Stanford Math Course coming, anyone else interested?

[mathwonk]

I have read some of the mathematician's lament, the stanford course description, and this thread,

 

but have not registered for the actual course.

 

 

(My favorite part of the Lament, was the very creative discussion on pages 3-4

 

on the area of a triangle compared to that of a box.)

 

 

I don't want to criticize a course I have not seen, but some reactions here suggest

 

that for some of us, it may not be enhancing our confidence in what we can do.

 

 

When I read the course description I was excited that it might help me learn to

 

accomplish all those goals.

[Chrysalis Academy]

Well, we've only seen half of the course so far - the first 4 weeks, which is identifying the problem, and is kind of preaching to the choir to most of us here.

 

I'm hopeful that the next 4 weeks will have some practical steps, more of a how-to once she's got her audience convinced.  We'll see, right?

 

I also really dug the area of a triangle thing.  This is one of those formulas my dd has a hard time remembering.  I want to go through this discovery-based exploration with her, and see if it makes a difference in helping her remember how to do this kind of problem - I suspect it works, she's one of those kids who really needs to grok *why* something works, or she has trouble remembering it.

 

ETA: also, on a positive note (I'm much less cranky today!  :lol: ) It really did reinforce my feeling of *not* doing speed drills with my girls.  They both hate them, and become really anxious.  Things like Xtra math, or sumdog, or timed flashcard programs, things a lot of folks use to drill facts, just don't work for us - my kids freeze up and get stressed.  Their least favorite thing about ps was Mad Minutes.  Jo's presentation really reinforced and supported my decision to avoid all that stuff with the kids, and to let them take their time.  My dd7 in particular has a really great conceptual understanding of math so far, but she just freezes if you try and rush her.

[Bang!Zoom!]

Mathwonk, I believe this course is being discussed in other sections of the board as well - lots of folks are viewing it.

 

Was it the first video she mentions that 20K have signed up for this course?

 

Here is another thread which it is being discussed on a sub forum for K-8 here: (mathwonk, you may find some of this discussion interesting also)

 

http://forums.welltrainedmind.com/topic/480900-stanford-math-course-what-are-you-learninggoing-to-do-differently-for-math/

[Pen]

Yeah, I think it's important to still do the summative assesments if the goal is teaching to mastery, for sure.  Maybe less often though? And instead of giving the paper back with xs through the errors and a score, give it back with errors highlighted, as an opportunity to "grow your math brain" by finding and correcting mistakes?

 

I'm also butting my head against this:  ok, this is all great, but there is still an amount of content our kids are expected to master, right?  I mean, when you say they have done Algebra 1 and gotten an A, that means something - it means they have covered a specific set of material and mastered it.  Right?  And how do you know if this is true if you aren't assessing in this way, at some level?  And unless you are already brilliant in math, how do you take the results of a formative assessment and say, "Ok, they clearly need more work on X concept so I'll do Y kind of open-ended problem?"

 

Yeah, I still see myself being dependent on someone else's great curriculum, here.  Thank goodness for AoPS and Zaccaro and stuff like that, huh?

 

 

I went to a number of schools growing up, 2 of which were ungraded and also tended not to have formal testing.  I learned a lot more at the ungraded schools.  

 

One way assessments can be done is to have the child self-assess...I think this is one of the best ways, but I am working hard to find a way to do it effectively and to figure out what to ask, and areas to ask about.  But back when I was doing tutoring one way I used that was just to say, okay, book shut, explain ______ to me, or how would you explain ______ to someone who doesn't know about it yet?  That is harder with ds because there is more of a tension in the relationship than with someone I was tutoring for pay.  

 

Boaler's book gave an idea that I may try to modify for hs, of having a green, yellow, red,  stop light idea for classes--where a child indicates green if they get it and want to move on, red if having major trouble, and yellow is in the middle.  

 

The other is one does have an idea based on ongoing work and discussion and questions, and one can respond with comments rather than grades and numbers.

 

Recently I have been watching ds do some Sumdog, and have been looking at what he has trouble with, and using that for a new lesson on what he needs to know to be able to get that type of question.  It also is a time for knowing what is in the "I get it, I just didn't do it fast enough" category versus the "No, I don't understand that yet" category.    And it has also been interesting to see what the computer thinks he is ready for.

[Pen]

PS to above--also doing the Sumdog is a good chance to practice it being okay to be wrong and make mistakes!  And for me to be okay with that.  

 

I wonder if freezing up on things like that could already be a sign of math stress and phobia building up?  I've noticed now as it is in summer that for the USA flagged avatars many more appear to be boys.  (From Australia where it is school year, they seem more even.)   But, like this is something boys like to do for fun (and thus maybe get better) while girls feel it as a stress?

[Pen]

...

One thing that bothers me about the class it seems to be pushing for mixed ability classrooms. She talks about research that indicates that mixed ability classes are better for kids but I am skeptical. I want to actually see the research. Perhaps there are other factors that are involved. When I was young the lower classes had the worst teachers. What exactly is causing the difference in student success? It's hard to tell. She says what she thinks but doesn't give enough information for the students to analyse the problem themselves. I find this somewhat ironic and frustrating.

 

...

 

If you look in the references section you'll see some for this topic.  In her book she gives examples from her own research, and that other countries (such as Finland) where ability grouping is not done are doing much better.  One example given is that the highest level kids actually improved the most in mixed ability grouping (I think that was an English school), while highest level tracked kids were particularly stressed and doing less well---maybe from some of the Mindset ideas where telling kids they are good at something actually impairs performance and causes anxiety rather than freedom to make mistakes and so on.

 

I do not think the idea is to group kids and then give them a poor lower tracked class teacher, but to group them plus with an excellent teacher capable of multi-level teaching, stimulating group work and so on.

[mathwonk]

It is interesting that the high level kids benefited more from diverse grouping. Could that relate to the chance they have there to explain concepts to other kids? That always helps me.

 

 

The rest of this post got trashed by the browser, about how in using Euclid for geometry,

 

one encounters that equi-decomposable idea from pages 3-4 of the Lament, and no area formulas.

 

 

Then the suggestion that one way to get a mathematician's point of view, if that is desired,

 

is to consult books by the best possible mathematicians, like Euclid, Euler, Hilbert, Lagrange....

 

There was also the usual disclaimer that this advice may not work for everyone.

[onaclairadeluna]

 One example given is that the highest level kids actually improved the most in mixed ability grouping (I think that was an English school), while highest level tracked kids were particularly stressed and doing less well---maybe from some of the Mindset ideas where telling kids they are good at something actually impairs performance and causes anxiety rather than freedom to make mistakes and so on.

 

Thanks for the references, I'll have to take a look at the book. This is completely counter intuitive to my experience and observation and also from everything I have read about PG kids which says that they do much better when radically accelerated.

 

Sometimes I am so glad I homeschool because it takes so much of the pressure off. 

 

 

Luckily I think I have (almost) made it (son entering sophomore year of high school, not harmed yet, knocking loudly on wood).

 

Here's a third option. Put gt kids in a separate class and tell them that they are there because they need to work harder (not that they are smarter). That's always what I tell my own kids. Not that they are smart, but that they need harder problems to learn. 

[Eagle]

It doesn't sound like gifted kids are specifically tracked in the stats. I can see the benefit of mixed classes for very bright students, but not for kids working 2 or more years ahead on material. Unless they are suggesting no child should ever be accelerated in math?

[EKS]

The course is starting to annoy me too.

 

First of all, I think kids should be able to deal with the reality that there are times that mistakes are necessary and important and valuable and there are times (like when you're taking a high stakes test) that they are to be avoided.

 

I do think it's important for kids to learn how to show mastery.

 

Also, the thing about speed--I have two kids who have official diagnoses related to processing speed (or lack thereof), so I get that speed is not the most important thing--but it is a proxy for automaticity, which is important, as freeing up working memory to think about the actual math of a problem (rather than the rote calculations involved) is critical.

 

And the thing about *all* students can learn *all* math.  Um, no.  I *do* think that the vast majority of students are probably capable of learning far more than teachers and administrators think they can, but when making statements like this I think it's important not to say "all" because it automatically makes the statement wrong.

 

I can't stand touchy feely stuff in education, and I'm getting quite tired of all the touchy feely messages in this course.

[Woodland Mist Academy]

The course is starting to annoy me too.

 

First of all, I think kids should be able to deal with the reality that there are times that mistakes are necessary and important and valuable and there are times (like when you're taking a high stakes test) that they are to be avoided.

 

I do think it's important for kids to learn how to show mastery.

 

Also, the thing about speed--I have two kids who have official diagnoses related to processing speed (or lack thereof), so I get that speed is not the most important thing--but it is a proxy for automaticity, which is important, as freeing up working memory to think about the actual math of a problem (rather than the rote calculations involved) is critical.

 

And the thing about *all* students can learn *all* math.  Um, no.  I *do* think that the vast majority of students is probably capable of learning far more than teachers and administrators think they can, but when making statements like this I think it's important not to say "all" because it automatically makes the statement wrong.

 

I can't stand touchy feely stuff in education, and I'm getting quite tired of all the touchy feely messages in this course.

 

Thank you so much for posting this and putting my thoughts into words!

 

 

[Pen]

It is interesting that the high level kids benefited more from diverse grouping. Could that relate to the chance they have there to explain concepts to other kids? That always helps me.

 

 

 

 

Yes.  In one situation that was a reason the children themselves gave.  Some interviewed said things along the lines of that they had at first resented the idea of being in a mixed group where they might be slowed down or having to explain things to others slower than themselves, but then learned that they got a much deeper level of understanding from that.

 

There was also that some were surprised that everyone had helpful things to offer.

 

Another major reason appeared to be that in the mixed groups open ended problem solving was used, which allowed the upper level kids to soar even more.   Example, a problem where a farmer has 36 equal length straight pieces of fence, and wants to fence in the greatest area--to find out the configuration that is best and area that it produces.  Some kids might be just at starting to discover that a square gives more area than a rectangle, and to try to figure out the area for those shapes.  The highest level ones might be learning how to use trigonometry to calculate the area of a 36 sided figure.

 

But apparently the mixed groups are better than single level groups even if open ended problem solving is used in single tracked level group, as I understand it.  So apparently the conversation, group attitudes, and different attitudes toward oneself and others play a role beyond just the type of problem used.  

 

Perhaps it is much bigger than that, with societal attitudes playing a role in places where tracking is not done at all or is illegal, or even a single school or teacher attitude where it is being tried.  I did not see this discussed, but I think it may be significant.  And that might not be something that could then be easily changed by arbitrarily imposing it as a "fix" for present ills.  If a student feels resentful to be with lower level kids, but then discovers it is actually helpful, that is one thing, but if a teacher were resentful of (or just incapable of) teaching in this multilevel way (or getting the group to work well together), that would be another thing, and likely to cause a failure, I think.

 

This might be interesting for people homeschooling a number of children, where I think often things like history or art are combined, but not usually math.  Maybe math could be also in a useful way.

 

 I have only one dc, but it makes me wonder if with 20K or so in the course, there might be some in my area who would be interested in forming a group to try some mixed level learning like that.

[Pen]

... This is completely counter intuitive to my experience and observation and also from everything I have read about PG kids which says that they do much better when radically accelerated.

 

...

 

From examples like this, (and apparently from what is going on in Finland and Japan where this is used ?) it appears that the high level kids are radically accelerated, and in fact are able to radically accelerate more than in a typical traditional class set at what seems to be their level:

 

(quote from my own pp)

 

"Another major reason appeared to be that in the mixed groups open ended problem solving was used, which allowed the upper level kids to soar even more.   Example, a problem where a farmer has 36 equal length straight pieces of fence, and wants to fence in the greatest area--to find out the configuration that is best and area that it produces.  Some kids might be just at starting to discover that a square gives more area than a rectangle, and to try to figure out the area for those shapes.  The highest level ones might be learning how to use trigonometry to calculate the area of a 36 sided figure."

 

 

Of course, it would take an extremely capable teacher to be able to walk around the room and see that one child needs help with how to calculate the area of a square, while another is ready to learn some trig.--and be ready and able to teach the child ready for trig. how to do it in a way that makes sense.       And it also would take a lot of ability to be able to generate these sorts of problems that would have the range to cover the kids in the class and perhaps have potential beyond the highest level ones current abilities so that there were always soaring beyond expectations possible for all the kids in the class from highest to lowest.

 

I can imagine that there might be some situation where some kids could be just beginning to get the idea of solving for x, while others might be figuring out how to solve a quartic equation on their own.  But how to achieve that, I have no idea!

 

Or are there books of such problems?  Even if there were it would still take a good bit of teaching skill to pick a right one for a mixed group of kids to always offer something that could highly challenge everyone.

[Pen]

Another thing that struck me was in a male / female difference area:  that women/girls more often tend to want to know why the math works the way it does, while boys/men care about that less often.   I had tended to interpret my ds's lack of interest in the why as a lack of interest in math.   (As in the thread on here about kids who are good at math, but just not especially interested in it.)  Now I am wondering if that is my own female bias to think of it like that, and that in him as a male he is just in the 2/3 group who does not care about the why, and that it may have nothing to do with "interest" in math.   

[Pen]

I guess the thing I'm grappling with the most right at the moment is the idea that regular testing - summative assessment rather than formative assessment - is counter to the creation of a mistake-friendly culture and a growth mindset.  I get what she's saying, and why, and I don't disagree, but at the same time if my goal is to teach to mastery, how do I know when I've gotten there without doing summative assessments, on which I expect very few mistakes?  Do you guys get what i mean? I love the idea of valuing and celebrating mistakes and persistence on problem-solving tasks, but when I contemplate throwing out all tests until a final end-of-year assessment, I get kinda freaked out.  

 

The other thing that reading The Mathemetician's lament made me think was that I need a mathemetician to come teach my kids math - I have no business teaching them math, because I really don't see the deep, artistic underlying beauty of it.  Just like I have no business teaching them to play an instrument or create art.  But then I'm hosed, right? Because how many people do see math like a mathemetician?  None of the teachers my kids have had at the local elementary schools, that's for sure.  Where am I supposed to find such a person? I think I'm the best chance they've got, so I continue to struggle to turn myself into such a person, but it is a challenge.

 

I have to say, I do feel very shaken, that somehow what I've been doing and what I'm planning for next year isn't quite right, but I don't know exactly what I should be doing instead.  I need to spend a few hours on the MAP website I think, and see if that clarifies things at all.

 

In my case, my grandmother was a mathematician.  While I think it helped that I got some early exposure to things like numbers in Base 2, and a few things like that.  I am under no illusions that she would be a good math teacher, since she had about zero patience.  

 

I think the Mathematician's Lament is thought provoking, but I do not know if his he is entirely right.  

 

I would say it is one of those take what you can use sorts of things.

[Pen]

I feel the same. In my case, it's quite clear that I have no business teaching my kids math. I mean, I was diagnosed with dyscalculia and my eldest, who is only just seven, is going to surpass me in mathematical ability very soon indeed. I can barely do mental addition and subtraction with carrying, and I never even touched higher level math. (I went to a "great" gymnasium, and think I might have enjoyed higher level math — however, since I didn't have a good basis in the four operations there was just no way. I spent high school math classes reading novels and my history textbook.)

 

Yet, that very same child would just be learning to count to 10 here, if she went to public school locally (we're in Europe, and that's what they do in this country). I am a crappy math teacher because I do not know math, but if I hired a tutor she'll get the same kind of "teaching" I also received. This makes me feel very conflicted indeed. 

 

I love the concepts being introduced so far. I do hope there will be more than four sessions? I will definitely be poking around the various websites and references provided. Part of my problem is that any math beyond what DD is working on right now is way outside of my comfort zone. I have PTSD from a trauma that is widely recognized as being traumatic (childhood sexual abuse), and I am not kidding when I say math is also kind of traumatic for me. Just seeing that problem with the angles and mini golf gives me flashbacks. 

 

So, what's next? There is only one redeeming feature in my math phobia and lack of knowledge. DD and I will be exploring concepts (that I can, I am sure, master) together — taking a Miquon approach, if you like. We have no choice but to approach math as a common learning experience, because indeed I am learning right alongside her. I will never be laying down the One Right Way, and that's good I think. But.... 

 

Just sharing my possibly shocking and highly inadequate experiences/thoughts. 

 

 

8 sessions, I think.

 

Would playing with math puzzles together, or things like Sudoku help do you think?

[onaclairadeluna]

 

 

So, what's next? There is only one redeeming feature in my math phobia and lack of knowledge. DD and I will be exploring concepts (that I can, I am sure, master) together — taking a Miquon approach, if you like. We have no choice but to approach math as a common learning experience, because indeed I am learning right alongside her. I will never be laying down the One Right Way, and that's good I think. But.... 

 

Just sharing my possibly shocking and highly inadequate experiences/thoughts. 

 

From everything you have written, I have complete confidence in you. 

 

I assume you have seen this site. Not sure where you are in math right now, but I kind of love this for elementary math.

 

http://www.educationunboxed.com/

 

My son is  really good at math. He is also dyslexic and could not learn any standard algorithm. I had to do math completely out of order with him. He learned all the cool fractions and ratios and algebraic things first. He was also pretty good at multiplying in his head using little math tricks. (7*8 oh that's easy you just take 8*5 which is 40 and add 8*2 which is 16 and you get 56).

 

He could even figure out weird ways to solve bigger problems  

 

Disclaimer don't fret if you don't follow my sons circuitous math reasoning, I realize that you are discalculic however I am writing this to illiustrate there are ways to get around not being able to do column multiplication. So say I asked him to calculate 485x38 this might be how he would do it.

 

485 *38 ok  that's just 500*38-15*38

 

500*38 that's half of 1000*38 that's 38000 ok you need half of that what is that uh 19000.

 

Then you have to subtract 15*38 let's see that's a tricky one.

 

You probably need to take 20*38 what's that uh  760 and subtract  5*38 which is half of 380 ok that's 190   760-190 is 10 more than 760-200 so that would be 570

 

What am I subtracting that from...oh yeah 19000. So that's 19000-570

 

That is 30 more than 19000-600 so you get 18400+30 or 18430.

 

But if you gave him a piece of paper with 12+26  he would write 37 and cry if you pointed out his mistake. And it took me two years to figure out that this was a learning disability but you know. I did figure it out eventually.

 

I know this way of thinking might be confusing for some people but for him it made everything easy. I think it's kind of like how the spelling I am teaching my kids is really convoluted and confusing for me (non dyslexic mom) but it is the only way that they can learn. 

 

That's kind of extreme but this is what he would do. As a disclaimer this type of math thinking is really confusing to me. It is so much easier for me to just carry number and add things together but this is the way my son had to do elementary math and he got really good at math because I never forced him to do things the way other people did (I did eventually make him at least understand the standard algorithms and he admits that he still doesn't have any good tricks for long division)

 

Anyhow what I am trying to say is that just because you can't do math the way other people can does not necessarily mean you are bad at it. Mathematicians are notorious for being out of the box divergent thinkers. So teaching kids different ways of thinking about math will only make them stronger at it.

[onaclairadeluna]

For me, I think speed has it's place. I don't think it should be completely dropped, but it should be carefully placed into the curriculum once a child has mastered the problems. When they are still struggling slightly to remember the math facts, it's not a good time to work on speed.

 

I think one good way to handle this is to make speed work optional (but maybe encouraged for some). Having fast math facts can be really helpful for some kids and paralyzing for others. My daughter was able to do a little speed work with math facts. She doesn't love it and it is hard for her but she could do it in small doses and I think it was helpful for her. DS not so much. If you try to get him to do something fast he gets slower. It's weird. Only way I got him to get fast enough with math facts was to give him AOPS algebra which was so fun he wanted to work on math 2 hours a day and after a year of this he got fast enough that math facts didn't bother him much. 

 

He can hold more in his head than her but is not as good at unpacking it and getting it out. Does that make sense. So for her having a little bit more automaticity before she gets to more abstract work is more helpful/necessary and for him it would have been more of a hindrance. I am a "by any means necessary" sort of math teacher. I mean, if it works...?

[onaclairadeluna]

I just found these two links and want to share. I don't really have an opinion yet except that the whole thing reads like a soap opera. 

 

http://www.stanford.edu/~joboaler/

ftp://math.stanford.edu/pub/papers/milgram/combined-evaluations-version3.pdf

 

and there is more

 

http://math.stanford.edu/~milgram/Jo-Boaler-reveals-attacks-AccusationsResponse-trans.html

 

 

There seems to be a bit of a skirmish between the math department http://math.stanford.edu/~milgram/ and the education department. https://ed.stanford.edu/faculty/joboaler  

 

:lurk5:

 

 

 

Stay tuned for the next riveting episode "Math Wars" 

 

https://www.youtube.com/watch?v=UREJJfLzBfM

 

This is in Spanish (which I don't really speak) with the audio removed but still...

[Pen]

 

 

 

 

That's kind of extreme but this is what he would do. As a disclaimer this type of math thinking is really confusing to me. It is so much easier for me to just carry number and add things together but this is the way my son had to do elementary math and he got really good at math because I never forced him to do things the way other people did (I did eventually make him at least understand the standard algorithms and he admits that he still doesn't have any good tricks for long division)

 

 

This sort of thing fascinates me.

 

If you had 18430 divided by 38, you could say that 38 is close to 2 x 20 or to 4 x10.  Suppose you first divide by 10 and get 1843, and then divide by 4 and get 460.75--  that gets to a right ball park, which in a lot of situations is all one needs, but for precision, which in other situations one does need, it  is 24.25 off, I think.   Maybe he could figure out how to get that last part, and if so, please share it with me!

 

That sort of thing is where I think a group to work on problem solving could be a lot of fun.

 

If the number had been an easier one like 36 which could have been factored into more easy to manage 3's and 2's it would have been easier to get to a correct answer, but that 19 to me makes it hard.  I have a feeling there is probably a mental math solution though.

[Pen]

I just found these two links and want to share. I don't really have an opinion yet except that the whole thing reads like a soap opera. 

 

http://www.stanford.edu/~joboaler/

ftp://math.stanford.edu/pub/papers/milgram/combined-evaluations-version3.pdf

 

and there is more

 

http://math.stanford.edu/~milgram/Jo-Boaler-reveals-attacks-AccusationsResponse-trans.html

 

 

There seems to be a bit of a skirmish between the math department http://math.stanford.edu/~milgram/ and the education department. https://ed.stanford.edu/faculty/joboaler  

 

:lurk5:

 

 

 

Stay tuned for the next riveting episode "Math Wars" 

 

 

This is in Spanish (which I don't really speak) with the audio removed but still...

 

 

Very very interesting!  Thank you for posting those links!

[Pen]

I am now looking at what I can find on Finland, since there is less controversy over that it did improve.

[EKS]

I just found these two links and want to share. I don't really have an opinion yet except that the whole thing reads like a soap opera. 

 

http://www.stanford.edu/~joboaler/

ftp://math.stanford.edu/pub/papers/milgram/combined-evaluations-version3.pdf

 

and there is more

 

http://math.stanford.edu/~milgram/Jo-Boaler-reveals-attacks-AccusationsResponse-trans.html

 

 

There seems to be a bit of a skirmish between the math department http://math.stanford.edu/~milgram/ and the education department. https://ed.stanford.edu/faculty/joboaler  

 

Some more about this :  http://kitchentablemath.blogspot.com/2013/01/educational-malpractice-for-sake-of.html

[onaclairadeluna]

This sort of thing fascinates me.

 

If you had 18430 divided by 38, you could say that 38 is close to 2 x 20 or to 4 x10.  Suppose you first divide by 10 and get 1843, and then divide by 4 and get 460.75--  that gets to a right ball park, which in a lot of situations is all one needs, but for precision, which in other situations one does need, it  is 24.25 off, I think.   Maybe he could figure out how to get that last part, and if so, please share it with me!

 

That sort of thing is where I think a group to work on problem solving could be a lot of fun.

 

If the number had been an easier one like 36 which could have been factored into more easy to manage 3's and 2's it would have been easier to get to a correct answer, but that 19 to me makes it hard.  I have a feeling there is probably a mental math solution though.

I'll have to ask him when he gets back from camp. He is kind of beyond the long division thing but I am sure he would humor his math education minded mom. I have yet to really make long division fun. I can make it less painful but fun I haven't achieved. "We do a little long division and then we can go do fun things like fractions and modular arithmetic." That's kind of how we roll here. Spoon full of sugar and all.

 

I think he'd probably go this route. Factor 38 what is it 2*19 ick that's no good at all. Ok well whatever. 18430/2 is 9215 right? (I'm a little tired) Ok now we have 9215/19. Which isn't any better than what we had before. At this point I think he is probably tempted to use some infinite dimensional imaginary vector (I don't know I am just making this up, he is gone at mathcamp and I miss him a little) OK putting back on my serious hat ....I am sure if he had his druthers his answer would involve a fraction not a decimal so he'd probably figure out that 19*4=(20-1)4= 80-4= 76 and (7600+1615)/19 so now we have 400 and 1615/19 and just about now I can see why he still likes the long division algorithm better than this. Um probably 80 here what 80*19 is 80*(20-1) = 1600-80= 1520. I am not sure I even know what I am doing now but I have been working with him for so many years that I am pretty sure this is where he would go. Um 1520+95/19 so what the heck do we have now um 480 and 95/19ths so now we have to only have 4 more so it has to be 484 and oh man I have to figure out how many left over 4*19 is 4*(20-1) 80-4=76 so that woud be 76+19/19 wait a second that's 1 so what is it 5 ...485? But I think I just did the long division algorithm but with fractions. Something tells me he would have liked it better that way when he was little or maybe he would have tried to factor 9215 to see if it had a 19 in it um 5*1843 (that was a little annoying) um not 2,3,5, sheesh 7? I hate dividing by 7. I think at this point he'd go back to the long division like fraction thing.

 

OK putting aside my laziness I see pretty easily that 7 doesn't work. Maybe he wouldn't give up so easily. um 11? I am sure there is a neat trick he knows that I don't (at this point I use my calculator and cheat, I am not proud and after all I know how to do long division the other way just fine) I am thinking it is going to be something kind of annoying so I keep cheating ...19*97 totally annoying I am glad I did. But this is where he would go and after checking 13 and 17 he would still have to divide by 19 which is still almost as irritating as it was before. But I think this is where he would go if he had to but I don't worry about this much anymore since he can do it he just hates arithmetic. I have let him move on.

 

Oh wait I have to put this all together what do we have 

19*97*5 is that right so our answer is 97*5 or um 500-15=485 thank goodness. Oh man, I would not have wanted to have to go back and check my work. 

 

I could be wrong but this is kind of how I remember it. He usually checks to see if factoring works before he tries to divide. I am not sure at what point he'd throw in the towel and just do regular long division. I am positive he would not use a calculator like I just did to cheat. Even if he really hated the whole thing. He would feel compelled to do it himself.

[anabelneri]

Interesting!

 

I've started the course but I'm following the discussion here to see if it's worth investing my time.  So far it seems like she hasn't said much that is new and/or that impacts us much as homeschoolers.

 

Has she mentioned Liping Ma yet?  She was at the Stanford school of education too, though I can't seem to find the dates.

 

:)

[regentrude]

"The enemies of a mistake-friendly culture are regular testing and an emphasis on speed."

 So, what do you think?

The beauty of mathematics is that there is no room for debate whether something is correct. It does not depend on the personal preferences or political views of the evaluator - a solution either is correct or wrong. In that respect, a "mistake friendly culture" is the last thing I would want for my child's math education.

Yes, making mistakes happens and is a way to learn, and a student should be encouraged to make different attempts and play around with the problems - but IMO math education is not supposed to be "mistake friendly": it is supposed to teach creative problem solving that arrives at a *correct* solution.

 

It needs to be specified what the author considers as "regular" and what qualifies as "testing". Having a student do math problems is only useful if the results are evaluated on a continuing basis: EVERY problem is checked EVERY day. To me it makes no sense to have the student work on math and NOT verify that he is doing the problems correctly (this can be self-checking by the student). Now, what distinguishes "testing" from daily "checking"? The author needs to be precise.

Daily checking is absolutely essential. Timed tests under pressure can be done infrequently, but must be done, in sufficiently large intervals. Since an evaluation of math mastery only makes sense if *long term* mastery is evaluated, comprehensive examinations at semester/year end are much more valuable than weekly quizzes that test only the concept currently studied, but not whether last months' material can still be applied.

 

Speed: speed can be overrated, but speed is not obsolete. A student who lacks a quick recall of math facts will struggle in higher math when he is distracted from the actual problem by being hindered in numerical computations. Math is not *about* speed, but an automatic recall of facts frees the student's mind to ponder the actual problem and not be mired in arithmetic. Of course, quick rcall does not guarantee a ggood mathematical understanding and thus speed drills should not be emphasized over probelm solving. However, I am cringing again at the word "mistake friendly": math should be precise, not fuzzy.

 

When I hear statements like this, I am glad I am staying away from courses presented by math *educators*. Much of what is problematic in math education comes from the education departments. I'd rather listen to a real mathematician like Lockhardt, with whose "Lament" I agree ecompletely.

[Woodland Mist Academy]

 I'd rather listen to a real mathematician like Lockhardt, with whose "Lament" I agree ecompletely.

 

So you agree with his thoughts on art, poetry, and creativity?

 

I would be interested to hear how you approached art and poetry with your children using the methods he describes.

 

The rest of your post made sense to me, but your complete agreement with the "Lament"  left me with questions.

[regentrude]

So you agree with his thoughts on art, poetry, and creativity?

 

I would be interested to hear how you approached art and poetry with your children using the methods he describes.

 

The rest of your post made sense to me, but your complete agreement with the "Lament"  left me with questions.

It has been several years since I read it, and I recall agreeing completely with all his thoughts about MATH.

 

I was trying to find the article again to reread it so I could address your specific questions about art, poetry and creativity, but my link no longer works, since he has published it and the pdf is no longer available. So, I'll add a disclaimer: I remember agreeing with everything he stated with respect to math eductation. And I remember agreeing with his hilairous parallel description of art and music education *if that were done in the same unsuitable way as mathematics is taught*.

 

I have no recollection what he advocates about teaching music and poetry, but I can tell you how WE approach music and poetry:

I started by singing with my kids from infancy, exposing them to the folk songs of my home country and our country of residence. We listen to music and attend as many live performances as possible. We have a piano that is available for open experimentation and improvisation, and both kids had a few years of instrument lessons. DD has been singing in choir for many years. In high school, we study music history in a more formal way.

As for poetry: from infancy, I read to my kids, they grew up with nursery rhymes. We made silly rhymes, read poems, both English and German ones, recited ballads out loud. Both kids have, on their own, begun to write poetry that they sometimes share with me, more often with their friends and online writing groups. In high school, we do a formal study of meters, verse forms, analyze some poems, see how it is done - before that, we simply enjoy and create.

As far as fostering creativity goes: I found that having liberal amounts of free unstructured time available, if possible outdoors, is a great way to foster creative play. Toys like playmobil encouraged open creative fanatsy play; my kids invented worlds with rituals, customs, mythology. Reading stories forsters creativity. We always encouraged creative writing, and both kids have been writing a lot; DS has been working on his fantasy novels since he was 8. I rarely get to see the writing, it is soemthing they do for their own enjoyment.

 

I hope this answers your question - even though I do not know what strategies for studying music and poetry and fostering creativity Dr. Lockhardt had recommended. I don't recall he did recommend any, but then, I was focused on what he had to say about math. If you wish, let me know if what we are doing would be what he would suggest, I am curious.

[Woodland Mist Academy]

It has been several years since I read it, and I recall agreeing completely with all his thoughts about MATH.

 

I was trying to find the article again to reread it so I could address your specific questions about art, poetry and creativity, but my link no longer works, since he has published it and the pdf is no longer available.

So, I'll add a disclaimer: I remember agreeing with everything he stated with respect to math eductation.

And I remember agreeing with his hilairous description of art and music education *if that were done in the same unsuitable way as mathematics is taught*.

 

I have no recollection what he advocates about teaching music and poetry, but I can tell you how WE teach music and poetry. I started by singing with my kids from infancy, exposing them to the folk songs of my home country and our country of residence. We listen to music, attend as many live performances as possible. We have a piano that is available for open expoerimentation and improvisation, and both kids had a few years of instrument lesson. DD has been singing in choir for many years. In high school, we study music history in a more formal way.

As for poetry: from infancy, I read to my kids. they grew up with nursery rhymes. We made silly rhymes, read poems, both Englisha nd german ones, recited ballads out loud. Both kids have, on their own, begun to write poetry that they sometimes share with me, more often with their friends and online writing groups. In high school, we do a formal study of meters, verse forms, analyze some poems, see how it is done - before that, we simply enjoy and create.

 

I hope this answers your question - even though I do not know what strategies for studying music and poetry Dr. Lockhardt had recoimmended. I don't recall he did recommend any, but then, I was focused on what he had to say about math.

Thank you, this is helpful. It sounds similar to my approach. 

 

Perhaps I need to read the "lament" again, but this is not the image his writing left with me.

 

 

[Woodland Mist Academy]

 

I hope this answers your question - even though I do not know what strategies for studying music and poetry and fostering creativity Dr. Lockhardt had recommended. I don't recall he did recommend any, but then, I was focused on what he had to say about math. If you wish, let me know if what we are doing would be what he would suggest, I am curious.

 

He mentions the appreciation of poetry comes from writing, not memorizing. I strongly disagree with this. That is a personal preference.

 

Regarding art--My daughter was in an art class similar to the one he seems to prefer with the students being given a blank canvas to create with little instruction. That art class was not a positive experience in the least. It was actually a pivotal point in my dd's education. It showed me what NOT to do.

[regentrude]

He mentions the appreciation of poetry comes from writing, not memorizing. I strongly disagree with this. That is a personal preference.

I found that memorizing poetry comes as a *result* of appreciation. I would never have attempted to memorize 40+ verse long ballads had I not already loved poetry and wanted to "own" it.

I believe that many kids instinctively begin to write if they have been exposed to models, i.e. read or listened to poetry. In my experience, the act of writing poetry can lead to independent discovery of stylistic means, improvement, analysis - all without formal instruction. So I guess, I'd agree with him there, too.

I see the value of memorizing poetry, but more as a peg, as memory training, to create a repertoire of memorized poetry. It has NOT been my experience that forcing children to memorize poetry develops an appreciation for it. In my generation, we frequently were assigned memorizations (much more than nowadays), and the result was a universal *dislike* of poetry among the students who did not already love poetry anyway.

 

I can't comment about art; my active art education is sorely lacking - there I am solely an appreciative consumer and museum goer.

[Pen]

...

 

I think he'd probably go this route. Factor 38 what is it 2*19 ick that's no good at all. Ok well whatever. 18430/2 is 9215 right? ....

Yes the factors are 2 and 19 and it is the 19 that presents the problem for it being done as mental math, at least within my abilities with mental math.

 

If dividing into a much smaller number--say 100, then I would use 20, and get 5, and see that for each 5 I would also have a remainder of 1, so it would be 5 remainder 5.   By a number like 18430 I cannot keep track of all that: remember my quotient and then figure out how many 19's there are in the remainder that would need to be added back as wholes.

[Pen]

Honest answer? The very thoughts makes me experience panic. I wish she would address how to overcome that once you've got it in a later session.

 

I'm sorry.  I hope she will come up with an answer.  Maybe the answer is exactly what you are doing, namely learning it along with your child now, from the beginning, in as nurturing a way as possible.

[Woodland Mist Academy]

I found that memorizing poetry comes as a *result* of appreciation. I would never have attempted to memorize 40+ verse long ballads had I not already loved poetry and wanted to "own" it.

I believe that many kids instinctively begin to write if they have been exposed to models, i.e. read or listened to poetry. In my experience, the act of writing poetry can lead to independent discovery of stylistic means, improvement, analysis - all without formal instruction. So I guess, I'd agree with him there, too.

I see the value of memorizing poetry, but more as a peg, as memory training, to create a repertoire of memorized poetry. It has NOT been my experience that forcing children to memorize poetry develops an appreciation for it. In my generation, we frequently were assigned memorizations (much more than nowadays), and the result was a universal *dislike* of poetry among the students who did not already love poetry anyway.

 

I can't comment about art; my active art education is sorely lacking - there I am solely an appreciative consumer and museum goer.

 

Reading this furthers my belief it is personal preference.

 

It seems we disagree in regard to the order and/or effect that writing and memorizing have in relation to appreciation. I think that both memorizing and writing have the potential to lead to independent discovery of stylistic means, improvement, and analysis. I also think that both memorizing and writing can lead to appreciation--or away from it. 

 

I agree that often the appreciation comes first, but I don't think it always does.

 

I do think we agree that exposure comes before appreciation. I didn't see that coming through in the "lament".

 

As someone who describes herself as being without an active art education, what approach would be best for you to learn to draw or paint?

 

Do you think everyone would learn best that way?

 

Has the art you've seen not led you to want to create?

 

I don't believe appreciation always leads to creating or even the desire to create.

 

ETA: Perhaps I should have said "I've seen" instead of I believe or think. Obviously we've not met the same people, as my experience has been quite different. It seems to me there are many possibilties for the order of memorizing, writing, appreciation, etc. That belief comes from seeing it.

[Pen]

In one of the criticisms of Boaler's studies, I found a reference to the program actually being used and went to its website to look at it.  I could not get into much content, but one thing I noticed http://www.cpm.org/pdfs/ordering/2013%20Integrated%20Order%20Form%20050413.pdf was that the high school materials are only available for the first time for this next year.  So was that "Railside" school using middle school materials--which seem to have been out longer-- for high school, I wonder?  If so, that too could help explain why the students seemed to the mathematics dept evaluators to not be doing high school level algebra.

 

Another set of materials for an integrated type of math program that I was better able to see samples of, looked like it had a problem common to many project and craft type work in schooling (of any subject) namely a lot of "doing" compared to the math involved--that is drawing and cutting out and coloring...sort of the math related version of mummifying a chicken to learn about Egypt, or making biscuits to learn about life in the USA in the 1800's.     So much "doing" that I think I'd be very frustrated in such a class.   It did not seem to me to show that there is  necessarily a problem with group work or project based learning--done excellently, it may well be better than "traditional", but exactly how that is being implemented may be a major problem.

 

The bits I have been able to glean now from Finland is that it utilizes open-ended project oriented work, but maybe that more has gone into finding more "mathy" project and group work--and maybe at that level it takes cooperation from the mathematicians to design the problems and projects.  

 

And from somewhere (Japan? Russia? Finland?) an idea of group work on challenging math problems, where there is not so much competition and shaming as tends to be the case here in USA, but rather collaboration, an effort to have everyone help everyone figure out and understand how to solve the problems--and a cultural sense that it is good to do this both as to the collaboration rather than competitiveness and also as to the math being something good to do, not "nerdy" or whatever.

Edited July 21, 2013 by Pen

[Woodland Mist Academy]

 

 

Another set of materials for an integrated type of math program that I was better able to see samples of, looked like it had a problem common to many project and craft type work in schooling (of any subject) namely a lot of "doing" compared to the math involved--that is drawing and cutting out and coloring...sort of the math related version of mummifying a chicken to learn about Egypt, or making biscuits to learn about life in the USA in the 1800's.     So much "doing" that I think I'd be very frustrated in such a class.   It did not seem to me to show that there is  necessarily a problem with group work or project based learning--done excellently, it may well be better than "traditional", but exactly how that is being implemented may be a major problem.

 

This reminds me of a teacher who teaches the advanced math class at a local middle school. She lectured the students one day on the math projects they had recently presented.

 

"I want more glitter!"

 

She meant that literally. She was happy that there was more glitter on the posters for the next class presentation. She considered it a success--these were, after all, the advanced kids. (and thus more capable with glitter?)

 

:confused:

[Pen]

He mentions the appreciation of poetry comes from writing, not memorizing. I strongly disagree with this. That is a personal preference.

 

Regarding art--My daughter was in an art class similar to the one he seems to prefer with the students being given a blank canvas to create with little instruction. That art class was not a positive experience in the least. It was actually a pivotal point in my dd's education. It showed me what NOT to do.

 

For me I prefer to hear poetry read, or to read it and hear it in my own mind.  Yet a different personal preference.  

 

For art I think that having  times to do whatever one wants is great, but I expect an art class to actually TEACH a way of doing something, to help me advance my skill over whatever level it was at before.  I can then take those new skills and apply them on my own to my own creative projects, and might also see ways of advancing yet more skills on my own, until I am another "stuck" point, and then I want help again to advance further--if such help is available.  Sometimes one gets to a level where one does have to figure it all out for oneself, because no one around is at a higher level to help.

 

I think the analogy there for me works fairly well, because at first one needs to work on techniques and basics, and then bit by bit one gets better and can do more with the skills one has.  And in math too, I think one needs the basic skills in place.  But at the same time, maybe one can have fun doing the art or math even if not very good at it yet.  And maybe one can look at masterpieces and learn to appreciate them even if one is not there yet to make an art or math masterpiece and perhaps never will be.  

[Woodland Mist Academy]

For me I prefer to hear poetry read, or to read it and hear it in my own mind.  Yet a different personal preference. 

 

Overall this is my preference as well, although I do enjoy some memorizing.  Writing poetry? Not so much. At. All. ;)

[Woodland Mist Academy]

For art I think that having  times to do whatever one wants is great, but I expect an art class to actually TEACH a way of doing something, to help me advance my skill over whatever level it was at before.  I can then take those new skills and apply them on my own to my own creative projects, and might also see ways of advancing yet more skills on my own, until I am another "stuck" point, and then I want help again to advance further--if such help is available.  Sometimes one gets to a level where one does have to figure it all out for oneself, because no one around is at a higher level to help.

 

My dd's beginning art class consisted of a brief overview of a technique, a sheet of blank paper, and lots of time. The expectation was a finished drawing of a certain style. She was admonished to look at the paper and try--just try. Over and over--just try. Create!

 

No further instruction was given.

 

ETA: This approach resulted in neither art nor appreciation. I take that back. The unrelated doodling she started to do during class instead might be considered art. ;)

[regentrude]

As someone who describes herself as being without an active art education, what approach would be best for you to learn to draw or paint?

 Do you think everyone would learn best that way?

 Has the art you've seen not led you to want to create?

I would have to WANT to learn drawing or painting, LOL. I have never, even as a child, felt the urge to create visual art (aside from some generic drawing as a small kid).

I have written lots of poetry, even entered competitions. I am a singer and have been performing both as a soloist and with choirs for decades. Music and language are the art forms that most speak to me and in which I feel the urge to be an active participant.

I appreciate looking at art and have visited great art museums in the US and Europe, but have never been inspired to create visual art myself. I do not believe that all people have the inherent need to express themselves through ALL art media, and I do not believe this is a question of education, but rather of inherent preference.

[Pen]

The beauty of mathematics is that there is no room for debate whether something is correct. It does not depend on the personal preferences or political views of the evaluator - a solution either is correct or wrong. In that respect, a "mistake friendly culture" is the last thing I would want for my child's math education.

Yes, making mistakes happens and is a way to learn, and a student should be encouraged to make different attempts and play around with the problems - but IMO math education is not supposed to be "mistake friendly": it is supposed to teach creative problem solving that arrives at a *correct* solution.

 

It needs to be specified what the author considers as "regular" and what qualifies as "testing". Having a student do math problems is only useful if the results are evaluated on a continuing basis: EVERY problem is checked EVERY day. To me it makes no sense to have the student work on math and NOT verify that he is doing the problems correctly (this can be self-checking by the student). Now, what distinguishes "testing" from daily "checking"? The author needs to be precise.

Daily checking is absolutely essential. Timed tests under pressure can be done infrequently, but must be done, in sufficiently large intervals. Since an evaluation of math mastery only makes sense if *long term* mastery is evaluated, comprehensive examinations at semester/year end are much more valuable than weekly quizzes that test only the concept currently studied, but not whether last months' material can still be applied.

 

Speed: speed can be overrated, but speed is not obsolete. A student who lacks a quick recall of math facts will struggle in higher math when he is distracted from the actual problem by being hindered in numerical computations. Math is not *about* speed, but an automatic recall of facts frees the student's mind to ponder the actual problem and not be mired in arithmetic. Of course, quick rcall does not guarantee a ggood mathematical understanding and thus speed drills should not be emphasized over probelm solving. However, I am cringing again at the word "mistake friendly": math should be precise, not fuzzy.

 

When I hear statements like this, I am glad I am staying away from courses presented by math *educators*. Much of what is problematic in math education comes from the education departments. I'd rather listen to a real mathematician like Lockhardt, with whose "Lament" I agree ecompletely.

 

 

Re, mistakes and speed.  I guess there may be differences of opinion on what is "mistake friendly"--   I consider Sumdog, which my ds has been playing on this summer to be very mistake friendly.  You get credit for right answers in some way (virtual coins and new levels and stuff you can put on your avatar), wrong answers get a lit up and getting bigger to call attention to itself right answer correction shown to you immediately for feedback, and then also a list of what you got wrong and the correct answers at the end of the game--plus perhaps you do not do as well as opponent or your virtual spaceship gets blown up or something.   But you do not get anything really upsetting like mom looking unhappy, but trying not to let it show, with less than 100%, say.   What I have seen is that the friendliness of the program toward mistakes is hugely helpful in gaining speed...  (ETA, for my ds, it is so, someone else here says it causes her children to be in tears)  and interestingly at the same time gaining the ability to do a particular type of problem correctly increases.   When I have used more traditional timed drills  on paper or orally, there is a conflict between trying to get problems all correct on the one hand, and on the other trying to go as fast as possible, which seems to put my ds in a double bind.   On the computer game, psyching out the way it works to see how much one can miss, while going fast as fast as possible, to keep oneself still scoring points as longs as possible, seems to be part of the strategy.  I was concerned  that it might be training a sloppy approach or getting wrong answers in some cases, but overall that does not seem to be happening.

 

Re: testing:  I do not like grades, and am not especially fond of "testing"--we have state required testing and I also test in some way to decide if ds is ready for the next book/subject or needs to review past material.  But I do strongly believe in correcting every problem and in redoing the incorrect ones.  I also not only like to correct daily, or increasingly to have ds do it himself, but when something is new, I like to correct each problem till he seems to get how to do it, since otherwise I think one can practice a whole slew of problems wrongly and start strengthening the brain connections to the wrong answer or the wrong method.

[Woodland Mist Academy]

I would have to WANT to learn drawing or painting, LOL. I have never, even as a child, felt the urge to create visual art (aside from some generic drawing as a small kid).

I have written lots of poetry, even entered competitions. I am a singer and have been performing both as a soloist and with choirs for decades. Music and language are the art forms that most speak to me and in which I feel the urge to be an active participant.

I appreciate looking at art and have visited great art museums in the US and Europe, but have never been inspired to create visual art myself. I do not believe that all people have the inherent need to express themselves through ALL art media, and I do not believe this is a question of education, but rather of inherent preference.

 

I agree. It's also part of the reason I disagree with that section of the "lament".

 

Which makes your agreeing with it interesting to me.

 

Thanks so much for sharing your thoughts and experiences!

[Pen]

I would have to WANT to learn drawing or painting, LOL. I have never, even as a child, felt the urge to create visual art (aside from some generic drawing as a small kid).

I have written lots of poetry, even entered competitions. I am a singer and have been performing both as a soloist and with choirs for decades. Music and language are the art forms that most speak to me and in which I feel the urge to be an active participant.

I appreciate looking at art and have visited great art museums in the US and Europe, but have never been inspired to create visual art myself. I do not believe that all people have the inherent need to express themselves through ALL art media, and I do not believe this is a question of education, but rather of inherent preference.

 

That can be considered okay in our society.  Drawing and painting are not considered essential (though I would argue that art in some form is--that is another subject).  

 

But what about with math where many children do not want to learn it, and yet we do consider some level of it to be essential, and other level to be very beneficial even if not essential?  One of the Lockhart article issues had to do with the problem of making math mandatory--at least as it seemed from his analogy to if music were made mandatory.  Do you think it would be better to just let it be like painting and drawing, something that some people do and others perhaps most others choose not to?  In a sense, that is what we have now, it seems, just that they spend time in classes officially learning it, but many do not actually learn any math.

[regentrude]

One of the Lockhart article issues had to do with the problem of making math mandatory--at least as it seemed from his analogy to if music were made mandatory.

I did nto understand his essay to mean that he thinks math should NOT be mandatory. I think he meant to show that if art and music were under the same pressure of performance and that if the schools of education messed around with arts and music education the way they do with math, the outcome would be as he described.

 

Do you think it would be better to just let it be like painting and drawing, something that some people do and others perhaps most others choose not to?

No. As much as I love educaton for education's sake, I also believe that one purpose of education must be to prepare young people for a productive adult live and give them the skills needed to manage their daily life, obtain job training, be qualified to enter higher education.

It is simply reality that, in order to function in today's society, people need a certain set of math skills to manage their daily lives, be financially literate for example. (People might not be in so much financial trouble if they had a clearer understanding of exponential growth and compound interes)

The other aspect is that there are vastly more jobs that require a math background, starting from basic algebra and geometry needed by a carpenter or plumber to advanced calculus needed by scientists and engineers, than there are jobs that require art proficiency. For many more people art will be a source of enjoyment, creative expression and spiritual growth than a tool for their daily work. Any education needs to reflect this. So no, I do not believe one can, or should, make math education voluntary.

(Especially not in a country that lacks behind in math education compared to many other first world nations.)

[EKS]

I think that math education (and all formal education) should be voluntary beyond a *solid* 8th grade level (prealgebra level in the case of math).

[onaclairadeluna]

The beauty of mathematics is that there is no room for debate whether something is correct. It does not depend on the personal preferences or political views of the evaluator - a solution either is correct or wrong. In that respect, a "mistake friendly culture" is the last thing I would want for my child's math education.

Yes, making mistakes happens and is a way to learn, and a student should be encouraged to make different attempts and play around with the problems - but IMO math education is not supposed to be "mistake friendly": it is supposed to teach creative problem solving that arrives at a *correct* solution.

 

It needs to be specified what the author considers as "regular" and what qualifies as "testing". Having a student do math problems is only useful if the results are evaluated on a continuing basis: EVERY problem is checked EVERY day. To me it makes no sense to have the student work on math and NOT verify that he is doing the problems correctly (this can be self-checking by the student). Now, what distinguishes "testing" from daily "checking"? The author needs to be precise.

In my experience as a math teacher, many children (and adults) are math wounded. They sit there in class and think the wrong answers to themselves silently cementing magical, wrong, answer grabbing, panic stricken ideas. They are so terrified that they are wrong that they won't say anything. But they are thinking wrong things that don't get corrected. When they get wrong answers it is really tough to get them to talk about their reasoning unless you get on your cheerleader-counselor hat on and say "It's ok, lots of people make mistakes, let's just think about this a little" So much hand holding. So I think the idea of having a mistake friendly culture is to have the children not freak out every time they get an error but instead to calm down and analyze their mistakes.

 

I completely agree that mistakes need to be fixed and that students need to be evaluated but you need to get students to feel comfortable so that they can show you what they are thinking. Not just correcting the answer but correcting the wrong thinking. 

 

How many kids do you know that can take the reciprocal and multiply but have no idea what that means? What happens later in life is, if they ever have to take math again they wind up flipping the wrong number or doing some other thing that is close but complete nonsense. So right answers aren't really the holy grail in math. Often mathematicians will try to prove something by contradiction and discover a whole new branch of mathematics. 2+2 doesn't always equal 4. Of course you probably don't want your first graders to be answering that 2+2=1 but this is something that my son would have probably done if he was put in a mixed group class. He'd be so bored he'd answer everything mod3. But that's another topic entirely.

 

Anyhow I am pretty sure what is meant by a "mistake friendly culture" is not encouraging a test with only 40% correct.  I think the idea is to encourage kids to look critically at eachother's mistakes so that they understand what they are doing (and perhaps what they were doing wrong).

 

One of my big qualms with this course is that I am afraid much of this will be misunderstood by the teachers and that their take away will be to lower standards. I really don't think this is the intent, but I could easily see someone getting confused.

[regentrude]

How many kids do you know that can take the reciprocal and multiply but have no idea what that means?

I would say the vast majority of people I encounter have no idea WHY that is done. Because their teacher never explained it, and in some cases did not understand herself (no disrespect to you, I am sure you are proficient in your subject; many math teachers are not and even dislike math.)

 

 

What happens later in life is, if they ever have to take math again they wind up flipping the wrong number or doing some other thing that is close but complete nonsense. So right answers aren't really the holy grail in math.

I am not talking about just coming up with a correct number; to me, a right answer also implies a correct procedure.

 

 

Often mathematicians will try to prove something by contradiction and discover a whole new branch of mathematics. 2+2 doesn't always equal 4.

 

Agreed. But even there, there are objective criteria on what is considered clean math. A proof by contradiction has to be logically correct. Your new theory has to base on clear axioms and your theorems must be proven. Not every out-of-the-box-idea satisfies mathematicians' criteria for a logical extension of a concept. One can't just invent random stuff and call it "new math".

 

 

I completely agree that students should not be afraid to make mistakes and that they should be encouraged to learn from them. Unfortunately, with much that I have seen about math education, "mistake-friendly culture" sounds dangerously close to the attitude that conceptual experimentation is more important than actual proficiency and that feeling good about math is more important than arriving at the correct result.

[Pen]

...

It is simply reality that, in order to function in today's society, people need a certain set of math skills to manage their daily lives, be financially literate for example. (People might not be in so much financial trouble if they had a clearer understanding of exponential growth and compound interes)

The other aspect is that there are vastly more jobs that require a math background, starting from basic algebra and geometry needed by a carpenter or plumber to advanced calculus needed by scientists and engineers, than there are jobs that require art proficiency. For many more people art will be a source of enjoyment, creative expression and spiritual growth than a tool for their daily work. Any education needs to reflect this. So no, I do not believe one can, or should, make math education voluntary.

 

 

I agree with that.  And then, I think, there comes the big question, how does one make it effective, so it is not just occupying a chair in a room, or filling out workbooks which somehow do not then lead to being able to use it in life.

 

And I think good use of math and science to do good research and figure that out would be helpful.  

 

As I read both Boaler's work and the Milgram criticism, I think they both have validity.

 

From feeling very frustrated with the whole thing, like when I read the Lament and felt like okay, I agree with at least a lot of that (not all in my case), but it does not tell me what to do instead.  Though I have ordered his Measurement book.

 

I do think the project and group approach would be helpful, but it has to be the right sort of project.  The bits of "look inside" on the type of project maths that seem to be getting generated by this new reform approach look like it might result in, say for exponential growth, something like drawing rabbits  multiplying, where the time and memory of the student is going to go too much to the drawing of rabbits and too little to the point of the exponential growth.   Though if it did work to help memory about exponential growth, that would be something, and maybe worth it taking more time than seems needed--rather than to do things that seem more streamlined, but went in one ear and out the other.  Or result in a series of problem sets completed, but no connection being made to life.

 

A project that went straight at a problem based on finances might be a lot better.  And if it needed graphics, maybe a dollar or euro etc. symbol that is easy and fast could show some amount of money, rather than lots of energy placed on the art and craft aspect.  

 

I think the current way is not working, but rather than an overhaul based on conjecture, I think perhaps it would take significant trials and evaluations not just of the idea of "traditional" versus "reform" but exactly what, even exactly what projects or open ended problems prove out as being helpful in the long run and in reality of the world--beyond  the end of book tests and SATs and APs.

[regentrude]

I agree with that.  And then, I think, there comes the big question, how does one make it effective, so it is not just occupying a chair in a room, or filling out workbooks which somehow do not then lead to being able to use it in life.

 

To me the key would be teachers that are actually proficient in math and enthusiastic about it.

As long as math teachers in this country are frequently lacking subject expertise (if you want to teach level x of math, you need a thorough understannding of several years of material beyond that level), nothing can possibly change. New curricula and methods are bandaid solutions.

 

I do think the project and group approach would be helpful, but it has to be the right sort of project.

And a competent teacher. Doing meaningful group projects requires even more insight on part of the teacher than the traditional frontal method. It is easy to spend a lot of class time on projects and feel great about it - without resulting in any meaningful learning.

 

IMO, the quality of math education stands and falls with the subject expertise of the math teachers. No amount of education reserach, new methoids and gimmicks will compensate for underprepared and incompetent teachers. I would prefer teachers had to first obtain a math degree and then a teaching license; without being proficient in content, learning "how" to teach is utterly useless.

[onaclairadeluna]

 

 

I completely agree that students should not be afraid to make mistakes and that they should be encouraged to learn from them. Unfortunately, with much that I have seen about math education, "mistake-friendly culture" sounds dangerously close to the attitude that conceptual experimentation is more important than actual proficiency and that feeling good about math is more important than arriving at the correct result.

This is precisely the problem I have with the course. I am afraid that this will be what the teachers take from it.

 

However if we are dealing with teachers that don't understand the algorithm for fraction division (and I think you are right this is often the problem). I am not sure what can be done. 

 

It is astounding how low the standards are for math teaching. But I think most of this is coming from elementary schools where the kids don't have access to a real math teacher. So maybe having specialists at the elementary level would improve things a bit. I am not sure. 

 

To me the key would be teachers that are actually proficient in math and enthusiastic about it.

As long as math teachers in this country are frequently lacking subject expertise (if you want to teach level x of math, you need a thorough understanding of several years of material beyond that level), nothing can possibly change. New curricula and methods are bandaid solutions.

 

This! Absolutely.  I would say probably worse than a bandaid, a complete waste of time money and resources and sometimes completely counter productive.  Kind of like the connected math stuff from 10 years ago. Nobody knows how to teach that way and you send a bunch of confused kids home to confused parents.

 

That said I love some of those problems. But they are to be used with great caution. Kind of like chemistry experiments in kindergarten.

 

Anyhow I think you are spot on. The real elephant in the classroom (to borrow the title of her book) is that the teacher has no understanding and enthusiasm for mathematics.