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Sharing Ideas
Counting Blocks in a Mirror
Sometimes we can arrange materials in such a way
that they almost “beg” to be counted. Try placing
a strip of polished mylar on a table covered with blocks. The
mirror instantly “doubles” a stack of blocks when
you look into the reflection. Lets watch Ria, a clever 5 year
old, count her blocks both in the mirror and the “real” blocks
on the table. Click here to see clip.
Notice how Mary Beth, the teacher, invents a reason to talk
about subsets. After Ria counts to 6 for both reflected and
real blocks, Mary Beth asks Ria how many is it “really.” Ria
counts just the blocks on top of the mirror. As it turns out
this game of “looks like” versus “really” plays
host to several interesting math concepts, as you will see.
Now Ria decides to add one more layer of blocks to her structure.
Before this new layer is placed, Ria asks Mary Beth if she knows
how many that will be. Mary Beth says, “I wonder.” Ria
assures Mary Beth that “you don’t have to wonder,” implying
that she, Ria, can tell Mary Beth for sure. Mary Beth accepts
the aid and asks Ria how many. Notice that Ria does not recount
the three blocks on the table. Rather she says, “If you
have 3 and you add one more, you have 4.” Watch
the clip. [Further explanation of Counting On]
So, did you notice when Ria said “You have four” that
she held this one block up for Mary Beth to see. What could this
mean? Why say,“You have four” and then hold up only
one. In this case the one block in her hand does not symbolize
a one; it symbolizes the last block in a count of four. In other
words, this block stands for the cardinal value of the set. Cardinal
value is the total number when all objects have been counted
only once. Watch clip again. [Further explanation of One as Four]
Mary Beth wants Ria to consider two sets of blocks, the real
ones and the reflected one. Ria complies and begins to count
both blocks in the mirror and the wooden blocks. She counts 1,2,3
for the blocks that she sees in the mirror and 4, 5, 6 for the
wooden blocks that have been placed, but she also says 7 and
points to the space where the new block will be placed. She did
not think to count a similar space in the reflected stack. Watch
this now.
Did you hear how Ria corrected Mary Beth? Mary Beth said, “there
will be seven altogether” and Ria was clear to say “It
will look like seven.” Ria enjoys this joke that deals
with the difference between the apparent and the real. While
this distinction between the apparent and the real has an important
developmental path, it is a different path than number development.
However, you can see how the teacher uses the child’s interest
in illusion to motivate the discussion on sets and subsets. [Further
explanation of The Empty Slot]
Now Ria has placed her extra layer of blocks to her structure.
Mary Beth sees this as an opportunity to check out Ria’s
prediction of 7 for the total (reflected and real). Mary Beth
asks how many in the mirror and the real ones altogether. Notice
how Ria counts the four in the mirror and announces that total.
But when she wants the number altogether, she recounts those
same 4 in the mirror. She could have simply started with the
number 5 as she pointed to bottom block in the real stack. Watch
this clip.
Ria did not “count on” from the total value of
the mirrored set to shift to the next number in the sequence.
Click here if you would like to read more about “counting
on.” [Further explanation of "Counting On"]
] But what Ria did do was to pause after she had counted the
four in the mirror. She structured the game one of counting two
separate sets. In so doing she exhibits some high level thinking.
She knows that this is not just a counting game, it is an adding
game. And for something to be added, you first need an original
set and then more added to that. You don’t just count all
the objects as if they belong to the same set. Notice the pause
this time when she says “its four it looks like in the
mirror, but if you add em up altogether it gets …..” Watch
clip again.
Just to bring closure to this little game, Mary Beth asks Ria
to remember her prediction, which was seven. Mary Beth asks, “Remember
how many you guessed before?” Ria had just counted them
to 8 so lets assume that she understood Mary Beth to mean the
total of reflected and real blocks, even though Mary Beth did
not say that. Ria says “4” Watch this a see if you
can figure out any reason why Ria would say 4. Watch clip now.
It is true that Ria guessed “7” when she pointed
to the empty space on the top of a stack of 3 wooden blocks.
Maybe she was thinking that a “guess” was a prediction
for the wooden block yet to be placed, and that would have been
block number 4. We don’t know for sure, but it is reasonable
to speculate that her memory had a firm grasp of the blocks that
she handled and the memory of the relation to the reflected blocks
was less dominant. But then when she hears the questioning tone
in Mary Beth’s voice, she thinks more carefully about the
original question that defined her guess. She changes her answer
to “7.” Then Mary Beth nails the case shut with the
final remark, “and there were 8.”
If the timing had been right, Mary Beth could have asked Ria
why she had guessed seven. With the teacher’s help Ria
might have enjoyed the joke that she had forgotten to consider
the reflected block to be added as well as the wooden block.
Often it is helpful to encourage children to revisit a guess
in order to understand the thinking that lead to the guess. Two
things are accomplished by doing this. One, the child will learn
more about how the game works (the mirror doubles the count).
And two, the child will realize that an error is not random or
careless. Errors often have their own logic and rules. If children
can revisit their work they ultimately have more respect for
the their thinking and they will ultimately treat mistakes as
useful information as opposed to failures.
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© 2003 Learning Materials Workshop
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