The abacus

In my other rant about math enrichment I think I mentioned that there isn't a whole awful lot to enrich at the arithmetic level. For the most part I say arithmetic is a matter of learning the mechanics and practicing them until you can use them for the fun stuff later. Enrichment is something more like sneak peeks at what's to come, with one great big exception. The abacus.

Woohoo!! abacuses! Or abaci. I'll go with abacuses although in real life I just avoid referring to them in the plural... which is difficult when you have such a collection....

This one is just a counting frame, ten beads per rod. I made it out of foamcore board (check the art supply store and make sure your xacto knife is good and sharp!) and if I could find the picture of it half-made, you could see that the wires serving as rods are wound around popsicle sticks inside two layers of foamcore. The sticks are jammed into carved-out hollows in the foamcore and then all taped together. It's definitely of the "jerry-rigged" school of arts and crafts, but at the time I had a little one who was this close to grasping place value and why a one in the tens place was different from a one in the ones place and I used what I had.

This abacus is perfect for that one lesson (and fine for several others I'm sure, but just exactly right for this one!) Start with a number, say seven, and slide over seven beads on one rod. Then count up... eight, nine, ten as you slide over each bead to match. When you get to ten, you have one rod full, and no spare beads. That's 10 -- the one and the zero. You can count up to fifty and point it out all the way up, and then you can work with adding and subtracting across a full ten -- like 8 plus 6, 11 minus 3, 5 plus 6, etc. Try counting up if it's confusing for the kid, but then try asking "how many beads do you need to make a full ten? and how many more than that are you adding? (or subtracting?) So 8 plus 6 can be thought of as 8 plus 2, plus 4. And 11 minus 3 is 11 minus 1, minus 2. Play with it, give each other problems to solve, and try thinking in terms of getting to ten, backwards and forwards.

Next! This one was just for fun... We were studying Ancient Egypt and their counting system, and just couldn't resist. This one we had a little more time on, and help, since the aforementioned child was not in desperate need of any mathematical illustrations and was old enough to be a bit handy. We used a cheap picture frame from the craft store, flexible beading wire from the bead store (everyone has a bead store, don't they? Well try the internet - the type of wire is really not terribly important, but I like the flexible stuff, and with crimping beads it holds up to use rather well.) The beads are made of Sculpey clay. This part was my job -- he may have been handy, but sixty identical blocks of clay was still a bit over his head, not to mention the risk to little fingers trying to make bamboo-skewer holes. I made the blocks, we baked them as directed on the Sculpey package, and then drew the various hieroglyphic symbols for the numbers -- 1, 10, 100, 1000, 10000 and 100000. When you get to a million the number symbol is a scribe throwing his hands in the air. I'm not making this up! We accidentally strung the beads upside down, so you read it from the bottom, so in this picture it's showing 103,264. If I were to do it again, I'd only make nine beads per rod, since you never really need the tenth.

This one is my favorite, and the one we actually use on a regular basis. It's our grocery store abacus. This abacus is modeled on the Japanese soroban, which is different from the counting frame in that the beads don't all have the same values. Below the reckoning bar (in this case a pipe cleaner...) the beads are 1s in whatever place they are (1s, 10s, 100s, etc.) Above the reckoning bar they're 5s (5s, 50s, 500s, etc.) For the grocery store, we use the darker bead rods (those two on the right) for "cents" -- that is, ten cents and one cents. Then the lighter bead rods (three rods on the left) are dollars (100s, 10s and 1s, reading from left to right) So if you were to write an amount, say $142.68, it would read exactly the same way on the abacus. This is true of the regular soroban too, but the type of soroban one usually buys is not made with pipe cleaners. Of coures the type of soroban one usually buys is made with a nice flat non-moving desk in mind. Not a grocery cart. The pipe cleaners keep the beads from clearing every time you turn a corner, and are therefore much less frustrating if you're a kid in a grocery cart, assigned to add up the bill for your mom. Why the hundreds place? Well we used to have a membership at Costco...

One thing you figure out with a soroban-type abacus (by the way, if you want to make one for the grocery store, the frame in this case is a piece of mat board. If you don't have a mat cutter, you can get them done at a framing shop, or maybe even pick one up cheap from their mistakes bin) is that now you've moved up from "how do you get to ten" when you add and subtract, to "how do you get to five?" Subtle difference, but one of the things I really adore about this type of abacus. It's efficient in its use of beads, matches rather well with our own monetary system (pennies and nickels, $1 bills and $5 bills, dimes and... okay we don't use a fifty cent piece much anymore but the idea is still good!), matches even better with the Roman numerals that you learned in elementary school and still don't know why they matter... but wait! there's more! Look at your hands! Four fingers and a thumb? Ringing any bells here?

Your fingers are the ones and your thumbs are the fives. So each hand is like a rod of an abacus, and you can count up to 99 before you run out of fingers to count on. Check out Fingermath by Edwin Lieberthal for details, or google "Chisenbop" which is apparently a Korean word meaning "finger counting."

Got a couple more here.... This one isn't an abacus in the traditional sense, but it bears some similarity. It's actually a string of 1000 beads. What's it for? Counting to 1000. Is this a trick question? Okay we made it when DS was just catching on to the whole pattern of numbers, where 20-29 is followed by 30-39 and 40-49... and where after 100 you just start back with (a hundred and) 1 and go back through the whole sequence again... Once you've figured that out you can pretty much count to anything! So he did. It takes a long time to count to 1000, so we had those cards -- they're actually someone's discarded business cards -- on which he wrote the number he left off at, so he could pick up again later and finish. You'll note that it took five sittings to finish... The string is elastic bead cord and the beads are cheap plastic faceted things (if you need 1000 of something you're not likely to be going for the $5-each art glass... or at least I'm not.) I bought two boxes of multicolored beads and one box of red, and I only used reds for every 10th bead, so it was easy to keep track. I figured there couldn't be anything more depressing than counting to 999 and realizing that one had been off by one bead somewhere back there. This was what, five years ago? I still find extra red beads around the house. You've been warned.

Oh look! It's the rest of the red beads! You know they come in a box of like 700, and if I got two boxes of multicolored and one of red, I could only really have used 100 red beads in that whole 1000-bead project, which means 600 beads left over! So this was just a little thing we did. It's really as far from "abacus" as I can get without making a new post... but here's the deal: these are base ten rods. Each wire holds ten beads (to show the very-important "set of ten") and then there are some spare ones to use for the ones place. This is great for a couple things... first off, place value (which we used the first abacus all the way up there for) -- same general idea. Second, adding with carrying and subtracting with borrowing.. you can set up a number -- like what's shown in the picture is 38 (3 rods, 8 units) and add 24 (2 rods, 4 units) and show that now you have more than 10 units for your ones place -- you have something like "fifty-twelve". Ten unit beads from that twelve can be traded in for a nice neat rod, to make your fifty-twelve into sixty-two. And for subtracting, take your 62 and subtract 8. How do you get eight units out? trade the nice neat rod for ten individual unit beads (fifty-twelve again!) and then take out the eight.

If you're beyond the adding and subtracting stage, the other thing I absolutely couldn't have taught without these is long division. But that's more complicated... so I'll give it its own post instead of hijacking my own abacus soapbox for the purpose.

Mmmmmm.... Radioactivity

Alright everyone, get out your geiger counter.... You do have a geiger counter, don't you??? What? No? Okay nevermind that part. Get a big bag of mini marshmallows -- the freaked out pastel colored kind. You can't beat a science lesson that calls for a whole bag of mini marshmallows and who can resist a science project that only takes one trip to the grocery store?

This is what you're making. It's the nucleus of a Uranium-235 atom with a loose neutron lurking in the wings, just waiting to blow it all to smithereens. How do I know it's Uranium? 92 protons. In our model they're green... I would have liked them to be pink -- something about Pink Protons just appeals to the ear, but there was a misjudgement in someone's estimate of how many mini marshmallows might be eaten before we started counting out the ones needed for the model, and well... we were stuck. That or we'd have to go back to the grocery store and I already mentioned how much I appreciated that this project only took one trip! So my advice? Buy two bags... maybe three.

So back to the protons. Here's the big thing about protons. They are the one single thing that tells you the most about what you've got. If it has 92 protons then it IS, absolutely and positively, Uranium. If it's 93? Not Uranium. 93 is Neptunium. 94 is Plutonium and I have to admit here that I only just now realized that they're named after the planets... That might have been a useful hint back in 6th grade when I had to memorize the periodic table, but at my age it's probably only good for bar bets. But I digress.

The point is, if you had a molecule of Oxygen (8 protons) and you somehow managed to remove a proton, you would then quite suddenly have Nitrogen (7 protons) with all of its different properties. So 92 protons is Uranium. Now why Uranium-235? The 235 means that there are 235 total thingies in that nucleus. And by thingies of course I mean protons and neutrons (since electrons are all floating around outside the nucleus). So that means 235 mini marshmallows. This is why you should always look up the details before wheeling your shopping cart up the candy aisle and saying "hey, we'll just pick up some marshmallows and have a nice little spur-of-the-moment lesson on nuclear fission - I'm sure one bag is enough!" Two bags. Trust me.

So if there are 235 thingies, and 92 of them are protons, that leaves 143 neutrons. Now you see the pink problem. It turns out there are more pink mini marshmallows in every bag I've opened so far. I don't know if that's on purpose or not, random (we don't buy that many -- it's a really small sample size!), regional, or what, but not only are there more pinks, there are orange ones too, which look an awful lot like the pinks and you can fake it a little bit in that direction if you're running a little short. I personally think they're short of greens because the greens taste vile, but then I don't think I'm really their target demographic anyway. In any case. For our model we used green for protons and pink (or orange... just a couple) for neutrons.

Pile up all those marshmallows on a plate. Then get your hands a little wet and start squishing. You don't want a homogenous mess, you just want them damp enough to stick together, but maintain their identies as protons and neutrons. Once you have it reasonably stuck, take the spare neutron. This is the one the nuclear reactor fires at the uranium...

Let your enthusiastic kiddo do the firing (if your house is anything like ours this part is way more fun than anything else, even the part where they're allowed to eat a few subatomic particles). The uranium splits, more or less in half, and what comes out? More neutrons! Now if you had a whole class of kids you could illustrate a chain reaction. One neutron blows up one nucleus, which then releases let's say three neutrons, which if they're lucky run into three more nuclei, which split themselves and go on to release more neutrons, and soon you have a few dozen sugared-up children running around the lab flinging marshmallows at each other. Now you know what a nuclear bomb looks like.

One thing you have to mention, although I'm sure it gets lost in the fun of blowing things up, is that most things don't do this. In fact most Uranium doesn't do this, because most Uranium is actually not Uranium-235... it's Uranium-238. More neutrons (still 92 protons though!!) Uranium-238 doesn't go around splitting in half and flinging neutrons at its classmates. It's much more likely to just lose a piece here and there... a couple few parts here and there, even entire Helium neutrons (2 protons with 2 neutrons), which are called alpha particles, and electrons and positrons, which are beta particles, and which can make a proton into a neutron or vice versa. Of course if you're changing the number of protons (which each of these bits falling off is doing) then you don't end up with Uranium in the end. Actually it goes through a lot of different steps before it ends up as Lead. This is the whole half-life thing that radioactive elements go through. They lose pieces at a reasonably predictable rate, turning into other things on the way to something stable. What we were doing with the marshmallows is nuclear fission. Start with Uranium, end with two molecules of something completely different and about half the size.

That's a rather big difference.... and why you can still get old red Fiestaware with the Uranium glaze. Now I'm not saying I'd eat off them myself (although my grandmother did for years and lived well into her nineties), but your vintage salt and pepper shakers aren't going to blow each other up in the china cabinet. They will however lose enough particles to keep your geiger counter clicking.

If I had to homeschool on a desert island....

Nevermind the perfect curriculum, the magic math-fact-memorizing trick, or the schedule that somehow makes everything happen. They don't exist. But who cares about them because Santa (well okay Grandpa) brought a Digital Blue USB microscope, and what could be more fun than anything made 200x bigger. If you click around Amazon you'll find a few different versions under slightly different names, and three types of reviews: either "I love it and can't live without it" or "it's junk" or "the guy who says it's junk doesn't know what he's talking about".

Is it a professional quality microscope with everything you'd ever need? Of course not. But you know what? A four year old can use it. It's plastic so when you get frog guts on it you can squirt bleach all over the place. It plugs into the computer so you don't have to squint to see anything and it's easy to point at one thing or another, it takes pictures and movies, and it's super-easy to set up (and leave up!) so you can throw anything under there anytime. I have more pictures of 200x-sized ants than anyone could possibly want.

In deference to the insectophobic or otherwise squeamish, here's a picture of snapdragon seeds instead. Because you know, gigantic ants really aren't everyone's cup of tea. And don't even get me started on frog parts... but oddly enough the kid set really goes for that sort of thing, and if you set them loose with something "icky" and the means to take really really high-magnification pictures of it, you get all kinds of neat stuff, and a whole list of people who refuse to visit your photobucket page just on principle.

Speaking of principles... We don't have any qualms about dissection in general here (although we do generally stick with invertebrates until we get past the "hacking" stage in learning to handle a scalpel), but for anyone who wants the experience and something to throw under the aforementioned microscope without sacrificing animals, flowers make excellent dissection specimens. Lots of parts to look at, no guilt even if you end up chopping the whole thing into tiny bits, and you can get excellent specimens from the clearance table at the local nursery or wherever it is you buy plants. The snapdragon whose seeds are pictured above came from Home Depot and cost a dollar. It looked pretty sad, as plants go (being an annual, and right at the end of its blooming season), but it had both flowers and seed pods to compare, which was really very cool.

Math Enrichment

I have to say I'm not orderly about enriching math, so this is all very scattershot and random, but someday I aspire to orderliness..... Right now this is working for us. Also just as a general thing, I should admit that I have very little interest in enriching arithmetic. As far as I'm concerned, arithmetic is arithmetic and (provided you are really understanding it) if you can get it done fast then definitely do so. There are some things that will contribute to the understanding (I'll save my essay about the abacus for another post), but it's only once you have that basis that I'm really interested in all the options, because SO MANY suddenly open up when you have the tools to figure through them.

1. Lots of books... I really like Martin Gardner's books of puzzles, at least in part out of nostalgia, since I read them as a child. And there's something about learning card tricks and calling it schoolwork that lets a kid think he's getting away with something. George Gamow wrote about math and physics in fun and interesting ways (very dated, both in the style and the content, but I like them anyway!), Lewis Carroll, The Number Devil, Flatland... someday I want to read Godel Escher Bach, but I haven't gotten to it yet.... That sort of thing. We don't actually read any of these on any schedule or with any kind of linear progress... they just get picked up and put down and found and discussed and whatever. I've only flipped through Mathematics: A Human Endeavor at the library, but it did look good, and I really do like Jacobs' books in general. TOPS science has some math kits too, but I've not used them myself (I do like their science though!)

2. Proof... Anything worth doing (mathematically) is worth proving. So even with arithmetic, if you are learning about dividing by a fraction, can you PROVE that it works in every case (multiplying by the reciprocal). I print out "proof paper" in the two-column format just because I'm such a geek, and we write it up with steps and justifications. Once you have a couple rules under your belt you can put them to really good use. Of course geometry is the classic application (and we have our copy of Euclid and a study guide, and a really good pointy compass to go with it) but Algebra can be just as "proofy" -- we have a copy of Algebra by I.M. Gelfand that I adore.... Very thin but VERY dense, and it does two things for me. First, it asks for proof -- will this work for all values of x? And second it dumps you right in the middle of seemingly impossible problems and asks that you just start digging your way out. We do that one more-or-less collaboratively. .. insofar as he does the work but we discuss it at great length, before, during, and after.

3. Statistics -- This isn't really enrichment as much as it is just my little pet peeve... statistics education is severely lacking in the standard curricula and no one is getting out of my house without understanding enough to read original scientific research and enough so you'll know, when the news report cites a scary-sounding number whether you really need to be terrified, or amused, or just sensible. Annenberg has a telecourse called Against All Odds, which I like. The textbook is cheap because it's a few years out of date, and we got some extra books to go with it that set up the lessons and give extra exercises in using Excel and SAS. Someday we'll buy a SAS student edition and DS will find out how much he could make as a programmer and I'll have a heck of a time convincing him that he needs anything else to make it in the world... so maybe I'll put that off for a while.... But we're surrounded by statisticians among our friends, so it's not just practical education; it's necessary to keep up with the dinner conversation.

4. Random other stuff... Every so often something just comes up. We listen to a CD in the car and go off on a long tangent about rhythm and fractions and meter in poetry and somehow I end up at the sheet music store... Origami makes for interesting topological problems... There's a whole long list of math stuff related to handiwork -- knitting and crochet can be fantastic if you're discussing non-Euclidean geometry (google crochet hyperbolic plane, or check out this article), and there is a Japanese decorative craft called Temari that's perfect for geometry on spherical surfaces. Spanish blackwork embroidery has a lot to do with fractals. Whatever comes up can frequently be expanded on: when you learn about pi there are dozens of ways to try to find it yourself (and learn about irrational numbers and approaching a limit at the same time!) We just had a simple statistics problem that took on a life of its own when I asked DS in the car what the chances were that I'd blindly grab two identical jelly beans from a bag of 50% red and 50% yellow... and how it changed if you started out with 4 (2 of each) or 4 million.... Opportunities present themselves all the time.

Timeline

I remember absolutely nothing from my own history education before high school except for one soap carving of the Parthenon, and from what I remember of that project is that it's a darn good thing that I've never had to depend on soap carving to make a living.

I don't know why history gets the life sucked out of it, but my cure has been to add a much more interest-led layer of projects. Pick something - anything - and find out its history. It doesn't have to be the Liberty Bell, or anything so textbook. How about the history of horses? or just the history of wild horses in the outer banks of North Carolina? History of Legos? Rocketry? Hawaiian shirts? Brain surgery? Poodles?

My only rule for history projects is that they go on the timeline. We plod through our regular studies (and really they aren't that bad) just to give a framework for everything. Ancient Egypt was before the American Revolution. Pompeii was before the Magna Carta. So once we've been through once, all those projects can have a context.

I printed my own timeline book on ledger sized paper (65lb coverstock so it stands up to use), 3-hole punched and folded so each page makes a kind of pocket for extra bits and pieces (coloring pages, reports) If you want to do one just like mine, I've uploaded it to lulu.com as a free download. It requires 141 sheets of 11x17 paper and at least one 4" binder. I've split ours into two 3" binders because we end up sticking a lot of extra stuff in the pockets and it keeps getting bigger. I would have been lazy and just bought a blank timeline, but honestly I couldn't find one that actually went back as far as I wanted. I really wanted all 4.55 billion years so we could put "first bacteria" on the line just for perspective...

The Basics

I can type for eight billion years, babbling on and on about knitting, math and science, homeschooling schedules, favorite books, television, movies, the pros and cons of standardized testing, but faced with a great big blank blog... well I've been sitting here for two hours now staring at the screen and going uhhhhhhhhhh. Mind you I was also watching TV, eating a candy bar (3 musketeers mint! woohoo!), changing the laundry, feeding the cat.... Okay so I'm typing now.

We homeschool, and in homeschooling circles one is expected to have a "style" or "method" that one adheres to, if only so that you can decide which messageboards you're going to frequent and which you're going to troll when you're bored. I don't have a proper homeschooling method... all the unschoolers think I'm terribly school-at-home and all the school-at-homers think I'm a lazy bum who ought to be able to stick to her own schedule. Our homeschool style is more along the lines of "Ooh look something shiny!" On the plus side, we have very interesting days -- you never know how it's going to go! But on the minus side I can't find our spelling book. Actually two of them. I think we had three at last count, and I'm sure they'll turn up when I least expect it. But going back to the plus side (I really like the plus side -- much cheerier!), one gets so many more interesting ideas when one is forced to think on ones feet, as it were.

There's this popular image of the homeschool mom, serene and organized, leading her little ones in a poetry study over tea... That's not us. Our homeschool is something more in the line of DS lobbing questions over the bunker at me like so many hand grenades -- "Well what was there before the big bang?" "All the numbers added up together can't be infinity because infinity isn't a number and if it's not a number it can't be the sum of numbers!" "Can I get a pet bird? I read a book about them" -- and me alternately ducking: "We'll look it up at the library next time we go" or firing back: "Alright smartypants, how about you just sit down and start adding up all the numbers and see if it doesn't take you an infinite amount of time, eh? And birds are noisy!"

So for all those metaphorical grenades flying around our house (explains the mess anyway) I'm sure a handful are duds, but there have been a few live ones, and I promise I'll share. Math and science are the favorites around here, but that won't stop me from weighing in on anything else that catches my eye.