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Math learning through evidence/ proof vs just a list of things to teach


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Seeing Struggling Math Learners as 'Sense Makers', not 'Mistake Makers'. (Katrina Schwartz, Mind/Shift, Aug 4, 2015)

 

 

“A major goal of math classrooms should be to develop people who look for evidence and try to prove that things are true or not true,†Wees said. “You can do that at any ageâ€

 

No student solved the problem — in fact, the mathematician Leonhard Euler proved it was impossible. Wees showed his students Euler’s proof, and pointed out how similar their graphing was to his. Wees said kids were a little mad when they discovered there was no answer, but they enjoyed the experience and along the way realized that learning is about the process.

 

Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.

(Thank you Mr Lockhart!)

 

When doing his master’s in education technology and the pedagogy around it, Wees learned to categorize the three kinds of questions students ask and changed his teaching practice entirely. Kids ask questions: 1) to find out if they did the problem right; 2) because the teacher is standing near them and they can, and; 3) occasionally they ask “I wonder what if†questions, which show they are thinking about the math. Wees took to not answering the first two kinds of questions and encouraging the third.

 

“We have generations of math-phobia,†said Laura Thomas, director of the Antioch Center for School Renewal. “A lot of teachers who teach math are second- and third-generation math-phobic, so our system is really calculation-based as opposed to applying in context.â€

 

 

Wees is frustrated at how linear math learning has become. “The standards are a list of things the kids are supposed to do, not a list of things you have to teach,†Wees said.

In other words, many standards can be embedded in a problem so that students are exposed to lots of ideas in different ways. When teachers focus on clusters of standards as opposed to individual ones, “that kid who doesn’t get one idea on Thursday is going to get 10 or 12 other ways of looking at the idea in the unit,†Wees said.

 

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Clock still amazes me. How can one claim to teach clock with just rote memorization of hand positions. It would be much more effective to use science and social studies to show measurement history and how tne clock works.

 

In exclusion of teaching hand positions?? Just how old do you want children to be before they can just, you know, tell time? :ohmy:

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4-6 is fine with me, and that seems about when kids match the digital to the analog, if they have access. But teaching an 8 by rote memorization of hand positions is cheating him out of learning why.

 

So, you think that "It would be much more effective to use science and social studies to show measurement history and how tne clock works", even for children as young as 4-6? Really? Effective??

 

Huh.

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Clock still amazes me. How can one claim to teach clock with just rote memorization of hand positions. It would be much more effective to use science and social studies to show measurement history and how tne clock works.

 

I have never heard that parlance before. "Teach clock."

 

1. Teaching a child to tell time in the first place.

 

2. Showing and explaining how clocks work, and studying how people have measured time throughout history and around the world.

 

You're setting these two concepts up as different paths to be chosen, but everybody I know teaches all of it, one way or another. Nobody chooses between teaching a kid to read a clock and telling him about the history of timekeeping. It's all to be learned.

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I think a large part of the problem is that most math teachers - especially at the elementary school level - don't understand math well enough to be able to teach math through evidence and proofs. I would also fit into this category, but luckily my dh is a math guy and can show us a better way to understanding and enjoying math.

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Referring to the comment about figuring out the why - I just wanted to plug something I've been reading lately - it's How to Bake Pi. The first half is really about what math is for, how it works, and the kinds of problems mathmeticians think about.  The second half is about category theory - I haven't tackled that part yet - but I think this book is written so clearly that it would be really helpful for parents, teachers, and middle grade and up kids who are really struggling with that question of what math is for, what it's about, what mathemeticians do, and why it's worth caring.

 

http://www.amazon.com/How-Bake-Pi-Exploration-Mathematics/dp/0465051715/ref=sr_1_1?ie=UTF8&qid=1438783917&sr=8-1&keywords=how+to+make+pi

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[...] I will bow out here, as I am not interested in figuring out your puzzle.

 

As a parent volunteer for elementary, clock is the topic that I always have to show evidence/proof to little ones to have a breakthrough in understanding. For those whose parents do not teach it, its not obvious, and they benefit greatly from understanding. Many do not have access to any clock, digital or analog, in the home.

 

Heigh Ho, as OP, I apologize that you feel the need to bow out. Personally, I am very interested in everything one has to say as long as it is expressed respectfully. Thank you for showing that respect.

 

I can relate very much to your experience.

 

Teaching time was one of the most challenging areas for my son. He just couldn't get it. We had to wait till he was 8, closer to 9 for it to click and I think learning about degrees, modular math, history of clocks etc plus that thing that one just cannot control (i.e. a child's readiness to understand) finally did the trick. I added a challenge too...he had to tell me when it was time for his favorite TV shows (evil grin). We had both analog and digital clocks at home. We even tried making our own clock. Teaching him just based on hand positions did not help him at all. I have a highly asynchronous child and we both felt a level of frustration with this that might not be common.

 

Please come back Heigh Ho.

 

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Referring to the comment about figuring out the why - I just wanted to plug something I've been reading lately - it's How to Bake Pi. The first half is really about what math is for, how it works, and the kinds of problems mathmeticians think about.  The second half is about category theory - I haven't tackled that part yet - but I think this book is written so clearly that it would be really helpful for parents, teachers, and middle grade and up kids who are really struggling with that question of what math is for, what it's about, what mathemeticians do, and why it's worth caring.

 

http://www.amazon.com/How-Bake-Pi-Exploration-Mathematics/dp/0465051715/ref=sr_1_1?ie=UTF8&qid=1438783917&sr=8-1&keywords=how+to+make+pi

 

Thanks so much for this Rose. I love how you manage to suggest a delightful book into your thoughtful responses.

 

I really hope articles like this one will help mitigate some of the math phobia I see and have felt myself (and still feel). My take away is conversing about math and about not having to solve every single problem correctly. I used to think about this very differently but now I feel happy when we can't solve a problem. There is a very deceptively simple problem on our whiteboard right now that both DS and I can't solve (it's been almost 2 months). We can very easily google the answer but we don't want to. And no, the answer is not going to miraculously help us in our every day lives. We are just curious. We want to see if the aha moment will come, when, why and how. What led to it. And why it just doesn't seem obvious right now.

 

I suddenly feel like I am solving a mystery when this happens. I've loved mystery novels all my life. Fighting with math like Poirot would tackle a particularly pesky poisoning or serial murder...it's SO satisfying!

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,

Teaching time was one of the most challenging areas for my son. He just couldn't get it. We had to wait till he was 8, closer to 9 for it to click and I think learning about degrees, modular math, history of clocks etc plus that thing that one just cannot control (i.e. a child's readiness to understand) finally did the trick.

 

My DS9 have to look at the hour glasses and his analog watch and watch the second hand, minute hand and hour hand move to get the idea.  I don't think schools would have allowed him to stare at his analog watch and do nothing else for more than an hour but it works for this kid :)

 

I read the book that Rose recommended :)  I kind of read every book in the libraries in the math and science section :lol:

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There is a very deceptively simple problem on our whiteboard right now that both DS and I can't solve (it's been almost 2 months). We can very easily google the answer but we don't want to. And no, the answer is not going to miraculously help us in our every day lives. We are just curious. We want to see if the aha moment will come, when, why and how. What led to it. And why it just doesn't seem obvious right now.

 

I suddenly feel like I am solving a mystery when this happens. I've loved mystery novels all my life. Fighting with math like Poirot would tackle a particularly pesky poisoning or serial murder...it's SO satisfying!

I love this idea! I think I will have to try this with my kids. :-)

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  • 2 weeks later...

Referring to the comment about figuring out the why - I just wanted to plug something I've been reading lately - it's How to Bake Pi. The first half is really about what math is for, how it works, and the kinds of problems mathmeticians think about.  The second half is about category theory - I haven't tackled that part yet - but I think this book is written so clearly that it would be really helpful for parents, teachers, and middle grade and up kids who are really struggling with that question of what math is for, what it's about, what mathemeticians do, and why it's worth caring.

 

http://www.amazon.com/How-Bake-Pi-Exploration-Mathematics/dp/0465051715/ref=sr_1_1?ie=UTF8&qid=1438783917&sr=8-1&keywords=how+to+make+pi

 

Rose, I'm going to have to move in with you in my old age as you're causing me to draw down my retirement buying books! At least we'll have good stuff to read.  :D

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He found himself often asking the same question, whether a student had gotten the problem right or wrong. He’d ask them to explain their answer or how they could check to see if they were right or wrong.

 

“I became better at having a poker face so I wasn’t communicating whether they were right or wrong,†Wees laughed. When students asked questions because he was nearby, he deferred them to their peers, who often explained the math quite well.

 

I like to do this to my kids, but it's funny how they always almost immediately start erasing.  Like, if I'm asking, they must be wrong.  Until reading this, I wasn't completely conscious of this.  I think it may be one of those things that if we address it, I might have better outcomes.  My kiddos are just little, but somewhere along the line, they've gotten to thinking that math is about always being right, and being right before your twin!

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Miquon

 

Discovery method math isn't new.  It's just not trusted.  Parents like to brag that their children are getting 100% on their math tests.  I don't often hear anyone brag that their child spent a whole hour trying to figure out one problem.  (That sort of thing is assumed to be a discipline problem and met with frustration and punishment.)

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