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Below is an example of Jacks work on the last day of the case
study. When compared
to the work above, this example shows that Jack is capable of
grouping ten ones together as a
cohesive unit without drawing each individual one.
I asked Jack if he could solve the second problem (30 pencils,
29 more pencils) in his head; he thought
for a moment, said no, and proceeded to draw this:
2. On the fifth day of the case study, the group read a book
titled Only Onethat was about
how multiples of one object can be a unit. Although the students
had some trouble with the
problems for the rest of the day, they all seemed to grasp the
concept and were able to use the
experience as a reference point later. One problem that seemed
particularly productive was
problem number 3 on Day 6. This problem was about cookies in
packages and some extra cookies.
Even though Jack had started by drawing each cookie in the
packages, he eventually remembered
that he could just write the number and used that strategy with
an impromptu problem. Sunny
began by drawing then switched to unifix cubes. However, she
didnt count the cubes by ones to
reach her answer; Sunny counted by 10s.
3. Next time, Ive decided to write problems with numbers that
are more conducive to using
tens in the strategy. For example, if the problem is something
like 20 pennies and how many more
pennies to have 45, it will be easier to build up from 20
pennies to 45 using unifix cubes in sticks of
ten. (Of course! I now say to myself.) This teacher reflection
was particularly interesting to me
because of the part in parentheses. However, the whole utterance
was insightful as both an
example of decision-making and reflection. As the saying goes,
Hindsightis 20/20 (or something
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like it) and this professor was no exception. She knew she
should have used other numbers to aid
the children in their problem-solving and used this hindsight to
influence her numbers for the next
day. Teachers always hope to get it right the first time but
sometimes we skip or forget some baby
steps that the students need in order to have a deeper
understanding. We just have to remember
that it is ok to go back or work a little slower depending on
the students needs.
4. Next Problem: Danielle has 62 beads. She wants to make
necklaces with 10 beads on
each. How many necklaces can Danielle make?
I believe that this problem would be good for the group because
it is similar to one of the
problems from the last day of the study and would provide more
insight into how fragile their
thinking of 10 as a unit might be. The number is also capable of
being used with unifix cubes but
would get tedious when drawing ones as opposed to a ten
chunk.
