Intuition Math - Please Help

[Iskra]

I love threads like this!  Good thing I checked the boards today:).  I'm following this for sure.

[Iskra]

Ruth,  your "hints" are awesome.  I wouldn't call them hints even.  A hint to me means giving away a part of the solution (even a small part).  What you described encourages the kids to think of new ways to attack the problem, without giving away those "new ways", if that makes sense.  I was lucky enough to grow up in a country where this was the way of teaching for most teachers in most schools (well at least in mine).  So this kind of approach comes to me very naturally.  Sometimes it drives my kids batty that they won't get a straight answer from me (and not just math, but anytime they get stuck on anything), but oh the spark in their eyes when they figure it out for themselves a(fter my question/comment to them) is priceless!!!! and they can take ownership of the solution and are not left with "Oh yeah I get it, but how do I come up with that myself?" which I think is the problem with giving them a hint of the solution. 

Like I said, I didn't need to train myself to learn to talk this way.  This is the only way I know how to teach, because this is what was modeled to me my whole school life.    However,  if it doesn't come naturally to talk in this way for someone, I would like to whole heartedly suggest  Polya's "How to Solve it". He tells you what to say to your students in what situation and the reasoning behind it.  His choice of examples for the problem solving strategies ranges from simple ones to Calculus, but you can get a lot out of the book even if you don't know any calculus.    Also  "Creative Problem Solving" by George Lenchner paraphrases in a way that's easier to understand some of the main points from Polya in the beginning section of the book and this one is an easy read  (but it's for teachers of younger pre-algebra kids, meaning the sample problems are from pre-algebra material).

 

And last but not least,  the final step in helping kids learn how to intuit or come up with the solutions themselves came to me from none other but Andrew Pudewa from IEW.   He was talking about violin teaching and the many questions that he has to ask his students at the beginning of each lesson in order for them to position the instruments and fingers properly etc and he said he knows he has done his job well, when the students start asking themselves these questions without him because then they can do it right even when he is not around.    In the same way I think that's the ultimate goal in problem-solving.  That at some point when the kids get stuck, instead of "mom I need help" they will have a conversation in their own head and ask themselves the very same questions that you would have.   This happens in our house sometimes this way.  One of them will yell "Mom I need help".  I go to him and he tells me what he's stuck on and instead of my even asking the good questions all I say is ,  "Now what am I going to ask you?"  or "Now what am I going to say to you?"  and then he takes over and asks himself the things he has heard me ask him a million times:). 

[lewelma]

Ruth,  your "hints" are awesome.  I wouldn't call them hints even.  . 

 

Good point.  It is the direct hints that my older wants to steer away from. Perhaps what I am after is teaching problem solving skills without giving direct hints to specific problems.

 

 

I'm sorry that I'm going camping.  This is such an interesting thread and has really helped me think things through.  I still have a long way to go with my younger.

 

See you guys in a week. :001_smile:

 

Ruth in NZ

[quark]

Ruth,  your "hints" are awesome.  I wouldn't call them hints even.  A hint to me means giving away a part of the solution (even a small part).  What you described encourages the kids to think of new ways to attack the problem, without giving away those "new ways", if that makes sense.  I was lucky enough to grow up in a country where this was the way of teaching for most teachers in most schools (well at least in mine).  So this kind of approach comes to me very naturally.  Sometimes it drives my kids batty that they won't get a straight answer from me (and not just math, but anytime they get stuck on anything), but oh the spark in their eyes when they figure it out for themselves a(fter my question/comment to them) is priceless!!!! and they can take ownership of the solution and are not left with "Oh yeah I get it, but how do I come up with that myself?" which I think is the problem with giving them a hint of the solution. 

Like I said, I didn't need to train myself to learn to talk this way.  This is the only way I know how to teach, because this is what was modeled to me my whole school life.    However,  if it doesn't come naturally to talk in this way for someone, I would like to whole heartedly suggest  Polya's "How to Solve it". He tells you what to say to your students in what situation and the reasoning behind it.  His choice of examples for the problem solving strategies ranges from simple ones to Calculus, but you can get a lot out of the book even if you don't know any calculus.    Also  "Creative Problem Solving" by George Lenchner paraphrases in a way that's easier to understand some of the main points from Polya in the beginning section of the book and this one is an easy read  (but it's for teachers of younger pre-algebra kids, meaning the sample problems are from pre-algebra material).

 

 

Just liking this wasn't enough. :thumbup: I found myself saying "yes, yes, yes" with every line. This is us too.

 

I'm sorry that I'm going camping.  This is such an interesting thread and has really helped me think things through.  I still have a long way to go with my younger.

 

See you guys in a week. :001_smile:

 

How fun! Happy camping!  :seeya: 

[Luckymama]

Since I am physically with my son, I give different types of hints/help than these. They are more help on problem solving skills, rather than help on the specific problem:

 

1) Let's go through your book and see what you have learned recently. Perhaps one of those theorems will help. And then we sit together and evaluate each one, one at a time.

2) Can you work backwards? Lets think about what the last step would look like?

3) Can you draw an image? Can you color code it? Should you draw it again but in a different way so you might see things differently?

4) Can you solve a simpler problem? How can we make it easier? What have we learned by solving the easier one?

 

.

.

.

 

The other thing I always do is make sure my kid generalizes the hint/help. After the problem has been solved with a direct hint, I ask 'what did you need to see that you did not see'. 'How can you make sure that you never need this type of hint again.' 'Name the different approaches you can use when you get stuck.' I think that this consolidation is really important. Otherwise, kids never take the time to reflect..

This is exactly how I treat math with dd. When she gets stuck (like, really stuck) I lead with "So what ways have you tried to solve it?"----which means have you checked your arithmetic :rolleyes:, have you forgotten something basic, have you drawn a picture (for those types of problems), are you trying to prove forwards when it's better to prove backwards and rewrite, can you answer a simpler problem, etc.

 

At times, she does have difficulty in finding the simple problem. I have stopped her work to show her (with another problem) how to go about the process. I have always, always, problem-solved this way so it's very natural to me. Dd is much better at visualization.

 

The "debrief" afterward is very important. I catch dd going over problem issues herself, which shows she has internalized this type of self-check and -reflection :)

[Luckymama]

We have a lot of "I don't want to be alone" in our house... when dd13 is working on geometry, she wants me to sit with her. Then every time I open my mouth to speak, for any of the aforementioned hint reasons, she commands me to not say anything at all, and I have to sit there and bite my tongue, probably a good idea in any case (never mind that it's late and I want to go to bed!). Every time I try to get up, she commands me to sit back down. Oh, and I'm not supposed to browse the forums either. I'm supposed to pay attention and "help" but say nothing, LOL.

 

Eta, I think for younger kids, working together on a white board is one of the better ways, especially if it can be light and fun and puzzle-ish, which may mean few problems in a sitting, or just one. Buddy math. "What should we do next?" The thrill of figuring out the puzzle can be shared.

 

For dd, I have in fact said, "what is the topic of the chapter?" if I'm allowed to squeak out some words. On the proofs, I get a lot of "how should I start? NO don't tell me what to say but what should I say?" Um.

:lol:

 

God forbid I leave the kitchen/family room/ home office area of the house when dd is working on math!

 

But

 

She'll sigh heavily, say "Mom...?", yet as soon as I open my mouth I am faced with the Hand of Silence and the Scowl of Leave Me Alone I Wasn't Talking to You.

[Mukmuk]

:lol:

 

God forbid I leave the kitchen/family room/ home office area of the house when dd is working on math!

 

But

 

She'll sigh heavily, say "Mom...?", yet as soon as I open my mouth I am faced with the Hand of Silence and the Scowl of Leave Me Alone I Wasn't Talking to You.

I had written but lost my post. Wapiti's situation is what I have at home, together with The Hand and the The Scowl. Hours go by as I sit with the computer. I AM allowed to browse, whew! Or read a book or go cook. I have been known to get away with, "How about reading the text book again?" once in a while when noises from him grow louder. :ph34r:

 

I think I'll be obnoxious and ask, "so what approach did you take to finally crack that?". Ooh that's living on the edge :D.

[daijobu]

Just in case anyone else is reading this and wondering whether AoPS is a guarateed source of pain and frustration, I wanted to add my 2 cents.  Or maybe it's because the kids on this thread are at a higher level (olympiad geometry) than my dd's (algebra and regular geometry).  

 

But I can say for my own kids AoPS is mostly fun for them, without frustration.  Usually the problems they can't solve right away are the challenge problems and I don't sweat those too much.  We do go over any problems (challenge or otherwise) they can't solve on their own, but they don't work for hours trying to solve them, and I don't make them.  But then, in all likelihood I'm educating a couple of engineers, not mathematicians.  

[wapiti]

Just in case anyone else is reading this and wondering whether AoPS is a guarateed source of pain and frustration, I wanted to add my 2 cents.  Or maybe it's because the kids on this thread are at a higher level (olympiad geometry) than my dd's (algebra and regular geometry).  

 

But I can say for my own kids AoPS is mostly fun for them, without frustration.  Usually the problems they can't solve right away are the challenge problems and I don't sweat those too much.  We do go over any problems (challenge or otherwise) they can't solve on their own, but they don't work for hours trying to solve them, and I don't make them.  But then, in all likelihood I'm educating a couple of engineers, not mathematicians.  

 

FWIW, our house is like yours, in that my dd isn't going to spend hours and hours on a problem (though a proof I can guarantee more than an hour of thinking about it, though it seems to me it's more efficient to work for a little while on it and come back to it later after there's been time for the problem to marinate a bit, but that would require actually getting started before the night it is due).  Eventually, she might let me help her, especially if she's in a hurry.  *If* I even can.  And we're only talking regular geometry.  But, she's especially proud and happy when she solves problems without any help at all.  I want to cultivate that sense of accomplishment coupled with the thrill of figuring out the puzzle.

 

I think my kids feel about AoPS a bit like how I feel about running - I feel really good once I've done it (especially in the homestretch, LOL) and often it's fun, but getting myself out there in the first place is the hard part.

[daijobu]

 But, she's especially proud and happy when she solves problems without any help at all.  I want to cultivate that sense of accomplishment coupled with the thrill of figuring out the puzzle.

 

 

 

"Mom!  I solved a challenge problem!"

 

"Mom!  I solved a starred problem!"

 

Music to my ears.  

[Chrysalis Academy]

"Mom!  I solved a challenge problem!"

 

"Mom!  I solved a starred problem!"

 

Music to my ears.  

 

 

And I love that some of the starred problems in BA aren't actually that hard.  At first I thought that was a bug, now I realize it's a feature.  When Morgan solves one of those, and says, "That wasn't that hard!" it's a good thing - it shows her not to be afraid of tackling a challenge, while making her feel good that she's doing "hard math" successfully.  It makes her more likely to persevere when the problem actually *is* hard.

[Arcadia]

Both my boys prefer curriculum that make them cry than make them bored. So YMMV.

Hubby thought it was too hard since they cry but they don't want to change.

My older found the challenge sections easier than the main sections sometimes. His challenge sections has no careless mistakes but his main section does.