I’m quoting here from Keith Devlin’s book The Math Gene.
First, a experiment from Piaget:
… Piaget believed that children do not have a number sense until they acquire it at around four or five years of age. In one of Piaget’s experiments, repeated many times by different groups, a psychologist would show a four-year-old child two equally spaced rows of six glasses and six bottles and ask whether there were more glasses or more bottles. The child invariably answered that there were the same number. Presumably the child observed a one-to-one correspondence between the rows. The experimenter then spread out the glasses to form a longer row and asked the child again whether there were more glasses or more bottles. Now the child would answer that there were more glasses, apparently misled by the longer length of that row. “Obviously,” Piaget concluded, “this shows that the child does not have a properly developed number sense.”
This conclusion was overturned later in a spectacular fashion by James McGarrigle and Margaret Donaldson:
Like Piaget, they started by aligning two rows of the same number of objects and asking the child which row had more objects. After the child responded correctly, the experimenter pretended to look away while a teddy bear puppet lengthened one of the rows. Turning back, the experimenter exclaimed, “Oh dear, that silly teddy has mixed up the rows. Can you tell me which row has more objects again?” Children from two to five invariably gave the correct answer. Since the teddy bear had rearranged one of the rows, unseen by the experimenter, the child presumably found it reasonable for the adult to ask the same question again. Yet when the experimenter repeated the process with the same children but rearranged the objects him- or herself, the four- and five-year-old children responded exactly as they had for Piaget, basing their answer on length.
In other words, the children modified their answer based on the expectation of what they thought the adult wanted to hear.
Filed under: Education, Mathematics |
The Number Warrior 
That is awesome.
Hi Jason,
I am writing to say it is great to see you appreciate my brothers experiment which I tried to understand all those years ago. He died unfortunately while doing this work and I cannot have a conversation with him about it now. I understand it better. Margaret Donaldson’s book does great service to him as she was his supervisor on his phd. I have a letter from her she sent to our parents. James was my older brother and I looked up to him. I now lecture in early childhood and try to promote his work. Good luck with your work.
yours
John McGarrigle
Amazing, but we take kids into schools who tell us what they think, and 12 years later graduate students who tell us what they think we want them to think.
[…] post by Jason Dyer regarding the findings of Piaget being re- interpreted by James McGarrigle and Margaret […]
Interesting!
Not knowing all the details here, and with all due respect to both Piaget and McGarrigle/Donaldson, it seems like a lot more needs to be done here to conclude that the second experiment refutes the conclusions of the first one or that THE correct interpretation of the second is “the children modified their answer based on the expectation of what they thought the adult wanted to hear.”
I’ll look forward to checking out the work of the later experimenters, but since I can think of other reasonable interpretations of what happened in their first experiment, I’m hoping that there is a good deal more detail to explain how they came to their conclusion. In particular, I hope they looked at alternatives and that they spoke with the children afterwards. Otherwise, there’s a distinct danger of simply replacing one inadequate viewpoint (Piaget’s) with another.
Michael,
I am troubled by the “inadequate viewpoint” comment. Rather than disagree that it is incomplete in someway–imagining that some “universally true theory of knowing and learning” might exist–it seems much more productive to understand well what one particular theory works well to do.
It strikes me as rather dismissive of Piaget’s work, when it is such a profound and thorough theory that is still hardly understood in Math Ed research, and to a massively greater extent in the practice of mathematics education.
The EXACT same should be said of theories that account for the role of power and privilege in the social interactions of learning.
I think I am saying that we could be more effective in better utilizing these theories as we currently know them, than to critique and seek the “theory of everything.”
@blaw: Trust that I don’t believe in a ‘theory of everything,’ either. My use of “inadequate viewpoint” was probably less pointed than my usual writing. That is to say, my irony was intended for the triumphant belief that Piaget’s view was wrong and that that of the more recent investigators was right. Rather than go in with hobnailed boots, I tried a somewhat more polite tack and managed to imply something I don’t believe: that there are final theories or completely objective truths to be explained by them. That’s likely a reach in physics, but an utter chimera when it comes to human behavior.
That’s what I get for writing “out of character,” I suppose.
LIke Michael, I am concerned about the certainty of the conclusion, “the children modified their answer based on the expectation of what they thought the adult wanted to hear.” It is a viable theory, but not satisfactorily confirmed (imho). It seems another viable counter theory is that the child felt empowered by first answering “equal number of bears in each row” and receiving some sort of confirmatory response from the adult. Then, when asked again in these slightly modified conditions, the child repeated the prior answer based on what they felt they had been acknowledged for previously. A small nuance from the expectations conjecture, but one that foregrounds the social capital of “correctness.”
It is interesting to me, that Piaget looked for child’s ways of reasoning, and the second experimenter seemed to focus on the effects of the social interactions.
Maybe this is what MPG means when he identifies 2 different “inadequate viewpoints.”