Amy Gaisser
Dr. Bloom
ED 316
Reflective Writing About Session #4 on October 11, 2011
For this lesson, I had planned to focus with Yadira on the subtraction strategies of
“removing” and “adding on.” We were also going to work on explaining our thinking. And we
would do these things in the context of a pizza business. I created five pizzas and four pizza
order forms (the fifth one Yadira would create). The amounts of toppings that were requested on
the order forms for each pizza were strategically chosen in a “string” that, I hoped, would
naturally lead Yadira to utilizing these strategies. The first couple of orders would be conducive
to “removing,” so I knew we could explicitly talk about that strategy. The third order would be
conducive to “adding on.” The fourth order would cause Yadira to think about which strategy
was best to use in different cases. Finally, Yadira would create the fifth order and continue
analyzing the situations to determine the most appropriate strategy. I was also curious to see if
she would group and skip count when finding the original amounts of toppings in her inventory,
because I was not entirely sure where she stood with those concepts. I had planned to assess her
understanding of these concepts informally through conversation and the written work she did.
Though these were my plans, they did change, as plans usually do. We ended up having to
stop after starting to make the second pizza. To begin the lesson, I explained the context of what
we would be doing to Yadira, and I made sure that she had a basic understanding of what the
word “inventory” meant. (This actually ended up being a vocabulary lesson of sorts, because she
really connected with that word through experience, seeing it used in context!) Then, I gave her
the three types of toppings and asked if she could figure out how many of each of them we had.
For the mushrooms, she counted by ones. There were only ten of them, so it was a reasonable
strategy. However, I anticipated that she might also use that strategy for the tomatoes and
pineapples, even though there were more sizeable numbers of them. Thus, when we got to the
tomatoes, I asked her, “Is there any strategy we could use to count these quicker?” Yadira
thought about this for a minute. Then, I noticed her grouping the tomatoes in rows of ten. I asked
her to explain her thinking and, without any hesitation, she told me that she had grouped the
tomatoes into five lines of ten and counted the lines to get to 50. She had used grouping and skip
counting – this was so exciting! I had tried to lead her to using these strategies before, but she
had not until now. I mentioned this to Mrs. Harden in an email and asked if the children had been
working on these strategies. She said they had been working on number sense and place value
since the beginning of the school year (and would continue to do so). “After next week,” she
said, “we will start addition with and without regrouping with just 2 digit numbers.” She
indicated her hope that the kids’ foundation in place value would help them transition to using
this strategy. I would say Yadira is already well on her way!
For the pineapples, Yadira did not naturally seem to generalize what she had just done with
grouping and skip counting to this new situation. She seemed to be counting by ones again in her
head. She told me there were 29 pineapples. I asked her to explain her thinking to check if she
really did count by ones. Then, I prompted her by asking if there was a strategy we could use to
help us count them quicker, maybe something similar to what she had done with the tomatoes. I
gave Yadira a significant amount of wait time, but she did not mention or try another strategy.
Thus, I got more specific and said, “Could we make lines of five?” I had tried to leave this as
open-ended as possible, but I knew she needed a bit more structure in this case. She decided to
try this method to see if it could be done. She made lines of fives and counted by fives, arriving
at 30 for her answer. She realized that this answer was different than her first. We tried to
consider which one might be more accurate. Since she is a visual learner, I think seeing the
pineapples organized in that way helped her to realize that her second answer was the accurate
one. Even though she did not generalize these strategies on her own, she was able to utilize and
understand them, and she was also consistently able to explain her thinking, which she continued
to do throughout this lesson. This was a big breakthrough!
I was curious to know why she had suddenly started explaining her thinking to me almost
every time I asked her about it, so Dr. Bloom had suggested that I email Mrs. Harden to gather
her insights. He thought she might have noticed something about Yadira that day. I shared some
possible theories with Mrs. Harden: “(a) she has learned how to go about explaining her thinking
and has been working on that, (b) she is getting more comfortable with me, and/or (c) she was
having a great day on Tuesday.” Mrs. Harden shared with me that the students work extensively
in class on explaining their mathematical thinking, though it is often still hard for them to put it
into words. She also thought that Yadira was becoming more comfortable working with me. And
a final factor of which she reminded me was that Yadira is “a very sweet girl who gives her best
daily. That too is helpful.” I could not agree more! Yadira is a great student and person.
With the lesson starting out so wonderfully, I was surprised by how it continued. What I
realized was that Yadira, when counting backwards, did not always remember what number
came next. On the first pizza, the order asked for 2 tomatoes. Therefore, Yadira used two
tomatoes and was trying to figure out how many she had left in her inventory. This called for the
“removing” strategy. Yadira used this strategy in her head and said the answer was 47. I thought
that, since she was so close, she must have just counted backwards one too many. I drew a
diagram and, starting at 50, asked her to count with me backwards by ones. We also tried this
with our fingers to see what happened. Yadira still counted “50, 49, 47.” She really thought 47
came after 49, and I know she was wondering why I was still having us think about this. I did not
want to explicitly tell her the answer was not right; I had hoped she would realize that by herself.
For the pineapples on that pizza, Yadira said the answer (for 30 – 2) was 28. The reason she
got this so quickly and accurately was that, unlike the other toppings, she still had the pineapples
laid out in rows of fives. She could visualize removing two (or only counting three) from one of
those rows. I had noticed that she seemed disengaged when working with the tomatoes, but this
time, things moved more quickly. I think that helped to keep her interest and focus.
For the mushrooms, Yadira said the answer was eight. She was subtracting one from ten. I
asked Yadira what came right after ten when we count backwards; she said eight. I first
represented her thinking by drawing a diagram that went from ten to eight, removing one. I also
tried drawing a diagram of one up and one back from ten, to see if that jogged her memory. We
tried thinking about what came before ten when counting up. I think it was hard for Yadira to
picture the numbers in descending order, without having the visual of a number line. I could tell
she was wondering why I did not just accept her answer and move on. Again, I was trying not to
explicitly tell her the answer was not right, but at the same time, I also truly did want her to know
which number came after 49.
The final situation we worked on was how many tomatoes would be left after using three on
the second pizza (i.e., 48 – 3). I noticed Yadira, for the first time during this lesson, taking up her
marker to help her think about the problem. She made four tallies together and eight tallies
together. Then, she crossed off three of the eight tallies. She counted up the remaining tallies and
said the answer was nine. What I observed she did know was that one should remove the three
ones from the eight ones. What she did not realize, however, was that she had counted the four
tens as four ones. I specifically asked her what the “4” in “48” meant. She still thought it was
four. This caused me to consider her understanding of place value. I drew another diagram (like
the ones in our textbook) to try to help Yadira picture the backwards counting we were doing.
Nevertheless, none of the diagrams I drew during this lesson seemed to help her! I realized that
these diagrams are probably just for me to represent her thinking as I hear it, if indeed she is
thinking that way. But often, she seemed to come up with her own unique ways of thinking.
I really wondered if I should have made the task more open-ended, with more than one
possible answer so that she could be more successful. On the opposite side of things, I had
wanted to make a specific “string” that would lead her to discovering and using certain
strategies. What I realized after this lesson is that “strings” are more successful with a group of
kids, so that there are a variety of answers to consider (one of them likely being the right one)
and a variety of students to consider them (rather than just the teacher and one student). This is a
more positive experience, because the kids all work together to figure out which answer is right.
As a team, they are successful. Yadira did not have a team of students, and I did not want to
guide her too much. I wanted for her to be able to take the lead.
Thus, I realized Yadira was going to need more scaffolding next time. Instead of team
members, she would have extra support that would help her to be successful and to take the lead.
I would provide her with a number line to reference when counting backwards, so she could
become familiar with the numbers in reverse order. I would also have her group the remaining
toppings in some way that would be helpful to her when keeping the inventory. These things
would also help the lesson to move quicker, so that she would stay engaged. Additionally, we
would get on to considering the strategy of “adding on” for the third pizza, which I anticipated
Yadira would really connect with, given her strength with addition. This would help her to see
how addition and subtraction are connected. She could think about which strategy to use when.
But first, as Dr. Bloom suggested, I would involve Yadira in thinking about “adding on” with
some smaller numbers (i.e., 7 – 5, 9 – 7). I could create a “string” that would familiarize her with
that strategy, and then guide her in generalizing the strategy to the pizza-business context.
The lesson had started off positively – Yadira had even drawn a beautiful picture of a
butterfly to give me when I arrived! Additionally, she seemed a bit sad when we had to leave our
work with the pizzas unfinished at the end. We were building rapport and having fun. I hoped we
had not lost that rapport when Yadira became disengaged at times during this lesson. These
supports, I hope, will help her to show what she knows, be successful, and enjoy this math
experience. (Note: I am also going to draw a special picture for her in an effort to keep building
rapport. She was so thoughtful to do that for me. I want to do something nice for her, too!)