Constance Kamii, author of Young Children Reinvent Arithmetic: Implications of Piaget's Theory offers her belief that children cannot be taught place value.
Her conclusion is based on a post-test she created in which most children, after direct instruction in place value, fail to demonstrate understanding of place value.
Here's the test from her book:
1. I put 16 chips out and asked the child to count them and to make a drawing o f"all these." The children made drawings either in a line or in a bunch.
2. The child was asked to write "sixteen with numbers" on the same sheet to show that there were 16 chips.
3. The child was asked what "this part" meant as I circled the "6" in "16" as shown. The answer was to be indicated by circling some of the chips drawn on the paper.
4. The child was asked what "this part" meant as I circled the "1" in "16." The answer was again to be indicated by circling some of the chips drawn on the paper.
5. Finally, the child was asked what "the whole thing" meant as I circled the "16" and probed into the relationships among "16," "6," and "1." For example, when a child circled [one chip instead of ten] I asked why "these" (the six chips circled) and "this" (the one circled) were circled but not "these" (the 9 leftover chips).
Kamii found that none of the children in her class said that the "1" in "16" stood for 10.
But there's a problem. I tried the test myself, and see that it's faulty. A loose collection of ten does not, indeed, represent the tens place in base ten.
The first two pictures depict Kamii's test. The third picture depicts a more appropriate test.
Tester circles digit "6" and asks "What does this part mean?"
Child selects the correct number of "ones"
Tester circles the digit "1" and asks "What does this part mean?"
Single chips don't represent the tens place, so the child becomes confused and circles one chip instead of all ten
Here's a better test
This test offers a place ten chip to circle. Perhaps the child can be given several, from which she should select just one to represent the "1" digit in the number "16."
From my preliminary research, it appears that Kamii's faulty test is the reason why reform math purists don't teach place value. Constance Kamii seems to be cited a lot, including by the publishers of TERC Investigations.
Teachers trained in this method won't use base ten blocks or any form of manipulatives to assist in understanding.
Instead, they offer example after example of equations using numbers with two places such as (20 books plus 15 books is 35 books) but without explanation of what the numbers mean. And they wait for a child to "understand" internally.
The reform math purists (of which some teacher are, while others are eclectic), tend to be very rigid about not teaching place value by any direct means at all.
As one pre-service teacher instructor put it:
"Have you seen the new TERC Investigations books? They have totally caved to some pressure and I wish I knew where it was coming from.
"They are now advocating the use of the base ten blocks to teach regrouping (long rods that represent 10, single cubes that represent 1, and flats that represent 100). I was shocked and deeply disappointed."
Disappointed? Shocked? To believe that teachers must not teach place value is a far cry from the conclusion that children don't understand place value when taught using base ten blocks.
And the basic premise, that children don't understand it when taught it, is based on a flawed test. But worst of all, to suppress all direct instruction in the belief that children can't learn place value by direct teaching is nothing less than cruel and myopic.
Good for TERC Investigations Version 2 for "caving." Still, I don't want TERC in my backyard.
A loose collection of ten does not, indeed, represent the tens place in base ten.
ReplyDeleteI was going to completely disagree with you here, but I think I realize what you're saying--so I'm going to only partially disagree.
It's true, every digit in a number is a counter of sorts (from 0 to 9) of the unit value of the place that it is in. So, the digits in the ones place count the unit value of the ones place--ones--and the digits in the tens place count tens, and so on.
However, in your example, the value of the 1 in the tens place is 10. The fact that students didn't understand this is a pretty good indication in my opinion that they didn't understand place value.
But putting differences of opinion on this aside, you have to admit that Kamii's conclusion is kinda funny.
To see the hilarity, you have to forget for a moment that Kamii is a researcher. Instead, let's imagine she's a teacher and read her conclusion again:
My teacher, Miss Kamii . . . [believes] that children cannot be taught place value.
Her conclusion is based on a post-test she created in which most children, after direct instruction [by her] in place value, fail to demonstrate understanding of place value.
Hilarious, right!
My instruction didn't work. Heh, must be the kids' fault.
The third version that you depicted is how Singapore Math teaches it. Just curious, what were the control groups that Kamii used in her "experiment"? What? There wasn't a control? Oh, OK.
ReplyDeleteYou can find the actual pages in her book on Amazon.com, using "search inside this book."
ReplyDeleteIt's on pages 79 to 84.
No control groups. Just informal observation after someone else taught place value.
I thought her test was poorly executed, and prone to misinterpretation. It almost seems like she was setting the kids up to fail, so that she could make the case for discovery learning as the only path to understanding.
What irks me is this very informal
observation and flawed test seem to have led to anti-teach-place-value mania in reform math circles.
But what's even worse is that while they're waiting for the kid to discover place value, what they're doing in the meantime is offering endless examples of ways to express a number, as though some magic is supposed to suddenly happen.
And if they're offering two digit numbers in real-world contexts without teaching place value first, then they're leading the students further away from understanding. The number 25 turns into heiroglyphics rather than meaningful representation.
They're doing things in the wrong order.
Clearly, Kamii hasn't visited Montessori schools. Kids learn place value very early on in Montessori classrooms. At age 4, my children were playing the "stamp game," using tiles that represented ones, tens, hundreds, and thousands and understood that they could trade ten ones (or "units") for a single ten tile. It is not unusual to see 4 and 5 year olds doing addition and subtraction with 4 digit numbers.
ReplyDeleteYes, isn't it interesting that reform math, which is so big on games, disallows games to teach place value. Instead the games are about chunks of amounts, but no attempt at representing hundreds, tens and ones.
ReplyDeleteInstead, the whole idea of hundreds, tens, and ones must be guessed by children, based on games that have nothing to do with place value.
It's outrageously misguided.
Kids learn place value very early on in Montessori classrooms.
ReplyDeleteInteresting.
I MUST read her biography. I've long admired her from afar, because she was - or so I understand - one of the first people to say that mentally retarded children could learn.
That's what I love about behaviorists, too, btw. They are intrepid souls.
Dipping in - make that diving in - to the fluency teaching material has been like going home.
Back to ABA!
You bet they can learn. My son is very cognitively impaired, but reads very well. And he makes connections to the real world from what he reads.
ReplyDeleteIt's no easy task to sound out words phonetically, especially with all the exceptions in our language, but he does!
I'm glad somebody cared enough to teach him to read when he was six years old.
My Montessori experience can be summed up as this : "they never put a ceiling (even a glass one) on what my child could learn"
ReplyDeleteFunny how public education is all about those ceilings -- when learning the skill of long division so that a child can do polynomial long division (ie, higher math skill) a principal of a fuzzy mindset remarked "well we dont teach polynomial long division here"
well duh of course not but what you do teach will affect whether the kids can LEARN it or not
Instead, they offer example after example of equations using numbers with two places such as (20 books plus 15 books is 35 books) but without explanation of what the numbers mean. And they wait for a child to "understand" internally.
ReplyDeleteThe other issue here is that children are "hyperspecific" - as apparently we all are when we're first learning novel material.
This is the theme of Animals in Translation: animals are hyperspecific. To an animal, a man walking on foot IS A DIFFERENT THING than a man riding a horse.
To a child, 10 single chips are going to read differently from "tens place."
It almost seems like she was setting the kids up to fail, so that she could make the case for discovery learning as the only path to understanding.
ReplyDeleteWickelgren makes this point.
He says that constructivism is based in a belief that young children can't do any kind of abstract thought, which is why everything has to be concrete and "hands on."
I had never been able to track this down; it doesn't appear to be in the Standards.
I bet it comes from Kamii.
The reform math purists (of which some teacher are, while others are eclectic), tend to be very rigid about not teaching place value by any direct means at all.
ReplyDeleteI had no idea this was the case.
I knew they had stopped teaching long division; I'd never heard they'd also stopped teaching place value.
Of course, to understand place value you have to understand division & you have to understand that division and multiplication are inverse operations.
I spent quite a lot of time "contemplating" place value when I was reteaching myself K-6 math. It was incredibly helpful.
I've reinforced place value many, many times with Christopher. Basically any chance I get I remind him, or have him remind me, that when you "move to the left" you are multiplying by 10; when you move to the right you are dividing by 10.
This may be one concept he has down cold.
Let's hope.
btw, since I can't reteach & preteach math coherently around here one thing I do is just to repeat myself at every chance I get.
ReplyDeleteIt's probably a decent workaround.
This summer I may have decided (not sure!) to go back to Saxon 8/7 and review/reinforce material he knows just to establish it as firmly as possible inside his head.
ReplyDeleteThat's exactly what I'm doing. I have a seventh grader whom I've been teaching Saxon Algebra, but just this week decided to set the Algebra aside, in favor of Saxon 8/7.
ReplyDeleteThe newest Saxon 8/7 is very comprehensive, and although it has a core of pre-algebra, covers a much broader range of topics as well.
My 9 year old is also halfway through 8/7, while dabbling in Algebra because he can't wait. LOL
If you're looking for a really thorough preparation for higher level math before embarking on Algebra, 8/7 does a good job.
ec,
ReplyDeleteI loved Saxon 8/7. I took it myself just to make sure I could teach my special ed son better. There were things in there that I had never been taught (but should have been.)