Tricotomania and more

The ongoing conversation about what unschooling means and whether it is a useful label is continued by Cindy. She hadn’t been blogging for quite a while but this post, sparking off Elsie’s, is a worthwhile contribution to the conversation so I wanted to make sure people saw it.

I have finally joined the Living Math forum on Yahoo groups. Although, as one recent post pointed out, it is dominated by people with younger children learning elementary math, there are some good discussions and good people on there. I am hoping that it will be a useful resource for me as I move forward with Tigger.

Just yesterday someone posted a link to this amazing site: Mathematics Illuminated. I need to explore this more but it is a course for teachers and adults. There are video courses and PDF files of both teacher and participant materials at the “course materials” link. There are also interactive activities, and a timeline that shows the relationships between topics covered in each unit (the mathematics family tree). I’m going to put that link in my “useful homeschooling links” list but I wanted to post about it here because I thought some of you might be interested in it.

One thing that did get discussed on the forum, in response to the query about resources for teaching high-school level math, was the fact that parents need to be willing to learn or re-learn the material together with their children. This resonates with some of what I’m discovering about unschooling and with my own experience of what works well with Tigger. If I think that this material is important for her to learn, then it must be important enough for me to learn, too. I don’t value educational models that look like hoop jumping. Even though we might forget things we learned earlier in life, if they are important and interesting, we will pick them up again when we need them. Needing to teach her constitutes a valid need.

Also, I am genuinely enjoying reading some of the math books that I’ve picked up. They are slow going. And I don’t understand everything. But I’m liking it. (I will get to the review of Devlin’s Language of Mathematics when I’m done.) I think this online course will be an interesting part of my own mathematics education. If some of you think it looks interesting but are a bit daunted by doing it on your own, I’m happy to consider doing it together using the facilitator materials provided. That might give me a bit of a structure to actually do it, too.

BTW, the other interesting thing going on at the Living Math forum is a spin-off forum of facilitated discussion of Jacobs “Mathematics a Human Endeavor”. It started in February and since I’m in the middle of a different math book right now, I don’t think I’m going to join it. But the model is interesting. I suspect similar things could happen again in the future.

We sat down this morning and looked at what was coming up in the next couple of months. It seems that while others are gearing down and looking forward to the end of their school year, we are gearing up. That’s the kind of thing that makes me realize that I really am an unschooler. 

Earlier in the spring there was a little flurry of organizing being done by other moms. Our local homeschooling e-mail list was a-buzz with various field trips, classes, etc. Folks around here hibernate a bit in the winter and don’t like to have too many things out of the house that will either be cancelled due to bad weather or involve driving in dubious conditions. So once March rolls around, all the moms who like to organize things get busy. I appreciate this enormously.

So Tigger has been doing a geometry and art class once a week for 3 weeks now and there are 5 more sessions. She’s really liking it. And we carpool with another family so it works well for the moms, too. The same mom who runs that class has also organized some sessions at the War Museum. Tigger will be there for a morning and afternoon session this Thursday.

One of the things we were interested in seemed to be harder to get organized than it should have been. Getting the minimum number of kids took a while and then we’d lost the original dates. So that is only starting on May 9th (yep, it’s on Friday mornings) and due to a previous commitment of the teacher’s on one Friday, it’s going through to July 4th (which is not a holiday here; our day is the 1st). This is a science class loosely based on the Grade 9 curriculum (his description) and covering a bit of physics, a bit of chemistry, and a bit of biology. Tigger is excited and I think it will be a good intro to a range of things that could lead to more detailed study in the future.

The botany study with her dad is going well. So far it has mostly been reading the Elpel books, doing some sketching and talking, and checking out plants in the garden and the neighbourhood. But it is going to expand to include an exploration of local conservation areas and parks. He’s buying a map this evening. And someone at one of the conservation areas has organized a couple of homeschool field trips that they are going to participate in.

We are also trying to figure out when to go to Toronto to see the Darwin exhibit at the ROM. I think we are going to combine it with a visit to the Science Centre and just seeing some friends. But I need to look at train times and work out where to stay and figure that out. Probably the 3rd or 4th week of June, early in the week (so we are back for science class).

I’m one of those people that sees the activities organized for school kids in the holidays as great learning opportunities. One such is a Shakespeare camp run by a local theatre group. That’ll occupy Tigger for 3 weeks of July. They are doing As You Like It. She is really looking forward to it.

So, as you can see, there is no winding down or end of school year around here. We just transition from one set of activities to another. For awhile I was vaguely worried that we weren’t doing any science and now it seems like that’s all we’re doing. We seem to go in waves. I have been trying to keep some sort of record of what is going on, mainly by recording things in a day planner (after the fact). This is haphazard and not at all detailed but I think it is probably a useful exercise. I am still very glad I don’t have to report to anyone.

Ok, Em, you can get up off the floor now. Yes, you are at my blog.

Sarah B. posted a video today with instructions on how to make a cute little handbag out of an old book with a cool cover. (Someone get some CPR for those who think that cutting out the pages of a book to make a handbag should be a capital offense.) I bet there are good candidates at the thrift shop.

Anyway, I’m not a big user of handbags, particularly not cute little ones (if you can’t get a book and my knitting in it, it is useless to me) but I’m sure some of you are. And I bet teen girls would find this project a win-win: cool craft and cute handbag.

Have fun. I apologize for all the heart palpitations

Elsie had a post about Goals, Objectives, Standards and the like that raises some interesting points. It made me realize that part of the problem is the woeful lack of understanding of statistics both in the general population and in professions who are using statistical information to provide advice (or even design regulations). So here is a little Statistics 101 conceptual refresher to help you deal with educational standards, like the definition of “on grade level”.

Developmental milestones, in health and in education, are based on statistical data on very large populations of children. They are averages of some sort. Those numbers are very meaningful for large populations, which can be expected to have about the same level of diversity as the even larger population from whom the figures are derived. So, for example (and using made up numbers), if you were to say that “on average” children learn to read by age 6, this would mean that in a very large population when you add up the ages at which children learn to read and divide by the total number of children, 6 pops out (the mean). That number could be the average even if NOT ONE CHILD actually learned to read at age 6.

You might actually use a different kind of average, the median. If the median age at which children learn to read is 6, that means that 50% of children learn to read before (and including) 6 and 50% learn to read after (and including) 6. If you strung all the kids in the population out in a line ordered by the age at which they learned to read, the half-way mark of the line would be a kid aged 6. Or between two kids aged 6. Or between a kid age 5 and a kid age 7 (unlikely to happen but it illustrates the point that this could be the right median even if no individual kids actually learn to read at 6).

With either measure you can see that those numbers can be correct, for a population, even if large numbers of children learn to read at ages ranging from 3 to 12. What counts as “normal” is not the same as “average” (by either definition). If the range of ages at which children learn to read and the percentage of children learning to read at a particular age remains relatively stable over a period of time, then learning to read at any age within that range is “normal”. Unfortunately, those statistics are much less interesting and harder to understand so they don’t get talked about in the media and by politicians very often.
So what use is the average? It might be nice to think that statistics lie, but that wouldn’t be true. A statistical average can be a useful measure when comparing large populations. So, for example, if you wanted to compare the quality of education between one state and another state, you could look at the average age at which children learn to read (or some other measure of attainment, ideally several). There are enough children of each age in each state to make that comparison meaningful. With that size of population, you could assume that the variation should be within the same range. So if it is not, you would want to figure out why. That might be because of the quality of education, but it could also be because of the quality of library provision, or the percentage of the population with English as a second language, or something else.

It probably also makes sense to use these kinds of statistical averages to compare school districts though some might be getting small enough that the differences from the average could be the result of random variation (in other words, not problematic). At the level of individual schools, the populations are probably not big enough for the statistical averages to be significant. Having one kid out on the far end of the range could skew the average significantly even though it is “normal” for a certain percentage of the population to be at that far end of the range. That doesn’t stop many jurisdictions, including the UK, from using school level standardized test results to rank the quality of schools. The fact that some authority does it, doesn’t make it meaningful.

Statistical averages tell you NOTHING meaningful about an individual. And if someone tells you that the quality of education you are providing is poor because one student is performing below average then they are misusing statistics to lie to you. If you know the normal range then that information can help you work out where an individual fits in relation to the population as a whole. Sometimes that information will be expressed as percentiles or deciles. So if your child is in the 60th percentile on some standardized test, it means that her attainment is higher than 60% of all the children who took the test. The median is the 50th percentile. Deciles cut the population into 10 groups. It is also helpful to know something about how the attainment varies between each decile because most of the population is likely to fall in a relatively narrow range, with fewer people at the outliers (the famous belled curve). So variation within the 5th decile might be minimal, but within the 1st and 10th decile might be considerable. The number of people within each decile is the same.

Do not get confused. If the results of some test say that your child is in the 60th percentile that DOES NOT mean that your child scored 60% on the test. It means that whatever your child scored was higher than what 59% of the test takers scored, and lower than what 39% of the test takers scored. Ideally, those percentiles will be calculated over a large enough population, like the whole state, or all individuals who took that particular test over the past 5 years or something.

Broader notions of grade levels are also based on some understanding of statistical average attainment for kids of a particular age. Ideally, those understandings should take into account the normal range of attainment within a particular age group. A good teacher should be able to work with reasonably diverse group in terms of ability and attainment. Unfortunately, the rise of standardized testing (and funding linked to performance in standardized tests) and ever more detailed “state standards” or “provincial curriculum” make it difficult for teachers to do this, as Elsie points out. It is important to remember that most school systems are graded by age, not by attainment. It is only a very small minority of children who will be held back; and a similarly small minority that will be accelerated. The combination of that with the narrowing of the acceptable range means that ever more kids need to be diagnosed with some “learning disability” in order get even some of their learning needs met. But that is another story.

For homeschooling parents, these notions are practically meaningless unless you are actually going to put your kids into school and need to know whether they are going to be ahead or behind the kids in their grade. What is important at the level of the individual, as opposed to the population, is whether a child is developing. Does your child know more now than she did 12 months ago? Has your child developed new skills or improved any skills? You only need to be worried if your child is stagnating or is going backwards. And that would be a trend over a period of time, not a few days. Development is notoriously uneven. You might also be worried if development is particularly slow. Slow development or lack of development might indicate an underlying problem that could be addressed. Development in some areas might happen more quickly or at different times than in another area. A certain amount of judgement, and perhaps advice from trusted others (including doctors, psychologists, etc) is required to work out if this is a “problem” or just what normal looks like for your kid.

The other thing this means is that, politically, it is in our interests to ensure that statistical measures are not being used inappropriately to judge us or to judge schooled children. If education policy misuses statistics to develop methods of working, regulations, test, funding formulas or whatever, that misuse will impact us. It may take a while. But if other kids are expected to meet certain milestones by a certain age, then sooner or later someone is going to ask that your kids be required to meet those milestones. And if it isn’t good for your kids, then you can bet there are at least some kids in the system who are also disadvantaged by it. Even if we don’t use them, public schools are provided on our behalf to provide an education for anyone who needs it. How much we do about it is up to us. But if we have opportunities to point out the misuse of statistics in formulating policy, we should take them. Even explaining the problem to relatives and friends increases the general understanding in the population. And challenging elected representatives who put forward these crazy proposals is probably a good idea.

Today’s Unshelved blog post reveals new t-shirt designs. And one of them adds a new possibility to all of our discussions about unschooling, homeschooling, and generally what to call ourselves.

Library Schooled

Maybe we need to bulk order some of those.

I got to borrow the Jacobs textbooks (both Algebra and Geometry) from a local homeschooler. These were in my list of possible math resources. Her son is using the Algebra right now so I only had the book for the weekend. Although reviews of these texts on homeschooling sites are good, I wanted to take a look myself because they aren’t cheap. (I’ve been watching a few e-bay auctions and even 2nd hand copies usually go for over $50; sometimes well over.) I’ll just talk about the Algebra text in this post.

On the positive side, this textbook is very thorough. It is written by one man and has a coherent approach. In an age where textbooks are often written by committee, with consequences that remind one of the joke about the camel and the horse, having a coherent treatment of the subject is refreshing. It is also clear that this textbook has been around for a while, even though there are current editions. There are no “boxes” with sound-bites of “relevant” or “exciting” information. There are cartoons in every chapter, germane to the topic, but that’s about it. No colour plates. The presentation is straightforward and seems to be based on the assumption that algebra can be interesting in and of itself. (Complaints about these general features of contemporary textbooks can be found here (mine) and here (Steph’s). Becky raised the committee issue in a comment on that first link.)

The other thing that indicates that this is an older-style textbook is that most textbooks are now designed to fit with “state standards” and “provincial curriculum”. Since most of these split Algebra over 2 years — Algebra I and Algebra II — the Jacobs text stands out for treating the whole subject as one coherent whole. You would need to do the work of correlating it to your state standards if you are producing a transcript, but that might mean taking 2 years to go through the book and providing a double credit at the end. Certainly feasible. And the coherence of the program might make this an attractive option.

Although a teachers guide (which I didn’t see), transparency masters (ditto), and test masters (I did see these; seemed very useful if you like that sort of thing) are available to accompany the text, the text is reasonably self-explanatory. The concepts are broken down and sequenced in a sensible way. And each section has lots of practice problems divided into 4 sets of increasing complexity. Set IV is almost always just one or two really interesting problems. The answers to Set II problems are included in the back of the book (the others are in the teachers guide) and this is a level that seems reasonable for a self-taught student to master.

Those of you who know that I hang out (in cyberspace at least) with the creative learner types will immediately see what the problem with the text might be. It is very sequential. And breaks things down in small steps. I found myself trying to work out what we would skip and how we would decide that that was okay. I know Tigger is already quite familiar with some of this material and while I can see the logic of reviewing it as a set of building blocks to something more complex, there would be a very real danger of turning her off. A kid with even more right-brained learning style might find this approach very off-putting. In addition, from this perspective, the text book is very text heavy. The cartoons provide light relief. They are not integrated into the instructional text. Nor is there much else in the way of visual approaches to the concepts.

The other main problem from our perspective is the number of problems. The Set IV problems look interesting (but we’d need the teacher’s guide to get the solutions). I like the idea of doing a few basic problems to warm up, and then doing some of increasing complexity, but I’d have to really pick and choose. It would be unreasonable to ask any student to do all of the problems in all 4 sets for each lesson. I guess you’d have to work with it and see how it works. I suspect that Set II is a good level for most kids. And you could use a couple of Set I questions as a warm-up or, if the student was having some difficulty grasping the concept, you might use Set I to work through it. For a bright kid like Tigger, I’d certainly want to be challenging her with the Set III or Set IV questions. But she is a 5 questions a day kind of kid (Though that is with the questions in Challenge Math, which are all complex.) so the total number seems really daunting.

From our perspective, Jacobs Algebra would not be worth the money, particularly since I’d need the teacher’s guide as well as the student text. It seems that the Key To Algebra series probably provides sufficient problems to work through in order to learn the concepts (thus Sets I and II would be redundant) even though the concepts are presented in a different order. It is similarly self-teaching friendly. I am glad that I know someone locally who has the Jacobs, though, because it might be a useful source of more complex problems (even without the answers). Or, as Tigger matures, she might decide she likes that kind of sequential approach for self-teaching.

If you have kids that like a sequential approach and take in information well through reading this textbook might be a very good fit for you, especially if they prefer to work independently. I suspect the teachers guide would be handy to have, not only for the other answer keys, but also for ideas on how to extend the explanations in the student text when required.

Spring has sprung. There are still heaps of snow, particularly next to north facing walls, but the birds are about, plants are poking up through the soil, and the weather is lovely. I have snowdrops and crocuses in my front garden and they make me smile. We need to plan more in the back garden in the corner that melts first so we can see that kind of joy from the window. The cats are very happy (though not this afternoon because I’m keeping them indoors so I can find them for their vet appointment later) and have tried to bring birds in. I draw the line at wildlife in the house, dead or alive. Mostly they don’t bring it to me but catch birds (mostly sparrows, I am not worried about the general state of the population), chipmunks and other small wildlife for their own purposes.

My spring plan was to study botany and now that we can see the ground we have started. My partner, Mat, is leading on this because I will be travelling a lot in May. But when I’m in town, I’m joining in. Overall, my objectives are pretty simple.

• Develop the habit of taking nature walks.
• Observe nature carefully.
• Learn something about botany.

We are using the Elpel books as a guide. Mat and Tigger started reading Shanleya’s Quest this morning. He has been reading Botany in a Day at bedtime, giving me highlights and interesting facts. Also, Mat loves to draw and his Christmas present was a drawing class at the local art school, which he loved. So he is a much better candidate for the whole sketching side of this study.

We actually started on Wednesday. We went for a walk in the little woods that borders the Experimental Farm (an AgCanada site near our house). There was still snow in the woods so not much coming to life. We ended up collecting different kinds of pine cones and needles (ones that had already fallen to the ground). We came home and sketched what we collected (well Mat and Tigger did) and talked a bit about how conifers work and where they fit in the evolution of plants. I found a field guide online but it wasn’t designed in a way that makes it helpful for the kind of identification we are doing right now. We need to get some field guides that are organized by family so we can look at the characteristics of the plants we are observing and relate them to classification and whatnot.

Other things that happened included a rather detailed discussion of the plot of the Agatha Christie novel they are currently reading (Death on the Nile), with Tigger putting forward her thoughts on who might get murdered and who might do the murdering with her reasoning. Very interesting and worthwhile. Sometimes just going out walking offers an opportunity for this sort of thing in a more natural way. Since one of our learning goals is for her to develop her ability to discuss what she reads, I’m very happy that this is starting to happen spontaneously.

And while my goals are vague, I did decide to push her a bit to go out in the garden and actually sketch something related to the reading on the mint family that they did this morning. They had both gone out into the garden after reading the chapter but hadn’t done any sketching. I think I was right to give that gentle push because when I came back from my fitness class she had done some drawings and explained to me what the characteristics of the mint family were (clearly illustrated in her sketchbook). It didn’t take her long and she wasn’t bothered by it at all. The trickiest part of this whole homeschooling thing is knowing when to push (gently) and when to just drop something.

So we are off to a good start. Our plan is to go for 2 or 3 short nature walks every week. I suggested to Mat that a further objective of this unit could be

• To explore local conservation areas, parks, and other related amenities.

We like to go hiking. We like nature. We enjoy cycling. But we don’t get around to getting out and doing those things as often as we might like. And we’ve lived here for over 4 years now and haven’t been to some of the parks and conservation areas that are really close to us. So this might be part of objective 1 above. We need to get in the habit of going out and doing these things more often. And we need to have a sense of the places that we could go. We seem to be on our way.

Thanks to Shaun for sending me over to read something on Elsie’s blog. Elsie has written a few posts recently that get into some of the fundamental questions about the politics of education. She has been a school teacher and now homeschools and has some very thoughtful posts on the relationship between those two roles.

Reading some of these have reinforced some ideas I’ve had for a while. It would be good for homeschoolers and teachers to try to understand their points of commonality and also their differences and then to work together to address some of the bigger problems with education in many of our countries. Because the problem isn’t fundamentally about the teachers. It is about the mistrust of teachers, demands for accountability of a particular sort, and larger political decisions about both the funding of schools and how we will make schools accountable. Teachers as much as homeschoolers suffer from too much emphasis on standardized testing. And many of the good teachers have left the system because they don’t think “teaching to the test” is good teaching.

I learned something today. Something really cool. And I thought that Angela, and maybe some other folks, might like to know about this, too. It is part of the beauty of math thing. Everyone who already knew this and thinks, “well, duh” can please keep it to themselves and just reflect on how even those of us who were good at math and liked it missed some really important things.

So, get out your compass, a little ruler and a pencil. And a piece of plain paper. I’ll wait. Got it?

Okay. Draw a circle. Keep the compass set at the same radius. Put the pointy end on the edge of your circle somewhere and draw an arc that goes inside your circle from one edge to the other. Now, with your pencil end still on the edge of your circle, move the pointy end to another point on the circle (there will only be one place you can put it). Then draw another arc. You’ll have a sort of petal shape with 2 curved lines. Repeat (move the point while keeping the pencil on the edge). Now you have 3 petals. Put your point on the end of one of the petals (on the edge of the circle) and draw another arc. Do that thing where you move the point while keeping the pencil end on the edge. Twice. Now you have 6 petals that meet at the centre of your circle and are evenly spaced.

This means that you can draw a perfect diameter in 3 different places. You can also draw a regular hexagon with sides equal to the radius of your circle (use the ruler to join the tips of the petals). And equilateral triangles with sides equal to the radius of the circle (join tips of petals to each other and to the centre).

How cool is that?!

I learned it from String, Straightedge and Shadow by Julia Diggins, a very cool history of geometry book that Tigger and I are reading together. Apparently it is this simplicity of dividing a circle in 6 that is at the root of the fact that we divide them in 360 degrees or, in the case of clocks, into 60 minutes. I still haven’t figured out how you accurately divide those sixths in equal parts. Though I must admit that I haven’t tried very hard to figure it out.

I started reading The Language of Mathematics last night. And the key point he makes at the beginning is that mathematics is about patterns, usually abstract patterns. This discovery is making that kind of obvious. I’m liking it.  (I’ll review this book when I’m done.)