Natalie Mabile
Problem Teaching Critique
During my practicum, I presented 3 of the problems from my problem log to
a small group of students (ranging from 3 to 5 students on any given day) from my
two classes of fifth-grade students. They are selected for this group because they
need additional assistance with mathematics.
Problem 1: “Dad bakes some cookies. He eats one hot out of the oven and leaves the
rest on the counter to cool. He goes outside to read. Dave comes into the kitchen and
finds the cookies. Since he is hungry, he eats half a dozen of them. Then Kate
wanders by, feeling rather hungry as well. She eats half as many as Dave did. Jim and
Eileen walk through next, and each of them eats one third of the remaining cookies.
Hollis comes into the kitchen and eats half of the cookies that are left on the
counter.
Last of all, Mom eats just one cookie.
Dad comes back inside, ready to pig
out. ‘Hey!’ he exclaims. ‘There is only one cookie left!’ How many cookies did Dad
bake in all?” (Answer: 22)
For this problem, I anticipated that my students would start to solve the
problem by working backwards, knowing that there is just 1 cookie left at the end.
To help them, I asked prior to their working on the problem, how many cookies
were left after everyone ate cookies for the purpose of clarification and reminder. Of
the four students to whom I presented this problem, only one found the correct
answer on her first attempt. Two students, including the one who solved correctly,
did work backwards.
The first student who did not work backward, who I will refer to as Student 1
for this problem, opted to guess and check. Her first guess to start with was 10. I
allowed her to use this number and tokens as manipulatives until she reached Kate
eating cookies and realized that the number did not work. Student 1 wanted to next
try 11, but I hinted to her to try even higher a number since 10 did not leave enough
cookies for the remaining family members. She grew more frustrated when she
found she couldn’t divide her tokens into thirds when she started with 20 cookies. I
struggled not to tell her what number to try, and encouraged her to try again with
another number. Her answer was 25 cookies, and though she got to observe other
students sharing how they reached their answers, she still did not seem to
understand how hers was incorrect. In working with her again, I would try to help
her write out the numbers she’s getting as she works with the manipulatives to keep
herself organized and not make mistakes in her calculations, which she was as she
showed me how to move the tokens during the parts involving fractions.
Student 2 began by using the numbers he could figure out from the problem,
starting with Dad eating 1 and Dave eating six, as the student stated that half a
dozen is 6. He then added 3 for Kate because half of six is three. At this point,
Student 2 had added up 10 cookies. The part with Jim and Eileen is where Student 2
hit a problem. First, he did not seem to focus on the word “each” in the problem, as
he incorporated Jim and Eileen as though they were just one person eating one third
of the cookies. Additionally, he became frustrated because he tried to add one third
of the cookies he had calculated being eaten (10), rather than from the cookies that
remained. I encouraged him to reread the problem and highlight important words;
however, he failed to highlight the word each, so I pointed it out to him and asked
him what it meant for the problem. In re-attempting the problem, he now opted to
work backward. He added 1 + 1 for the remaining cookie and mom’s share. He
became confused when it came to halves and thirds, however, most likely because
he was now using the opposite operations from those he was using originally due to
working backward. To help, I modeled for him how to add back half of an amount
with the token manipulatives. I am not sure he completely understood why we were
adding the tokens in the amount we were, but in working with him using the
manipulatives, he was able to get the correct numbers of cookies to add back for
each step, except that he again grouped Jim and Eileen together in eating one third
of the cookies, rather than one third each, so I had to point this out to him again in
order for him to reach the correct answer of 22.
My third student reported an answer of 30 cookies. He added dad’s one
cookie to Dave’s half dozen (1+6 = 7). Student 3 then noted that half of Dave’s
cookies would be 3, so Kate ate 3 (7+3 = 10). Student 3 then began to get the events
slightly out of order, which messed up his calculations. He stated that mom eats one
cookie, and dad finds one cookie at the end, so there are then 12 cookies (10+1+1 =
12). He then determined that Jim eats 1/3 of the 12, and Eileen eats 1/3 of the 12,
and 1/3 of 12 is 4, so there are 20 cookies (4+4+12 = 20). Then, because Hollis eats
half the cookies, Student 3 concluded that there were 30 cookies, because half of 20
is 10 (20+10 = 30). I would encourage him in the future to make a chart, putting the
family members in the correct order of when they took cookies to stay organized
and avoid the errors he did, as his calculations would have been correct if he could
have kept them in order. It also seems to me that he may have put the events with
fractions later because they were harder for him to use, so I would simply state for
him that this problem requires the events to be kept in order, and that doing so
would make the fractions easier to incorporate.
The student who got an answer of 22 from the start did made a chart prior to
working backward to keep herself organized. On her chart, she wrote: Dad – 2
(which is not entirely correct, but did not seem to be problematic, as she still began
with 1 cookie in her calculations); Mom – 1; Dave – half-dozen; Kate – half of Dave’s;
Jim - 1/3 of what’s left; Eileen – 1/3 of what’s left; Hollis – half of what’s left; Mom –
1). She asked me in beginning her problem whether Jim took cookies before Eileen
or at the same time, because that would change the fractions. I was unsure whether I
was supposed to tell her, but it did seem confusing, so I told her they took cookies at
the same time. She then clarified some of the numbers on her chart as she worked
through the problem, changing Dave’s half-dozen to 6, and Kate’s portion to 3. She
would do the addition on a separate paper and then changed the numbers on her
chart to match. It would have helped her to do her addition in a new column of the
chart to ensure she did not make any mistakes in going in order, but her strategy
seemed to work regardless. She started addition with 1 (presumably the remaining
cookie at the end), then added mom’s one (1+1=2). She then found that she needed
to double the remaining cookies to account for Hollis taking half (2+2=4). Then,
knowing that Jim and Eileen each took a third of the cookies at the same time, she
recognizes that 4 would be 1/3 of the total cookies at that time, so there were 12
total cookies (4x3=12). She then added 9 to represent the cookies she previously
determined were eaten by Kate and Dave (6+3=9; 9+12 = 21). Lastly, she added the
one that dad ate to reach the answer of 22 cookies (21+1=22).
Conclusion: Overall, the biggest issue I had with presenting this problem
was that it was difficult to resist telling students to work backwards and let them
use their own approach, particularly when they seemed to be getting off track with
their approach. I know, however, that it is important for students to work through
problems on their own to hone their skills and see which strategies work best for
them. For the future, I would make sure that all students have the option to use
manipulatives and have background knowledge about a variety of strategies, such as
making charts or drawing pictures, that they could utilize to solve this problem.
Thus, if they became stuck using one strategy, they could turn to another rather than
becoming frustrated.
Problem 2: Mr. Lincoln’s 32 fifth grade students are standing in line to go to lunch.
The 3rd child, and every 3rd child after, is a boy. The 5th child, and every 5th child
after, is wearing glasses. The 4th child, and every 4th child after, is carrying a lunch
box.
1. What do you know for certain about the 20th child in line?
2. What do you know about the 24th child?
3. What do you know about the 30th child?
(Answer: 1: the 20th child is a girl, has a lunchbox, and wears glasses; 2: the 24th child is
a boy, has a lunch box and does not wear glasses; the 30th student is a boy who does
not have a lunch box, and wears glasses.)
At first, all three students who completed this problem were confused by the
original wording of the question, so I changed it slightly to ask the students whether
each of the three particular students targeted in the problem 1) was a boy or a girl;
2) wore glasses; and 3) carried a lunch box.
My first student made a list by writing out each number from 1 to 32. She
then added a B next to every third student to indicate that the student was a boy.
Last, she counted every fifth child on the list and put a G next to it to indicate that
the student wore glasses. By scanning down her list and seeing which letters were
placed at 20, 24, and 30, she correctly identified that the 20th child was not a boy,
but did have a lunchbox and glasses; that the 24th child was a boy and had a
lunchbox, but did not wear glasses; and that the 30th child was a boy who did not
have a lunchbox, but did have glasses.
My second student wrote out all the multiples of 3 up to 32 to determine
which students were boys. He then wrote out all the multiples of 5 up to 32 to
determine which students wore glasses, and wrote out all the multiples of 4 up to 32
to determine which children carried lunch boxes. Looking at the multiples of 3, he
concluded that both the 24th and 30th students were boys. Based on the multiples of
5, he concluded that the 20th and 30th students wore glasses. Lastly, he found using
the multiples of 4 that both the 20th and 24th students had a lunchbox.
My last student started making a list of multiples similar to that of my second
student, but stopped halfway through and instead started listing just the factors of
each of the numbers relevant to the problem (20, 24, and 30). This strategy most
likely occurred to him because we have been recently discussing Greatest Common
Factors in class, which requires students to write out factors. Using this strategy, he
also found the correct answers.
Conclusion: This problem, though advertised for grades 3-5, may be below
the level of my students. If I were to use it again for students of a similar level, I
would adapt it to make it more challenging. However, I did learn from presenting
this problem how valuable it is for students to be provided with a range of resources
with which they can work on the problem. Each student solved this problem
differently, but all received the correct answer while feeling confident because they
were more comfortable with their personal strategy.
Problem 3: “Wendy builds wooden dollhouse furniture. She uses the same kind of
legs to make 3-legged stools and 4-legged tables. She has a supply of 31 legs and
wants to use them all to make stools and tables. Find all the possible ways she can
use all 31 legs.” (Answer: (1) 4 tables and 5 stools; (2) 1 table and 9 stools; (3) 7 tables
and 1 stool)
My first student concluded that Wendy could make: 4 tables and 0 stools, 1
table and 7 stools, and 0 tables and 12 stools. She started by drawing the 31 legs to
solve the problem. She had many circles and erasure marks on her paper, explaining
that she had grouped them in different ways to solve. However, I could not figure
out her strategy outside of this, nor how she came up with the particular numbers
she did.
My second student said that Wendy could make: 10 stools and 0 tables, 9
stools and 1 table, 8 stools and 1 table, 7 stools and 2 tables, 6 stools and 3 tables, 5
stools and 4 tables, 4 stools and 4 tables, 3 stools and 5 tables, 2 stools and 6 tables,
1 stool and 7 tables, and 0 stools and 7 tables. She wrote out a chart starting with
the 10 stools and going down to 0 stools, with the right side of her chart showing the
number of tables she found to correspond to that number of stools. She used
division and subtraction to find these numbers. Based on her answers and strategy,
it appears that while she did not recognize the part of the problem that stated that
Wendy wanted to use all 31 legs, since many of her solutions would leave extra
remainder legs unused.
My last student in this group grew extremely frustrated with this problem.
When I asked him for his answer, he told me “4.” I asked him what 4 meant,
specifically whether 4 referred to a number of tables or chairs, he paused and
eventually said tables. Looking over his attempt at the problem, in which he drew
the legs and then started drawings them into tables and stools seemingly randomly,
I am not sure he understood what the question was asking, and this may have been
the cause of his frustration.
Conclusion: The main issue I noticed my students experiencing with this
problem was in interpreting the problem and developing strategies that fit. To help
with this, if I were to reteach this problem, I would have the students use their
highlighters to focus on important numbers and words in the problem, and I would
ask them to rephrase the problem in their own words to check for understanding
before allowing them to proceed.