzle, I can only imagine what a grade-schooler would feel. More mathematical self-confidence than they would get from solving a sheet of 100 addition problems, that's for sure! Even if the lesson doesn't "stick" and teach the kids how to use proof-by-cases in general, it looks like a win in my book.


But really, teaching the 5 cases version is also not quite right. Maybe even numbers are "a bunch of 2s added up" and odd numbers are "a bunch of 2s and a leftover one". Then you just deal with the cases... two bunches of 2s is still a bunch of twos, if you have two of the leftover ones you get another two...

The point is, both lines of reasoning are fine. The value comes from having the students struggle through any of many, many ways of explaining it, comparing their answers, and thinking about the aesthetics of the models, and getting a tangible feel on their isomorphism. All of which you can do before needing to even have memorized your multiplication tables.

That said, 100 addition problems can be useful: jamming the addition and multiplication tables into your memory so you can do the needed symbol manipulation later. But similarly massive problem sheets for, say, remainders or algebra is just silly.


Exactly - I like the idea of trying to get kids to understand the difference between "I can' think of any counterexamples" and "There can't be any counterexamples because I can prove it always works this way". But I think two inches of space on a homework worksheet is probably not enough space for a child of 8 to explain adequately why the latter is true.

My suggestion would be that the child should write "I have an elegant proof of this but this worksheet is too small to contain it" in the space...


The "Why?" wasn't a prompt for a formal proof, and probably not a prompt to regurgitate what was learned in class. At least back when I was a kid, it was clear to me that "Why?" was a way of forcing the student to think about the problem. To attempt to justify your yes/no answer requires thought, whether or not you are right or wrong! If you only write yes/no, the teacher cannot tell if you actually thought about it or guessed.


A proof need not be "formal" to be a perfectly good, elegant, insightful proof.

For example, one could observe that an even number of quarters can be formed into two rows of the same size, and if you do this with two piles of quarters, you can put the rows together to make two big rows.

This is as good as any "formal" proof, and I sincerely hope it's the kind of proof the teacher shows as an example of a good answer.


We have to reference Tom Lehrer and the New Math era of the 60s:

"But in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer."

  http://www.lyricsfreak.com/t/tom+lehrer/new+math_20138395.html
I did new math in the 70s and it did prepare me for understanding unix file permissions. chmod 0755 foo.pl!


I love Tom Lehrer dearly, but I don't see how it's appropriate here. Do you think that trying to teach kids to understand what they are doing is wrong? Nowhere am I suggesting that I have the impression that the child writing down "When you add two even numbers you always get 7" would be considered a valid answer on this worksheet - clearly they're trying to get kids to comprehend parity.

Or are you suggesting that the answer would be different if you were missing two thumbs?


Math gets a lot of attention and therefore a lot of criticism from people who think math education is substandard and from people who think it is overemphasised to the detriment of creative thinking and such.

I think that overall, our math education is surprisingly good, possibly better than any other subject. Especially if you consider the level of proficiency a high school graduate at below average achievement has. Reading, writing and arithmetic. Those are still the things our education system is best at.

I don't mean that the average person will use trigonometry confidently, but they will use basic algebra. That in itself is an achievement. The same cannot be said of foreign language education, most science classes or anything else that requires cumulative learning. Most people that learn a foreign language (apart from english) for years in school do not come out with even a basic ability to communicate in it.

I think part of the reason is the very long tradition of teaching the subject. But I think another reason we're good at teaching math is the sort of fundamental property of the subject. It's unforgivingly right or wrong, which helps cut through the cynicism of a youth that doesn't want to be in that class. The problem with the "holistic" (scare quote intended) approach to subjects is that it opens the door to nonsense of the bikesheding type. It's hard to distinguish nonsensical but enthusiastic from genuinely thoughtful.

The hard right-wrong distinction in maths gives kids a feedback loop of positive and negative reenforcement. At the end of the year it's glaringly obvious if the student now has ability that he/she didn't before.

I don't disagree that the ability to reason is important. But, I don't think teaching "hard" maths is debilitating to developing that ability.

I'm not sure exactly where I'm going with this. Maths gets a bad rap. Teaching kids "real world reasoning" within the fake world of school is hard. No one broke the kids. They are just reacting to you changing the rules.


> It's hard to distinguish nonsensical but enthusiastic from genuinely thoughtful.

English teachers successfully do this every day, as do history teachers and teachers in most other subjects. Why can't we let mathematics teachers do this when their students provide proofs via creative thinking? Certainly anyone with sufficient training in mathematics can do this.

But overall your comment seems to fall short of reality. Math education in the US produces students that seriously underperform compared both to other countries and in light of the skills and knowledge required for technical careers. Contrary to your belief, high school students still generally can't do algebra. In my experience teaching the subject they don't learn algebra until they take pass calculus, and even then it's because it's a cumulative learning process, not in spite of it. Even if we teach reading, writing, and arithmetic well, that's just not enough.


What I'm claiming is that English teachers both succeed and fail at this every day. Of course, you can disagree. No one wants to tell kids their idea is wrong when the subject matter is somewhat subjective (like English or History) if they appear to be trying. Math has built in quality control because of the objectivity. It creates a constructive feedback loop.

I'm not from the US, I meant "we" as a whole. Obviously some places are better than others, but I think math education in 7-18 schools is better than physics, biology, history and most subjects. Obviously it's hard to compare.

Your point about algebra-calculus an interesting one. I've heard several time an interesting rule of thumb that to be a teacher you need to have studied about five years beyond the level you teach. I suspect that to be able to confidently use knowledge like maths knowledge, you need to have studied x number of years beyond. So, a person who never studied math past year 12, can confidently use maths learned around year 9-10 or somesuch.

In any case, I'm not claiming that we reading, writing, and arithmetic are all we should teach, just that we're relatively good at teaching these.


http://www.psychologytoday.com/blog/freedom-learn/201003/whe...

"Benezet showed that kids who received just one year of arithmetic, in sixth grade, performed at least as well on standard calculations and much better on story problems than kids who had received several years of arithmetic training."

I've wondered if it was a big waste of time to teach arithmetic in elementary school, if kids that age mostly just aren't ready and they'd catch up fine if you waited. This experiment seems to say that it's worse: it inculcates cargo-culting operations until you get the 'right answer'.

(I haven't read the original paper; the reference came from http://slatestarcodex.com/2014/05/23/ssc-gives-a-graduation-...)


After reading most of the replies in this thread everyone who thinks that maths is about answers needs to read A Mathematicians Apology: http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%2...

Mathematics has nothing to do with answers. It has everything to do with the most sublime and subjective beauty there is. The fact that this purely aesthetic system has says anything about reality should be more shocking than if we lived in a world where playing Beethoven's 5th symphony made it rain food from the sky:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html


When did we brainwash kids into thinking that math was about getting an answer?

That's what school is about. It's not math in particular. Doing well in the US mandatory education system is the art of finding the right answer.


Where finding the "right answer" is really more like guessing the teacher's password:

http://lesswrong.com/lw/iq/guessing_the_teachers_password/


He is talking about the same problem that is discussed in Lockhart's Lament [0]. If you've never read this piece of writing, I encourage you to do so now as it presents our math education system in a wonderful and critical light. I can't speak as elegantly as Lockhart, so I won't damage the piece by summarizing it or discussing it in the post; rather, I'll leave you to read it and draw your own opinions.

[0]: http://www.maa.org/external_archive/devlin/LockhartsLament.p...


IMO, this is one teacher who gets it. There is a lot of math that is entirely about answers that are right or wrong. Most of the time that includes how you arrived there. Given the references to MAP and Common Core in the blog, I'm sure there's a lot of that in the classrom.

However, there is so much of the K-12 education that arguably should be about exposure. Discussion about probabilities of pictures and phrases on a number line is a great example. Have kids make their case among peers regarding why 'It will rain tomorrow' is .9 in one group, but .75 in another.

For most, you don't get to pick your desired area of study/career until College. So why not use K-12 to expose kids to all sorts of different things without standardized tests as the only goal?


When Feynman was a boy he'd always relate the math problem to a real world puzzle. With trig problems, he might imagine being given a riddle involving a flag post and rope and calculating distance. One got the sense that he was, in effect, using math to prepare himself to solve future, real-life riddles. Or rather, math was giving him the tools to (correctly) answer questions he otherwise could not have answered - and Feynman's great preoccupation was his obsession with solving puzzles of all kinds, and he would enthusiastically (perhaps greedily?) grab at all the tools he could. (Physics leans heavily on trig almost constantly, so a total mastery of it would be quite handy for a budding young physicist).

It's also interesting to me that he was, at an early age, concerned with the usability of math, and was unafraid to create his own notation that was more comfortable for him (he liked the square root symbol and created analogs for sin(), cos(), etc.)

Frankly, I think this is a fantastic way to approach learning. After all, it feels good to solve puzzles; if you solve enough of them, the way they fall together, the way they relate to each other (sometimes in unexpected ways) become useful insights in themselves. With a large, solid core of puzzle mastery, you might even be able to turn your attention to the more difficult puzzles of "how to teach". (Of course, the greatest thing you can teach is the love of solving puzzles!)

I can't hold back anymore: what a foolish teacher! To get emotional over kids asking if their answers are right! In general, the yearning to be correct in one's calculations (and ordering cards is a calculation) is a good instinct, not to be beaten out of them.


I'd say it's the parents. Listening to parents of elementary school kids they seem to get confused or angry if there isn't one single right answer to an assignment. For example drawing shapes one line at a time and listing the lines as "commands" to a computer. There might be multiple ways to form a shape.

No use trying to argue that it might be valuable to understand how to think with algorithms, even though it might produce several different answers.


You hit the nail on the head here. Most of my family is entirely and strongly anti-education. My brother and I (32 and 29, respectively) were the first and so far only people in our family to ever go to college. My grandparents championed it, but our father has gotten progressively more and more anti-intellectual (he's what you might call a "crazy redneck"). Upon our last meeting he actually asked me if I am "actually smart" or if I am "just good with the computer." The sad part is he had a chance to go to college, my grandparents were going to pay for it if he would have just gone. But he's paranoid and thinks college is where they convince you to not love Jesus and to get rid of your guns.


That just moves the goal post, doesn't it? Such a parent is as much a broken kid as their offspring in this respect.


The parents might be afraid that their child could "fail" such an assignment, even if it's elementary school and the children aren't graded.


I've lost count of the number of times that Colin has posted an interesting article on mathematics education on Hacker News over a weekend. This submission here is particularly good. The problem indicated is quite stark, and very commonplace. Most pupils in a mathematics lesson in elementary school swiftly learn that "getting the right answer" is the point, and some check out and soon begin to doubt their own ability to REASON to the right answer.

The author of the submitted article writes, about a seventh grade class, "The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability." As the author makes clear, the particular lesson arises from the new Common Core State Standards in mathematics,[1] which are only recently being implemented in most (not all) states of the United States, following a period of more than a decade of "reform math" curricula that ended up not working very well. I am favorably impressed that the lesson asked students to put their numerical estimates of probability on a number line--the real number line is a fundamental model of the real number system and its ordering that historically has been much too unfamiliar for American pupils.

The author continues by elaborating on his main point: "When did we brainwash kids into thinking that math was about getting an answer? My students truly believe for some reason that math is about combining whatever numbers you can in whatever method that seems about right to get one 'answer' and then call it a day." I like the author's discussion of that issue, but I think she misses one contributing causal factor--TEACHER education in the United States in elementary mathematics is so poor[2] that most teacher editions of mathematics textbooks at all levels differ from the student editions mostly just in having the answers included[3] and don't do anything to develop teacher readiness to respond to a different approach in a student's reasoning.

What I LOVE about the Singapore Primary Mathematics series,[4] which I have used for homeschooling all four of my children, is that the textbooks encourage children to come up with alternative ways to solve problems and to be able to explain their reasoning to other children. The teacher support materials for those textbooks are much richer in alternative representations of problems and discussions of possible student misconceptions than typical United States mathematical instruction materials before the Common Core. Similarly, the Miquon Math materials[5], which I have always used to start out my children in their mathematics instruction before starting the Singapore materials, take care to encourage children to play around with different approaches to a problem and to THINK why an answer might or might not be correct. (Those materials, both of them, are very powerful for introducing the number line model of the real number system to young learners, as well as introducing rationales as well as rote procedures for common computational algorithms. I highly recommend them to all my parent friends.)

I try to counteract the "what's the correct answer" habit in my own local mathematics classes (self-selected courses in prealgebra mathematics for elementary-age learners, using the Art of Problem Solving prealgebra textbook[6]). I happily encourage class discussion along the lines of "Here is a problem. [point to problem written on whiteboard] Does anyone have a solution? Can you show us on the whiteboard how you would solve this?" Sometimes I have two or three volunteer pupils working different solutions--which sometimes come out to different answers [smile]--at the same time. We DISCUSS what steps make mathematical sense according to the field properties of the real numbers and other rules we learn as axioms or theorems in the course, and we discuss ways to reality-check our answers for plausibility. We don't do any arithmetic with calculators in my math classes.

[1] http://www.corestandards.org/Math/

[2] http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma...

[3] When I last lived overseas, I had access to the textbook storage room of an expatriate school that used English-language textbooks from the United States, and I could borrow for long-term use surplus teacher editions of United States mathematics textbooks. They were mostly terrible, including no thoughtful discussion at all of possible student misconceptions about the lesson topics or of alternative lesson approaches--but they were all careful to show the teachers all the answers for the day's lesson in the margins next to the exercise questions.

[4] http://www.singaporemath.com/category_s/252.htm

[5] http://miquonmath.com/

[6] http://www.artofproblemsolving.com/Store/viewitem.php?item=p...


My kids are now in middle school. The curriculum that they used in K-5 encouraged kids to come up with their own approaches. I'll admit that I was rather shocked, as it seemed quite messy and inefficient, but I discussed it with a neighbor who's a high school calculus teacher. He told me: "Be patient, just you wait, they'll surpass you in math." And he was right!

Also, our school kept several sets of the Singapore Math books on hand, and loaned them out to families who wanted to give it a try at home.

So I think the "correct answer" is not the problem, because that's just math. The "correct algorithm" is the problem. Or maybe, the algorithm is the answer. Oddly enough, the one course where the algorithm is indeed the answer -- high school geometry -- is the one that most people remember fondly.


Most pupils in a mathematics lesson in elementary school swiftly learn that "getting the right answer" is the point, and some check out and soon begin to doubt their own ability to REASON to the right answer.

It's been some years since I was in K-12 (I'm in my 30s), but I remember the opposite being the case: almost all the points on homeworks and exams were typically awarded for "showing your work". A correct answer with no work shown got few to no points; meanwhile an incorrect answer could get close to full credit if it showed the correct reasoning but ended up with the wrong result just because of a trivial addition/etc. error.

People tended to criticize this from the opposite direction, arguing that "in the real world" getting the right answer is all that matters, and that giving kids near full credit for incorrect answers was coddling them.


Homeschooling as in not sending your kids to school and teaching them yourself at home? This is such an exotic concept to me as a european, would you mind telling my why you'd do that?


My personal reason for choosing homeschooling for my four children, after discussions with my wife about our childhood educations in two different countries, was to ensure flexibility so that we would never be slowing down our children's education. That has allowed them to learn more at younger ages than is typical in United States schools, even in a good school district like the one we live in. This school year, all three of my children who are still minors are actually enrolled in public school programs (one full-time at the local high school, and the younger two full-time in an online public school). Next year we will try something different again. Flexibility and the opportunity to mix and match learning resources with children's needs is why we homeschool mostly, relying on outside resources as needed.

My badly neglected personal website[1] (I've hardly updated any page on it for YEARS), provides some more details of my thinking about homeschooling, current to a decade ago and perhaps not fully representative of my motivations today.

[1] http://learninfreedom.org/


I can't answer for the OP, but I was homeschooled and I was around a lot of people from other cultures that were also homeschooled.

In the case of the Americans, they were homeschooled primarily because the parents felt they could do a better job than the US school system, and because they wanted to use Christian-friendly learning materials. For social contact and 'real-life' social education, they'd put their children into a variety of (sports) clubs.

In the case of my parents, it was a practical necessity, as we lived in a developing country. The nearest international school was a long drive away, it was expensive, and reintegration into the Dutch school system would have been much more difficult.

Instead, their solution was the get a just-out-of-school teacher to have one or more exciting years volunteering and practicing their craft on me and my siblings, which my parents would augment with personal attention and long-distance help from retired teachers back in Holland. The teacher would use a Dutch 'long-distance learning' approach that was the defacto system for Dutch expats. For high-school material, this meant a book that explained, day by day, what should be read and what exercises should be done.

Now, I personally feel the results were so good, that I would do everything to make it possible to either home-school my children (if I have 'em), or to get them into some alternative school (montessori, etc.). Aside from some difficulties adjusting to Dutch society, my siblings and I did quite well, and I believe on the reasons we are all active learners and autodidacts is a direct result of not having been part of the normal system.

The primary argument I hear against home-schooling when normal schooling is possible is that it hampers a child's social development. I never bought that. Kids get a lot of that from playing out in the street, or being part of clubs. Furthermore, for a significant portion of kids I believe their 'normal' schooling can have a negative social impact, through various degrees of bullying, and the group pressure to believe that learning is stupid. I did not have a clue that voluntary study was stupid and nerdy until I returned to high school at age seventeen...

(of course, I also understand that for many people home-schooling isn't a practical option. I think that's a shame. And I also understand that some parents might not be any good at teaching, and that a mandatory educational system can be a benefit in that case.)


The most common reason around here are parents who don't want their kids religious beliefs distorted by the general public and secular school system. There are religious specific schools but they tend to be expensive and far apart. It is easier to just pull your kids from school and have a parent teach them, supplemented with clubs and special activities.

Another reason is parents might not want their kids to have to cope with the social pressures of the school resulting from both income disparity and bullying.

Some kids just don't do well in large group activities with generic and/or dumbed down lessons. 1:1 teaching and custom tailored educational experiences can really help some kids excel. When the household has enough money and a well enough educated parent, to support a parent staying home doing full time teaching, these kids get educational opportunities they wouldn't get in a large public school system.

Similarly, for whatever reason sometimes the family lives in a school district with either terrible schools or schools with such a cultural mis-match that it doesn't make sense to even bother sending your kids there.

Schools in the US are not all equal. Even a few miles apart, schools can be VERY different. And kids are all different too. We have the opportunity to be flexible about educating our children, which I think works out for the best for the majority of such special cases in the long run.

I have seen home schooled kids who have serious issues with: 1) Deadlines 2) Structured Schedules (like showing up for work on time). 3) Coping with social situations that are outside their comfort zone. 4) Separating from their parents. 5) Recognizing that not everyone learns the same way or at the same speed or is as smart as they are.

Sometimes they end up in trouble and unable to be self-sufficient in our society. Sometimes they find a niche that works for them and are very happy. Sometimes they have none of these issues and you'd never know they were home schooled.


School did.

When I was a very young boy, 2 or 3, my parents did everything in their power to ensure I could read. They would read books to me, I would read books to them. It is the very reason I am so well spoken and intelligent to this very day. Books were awesome. I loved books.

Note the past tense. Loved. I have not been able to will myself to read a book on my own since the third grade. It was Lois Lowry's The Giver, that is the last book I can confirm to you that I read, in whole, because I wanted to.

It was in the second grade that I had to do my first book report, and the entire concept of reading a book not for joy, but for work's sake, was a concept I could never rationalize. Even while I was reading The Giver, a book I had selected out of the elementary school's own library through my own volition, I received snide and discouraging comments from the library staff and teachers. "He shouldn't be reading that book." "That book is too advanced for his age." And on, and on.

Something changed. I lost my will and my zest for books. Even books I would've very much liked to have finished and I found interesting, such as Mick Foley's Have a Nice Day!, his autobiography about breaking in to the wrestling business, I have not been able to finish. I have no will left inside of me to crack open a book, to even lift one off of the shelf. I feel like my love for books was steadily beaten out of me at a young age by the very institution whose job it was to educate me with them.

At least this author seems to be on the right track. In all of my years of watching society ask "What is wrong with the kids today?" so few have bothered actually trying to ask the kids themselves.


If you have trouble finishing books, try audiobooks. That's what has worked for me, and I got subscription at audible.com, which is great.


In his “Intuitive Explanation of Bayes’ Theorem”, Eliezer Yudkovsky wrote:

It’s like the experiment in which you ask a second-grader: “If eighteen people get on a bus, and then seven more people get on the bus, how old is the bus driver?” Many second-graders will respond: “Twenty-five.” They understand when they’re being prompted to carry out a particular mental procedure, but they haven’t quite connected the procedure to reality.

I was awe-struck, so I asked a friend who sometimes teaches second-graders to try this. 11/18 wrote “25″, 5/18 wrote “25 passengers on the bus” and 2/18 returned a blank note.

I think this is a big part of the explanation. If you’re taught addition as a process that happens in a notebook, not in reality, then you have no way to separate answers that make sense from those that don’t. You also have no way to connect math to things you experience in your life, and I think the most common way to develop an interest in something is to find out it’s related to something else that you’re already interested in.

In the last Super Bowl, the Seattle Seahawks scored 8 points in the first quarter, and 14 in the second quarter. Who won the match?


3 + 4 = _____

There's your answer.

When kids are measured by "correct answers" and pressured to find them, then that is all they will care about. They aren't, then, doing math— they're generating correct answers.

When you add multiple choice computer-readable tests into the fold, the problem compounds on itself.

That is what broke our kids.


So the author asks a room full of kids to arrange numbers on a number line, then gets mad when the kids want to know if their answer is right? As far as I could tell, it was an exercise with one right answer. Why get mad at the kids for wanting to confirm whether they understand the material?


He's not upset because they want to know if they're right, he's upset because they just grab some options, throw them together, and check with him. There's no thought going into it and I'm sure if you ask the kids why they chose the locations they did, they'll have no response.

I recently graded some of the math final tests for my wife's second grade class, and the students who are doing well all did well because they extracted the parts from the problem that were required (even simple things like how many jugs of milk are shown below), while the students with a history of not doing so good in math instead picked numbers off the paper and either added or subtracted them, assuming they just needed an answer. There was no thought about the process, just the end result. Got some numbers? Good enough.


What's the exact probability of "it will rain tomorrow" or "You'll visit the beach sometime"? Maybe the point was to figure out that there are some known and unknown factors that affect both probabilities while you could still be pretty sure that you'll go to the beach sometime.


Some of them weren't numbers, there were symbols and natural language statements which required interpretation.


Because there is a deeper underlying problem in education - we have been teaching kids to pass tests for quite a while now. And the growth of the Common Core standards is now teaching them to pass that that use new terminology coined from those standards.

These goals are creating changes in curriculum. Some schools are devising new curriculum that is working well, some are not doing so well.

You can compare it to coding in some ways - if the entire industry was told to drop their current toolkits and use new ones, how would it go? There would be some successes, some failures, and a few years of trial and error until new best practices come about.

The Common Core is making this happen within education. Some educators think it is great, some educators think it is horrible. But it definitely is impacting the kids.


I must have been taught critical thinking by my parents as even as a child, I could see that the education system was mostly about regurgitating not even the "correct" answer, but the answer expected by the teacher. If you want to see who "broke the children," you need not look much further than that. I am glad that teachers occasionally realize that there is value to being able to reason about a problem rather than just returning the answer by rote. It is rare that grade school teachers even notice there is a problem.



If I were a teacher/professor, the words "Will it be on the test" would be criterion for an an F, or a detention, or something along those lines. But somehow we've codified that mentality into law? Jesus.

We've broken school by making it all about getting a piece of paper with a number on it, rather than by being about being able to do things better each day. Learning is about answering "Can you do more, or do things better today than you did yesterday?" Repeat this process and eventually you're good at things.


> If I were a teacher/professor, the words "Will it be on the test" would be criterion for an an F, or a detention, or something along those lines.

So I assume you're not giving tests then? Because jesus, if you are, and those tests are consequential to the kids future, that's a pretty damn relevant and reasonable question. (Also: seriously, you'd punish a kid for asking a question?)

The only way that attitude even remotely makes sense is if your students are there voluntarily, but lets be real: they're probably not. Either they're forced to be there (K-12), or they're compelled to be there for job prospects (college). You're probably lucky if even %10 are there because they want to be there.


> those tests are consequential to the kids future

Most of them aren't. Especially in college.

> lets be real: they're probably not

Welcome to the REAL problem in education. We fight wars over the right to education, and then when we have it, we piss all over it?

> You're probably lucky if even %10 are there because they want to be there.

If we stopped obsessing over the damn tests, we might make some progress on this. It's a failing of culture and of teaching that students don't want to learn. I'd much rather have a country full of people who enjoy learning than people who learn only at metaphorical gunpoint, and spend their lives trying to avoid it. If they're a couple years behind at 18, so be it. I'll take the learners at 25 any day.


Not the parent commenter, but:

>Because jesus, if you are, and those tests are consequential to the kids future, that's a pretty damn relevant and reasonable question.

The point of a test isn't to test every single detail of what you were taught, it's to hit a random distribution of topics and examples of what you were taught. This gives a pretty good indicator of how well you learned the subject overall.

Asking "is this going to be on the test" shows that you have no intention of actually learning everything you're taught, which is the point of the class, and instead you're only interested in memorizing the minimum to pass a limited test, which is not the point of the class.


> Asking "is this going to be on the test" shows that you have no intention of actually learning everything you're taught, which is the point of the class, and instead you're only interested in memorizing the minimum to pass a limited test, which is not the point of the class.

I think you have it backwards. The point of the class is to make kids pass tests. The fiction is that it's so that they learn the material. Learning is ancillary to the systems actual purpose. The fact that a lot of people that are part of the system are well meaning and don't realize this doesn't negate the design of the system itself.

I realize that sounds like some sort of crazy conspiracy theory, but keep in mind that "schooling" and "education" are related but separate things. Public schools are a relatively modern invention from around the industrial revolution, and the inventors of the public school system explicitly designed it to create people that were obedient. They weren't even particularly secretive about it -- a lot of the people that helped found what we think of as our modern school system said as much directly.

Not the most elegant page, but here's a good book on the subject from a former school teacher: http://www.johntaylorgatto.com/underground/toc1.htm (I think at one point he even won a teacher of the year award, or something along those lines, so he's not necessarily just some crackpot)


I wrote so many tests in university that were poor reflections of what was taught in the course that I couldn't ever fault someone for optimizing. There's not enough time in the world to study every word that was said in every course in detail. Students have to try to pick out the important stuff and focus on that.


Knowing what a test is going to be like is pretty useful if that test matters at all. Otherwise there is a higher chance that the people who fare well on the test will be those who happened to work more on the topics or skills which come up. I'm happy my lecturer for group theory in university let me know which of the 5+ page proofs of theorems from the course wouldn't need to be known for our exam - it wouldn't have been very good use of my time to fully internalise all of them. However they were interesting and I think it added value that they were in the course.

I think you just have an idealised picture of how tests work. In the best maths exams I have taken, you know all you could within reason about what you need to know and what will be the content of the test, and yet the questions still surprise you and cause you to think in new ways.


Knowing what a test is going to be like is pretty useful if that test matters at all.

Knowing and understanding the subject matter is likely to be pretty useful if the test matters at all. This is why you study and understand the entire course, which is the point of the course. However, being told explicitly which portions you will be tested on goes against this goal.

which of the 5+ page proofs of theorems from the course wouldn't need to be known for our exam - it wouldn't have been very good use of my time to fully internalise all of them

What? If you're doing mathematics correctly, you are ably to derive the proofs on the fly. You don't have to memorize "five pages of proofs". You understand the concept, you understand the mechanics, and you derive the proof as necessary.

I think you just have an idealised picture of how tests work.

I'll refrain from saying what I think about your "picture of how tests work".


Maybe I worded my post badly; I was trying to say that your idea of a good test isn't in line with how tests are designed in my experience of education. If you agree with the sentiment that children should be punished for trying to optimise for exams, then yes I think your picture of how tests work (at the moment, in school especially and somewhat college) is wrong. I see nothing wrong with a well designed test, made to cover a whole course and to be fruitless to optimise for. At least in the UK in school, all STEM type exams are not like that as far as I remember. Humanities exams are much closer in style to this.

I do think GCSE and A-level science exams could be much better designed. At college level though I'm not sure the way you describe is the only good way to do things.

I'll refrain from saying what I think about your idea of how to "do mathematics correctly".


I think learning comes naturally to all of us when we are trying to find an answer or solve a problem.

Perhaps a good way to interest kids in math is to offer them the opportunity to make something that actually requires learning some math.

Let them try to empirically do something that can really only be done with a formula. Show them the power! Without it, the world as we know it would not exist.

If you're really motivated, you'll spend years seeking the answers. We should be motivating without coercing, providing tools for discovery, then just stay out of the way.


If I have to solve exercises and after that, I have no way to know whether or not I did the right thing, I don't learn anything at all. The point of exercises is to learn something new and uncover your misunderstanding of the subject. Your wrong solution clearly points to the things you have to work on. No solutions = no learning. Mathematics is about getting answers. It's a formal way to declare problems and seek for answers. Right answers. I'm glad I wasn't taught maths by this person.


The article's point is perhaps that it's better to try and gain some intuition for whether or not you have the right answer, rather than checking the answer key. Relying on the answer key (or waiting for the teacher to say "you are right!") is an extremely poor habit and we should avoid teaching students to depend on that (in maths class or otherwise): in real life, there is no answer key.

One of the philosophical ideas of maths is that there are multiple ways of looking at a problem, and those multiple ways should feed back onto each other to confirm your answer.

Maths is not just a bunch of rules that have to be followed without question: in this case, maths is modelling something practical and the answer needs to match reality. Without that, the probability model could not even have been invented. If you can't get beyond the stage of do_question+check_answer, you are never going to be capable of anything more than grinding out other people's mathematical results.

I'm sorry you weren't taught maths by this person.


Before you've gotten a firm grasp on the subject, you might be convinced that your answer is right but it's not. Especially when the maths becomes more complicated. If nobody tells you it's wrong, you will incorporate more and more wrong things in your understanding.

I notice this horrible effect every time I'm solving exercises for uni and there are no solutions. I think I've done good work. Sometimes, everything really was right, other times, I made fundamental mistakes all across the exercises, showing me that I didn't understand something important.

Of course, when you're in research, there's no key. That's why it's so difficult. But it doesn't make sense to reinvent the wheel, as in, researching things that are already well known.


In real life there is no answer key but there is still validation, which is all your parent is asking for and probably what those kids were really looking for.


Why? Because students today are focused on "doing well". That means getting good grades. Look at your tests, be them locally-created or standardized. How are they graded? Do students get good grades for debate or mechanically computing the right answer?

Who? An American culture that constantly enforces a science v. arts dichotomy. I say American because in no other country is this split enshrined in language: Metric units for the scientists and English for everyone else. I say American because few other countries produce so many millionaires with zero science education. Look at your student's role models. Athletes. Celebs. Seth McFarlin. Politicians. Donald Trump. The American system does not reward scientific creativity in the same way it does social aptitude.

Change would be painful, so painful that it cannot happen for a generation or two Just try suggesting highway signs be changed from mph to kph. And that's easy part.


I think that the obsession with "doing well" is by no means an exclusively American trait. Just look at the Indian obsession with IIT-JEE, or Koreans with the CSAT, or the English with A-levels, or the French with the bac. American students, in my opinion, don't really obsess more about our system that anyone else.


I recommend a simple, fun solution that kids of all ages LOVE:

1. Download DrRacket http://racket-lang.org/download/ 2. Show the kids cool functions from the 2htdp/image library, like (circle 10 'solid 'red) 3. Show them the animate function from the 2htdp/universe library 4. Leave them to their devices and instruct when they have questions

It won't be long before they want to make a game, and they'll need to learn all sorts of things about coordinates, shapes, colors, animations, and functions. Kids love functions. They'll have no idea how much math they're doing, yet at the end of the day they'll actually get it.

For probability, check out the (random) function.


Whatever broke the kids, it's fairly predictable that Common Core will break them more.

Hey, let's do a probability problem? How many teachers will turn this lesson on it's head as this teacher did transforming it from something useless to students and painful to teacher? My guess is somewhere south of 1%. In the spirit of the true lesson here, I welcome well-supported alternate guesses.

Centrally directed (er, suggested) curriculum will, I predict, relieve teachers from their responsibility to get results. How can you fault the teacher in the next classroom for not having had a meltdown and simply suffering the through the banality of the exercise as formulated by elite geniuses who have surely figured out the ideal pedagogy.


I think the point the author makes is fundamentally correct: the correctness of answers comes from human reasoning, and is subject to discussion and debate. It's not something completely external, and education shouldn't be a game to guess what these external decrees are. As the author says, what should matter is what the student thinks is correct, not whether they think their answer will be accepted as correct.

However, the author uses language that is belittling and rude. I think that teachers who use students as an emotional outlet for their frustrations are very harmful. I would rather see the teacher find another way to get the students to see her point of view, than emotionally browbeating them.


These conversation and facilitation skills among kids are really great, and I agree the teacher should be encouraging them, but I have some serious questions as to whether this teacher and I have the same definition of "math".

To me, math is a skill involving the manipulation of abstract symbols. It doesn't involve sharing, being nice to other kids, helping each other out, or any of that. Those things are great, but they ain't math.

Now "teaching math in a primary school setting" may involve the combination of all of those things, and that's fine. So I guess we agree. But this essay was terrible. If this teacher came to me with this definition of math as an answer on a test, I'd fail them.


> To me, math is a skill involving the manipulation of abstract symbols. It doesn't involve sharing, being nice to other kids, helping each other out, or any of that. Those things are great, but they ain't math.

Coincidentally the math stack exchanges are regarded as the most toxic parts of that community.


You know, the really sad part about this essay is, because the teacher doesn't know what the hell they're teaching, they miss the bigger picture. These kids have been learning-crippled. It's not just math. They're looking for a mechanical process for everything. That's a tragedy. But it's lost in some misguided rant about how math might be involved in what's going on.


Who the hell cares about definitions? Oh right, mathematicians, and they manage to make a beautiful subject terribly boring by insisting it's all about manipulating symbols, instead of the underlying structures they represent.

I mean, programming is about typing and manipulating text into a box, right?


>One student had seen the weather and knew there was a 90% chance of rain the other had not seen the weather and though the probability was 50% since it would either rain or not. They compromised and picked the middle but that’s not the part I cared about, I cared that they had a reasonable discussion about their thoughts.

Wow. One of those students was right and the other was wrong; if the teacher praised the discussion instead of pointing out AND EXPLAINING the right answer, then the teacher did a disservice to the entire class.

I've heard jokes about touchy-feely "Bill has five apples and John takes two, how do we feel about that?" math, but I honestly believed they were jokes.


Neither student was right or wrong. This is a Bayesian problem. Each student had a distribution of priors based on evidence. Each source of evidence had its own level of quality which stipulated the variance of the two sets of priors. The challenge is how to combine these priors to create a posterior distribution.

Another way to look at it is that, because "random variation" is how we refer to unknown influences, the perfect-knowledge probability was either 1 or zero. Neither student had perfect knowledge.


> [One student] though the probability was 50% since it would either rain or not.

That student might have the right answer, but their explanation is dead wrong. I hope that (common) error was pointed out to that student and the whole class. The story doesn't say that it was.

I should not have said the other student was "right." Their answer is just rote repetition of something they heard, without an understanding of the methodology, but their answer is more likely to be correct.

If, in the end, they decided to average the two answers... then their discussion was unproductive and did not arrive at the correct conclusion.


You may be right about that. Unless I have flubbed my calculations, if P(estimated chance of rain=.9|rain) = 1.4(P(rain)), then Bayes theorem leaves the updating at average(.5,.9). Also, an interesting reference about selecting priors: see 2.2 in http://www.stat.cmu.edu/~kass/papers/rules.pdf. Which is not to say that every set of procedures to calculate answers will produce equal errors. There are provably better procedures and provably better answers, given a set of assumptions.


Technically, lacking in any evidence one way or the other, 50% is a correct prior. There were two mutually exclusive possibilities, the laws of probability demand all mutually exclusive propositions add to 100%, and we become subject to various biases if we refuse to assign uniform priors in this case.

Source: Probability Theory by E.T. Jaynes


What exactly is the correct conclusion for them?


The right answer here is not a single correct number, but a handful of correct technique or methodology. Counting the number of rainy days in the last x days would be one possible method of prediction. There are many wrong ones too, like "rain has four letters, so it's 4%" or "there are two choices, so it's 50%."

I hope the teacher taught the students which answers were right and which were wrong. If they did then I retract my criticism, even though that wasn't clear in the article.


Which is then followed by:

> Why Constructing a Viable Argument and Making Sense of the Reasoning of Others is Crucial. This may seriously be the most vital mathematical practice.

Where the emphasis is on developing an understanding, a viable one (ostensibly leading to a correct answer).


Where the emphasis is on discussion and understanding each others feelings.

>We brought it back together as a whole class to follow it up and each group shared the most interesting conversation that they had.

Where's the part about explaining a correct methodology and the correct answer? I don't believe that rote memorization is the right way to teach math, but discussion and actual understanding are not the same thing. Discussion can LEAD TO understanding, but should not be mistaken for understanding.


I disagree: The teacher's job is to teach mathematical reasoning, not to teach students to retain meteorological trivia from earlier in the day.


That's why I hope the teach corrected this student:

> [One student] though the probability was 50% since it would either rain or not.

His or her mathematical reasoning was wrong; having two options does not mean there is a 50-50 chance.


She is a teacher but as you expect she is not a statistician. If she would have studied statistics in statistics department for a year she would know what to teach. First you start with a coin, you have 1/2 chance to get this and that etc. This means 50%, then you repeat your move 100 times and you say see, the output is 53/100. Then you repeat and the output is 502/1000. So they start to get what 50% means. Then you do it with die and so on. However she has started it in wrong way. And I'm sure that most of the kinds do not know what 50% chance rain means.


It wasn't until a 10th grade class in world history that I encountered a teacher who wanted you to not only know the answer, but why the answer was correct. He ended up being one of my favorite teachers ever.


> If You Can Type the Problem into Wolfram Alpha and Get an Answer You Aren’t Doing Math

If something can be computed with WA, it's somehow a waste of human time to do by hand. I could only hope, eventually, someday, we'll teach those aspects of math somehow combined with algoritmization and programming. Like, "Read on algorithms that could solve this class of problems, understand those (how they work and why), implement them, run the problem sets to see how it works in practice."


It's unclear to me what the problem is that this teacher encountered. The students were given an exercise to learn something. They wanted to validate whether they learned something, by validating their solution. This could be achieved by reaching agreement with other students or by asking the teacher. So the problem is that only that ne was unclear that they were supposed to reach agreement instead of asking nir?


> Who or What Broke My Kids?

From the OP, sounds like heavily that teacher did. Let's see why:

> understanding that all probabilities occur between zero and one

Better would be, "a probability is a number between 0 and 1 with both 0 and 1 possible".

Try not to use "occur" here because if event A is "it rains" and it does rain, then we say that "event A occurred". So, it is better wording to say that a probability "is" than to say that it "occurs".

> differentiating between likely and unlikely events

Using the word 'differentiating' here is not good because (1) it is crucial in calculus where it has a quite different meaning and (2) it is just too long for the simple concept of, say, 'identifying' likely and unlikely events.

The worst is,

> The basic premise of the activity is that students must sort cards including probability statements, terms such as unlikely and probable, pictorial representations, and fraction, decimal, and percent probabilities and place them on a number line based on their theoretical probability.

I have no idea what is intended here! Probability theory was the main foundation of my Ph.D. in engineering from my research in stochastic optimal control, and I can't make much sense out of the teacher's statement. Calculating probabilities of poker hands would have made much more sense. Or, just for a start:

We shuffle a standard deck of 52 cards and pick the top card. What is the probability

(a) the card is the queen of hearts?

(b) the card is a king?

(c) the card is diamond?

(d) the card is a spade?

(e) the card is a face card?

Suppose that first card was none of (a)-(e), and we pick the next card from that deck. Now what is the probability of each of (a)-(e)?

Or:

We flip a fair coin five times. What is the probability we get Heads exactly three times?

> where they can solve any computation problem with technology with no issue.

I wish that were so! I guess the teacher has not heard of the problem in the set NP-complete and how common such problems are in business planning and scheduling.

> the probability was 50% since it would either rain or not

Uh, can we place some real money on that!!!!!!


Your questions are on discrete probabilities with knowable objective answers (cards/coins/dice), but the teacher also needs to teach the sense that there are continuous probabilities with fuzzy subjective answers (weather/real life events).

This may be one source of the teacher's frustration: most likely the students are answering the question "what's the probability of rain tomorrow?" as though they are guessing the teacher's password "umm... is it 0.5?", as the teacher has effectively taught them to, when in fact they are being asked to have their own opinion.


For your first paragraph, I was trying to teach a little about probability. For the probability of rain, I mentioned that, and my mention implied the good lesson there: We can assume that there really is a probability of rain, that is, it definitely exists. Now, especially for deciding to place a bet, we could get an estimate of that probability. The 50% is a first cut estimate with some curious properties about 'total ignorance' but otherwise poor. So, how the heck to get an estimate? Likely the best practical way was by the student seeing a weather report on the news. Otherwise, for, say, the probability of rain of any day in, say, August "We might, what, class? Anyone? How might we get an estimate of that? ...?"

Or, "Billy, what company makes iPhones? Billy, you can't get the answer by teasing Sandra. Mary? Right, Apple! Okay, suppose we buy 100 shares of Apple stock today and sell it seven days from now. What is the probability we make over $1000? Assume that this probability exists and we want to estimate it. People especially good at such estimates can become among the richest in the world, fairly quickly. So, how might such an estimate be made? Information? Any ideas? Anyone remember the motorcycle, airport, and airplane in the movie 'Wall Street'? What was the role of 'information' there for an estimate of the probability that the stock of the steel company in Pennsylvania would go up a lot, soon? Anyone? Sandra?"


The point of the lesson/activity is to evaluate whether the students understand that 1/8, 0.35 and 50% are comparable things. It wouldn't be a great idea to assume that the preponderance of 7th graders have mastered this.

If you click through to the pdf, the language in the standards is more careful, so it's a little easier to make sense of.


> It's been a while for me, but I thought that that was fifth or sixth grade!

For the seventh grade I remember: "A race track is 2 miles around. A car travels the first half at 30 MPH. How fast does the car need to travel the second mile so that the average speed around the whole track is 60 MPH?"


Infinte speed! Mathematically but not physically possible.


You got it! Yup, it was a trick question! It took the class about 40 minutes before the first student got it! But the class didn't have any trouble comparing 1/8th and 50%.


Are you disagreeing with my assessment or just riffing on how much better your class was doing?

I wouldn't want to speak real confidently about things that happened 20 years ago, especially my 10 year old assessments of the mental states of other 10 year olds.


You wrote:

> The point of the lesson/activity is to evaluate whether the students understand that 1/8, 0.35 and 50% are comparable things.

It appeared that the teacher, in the 7th grade, was trying to teach probability, and your point about "1/8, 0.35 and 50%" is not not significantly about probability. Further your point about "1/8, 0.35 and 50%" is too elementary for the 7th grade. Uh, at one time I was in the 6th grade, and the 7th, and even, could one believe, the 8th and do remember some of it!

My 7th grade teacher was quite comfortable giving that little puzzle problem and letting the class spend the whole period on it. It was a good little lesson.

It appears from this thread and some other sources, e.g., from North Carolina, that probability is too difficult for most of the K-12 faculty. So, I gave some easy lessons on how to teach elementary probability.

I've been a student, am well qualified in probability, and have done a lot of teaching. I know fairly well what the heck I'm talking about.

Sorry, but that 7th grade teacher was not doing well.

Also the remarks above in this thread -- I tried to be nice and not mention them -- on "continuous probabilities" and "fuzzy" need to be flushed: Some probability distributions are continuous, but the 'probabilities' themselves are not. That is, we know in very fine detail just what 'continuous' means and what a 'probability' is, and 'continuous' does not apply to a 'probability'.

Further, 'fuzzy' is from a quite different subject. Saying 'fuzzy' seems to imply that maybe some probabilities do not really exist; no, this is a fundamental error, and quite serious. Instead, with meager assumptions we can assume that probabilities always exist; the issue, then, can be finding numerical values for the probabilities, and for that we usually have to make an 'estimate' and that is mostly in the field of statistics and usually makes use of at least the weak law of large numbers. That we have to make an estimate does not justify 'fuzzy'.

From the OP, I can believe that the 7th grade students are up to understanding probability but that the OP is not and is not up to teaching it. Sorry 'bout that.

It appears that the K-12 teachers have been in an echo chamber far too far from the solid material in US universities and now are frequently circling their wagons to defend themselves from attacks on their, let me pick the right word, incompetence. Instead, the teachers should learn some probability. I suggest texts by M. Loeve (long at Berkeley), J. Neveu (from Paris and a Loeve student), K. Chung (long at Stanford), L. Breiman (a Loeve student and long at Berkeley), or E. Cinlar (long at Princeton)? To read these texts, start with, say, W. Rudin, 'Principles of Mathematical Analysis' (at least once used in Harvard's Math 55), the first half of W. Rudin, 'Real and Complex Analysis', and H. Royden, 'Real Analysis'. Royden was long at Stanford. To read those books, be a college math major in a good program.

Then we can talk more about probability and how to teach it. In the meanwhile, if I had a student in that 7th grade class, I'd insist to the principal that the class receive some competent instruction in probability from a different teacher.

Probability is a nice subject, and it is possible to do a good job with an elementary introduction, but the OP showed that the content of the teaching was a mess. E.g., there was "percent probabilities" and "theoretical probability", and the informed mind boggles. With such teaching, probability will be several times more difficult than necessary; with such teaching, the students are done real harm and later will have to "unlearn what you have learned". So, the teaching is not just babysitting but harmful babysitting.

And we pay real estate taxes for such garbage?

Am I making myself more clear?


You were clear enough before. You still hand wave aside the question about whether the majority of the students are ready for the material (this is separate from the question of whether they should be).

(I agree that the comparison is not probability, but understanding the comparison is what makes probability useful in everyday life...)


> hand wave

You are are hand waving that 7th grade students are not yet ready for 5th and 6th grade work.

Why the heck not?

If we take your hand waving seriously, then there are problems much more difficult and serious than elementary probability.

I have no patience with your assumption: If I had a child in such a class, they'd be home with home schooling or some such alternative before the teacher first touched chalk to the board. No patience at all. Can't build a strong house on a rotten foundation.

The school I went to expected the students to keep up with the right material. I have no patience with lowering the standards to meet poor students.

> understanding the comparison is what makes probability useful in everyday life

Where can one get the really strong funny stuff someone needs to be smoking to assume that such a triviality is "useful"? That "comparison" is just triviality from trying to bend education down to students not ready for 7th grade work. You are welcome to pursue such things if you want, but any of my children would be out'a there and home schooled, charter schooled, privately schooled, etc. in a nanosecond. The teacher, the principal, and the school board would all hear from me loud, clear messages.

There are some good reasons for a local school board: Parents can scream bloody murder at incompetence and trivialities. E.g., my father in law was on his local school board, and my wife was Valedictorian, PBK, Woodrow Wilson, 'Summa Cum Laude', two years of NSF fellowship in one award, and Ph.D. There wasn't a lot of patience with grotesque incompetence.

My father got transferred to a new city and for a house first he looked at the schools. He picked what was by a wide margin the best school in the city and then picked a house in the district of that school. It was a decently good school although could have been much better.

We need much less patience with incompetence. "No child left behind" can't be permitted to mean leaving all the children behind.


You seem to be reading positions into my statements, positions that I don't mean to put into them.


How could any student not be fall-right-over excited about learning standard 7.SP.C.5 ?


So many schools seem to "teach only towards the test" because of the heavy reliance on standardized testing. I don't claim to have any answers, but society has pushed for this and it's a logical result IMO.


John Taylor Ghatto & John Holt have already laid down in great detail of the problems of government schooling and solutions to those problems as illustrated in many of the comments in this thread.


Tom Lehrer on 'New Math':

http://www.youtube.com/watch?v=UIKGV2cTgqA

Enjoy the 'octal' bit.


> Who or What Broke My Kids?

Rigid Institutions, Rigid Hierarchies, Social Stratification, Judgement, Power & Control, Special Interests, Lack of Freedom, Lack of Autonomy


You hear about this all the time these days. Is this issue prevalent in all schools now from public to private and low to high socioeconomic status?


What cost effective alternatives are there to public schooling in the U.S.?

Is it possible or feasible to home school your kid with success?


You're calling your students "broken"? Can we start there?




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