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Have I been doing it all wrong?

post #1 of 40
Thread Starter 

So I just re-read - and really read for the first time in years, Lockhart's article A Mathematicians Lament, which he has also published as a small book.  The online pdf is here:  http://www.maa.org/devlin/LockhartsLament.pdf


After reading this I am starting to understand the people who believe unschooling is best done ALL the way.  Can't just say, well we only do writing or math practice for a little while, the rest of the time is free. It's the presentation of math or writing or whatever for that "little while" that can make so much difference.   And then the writing and math will be free too.


My dd LOVES math and sees it everywhere - when listening to music,when looking at jewelry, she just picks up on the patterns and makes comparisons using terms from math.  She can't necessarily "do" what one might expect from someone of her age - this is getting more so now that she is almost 10 and the "expectations" are rising rapidly.  So I have been trying to ensure that she can do various things and have been pretty happy that she has gone through the Singapore Math book, though lately it has become more of a chore and I don't like that.  But at least it is helpful to learn the skills ... or so I thought. 


Now after reading Lockhart's article, and seeing that he says so emphatically what I have long believed and observed, I am wondering if I am taking the beauty, the art, the struggle and the epiphany away.  Taking away the chance to discover and to be amazed. 


Sure, she can master the skills ... but are we cutting out the mathematical experience in the process? 


This may not make sense if you haven't read the article ... I will try to come back with some relevant quotes, but it would of course be better just to read the article. 

post #2 of 40

rumi I really just wanted to say thank you for posting this. I've skimmed it and broadly agree, but I want to go back and give it a proper read when I have time.


My own thoughts on math education are complex. My main issue is that I do think, for a number of reasons, that in this society, and especially for our girls, maths can be one of those subjects that kids can find it too easy to dismiss as not for them, and I have to say, for that reason, I've always been very proactive about encouraging my kids to gain basic mathematical skills. And I've also felt that working through an efficient and creative program like (for us) Miquon/Singapore/Beast Academy can be enormously helpful in making sure that the basics are covered and giving a child the tools and vocabulary to engage with the discourse. I see it in exactly the same way as I see them learning scales and studies and so on on their musical instruments-just things that need to be done to gain and maintain proficiency. I'm coming at this from the perspective of someone who loves math (and as a chemistry student), who is married to a mathematician, and my kids have two mathematician/physicist grandfathers so there really is an element of them needing (and wanting) to gain these skills to participate in family discussions. Its not normal to be math-illiterate.or hate math in our family! We're possibly unusual in that math, for our family, is a social thing. Right now my 5 and 7 year old have spent much of the weekend so far playing math games, and my 9 year old is off reading a book about programming, I think.


But I need to read the article and report back!

post #3 of 40
Thread Starter 
Originally Posted by Fillyjonk View Post
 Right now my 5 and 7 year old have spent much of the weekend so far playing math games, and my 9 year old is off reading a book about programming, I think.

What games?

post #4 of 40

oh dunno really, just whatever. Off the top of my head there's been snakes and ladders with some weird rules involving counting by twos and threes, some Timez Attack, Uno and Yahztee. 


I otoh have spent an incredibly dull weekend revising for an exam (and escaping on here) so I'm not that sure what exactly they've been up to really. They just present themselves for food every so often!

Edited by Fillyjonk - 4/21/13 at 9:32am
post #5 of 40



Love this:





Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.



post #6 of 40

Also writing to thank rumi for the wonderful food for thought. I'd read Lockhardt's Lament before, but it had been some time. Of course I can point out plenty of flaws in his metaphors, but that's just the nature of metaphors. 


Overall I feel pretty comfortable when I read that essay. I don't feel we've approached math, music and art radically differently. We've made use of formal learning approaches where it's suited us -- perhaps more than some unschoolers because my kids seem to enjoy formal intellectual challenges -- but the music lessons, art classes and math workbooks or whatever have always rested on a foundation of delight-directed learning. Occasionally I've worried that systematic curricular-type learning has begun to take precedence over more natural, informal, experiential forms of learning, particularly in the realms of math and instrumental music. But before I get too worried, my child will do something brilliantly self-directed and interest-led to reassure me. Just yesterday dd10 went from a workbook-based introduction to algebraic functions to a self-led exploration of dependent and independent variables in real life, chattering away about it on and off all through dinner ("I just thought of another function: tomato plants growing taller each day -- and the height would be the dependent variable, right? That one would be curve that would get steep for a couple of weeks and then plateau after a month or so."). As a parent I may see formal and informal learning as distinct and worry about favouring one over the other, but as unschoolers my kids haven't really made any such distinction. 



post #7 of 40

Ok I've skimmed the article now, feel absolutely free to tell me to go back and look again. (my computer is very slow right now-happy to go back and look again its just that I'll be gone a long time!)


Here's the thing for me. I think yeah, great for any kid growing up in a math-interested household. TBH to some extent this is what we've done, though like I say we've made systematic math materials available and strongly encouraged the kids to work with them.


But we are two parents who are pretty comfortable with math, who model math use habitually, who find math exciting, and, (ok I know this might be a bit sad) comforting. It is, sadly, far from unusual for a dinner conversation in our house to revolve around how a statistical test has been misapplied. Therefore it seems quite likely that our kids will see math as something interesting and relevant and so far this has proven to be true. 


My question is, - and this is a genuine question-some people do not love math. Some people have not studied math beyond, say, age 16, if you didn't like it then, if you were afraid of it, if you have basically spent the years since trying not to use it. Some of these people are homeschoolers/unschoolers. There was a recent facebook meme along the lines of "click like if you haven't had to use algebra today.". If you are one of these people, if your kids are being brought up in a home where love of math is not a given-what happens then? If you have no one who can introduce you to the beautiful riddles inherent to math, in language that you can understand...what happens then? Some kids stay interested, no doubt about that-but what of the rest?


Genuine, genuine question. Not having a go at anyone. To put this into context, we are a math/science/music/literature loving family, I think, and I have a lot of guilt about my near-complete failure to expose my kids to languages despite actually living in a bilingual area.. So I am totally open to the same criticism myself and in my case, perhaps more because my sense is that I've stacked the odds against my kids being anything other than English-monolingual.

post #8 of 40

He did speak to the idea of some kids never hopping on the path to mathematics and being OK with it, and he even admitted that anything would be better than the current system, but you are right that he addressed this issue outside of the homeschooling context.  However, to jump back to his analogy with required painting or music, I imagine he would drag those back into the HSing context.  And I would add that this question is the same as the question of qualifications to homeschool.  Some math can be important to daily, or at least weekly, life, but mathematics...... and I think that HSing parents can learn quite a bit from this article.  So, how would a HSing parent interest kids in things they are not adept at?  Art?  Music?  Literature?  Writing?  Computer programming?  Then mathematics is the same, according to the author--and I agree!  


I'd say that my family is curious, fascinated, but not particularly mathy.  I think that is enough.  Eventually, my girls will meet others who play with math more skillfully, just as they will meet potters and writers and botanists and farmers and software engineers.  The issue of math, says the author, is no different than that of the arts, because that's what he says it is.  I think he would agree with my arguments to extend his thoughts to the HSing world.  (I'm sure he would say something like, "You certainly couldn't do any worse than what they do in schools right now!"

post #9 of 40

Thanks SweetSilver-that makes sense. 


I think on balance my trouble is that I don't think math actually is treated like painting, art, writing, etc by society, which I think might possibly mean that kids don't get that casual exposure to it , especially the exciting problem side, unless they have parents who actively model this (I could be wrong, and of course this will be kid and aptitude dependent). I don't think we have lots of hobbyist mathematicians in the way way have hobbyist potters, its not something people take up. And those that do like math or work with it have don't really talk about their work. Our HE group if we found out a parent was a potter would probably be hugely excited. Maths? Not so much.  And I think, for various reasons, an enthusiasm in math can be hard to convey to another person, especially if that person has no prior grounding in arithmatic. My partner is pretty good at explaining math and loves it and even has a specialisms that are quite interesting and accessible (cryptology and computing) and yet he struggles to explain a lot of what he does or the puzzles that interest him to our kids-and our kids are actually pretty good at arithmatic and good at using math to solve problems. And the other issue is that, honestly, if you are into science (pos aside from medicine) you learn very fast that people are very seldom actually interested in what you get up to. So its not that much talked about anyway.




I guess my question, and its a real question, not a rhetotical one, is that if one doesn't proactively expose one's kids to math, will this exposure be likely to take place? Real, genuine question, though like I say I kind of feel that the odds are against it unless you have that kind of family. (and mathematicians also tend to be shy, that's another thing. Its rare to find an extroverted mathematician IME!)




And I don't know what the solution is! Maybe its that we do make some choices for our kids. I've chosen, really, to stack the odds against having trilingual fine artists and you could certainly argue that trilingualism and possibly fine art are more useful in the pursuit of happiness and so forth than mathematical facility. I've always struggled with this idea that we are showing our kids quite a limited world by homeschooling... I know that's irrational though-school, the only other option, doesn't show them a wider world, IMO.

Edited by Fillyjonk - 4/24/13 at 11:45am
post #10 of 40

Fillyjonk, this is something I think about a lot too. In our homeschooling community, I've found there to be lots of cool opportunities for exposing kids to the arts (pottery, crafts, painting, dance, drama, music etc), to outdoors activities, and to sports-- but I've had to work harder to find or create opportunities related to science and math.


One thing that makes me feel better about the questions you raise is that it seems to me that kids still manage to find and express their interests surprisingly well- even without a ton of exposure. I'm more arts oriented (I write, mostly teen fiction) though I do love science... but my son is far more drawn to science and computers and engineering and anything to do with how-things-work, and is not very interested in the arts in general. And he expressed that preference very strongly and very early (as a toddler, really). He does love opera though, and is interested in technical aspects of drawing- perspective etc- intersections of art and science and math are everywhere.


Mostly, I try to share out stuff I think is cool and stuff that I think he might find interesting, and trust that his curiosity will lead him to find more. I do find it reassuring that his interests are so clearly his own and so different from mine (though actually, he's influenced me a lot in this way and I'm much more interested now in topics I used not to think much about- so I guess this works both ways).


My other concern about math is that a lot of (schooled) kids seemed to get turned off by rote learning and repetition long- or conclude they are not good at math-- before they get to discover the more interesting stuff. So it is interesting to hear from other unschoolers about how their children engage with math. 

post #11 of 40
Originally Posted by Cassidy68 View Post

 So it is interesting to hear from other unschoolers about how their children engage with math. 

My girls know Pythagoras because of the beans and farting thing.  Sheepish.gif  It creates a powerful image!  I know I would have remembered him more readily if I had known that.  I do remember Archimedes and the bathtub, but that was introduced in science, and science curriculums include more history.  I am glad to be just keeping up with finding the fun stuff as we go.  


Yes, my girls are going to have in-depth knowledge of gardening and chickens, not so much computers, etc.  It's just the luck of the draw, I guess.  Not much different from the fact that my American History teacher was beyond incredible, my chemistry teacher a bore, and many, many forgettable teachers in between.  How many kids have been inspired by that one teacher that they might not ever met?  Or, like me at twenty-something, meet someone involved in something that sparks their imaginations and gets them moving in an entirely different direction?  There is a bit of randomness about it all, isn't there?  The very fact of our choice of residence, our nations, all those things get wound up in who we become.  We cannot possibly give children every possibility in the whole world for them to choose from.  

post #12 of 40
I think when we don't expose our kids to something sufficiently, we make choices for them by default. I also believe mathematics is like learning a language. It requires a certain level of basics to be there in order to move to a higher level. Just like learning a language, the basics require not just understanding but also good old boring rote memorization.

Just like learning languages, frequent, daily exposure is important. Another parallel between math and language learning is, it is easier to be fluent as a grown up if you start as a child.

While not knowing a second language or fine art may indeed curb future choices, I think weakness in math does more so. There are just so many interesting/more accessible career paths that require strength in math than in second language/fine art.
post #13 of 40

Really interesting.


Can I share my other, big, big concern. I have two daughters, as well as a son. I'm becoming very aware that society at large does send girls a message that math isn't for them. In so many, small, subtle ways. Even in the home ed community. My girls do like math, they see it as a tool as well as a fascinating puzzle, but I'd say we've had to work hard to counter the ideas coming from society to make them see math as something important and relevant to them. I feel I've had to actively work to give my girls a self identity as "people who like math". Whereas with my son, its just seemed natural and easy, not because he's better at math or more interested than they are, not at all, but because in so many subtle ways he gets the message that math is something for him. Seemingly tiny things like that the adult men around him tend to be into practical science stuff and use math in that, whereas my kids have heard adult women saying things like "I can't knit something like that because of the math.". When men are together with a small boy, for whatever reason, the activities they seem to choose or things to discuss are often vaguely sciencey. Girls, well, don't tend to be alone with adult men so much outside their own family (unless you specifically bring it about as I do).


Another issue I think could be that girls often show a strong early preference for literature and writing and creativity, which might or might not be inbuilt. I certainly cannot deny that I model non-mathematical creativity a lot more than their father. So they have this strong interest and that is encouraged as of course it should be...but then I think society does tend to see people as EITHER arty OR science/mathsy, which I think compounds the problem. 


Emaye I think thats an interesting parallel you draw between learning languages and math. I can see the parallel. I mainly learnt my higher math as an adult which might or might not be relevant. I'm going to ponder on it today. 


One thing, tangential to this. I grew up primarily identifying as an arty kid and got into math/science in my mid twenties. That switch was harder than just learning a bunch of new stuff. One big issue was that my fellow students, mainly male, had just way more peripheral background than me. They'd spent a childhood vaguely puzzling on certain stuff whereas I'd learnt early that if I didn't find this stuff interesting, that was fine, there was no expectation from my friends or parents that I'd be able to have a conversation about this stuff or assemble a go cart or whatever. I really remember vividlly doing a sailing course a year after finishing two years of math/physics and seeing very clearly how the expectation was just there that I wouldn't really get stuff like momentum, torque etc-and how people struggled to place me once they realised I did understand these very basic physics concept. Up to then I guess I'd been treated as science/math illiterate and not even noticed.

post #14 of 40
Generalizations and pigeon-holing are two problems. The belief that boys are more science and/or math inclined, while girls are more language oriented is too common. And the belief that it's best to learn it when you're young stops some from trying to learn in their teen or adult years. So what if you have a bit more work to do to catch up? If you start where you are and move forward you'll make progress. Stop looking at how you compare to others and focus on how you compare to your past self. That's what's important when learning later in life.
post #15 of 40
Originally Posted by Emaye View Post

I think when we don't expose our kids to something sufficiently, we make choices for them by default. I also believe mathematics is like learning a language. It requires a certain level of basics to be there in order to move to a higher level. Just like learning a language, the basics require not just understanding but also good old boring rote memorization.

Just like learning languages, frequent, daily exposure is important. Another parallel between math and language learning is, it is easier to be fluent as a grown up if you start as a child.

While not knowing a second language or fine art may indeed curb future choices, I think weakness in math does more so. There are just so many interesting/more accessible career paths that require strength in math than in second language/fine art.


I think pointing out mathematical things is important--patterns and the like.  But doing equations every day?  That I would I disagree with.  The point of the essay was to say that encouraging enthusiasm and excitement for mathematics is a far more important foundation than doing the equations.  Oh, and have fun with along the entire way, not just when you "learn enough". 


In regards to language: you can learn to speak languages in 2 different ways: immersion and the typical foreign language class.  The latter is vastly inferior.  Having one parent speak a second language at home is showing to have some noticeable benefits--but we can't all speak 2 languages at home.  So, we get along as best we can with the cards we were dealt.


And if we speak of babies, well, babies start playing with language from as soon as they have even the most basic control over their lips and tongues. B-b-b-b-b-b.  Phlllbt! They play with their mouths constantly.  And nothing stops them from using it if they use it incorrectly.  They don't wait until they have proficiency to start using it in fun and useful ways.  The joy of their increasing mastery is apparent.  If only mathematics could be learned in such a way!


In regards to literacy:  there is exposure to language, and there is literacy.  As unschoolers, we are all familiar with the criticisms about reading practice. "It's about learning the love of reading."  "The joy should ideally come first."  Again, back to the point about babies learning to speak.  Kids should not have to wait until they achieve mastery to read a story with mom or "write" their own, just as kids should not have to know the types of brush strokes in order to enjoy painting.  For both reading and painting, what do parents do?  They hand them a book, they hand them a brush and some paint and they watch while their toddler immerses himself (quite literally in the case of painting).  Why not math?


If we are comparing it to languages, then we need to explore not just how often we are actively learning language and literacy, but how.  Most people's idea of "daily math" is frankly entirely dull (except for those kids for whom it isn't).  How can we learn mathematics every day?  Comment on patterns, especially in the natural world where cool mathematical patterns abound.  Read stories about mathematicians.  Play with it.  


I think Miranda wrote an awesome post once about how you can incorporate mathematics into your life, and I wish I knew which thread that was in.... it was wonderful, and I'd like to copy that for my fridge.

post #16 of 40

I was reminded that mathematics was happening for centuries before our modern equations.  Vi Hart has a fun proof of the Pythagorean theorem, minus the mathematics that was invented long after his time.  In math class, we are taught this theorem using techniques and numerals that simply did not exist in his day.  That didn't stop anybody from exploring!




My girls get a kick out of the video, even though they aren't quite understanding the proof.  

post #17 of 40
I am typing on my phone so I can't be as through as I want to be in my response. But wanted to say that babies learning to speak are driven by the very human compulsion to communicate. Once that is achieved, learning a second or third language is harder. I was comparing math to second or third language learning -- that is, with the extreme desire to master it for survival removed. Sorry that wasn't clear.

Of course kids can play around with math all they want. No need to master it to play. I am more talking about math within the context of subjects like calculus. You need a solid background in math to do advanced calculus. Just like you need a certain level of language mastery before studying literature in that language.

There are people who are math inclined by nature just as there are people who learn languages faster. However, for most of us, it takes a massive amount of effort. That effort, if it is divided up over years of practice, then it wouldn't feel as daunting or discouraging.

I think it is actually easier for unschoolers to pick up reading -- that primal drive applies although not as strongly. Besides it is still in a familiar language. For math, beyond shopping and other practicalities, higher levels require pursuit. That pursuit requires the basics to not only be there but to be mastered. Otherwise, future stumbling might be inevitable. Of course there are exceptions but in general, I think this is the case.
post #18 of 40



Delete.  Need more time to think  :)

post #19 of 40

Emaye just want to respond to this one point, having given this much thought ROTFLMAO.gif


Another parallel between math and language learning is, it is easier to be fluent as a grown up if you start as a child.


I think its true to say that generally, people who are fluent in languages as adults are those who have started as kids, but my understanding is that this is not actually necessary. I've recently been following this guy http://www.fluentin3months.com/ who believes (and has demonstrated) that adults can aquire fluency in any language very fast, actually I think possibly faster than kids. What especially interesting is the method he used is basically what kids do. They decide that they want to communicate and get on with it and keep trying. Yes they fail- a lot. But tbh, they keep on because they have no real choice if they want to be understood. His argument is that if an adult approaches languages with the same mindset, they will also manage to learn.


So related to this, another thing occured to me. I mentioned that I'd started learning math in my twenties and it was hard. It hit me that actually, I'm working on the assumption that for kids, is isn't hard, that they don't fail a lot and don't experience the mental whiplash. Again, I think part of this might be me underestimating how much kids just assume they need to keep going and not complaining. So quite possibly the idea that I had that kids need to start early and often really to be good at math when older was actually based on a false assumption-that its not simply that starting to learn math is hard at any age and the difference with being an adult is just that you've kind of got used to things not usually being hard.


As I say my concern is more that I feel society hands our girls (in particular) a message that math/science is difficult, dull and unnecessary so I see part of my task as to curb that. But to be clear, although my older kids have elected to follow certain programs of math at times (and then we have long fallow periods like now -which is fine, by the way), I see my task as to enthuse them and stop them from in any way thinking math isn't for them. I have also told them that, IMO, its not a bad idea to leave childhood having a facility with basic math and when they express a serious interest in a career path I've encouraged them to find out what's needed for that. (I did do that when my son wanted to be a cook as much as when he wanted to work at CERN.ROTFLMAO.gif ). I actually have always taken my cues here from my dad, who trained as a math teacher but got out because he honestly couldn't stand it. His particular bugbear is that math was set as a punishment. For someone who loves math and has spent their life on it, having learning about your subject be a punishment is really quite insulting. I need to send him a copy of that article because he would agree with every word.


ETA oh and Vi Hart, awesome woman. Love her stuff. 

Edited by Fillyjonk - 4/25/13 at 10:03am
post #20 of 40

It's still a little different-- at least the way I am thinking about it.  When I learn a second language, and go use it, immerse myself in another country, (especially in regions where English is poorly understood and they won't get frustrated with me and start using English) I can still be understood to some degree, even if my skills are less than perfect.  They might help me say it right, or they might simply point the way to the bathroom.  


You're right, mastering it requires regular practice, but..... these are my thoughts:


For years, I resisted picking up the guitar, because I expected myself to achieve mastery.  It eventually prevented me from touching it for decades.  Then I became pregnant with dd1, and I finally decided, screw it, I'm just going to play anyway and use it to sing her songs--I wasn't going to let my perfectionism get in the way of the joy of music anymore.  Yes, at some point I did practice everyday, or at least frequently, while the promise of mastery still figured large in my imagination, but I stopped.  Completely stopped because I mistook the point of playing my guitar.  It paralyzed me for the longest time, until I was able to give up my preconceptions about what I needed to achieve.  I think that same feeling prevents many people from exploring making their own music as they get older (and of course, nothing wrong with appreciating beautiful music from those who have made it their lives).


I would say the same happened with my love of learning other languages.  I did practice every day, but again, I was discouraged because of the the amount of effort it took to achieve fluency.  Then, decades later and facing a trip to Germany to see my sister, I picked up a refresher course through university "experimental college" and had the revelation of--screw it, I just want to get around and be understood!  


I think, for me, the idea that my early math was supposed to be a foundation for higher mathematics might at some point paralyze me, because that's the person I am, and that has been my experience in other things.  Perhaps a less lofty expectation-- that I can use math to play games and have fun and use a bit in cooking and knitting and finances--then great!  


And that's where I am now.  I wasn't interested in math at school, for the most part, because I was destined to be a theater major in college.  What was the point?  This stuff was dull!  Now that math is mostly entirely removed from any actual equations in my life, I have a lot of fun with it-- I like learning about it, playing with it.  I was completely misinformed growing up--this stuff is cool!

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