Amy Gaisser
Dr. Bloom
ED 316
Reflective Writing About Session #7 on November 1, 2011
For this lesson, I had planned a string of problems that I thought would help Yadira to think
about subtracting by making “jumps of ten.” I thought she would really connect with this
strategy, because she had told me the week before that counting by tens was one of her favorite
strategies. Also, since she had drawn several pictures for me, I wanted to make something special
for her. I came up with the idea that the minuend, subtrahend and difference of each problem
could be circled (or enclosed in a heart) on the page. I had put various numbers on the page, so
that Yadira would not know ahead of time what the pattern would be, so that it would be a
surprise. I arranged the numbers I would circle in a way that was symmetrical on the page,
because I knew that Yadira loves symmetry. Finally, after becoming familiar with this new
strategy of making “jumps of ten,” I was going to have Yadira create some problems of her own
and add this new strategy to her list of subtraction strategies.
We were also going to begin the lesson by having a quick “running review” of the strategies
we had been using. When I asked Yadira what strategies we talked about the last time, she was
silent for awhile and then said she did not remember any of them. Therefore, I thought I would
start the conversation by mentioning a strategy I remembered: using the number line (for “adding
on” and “removing”). Then, I asked Yadira again to tell me one strategy that we had used. I had
been hoping she would bring up counting by tens, because that would have led in nicely to
making “jumps of ten.” Nevertheless, I had to end up being the one to bring up this strategy. I
reminded Yadira that, in our previous lesson together, she had said that counting by fives and
counting by tens were her favorite strategies. She did not acknowledge any recollection of this,
however. She was pretty quiet today, more so than usual.
I did not even ask whether she had used any of the strategies since I last saw her, because I
could see that she was not thinking of them at the present. Thus, I mentioned that we were going
to learn a new strategy this time, and told her about what we would be doing. The first thing I did
was tell Yadira that there were 100 pennies and I asked her how much that many pennies was
worth. She said, “One,” which I interpreted to mean one dollar. Next, I said that we had ten
dimes. I asked her how much ten dimes was worth. She thought for a bit, and then said 100,
which I interpreted to mean that she saw ten dimes as 100 pennies (rather than as a dollar). I
went off of this and asked, “So, a dime is worth how many of the 100 pennies?” Yadira said that
there were ten pennies for each dime. Though she did not remember later that she said this, I
knew that she knew it. I hope it was okay to ask her these questions. I could have just assumed
she knew these things, but I wanted to make sure it was all familiar to her.
Then, I presented the first problem: 50 – 20. Instead of just circling the numbers in the
problem, I decided to discuss with Yadira the names of the numbers in the problem (for example,
minuend and subtrahend). She said she had not heard those words before, and I thought it would
provide a further learning opportunity to have her refer to them by name, rather than just saying
what numbers I should circle on the paper. (I let this go, however, because she did not seem to be
connecting with it and there were more important things to focus on in the time we had.)
As Yadira started thinking about the first problem, she became restless, moving around on
the floor as she sometimes does. She actually started taking the pennies in her hands and jingling
them around. When I acknowledged that she was making music, she smiled. I knew I would
need to give her more guidance in this instance. The wait time I was giving her was actually
losing her. She needed some more structure in order to be successful. Thus, I gave her specific
prompts along the way, such as, “How are you going to start?” For this problem, Yadira grouped
five dimes together almost right away. After doing so, she did not seem to know what to do next.
In prompting her to think about the twenty, she ended up taking two dimes away from the group
of five dimes. I could see that she really understood the concept of a dime representing ten. After
she separated out the two dimes, I asked her how she knew how many cents were left. (She had
already said the answer was thirty.) She did not respond, so I rephrased my question as, “Did you
count ten, twenty, thirty [pointing to each dime as I counted] to see what was left, or did you
count all three [of the remaining] dimes at once?” Yadira said she counted them all at once and
knew that three dimes was thirty, which tells me that she was not only making “jumps of ten,”
but that she was also able to make those jumps in increments greater than ten (or in tens). At the
end of this problem, I asked Yadira why she had chosen to use all dimes and no pennies. (How
she decided what coins to use was left very open-ended). She responded, “It’s easy,” which says
to me that she recognizes that counting the dimes was more efficient than counting the pennies.
The next problem was 70 – 60. I had been curious to see if she would jump in larger
increments, or if she would “add on” (since sixty was close to seventy). What Yadira did instead
was even more wonderful! She began by grouping seven dimes together. Then, I asked her
something like, “What are you going to do now to find the answer?” She did not answer that
question, but gave me the answer to the problem: 10. Naturally, I asked her about her thinking.
She said that she took “minus six.” I tried to ask more about this to verify what I believed she
had done. I questioned whether she had taken 70 – 6 or 7 – 6. At first, she agreed that she had
taken 70 – 6, but then when I mentioned 7 – 6, she changed her mind and said that this was what
she did. I cannot be sure, but I believe that she was “canceling out a common number.” I believe
she decided to ignore the zeros, since both numbers had them, and change the place value of the
seven and six to ones (from being tens). I showed her what I meant by her canceling out a
common number, and said that that is something mathematicians do, hoping that this would be
another opportunity for her to grow to see herself as a mathematician. This was very exciting!
The next problem was 70 – 57. I had wanted to see what Yadira would do with a subtrahend
that was not divisible by ten. Yadira started again by making a group of seven dimes. Then, she
took five out of the group of seven. When I asked about her thinking this time, she must have
thought she did not do the right thing and started to move her dimes back. I tried to assure her
that I could see she was doing mathematical thinking and I was curious to know more about
those thoughts. After that, she still did not seem sure of what to do. I tried to prompt her to try
something, and I asked her what she was thinking when she started working with the pennies.
She told me the answer was twenty-three. I believe she had intended to take fifty-seven pennies
away from seventy pennies, but had somehow missed taking away a group of ten. I did not say
anything about her answer, but proceeded to ask if she could also go about figuring out the
answer by starting with the seven dimes she had laid out.
Sometimes, it was hard for me to tell if she was thinking about the problem or something
else, if she was being shy or did not know. She was not as outgoing today as she had been in
other lessons. Nevertheless, I observed that she started to seriously work with the pennies again
(rather than the dimes). I could see she was counting them, and I thought I should not interrupt
her train of thought to ask about her thinking. When she was finished, all the pennies were in one
pile together again. She had found that there were 100 pennies. I tried to guide her back to the
task by asking, “How many of those 100 pennies do we need to make a dime?” Yadira either did
not remember what she had said earlier or did not want to say it once more. For the number 57, I
drew a line going to fifty from the five and a line going to seven from the seven, to try to help
her see that she had seven still left to take away. We were not connecting, however. I could see
she was getting anxious and needed to move around. Even though we had not yet had a chance to
explicitly talk about making “jumps of ten,” I decided that a sizeable change had to be made.
This was a turning point for me as a teacher. I have changed many aspects of many lessons
before as I have seen that there is a need to do so during a lesson. But I have never completely
thrown a lesson out the window to try something different! I could see that Yadira was shutting
down and not engaged with the coins at that time, so I said something like, “Why don’t we
stretch out a minute?” I did not know yet that I would not come back to the lesson plan. I just
thought it would be good for her to catch her breath and relax a minute, and get back into the
mindset. We stretched our arms over our heads and behind our backs, we reached down and
touched our toes, and we did some jumping jacks – or, at least, I did some jumping jacks. Yadira
smiled as soon as I suggested stretching out, but she was a little reluctant to do the jumping
jacks. At that moment, a thought came to me: I could do what I had observed Ms. Estridge do at
School 60 and make numbers into math problems! Instead of just saying, “I’m going to do four
jumping jacks,” I said to Yadira with a smile, “I’m going to do 1 + 3 jumping jacks!” Yadira
watched me as I counted up from one to four. She smiled and seemed engaged.
I had Yadira give me the next problem for my jumping jacks. She did not have to participate
in the jumping, but I was hoping to get her to participate in the math. I was not sure what Yadira
would do when asked to create her own problem, because we had not yet gotten far enough in
our plans to do that. Well, Yadira seemed to love it! Her first problem for me, the creation of
which I left totally open-ended, was 6 + 1. Yadira watched as I did 6 + 1 jumping jacks. Then, I
asked her to give me a subtraction problem. This time, she not only gave me the problem, 5 – 2,
but a story to go with it! (Unfortunately, I could not hear the name of the person or the object she
mentioned in the story. She was pretty quiet, even when I asked her to tell me again.) I thought
out loud about how I could “remove” starting at the five and counting two backwards, or how I
could start at the two and count up to five. I decided to count backwards from five when I did my
jumping jacks. Yadira’s next problem was another story: Someone [I could not hear the name]
had ten cookies. She gave away one cookie. How many cookies did she have left? I modeled my
thinking again, saying that it made sense to me to start at ten and count one backwards. I told
Yadira my answer and asked if I was right. She said to me that she was not sure. I think she was
probably more focused on challenging me with the question than on what the actual answer was,
but that was okay. She was using her knowledge and higher-order thinking to create the question.
The last problem we had time for was 20 dolls + 5 dolls. I had asked Yadira to give me a
problem that involved numbers greater than fifty. Although these numbers were not greater than
fifty, they were greater than the numbers she originally used in her problems (which were ten or
less). For this one, I asked Yadira if it would be better for me to count up starting at one or to
count on starting at twenty. She said that it would be better for me to count up from twenty. This
told me that she realized that she did not need to count all the numbers in twenty, but that she
could just start with the last one (i.e., twenty) and all the others would be included within that
number. In other words, she understood hierarchical inclusion. I was ecstatic about this! And,
Yadira did the jumping jacks with me this time around.
We had to stop there, but I did finish my picture with the hearts and circles (and a star) for
Yadira. At the beginning we had talked about the complementary colors Yadira had used in her
drawing for me from the previous week. I said that I hoped Yadira would like my picture,
although I thought hers was cooler. She then proceeded to show me her own unique design that
she created, which looked something like an “S.” She was obviously very proud of her artwork.
We talked about the symmetry of the drawing I had made for Yadira, and I mentioned how I had
done that because I wanted to create symmetry like I have seen her do before. I wanted to
validate that she was an artist with cool ideas that I wanted to try, too. I am constantly trying to
keep building rapport with Yadira, and I think our common interest in art is one way to do that.
I learned so much from Yadira today! I hope she is learning from me, as well. She taught
me that it is okay to throw all my plans out the window, if it is for the student’s best interest.
This calls for improvisations sometimes. I had been thinking for awhile about how I could
involve Yadira in subtracting in a more active way, but had drawn a blank as far as ideas go.
Then, today, the idea just came to me as we were working together. By being attentive to and
aware of our students’ needs, we can learn what it is that we ought to do next. Our students truly
help us to re-plan our lessons as we go. Their needs take precedent, and when we follow them, it
will lead to great things. Being flexible led to more learning and enjoyment for Yadira today.
Though I have already been working on a homemade book about making “jumps of ten” to
share with Yadira during our last time together, I see that it will be more meaningful to go in
another direction. We did not really get to talk about making “jumps of ten,” but we have talked
about having a variety of strategies in our toolbox, and we have especially focused on using a
number line to “remove” and “add on” (although, I do not think Yadira really connected yet with
the “adding on” strategy). Another major piece of learning we have been working on is
explaining our thinking. To assess Yadira in an authentic way next time, I plan to have her put
her artistic talents to use and make a small book. In this book, she will include some subtraction
problems that she creates. When she does each problem, I will ask her what strategies she could
use to solve it, and have her try them out and explain what she does. I will have a copy of the list
of strategies she made, so that she can remember them better this next time. This assessment is,
in part, an aspect of the closure, as well. I think it should be a fun way to see if Yadira can put
her strategy learning to use, and tie her love of art into math to create the book. Additionally, I
also want to create a picture for Yadira (perhaps on Photoshop) with some words that share the
mathematical thinking I have observed and heard her do. I want to leave her with that something
special from me, a reminder of the fact that she is and should view herself as a mathematician.