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I wonder what the simplest actual-practical-use-of-math is? Balancing a checkbook? Counting sheep? (I really ought not cite an essay I dislike so much).

P.S. At least math has basic facts that are worth memorizing. I'm still confused on what can usefully be learned in elementary school history class.

[anonymous]12y00

I wonder what the simplest actual-practical-use-of-math is?

Despite the rambling nature of that "essay" the counting sheep thing is a really good example.

Might want to make it connect to their reality. I wonder if there is a way to use math on becoming popular or whatever elementary kids care about.

Pokemon was an epic driver of (nonuseful) study for people in my generation. It was an RPG, too, so you could squeeze a lot of math out of it. I wonder if the schooling establishment has the ability to keep up with what kids are doing?

I'm still con

... (read more)

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35 How to Teach Students to Not Guess the Teacher’s Password?

by Petruchio

4th Jan 2013

1 min read

35

As a teacher, I wonder if it is possible to instill this skill into students the skills of rationality and critical thinking. I teach the third grade, and it is not immediately apparent how to apply this with my own class.

The problems I foresee are as follows:

Young children often do not know the basics on the subject which they are learning, be it math, science, art, religion, literature etc.
Many children are very shy, and try to give as short of an answer as doable to a verbal prompt.
Written prompts are arduous, straining the attention span and writing capabilities of the students. This is not a bad thing, but it presents difficulties in the economy of time and material to be presented.
Attention spans in general are very short.
Experiments can be very infrequent, and nigh impossible with certain subjects.
Children, at this age, are likely to take the words of a parent or teacher at face value, and naturally parrot it back. This may be a hard habit to break.

In the sequences, it is suggested teachers should drill into students words don't count, only anticipation-controllers. How practical is this for an elementary school level? Also appreciated would be any ideas or experiences on how to do this, or how to combat the above problems. Hearing from other teachers would be excellent especially.

Anticipated ExperiencesEducation

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How to Teach Students to Not Guess the Teacher’s Password?

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[-]Vaniver12y270

As a teacher, I wonder if it is possible to instill this skill into students the skills of rationality and critical thinking.

It seems easy to disincentivize it. I'm not sure it's that easy to instill it. This is relevant, but targeted at college students.

One of the big things mentioned in the education literature for gifted children is "encourage asking questions," which is different from the default result for most people. A relevant Sagan quote:

But there’s something else: Many adults are put off when youngsters pose scientific questions. Children ask why the sun is yellow, or what a dream is, or how deep you can dig a hole, or when is the world’s birthday, or why we have toes. Too many teachers and parents answer with irritation or ridicule, or quickly move on to something else. Why adults should pretend to omniscience before a five-year-old, I can’t for the life of me understand. What’s wrong with admitting that you don’t know? Children soon recognize that somehow this kind of question annoys many adults. A few more experiences like this, and another child has been lost to science.
There are many better responses. If we have an idea of the answer, we could try to e

... (read more)

[-]fubarobfusco12y240

One of the things I'm most glad that my parents taught me was to look things up. Even if they knew the answer, they'd give me a book or send me to the library instead of just giving the answer. What does "vague" mean? Look in the dictionary. Where does rain come from? Here's a book about weather.

6mwengler12y

I'm glad to see this quote. I often ask my kids what they think, and try to get them to figure out the answer, but very often I simply give them the best answer I can come up with. The result is they are totally willing to ask questions of me on virtually anything. I bend over backwards not to "teach" them my opinions, or at least to flag them as opinions when I mention them. Things like existence of god or political questions get flagged. Or even whether kids should be hit.

3ChristianKl12y

Do you thnk that flagging works?

0mwengler12y

My kids do have opinions that differ from mine, and they do it matter of factly, without thinking it is a particular sin or threat to our relationship for sure. At least one of them, when I explained a cryonics thread, allowed as she believed in some sort of life after death and thought cryonics was irrelevant and even a negative in that light. So yes, I think flagging "works" along with the other things I do which amount to treating my children as though they are independent intellectual actors. Just as I think flagging "works" in any conversation where values, judgements, and observations might get confused for one another, possibly even by the people who express them.

-1Vladimir_Golovin12y

Regarding the Sagan quote: these days, when everyone has Wikipedia in their pockets, "I don't know the answer" is not a valid excuse (for most questions that a child may ask).

[-]handoflixue12y160

"I don't know, but let's look it up" is an awesome answer!

It teaches the kid what their resources are, and gives them a handle on how to look stuff up independently in the future.

Plus, if you're going with (shudder!) Wikipedia, it means there's an adult to translate the ridiculously obtuse language that Wikipedia uses for all things science:

The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields.[12][13] It has a diameter of about 1,392,684 km,[5] about 109 times that of Earth, and its mass (about 2×1030 kilograms, 330,000 times that of Earth) accounts for about 99.86% of the total mass of the Solar System.[14] Chemically, about three quarters of the Sun's mass consists of hydrogen, while the rest is mostly helium. The remainder (1.69%, which nonetheless equals 5,628 times the mass of Earth) consists of heavier elements, including oxygen, carbon, neon and iron, among others.[15]

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

Any parent who STARTS with that is probably not helping any more than Calvin's dad explaining "Old photographs are black and white because the world didn't gain color until sometime in the 1920s" :P

[-]roystgnr12y140

It's not a complete improvement, but for young children the Simple English Wikipedia is at least a little more comprehensible.

[-]ChristianKl12y200

For science problems that involve math, always add more number in the question than necessary.

"A 1.80m tall man runs 100 meter in 8 seconds. What's his speed in m/s" is a better question than "A man runs 100 meter in 8 seconds. What's his speed in m/s"

[-]CronoDAS12y190

This is just throwing an idea out there...

I think the issue might be a matter of context. Ideally, you want the students to use information to complete a challenge instead of answer trivia questions. Video games, in particular, tend to be very good at this - this video [NSFW for language] demonstrates just how much the game Mega Man X teaches players within the first few minutes of playing, and none of it is "password" knowledge, because it gets reused almost immediately in slightly different ways.

Further reading: Digital Game Based Learning by Marc Prensky. Chapter 1 is available online and is probably enough that you won't need to buy the book.

Professor James Paul Gee has also written extensively on the subject of video games and learning.

[-]NancyLebovitz12y170

John Holt (a founder of home schooling) thought that guessing the teacher's password was caused by grading.

What started him on the path to home schooling was seeing third and fourth graders (9 and 10 year olds) so concerned with getting the right arithmetic answers that they weren't thinking about arithmetic.

0DanArmak12y

I don't understand what you're saying. "Thinking about arithmetic" is supposed to be the tool that helps them "get the right arithmetic answers". If they aren't getting the right answers, shouldn't that stimulate them to work harder at thinking about it?

7NancyLebovitz12y

I'm sure that was the hope. Holt's opinion was that grading (mere grading-- the school didn't use humiliation or corporal punishment) was enough to crank up their anxiety so far that they weren't paying attention to anything else. Here's another clue about grades. I took a college level Alexander Technique course where the students were graded on attendance. Alexander Technique is a way of releasing habitual tension [1] so as to be able to move more efficiently. The teacher said he'd found that if students were being graded on their ability to not pull their heads down and back, there was just no hope-- they'd go into the pattern of tension. Since he had faith in his ability to teach anyone who showed up, he graded on attendance. He mentioned another teacher who graded on quality of class notes. My hypothesis is that someone who'd absorbed Alexander Technique very deeply would be able to be graded on it without pulling themselves down-- in general, Alexander Technique is about not having that habitual response to a stimulus-- but it's too much to learn in a semester. [1] Tension isn't a static thing-- it's frequently a habitual response to a stimulus, including much milder stimuli like getting out of a chair or starting to speak.

0ikrase12y

No grades is likely a limited thing, might work badly with unmotivated students.

2handoflixue12y

In elementary school, we had to memorize the multiplication tables (1-10 times 1-10). We were then quizzed on this. Being able to answer "5x5 = 25" meant we had memorized the tables The teacher then asked what "5 x 12" was, which checked to make sure we actually understood multiplication. Hopefully that clarifies the difference :)

0DanArmak12y

I understand the difference. What I don't understand is the odd conjunction. I parse this as saying: because they were so (strongly) concerned with getting the right answers, they weren't thinking. If they had been less concerned with getting the right answers, they would have been thinking more. What is the idea here? That if they didn't spend so much time, effort, or worry on memorizing, then they would have had more time to think?

2asparisi12y

I think the worry is that they are only concerned about getting the answer that gets them the good grade, rather than understanding why the answer they get is the right answer. So you end up learning the symbols "5x5=25," but you don't know what that means. You may not even have an idea that corresponds to multiplication. You just know that when you see "5x5=" you write "25." If I ask you what multiplication is, you can't tell me: you don't actually know. You are disconnected from what the process you are learning is supposed to be tracking, because all you have learned is to put in symbols where you see other symbols.

0DanArmak12y

But surely in math, of all subjects, it's easily possible to construct problems that cannot be solved without thinking and understanding, that do not reduce to mere memory and recognition of a known question "5x5". Then students who want the "right" answer will be forced to understand. This isn't absolute, of course. When learning elementary multiplication, pretty much all you can ask about is multiplying, and there are only a few dozen pairs in the 10x10 multiplication table, and students generally just remember them, they don't calculate. But by the time you're up to arbitrary size multiplication or long division, you need to apply an algorithm; that's a step of understanding, because you can see that the algorithm also produces the results you memorized earlier. And so on. When students are at the 5x5 level, I don't think there is an answer to "why is 25 the right answer?" that they could understand - it just is the right answer, a brute fact about life, just like the sky is blue and sun comes up every day. But that doesn't continue forever. In my personal experience schools go way too far in the other direction, and keep asking for rote memorization when it's already possible to ask for understanding.

4asparisi12y

That depends on the level of explanation the teacher requires and the level of the material. I'd say that at least until you get into calculus, you can work off of memorizing answers. I'd even go so far as to say that most students do, and succeed to greater or lesser degrees, based on my tutoring experiences. I am not sure to what degree you can "force" understanding: you can provide answers that require understanding, but it helps to guide that process. I went to a lot of schools, so I can contrast here. I had more than one teacher that taught me multiplication. One taught it as "memorize multiplication tables 1x1 through 9x9. Then you use these tables, ones place by ones place, ones place by tens place, etc." One problem with this approach is that while it does act as an algorithm and does get you the right answer, you have no idea what you are trying to accomplish. If you screw up part of the process, there's no way to check your answer: to a student in that state, multiplication just is "look up the table, apply the answer, add one zero to the end for every place higher than one that the number occupied." Whereas I had another teacher, who explained it in terms of groups: you are trying to figure out how many total objects you would have if you had this many groups of that, or that many groups of this. 25 is the right answer because if you have 5 groups of 5 things, you generally have 25 things in total. This is a relatively simple way of trying to explain the concept in terms of what you are trying to track, rather than just rote memorization. Fortunately, I had this teacher earlier. The point being that you can usually teach things either way: actually, I think some combination of both is helpful. Teach the rote memorization but explain why it is true in terms of some understanding. Some memorization is useful: I don't want to actually visualize groups of objects when I do 41x38. But knowing that is what I am trying to track (at least at the basic level o

0DanArmak12y

Yes, but this only really works if, when the student is presented with an example they didn't memorize, they can still solve it using their understanding. And to make sure they do understand, after they've practiced on the simple cases they can memorize, you routinely set problems that require understanding. You can't start with understanding because when solving a few simple cases (like 5x5), memorization really is effective, and students may choose to memorize even if you don't explicitly tell them to.

[-]ChristianKl12y110

There a lot of learning where students need to give a specific answer to a specific question. It's useful to learn multiplication tables by heart. The most efficient way to teach that stuff is spaced repetition. Every kid should use Anki for those problems.

If you solve those topics with Anki you have more time for teaching critical thinking.

Let students evaluate the work of other students. Every student writes a text. Alice gives her text to Bob. Bob reads Alice's text and searches for spelling errors. Afterwards the two students discuss the erors that Bob found in Alice's text.

If Alice and Bob can't agree on whether something is an error they ask the teacher.

In art class students should evaluate and discuss the work of other students. To prevent concerns of bullying I would advocate to randomize groups and use mostly groups of two students.

Create a enviroment where the kids can give each other constructive feedback.

Experiments can be very infrequent, and nigh impossible with certain subjects.

Then how about increasing the amount of experiments?

Experiments in itself are however not without issues. One of my worst experiences in school was where we did an experiment about gra... (read more)

[-]maia12y130

I sticked with the values that I measured and got a bad grade. The teacher didn't explain to me why I did get the results I did.

That's terrible. In all of the labs I had in high school, our teachers specifically assured us we wouldn't get worse grades for getting unexpected results, as long as we documented everything well and provided some possible explanations for those results.

Teachers absolutely should not incentivize students to do things that would send them to Science Hell.

7ChristianKl12y

I think the problem did lay in the possible explanation for those results part. I couldn't think of a possible reason for getting those results. I listed a bunch of reasons I could think of. I didn't believe in them. The teacher didn't gave me any reasons that I missed and just noted that my reasons were bad. If the real motivation would be to do science, I should have repeated the experiment to see whether my results replicate. There was no time for a repeated experiment. Getting results that you are unable to explain is very frequent if you do real science. In university we did a group PCR experiment and for some unknown reason it didn't work in any group. The person leading the experiment made some changes the next day when another group did the experiment and it still failed. She eliminated possible reasons and in the end didn't know why it failed. She was quite embarrased about it. In real science you frequently have say: "I don't know why the experiment I did produced the results that it did." In my own Quantified Self experiments I have frequently data where I have no explanation what happened. If you really want to teach the scientific method you have to confront students with experiments where the students are allowed to trust the results of the experiment over a theory that tells the student how the experiment is supposed to end. Most of the physics/chemistry experiments are poor in that regard. They teach cargo cult science, in which experiments that don't validate theories as right or wrong instead it's the other way around.

[-]maia12y120

That is unfortunate. I think my teachers would have accepted something along the lines of "Here are some reasons I thought of; I don't think any of these reasons are very good, though, so I actually don't know what happened."

Maybe a teacher could introduce an experiment where the students have been taught a "lies-to-children" model that doesn't quite work in the experiment, then have them do the experiment, and then after some agony on the students' part, explain that the results were because the model is actually wrong and now they need to learn a newer one. As a sort of live-action science retelling. Still a little bit cargo-culty because there is still an answer at the end of the tunnel, but might give them a better idea of how things are supposed to work.

6CronoDAS12y

Indeed. The results of experiments done in first-year physics and chemistry labs routinely demonstrate that the laws of physics as we know them are clearly wrong. ;)

5cousin_it12y

Yeah, here's a nice article on the topic :-)

5ikrase12y

I agree that experiments in school tend to be downright TERRIBLE. THere is never, even in a wealthy enviroment, enough time, equipment, or whatever to let students make any more than trivial choices in experimental design -- and I would say that knowing what experiment to perform is downright essential. My main opinion is that we should not teach so many scientific facts in high school, and more rationality and scientific method, using that as a hook for specific disciplines. Chemistry is pretty good because you can get into some nice empirical predictions and deconstructionism without much difficult math. A lot of labs in high schools are not even all that experimental, and the hypothesis is always crystal clear.

2ChristianKl12y

We have art classes in which there's plenty of time for the students to create artwork. In a similar context there no reason why there shouldn't be experimention classes that teach students to experiment and give them plenty of time for the exercise. School wastes massive amounts of time by teaching facts without SRS that get forgotten by the students a few years after they take the class. Learning the paradigm of experimentation might be more important then teaching the way the periodic table is arranged.

0ikrase12y

In high school, most studio art classes have a lot less time devoted to verbal learning or whatever and much more to practice. Plus mundane art materials are both inexpensive and very safe to use. (and may already be in the classroom. Much can be taught with pencil sketches.) Chemistry (especially with modern-day, possibly excessive safety consciousness) is not, and requires more specialized space and has a hard time admitting of experimental design by students beacause they might design a bad experiment. I don't know about mechanical physics. My school had an 'applied physics' class meant to go alongside the normal class, in which actual freaking predictions were made but it was more engineering than physics and more shop class than either. Plus the kind of rudimentary stuff made by students such as air cannons and the like tends not to be regular enough in i.e. firing velocity for math analysis to work very well. OTOH, chemistry often isn't either at the high school level. Damn three-beam balances. I don't know how to deal with the issue of governmentally imposed tests that must be taught to. It seems to me that one thing that should be done is to develop a way of testing this sort of thing, but on the other hand there might be

0ChristianKl12y

That's the point. Setting up real experiments is scientific practice. I don't really see the point of most chemistry experiments that I did in school and university. The last batch at a university introductionary chemistry course just showed me that my finger coordination was really bad. Afterwards I learned some card magic to get finger coordination.

3Desrtopa12y

I don't expect ordinary elementary school students to be mature and competent enough to do this without devolving into bickering and status posturing. Keep in mind the age of the students in question. Feedback from eight year olds on academic work is likely to be unhelpful even to other eight year olds.

0ChristianKl12y

In some schools teaching the skill of giving and receiving feedback without going into status fight will be a challenge for the teacher. On the other hand it's a skill that students should learn as early as possible. You are right that eight year olds can't give each other the "correct" passphases to academic work in all cases. On the other hand that doesn't mean that the feedback isn't useful. As far as spelling errors go, I see no reason why eight year old should be unable to find them. As far as style issues go, a eight year old has a much easier time to understand feedback from another eight year old than to understand feedback from an adult. It's also not only about getting feedback but also about giving it. If you give feedback then you think about what you consider to be good and what you consider to be bad. That process is very important. Verbal feedback is about more than giving someone a correct way to do things. It's about direction attention to an area over which the person might learn something.

0Desrtopa12y

I think you may be overestimating the writing abilities of the average eight year old. When it comes to style issues, more third graders are in need of learning basic coherence than stylistic elegance. I think that most of the Less Wrong population was probably at least somewhat precocious, which may make it hard to relate to what the average third grader needs to learn, as opposed to exceptional ones. Having third graders attempt to correct each others' work sounds to me like trying to get white belts to try to correct each others' form in a martial arts class. Low level students may practice together, but you don't try to put them in a position of giving each other instruction until they achieve some measure of basic competence.

0ChristianKl12y

I haven't used the word 'correct'. A third grades should be able to give you an answer to: Does this sentence look coherent? Why do you think so? I see no reasons why he shouldn't be able to share the answer to those questions with his classmates. If the teacher wants he can offer the correct solution afterwards.

0Desrtopa12y

I think it would probably be more effective to have the students work on samples provided for them by the teacher, perhaps ones written by students from other classes, so as to avoid bickering and posturing between students. Students are used to receiving corrections and critiques from their teachers. Having a peer critique one's work and offer corrections, and not take it as a status affront, is and act that requires some maturity, and many people never learn to do it at all. Learning to receive correction from peers gracefully is an important skill which is probably worth taking the time to teach to students, but I don't think that paired peer critiques would be the best way of doing this. Some students will naturally produce better work than others, and thus have more criticism to offer and less to receive compared to their peers. The exercise would provide an element of "smart people don't have to receive criticism from their peers, ordinary/stupid people do," which the teacher would have to actively work against. To teach students to accept correction from peers without taking it as a status challenge or an affront, I think I would try an exercise along these lines. Split the class into two groups. In each group, students work solo on a task provided by the teacher. However, the teacher provides different information to the students in each group. Neither group has all the information needed to perform the task correctly. After the students have tried and failed for a while, integrate the groups and pair the students up together, and tell them to pool their information.

1ChristianKl12y

Another way to get students to accept critizism would be to let each student rewrite his text after the feedback and then grade the revised version. Saying many people never learn the skill at all is just another way of saying that today's schools are very bad at teaching the skill. To me the skill seems on of the most important that children should learn in school. I think it makes sense to try to teach the skill as early as possible. Of course, when trying something new it's importance to see how it works in practice and calibrate on the reactions of actual students.

0Desrtopa12y

But this runs into the problem I mentioned before, that some students will have plenty of correction to offer others' work, but little correction to receive from their peers. So instead of "everyone needs to learn to accept correction gracefully," you risk teaching them "regular/stupid people need to learn to accept correction from smart people." If I had been put through such activities at that age, I suspect it would have amplified the arrogance that I was already developing at that time. In third grade, I tested as having a twelfth grade reading level; it's highly probable that no student in my class at the time could have offered any useful critique on my work, whereas I could have offered critique on all of theirs. It was already hard enough to unlearn the habits of thought that sort of disparity cultivated in me without participating in class activities which reinforced it.

0DanArmak12y

Which is a good and useful thing for stupid people to learn.

0Desrtopa12y

Better than their thinking they don't need to accept correction at all, but then, if you don't teach them measures for determining whether "corrections" are actually correct, they're liable to get their heads filled up with garbage. And teaching the stupid students to learn to accept correction gracefully from smart students is a lot less good and useful than teaching all the students to accept correction gracefully from their peers whenever they're mistaken and receive it.

0DanArmak12y

Well of course, if after hearing an explanation they realize they were wrong and why, then they should accept that correction no matter where it came from. And they should be open to listening to such proposed corrections from all sources. But often even after the error is pointed out and explained, they still don't understand why they were wrong. The one correcting them needs to know more than the minimum to be able to teach others, and needs to invest the time to explain it. There's a long way from "remember correctly the right answer" to "be able to explain why this answer is the right one". Most children are not able to understand why what they are taught is right, and even if they are, they are rarely taught these reasons. So if children know some other, smarter children are pretty reliably right, then they should definitely use a heuristic of deferring to them whenever they disagree about the study subject matter. This is easy to calibrate empirically based on previous experience, even for children - people are good at tracking who tends to be right.

0Desrtopa12y

I don't think students tend to be particularly shortchanged with respect to that sort of lesson by the current system though. Most students are aware of who the top students are, and that the top students are more reliable in their areas of expertise than the lesser ones. Should they defer to the top students in matters outside of what they know to be those students' areas of expertise? Not necessarily; a lot of smart people are not particularly good at getting right answers outside their areas of expertise. For a not-so-smart person, simply trusting that a smarter person knows what they're talking about on any given subject, even when they don't know that person to be an expert on the subject, is not a very trustworthy heuristic. Trusting people who're more expert than you in a particular field over your own judgment in that field is something that most students already learn. Being unable to assess who's more expert in domains in which they're not well trained is natural and probably unavoidable, and students can't be trained to be expert in everything.

0ChristianKl12y

There's a corollary: "Smart people need to learn to give feedback in a way that doesn't annoy stupid people but let stupid people accept their feedback." You probably would have profit from learning that skill in third grade.

1Desrtopa12y

I did learn that skill much earlier than I learned to accept correction gracefully from others. I had much more occasion to practice tailoring my feedback to others so that it would be accepted than I had to practice receiving and incorporating it. I don't think smart students are as shortchanged when it comes to learning that skill as the reverse, with the current status quo.

-2DanArmak12y

"Don't write like that, it's stupid! You're a stupid if you write like that!"

[-][anonymous]12y80

Teaching by negative example addresses all your concerns.

No basic knowledge required. Logical guesses will serve.
Short answers are the right answers.
No writing required.
Story-problems can hold attention longer.
Experiments can be described as well as shown.
Opportunity to disagree with the teacher, a lesson in critical thinking in itself.

Here is what I mean in the abstract by teaching by negative example. Set up a situation that can be resolved or addressed using logic. Suggest or demonstrate the wrong answers. The students will (with gusto) tel... (read more)

1Emile12y

That sounds interesting, could you give a couple concrete examples of "wrong answers" that worked particularly well? Do you mean things like "All birds fly, a bee flies, therefore a bee is a bird"?

3[anonymous]12y

More like this: Q what is a bee? A A bee is a bee. And there is stops because questions about groups or kinds of things (1) are lacking in background knowledge (2) are viewed as another dumb English thing - English has too many words for some things and not enough for others. So I try this: Q Is a bee a food? A no Q is a bee a place? A no Q what is a bee? A a bee is a bug. Logical structures that are factually false as you describe can be helpful too.

[-]DanArmak12y80

How to Teach Students to Not Guess the Teacher’s Password?

By making it impossible to guess!

When I was in school, questions were 100% memorization. You were supposed to remember the answer, not understand it. Math required calculation, but that just meant remembering the algorithm of calculating. I never needed to think about how to solve a problem, just to recognize it as one of a very few known types and remember the method that was taught to solve it. (If I used a different method to get the right answer I would get points deducted.)

If you ask questio... (read more)

0Petruchio12y

Good catch on the omitted word. Corrected. Up-vote for you.

-1OrphanWilde12y

I don't know about this "What without the how and why." Every time it happened to me - finding the inflection point of a quadratic equation, for example - I was pissed off about how much time I had wasted on stupid crap. Re-deriving Calculus would have been entertaining and educational, and loads more useful than memorizing a formula so I could draw ugly graphs over and over and over again.

0Decius12y

What about 'this is what an inflection point is'; here are some ways to find them on specific equations. Now, here is how to find the inflection points of arbitrary equations.

0DanArmak12y

The great majority of students aren't capable of re-deriving calculus, even with guidance, let alone in the third grade. There's a difference between letting the capable ones do it, and asking a whole class to do it.

0OrphanWilde12y

I wasn't graphing quadratic equations in third grade. Actually, I never did third grade. In fourth grade I was doing fractions, and that only because they let me take math classes with the fifth and sixth graders. It's not important that they succeed. It -is- important that they try.

[-]Desrtopa12y70

Children, at this age, are likely to take the words of a parent or teacher at face value, and naturally parrot it back. This may be a hard habit to break.

I'm not sure that third grade is an appropriate time to try to break that habit. That's an age where at least some of the students are probably still in the preoperational stage. Kids naturally start to question adults' reasoning more when they develop the capacity to manage those sorts of thoughts effectively.

Teaching students to grasp a complex topic will generally require walking them through vari... (read more)

1Fronken12y

But then they don't, so we need to try another method, yes?

0Desrtopa12y

We need to do something differently, but we need to make the right changes at the right places. If you want kids to better understand rationality when they grow up, you don't want to start by teaching them things like the content of the Sequences, you start with something like "what did you see?"

[-]Xachariah12y60

Are you familiar with Dan Meyer's "Math Makeover" Tedx Talk? I'd highly recommend it.

It covers almost exactly this topic on the subject of math. He's also got curricula on a link on his blog (though it is a bit hard to locate in the sidebar). Also, in his talk he mentions people send him all sorts of stuff for different subjects, and a cursory look at the blog shows a ton of links, so you may be able to do more than just math work with pre-made lesson plans that require minimal tweaking.

1NancyLebovitz12y

That's an excellent talk-- Dan Meyer has something to protect. He's going to be living in the future made by the people who are learning (or not learning) math now. Sometimes scope insensitivity is useful. He's evoking intrinsic motivation rather than trying to use grades as a substitute for it.

1Xachariah12y

He's doing that and more. He's showing the children that math is more than the memorization of it's parts. Math is not an arbitrary set of rules that we learn because... adults make them learn it. He's showing them that the universe runs on predictable rules. Math just happens to be how we express those rules. Even more important than building intrinsic motivation is that they get to see what a reductionist universe looks like.

[-][anonymous]12y50

Well partially it seems that if they're just parroting stuff back, they haven't really learned anything, so it's not much of a loss if you change your methods and they still don't learn. Basically, you haven't got a lot to lose. Mind you, I'm an engineer, not a teacher, and this is just my impression. I would defer to the actual experts.

I'd say just burn them over and over for parroting, and reward them over and over for thinking. If you ask questions of the class as the normal thing, a student will feel good about themselves for saying what you want to he... (read more)

3maia12y

The tricky part here is not to shame them too much; otherwise, they might stop answering questions at all. It's important to encourage trying things, at the same time as you try to discourage trying random, haphazard phrases. Sometimes it might be hard to tell the difference.

0[anonymous]12y

Yes, very tricky balance. With people in as fragile a situation as students, even explicitly calling them out at all might be too much. That's why I meant that the unreward should come from within the student. Maybe it's just a matter of not rewarding it. I don't know. It's much easier with adults. At least with adults they will sometimes tell you what they are thinking.

0TimS12y

For some introductory material, memorization really is required. 5 x 5 = 25 is a true fact that I expect a young student to simply memorize. Looking back, we can understand why it is important to learn those types of things - but the explanation of why is often far more complex than the object level fact.

4[anonymous]12y

I recently spent a bunch of time realizing that memorizing multiplication tables, formulas, and random numerical facts about the world is a really good idea. (fermi estimation). I notice you need some theory to actually connect the memorized facts to reality, though. What use is 5*5 = 25 is you don't know how to use numbers and recognize when multiplication is appropriate? Maybe the trick is to get students to solve mostly complete world-connected problems so that they develop the skills of connecting the memorized procedures to something useful. Still, as you say, at a sufficiently early stage in development, the students might not even be cognitively capable of connecting the facts to reality, but they need to memorize them for later. I certainly learned "six times four is twenty four" before I learned anything approaching actual practical use of math.

0TimS12y

I wonder what the simplest actual-practical-use-of-math is? Balancing a checkbook? Counting sheep? (I really ought not cite an essay I dislike so much). P.S. At least math has basic facts that are worth memorizing. I'm still confused on what can usefully be learned in elementary school history class.

0[anonymous]12y

Despite the rambling nature of that "essay" the counting sheep thing is a really good example. Might want to make it connect to their reality. I wonder if there is a way to use math on becoming popular or whatever elementary kids care about. Pokemon was an epic driver of (nonuseful) study for people in my generation. It was an RPG, too, so you could squeeze a lot of math out of it. I wonder if the schooling establishment has the ability to keep up with what kids are doing? Propaganda and indoctrination? You might be able to tell them that the past really sucked and most of the world still sucks, and it's their destiny to grow up to make the world awesome.

0ikrase12y

Probably, as well as attempt some really quick immunization against fascists or whatever. Also, you might drill into them that things are complicated and happen You also could give a really quick outline of the world. I know that in 5th grade I did not know any of the following: * The world is really, really big. * A lot of the world outside of America has a standard of living similar to America, but even more of it doesnt. Its getting better. Not sure about Pokemon or making studying fun. Attempts to make educational games often fail miserably, and children quickly learn that things that adults tell them will be fun won't be. Some of these were basically just 'guess the teacher's password' games. * * Democracy has been around for a long time and most countries are democracies, or pretending to be. Teaching people that the struggle of awesomeness is on their shoulders is easy to sound ridiculous, may seem absurd to people who are about to go into low-status or even just unglamorous jobs, and tends to get really trite really fast so I am not sure how to deal with it in a to-be-mainstream school.

[-]John_Maxwell12y30

Some thoughts: Ask yes or no questions, or questions that have a finite answer set. Learn to ask the questions in a way that your vocal intonation doesn't give away the answer immediately. Call on a specific child and ask him/her to "take a guess" at the answer to the question if it's too hard for them to know (thinking about questions can be really useful even if you aren't going to find the answer), or say "here's a challenge for you: can you figure out...", or "given that X, what do you think Y is going to be". Find a sy... (read more)

[-]CronoDAS12y20

In response to other comments about math in elementary school:

Did anyone else watch the program Square One?

[-]jdinkum12y20

You could start off by overtly letting the kids know that "guessing the password" is how their success in school is measured and you're not going to be able to change that reality, but you could introduce "alternative" ways of thinking.

How about a game where each student writes down their answer to a passwordy type question and scores a point for every other student with the same answer. Lowest score wins. But they have to justify their answer.

If a teacher asks the question: "Who discovered America?" The password is: "Chr... (read more)

6Desrtopa12y

This sounds like a solution to something, but I think it's a separate problem, and it's also potentially an introduction to another problem. In fields that aren't constrained to a single right answer, students frequently learn to optimize for being interesting and creative over coming up with the best supported answer they can. To quote Dave Barry This sort of thing is good for stretching students' creative muscles, but bad for preparing students to grapple with tasks like "read these arguments for opposing conclusions and try and determine which is actually true, given that we can go out and check the answer objectively," or "Try and figure out whether this proposed design would work."

2ikrase12y

There is a problem with that: It assumes multiple valid responses and deals too much with what 'discover' or 'america' means. It wouldn't work for the 'Why is this steel plate hotter on the side away from the fire' question. Hermenueutics-like games are risky since I think they teach contrarianness, thinking up unique-non obvious answers w/o regard to correctness. They teach the kind of reason that is rightly accused of being able to argue for atrocities. I think that's a kind of terrible lesson. It might work well to come up with a whole bunch of questions that are not to trick-questiony, but in which guessing the password is spectacularly wrong. For the sciences, a better method might be to set up things where you have to make simple predictions.

[-]Shmi12y20

Unless you are dealing with unusually bright children, you are out of luck. Parroting is how most children learn. You might luck out and get an occasional Wiggin or HJPEV in your class, but they would hardly need your instruction to begin with. In addition, teaching young kids to question "because I said so" from parents and other teachers might land you in some hot water. I'm guessing that an average child would not be receptive to critical thinking training at least until adolescence.

6DaFranker12y

Indeed, this seems to be a problem. Even with unusually bright children, it seems deception still remains the only charitable option, otherwise you're pretty much condemning them to an early dose of "Everyone expects X of you. You must do X. We both know X is wrong, and stupid, and your next ten years will be a waste of time and effort and resources, but you must do X or be treated like a demon." and all the subsequent depression, narcissism, detachment and unhappiness. Encouraging them to obtain information on their own and keep asking questions seems like the most worthwhile strategy. Children are often visibly treated more like pets than people, at least in north american society. When a child asks a scientific question that upsets religious creed, receives a dogmatic answer and instructions to never speak of it again, and then loudly rejects this answer in light of obvious evidence, what happens isn't a discussion or an argument with the person, the child themselves... What happens is an angry parent screaming "WHAT THE F*** DID YOU DO TO MY CHILD?!", in similar manner to how someone might yell at a pet-keeper upon finding out that the cat was taught to scratch itself and eat rotten food when it was left in their care during the owner's vacation.

3DanArmak12y

Where X = going to school, for instance. On the other hand, if you don't tell them, most of them will come to that conclusion anyway. Then they will feel just as depressed, but also alienated from the oppressive adult caste.

2Fronken12y

I find most avoid considering the question.

1DaFranker12y

Something like that, yes. What I had in mind was mostly stuff related to / described in: Gatto's Lessons and Graham's essay on nerds. Preparing children for dealing better with the situations and problems described in both of those seems like the best thing a rogue teacher can do for their students, at least at the ages mentioned by the OP. It seems like organized support from the parents, school board, school personnel / other teachers or preferably all of those would be necessary to really achieve more.

2mwengler12y

I don't raise my kids this way and neither does anybody else in my large community. I'm not saying everybody everywhere is enlightened, but there are large swaths of civilization where kids are trained to think. In my own case, i constantly answer my children's questions with questions, and have never yelled at anyone for the way they have answered my kids questions. I have certainly undermined some of their answers, but not most.

1Fronken12y

Upvoted for quote, though unsure on conclusion. Has this been tried, that you've seen?

0OrphanWilde12y

From my memories of childhood, the average child lost critical thinking skills throughout the process of "education".

8Desrtopa12y

Can you give examples of this from your observations?

7OrphanWilde12y

The children in my elementary school classes were curious and asked questions. In a biology lesson in which some sort of beetles were raised from larva, every student was -fascinated-. These same students, three years later (discontinuity after that point - I changed school districts), were bored speechless by dissections, and wouldn't even answer questions, much less ask them, in lessons. It was a lot more obvious to me, because I typically dropped out of public education less than halfway through the year, bored. So the changes weren't slow and subtle - I'd come back with the new school year, and the students would be noticeably more apathetic. I don't know for certain that the apathy translated into reduced critical thinking skills, but certainly they weren't using them in the lessons anymore.

0DanArmak12y

So there was a change over time, but that doesn't establish that school was the cause. It doesn't even show a correlation as compared with different styles of education.

5prase12y

I can remember believing very weird things before school age, just as when my grand-mother told me that gravity is because of Earth's rotation. I tried to verify that experimentally with a globe, but in spite of the failure to attract things to the globe I continued to believe the explanation for some time, concluding that Earth has to rotate very fast. I am, on the other hand, not aware of losing any skill at school. Not sure about others, but in the third grade they didn't seem any more stupid than in the first. But of course, I might have had lost even the ability to observe critical thinking during the time.

2mwengler12y

I agree I gained critical thinking skills throughout my childhood, much more aided by school than impeded. Science was science. I had a 5th grade science teacher who was an idiot, but when I argued with him over his stupidities, he didn't shut me down, he argued back. And all along I was learning that this was a description of the world and the world, not some authority, got the last say. Not all kids are going to be as good at critical thinking as all other kids. This is not a failure of the education system, it is a failure of the human race. The best a system can do is add a delta in the right direction, on average, to most of us. My kids are pretty normal girls, but they are reasonable arguers and don't believe stupid stuff. This latter from training if I do say so myself. They think their opinions matter and so they put some effort in to them.

9DaFranker12y

This is very atypical, from what I know. I could name several local high schools and at least two cégeps where not a single teacher would consistently respond with a "the world and reality have the last say" attitude or be willing to argue rather than use the Authority and That's How It Is cards. I know of only one place, a private college, where about 20% of teachers would be as reasonable as your education seems to have been. The typical interaction between me and teachers during my own time in high school went more along the lines of: Me: How does gravity work? Where does the seemingly-unlimited energy for the force come from? (Physics) Teacher: Potential energy, due to the work that was put in raising the object to a certain altitude. *shows me a "standard" equation for calculating that* Me: But that doesn't work! How does gravity first happen with planets and in space, then? And also, here, *points at some places where those equations are just completely disconnected from reality and explains why there's something missing* Teacher: This is what is taught. You do what is taught, and it is the Right way. Don't ever mention this again. No, I'm not exaggerating. This actually happened. It's only months later that I learned (by reading an arxiv physics paper, heh) that both of us were completely off the mark with regards to the (then-)current best understanding of how gravity works. From what I gather, in general, most of north american education seems closer to my experience than to yours. This is horrible, if true (which I'm very convinced it is).

1ikrase12y

I've never encountered a teacher who was hostile like that, but many who were decidedly unhelpful.

1mwengler12y

There are things which are mysterious and difficult. There are things which even many experts in a field might not understand. I suppose my "good" attitude towards my teachers and my education may come from my appreciation of the facts that there is a lot to know and at the same time, no one knows close to everything about anything. I am a physicist and I read your description above, and I can easily put myself in the place of the teacher. The total energy in the gravitational system is indeed conservative. Whether or not the total system in our actual universe is open or closed is a real question. How it got that way? Who knows how the universe started, even the big bang doesn't give a hint what was going on a second before the big bang started. I can imagine telling a student who persisted in asking about these puzzles that they are indeed puzzles and what we could know is what we did know about the equations, and we were not going to use any more class time noticing they were puzzles, we were going to move on. I'm sorry if that doesn't work for every student. I'm not sure though that there is anything that WILL work for every student.

1Fronken12y

This seems lucky, from what I've seen the standard is lower.

1OrphanWilde12y

When individual students were called upon, which group was more likely to hazard a guess, to try to reason through an answer?

1prase12y

Are you asking whether the first graders were more likely to guess than the third graders? I don't know, it's a long time ago and I haven't consciously monitored the guessing frequency. But guessing was the obvious choice when someone didn't know the answer, as it was always better than simply saying "I don't know".

[-]HonoreDB12y10

I came here to refer you to John Holt, but since User:NancyLebovitz already did that, I'll just add that I'm amused that your handle is Petruchio.

[-]jooyous12y00

I've been thinking about this a little bit and for math problems, making an activity of writing out what they did wrong and correcting their work might be more useful than just giving back a scored assignment. Maybe, for written prompts, you could have it span a few days or weeks? Like you give the first prompt and then they answer with something short, so then then you write them a message back and have them answer that? So it's like a written conversation! And then make assembling the sentences they produced into a paragraph into an activity as well?

0DanArmak12y

Sounds like something that could devolve too easily to a shame-based system where one must submit self-criticism. Children would get an even more negative association with the subject.

0jooyous12y

Oops, that's not what I meant at all. I just meant identifying the type of error and noting it down with absolutely no shame. That's what you do after you've made an error -- you figure out what you did wrong and what extra understanding you need to fix it so you don't make the same error again. I'm talking about normalizing that process early without shame. Maybe you still don't understand how long division works. Or maybe it's just a tic error that you keep making everywhere and you just need to remember that 3 × 7 = 21 and not 24 and then you'll be fine. If the teacher who posted this question facilitates the process and doesn't shame anyone, I'm pretty confident the kids at this age will take their cue from their teacher and won't start shaming each other randomly. What happens in schools now is the teacher gives you your test back and if you have a good score then you feel good and if you have a bad score then you feel bad and go home and report your bad score to your parents, who act disappointed and no one ever really says "Now given that you have a bad score, what are you going to do to have good scores in the future? And how can I help you do that?" And if the kid is writing "What I did wrong is being a stupid person and sucking at math," then this child probably need help with things other than math.

[-][anonymous]12y00

Be genuinely interested. Since it is hard to be interested in the result of basic arithmetic operations, you need a proxy. So be interested in the kids’ learning process. Because there is no parental link and you don’t see them evolve long, eventually you will loathe the little never-learning idiots. Take a break then.

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