My six year old daughter has many, many questions and observations that can be filed under the catch-all category of 'spontaneous math'. In fact, most of the math thinking she does exists primarily in the verbal realm and in relation to the moment she's in. We seem to have some kind of math related conversation every day.

She has been known to enthusiastically break into a chorus of counting to twenty by ones, then to 100 by tens, forward and backward. She likes chant counting by twos as well.

She has two cosmo flower plants in her garden that she has nurtured from seed. They have not yet flowered (and it's mid August!) but they have been a daily source of measurement since early summer. "The cosmo is taller than me! It's up to Papa's chin now!" she cries as we turn towards autumn.

She agrees to fold a basket of laundry for 50 cents. She says, "Will you give me fifty more cents if I put it away?" I say, "No, but I'll give you a dime. How much is that all together?" When it comes time for payment, I say, "How many quarters should I give you? How many pennies is a dime? How many other ways can you make ten cents using pennies and nickles?" She starts to put her hard earned 60 cents into her savings jar and then dumps it all out on the counter. "Let's see if I have enough change to make a dollar so I can exchange it for a dollar bill!" Only eight cents short.

[A side note: I'm really excited to be having this kind of conversation with my daughter. I really, really, desperately do not want her to end up like the college student behind the cash register with whom I recently interacted. She said, "That'll be $3.73." I gave her a twenty dollar bill. She rang the cash tendered into the cash register. In the middle of that I said, "Oh, I have the correct change," and gave her 73 cents. She looked at the register, looked at the $20.73 in her hand and went to find a cal...cu...la...tor.....]

Back to my daughter...who is curious about how much things cost, and how that compares to other things we buy. It's a little embarrassing when our neighbor brings her a birthday present and she asks how much they paid for it, but she's there's something she's trying to understand, even though I'm not quite sure what it is.

Over Christmas, she and her Aunt Karen and Uncle Arlen measured the sunroom in a hilarious series of "Arlens". The sunroom is almost exactly four Arlens long. Isobel, it turns out, was two Arlens long. The couch was one Arlen, and five of pet cat Lucy.

We've read Sir Circumference the first Round Table a number of times. Now she has a game she made up where she leaps towards her blow-up wading pool in what she calls the "diameter jump' -- I hold my breath every time as she leaps, finger tips to toes stretched out in one long line to touch the front and back of the pool at the same time, literally flying, flopping almost on the other side of the pool.

She meets a four-year-old in the grocery store. She says, "You're four, I'm six. I'm two years older than you." Or, a three year old. "I'm twice your age." She has a pretty good experienced-based understanding about basic fractions, and has even been known to figure out something via multiplicative reasoning, but these episodes are fleeting, here and gone, hard to keep track of.

She asks, "Who is older, you or Papa?" When I tell her Papa is 53 she is silent for a minute, looking at her fingers. Then she exclaims, "In five years he'll be a HUNDRED!" "No," I respond, "You're counting up by tens. You need to count by ones."

We took a city bus not long ago. Got on a block from our house, followed the route south, then north again, to the center of town and out to the west side of town. It took a full two hours. Two thirds through our trip she asked for the route map in my lap and started tracing our route in real time. Every time we turned a corner or went around a curve, there her finger was, following right along.

And, not that I'm at all surprised, she dances through the grocery store to the satellite radio oldies saying, "I use the square tiles to show me where to go, they help me make up my dance steps!" I swear to heaven above that I did nothing to overtly influence that observation.

At some point I expect we will bring all this math down to the page. I almost said bring the math 'back to the page' but that's not what is happening. The math is in her head, thoughts and daily maneuverings first. You can't bring something back where it never was in the first place. There will be plenty of time for learning a formal notation system in the future, but it's still August and she's only six.

It does strike me, though, that much of her mathematical thinking and reasoning right now seems to center around numbers, calculation and measurement. Ever since they started adding the dots on dominoes in Kindergarten this past winter she has really been taken by the process of adding together and taking away. I know math is not all about numbers but, no matter. These are the concerns of her mind, not mine. I do celebrate, however, that it is, apparently, very easy to find math all around us and I 'm excited to see what we find next!

Thank you. It was fun to see you and your daughter talking math.

ReplyDeleteA few thoughts. First, did you actually check that the clerk was a college student?

ReplyDeleteDid you ask the clerk to pause and THINK in lieu of resorting to a machine to solve that "problem" (which apparently counts as a problem for the clerk, though not for us)?

Are you familiar with the elementary math curricular work of V V Davydov and D B Elkonin? Your daughter seems to have picked up on measurement as well as counting. I would have wondered which came first and how number got into her thinking - pre-school, that is. Probably too late to ask.

Hey Michael! Well, we live in a college town and classes are starting soon so...I'm assuming she is/was a college student. She looked kinda on the young side, so maybe just out of h.s.? Still, you'd think you could do that kind of mental math with a high school diploma, but maybe I'm silly to assume that as well? lol.

ReplyDeleteAnd, really, I was just stunned. It happened so quickly I didn't have any time react or respond. I do think, though, it was not just her problem but also a problem for the entire society summed up in one small moment/event. ;-)

As to my daughter, she has had a lot in the way of preschool and kindergarten, so it probably is hard to tell how much and what kind of math she was exposed to. I would hazard the guess that it was sort of the 'normal' kind of math, e.g. not the Montessori model for example. I'm sure there was lots of work with numbers there, but she never seemed too interested them until recently (last half a year).

What I do know is that you only have to show her something once and she is off and running, creating a teacher's dream integrated lesson plan through her own play and exploration, so it probably wouldn't take much exposure to get her thinking about this kind of stuff.

I've noticed, though, that she is almost 100% experiential/verbal/kinesthetic in her math understanding. For a while she was really into writing down little number sentences, probably b/c that's what they showed her in K. But the math that has emerged this summer is much richer and deeper than I've seen before; a combination of discussions (prompted by all sorts of things we're doing and reading) and the use of all her fingers.

So...my sense is that measurement came first for her -- she was slow to counting and one-to-one correspondence, but she has always been aware of and comparing the size and shape of the world around her. Only recently, though, has she started putting words to her math thinking, which is really fascinating for me. Ultimately, it is probably too late to figure out which came first, but I hope this answered the question somewhat.

I'll look into Davydov and Elkonin -- thanks for the tip! Developmental teaching/psychology seems right up my alley in terms of interest. Thanks for checking in!

I recently was gifted with a set of grade 1 - 5 materials for students and teacher of one of the math programs that were developed in Russia by colleagues of Davydov and Elkonin. However, they are in Russian. My Russian isn't too strong, to put it mildly, but I can likely do some translating with help of the internet. What I knew going in was that those influence by Davydov & colleagues start with measurement, not counting, and build arithmetic from that. There are articles on-line by folks like Barbara Dougherty (who worked on the Measure Up! math program at U of Hawaii) and Jean Schmittau, who is supposed to have translated a set of grade 1 - 3 books by Davydov & Elkonin and used them with kids in Binghamton, NY. However, she will not respond to my repeated inquiries to see the materials, so that's why I've looked for over two years for other sources, and finally someone was kind enough to oblige.

ReplyDeleteYou should also see if any of what Susan Addington has developed using a measurement approach to arithmetic for education students is accessible on-line. I have a set of pilot materials in pdf form she gave me access to about a year or so ago.

Hi Michael --

ReplyDeleteThis is all *very* fascinating to me.

Does measurment basically mean more/less, bigger/smaller, longer/shorter, etc.? If so, I would expect the young child is hard wired to explore those concepts kinesthetically. Davydov et al were probably the only ones who were paying attention to that.

I looked for Susan Addington online -- was it called "measuring the world"? At http://www.quadrivium.info/MtW/MtWindex.html I found this passage particularly interesting as it relates to the kind of experiences and interests my daughter has had for most of her life -- "cooking and sewing, carpentry and gardening, travel and telling time, music and decorative arts". The passage says:

"Until fairly recently, most people learned about measuring from family and on the job. Activities in daily life---cooking and sewing, carpentry and gardening, travel and telling time, music and decorative arts---all required measurement. Sometimes precise measurement with tools and units were needed, but often estimating and common sense were more important. In our modern urban technological society, most of our food, clothing, furniture, music, and decorative objects come ready-made; making these things is often just a hobby, and is not done out of economic necessity. Measurements are commonly done digitally, in a way that the physical measurement process is obscured. Children often do not develop the intuitions about measurement, quantities, and units that are the foundation of elementary mathematics."

I might need to see if I can get this book. Maybe the IU library system has one I can look at.

Thanks for all your thoughts. It's nice to have a better understanding of what I'm seeing my daughter do.

Yes, you've got the categories identified correctly, though there are others: the Russian books I have ask kids to compare "mass," though I don't know if that has a less technical meaning for them and is interchangeable with weight: heavier/lighter, in any case, would seem to fit hard wiring or near enough for jazz.

ReplyDeleteAnd yes, that's the Addington material I mentioned. Is it out as a purchasable book, now?

Have you searched for any free on-line articles? I think you'll find some worth reading are downloadable.

By the way, I don't get updates on the comments here, so you might want to e-mail me directly: mikegold@umich.edu