Tricotomania and more

ALERT: If you have come directly to the page for this post (say, from a feed reader) you are going to get the whole thing. It includes solutions to the problem. I tried to put this after a break that you would have to click through to get but that only seems to work if you read it on the main page. Click here to go there now if you aren’t sure you want the solution.

Yesterday AnimalGirl came around to tackle this problem with Tigger, which I blogged about a couple of weeks ago. I thought I’d provide some detail of how they worked on it because I know some folks are interested. First go look at the problem. The girls read through the problem and then focused on the map. They weren’t sure where to start but one of them suggested working out what all the possibilities were and going from there. This was probably the best thing to do and they worked out what all the possibilities were pretty quickly and made a decision about where the princess should go. I thought that looked like a reasonable solution but something was bugging me that I couldn’t quite put my finger on. I know that there is a solution provided on the nRICH site, so I went and got that for them. (There is a link at the top of the page I linked above.) Their solution was somewhat different from what the girls had come up with so the 3 of us tried to work out how and why they differed. In the end, they agreed that the other one was probably a better solution (and thought they should change their view of which room the princess should go in).

I pointed out that the difference between the two rooms wasn’t that great and talked a bit about how probability problems often come up with answers that require some judgement. Whichever room you choose, there is still a pretty good chance that the poor peasant is going to get eaten by the lions. This problem didn’t take them very long but they enjoyed it and it did make them think.

For those who want to know what their solution was and how we differed from the published solution, I’ve got that after the break.
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This is not a Poetry Friday post. I love the idea of Poetry Friday but I freeze completely at the thought of posting something. And I have to admit that I tend not to read the poems folks on my regular blogroll post on Fridays. I have some kind of aversion.

Literature is one of those subjects that I never “got” in school. You know how some folks say that they got good marks in math but feel like they don’t know anything about it? That’s me with literature. I read voraciously. But I never understood what was going on with literature class. Still don’t. I love Deconstructing Penguins but have absolutely no confidence I could guide that kind of a discussion about a book that isn’t already in there. (Seems I was going on about similar concerns this time last year.)

So. Poetry. Not sure how I didn’t get clued into it. I read a lot of A.A.Milne as a child and enjoyed it. I remember enjoying Dennis Lee as well. But I never really moved beyond kids poetry. I’ve had occasional forays into contemporary poetry — Jackie Kay, for example. And Mat got me interested in Attila the Stockbroker. But mostly I don’t know where to start and find that reminder that I haven’t got a clue to be kind of paralyzing.

One of the things I like about homeschooling is the ability to learn along with your kid. I didn’t know anything about art until Tigger got interested in it, either. So I figure this is my opportunity to figure out poetry. Or at least learn to be comfortable with it. I love Julie’s idea of teatimes. But having a teatime when you have only one child feels a bit weird. I’ve hosted one before with a few of Tigger’s friends and everyone loved it but that never became a regular thing.

No more. Our homeschool group has just had a meeting to work out what we’re going to do this term. I said, I’d like to do a teatime about once a month. So that’s what I’m doing. We each host 3 sessions between mid-September and mid-December and mine are pretty evenly spaced (and in the calendar). My plan is to go to the library and stock up on poetry books from the kids section so that I have a selection of things on hand when the kids get here. And bake. And make sure we have some nice drinks in (even for those who don’t want tea). And then we’ll just see what happens.

My first teatime is September 23. I might get some fall themed stuff. And I need something that our one lone boy might like (he brought a hockey poem the last time) so he isn’t overwhelmed by fairies or something. And I need to think of something to read to them while they have their tea…

Shaun has occasionally posted about the difficulties of educating her profoundly gifted daughter. One of the reasons she homeschools is because she just couldn’t get a school system based primarily around age to work for her daughter, particularly when she needed to be accelerated more than 2 grades in some subjects (which seemed  to be the limit the school would consider). Now Tigger is not really in the same league as Violet but she is pretty bright and if I had pushed I probably could have got her into the gifted program with our board. But homeschooling works much better. And one reason is because we can just work at the level that seems right without really worrying about levels. Since she doesn’t even really like textbooks, this is even easier because levels just never come up.

But Tigger is also a really social child. She loves being with other kids. And she loves learning from other people. So I like to sign her up for things that get organized on topics that interest her. Back in May/June she did a science class. It was loosely based on the Grade 9 curriculum and advertised as for 12-15 year olds. I knew the mom that organized it and talked to her about it. She wasn’t sure what the teacher would think but thought Tigger was probably at that level (from previous interaction) and said that if everyone who signed up was at the top end of the age range maybe not but we’d see who else was interested. In the end it was pretty hard to get the minimum number together and there was an 11 year old also interested and Tigger did it. She was fine. I had to find alternative readings a couple of times or go through the reading with her but she grasped the concepts well and did great in this lab based class.

At the end of that class the teacher had said that there was another class he planned to offer in the fall if the kids were interested — cell biology with lots of microscope work (Grade 9 level again). Even though Tigger and this other kid were younger than his usual target age range he made a point of saying to me and the other mom that they were both easily capable of doing that class and he’d welcome them. So this fall, I contacted him and then took it upon myself to organize some people.

Of course it is easier to get enough kids in September than in May So I had more than enough and did it on a first-come first-served kind of basis. I sent an e-mail to those that were in and the teacher (so he could take over) and listed what I knew about the participants at the end. Some I didn’t know ages and those I did seemed to be several 12 year olds and Tigger. One 14 year old boy then told his mom that he didn’t want to do a class for 10 - 12 year olds. AAACK!

I dealt with it all but I felt personally really awful. Here I am trying to get an activity at a good level for Tigger and because she is younger it gives the impression to others that the class isn’t really Grade 9 level as advertised. The teacher by then had more information about some of the kids and we confirmed with this kid’s mom that there were older kids in the group and that the level was right for him and that he wasn’t going to dumb it down for younger kids. He’s back in. Thank goodness.

But I ended up feeling like you can’t win. The age to level culture runs so deep that even homeschooled kids (and this kid is unschooled so I don’t think he does a lot of workbooks with levels printed on them either) immediately think that younger kids must be doing lower level work. Or maybe that it would be “normal” for a class to have kids all about the same age rather than having kids from 11 to 16 (as this class will). I’m sure he didn’t mean anything bad by it but it does seem to be an indication of how deep those assumptions go.

Which gets me back to my complaint that we, as a society, need a better understanding of statistics and probability. Because the variation in ability (all kinds of abilities) around the mean is significant for most things. Somehow we have a society that thinks “average” is where everyone should be and that both “below average” and “above average” are somehow “abnormal” in a way that needs to be corrected or reined in or something. This starts to happen from when we are measuring our babies’ progress in terms of “developmental milestones”. I have even met moms who misunderstand those growth charts thinking that if their kids in the 15th (or 85th) percentile there is a problem with their growth. Somewhere along the line we’ve lost the distinction between characteristics of a population and characteristics of an individual  member of that population. And it causes real problems.

My kid is well within the normal range of ability for a kid her age. But she’s 11 and capable of Grade 9 science. Her understanding of history and history of art is way beyond what anyone would expect of an 11 year old. But she still plays imaginative games with Playmobil toys and dolls and other “normal” 11 year old stuff. That’s probably normal too. Somehow we need to be able to recognize all of our normal kids, in all their variation, and help them learn. I would hope that classes with kids ranging in age from 11 to 16 were actually more common.

Whoever recommended Geraldine Brooks to me: Thank you! Year of Wonders was an amazing novel. I loved it.

Loosely based on the history of the village of Eyam in Derbyshire and set in 1665-66, the year plague hit that village, Brooks does a masterful job of bringing the characters and relationships in this time and place to life. The story is told from the point of view of Anna, a young widow who is a part-time servant at the rectory. We experience how the inhabitants of this village lived through a very trying time, imposing a quarantine on themselves to contain this horrible disease. The complexities of religious differences, class, gender, and education are all subtly conveyed in a gripping story. Brooks has mastered the language of this time and place, drawing you into life in the village.

I’ve already said that I seem to enjoy novels with rich characterization and this is another. Several of the characters are deeply and subtly drawn. And known facts of that period, such as the ways some individuals would blame witchcraft and scapegoat particular women in an attempt to eradicate the disease, are well described to give a sense of how people can be caught up in mob violence in stressful circumstances and yet others do not get drawn in. The main character is sympathetic and easy to identify with, at least for me.

The author’s note at the end of the book teases out the relationship to the historical facts in more detail. This is a fascinating addition to a book that stands on its own as just a good story. The insight into the author’s inspiration and process are very interesting.

Highly recommended.

One thing I love about blogging is how much I learn. I hope none of you were under the impression that I have everything figured out. I am a extrovert decision maker, which means that I need to talk about things with others in order to figure them out. And sometimes, just the word someone uses makes a whole bunch of stuff click.

“Probabilistic thinking” is a phrase Sarah used in her comment on my last post. (She also provided a link to a cool simulation so you might want to check that out.) I had been talking about how my general goal for math is to develop the skills needed to spend time working on tough problems and at least move towards a solution. But because the stuff we’ve been working on recently has been probability her response made me recognize a reasonable objective for the probability stuff: shifting to probabilistic thinking instead of “one right answer” thinking.

This ties in really well with the physics stuff I’d been reading because so much of physics relies on this way of looking at the world. As does so much of life, as I have also pointed out.

So now I have a clearer sense of what we’re doing and why. For now, anyway.

We’ve been working on more probability problems and this has forced me to recognize what the real goal of this approach to math is: learning to work for sustained periods on tough problems. The difficulties Tigger has faced in the past few days have not been about the math, per se. They seem to have arisen from the fact that the problem I have given her to work on is not one that she should be able to find “the answer” to in 5 minutes. And that even when she has been working for 20 minutes or more and doing good valuable work that contributes to finding the answer, I come an suggest other tacks she might take with the problem.

There has been some shouting, crying, and other frustrated behaviour. But we are making progress. We’ve talked about the importance of the process. About how math isn’t necessarily about solving easy problems in large numbers and getting all the answers right. We’ve talked about the ice-cream problem, how tough it was, the wrong alleys, and how we got to the answer. Also how good it felt when we figured out that formula after all those frustrating attempts.

Yesterday we said we’d put the problem away for the day and come back to it tomorrow. When she said “We’ll finish it tomorrow.”, I corrected her and pointed out that we might not finish it but we’d work on it some more.

Today, we worked on it some more. Together. She started falling into letting me do lots of the work and that led to some more serious discussion and frustration on my part. We talked it through a bit, went over the discussion of what the goals were, etc. We switched to playing the game the problem is based on. With dad instead of me.

Tomorrow, we’ll work on it some more but we might call it quits even if it isn’t “finished”.

Her friend is going to come over sometime next week to work on the Lady or the Lions. I am so glad she has a friend who thinks that coming over to do math together sounds like fun.

In the meantime we might work on the Birthday problem*. When I mentioned that just now, she said “But we know the answer.” and I reminded her that we don’t know why that is the answer. She thought for a moment and agreed that maybe it would be a good one to work on. Maybe we are making progress, slowly.

* How many people do you need to have in a room for the probability of 2 of them having the same birthday to be 1 in 2? The answer is 23.

Wow. iTunes now has a category for educational content. iTunesU aggregates audio files of lectures and stuff from various universities and other providers. Sounds like most of it is free. There’s a promo-video here. I haven’t had a chance to check it out yet, but it seems to have potential.

Both my daughter and Meg saw fit to award me this. Although, as Meg notes, I’m not keen on memes, I am happy to accept their praise of this blog. Thank you very much.

I think most of those that I would award it to are already on someone else’s list. There are lots of brilliant blogs out there. To keep my sidebar a bit less cluttered, those I read are listed in pages. So in case you haven’t found those blogrolls, I’ll link them here: homeschooling blogs & knitting blogs are listed separately (though I know a lot of folks do both).  I note that I haven’t updated either list lately. Apologies for broken links and the like.

That ice-cream problem worked so well that I thought I should keep going with that kind of approach. We haven’t covered much probability so I thought that I would continue in that vein.

This morning we worked a bit with dice talking about the chances of rolling a particular number, rolling it 3 times in a row, rolling it once in 3 rolls, etc. We went on to talk about rolling 2 dice and the probability of different sums, something that is important strategically in some games (including Settlers of Catan, which marks the probability on the number tiles).

But this was not grabbing Tigger’s interest as much as the ice-cream problem did. So I’ve been looking for some other problems. And then I remembered to check my own links page. Duh. The NRICH site is a treasure trove of interesting problems. I did a search on probablity and then looked at a bunch of the different detailed topics. I narrowed each set of results to Key Stage 2 & 3 + Problems. You could narrow it differently depending on your kids. (Steph, you might want to look at Key Stage 1 for your younger ones.) There is an explanation of the Key Stage thing somewhere on that site for those not familiar with the UK system.

The Lady or the Lion looks like something that might appeal to Tigger. I’ve printed it along with couple of other things.

There is even an article on the history of probability.

I’m also thinking that we could do worse than just go to that site and try one of the monthly problems 3 times a week or something. Rotate between topics. Whatever. Hmmm…

If anyone else has some cool problems they want to tell me about, shout out in the comments. I’m not buying any books though.

Taking more tips from Steph, we had some fun with math yesterday. I figured it would be more fun with an extra kid so we invited a friend over. The girls had a great time. Honest.

As I have recently discovered math is really all about patterns. Figuring them out is what mathematicians do. And what I’ve learned at the Living Math forum recently is that a good thing to do in the middle school years is a lot of problem solving. Real problem solving. Stuff that might involve really puzzling things out for a long time. So the 31 Flavors problem that Steph described seemed like an ideal activity. Our local ice-cream shop isn’t that same big multi-national chain though and they have 48 flavours!

I had explained the basic problem to Tigger the other day when I suggested inviting AnimalGirl over to work on it with us (with a visit to the ice-cream shop of course). So she’d been pondering the smaller numbers for a few days. Just after lunch she got out some graph paper to work on and the two girls sat and worked through some of the problem. I let them get started and was cleaning up from lunch and making a coffee behind them. At one point I suggested a table and got them started. They worked through the table to 10 flavours but were somewhat baffled about the pattern. They were trying some different things to try to work it out.

Once my coffee was ready, I joined them at the table and we worked on it together. I noticed that one of their early attempts (failed) might actually lead us in the right direction. They had put the numbers 1 to 5 across the top of a table and down the side but then filled in the whole box. I said that I thought they might be on the right track there and set out another grid. Then I filled in the diagonal because that was all the cones that were 2 scoops of the same flavour. We used their other knowledge of the options to fill in the appropriate part of the grid.  …

I’ll leave you with that because we found that once you got to that point, the visual gave us big clues as to how to figure out the formula we needed. Our first attempt wasn’t quite right but since we knew the right answer, we could work out what we needed to do to get there, figure out why that made sense, and then test the formula on a couple of other numbers. I let them use the calculator to do multiplication with bigger numbers.

We all were really pleased when we figured it out. And that math history came back when Tigger said “The Greeks were right. Everything is about geometry.”

While I finished my coffee they found out what time the bus was going to pass and got ready. We headed down to the ice cream shop, chose ice-cream and counted how many flavours. After we ate our ice cream we got the calculator out and figured out how many choices there are if you have 48 flavours. And since the woman behind the counter said that they make 100 flavours but only have 48 out at a time, we also worked out how many choices there would be if all 100 were available. Lots of fun.

And that last bit led to a discussion of the difference between plugging numbers into a formula and doing math. I emphasized that the math part is figuring out what the formula is. This is a great problem. I highly recommend it.