My question is this. How have other unschoolers handled math and what have you seen for the older unschoolers and their experiences around no formal math training. I don't think they need more than they will use out there in the world but are we doing them a disservice by not having them exposed to the possibilities.
The subject of math can be frustrating for unschoolers! ;-)
Math is as easy to get as reading.
But there are two things that get in the way of naturally learning math:
1) Most of us are familiar with school math and that's the only model we have. It seems impossible to get all that by living life.
Well, it is and it isn't. The problem is that schools teach kids to memorize how to do things. They don't teach kids to understand. The hope is that understanding will come from memorization but memorization alone is enough to be testable and show that learning is taking place. And it takes oodles of time to memorize things you don't understand. It take oodles of time to go step by step through something that doesn't make sense.
My favorite analogy is that learning math naturally is like how kids learned to speak English. They picked up and used the parts that they needed to get what they wanted. And they got better at it as a side effect of using it. It works like a charm!
Learning math in school is like how kids learn (or not learn!) a foreign language in school. The process in school is to memorize and acquire an abstract understanding in order to apply it. But it doesn't work.
Kids learn to make plurals by hearing lots of plurals being made in lots of different contexts, not because they're told to add s. They learn to do verb tenses by hearing lots of different verbs in lots of different contexts, not because they've memorized the rules. (Which are broken for lots of the verbs we use most often anyway!) They learn to put sentences in proper grammatical order by hearing oodles of different sentences and without even learning what nouns and verbs and prepositions and so forth are.
2) Most of us don't speak math. We don't point it out. We don't use it in front of the kids in ways they can understand. Mostly because we think math is what's found in textbooks.
So we need to learn to see math in real life to gain confidence that it exists. We need to speak math when they ask us a question. If they want to know how many days until their birthday, figure it out out loud without resorting to paper. Add numbers out loud without paper. It forces you to manipulate numbers to make them easier and kids can see how numbers actually work.
There are some math threads on the message boards at Unschooling.info that might help. If you think of the message we've been inundated with since grade school that math needs to be taught rigorously, it shouldn't be too surprising that most people need to read a lot about naturally learning math in order to counteract that message.
The only subject I can't think of how to do is math. When the kids are interested we measure in different ways and the money thing is on going but how will my kids learn multiplication and division and such?
Real life math probably bears the least resemblance to its schoolish counterpart than any other "subject". Because real life math is about discovering how numbers work rather than memorizing formulas to impose on numbers.
Real life math is, as an example, casually encountering percentages in a dozen different contexts and therefore slowly building up an idea of what percentages are and how they're used.
It's similar to the process of how we acquire new words. Usually when we're reading or listening to conversations we don't run and get the dictionary to look up a word we don't know. Generally we can get a good enough idea of its meaning from the context. And the next time we encounter it we add another facet to our understanding and the fuzzy impression of what the word means gets a bit more clear. And so on. The process probably accounts for our often not being able to define a word for someone else that we've not only read and heard dozens of times but even used.
For some reason people think multiplication and division are such difficult subjects that, after algebra, that's the one thing they question under "how will they learn?" But once a child realizes that multiplication is just a fast way to do repeated addition and that division is just a fast way to do repeated subtraction, a great deal of what causes math phobia in adults disappears. Multiplication and division aren't mysterious at all. They're just addition and subtraction short cuts.
One thing I've found helpful is expressing things in a couple of different ways. When we've come across percents, I've said "17% or 17 out of every 100," or "25% is the same as a quarter."
Another thing is solving problems out loud without pencil and paper so they can see how numbers can be manipulated. So for instance to add 138 + 53. (Hmm, a bit tougher than I normally pull off the top of my head! ;-) but kids do pick up on the process when they hear similar processes dozens of times.) 39 is almost 40 and 53 is almost 50. 40+50 is 90. But we added 2 to the 38 so we need to take away 2 from 90. And we subtracted 3 from the 53 so we need to add 3. That brings us up to 91. Then just add 100. So 191.
Any thoughts about strewing a little more math into our lives?
By being more conscious of math that's around you. And not just numbers! But also patterns, connections, and relationships.
By realizing that what kids need is to absorb the language of math by seeing, for example, percent used in a variety of contexts.
That said, board and card games, computer games, video games, puzzle books.
There's a Dorling Kindersley book that used to be called Comparisons but I think has a different title now?
But above all speak math. Walkthrough solving problems out loud. (Stick to problems he asks, like how long until Christmas, if you can't help sounding like a lesson when you figure out a tip. ;-) Compare things. Pick some standard of measurement to help him grasp relationships. (It's more meaningful to translate 18 feet into 3 Daddy's for instance. Some of the ones I use are a story which is 10 ft, 3000 miles across the US, 600 miles from Boston to Pittsburgh as units of measurement.)
The reason math in school takes so long is they need to substitute drill for understanding. It's really hard to do pages of problems like 7.5% of 182 when you don't know and don't care what percents are, let alone 7.5% of 182. If the understanding is there first, the details are much easier.
I have let go of the math study, confident that they will do it when they are ready.
Perhaps it's more accurately stated that they will acquire math as they need it. And they may study it formally if they find it interesting or want something a formal study could help them get.
Unschooling math looks more like playing games and celebrating Pi Day (March 14 to coincide with 3.14) and saving up for something they want.
If he should decide in 9th grade that he wants to be an engineer will he be able to jump into math and be where he needs to be when he wants to start college.
Math learned in a classroom is like a foreign language learned in a classroom. It's memorization of stuff that isn't being used or understood. Math taught with school methods is all about memorization mostly because it's really difficult to test for understanding and very easy to test to see if something's been memorized. It's hoped that understanding will come from memorization and practice, but that isn't the goal.
Math learned from life is how kids learn to speak their native language. It's just absorbed as they use it. They get better as a side effect. It's painless and effortless.
If he begins exploring engineering type things he'll use math as he used English as a toddler. He'll be able to pull the sense of it out because he'll see the context it's in.
Kids get math by providing opportunity and fun materials. By fun materials in math, I mean things like Tangrams, an abacus, a hand counter (those little clicky things), magnetic marbles, plastic frogs for sorting and counting, plastic magnetic numbers, books, puzzles, chess game.
Video games. Grocery store. Allowance. Computer art programs.
Even better than plastic counters are real things! :-) Leaves. Rocks. Acorns. Hot wheels! ;-) Anything specifically designed for sorting and counting has built in ways the kids are "supposed" to sort. The counters are deliberately limited so as not to be "confusing". That way when kids come up with the built in right answer teachers can check off that they've demonstrated an understanding of the concept.
Real things allow kids the freedom to explore surprising similarities that may not be obvious. Learning to observe is a much more useful skill to a real scientist or mathematician (or cook or automotive mechanic or ...) than being able to notice what everyone else has already noticed!
Do you consider doing math from a book "out of context"?
Yes, if it's pushed on a child. No, if the child chooses it.
Which is too simplistic, of course, because it's based on certain assumptions.
I have a degree in computer engineering from MIT and there are definitely prereqs. in math that I think my son would need for most math, science, engineering, or computer majors.
Okay, here comes the math spiel. ;-) I have a degree in Electrical Engineering from Carnegie Mellon University. I certainly agreed with your assumptions about math when I first started reading about unschooling. I, too, was a victim of contextless, rote-learned math. It really seemed the only way. There were specific ways to do addition, multiplication, division, and on up the math scale that just had to be explained step by step and sat down and practiced ad nauseam. And what child was going to put in all those necessary hours on her own?
It took me several years of reading what other unschoolers had to say but it really wasn't until I saw my daughter actually manipulating numbers without being specifically shown how that I understood how unschooling could work with math.
The problem with school math, and as far as I've seen all math curriculums, is they start kids off immediately with the abstract. A child may be able to see they have one brother and one sister and therefore have two siblings, or one gray cat and one yellow cat to make two cats, but put 1+1 on paper it becomes incredibly abstract. Why would anyone want to add 73+48? The process is meaningless. The answer is meaningless. It has no context.
Many math programs do have kids adding sorting bears or manipulating rods or any number of other hands-on things, but they're still basically meaningless because the emphasis is still leading a child to discover a particular concept or particular answer, not on the problem itself. Why does anyone want to know how many blue bears there are? Why are the red and blue bears being added together?
Now, on the other hand, my daughter is quite intrigued to find out how many Jurassic dinos she has versus Triassic. How many plant eaters versus meat eaters. (And whatever other classifications she can come up with, limited only by her imagination -- versus the 2 or 3 categories of the sorting bears.) How many years separated the various ages of the dinos. The heights and weights of them.
And though counting and graphing M&M's by number and color seems the same as doing these same things with the counting bears, it's not. She's gaining information in the form of patterns and relationships (that are often expressed as numbers) about her own world, things she cares about.
Obviously there's only so far counting will get you in life ;-) but we manipulate all sorts of numbers in her life and I make sure she's immersed in patterns and relationships between various things in her life for her to examine (or not). Like fractions in cooking and time: "Since the cup is dirty, how can I make 1 1/2 cups?" "The recipe calls for 1 Tablespoon but we're cutting it in half. And a Tablespoon is 3 teaspoons. So what would that be?" "It's an hour and a half or 3 Bill Nyes until Daddy comes home." "It's 20 minutes or a third of an hour until Xena comes on." Though learning to take 1/3 of 60 is more universally applicable, she can feel the 20 minutes wait out of 60 minutes and she can get the feel that fractions are ways of relating one thing to another. Decimals come up with money. Percentages come up with sales, tax, food labels, possibility of winning a contest, shrinking an image in a paint program.
She's gaining a feel for the contexts the various concepts are used in, she sees me manipulating them and helping her manipulate them. And in the course she's adding pieces to the puzzle of her world, making new patterns and relationships clearer.
Up until recently we've done zero in the way of formal math. Only a few months ago she wasn't totally consistent on her addition but I asked her if she knew what 8x5 was. She said that was 16 +16 + 8. Not 8+8+8+8+8, which would have been a good answer showing she understood the concept of multiplication, but she manipulated the numbers properly into something she could feel more intuitively.
Recently she has been doing paper and pencil math under protest. Sort of.
She wants to earn money for Pokemon cards. I buy the packs at any where from $4-$6 a piece, pull out the trainer cards and then calculate how much she needs to pay for each Pokemon card. (Or have her do it for a whole card, though that's still a bit beyond her true understanding even if she does get the answer right.) I suggested all sorts of household tasks for her to earn 25 cents or a dollar or whatever which were met with groans. (She even turned down $2 to clean out the floor of my car! ;-) I suggested she do pages in the Miquon math workbooks that have been gathering dust on the shelf at 10 cents per page. Being a low energy child (like her mother ;-) she usually opts for the math.
She's getting much better at the pages, but I can still see a huge difference between what she does on paper versus what she does with the real meaningful numbers in her life. She quickly calculates in her head how much she's earned and how much she needs and how much she'll have left over after buying a card, tells me how many 36 cent cards she can get with her $2 allowance versus how many 41 cent ones. (And she does this without drills and without pages of workbook practice, just from messing around with the numbers in her life in a very low key way -- the stuff she's doing in the workbooks is actually much simpler.) She told me the way she figured out 16+16 was it was just 10+10 then 6+6 which is 12 which is 10+2, so that was 10+10+10+2 or 32. She's discovering for herself how to break numbers apart and play around with them. And she knows why someone would want to do that. If it were taught in a book, it would take weeks and most kids would still be baffled about what the purpose of it was.
Pencil and paper math and head math are different. The pencil and paper math are a new language she's learning. And yet, I'm quite confident if we had gone on without much pencil and paper stuff (other than the normal things that come up in life) she would have caught onto it way quicker in a couple of years without the agony she was putting herself through.
But that's obviously a far way from algebra and trig and calc.
Someone pointed out that algebra is just figuring out what you don't know from what you do know. Now how did I get all the way through engineering school without realizing that insight? Maybe because I enjoyed identifying the problem types and figuring out which technique to apply to them. It didn't make any difference whether I truly understood why I was doing what I was doing. The fun was it worked. Because that's how algebra is taught. It's all about practicing manipulating different types of equations. It's not about what those equations mean. Or why anyone would want to write a quadratic equation let alone solve it. It's all just preparation for potential contexts. But the equations themselves have no context. They're meaningless. (Unless you're one of the "good" ones who rise to the surface through this bizarre math-teaching process just because you happen to like to manipulate equations for the sake of manipulating equations.)
Quadratic equations don't come up in real life often, but I can help my daughter to think algebraically when we tackle real life problems. (I may be doing it already unconsciously, but you'll have to wait a few years for me to be conscious enough of it to provide real life examples of her using it. ;-)
Of course that isn't enough to get her into CMU. Or into MIT either ;-)
Now, given the choice, I'm quite certain I wouldn't have gotten in enough math on my own to get into CMU. So what makes me certain my daughter will?
Well, I'm not certain, but what leads me to believe that my daughter's outlook will be different is, for one thing, I was the victim of force-fed learning. I needed to be force fed math because I'd always been force fed learning. I needed to be force fed school math because it had no relationship to my own world. I didn't need it. I can't imagine learning what I learned on my own because the only thing I have to base my imaginings on are the process I went through.
What I can imagine, though, is being so intrigued by something that the math gets learned because it makes what I'm interested in make sense. I can imagine forcing myself to learn something in order to achieve something else. (HTML comes to mind! Though that was more a combination of both of them.)
What my daughter has going for her is a different experience with math. Other than the workbook pages :-P, she's used to seeing math as a tool. She's used to using math because she wants the information it can give her. So when she gets to high school, she won't have the memory of 8 previous years of drudgework associated with math.
She'll also have a better foundation of understanding what she's doing. Though she might be behind her PS counterparts in calculation speed, she'll be ahead in understanding what the processes mean. (But the speed will depend on her. If she feels working around gaps in her multiplication tables is more annoying than learning the tables -- and if she knows that drilling them or doing other things will help her (and it's my job to help her learn to identify when a problem exists and to seek out solutions) -- then she'll learn them. If not, she won't. (I still have gaps in my tables.)
So she'll hit her high school years with a different attitude towards math and learning math. (And this really applies to all subjects.)
But will she be able to pick up all the math she needs to get into college just by living? Well, yes and no.
This is where it gets hard to explain because our thinking is based on oodles more assumptions.
It's so easy to project a schooled teen (which includes most of us) as a normal teen and assume all kids given the chance will watch TV and eat concoctions centering on sugar, fat and salt all day and want nothing more in life than 256 channels and a clicker in the hand ;-) That behavior is caused by the stress of school (and a lot of other factors. I have another rant about being forced to spend 12 years working towards a vague goal that someone else has chosen for you. ;-) But in an environment where the adults and everyone else in the family are curious about life, where everyone's interests are taken seriously (even the so-called non-educational ones), the kids are actively curious too. There's no reason for them to want to shut their brains down as a life's goal. (Which doesn't mean my daughter doesn't watch TV. At times she even watches a lot of TV. But she chooses it for other reason than shutting off the world. (Though that's a legitimate use too. It's just that she doesn't have to spend a goodly portion of her free time recovering from 6 hours of force fed learning in a high-stress environment everyday.)
Had unschooling been thrust upon me as a teen, I imagine I would have spent as much time as possible doing nothing. It's hard to imagine a teen learning on their own something that we ourselves would avoid. It seems obvious that given the choice most teens would avoid Shakespeare or American History or Algebra or whatever school made us hate because we know we'd avoid it. But, given a choice, would we have avoided it because it's inherently dull or because school made it dull? It isn't fair to assume the behavior of a schooled teen is normal behavior. The only experience schooled kids have had with most subjects is dull textbooks. The life has been sucked out of all subjects for them. Why would they pursue them on their own? Especially if they assume the only way to learn them in a worthwhile way is the way schools teach them?
There's no reason for my daughter to avoid learning because she's never been forced to do it. To her learning is something you do to find out more about what you're interested in and to become better at it. It's not something someone makes you do because they tell you you need it.
She will avoid learning in ways that aren't natural for her or don't suit her needs. Some kids like workbooks. That doesn't make them better learners than those kids who don't. It just means they learn differently. She will avoid learning anything that isn't relevant to what she wants to do or is interested in. Which makes parents nervous for two reasons:
1) What if she never gets interested? It's possible she won't on her own. But it's my job/pleasure to run as much of the world in front of her as possible. The broader her experiences, the more likely something will connect to something else in her life and be relevant. (Though I can't depend on when.) Everything is connected to everything else. And everything relevant is inherently interesting.
But it's also possible she won't get interested in something "important". Math? Writing? Chemistry? If she has absolutely no interest in it, then it's unlikely she'll be drawn to a profession that needs it to an extent greater than she can pick up by living. Though she won't leave the house without being able to figure out sales tax or write a letter to a friend or know that baking powder is important in cookies because she'll have used those. She'll have enough to get by. But it's possible she'll need higher math than she has. Or better writing skills. Or an entire chemistry course. Well, if it's just chemistry standing in her way, wouldn't it make sense for her to go down to the community college and take it rather than deciding on a different career just because of one course? And if that's too much trouble, how much did she want that career anyway?
But math and writing? Well, I hope something I'm saying here helps you see why I believe there's a middle ground between "no math" and 4 years of high school math from textbooks. And writing I talk about below.
2) And the second reason it makes parents nervous is supposedly there are things kids need to learn that they won't need until college. And supposedly it takes 12 years to learn them.
But does it? Does it take 12 years to learn math? Or does it take 12 years for schools to force feed a child math (and writing and history, et al) by the methods they need to use to force feed 30 kids at a time? Methods which are also limited to ways that can result in outcomes that can be tested to demonstrate progress. Also limited only to methods that must be progressive along a specific track so the next year's teacher can pick up where the previous teacher left off. Does math need taught that way? Or do schools need to teach math that way to satisfy the needs of schools as assembly lines?
In a way, school math is rather like learning to spell thousands of words and decline hundreds of verbs of a foreign language without hearing that foreign language spoken. The rationale being that once all the parts are learned, the whole can be built from that. But how many kids survive the rote process? How many kids conclude not before long that the language is useless because the parts have no meaning? My daughter is hearing the language and using it, without formally declining the verbs and learning the spellings. Even if she'd never been exposed to reading it (but already had the decoding skills from reading English) how long would it take her to learn to read that foreign language after having learned it from using it?
Once my daughter has a thorough understanding of what it means to do division, she won't need umpty gajillion problems to practice. Once she has a thorough understanding of problems with a range of potential solutions (programming and robotics come right to mind), and has encountered and understood powers and negative numbers she won't need years of practice to grasp algebra.
My job is to make sure there are reasons in my daughter's environment to need the skills and see them being used. (Just as I talked to her well before she could talk.) Though she finds a lot of uses for the skills on her own, given the freedom to do so. There's no reason for her to avoid writing or reading or math (until the workbooks) on her own because she's never been forced to do them. The hard part is waiting for her timescale. I need to wait until these things are internally important to her. I can't worry, well, she's 8 now and should be doing ... because natural learning doesn't have anything to do with calendars and time schedules. It has to do with needs.
If she has a goal in mind, she won't have anything except natural barriers between her and it. She won't have what someone thinks she needs to get there and someone else's way she needs to get it standing in her way. If she decides to become a vet, she'll know what colleges require for her to get there. If her desire is strong enough, she'll learn what she needs to learn because she wants what the learning can get for her. (Desire is an incredible motivator.) And most importantly she'll have better resources to achieve it than sitting down with a textbook and slogging through it. (Though that's an option too. Fortunately she won't have the history of slogging through textbooks putting up a psychological barrier for her.) She'll have a good foundation of understanding math concepts and will see it and other math being used (and use it herself) as she explores what it takes to be a vet: taking care of animals, working in a vet office or a horse stable.
So, if your kids aren't prepared enough to go to a university, then you assume that they will be motivated to study once they get rejected?
The answer to this one is probably obvious from the above. No, I don't expect rejection to spur her. I expect wanting to do something will spur her to do something. And perhaps that something won't even be college. I too had visions of my daughter going onto CMU or MIT. But now my vision has shifted from preparing her to be anything she wants to be to helping her be the best her she can be. Yet I'm not sitting around waiting to pounce on her interests to nurture them. I'm also directing things through her world that I think are important or I think will interest her. When (if ever) she picks up on them is up to her. The more important I think something is, the more likely I'll keep directing it in her path in a way that will interest her, or connect it to something she is interested in.
We do provide a very stimulating environment. We have books and materials everywhere. Lots of interesting folks float in and out of our home and office. While my 9 yo son likes to read and mess around with the computer, he wouldn't ever just open up a math book.
Nor would most kids. For a child to choose the more formal learning in a book requires an interest and need that the book can fulfill. The environment may be there, but he's not ready to ask the questions that the books will answer for him. Or he may be discovering the answers on his own through self-discovery or talking to people. Unfortunately for nervous parents, you can't put unschooling on a time schedule. You can't set up the environment and expect there to be a specific outcome at a specific time. (Though I can just about guarantee that if the innate talent or desire is in him for what the computers and people and books can provide, by the time he's 16 he'll have sucked the environment for all it's worth ;-) 9 is way too soon for most kids to be doing more than playing around with things and exploring broadly. They may be delving deeply into some things, but the cognitive development necessary to make them open a math book for information just isn't there until the teen years. (Of course there're always exceptions. But do the exceptions mean that the nonexceptions are falling behind? Or are the nonexceptions just learning other perhaps less obvious things? A homeschooled friend of my daughter's has at 8 read all the Little House books and all their sequels and is well into other historical novels. Am I jealous? Well, yeah, of course ;-) Yet my daughter is, less obviously, picking up bits and pieces of world and American history. She's gaining a broad overview of it all, expanding some bits here and there as she finds out more about someone or something she's heard interesting things about. And she's, of course, learning other things as she pursues her interests. Is one learner better than the other or are they just different?)
Studies I have read show that certain windows open for certain math concepts at specific times. There seems to be accumulating evidence for a certain scope and sequence for math too. I am talking primarily about getting skills so you can do higher level math.
The studies, of course, are based on kids whose basically only exposure to math is in school. Math to them is artificial, irrelevant to their own world. How many parents are helping their kids use the math that's all around them? Math, to most kids (and adults!) is just the stuff in math books.
But, my daughter is being exposed to math right now, using it in ways that are meaningful to her. She's using the skills she needs right now. I'm not sitting around waiting for her to pick up a math text.
So, yes, there probably is a window of opportunity for math knowledge. But there's no way to miss it if a child's curiosity is being fed and she is immersed in the language of math. There's a window for learning to speak too, but the only way to miss that is by not speaking to the child. As long as we speak math to our kids, they'll learn the parts they are developmentally ready for.
What if she chooses no math? How do you handle that?
Obviously she hasn't yet. It is possible she'll decide to be a painter and won't need math beyond consumer math and what's relevant to the science of color. But she'll have been exposed to fun stuff like Fibonacci numbers and probabilities and algebraic thought. But, honestly, how many people need algebra? Why torment a child with "what if" when it's more likely to cause them to dislike the subject than to learn it?
(As a follow up, I don't know how old Kathryn was when I wrote this post. At 14 she wants to be a writer so doesn't need math. Other than less than a dozen pointless workbook pages and 2 months of 2nd grade she had no formal arithmetic, only what she encountered in real life. But at 12 she and her father went through the Painless Algebra and Painless Geometry books because it sounded like something fun to do with him. And since her enjoying math and her time with him was his primary goal it was fun for her. At 13 he offered to let her take the college Statistics math class he was teaching which she greatly enjoyed. She had more one on one time with the teacher after class than the other students ;-) but again, because Carl's attitude is an honest "This is fun," and his goal was always to be sure that Kathryn was enjoying the class and not fear that she might not learn, she enjoyed it. Currently at 14 she's taking Carl's Contemporary Math and Algebra courses.)
Is it my role to lecture the benefits of the things I have to offer, but to back off if he doesn't want them?
Lecture? Ick. How important would you feel something was if your wife decided to lecture you about it's importance? What would come across is her needing to make you feel the same way she does about something. And personally, when someone's trying to make me feel some way about something, I tend to work up the opposite feelings.
So sorry. I should have read the whole post more carefully. My wife preached to me about that. OK. That is what you would do. I have a hard time with that one. I don't think you can play catch up in math and science all that fast. My opinion only.
But more than opinion, I have the advantage of seeing the same math being learned naturally way easier than it's taught and learned in school. I have the advantage of reading other people's kids' experiences with unschooling math. (Helen Hegener has a good story about her son and trigonometry as I recall.) I think it's natural to assume that years of school math is necessary. But reality doesn't match sense ;-)
As for science, ah, I have a rant about that too ;-) The short version is, I think way too much emphasis is placed on memorizing the answers to questions kids haven't asked and way too little time on fostering scientific thinking and fostering a wonder about how the universe works. Once kids are curious, they'll want the facts. Once they want the facts, they go in so easily.
Last updated: April 2009