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?
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.
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.
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?
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.
Report back when you're above 22.
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.
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.
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.
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.
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.
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.
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.
Thank you for the compliment.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
That's the part that counts, and is largely missing today.
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.)
I've read the Common Core (and the Australian equivalent) and they certainly require proving how answers were obtained.
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.
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?
"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).
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...
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.
At the basic level that has 4 choices, you can substitute X or Y in the top or bottom equation.
Take the Euclidian proof (#7). The order of the reasoning is important, because each step builds upon what was proved in the previous step.
(A+B) -> X, (A+C)-> Y
(X+Y) -> Z
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.
That's also a big part of why I got into Computer Science. Computer Science 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.
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.
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.
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.
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.
A: Because they can't tell their coffee cup and donut apart.
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 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.
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.
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.
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.
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.
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.
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.
What happens when you add together two even numbers?
Is that true for all even numbers?
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?
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.
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 
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'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.
Hilarious, another great article from Mr. Kun.
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.
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.
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...
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.
"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."
Or are you suggesting that the answer would be different if you were missing two thumbs?
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.
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.
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.
"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-...)
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:
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.
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?
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.
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.
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, 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 that most teacher editions of mathematics textbooks at all levels differ from the student editions mostly just in having the answers included 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, 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, 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). 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.
 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.
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.
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.
My badly neglected personal website (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.
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.)
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:
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.
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.
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?
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.
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.
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.
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.
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.
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.
>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.
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 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 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".
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".
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.
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.
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.
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.
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.
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.
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.
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.
Coincidentally the math stack exchanges are regarded as the most toxic parts of that community.
I mean, programming is about typing and manipulating text into a box, right?
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.
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.
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.
Source: Probability Theory by E.T. Jaynes
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.
> 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).
>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.
> [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.
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."
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)?
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
> the probability was 50% since it would either rain or not
Uh, can we place some real money on that!!!!!!
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.
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?"
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.
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?"
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.
> 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'.
'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?
(I agree that the comparison is not probability, but understanding the comparison is what makes probability useful in everyday life...)
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
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
We need much less patience with incompetence. "No
child left behind" can't be permitted to mean
leaving all the children behind.
Enjoy the 'octal' bit.
Rigid Institutions, Rigid Hierarchies, Social Stratification, Judgement, Power & Control, Special Interests, Lack of Freedom, Lack of Autonomy
Is it possible or feasible to home school your kid with success?