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Pam Sorooshian's review and comments on the article "Mathematics: The Most Misunderstood Subject," by Dr. Robert H. Lewis of Fordham University
I wrote an article on How Kirby learned math for the German magazine Unerzogen. It appeared in the August, 2008 issue. 😊 Here is the original English text, and a link to the magazine's site, to that issue, and to a PDF of the article in German: Mathematics / Unschooler und Mathematik - Wie war das bei uns? —Sandra
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I am collecting some of my favorite commentary on learning math naturally, written candidly by real parents of unschooled folk at various times. Joyce Fetteroll wrote:
Wild math looks different than domesticated math It looks more like conversations, using numbers to figure out something the child wants to know, video games, allowance, weighing things in the grocery store, finding the best deal among several choices—that is not as a lesson but what you would normally do—board games, figuring out "how long until?" when she asks, budgets, doing a rough estimation of the items in the grocery cart to see if you have enough money, baseball statistics, crafts, origami, wrapping presents ...There's a list at "A Few Words About Math":
Look at the things on there that naturally come up in your life. Don't look at the ones that don't! It isn't a checklist to see how many you can complete! It's meant to show you that there's more math in life than school math lets you see.
Holly Dodd in her mid-20's, reported by me (the mom):
Sandra Dodd, March 13, 2017:"Your math isn't wrong, it's just NOT math. 'Wrong math' isn't math. You're doing something that isn't math. It's art or something. People can't do bad math, it's just NOT math."That was Holly Dodd, just now at my house.
I told her it's because she didn't go to school, where "math" is 10:00 to 11:00 and you sit and write on whatever papers they put in front of you to write on. That's math.
We were both laughing. She is somehow confident that "math" is something solid and sensible, like there are patterns beyond human opinion. She's surely right, but it's still funny.
This subject is near and dear to my heart, so it jumped out at me.
How do unschooled children learn long division? What do they do to practice it?
Linda Wyatt:What do unschooled children need "long division" for? And could someone please explain to me what long division actually is, and is there such a thing as short division?
I know how to divide. Do I use "long division"? Beats me. Quite possibly not, since my elementary school experience was at a "free school" where we didn't have the usual types of "subjects" that most elementary schols have. I didn't take a math class until I was 10 and decided to listen in on a group of kids learning algebra. (Which I immediately LOVED, and have ever since!)
My point is that it's important to understand the concept of division, but HOW you do that is not so important. There are several different ways.
I'm going to say the next bit in all-caps not because I'm yelling at you, but because I really, really mean it and I think it's critically important and most people have no idea.
I HATE THE WAY SCHOOLS TEACH MATH BECAUSE IT TAKES ALL THE MEANING, JOY AND BEAUTY OUT OF IT!!!
People do NOT need to learn math the way it is taught in schools. In fact, they don't need to "learn math" at all. Math is INSEPERABLE from most everything else in life, and if you live a full life, you'll learn all the math you need because you need it. It's there. It's part of everything. You couldn't escape it if you tried really hard.
SOME people will love it and get interested and want to pursue things to a deeper level and learn more advanced math and do all sorts of cool things with it. MOST won't.
If you try to force school math on kids early on, and turn them off the subject- a VERY common thing that happens because most adults are so terrified of math because of their own awful school math experiences- they will likely develop an aversion, and at any rate, they are extremely unlikely to remember what they've "learned" anyway.
How much of the algebra you were taught in school do you remember and use? I'm serious. What do you still use? What turned out to be irrelevant? What could you easily look up and re-learn if you needed to? How likely are you to need to do that?
What about algebra and other higher mathematics?
First, I don't think algebra is "higher math" but I'll let that go. I think it's mind-candy.
But what about higher math?
Who needs it?
That's a real question. What kind of people use higher math? What do they use it for? When and how did they learn it? You might want to find some and ask them.
What I've found is this: most people who use higher math learned some in school, because that's where they were. But MOSTLY, they learned it through using it as they needed it. Like on-the-job training. They took it further than school does because it actually interested them. They had a use for it.
I remember the day I realized that calculus had a purpose. I was stunned! Why didn't they ever tell us this in high school? I had a rocky relationship with calculus in high school (along with a rocky relationship with just about everything, as most teens have!) but once I settled on a career path where I NEEDED it, it was suddenly easy and made sense and I relearned it all in a few weeks.
If I had not had that career path, I never would have looked at calculus again.
I no longer have that career, and calculus is unlikely to be part of my everyday life (although you never know) so I no longer use it and have forgotten most of it. Big deal. (Now, I use a lot more geometry.)
I'm reading a lot about the practical uses of what we learn, and I agree wholeheartedly BUT...the brain work that goes into learning and knowing how to do abstract number exercises like algebra, increases the ability of the brain to think about ANY subject.
YES!!! It does!
But it works the other way around, too. Learning abstract thinking any other way will make it easier to learn algebra, should you need to do so. Algebra isn't the only way to learn it. Just a fun one.
It's GOOD for the brain to "workout" by doing abstract stuff. And so, how would an unschooling family get algebra across to the kids? Or long division, for that matter? LOL
Well, to start with, we don't do "math" at all here. It isn't a separate thing.
But we use it all the time.
Like this:
Suppose your Dad is coming in the morning to pick you up at 10:00am.
You need about an hour in the morning to get up and shower and eat
breakfast, to be ready to go.
You need about 8 or 9 hours of sleep.
What time do you need to get to bed by?
(The cool thing about that particular need is that it includes using base-12 math as well as base-10)
My current favorite real-life mathematical figuring here:
We have a mini-van.
With all the seats in, it holds 7 people, and has a cargo space for two
bags of fencing equipment.
We have 4 people, and need cargo space for about three times what fits in
the back. Without the space, we had to shift bags of equipment in and out
of the car on a daily- or even twice daily- basis, to have what we needed
when and where we needed it.
I never made the shift from being 5 people to being 4 people, at least not
in a practical way.
One of my kids did.
He suggested one day that instead of shifting gear around and having to
move it in and out of the car all the time, why not take the back seat out,
have seating for 4, and then keep all the gear in the car permanently. Not
only that, but it could be organized so we wouldn't even have to shift it
around for any particular class, each thing could be reached individually.
He redesigned the layout of everything in the car.
We changed it.
A simple change that required a large conceptual leap-which he made, but the rest of us had not- and now, life is much simpler.
To really understand this and the effect it had, you should know that I teach fencing, 6 classes per week, plus I give individual lessons. Four different locations for the classes. Several different locations for the lessons. Different groups of people at each class, with different equipment needs. And all three kids fence, but they don't attend the same classes. I have equipment in the car for about 30 people, plus a variety of training equipment and record keeping stuff. We were really having an issue with having the right gear in the car.
Algebra is the art of taking the information you have, the things you know, and using that to figure out the information you don't have, that you need to know. It's a puzzle. That's all. You "expose" kids to it by doing it, by playing with it freely and uninhibitedly. By finding things fascinating and wanting to figure out what you don't know. By experimenting. By needing it.
I "make" my kids do their math...lying in a puddle of sunshine on the front room floor, wrap-ups in the recliner, Math Minute drills, etc. Right now, the two younger ones are playing dominos...MATH!
No, it's playing dominoes. Why not let it be that?
Why separate math out of everything all the time? Why do people do that?
The way to learn math naturally is to let it be a natural part of everything, like it is, and not make such a point of it all the time.
For example, what if, in writing and speaking, you had to stop and make a point of every time you used the letter "e"?
Look, I'm writing an e-mail... USING THE LETTER E!
How ridiculous is that?
Yet people do it to math all the time.
But would unschooling REALLY say not to give them seat work to practice their basic number skills, and on a consistent basis?
I wouldn't necessarily say it, but I also would NEVER do it. SEATWORK?!???
What on earth for?!?
And who came up with a term like that anyway? I find the very concept nauseating, I really do. Sit here and do this. Whether you like it or not. Whether you have any use for it or not. Stay in your seat, do your seatwork. YUCK.
This does not mean that we don't have a zillion math books around here, including workbooks and such. We do. I probably have more math books than most schools do. But it's because I enjoy them. I NEVER make or even ask my kids to have anything to do with them. It's my special interest, not theirs. Once in a while, they'll pick one up, look through it, put it back down. Sometimes, they'll use computer software that has math games on it- more so when they were very young, not so much now. But I have never and will never require it.
I will, however, discuss with them when I'm working on trigonometry for fun. It's interesting to me, and they might also think it is. So far, mild interest, no real excitement. That's fine. They know it's here. And they know something about what it's for. They can ask if they need to.
The best thing for my kids having to do with math is that I love it, and they know it. No math anxiety here. (Except for when my daughter had an issue with girls in her scout troop telling her she's not learning anything because she's homeschooled, and she came home wanting to "do math" and we had a little talk about that.)
People learn best by having resources available, and interesting things to do.
Learning is interest-driven. Joy-driven. Need-driven.
But please don't make it a command performance. ESPECIALLY with math, which is already handicapped by a national phobia about it.
The best thing you can do for your kids with math is to learn to love it yourself. Play with it. REALLY learn it. Not just how to calculate, but why it works that way. Why does borrowing work? Why does carrying work? What is the real meaning of "place value" and why? Why do we use base-10 math. Learn some other system. Learn to multiply in binary. Play with numbers. Play with shapes. Why are "odd" and "even" called that? Is it important? How many math words do you know? How many math words do you use every day? (Count them! Make a list!) How much math is there in your life that you are so comfortable with you're not aware of it?
And my very favorite math question:
How, exactly, do you know when it's safe to cross the street?
:-)
Play with patterns. Play with sets. Go outside and throw rocks and pay attention to the paths they travel. Drop stones into a pond and watch the ripples. Figure out why buildings don't fall down- or why they do. Ponder why the wind off Lake Michigan travels through the city of Chicago the way it does. And Oklahoma, where the wind comes sweeping down the plains... what's different in very windy places? How do you need to change things to accommodate that? Or other weather? Why are most of the roofs in places that get a lot of snow not flat?
I could go on and on and on and on. You can, too.
Question everything. Figure some of it out. Don't worry about whether you are learning math or not. You are. It's there. It's ALWAYS there.
If I waited for them to come to me asking how to divide...well, I sure wouldn't hold my breath!
Why would they not ask you?
If they needed to divide and couldn't figure it out, they woudn't ask for help?
Or is it that you believe they'd never need to?
Sometimes, we buy a box of popsicles. In our family, treats like that are portioned out by shares, so each kid gets a fair share. They're into this. But the youngest used to have to ask me how to figure out what her share was. She very specifically DID come to me and ask me to help her figure it out- to learn how to divide.
Then she thought it was cool, and figured out shares of practically everything in the house.
But NONE of my kids, as far as I know, can do division on paper, using the method taught in schools. They've never needed to. They do it in their heads. They estimate, when that's good enough for what they need. They use calculators.
And honestly, that's what almost everyone does in real life when they need to divide big numbers anyway, because few people trust their own figuring enough to use it without checking. I used to freak people out when I ran a food buying club and did all the invoicing and such by hand, no calculator.
Truly, just wanting to figure it out.
And that's what it's all about. Figuring stuff out that you need to know. Remember that.
Well. I kind of got going on this, didn't I?
Linda Wyatt
http://www.lightlink.com/hilinda/index.shtml
Learning everywhere, all the time
Algebra Before Breakfast
Linda sent an update in 2008 and there's a 2014 update below that!
In March, 2008, Linda wrote to ask me to ask my sons about a video game her son was considering. This came from that correspondence, and I was eager to add it here!
Just got home and spent a few minutes looking around on your site.Found some of my math stuff that you have posted prominently there (wow!)
I'd like to add a comment to it.
I wrote that in January 2003, just over five years ago.
Since then, some things have changed. But not everything.
We still don't "do math."But one day, a couple of years ago now, my oldest son, Simon (he's now 20, so he was around 17 or 18) was sitting across the room from me. He was using his computer; I was using mine.
He asked me a question about prime numbers. I don't recall exactly what it was.
Turns out, he was trying to create an algorithm to calculate prime numbers, and then to display them on his computer. I don't know why. For fun, I guess.
His method of display had something to do with a circle, and with sections of the circle, and he didn't know how to figure something out that he wanted to do. Again, I don't recall exactly what.
But basically, he needed to be able to figure out something about triangles.Oh, I said. Triangles are cool. Let me show you something.
So I showed him how to use basic trigonometry to figure out whatever part of a triangle, be it an angle or the length of a side, that he didn't know. Spent about half an hour, maybe, showing him what the trig functions look like on a graph and how, in a triangle, or a circle, you can see how they relate to each other. SOHCAHTOA. (https://en.wikipedia.org/wiki/Trigonometry_)
He thought this was very cool.
Which, of course, it is.
He asked if I had any books about it. And, naturally, I did. I gave him one.
In about two hours he taught himself more trigonometry, with a better level of understanding, than I gained in a year of school classes. Two hours. I wasted a whole year!!!Another time, he did some thinking and playing around and came up with something he thought was very cool. Turns out, it was Pascal's Triangle. (https://en.wikipedia.org/wiki/Pascal's_triangle) But he didn't know how to express it, how to write it down in any other way, and he thought there should be one.
We talked about factorials. I half-remembered how to notate a series. He asked if I had any more interesting books that would show this, and I gave him a calculus book. Not a particularly good one, but the one I had on hand.
He read it.
Learned about functions and derivatives and all sorts of yummy stuff.Started getting together with a friend who was taking calculus in college- to help her with it.
So.
In 2003, I wrote this: "I will, however, discuss with them when I'm working on trigonometry for fun. It's interesting to me, and they might also think it is. So far, mild interest, no real excitement. That's fine. They know it's here. And they know something about what it's for. They can ask if they need to."
Five years later, one of them HAS shown great interest. Has since taught himself more math than I am ever likely to need to know. And when we had a house fire, when we were allowed into the house to retrieve whatever could be salvaged, the first things he rescued were his math books. (We had to buy replacement dictionaries!!)
The other two kids?
No real interest.
At least.... not yet. :-)
Linda
So here is another update, from Sept 2014My kids are now 27, 24, and 21.
The oldest, Simon, who got excited about trig on that day all those years ago, now works as a paraprofessional math and science tutor at the local community college.
He decided he wanted to take some college classes at some point ( I don't recall exactly when) and he went and took the GED exam and the SATs cold, without any preparation or studying, just to see what was on them and how he would do. I didn't know about this until afterwards; he didn't ask for my help. He did extremely well on both.
He started classes at the community college, and I found his approach to that to be very interesting. First, he took a bunch of placement tests. Then, even though he "placed out" of some of the classes, he chose to take them anyway, because he was curious about how they would be taught, not having any experience of being in classes like that before. He quickly became very popular with all the professors, because he actively participated in class, asked lots of questions, and was more interested in the material than in the "social life" there.
Then he found that he was interested in more things than could fit into a normal class schedule, since several times, things he wanted to do met at the same time. He went and talked to the various professors, and was given permission to take simultaneous classes, going to each half the time. I had never heard of someone doing this before.
He took nearly every math class offered at the college, including a very interesting class designed for prospective public school teachers, about how to teach math. Interesting both because some of it was potentially useful (like how to figure out what misconception someone is having based on precisely WHAT "wrong answer" they give, sort of a "backwards engineering" concept) and some of it was horrible (relying on timed drills, class management, etc).
He also took advantage of the wide variety of opportunities, including a biology class that included a trip to Costa Rica, some hiking, backpacking and canoeing classes, and whatever he felt like doing for fun. He found some excellent teachers (who are now friends). He met his now-serious girlfriend in a philosophy class because she was one of the other people there who was willing to argue.
He didn't feel constrained to taking what the college "required" for a degree, and simply took classes he liked. He ended up with enough of the right distribution of credits that they gave him a diploma, but it wasn't a typical program or time frame, neither of which was important to him.
He was recruited by the tutoring center, and became first a peer tutor (while still a student), and then was hired as a paraprofessional after he graduated. He really likes working with the remedial students, because they come to him with so much fear of the "hard" stuff, of math and science, and he really enjoys helping them come to a better understanding so they can let go of that fear. His goal in working with them is understanding the concepts, rather than just passing tests. He tutors for all of the math and science classes the college offers, as well as various computer classes and a few odds and ends. Some of the things he tutors are things he has never taken a class for, but learned on his own. He also helps train the peer tutors. What started as getting together to help a friend with her calculus homework, became a career.
My middle child, Tim, is a musician, writer, photographer, and deep thinker. His interest in math has never been in a written form, math-for-the-fun-of-math, that his brother and I enjoy. It is an intrinsic part of everything he does. He has occasionally had some concerns that maybe he "should" know more math than he thinks he does, but whenever we've sat down and looked at it, he hasn't ever had any trouble doing whatever he's needed to do. It's a funny thing, in a way, how often kids who never go to school sometimes feel pressured by cultural things, just like many unschooling parents do, but in our experience, it has been only that: a culturally induced concern that turns out not to be validated.
Tim has not ever taken an academic class in anything. At all. He has no way to compare his experience learning without any formal teaching, to learning with it, because he has only ever experienced learning on his own, except for swordsmanship. And even that, though it is formally presented, is really learned through his own practice. I think he may take some classes, somewhere, at some point, which would ease his mind about the whole thing, but so far, he has been busy doing other things.
My daughter, Sarah, HAS had an experience that allows her to compare. Her first academic class experiences came in the form of firefighter training, and then in emergency medical training. She had no trouble with either, being at the top of her classes. Then she decided to take a higher level medical class, in a college setting. One involving a lot of anatomy and physiology, biology, chemistry and math- none of which she had ever specifically studied, formally or otherwise.
She was a little apprehensive, being the youngest in the class, with the least experience.
She rocked it.She was one of the few people to do well on both written tests and practical skills.
What she discovered was, I believe, very valuable to her.She expected everyone else in the class to have an easier time, since they HAD taken math, chemistry, biology, etc, in school. All of them had graduated high school, some had college degrees, and all had more medical experience.
But what she saw was that a) they largely had really negative attitudes about both the materials and about testing, neither of which she shared, and b) their performance did not indicate ANY "advantage" from having taken and passed all those classes in school. They didn't remember any of it, and experienced significant stress about all of it. She, on the other hand, really wanted to understand it, and also found that she HAD learned a lot of it, over the years, as part of other interests.
Once she realized that, she was no longer apprehensive at all, and really enjoyed the class. She is now planning to go to nursing school.
To summarize, all three have had no trouble whatsoever with math throughout their lives so far.
Two have found considerable advantages in not having had a standard public school experience of math.
And all three still have very different interests. :-)
Linda Wyatt, September 2014
Where now? Links are up above.
