Andrea raises a good point in the comments to my last post.
“it’s not that school doesn’t teach useful real-world things, it’s that they don’t teach that the things you’re suposed to be learning can actually be applied in the real world.”
Although I clearly don’t think that the only reason to teach math is because it is useful, it also makes sense to teach useful mathematics in ways that make that utility obvious. This is where many unschooling discussions are actually very good. If you have to learn fractions because you need them when baking and in order to double or half things like recipes, then it is much better to teach them by actually baking and doubling or having recipes or whatever than to sit down and teach fractions with a workbook/textbook while admonishing your child to pay attention because this is “useful”. And I think many of us get that even if we don’t always follow through.
Of course that probably means that you shouldn’t wait until your child is 9 or 10 to introduce fractions (which is usually when they turn up in the curriculum guidelines) but should introduce them in appropriate ways as you come across them in your daily life. Dividing up cake at dinner. Baking. Cutting wood for shelves. Buying fabric for sewing. If this stuff really is part of life, then children will figure it out, with your help, naturally as they are exposed to the ideas and come to use them in their own daily life. Might be hard to report, I suppose, but that is your job not your child’s. (Neither Andrea nor I have to report to anyone what our kids are learning, except at the point where past learning is a prerequisite for something like college. But I’m not sure why your children need to know about that. They just need to know how to use fractions, or whatever.) Maybe report the observed evidence that they have now grasped the concept, rather than the work they did to grasp it.
But I can see that some of us have more opportunities to teach the utility of math than others. I suspect that Ron and Andrea use, on a regular basis, more math concepts than I do. Hexadecimals come to mind, but there are no doubt others. And Becky keeps a whole bunch of farm records — budgets, feed records, etc. Her children have 4H projects that involve calculating how much feed a heifer consumed, how much weight she gained, the cost of the feed, the price the meat will get, etc. Angela‘s husband was a contractor and could hire his son to help on jobs, presumably giving him exposure to all sorts of calculations related to building as well as the costs and profits of that business.
Many of us do not raise animals, build things (for fun or profit), or build databases on a regular basis. Heck, some folks don’t even bake much. I certainly have few occasions to double recipes, whereas others probably have to do that every day.
Some of the activities that involve math are activities we don’t involve children in, particularly when they are young, due to cultural beliefs about the appropriateness of certain knowledge for children. I am thinking primarily of household finances. In this culture, money is considered very private and often not discussed much even within households. We may provide children with ways of learning about money — making small decisions, or handling an allowance — but we are unlikely to involve them in discussions of household spending nor even conduct these when they are around.
So teaching children math by involving them in using it in myriad practical ways seems impractical for many people. It would involve changing deeply held beliefs about the proper roles of children and adults. It might involve doing many more tasks for ourselves than we currently do and doing them with children at least watching if not participating. Apart from safety concerns, involving inexpert children often slows the whole process down and certainly takes us out of our comfortable rhythms.
But just because something is difficult doesn’t mean it isn’t worth while. There is much to be gained from breaking down the separation between the adult world and the child’s world within the home. Willa has some interesting reflections on styles of parenting (and home education) that are too focused on children. It might feel a bit like making them grow up too fast or something. But it is precisely because we separate children from these facts of everyday life, that we then need these techniques for teaching apparently useful knowledge that ring false (to us and to our children) and often don’t work.
Of course, there are also opportunities to teach math while we are teaching other subjects. Unfortunately, many of the resources available for teaching those subjects seem to purposefully strip the math out, particularly in materials for younger children. For example, physics involves a lot of math. Newton’s laws, which we now study in physics classes, were published in a book of mathematics. Yet in my investigations of materials for teaching about mechanics and simple machines, little is said about the math involved. The concept of mechanical advantage might be introduced, but formulas for calculating it, or even experiments for taking precise measurements so that children could begin to get a sense of the mathematical relationships are sadly missing. I was surprised not to find a spring scale or any kits for investigating these issues at the local educational supply store, but perhaps I shouldn’t have been.
It seems to me that math-phobia has permeated the professional elementary teaching profession to the extent that math instruction is kept fenced in. As long as you can get kids through these worksheets and they can pass these tests, most of which involve not real problems but lists of sums (in the British usage), you will be fine. We’ll keep the math out of the rest of the curriculum.
This is not helpful for the rest of us who might be looking for assistance in presenting this material to elementary age children. I don’t want to introduce Tigger to mechanics problems requiring the use of calculus (though I have done a few of those in my life; I took a 2nd year Physics class in university). But I would like to include some of the mathematics of simple mechanics in our explorations of that subject. It’s been a while since I’ve done any of this stuff. But I guess I need to go find some high school physics texts and then work out how to present it myself. (Edited to add: I’ve found a useful online resource for high-school physics that seems to explain most of what I’m looking for. I’ll have to work out how to relate it to actual physical demonstrations but that should be less of a problem.)
The same could be said for probability and statistics. This often makes little sense in the context of a math program. But how many of our children are interested in other topics which lend themselves to the study of statistics? Tigger has read a lot of historical fiction, particularly about the settlement of the prairies. At one point she asked about why so many mothers and babies died in these stories. I went onto the Statistics Canada website looking for historic infant and maternal mortality data. It was a little bit hard to find and did not appear in any of their pre-packaged lesson plans. But I did find some data and we did have some discussions just looking at the tables and talking a bit about how to read the tables and learn things. But we don’t find much on these demographic trends in the history books addressed to elementary and middle-school children, much less assistance in teaching them about the mathematical concepts of statistics and probability in ways that would enhance their understanding of history.
Which brings me back to more general thoughts on the teaching of mathematics. Math is useful. And thus is makes sense that we should teach it in ways that make that utility obvious to kids. That is likely to involve less direct teaching of mathematics, more involvement in the daily activities that involve mathematics, and the inclusion of the mathematical components of other subjects. The problem is that our ability and confidence, as parents, to do those things varies considerably. And yet there is little supporting material out there to help us. I don’t think we need math programs that include more real problems and applications. I think we need more guidance to parents about how to explain the mathematics we use in everyday life. And we need resources for teaching physics and other subjects that don’t strip the math out.
Perhaps folks can share what they’ve found or even share materials they’ve pulled together to teach in this integrated way. Thanks, Andrea, for spurring me on to think more clearly about this side of the issue.