In 2005 I created An Unschooling Life, a blog detailing our unschooling experience after adopting our three children. Over time, An Unschooling Life became a hub for unschooling support and advice. The blog has been featured in print and digital media and was home to the popular Unschooling Carnival. I’m in the process of updating and moving all the posts to this blog where they will be housed under the An Unschooling Life section. This post was originally published on January 11, 2008.
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When my daughter Jacqueline was seven years old, she asked if I could buy some stories that explained math. She was becoming more and more interested in how math fits into her world and had started to take notice of it in movies, TV shows and by watching my husband & I. She had a basic understanding of addition and telling time but she was more interested in math as a whole, not broken down into subjects.
Unschooling Math Ideas:
I’d like to share something that I had saved when I first began unschooling. It’s an exchange (from the old message boards at unschooling.com) about unschooling math between a member who was having some concerns (whose posts are in italics) and Joyce Fetteroll.
Be warned…..it’s very long, but for those who are interested, you’ll find a lot of great food for thought.
>>>>> 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. < <<<<
Joyce: 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 nauseum. 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. The teacher has created the problem and dumped it on the child. Why does anyone want to know how many blue bears there are? Why are the red and blue bears being added together?
Unschooling math resources
Joyce: 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, the 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 with her addition but I asked her if she knew what 8×5 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 anywhere from $4-$6 apiece, 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 leftover 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 is 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.
Related: Unschooling science
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 methodology 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 drudge work 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 reasons than shutting off the world. (Though that’s a legitimate use too. It’s just that she doesn’t have to spend a good portion of her free time recovering from 6 hours of force-fed learning in a high-stress environment every day.)
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 to be taught that way? Or do schools need to teach it 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? < <<<<
Joyce: 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.< <<<<
Joyce: 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 14 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 daughters 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 the 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. Is one learner better than the other or are they just different?)
>>>>> My son also wouldn’t write anything on paper, which I understand is fairly typical for boys. Writing skills don’t progress overnight.< <<<<
Joyce: Who says? Okay, not overnight, but does it take years of practice? Or does it take years of using the skills in ways that are meaningful for the learner?
>>>>> Are you saying that I should encourage, but not demand? I am still missing something in terms of how this unschooling plays out.< <<<<
Joyce: How well would you learn Hindi if someone decided it would be important for your future because they used Hindi in their lives and so made you practice for the next 10 years? Wouldn’t your goal be to learn as little as possible to satisfy them? But if you were moving to India, then wouldn’t learning Hindi take way less time?
What your son needs is being immersed in an environment where it’s important to communicate his ideas. He also needs to see others using communication in a meaningful way and to read and hear others communicating in various ways. When he needs to communicate using the written word, he will.
In the meantime, you can make sure he has access to the skills. Listen to a variety of things: conversation, books and books on tape, comic books, movies (reading the scripts of favorites is really cool), plays, puppet shows, poetry, folk tales, nonfiction, cereal boxes, TV Guide, political talk shows, lyrics, ministers, magazine articles, Nintendo magazine, science shows, letters to the editor. Anything as long as he’s interested. He needs to hear good (and bad) literature so his ear can learn the rhythms of language. I’ve pointed out to my daughter why it’s tough for me to read the Magic Tree House books out loud to her and she can now pick up on parts that sound awkward. (It wasn’t a lesson, just an outgrowth of a natural discussion. Which is probably the heart of unschooling: just talking naturally about things that happen along. Despite the fact that I’m not a great talker, some amazing things have come up in conversation.) It has probably inadvertently sowed the seed of her being more conscious of there being a range of how well written things are. She would have learned that anyway though perhaps unconsciously.
(That “happen along” part of unschooling is misleading. It’s not that I’m leaving things to chance, nor am I deliberately bringing something in as a lesson. I direct a lot of things her way and just from experience know that from the wealth of things, there will be unexpected learning. Nothing I can plan though. She learned more than anyone would imagine from a few weeks watching Gilligan’s Island. 😉
Writing is just talking on paper. You’re trying to see where someone mentally is relative to where you intend your words to take them and then you plan out a course to get them there. Talk to your son. Ask him to explain what he’s doing and ask questions to help him learn to order his thoughts and learn to see from the point of view of who he’s communicating with rather than from his own position. (But only ask if you’re interested. Kids have good radar for lessons masked as conversation 😉
Unless someone has gotten the idea that writing is hard by being forced to write before they are ready or need to, or being forced to write in ways that aren’t natural to them, once they realize it’s just talking on paper, that little extra step is hardly any step at all. There are additional skills they can learn, like how to organize their thoughts for something longer, but it’s not a skill that needs 12 years of practice. (A schooled friend of my daughters came over to play with my daughter and they decided to make books together. The schooled girl told her there were all these things you had to do: title page, a plan, and some other things. My daughter said “Oh,” and just made books. The schooled girl never did finish. Merely an anecdote that may mean nothing, but it is a piece of data.)
I think it only takes years to learn to write when people are forced to write things they don’t care about. Where does most writing practice end up? In the trash, right? Real writing should make a difference in people’s lives. Sure there’s project reports and documentation to write, but do we need to force kids to write boring stuff so they’ll be prepared to write boring stuff?
High school is when it’s more common for kids to feel the need to put words on paper. But, again, they need real reasons. Perhaps letters of complaint about a product, letters to the editor, a family newsletter, a pen pal, email, message boards, an article for the local paper, or one of the websites out there that kids can submit their writing to.
But many of these things can be “laying around” for him right now, suggested when it’s possible he’d be interested. And dropped when he’s not or carried as far as his interest carries him. As long as he sees writing as purposeful, then there won’t be anything other than natural barriers between him and putting words to paper.
>>>>> 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.< <<<<
Joyce: 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 test.
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? < <<<<<
Joyce: 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?
>>>>> If I tell my wife that I want to try this unschooling approach starting tomorrow, then what we would do at 8 AM? < <<<<
Joyce: Sleep? Eat? Watch TV? Go outside and enjoy the sun shining through the trees? Read a book?
>>>>> Would my son choose when he gets up? >>>>>
Joyce: Unless he stops breathing, he’s always weighing his options and making choices. They may not be the choices you’d want him to make. But, what if you knew your wife had an agenda for you and there were “right” choices in her eyes and “wrong” choices and you knew she was weighing the choices you were making against her idea of “right” and “wrong” and judging the quality of your choices? How would that affect your relationship? I assume there are some things you each do to please the other, but they are *still* choices. The more pressure someone feels from the other to make the choices the other wants them to make, the more strain there is in the relationship.
>>>>> Would he choose what he wants to learn? Should we let him mess with the Star Wars games on the computer all day? I am going to go out on a limb and guess you would say that he would eventually get bored and look for something else to do or that I should keep offering interesting tidbits he couldn’t resist? < <<<<
Joyce: Yup. If he’s interested, he’s learning. It may be hard to see how what he’s learning relates to what is “important” in life. In fact, it may only be relevant to his life right now. But it is relevant. It’s nurturing the person he is now. I think we concentrate too much on moving kids along to what they should become and preparing them for that.
>>>>> What if he says he never wants to do writing ever? < <<<<
Joyce: Well, what if? There’s plenty of professions where people don’t need to write. But do you really think that if he loves something that he will choose something else just because he doesn’t want to write?
>>>>> We just wait him out until he thinks he needs it < <<<<
Joyce: And why shouldn’t it be important that he write when he thinks he needs it? Why should it be more important that he write when you think he needs it? Wouldn’t that mean when all kids hit 12 months we should make them walk because that’s when kids need to walk, and we all know how important walking is so they should get started when we think it’s important? Unless there’s something physically wrong with them, or their environment discourages it, all kids do eventually learn to walk just because they feel the need to.
If someone made me write an essay on math and kids, it would be as short as possible to make them go away. But since I’m writing this “essay” to satisfy my own need to get all these thoughts in order, it’s as long as it needs to be for me.
>>>>> 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? <<<<<<
Joyce: Lecture? Ick. How important would you feel something was if your wife decided to lecture you about its 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. <<<<<
Joyce: But I do have the advantage of seeing the same math being learned naturally way easier than it’s being taught and learned in school. I have the advantage of reading other people’s kids’ experiences with unschooling math.
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.
>>>>> I need to read a book about the day and the life of an unschooler in my spare time.<<<<<
Joyce: Actually a day in the life of an unschooler looks a lot like summer days and weekends for other people. Unschooling isn’t so much in what unschoolers do as in their attitude towards life and learning and how they’re intertwined. Our conversations are our lessons without being lessons.
Every time my daughter spontaneously asks a question or tells me about an observation, that’s a “test” that shows me unschooling is working. She may not be learning a set group of facts that others think are important and can test, but her questions and observations show she’s thinking about what she’s learning. For example, it’s not so important that she learn that sound waves bounce off things because that can go in as a factoid without any real meaning or understanding behind it, but it is important that she bounced a ball off a wall and said that was like a sound wave. She’s making connections.