Against Answer Getting

[Chrysalis Academy]

This video - very much in the spirit of Richard R's discussion of math problem solving - seems relevant to several current ongoing conversations, so I thought I'd give it its own post. 

 

What's wrong with math teaching in the US?

 

Is it the educational goals/philosophy?  

 

Is it teacher quality/prep?

 

Is it the curriculum?

 

All, none, something else?

 

http://vimeo.com/30924981

 

 

[Chrysalis Academy]

Huh, I was surprised to wake up and find no comments about this video!  Is it just so obvious that I'm the only one having an epiphany? ;)

 

 I went to bed last night asking myself, "Am I really focused on teaching the math in each lesson, or am I just teaching them how to get the answer?"  I woke up thinking about it this morning.  Kind of a lot of soul-searching and self-examination of myself as a math teacher.  It's uncomfortable.

 

I have the intention of teaching the math, of exploring different paths to the solution, of talking through and making sure they understand what is going on in the problem and how the problem works.  I have to be constantly vigilant, though, in not just showing them how to do the problem when they struggle.  Sometimes I forget.  Because, that was how I learned math:  I learned all the goofy strategies he was talking about and never understood why they worked.  I was just like the speaker's smart daughter: I figured out how to get As and Bs in math without ever understanding what I was doing.  I've done all the strategies he discussed - I don't know whether to multiply or divide, so try both and figure out which one gives an answer that is logically possible.  Oh, ok, go with that one.  Why?  No clue.

 

 

My 6th grader is working on ratios.  I have to actively hold myself back from saying "Just cross multiply" and make her work through *what* a ratio is - what is the fraction the problem is talking about - and make her work through the steps of solving the problem.  On the plus side, I'm finally learning math, conceptually!  

 

Sometimes I feel like I have no business teaching math.  My consolation at this point is 1) I know I can learn this alongside (or ideally just ahead) of her for now, and 2) She wouldn't be getting better instruction in school: the curriculum is worse, the teacher is unlikely to understand the math better than I do, and the teacher sure as heck wouldn't care about my kid's individual understanding any more than I do.

 

One thing I gleaned from the video that I want to put into practice right away:  do a better job of using errors to uncover conceptual misunderstandings, to use them as a kind of a formative assessment to guide what I need to teach next, or again, or better.  My kids hate getting problems wrong, and I've been working hard to change that - not to make them happy to get things wrong, but to help them see errors as a chance to learn something.  But I think I can go deeper with that.  Of course, it involves changing my attitude, too - not letting them see that I'm disappointed when they make mistakes, but genuinely looking at it as a chance to improve my teaching and their learning.  I can improve this, I know.

 

Dang, this teaching thing is tough, no?

[MBM]

I just watched the video and took notes of some of the things Daro is mentioning.

 

In the US, we are fixated on memorizing procedures to get the right answer without necessarily understanding the underlying mathematical ideas, but we would be better to spend some time attempting to understand the mathematics by examining what was done to get a right answer or what was done to get a wrong answer. 

 

Interesting.

[sunnyday]

It took me a while to have 18 minutes to sit and watch a video. :) Very interesting though! Ties back to a lot of what we've been discussing elsewhere. I will come back with a few resources that this brought to mind...had to share an anecdote though. :) Recently my son hit a tough problem in his Beast Academy book, tried it his way and got it wrong, then at my gentle urging tried the suggested way and got it right. When he moaned about how "uncomfortable" he was to have gotten the wrong answer the first time, I got a little frustrated and snapped, "This is MATH. There are usually many ways to get the answer. Some are better than others -- they might be more efficient, or make it easier to avoid mistakes. Your approach was totally fine! But you just learned that the book's suggested method gives fewer mistakes. THAT was the math!"

 

A few days later in the car, he informed me that, "If you get the answer wrong, it's important to know WHY you got it wrong and to be able to show why it was wrong. That way you can tell someone else, and they can avoid that mistake."

 

So, yeah. This is definitely the approach I want my children to absorb, and this speaker did a nice job explaining *why* it's so important...

[sunnyday]

OK, a couple of things that had popped into mind...first was that this goes right along with the Coursera course I started a few months ago (it just finished and I didn't get very far, unfortunately, but I still hope to go back and watch some lectures) about Mathematical Thinking. It was kind of intended as a tie-in/elaboration on the professor's latest book, Introduction to Mathematical Thinking. He talks about the shift in the mathematical world that happened in the mid-1800s, which changed the emphasis of the field from calculation to understanding abstract concepts. Other than the botched New Math, 150 years have passed and we still haven't adjusted our instruction to get with the times! Pretty sad.
 
The other was this speech by David Coleman, the "architect" of the Common Core State Standards, in which he discusses how the data (acquired from various international high-achieving school systems) shows that success with a few limited subjects can predict success in math over the long run. For example...a firm grasp of fractions, predicts success in algebra. He also talks about the importance of treating teachers as individuals, and of making sure that the effective ones have a greater impact on more kids, and weeding out the ineffective ones.
 
Oh, and the last one, from the Beast Academy Facebook page, on the connections between arithmetic and algebra:

"In a recent web forum, parents of aspiring math beasts have been discussing the usefulness of some of the mental math tricks that we demonstrate in the Perfect Squares chapter of Beast Academy 3B. The first trick involves finding the square of a number that ends in 5, and the second trick involves multiplying a pair of numbers that differ by 2.[/size]

Why do we teach these tricks? Are they really useful[/size] at all?
For one thing, kids think it is really cool to be able to compute 34×36 in their heads. (Can you? The answer is at the end of the message). But, there is another important reason.

These "tricks" are often taught as if they were some sort of magical coincidence. At Beast Academy, they offer the foundations for algebra. Many students never see the connections between the arithmetic they learn in elementary school and the algebra they learn later on. Perhaps this is why so many students hit a wall in middle school math.

The trick for squaring a number that ends in 5 looks like this:
65×65
=(60+5)(60+5)
=60×60 + 60×5 + 60×5 + 5×5
=60(60+5+5) + 25
=60×70 + 25
=2025
Later, when students are asked to expand (x+5)(x+5), students will have a foothold to stand on.

The second trick looks like this:
81×79
=80×80-1
(Admittedly, the trick only appears in the workbook and we offer only a brief geometric explanation, but we will explore it more later). When students get to (x+1)(x-1), the "trick" offers a way for us to connect arithmetic and algebra.

Hope this helps further these sorts of discussions on the web."[/size]

 

Pretty random thoughts, but I have a pretty random brain. :lol: All that to say, that there's a lot wrong with math instruction in the US today. I blame testing and grading, as well as an entrenched and habituated belief that computation -- answer getting -- is the most important part of mathematics. I've got no idea how to fix it, except by using more conceptual curricula with our own kids if we have the choice.

[Chrysalis Academy]

Oh, I lamely dropped out of that course due to lack of time.  Sounds like I should watch it!  Thanks for mentioning it.

 

 

[Arcadia]

 I blame testing and grading, as well as an entrenched and habituated belief that computation -- answer getting -- is the most important part of mathematics.

So far from kindergarten math to pre-algebra,  my boys only had one public school teacher that understand and could teach math.  She happen to have her first degree in economics.  The rest are reading off the text/curriculum and won't even notice a typing error in the textbook or PowerPoint slides. How do you expect a math teacher to give open ended discussion questions if they need work out solutions supplied by the publishers like Pearson, Holt and the rest?  I'm staying in a neighborhood that majority afterschool, even those whose children are in private schools.  

 

My opinion is that having subject teachers at least from 3rd grade onwards would help alleviate the problem.  Most of the elementary teachers I know are hyper good in language arts but lacking in math and science.

[8filltheheart]

Honestly, Rose, I just watched it and I can’t imagine teaching the concepts any of the ways that he said is going on in schools.   I have never taught my kids math that way and I am in no way an enlightened mathematician.   It just didn’t seem revolutionary and seemed more obvious to me that that is NOT the way to go about it.  Guess things are even worse in schools than I imagined!!  :P

[Chrysalis Academy]

Honestly, Rose, I just watched it and I can’t imagine teaching the concepts any of the ways that he said is going on in schools.   I have never taught my kids math that way and I am in no way an enlightened mathematician.   It just didn’t seem revolutionary and seemed more obvious to me that that is NOT the way to go about it.  Guess things are even worse in schools than I imagined!!  :p

 

Well, I can't imagine teaching those techniques either, and I do worry things are worse than we think . . . but at the same time, I have to ask myself if sometimes I am guilty of focusing too much on getting the right answer.  I think that's the part that is really sticking with me: Am I, as a teacher, too focused on my child getting the answers right? Am I really taking advantage of the opportunities that their errors provide to deepen their learning? Why are *they* so upset when they get answers wrong?  And how can I help them to see math as something more than getting to the right answer?

 

I know I use solid materials, and I hope I am constantly improving my technique, but I know I don't get math as deeply as I want them to.  It's just something I worry about: I want to be the best possible teacher I can be, and teaching math is tough.  

 

Or, this is just my latest thing to feel insecure and self-doubting about? :rolleyes:  IDK.  I know I'm doing my best for them, I just so desperately want that to be good enough.

[kateingr]

Rose, you're definitely not the only one! This is something I try to keep in the back of my mind, but it can be genuinely hard to remember when I'm in the midst of my school day and start to get to focused on getting the work done for the day. The thing that helps me the most is to remember that the point of school isn't to complete all the worksheets, assignments, readings, etc. but for the kids to learn. (I actually have a little dorky joke I mutter to myself to remind me of this: "These worksheets aren't going to complete themselves!" It reminds me that perhaps the worksheets don't actually need to be done sometimes. Or perhaps they should be done in a couple weeks once the topic is understood more fully.)

 

 

Have you read The Smartest Kids in the World? One thing I found fascinating is that teachers in some of the countries believe that the work is too easy if kids can get everything right. My son was mastering everything I threw at him in Rightstart c, so I'm now mixing it up with some Beast Academy so he gets used to not being able to get everything right in math. It's been challenging for both of us, but I'm seeing the fruit of it in his attitude and growing desire to be challenged.

 

Kate

[8filltheheart]

Deep breath.   :)   Remember the Circe thread?   Even if none of Kern's thoughts strike a chord with you, one pearl of wisdom that he shared is a valid truth.......teaching is its best when we teach from a place of rest.   Why?   b/c then we embrace the fullness of what we are teaching with confidence and joy.   We are able to convey that to our children.  

 

Have you ever noticed that the days you are stressed or rushed it seems like Murphy's Law rules??   Everything that can go wrong, does.     But the days that you are most relaxed and laid back, you often seem to get even more done??  When I take my time and approach teaching in a joyful back and forth conversation, everyone engages and thinks deeply.  When I rush and stress, everyone shuts down and becomes careless and dumb.  ;) 

 

You care about what you are doing.  You are teaching your kids well.   Have confidence in yourself and approach education with that truth.   :)

[Pen]

...

 

Oh, and the last one, from the Beast Academy Facebook page, on the connections between arithmetic and algebra:

 

The trick for squaring a number that ends in 5 looks like this:

65×65

=(60+5)(60+5)

=60×60 + 60×5 + 60×5 + 5×5

=60(60+5+5) + 25

=60×70 + 25

=2025

Later, when students are asked to expand (x+5)(x+5), students will have a foothold to stand on.

 

Pretty random thoughts, but I have a pretty random brain. :lol: All that to say, that there's a lot wrong with math instruction in the US today. I blame testing and grading, as well as an entrenched and habituated belief that computation -- answer getting -- is the most important part of mathematics. I've got no idea how to fix it, except by using more conceptual curricula with our own kids if we have the choice.

 

 

Can't really follow this thread right now until I'm where there's high speed and I can see the video and get links.

 

For the moment I'm subbing.

 

And expressing confusion.  

 

It is supposed to be better to have kids follow along a system that results in thinking 65 x 65 = 2025, why?-- because it is better to get answers wrong for ultimate learning?   ???

 

That idea keeps confusing me.

 

If they heard that 65 x 65 = 4225 that would be ultimately of less value?  Why?  Because with the wrong answer they figure it out for themselves?     ????

 

It makes my head spin.

[kitten18]

Can't really follow this thread right now until I'm where there's high speed and I can see the video and get links.

 

For the moment I'm subbing.

 

And expressing confusion.

 

It is supposed to be better to have kids follow along a system that results in thinking 65 x 65 = 2025, why?-- because it is better to get answers wrong for ultimate learning? ???

 

That idea keeps confusing me.

 

If they heard that 65 x 65 = 4225 that would be ultimately of less value? Why? Because with the wrong answer they figure it out for themselves? ????

 

It makes my head spin.

I think there's a typo.

 

They have 60 x 70 + 25

It should have been the right answer. I don't know where 2025 came from.

[sunnyday]

It is supposed to be better to have kids follow along a system that results in thinking 65 x 65 = 2025, why?-- because it is better to get answers wrong for ultimate learning?   ???

 

:lol:

 

If you follow the steps, the last one is 60 x 70 + 25, which is clearly 4225. I have no idea how the authors managed to typo 2025, but...I think it's clear that isn't the point. ;) Getting the right answer is not the FIRST objective, obviously!  :D

 

But seriously, I tell my 7yo all the time that algebra is pretty much the same stuff he's been doing, just sticking an unknown in where a number would go -- using a letter or symbol as a placeholder -- and looking at the patterns that come out. How nifty to have a 3rd grade math program that makes this explicit!

[kitten18]

I really enjoyed the video. Thanks.

 

I don't have anything intelligent to add right now but I will be thinking about it a lot.

I will probably dream about it. A few weeks ago I was watching all of Rosie's fraction videos and working out the problems just like her girls. I had a big white board full of fractions problems. A couple nights I had dreams about teaching fractions.

[Chrysalis Academy]

Ahh, sometimes it is good to revisit old threads (thanks, 8!).  Here are some quotes from Andrew Kern from that old Circe thread that seem relevant to my current (chronic?) situaion:

 

 

Above all, I entreat you to note the fourth point: we must teach from a state of rest, not anxiety. if we are anxious, we will pass our anxiety on to our children. Anxiety does not lead to sound decisions or careful thinking. 

Shuck off the failed expectations and theories of the world you home school to escape. Obey the laws with joy, but don't let the establishment intimidate you. Even more, don't worry about the neighbors and their children or especially about this awful average quasi-child, non-existent child who you are supposed to use as your standard to assess your own child.

Take your time. 
Rest. 
Don't even try to catch up.

Identify the core skills you want your children to learn and find the best way to teach them (yes, Memoria Press is excellent for this).

Identify the knowledge you want them to know and teach them. They won't remember what is in a text book anyway, so don't worry so much about which text book you use. There are hundreds and thousands of good and great books. 

Identify the ideas you want them to think about (you could include this on the page about apprehension above) and think about them.

You'll make a thousand mistakes, but not as many as I have. But you'll learn so much you won't be able to stand the pleasure and you'll watch your children's souls flow to overflowing. 

And chances are, they'll pass the SAT test too. 

[idnib]

A few random thoughts...

 

I have complained in the past on this board about DS's problems with perfectionism and becoming upset and angry if he gets the wrong answer. It turns out this has been an actual blessing in disguise because it has forced me to emphasize the mathematical thinking more than worrying about the computation. Which is not to imply he can get the wrong answer and walk away, but rather I have to approach the wrong answer from a positive place because otherwise he becomes quite anxious and upset with himself.

 

For the same reason I have had to drill into him over and over again that if he is getting the answers all correct we are wasting time and should move to something harder. In retrospect, this thing I started off saying in a trite manner to appease him has actually informed our school work as we've internalized it, and not just in math. This make me happy.

 

In some ways I am not sure Americans are ready for the idea of incorrect answers as good things. I am somewhat neutral to negative about CC, but I have seen parents up in arms about how the "right answer doesn't even matter" and I'm pretty sure that's a misinterpretation of this idea of letting mistakes be used to teach mathematical ideas.

 

I don't find the ideas in the video all that new, perhaps because I've read most of Liping Ma's excellent book Knowing and Teaching Elementary Mathematics. I confess to not having finished it though.

 

I'm having trouble deciding how I feel about CC math. One of the factors giving me grief is the amount of teacher complaints. I really do wonder how much of the grousing is because the teachers don't really understand math themselves and therefore are having trouble teaching it in a new and different way. If today's younger teachers learned only one way, I can see how it's a problem to do it another way. Is that the source of some of this complaining? I don't know. Of course it's also possible that many of the complaints are coming from teachers who do understand mathematical concepts and are therefore not happy or supportive. It's hard to say, especially when it's so much easier for teachers to say something is bad than to admit they don't understand it.

 

Like I said, random thoughts.  :001_smile:

 

[Roadrunner]

Honestly, Rose, I just watched it and I can’t imagine teaching the concepts any of the ways that he said is going on in schools. I have never taught my kids math that way and I am in no way an enlightened mathematician. It just didn’t seem revolutionary and seemed more obvious to me that that is NOT the way to go about it. Guess things are even worse in schools than I imagined!! :P

This was my exact reaction, but I haven't studied in an American high school. I am having a hard time believing the instruction is as bad as this video shows, but then what do I know.

[8filltheheart]

Do teachers really teach fractions in such a way that kids can't do simple 1/2+1/3+1/4=   or division/exponents in such a way that x^5/X^5 = 0???   I am really shocked if those comments really reflect the state of math education b/c that is beyond pitiful.   It is inexcusable.     If students can't solve those problems, then what exactly are they doing that is called "math'????  

 

If this video is supposed to instill confidence that CC is the answer, it doesn't do that for me at all.   It simply leaves me unimpressed that actual teaching/learning is a REAL educational objective at all.   How on earth did we get to a pt that those simple problems are used as examples of what kids need to be able to do?  I am simple left astounded that it would ever be possible that they wouldn't be!!!!

[Roadrunner]

If this video is supposed to instill confidence that CC is the answer, it doesn't do that for me at all. It simply leaves me unimpressed that actual teaching/learning is a REAL educational objective at all. How on earth did we get to a pt that those simple problems are used as examples of what kids need to be able to do? I am simple left astounded that it would ever be possible that they wouldn't be!!!!

Apparently CC specifically indicates how a concept needs to be taught. Assuming teachers get training in correct methods, there is some hope of improvement. At least our district is engaged in extensive training.

Overall, I share your outrage.

[8filltheheart]

Apparently CC specifically indicates how a concept needs to be taught. Assuming teachers get training in correct methods, there is some hope of improvement. At least our district is engaged in extensive training.Overall, I share your outrage.

That is not reassuring to me at all. There are multiple strategies that can be employed for any given problem. One of the reasons I swore I would never teach in ps is bc of the restrictions on how to teach.....the how was a narrow dictate. "Explicitly indicating how a concept needs to be taught" is restrictive. It is a top down solution vs fixing the actual problem. Scripts don't allow for exploration and free flow discussion.

 

If the 2nd of the 2 problems Rose posted last night are indicative of what is going on in the avg 2nd grade classroom, my belief that we are heading in the wrong direction is underscored. If expecting 2nd graders to divide 134 in half without understanding division is the norm, then we are instilling in them nothing more than math is mystical vs logical either that or math is so confusing we can't figure it out.

 

I would much rather see teaching as teaching the needs individuals and allowing room for how individual students learn. For example, my 6th grader cannot visualize bar diagrams to solve MIF's word problems. But, she can set up the equations like she does for HoE and solves them. Is that wrong? Why is there only one way to explicitly set up and solve?

[Incognito]

Teaching for understanding the concept and not teaching to get the right answer is why I LOVE Life of Fred.  Good way to explain what is good about it - it isn't learning the pattern for the day or how to get the answer right, it is learning how to think mathematically.  And the practice or problems are called "your turn to play" and the answers assume kids get them wrong - explaining the meaning behind the answers is a big part of the point of the questions in the first place.

 

 

[Chrysalis Academy]

What interested me most about this video wasn't really the specifics of the (poor) teaching techniques, nor was it the "promise" of CC to change things.  What really hit me between the eyes was the first statement, about goals:  "I see the goal of the Japanese teacher as different from the American teacher.  The American teacher looks at a problem and asks, 'How can I teach my kids to get the answer to this problem.' The Japanese teacher asks, 'What is the mathematics they are supposed to learn from working on this problem? How can I get them to learn the mathematics?"

 

Part of why this really struck me goes back to that Stanford online class some of us followed this summer, the one by Jo Boaler.  A lot of us had mixed feelings about what she was saying.  One of the things she said that kind of stuck in my craw was that you should make your classroom a mistake-friendly environment.  That kids learn more from making mistakes than they do from getting the right answers, so that mistakes should be celebrated.  And while I got the point about learning from mistakes, I totally disagreed with the idea that "mistakes should be celebrated."  I mean, isn't the goal for the student to be able to get the right answer? How can we celebrate it when they don't?

 

So that's what hit me about this video:  I think he got right to the heart of it.  It's not that "mistakes should be celebrated" or that right answers don't matter.  But I was wrong, too.  The goal of my teaching should not be that my students get the right answer.  The goal is that they learn the math.  If they do that, they will show it by getting the right answer, which is certainly what I want.  But it can't be my goal.

 

Why?  Because your goal affects how you teach.   This is where re-reading the Circe thread last night really resonated.  A few pages in there was a discussion of exactly this: what is our goal when we teach our kids math? How do we teach math from this place of rest, this quest for truth and beauty?  I don't want to digress on to that too much here, but I just thought it was so interesting how connected that discussion was to what was going through my head already.  Serendipity.

 

If my goal is to have them get the right answer, I will probably not challenge them sufficiently with hard questions.  If my goal is to have them get the right answer, I will give them easy problems that they can solve, so that I feel good about reaching my goal.  If my goal is to have them get the right answer, I will move them along too quickly rather than lingering and grappling with hard problems.  

 

If my goal is to have them get the right answer, I will not consistently teach them the way that I want to.  I may have moments of grace, but when I'm hurried or distracted, I will fall back on the old habit.

 

So the epiphany for me was that I have to truly, deeply, change my goal.  And I have to been conscious and mindful about changing bad habits.  It's not about mistakes being bad or good, it is about the fundamental, core goal of my teaching.  This is something I can focus on.

[Chrysalis Academy]

That is not reassuring to me at all. There are multiple strategies that can be employed for any given problem. One of the reasons I swore I would never teach in ps is bc of the restrictions on how to teach.....the how was a narrow dictate. "Explicitly indicating how a concept needs to be taught" is restrictive. It is a top down solution vs fixing the actual problem. Scripts don't allow for exploration and free flow discussion.

 

If the 2nd of the 2 problems Rose posted last night are indicative of what is going on in the avg 2nd grade classroom, my belief that we are heading in the wrong direction is underscored. If expecting 2nd graders to divide 134 in half without understanding division is the norm, then we are instilling in them nothing more than math is mystical vs logical either that or math is so confusing we can't figure it out.

 

I would much rather see teaching as teaching the needs individuals and allowing room for how individual students learn. For example, my 6th grader cannot visualize bar diagrams to solve MIF's word problems. But, she can set up the equations like she does for HoE and solves them. Is that wrong? Why is there only one way to explicitly set up and solve?

 

I agree with you completely here, and I am both mystified and outraged at this "method" of teaching.  This math curriculum, and how it was being taught (or, really, not taught) by teachers who couldn't implement it was The Reason that I started homeschooling in the first place.  I have lots of other reasons now, but this was why I started.

 

 In all fairness, though, I want to say that this math program (Houghton Mifflin CA Math) is, IMO, truly atrocious but it is not a CC-aligned program.  The schools haven't had the budget to switch to a CC-aligned math yet.  I don't have hopes that the CC-aligned math from this publisher will be much better, but this bad curriculum isn't a result of CC, it precedes it.

[8filltheheart]

So the epiphany for me was that I have to truly, deeply, change my goal. And I have to been conscious and mindful about changing bad habits. It's not about mistakes being bad or good, it is about the fundamental, core goal of my teaching. This is something I can focus on.

I think this goes to the very heart of good teaching. I love teaching. I love the dialogue, the back and forth, the questioning, the epiphanies. I honestly still can't relate to the video bc I envision education as one big seeking out why's. The idea that the "whys" are found in a script leaves me cold. ....which is why I struggle in writing out how I teach in a curriculum bc for me teaching is a dialogue. Dialogue requires input/feedback. It is not simply "giving".....it requires listening, responding, redirecting, questioning. That is teaching. Telling someone how to explicitly teach something is a contradiction of terms from my perspective bc it removes the very elements that make teaching teaching.

[idnib]

What interested me most about this video wasn't really the specifics of the (poor) teaching techniques, nor was it the "promise" of CC to change things.  What really hit me between the eyes was the first statement, about goals:  "I see the goal of the Japanese teacher as different from the American teacher.  The American teacher looks at a problem and asks, 'How can I teach my kids to get the answer to this problem.' The Japanese teacher asks, 'What is the mathematics they are supposed to learn from working on this problem? How can I get them to learn the mathematics?"

 

This was the most interesting part for me too. I think I've mostly been doing the more "Japanese-style" teaching, having read Ma's book and using Singapore, plus dealing with my aforementioned difficulties with DS's perfectionism. But still it's good to get a concise way of thinking about this. The questions he posed above are a good shorthand for the more amorphous idea I've been trying to hold in my head while teaching.

 

The other interesting part for me was the fractions and how they are multiplied. First I nearly fell out of my chair because he said only ~20% of US kids could add those 3 fractions on the test. Seriously?!!

 

I then had an epiphany that perhaps my Southern CA late-70s and 80s math education wasn't as bad as I had thought. I never learned any of those answer techniques he discussed and the only arithmetic operation I could thing of where we purely learned an algorithm instead of the reasoning was long division. I'm not saying it was a fabulous math education, but I doubt anyone came out of 5th grade not being able to add 3 fractions. Good grief, what's happened here?  :crying:

[Caclcoca]

Can someone tell me how to find the old Andrew Kern/Circe thread? I must not be doing something right. It's not coming up. Thanks!

 

ETA: NM I found them right after I asked!

[Roadrunner]

First, I didn't say there were scripts, well at least not in CC, but unfortunately (or fortunately) some programs come with scripts. What I said is CC is specifically indicating that procedural fluency alone isn't enough and math needs to be taught conceptually as well. In practice this means CC is introducing SM style diagrams, which should be a welcome development. It also means the type of teaching in this video shouldn't be taking place in a classroom that claims to be aligned with CC.

 

When my kids were in school, they used enVision math and I have never since anything remotely as idiotic as what arose posted. Not all programs are created equal. My worry is that publishers will slap CC tags on their books without changing much the existing approach. My fear lies not with CC, but with CC implementation.

 

 

Also, I don't see how teaching to get the right answer and teaching to understand the math you need to solve the problem is somehow two different things. I had a soviet education. We were taught the math so we could get the right answer, which demonstrated to our teachers that we understood math concepts. Or maybe you mean teaching shortcuts to get the right answer without understanding? Then yes, I would agree.

[kateingr]

I had a soviet education. We were taught the math so we could get the right answer, which demonstrated to our teachers that we understood math concepts. Or maybe you mean teaching shortcuts to get the right answer without understanding? Then yes, I would agree.

I think this gets at the fundamental issue here: what does a right answer really show, particularly in math? There are so many possibilities:

 

1. The student understands the concept thoroughly and got the right answer.

2. The student learned a quickie shortcut and was able to use that to get the right answer.

3. The teacher led the student along step by step to the right answer, but the student is unable to retrace the steps and doesn't really understand.

4. The student read the teachers body language and guessed correctly.

And so on...

 

At a certain level, I wonder if it helps to be fundamentally skeptical of right answers, that the concept shouldn't be considered mastered until the student can prove otherwise. It's just so tempting sometimes to see a kid getting a couple right and then to assume that he or she really understands the concept.

[Arcadia]

I think this gets at the fundamental issue here: what does a right answer really show, particularly in math? There are so many possibilities:

 

1. The student understands the concept thoroughly and got the right answer.

2. The student learned a quickie shortcut and was able to use that to get the right answer.

3. The teacher led the student along step by step to the right answer, but the student is unable to retrace the steps and doesn't really understand.

4. The student read the teachers body language and guessed correctly.

And so on...

 

At a certain level, I wonder if it helps to be fundamentally skeptical of right answers, that the concept shouldn't be considered mastered until the student can prove otherwise. It's just so tempting sometimes to see a kid getting a couple right and then to assume that he or she really understands the concept.

 

That is where show all working and be able to verbally explain why comes in.

[8filltheheart]

I think this gets at the fundamental issue here: what does a right answer really show, particularly in math? There are so many possibilities:

 

1. The student understands the concept thoroughly and got the right answer.

2. The student learned a quickie shortcut and was able to use that to get the right answer.

3. The teacher led the student along step by step to the right answer, but the student is unable to retrace the steps and doesn't really understand.

4. The student read the teachers body language and guessed correctly.

And so on...

 

At a certain level, I wonder if it helps to be fundamentally skeptical of right answers, that the concept shouldn't be considered mastered until the student can prove otherwise. It's just so tempting sometimes to see a kid getting a couple right and then to assume that he or she really understands the concept.

This is why students shouldn't be given only plug and chug replication problems. Problems should require applying the concept to unique problem sets and word problems. Fwiw....I have seen this in curriculum outside of math, too. I seen the same issue in chemistry, etc.

[Roadrunner]

That is where show all working and be able to verbally explain why comes in.

Exactly.

[8filltheheart]

First, I didn't say there were scripts, well at least not in CC, but unfortunately (or fortunately) some programs come with scripts. What I said is CC is specifically indicating that procedural fluency alone isn't enough and math needs to be taught conceptually as well. In practice this means CC is introducing SM style diagrams, which should be a welcome development. It also means the type of teaching in this video shouldn't be taking place in a classroom that claims to be aligned with CC.

 

 

Also, I don't see how teaching to get the right answer and teaching to understand the math you need to solve the problem is somehow two different things. I had a soviet education. We were taught the math so we could get the right answer, which demonstrated to our teachers that we understood math concepts. Or maybe you mean teaching shortcuts to get the right answer without understanding? Then yes, I would agree.

I know nothing about CC and don't what to sidetrack Rose's thread into a discussion about it. I don't think we really disagree, but here are a few of my thoughts. But, no, I am not encouraged by the idea that SM diagrams are going to be taught. I am probably not going to convey my thoughts very well.....I haven't slept in 3 days and am drugged up on theraflu...

 

But, I have seen some horrible SM explanations. Even with an answer key with the explanation unless the teacher understands what she is explaining, it can lead to nothing but confusion (on both teacher and students' part). If teachers really understand what they are doing and really teach, it could be a good thing. But call me very, very skeptical that teachers that can't manage to teach kids 1/2+1/3+1/4 are going to leap into effective bar diagram teaching. :p

 

Beyond that, my pt was more along the lines that teachers should be encouraged to teach concepts multiple ways, not just one. Bar diagrams are just one tool and may not be the right one for all students. There are other methods to teach the same concepts and the freedom to find the right methods for her students is what allows real learning to take place.

 

I agree completely with your last pt.

[idnib]

Beyond that, my pt was more along the lines that teachers should be encouraged to teach concepts multiple ways, not just one. Bar diagrams are just one tool and may not be the right one for all students. There are other methods to teach the same concepts and the freedom to find the right methods for her students is what allows real learning to take place.

 

I agree completely with your last pt.

 

Completely agree.

 

I had the same point earlier when I said I wasn't sure how much CC complaining was the result of actual CC and how much of it was based on having to learn a new method. For example, I love math and while I have preferred ways of thinking about it, I feel I could teach in any method if I had to. I don't like some methods, but they all make mathematical sense to me, KWIM?

 

I honestly don't know the best approach now that it seems we have a large proportion of teachers who cannot make sense of math. It's one thing to have a teacher who understands mathematical thinking, has been forced to teach it one way, and is now being asked to use multiple ways. It's another thing if the teacher only knows one way herself and doesn't really understand that. How do you recover from that?

[Roadrunner]

Beyond that, my pt was more along the lines that teachers should be encouraged to teach concepts multiple ways, not just one. Bar diagrams are just one tool and may not be the right one for all students. There are other methods to teach the same concepts and the freedom to find the right methods for her students is what allows real learning to take place.

 

I agree completely with your last pt.

Actually we don't disagree at all. There is nothing better than a competent teacher who has the knowledge and freedom to teach the concepts and use various methods to address individual needs. If a kid doesn't understand the bar method, the teacher should be able to use a different strategy.

 

As somebody who hasn't attended PS in the U.S., I am the least qualified to judge what is going on in actual classrooms. My experience is very limited to early elementary my kids attended and I have mixed feelings and experiences. There is a reason we are no longer there and math instruction is among those reasons. I have high hopes that all the money currently being spent on training in our former school will bear its fruits. Our local high school is supposedly ranked in the top 2% in the country, and I am really, really, really hoping its one of those schools Bill talks about, but that remains to be seen :).

 

To go back to the video, I want to believe ignorance isn't as widespread as the person suggests.

[Pen]

I finally got to where I could watch the video.   I also was surprised that apparently a large percentage of American students are not taught fractions in a way that allow them to add:    1/4 + 1/3 + 1/2 =  ?  Is that really so?  I had never seen the butterfly method before, and the only program we had that might perhaps have had it to my knowledge was MUS, but we left MUS before fractions, so I don't know one way or the other.  But he acted like everyone would of course be familiar with the butterfly method, and I was wondering if most people actually are.

 

Oddly, in regard to mistakes and my misunderstanding about the typo in the RR material, I found later that I thought about it more than I would have thought about a correct answer that would have just passed by, and in thinking I wonder why they put 2025, came to realize that it would be the right answer for 45 x 45, so maybe in fact there is a learning benefit to wrong answers.  Did anyone else have that experience at all? 

 

OTOH, while it might help with remembering the "trick" I am not sure that having that kind of thing in elementary math is going to help with algebra so much as that maybe the kids who use BA are likely to be kids who are going to do well in algebra...  

 

I think having variables and solving for them can be introduced early and does help, however.

[idnib]

I finally got to where I could watch the video.   I also was surprised that apparently a large percentage of American students are not taught fractions in a way that allow them to add:    1/4 + 1/3 + 1/2 =  ?  Is that really so?  I had never seen the butterfly method before, and the only program we had that might perhaps have had it to my knowledge was MUS, but we left MUS before fractions, so I don't know one way or the other.  But he acted like everyone would of course be familiar with the butterfly method, and I was wondering if most people actually are.

 

I've never seen the butterfly method before. It seemed confusing, not to remember, but to understand why it works. But I guess if the goal isn't to teach why it works, it's okay.

 

I am not having any trouble believing many schools teach the butterfly method. I am having trouble believing schools are not teaching kids how to add 3 fractions. I'm wondering if the kids are taught the "easy" butterfly method for two fractions, resulting in no understanding of the most basic fraction problem. At a later time the are taught by a more legitimate(?) method to add more than two fractions, but because they didn't really understand the two-fraction model, the three-fraction non-butterfly method goes over their heads.

 

In other words, their inability to add three fractions is not because it was completely cancelled as a topic, but because they didn't really understand the simpler problem, they were out of their depth for higher numbers of fractions.

 

Or are schools really not even covering how to add more than two fractions together?

[Chrysalis Academy]

Maybe the butterfly method thing was more of a metaphor for teaching "tricks" without teaching an understanding of the underlying concept.  If you only learn the trick, then you can't handle any problem that isn't set up exactly the way you learned to apply the trick - so he gave the example of the FOIL method, too, and then there is PEMDAS, which kids apply incorrectly if they don't understand that multiplication and division are done in the same step, left to right, and addition and subtraction as well.  If they only have the acronym in their heads, but don't understand that M&D, A&S, are really just reciprocal operations doing the same thing, they think they have to go "in order" to solve the problem.

 

Whether the butterfly method is common (I'd never seen it before) or whether or not kids are being taught to add three fractions (I bet they are), I bet it's really, really common that kids are taught shorcuts/methods/tricks like these to get through the computations without ever learning the underlying concepts.  If this is the case, then they will hit a wall.  Different people will hit the wall at different times, depending on their processing speed, their memory capacity, or other cognitive factors, but everyone will hit it at some point.   I didn't hit it till Calculus in college. I had no clue what I was doing or why, but still managed to pull a B in class.  But it was the last math class (other than statistics) that I ever took, and I specifically chose a social science instead of a "hard" science to avoid having to do any more math,

 

 

[idnib]

Whether the butterfly method is common (I'd never seen it before) or whether or not kids are being taught to add three fractions (I bet they are), I bet it's really, really common that kids are taught shorcuts/methods/tricks like these to get through the computations without ever learning the underlying concepts.  If this is the case, then they will hit a wall. 

 

 

Yes, that's what I'm saying but you said it better than I did. I highly doubt the school are skipping the topic of adding 3 or more fractions together. I think it's more likely that if the kids use a shortcut for the 2-fraction problem they are less likely to understand the 3-fraction problem.

 

At least I hope that's the case. It's a much bigger problem if the schools are not even teaching it. 

[Pen]

It could be a metaphor for teaching a trick without teaching the underlying concept.  

 

And it also could be a logical "straw man" argument.  I think there is something wrong with education in math in US (and other subjects too), but I am not at all sure if what he said is the problem is  actually it.  For example, the problem could be that the teachers themselves do not understand the underlying concepts and so cannot teach them.  It could be that they are allowed only one approved method, and cannot reach all students with that method. Or it could be that by the time it would be time for adding 3 or more fractions it is time for that school year to end, and it does not get taught.    It could be any number of things.

 

Though I never saw it before, the "butterfly method" appears to me to be legitimate. I could give the conceptual reason for why it works, with, I think, no greater difficulty than to explain finding the least common denominator and multiplying by a fraction equal to 1, etc..   And I can use it to add up the 3 fractions (1/2, 1/3, 1/4) just as quickly (and get 26/24 which reduces to 1 and 2/24 = 1 and 1/12) as I can convert them all to twelfths as the LCD, add and then reduce.  In fact, with fractions where the denominators are that small and numerators are 1, I can do it in my head more easily using the butterfly method, while I find my working memory has a harder time remembering 6/12, 4/12, 3/12 at the same time as I am adding them.   So while I am supposed to be saying to myself something like, "oh horrors, what an awful method," I am instead thinking "Cool!"  I think I'll add that to my tool-chest of math methods.

 

If taught along with the concept of why it works, the butterfly method appears not to be a shortcut at all, and might be a way of keeping out the confusion that many children have with least common denominator versus greatest common multiple.

[regentrude]

 

 

I think there is something wrong with education in math in US (and other subjects too), but I am not at all sure if what he said is the problem is  actually it.  For example, the problem could be that the teachers themselves do not understand the underlying concepts and so cannot teach them.

 

This is without doubt one of the biggest problems, especially in math education.

 

 

If taught along with the concept of why it works, the butterfly method appears not to be a shortcut at all, and might be a way of keeping out the confusion that many children have with least common denominator versus greatest common multiple.

 

But the kids need to understand what the greatest common denominator or the least common multiple are - we can't just teach them a trick that eliminates the need to know, because they need it later. Butterfly trick works nicely as long as there are not too many terms, but once your sum consists of more terms with variables, not numbers, it becomes impractical. And once you hit algebra, knowing what the greatest common denominator is will matter very much, so we should not use the excuse that it is difficult to avoid teaching it.

 

I see college students with science majors who have not mastered fractions. They can't do algebra, they can't do intro science. Because they rely on tricks like these.
 

[fourisenough]

My two older girls have been in school two years (this and last), so collectively we've experienced just 4 public school math teachers; however, our experience is the exact opposite of everything I'm hearing on this board. These four women have been brilliant educators with a deep understanding of and an obvious passion for mathematics. Have we just gotten lucky? Or are many of us overestimating how bad math education in the US really is?

[8filltheheart]

My two older girls have been in school two years (this and last), so collectively we've experienced just 4 public school math teachers; however, our experience is the exact opposite of everything I'm hearing on this board. These four women have been brilliant educators with a deep understanding of and an obvious passion for mathematics. Have we just gotten lucky? Or are many of us overestimating how bad math education in the US really is?

 

The comments are being based on the video which stated that only around 20% of American students could answer 1/2+1/3+1/4 whereas in other countries it was around 80-90%.  

[Chrysalis Academy]

My two older girls have been in school two years (this and last), so collectively we've experienced just 4 public school math teachers; however, our experience is the exact opposite of everything I'm hearing on this board. These four women have been brilliant educators with a deep understanding of and an obvious passion for mathematics. Have we just gotten lucky? Or are many of us overestimating how bad math education in the US really is?

 

Right, we're reacting to the statistics presented on the video.  However, it mirrors my own children's ps experience:  in a combined 6 years of ps, we had one good math teacher.  So I think you are very lucky.

[Pen]

 

 

 

 

This is without doubt one of the biggest problems, especially in math education.  Agreed!

 

 

 

But the kids need to understand what the greatest common denominator or the least common multiple are - we can't just teach them a trick that eliminates the need to know, because they need it later.

 

Agreed!  But it could be taught as a method along with the understanding of Why and How it works.

 

 

Butterfly trick works nicely as long as there are not too many terms, but once your sum consists of more terms with variables, not numbers, it becomes impractical. And once you hit algebra, knowing what the greatest common denominator is will matter very much, so we should not use the excuse that it is difficult to avoid teaching it.

 

I see college students with science majors who have not mastered fractions. They can't do algebra, they can't do intro science. Because they rely on tricks like these.

 

 

 

bolded:  It is easy to get those words confused even for the extremely well educated in math and science fields as you are :)   -- seeing what I thought must be a correction and that I had been making a mistake made me worried I had mistaught my ds so I went and looked it up!   What I taught and meant to teach was LCD method.   bolded and underlined:   I suspect the problem is that it is taught or learned as a trick without the understanding of the underlying math being communicated--possibly because the teacher him/herself does not have that.  It could be taught as a method along with understanding Why it works, and how it can be generalized to harder problems, ones with more than 2 fractions and so on.   

 

I totally agree that they should have mastered fractions beyond "tricks" in algebra stage.  I have not actually thought up a problem where the butterfly will not work, but we have just begun algebra at this point and problems are quite easy.  I can certainly think of problems where it becomes extremely cumbersome. Probably you can think of ones where it just will not work at all.

 

But what they need for algebra is not necessarily what they need in 3rd grade when they start fractions (or at least that is so in the public school where I am--personally I think that itself is a problem and that doing fractions somewhat later  for most students would be better, but that is another issue). Though I realize that the man in the video was saying that it should be done from the start as it will be done for algebra.   Is there proof that that is actually so, however?

 

Students could, conceivably, learn fractions in a different series of steps than what will ultimately be used in higher math levels.  It could be similar to how multiplying fractions is often taught IME first by having the whole thing, total numerators multiplied, and total denominators, and only later is one shown that one can sometimes simplify or "cancel" numbers before multiplying.  

 

Adding of fractions could first be done by finding the total combined common denominator and the corresponding numerators (which is what the butterfly method is doing), explaining that what we are really doing is multiplying the fraction by a total of 1, thus keeping it the same value, but with a different form that helps us work with it.   Later they could be shown that finding the LCD is a way to simplify adding/subtracting fractions when the fractions are more complicated.  It is still a way of multiplying the fraction by 1.

 

 

 

For example, if given an easy algebra with fractions problem like 4/x + 6/2x = 1 which can be solved by either the butterfly method or by finding an LCD method, it would become clear that the LCD method is then probably easier.  That is, it is possibly easier to convert 4/x to 8/2x and then add and get 14/2x = 7/x, rather than to get to 14x/2x^2 = 7/x.    It might actually be easier for some students to go from finding an LCD for x and 2x to seeing that the same thing can also be done for fractions like 1/12 and 1/8.

 

 

 

It is in any case, another option, and without a scientific study done with matched sets of students and matched sets of teachers using the two ways, I am not certain that the one is actually better or worse than the other.  Or that all students would be better off with the same method.

 

I am not even sure that American teachers are not trying to teach the mathematics as a goal.  I am pretty sure that when I was in school they were trying to do that, but that the problem lay more in their own lack of math knowledge.  They had studied "education" whatever that was.

 

If your students who want to go into science fields are weak in math like this, I can only imagine that students in this country going into "education" are likely far far weaker.  And so very likely there is a downward spiral generation to generation.

 

 

[regentrude]

  It is easy to get those words confused even for the extremely well educated in math and science fields as you are :)   -- seeing what I thought must be a correction and that I had been making a mistake made me worried I had mistaught my ds so I went and looked it up!   What I taught and meant to teach was LCD method. 

 

I am sorry, I was thinking of greatest common divisor. I never taught nor learned elementary math in the English language.

I am, however, sure that there can be no such thing as a "greatest common multiple" because you can always find a larger one; one would definitely be looking for the least common multiple

 

 

Students could, conceivably, learn fractions in a different series of steps than what will ultimately be used in higher math levels.  It could be similar to how multiplying fractions is often taught IME first by having the whole thing, total numerators multiplied, and total denominators, and only later is one shown that one can sometimes simplify or "cancel" numbers before multiplying.  

 

But why on earth would one want to teach students a procedure that does not rely on conceptual understanding and that can not be generalized to more complicated cases? What possible benefit can there be?

Your example of multiplying vs canceling does not illustrate two fundamentally different methods; it is the same thing - canceling before only simplifies the arithmetic as opposed to canceling afterwards - but there is absolutely nothing different about the underlying mathematics.

 

 

Adding of fractions could first be done by finding the total combined common denominator and the corresponding numerators (which is what the butterfly method is doing), explaining that what we are really doing is multiplying the fraction by a total of 1, thus keeping it the same value, but with a different form that helps us work with it.   Later they could be shown that finding the LCD is a way to simplify adding/subtracting fractions when the fractions are more complicated.  It is still a way of multiplying the fraction by 1.

 

I have no objections to first finding the common denominator by multiplying the two denominators - THAT is not the issue. The issue is that most students will not remember (if they had ever been explained) that this is what they are doing. They simply remember the "butterfly", which is worthless without the underlying conceptual understanding.

I would guess most students taught with this method will have trouble generalizing the "butterfly" to three or four terms that are to be added - because it won't look like a butterfly anymore.

 

I am not even sure that American teachers are not trying to teach the mathematics as a goal.  I am pretty sure that when I was in school they were trying to do that, but that the problem lay more in their own lack of math knowledge.  They had studied "education" whatever that was.

If your students who want to go into science fields are weak in math like this, I can only imagine that students in this country going into "education" are likely far far weaker.  And so very likely there is a downward spiral generation to generation.

 

Yes. Students going into education have on average among the loswest standardized test scores. Just google the study for Chicago's public school teachers who have average ACT scores of 19, below the state average.

 

[Pen]

I am sorry, I was thinking of greatest common divisor. I never taught nor learned elementary math in the English language.

I am, however, sure that there can be no such thing as a "greatest common multiple" because you can always find a larger one; one would definitely be looking for the least common multiple.  

 

You are right!  Least common multiple and denominator, both.   Greatest Common Divisor and Factor (GCF)!  Luckily I only did that wrong in the post, not in my fractions teaching!  We used acronyms LCD and GCF.   Ugh.

 

 

But why on earth would one want to teach students a procedure that does not rely on conceptual understanding and that can not be generalized to more complicated cases? What possible benefit can there be?

Your example of multiplying vs canceling does not illustrate two fundamentally different methods; it is the same thing - canceling before only simplifies the arithmetic as opposed to canceling afterwards - but there is absolutely nothing different about the underlying mathematics.

 

 

 

I have no objections to first finding the common denominator by multiplying the two denominators - THAT is not the issue. The issue is that most students will not remember (if they had ever been explained) that this is what they are doing. They simply remember the "butterfly", which is worthless without the underlying conceptual understanding.

 

I agree that it would need the underlying conceptual understanding along with it.  I think we could have done it with also putting in a big 1 showing what we were doing--as we did when we did it with the Least common denominator method and showed whatever fraction would be multiplied  so as to get the denominators the same in the addends as an x/x within a big 1.

 

If I take 1/2 and in my mind add by  "butterfly" method it to 1/3, I get 5/6, then if I "butterfly"/add that to 1/4, I get 26/24.  This reduces to 1 1/12.  It is the same mathematics as finding the LCD of each fraction in 12ths and then adding and converting to a mixed number. The reduction/ simplifying step is done early in the LCD  case, at the end in the "butterfly".  How do you see the mathematics that underlies these to be different?

 

I think some children would do  better with the one method, some better with the other, some fine with either, some might need something else than either of those methods (perhaps manipulatives to work with it hands on, or pictures of pies).

 

I would guess most students taught with this method will have trouble generalizing the "butterfly" to three or four terms that are to be added - because it won't look like a butterfly anymore.

 

I do not know.  I am very "visual" and I can see it in my head pretty easily in the example of the 3 numbers 1/2, 1/3, 1/4.   It  looks like 2 butterflies and turns it into easy mental math.  (4 fraction addends would look like 3 butterflies, 5 would look like 4 butterflies and so on.) ETA:  But you see far more students coming out at the college end, so I am inclined to think you are probably right that it is not a good method for most, and so decided I probably will not show it to my son lest it confuse him.

 

Lots of things can be taught without giving an understanding of the Why and How, dividing fractions where the child does not know why to use the reciprocal is also often the case, and easily forgotten.   Or FOIL and PEMDAS were other examples given.  Borrowing and Carrying (renaming) in addition and subtraction of whole numbers too can be done with or without understanding.  I think understanding the Why should always be part of the teaching whatever the method.

 

 

Yes. Students going into education have on average among the loswest standardized test scores. Just google the study for Chicago's public school teachers who have average ACT scores of 19, below the state average.

 

Personally I think that is the key problem.  Not the standardized test scores necessarily, but that teaching is not getting teachers who themselves are at a high enough level.  The guy in the video dismissed that saying that the teachers in Japan and here are the same range of quality.  I do not know if that is so.  Perhaps it is.   But I think the Finland change to making teaching a profession for top students, not bottom ones was critical.

 

  Need to write something here for it to post.

[Pen]

This video - very much in the spirit of Richard R's discussion of math problem solving - seems relevant to several current ongoing conversations, so I thought I'd give it its own post. 

 

What's wrong with math teaching in the US?

 

Is it the educational goals/philosophy?  

 

Is it teacher quality/prep?

 

Is it the curriculum?

 

All, none, something else?

 

http://vimeo.com/30924981

 

 

How about an example known to have produced results?

 

What is different about the way Jaime Escalante (might have the name wrong--the teacher Stand and Deliver was based on) does things and what happened in the same school before that?

 

Different teacher, who actually knew math.  Number One importance IMO, no matter what the guy in the video says.

 

Higher goals.   Different philosophy.

 

Very hard work by both teacher and students.

 

Other?