think aloud case study
TRANSCRIPT
COVER PAGE
Using Collaborative Think-Alouds to Reveal Students’ Thought Processes During Math Problem
Solving
Sara Renae Mattson
PO Box 152
Pismo Beach, CA 93448
(805) 234-5272
* This manuscript is the result of an independent action research study conducted in the author’s 5th
grade classroom.
Using Collaborative Think-Alouds to Reveal Students’ Thought Processes During Math Problem
Solving
January 6, 2015
Abstract
Math problem solving is an abstract concept that many students find challenging.
Observing their problem-solving processes through structured think-alouds provide teachers and
researchers valuable insight into these challenges; as a result, allowing educators to tailor their
instruction to student needs. Conducting one-on-one collaborative think-alouds with four
students of varying math levels, the teacher-researcher for this study discovered marked
differences in students’ abilities to problem-solve a math problem involving Pascal’s Triangle.
Through this study, the teacher-researcher also learned the importance of students’
comprehension of mathematical language is it related to the assigned problem.
Using Think-Aloud Strategies to Reveal Students’ Thought Processes During Math Problem
Solving
As a first year teacher, the teacher-researcher struggled with teaching math to her fourth
grade students. Math had always come so easily to her that, when presented with the necessary
formulas and equations, she could not understand why most of her students “just didn’t get it”!
The teacher-researcher spent countless hours that first year of teaching in workshops and
trainings to help her become a better math teacher, to learn how to better teach the formulas and
equations that came so easily to her.
Her entire first year of teaching she was constantly inventing and reinventing herself as a
math teacher, looking for that “A-ha!” moment that would make her students magically “get it”,
but with little success. She felt like a failure to her students that first year but, as luck would have
it, she was presented with the rare opportunity to move up to fifth grade with that first group of
students, giving her a second chance to try and make up for her previous year’s shortcomings.
Not able to find the “magic cure” for teaching math, the only difference she made in her
teaching from the first year to the second was incorporating more cooperative learning
opportunities, in all subjects, not just math. It seemed to truly help many of the students who
“just didn’t get it” the year before. The teacher-researcher attributed the increase in students
“getting it” to the following:
• Some students were “teaching” the math concepts, which helped those students retain the
concepts better
• Students were hearing different ways to learn the math concepts from their peers
• Learning math in small group settings helped students to stay focused on the task, which
aided in their learning
What she did not realize, at the time, was that working in small, cooperative groups gave
students the opportunity to process their math problem solving out loud, and in doing so, allowed
them to self-monitor their own learning. The teacher-researcher began to realize the importance
of working through problems for the sake of understanding them and not just to find the right
answer. From a constructivist lens, “the focus is on the process, where students create meaning
and knowledge, and where the students are actively engaged” (Finlayson, 2014, p. 110).
Working cooperatively with others, it was no longer sufficient for students to simply find the
right answer; they were responsible for explaining how or why they got a certain answer.
It was not until she entered her graduate program that the teacher-researcher related the
cooperative learning approach to be a form of think-aloud, during which students verbalized their
thought processes as they problem solved. According to Koro-Ljungberg, Douglas, Therriault,
Malcolm, and McNeill (2012), using a think-aloud strategy enables the teacher to “identify the
types of information, processes, and contextual clues used during problem-solving”, also
providing a “lens into the ways in which different psychological and cognitive processes
facilitate or hinder the problem-solving process” (p. 738). Observing her students during their
collaborative math time, the teacher-researcher could glean valuable insight into what her
students were grasping, as well as collect the necessary information to create “instructional
strategies that specifically target the difficulties students face” (Koro-Ljungberg, et al. 2012, p.
740).
The teacher-researcher also began to understand the constructivist perspective on which
her earlier inquiries were based. In a constructivist classroom, activities are interactive and
student-centered, “encouraging students to take risks, to be responsible and to be critical and
independent thinkers” (Finlayson, 2014, p. 102). By using a constructive perspective, teachers
are able to “gain valuable insights into students’ reflections, self-evaluations, and ‘inner voices’”
(Koro-Ljungberg, et al., 2012, p. 739). Viewing her situation through a constructivist lens
brought the teacher-researcher to the following questions.
Research Question
When presented with a new math concept, how are students able to verbalize their thought
processes during problem solving through collaborative think-aloud strategies?
From this question arose others:
• Are higher-performing students of math better able to verbalize their thought processes
during a problem solving task compared to lower-performing students?
• If so, what are those differences and how do they help or hinder thought processes?
• How does student understanding of math language affect their ability to problem solve?
• What are the perceived differences between genders?
Review of the Literature
Very little research was found in the area of using collaborative think-aloud strategies
during mathematics instruction. Of the studies found, the majority of them were conducted with
college-aged participants, focusing on collaborative learning. There is impressive research on
how to teach students to problem solve (Jacques, 2005; Karpov, 2003; Smith & Marvin, 2006;
Yetter, Gutkin, Saunders, Galloway, Sobanski, & Song, 2006), as well as the potential benefits
of having students work in cooperative groups during problem solving (Yetter, et al., 2006).
However, very few of these studies shed light onto the “how” or “why” of mathematical problem
solving.
Using think-aloud strategies during math instruction
According to Wilhelm (2001), “think-alouds make invisible mental processes visible to
children” (p. 26). While math may start out concrete and tangible, focusing on basic addition and
subtraction, it can quickly escalate to higher-level abstract concepts (Evans, 2008). In their study
on using the think-aloud strategy in qualitative research, Koro-Ljungberg, et al. (2012) asked
their participants to “verbalize what they were thinking while attempting to solve the problem”
(p. 736). By doing so, the authors were able to “identify the types of strategies used by problem
solvers”, as well as how they were “acquiring and using information during problem-solving
strategies” (p. 736). Through their observations of student think-alouds “teachers get an
immediate sense of whether students are ‘getting it’ and can adjust instructions to more
effectively help students learn” (Evans, 2008, p. 20).
Teacher think-alouds are also important in that “sharing mental solution strategies at
school, your students are exposed to how math concepts are used and applied” (Silbey, 2002, p.
2). Some may argue that these higher-order thought processes are too complex for elementary
aged students, but Silbey (2002) posits that students can and should be exposed to higher-level
math concepts through teacher think-alouds. Silbey (2002) further encourages teachers to model
think-alouds because “by thinking aloud, you share a thought process that may be too
sophisticated for [students] to come up with on their own, but one which they are able to
comprehend by hearing” (p. 1).
Uncovering the mysteries of problem-solving
Problem-solving occurs when “concepts are learned through the need to solve a complex
problem” (Koro-Ljungberg, et al., 2012, p. 739). The fundamental problem with problem-solving
in mathematics is that concepts are taught as a sequence of procedures, skills, and formulas.
Jacques (2005) posits “problem solving has been made into a sequence of skills to be learned
rather than a process of mathematical thinking…taking the problem out of problem-solving” (p.
40). Essentially, students are taught how to add, subtract, multiply, and divide but are rarely
given the opportunity to explore “why” we add, subtract, multiply, or divide. Likewise, we
seldom teach students how these computations might relate.
Once students learn the basic computation skills, usually by rote-memorization, they are
then presented with more abstract math concepts that require them to “think”, but they have had
so few opportunities to “think” mathematically, it is almost like learning a foreign language. A
telling example of this would be the rote memorization of multiplication facts. Students are told
to memorize their multiplication facts, which is an important skill as it enables students to
process higher-functioning math skills more quickly; but, then students are presented with
problems such as 3 x n = 12, and are unable to see the connection that 3 x 4 = 12. Students are
rarely taught to think about why 3 x 4 = 12. According to Smith and Marvin (2006), memorizing
multiplication facts before “developing an understanding of multiplicative situations and their
quantities prematurely narrows students’ focus and gives students the wrong impression about
the need to understand what it means to multiply” (p. 46). Consequently, in order to “think”
mathematically, students need to know how to “speak” mathematically.
Mathematical Language as Knowledge
According to Ferrari (2004), mathematical language is considerably different from
“every-day life” language, which proves to be a major obstacle for many students learning
math.mMathematical language is considered to be its own unique register, defined as a
“linguistic variety based on use” (Ferrari, 2004, p. 384), possessing similar characteristics of
literate registers, such as academic communication and books. To be successful in the use of
mathematical language, one needs to have enough experience with it to recognize the unique
characteristics between the literate language, mathematics, and the colloquial language, or
“every-day life” language, to be able to switch between the two (Ferrari, 2004).
Based on this intricate process, translating math problems into understandable language
requires much practice and exposure. Krawec (2012) discusses “problem translation”, which
requires students to “read the problem for understanding and then paraphrase it by putting the
problem in their own words” (p. 104), and identifies the challenge for many students is their lack
of understanding of the mathematical language used. If students cannot understand the diction of
a math problem, how can we expect them to put it in their own words?
Likewise, if teachers cannot hear students’ thought processes during math problem-
solving, how can they determine if students understand the math problems? This is where think-
alouds come in to the equation, specifically collaborative think-alouds.
Collaboration as a form of think aloud
In working collaboratively, students are in fact participating in a form of a think aloud
(Yetter, et al., 2006). This idea falls under the cognitive elaboration view, just one of many
learning theories that support learner collaboration. Yetter (et al. 2006) posits that through
cognitive elaboration, students are better able to retain new information and transfer it to existing
knowledge. According to Evans (2008) “for students to retain the often complex and process-
oriented math knowledge that they are required to master, they must have adequate time to
reflect on their learning” (p. 20).
A second theory that supports the effectiveness of collaboration as a think-aloud is the
constructivist theory, which states that peer interaction facilitates complex reasoning (Yetter, et
al., 2006). “By talking about their thinking with themselves or with another peer, students can
get feedback from themselves and others that helps validate or refute certain ideas they have
formed about their learning” (Evans, 2008, p. 19).
A third perspective under collaborative math think-alouds is the social interaction
perspective. Under this perspective, collaboration is effective in that it requires partners to work
together for a common goal and when that goal is reached, they share in the success. Some
researchers have found that collaboration facilitates more learning in mathematics than does
independent learning (Reglin, 1990, as cited in Yetter, et al., 2006) because students are able to
“share their thinking, and work through a variety of steps to a solution” (Finlayson, 2014, p.
111).
Methodology
Participants
Four students were chosen from one fifth grade class to participate in this study.
Pseudonyms have been given to all four students to protect their privacy. The fifth grade class
from which the students were chosen is part of the Oxnard Elementary School District that
serves K-8th grade students. The ethnic make-up of the school is primarily Hispanic, with Asian
and Caucasian students making up the second and third largest ethnic groups, followed with a
small portion of Black students.
The students were chosen based on their performance on an End-of-the-Year math
assessment. The top scoring male and female, Kevin and Anna, were chosen, as well as the
lowest performing male and female, George and Bianca. The students chosen will best shed light
on the questions being pursued, the differences in problem solving between high-performing and
low-performing math students, as well as any possible gender differences. Both females and the
low performing male are Hispanic in origin, while the high scoring male is of Asian descent.
Study
Students in this study were given an unknown task and asked to think aloud as they
worked through the problem. Rosenzweig, Krawec, and Montague (2011) describe this as a
“concurrent” think-aloud, in which the participant engages in the think-aloud during problem-
solving (p. 509). The teacher-researcher used the think-aloud process on a regular basis during
math lessons, so students were comfortable with the process. Students were also familiar with
working in pairs. During math think-aloud activities, there were two roles, “think-alouder” and
note-taker, both students taking a turn at each job. The “think-alouder” was given a word
problem or other such math task that required cognitive processing. While this student processed
the task out loud, his or her partner recorded what was said. Note-takers were encouraged to
write down as much as they could but were not expected to record everything. Then students
reversed roles.
During these activities, students were allowed to help one another if the “think-alouder”
asked. The note-taker was instructed to be a silent partner until help was verbally expressed. The
teacher valued these exercises because they fostered mathematical thinking in a way that the
regular math curriculum did not, and they allowed for students to help one another through the
problem-solving process. These activities also taught students, namely the note-takers, how to
listen carefully to others.
The problem presented to the students in this study was a problem-solving question
involving Pascal’s Triangle (see Appendix A). Pascal’s Triangle was used for its logical pattern
base. Students had to use thought processes to figure out the rule-based pattern that makes the
triangle. The pattern is symmetrical, which allowed students to “see” a relationship within the
triangle. According to Barton (2003), “the numeric form of Pascal’s Triangle is a mathematical
wonder” (p. 35), making it an ideal problem solving activity.
For this study, the teacher-researcher served as each participant’s collaborative partner.
Before continuing with the think aloud, the teacher-researcher asked each student if s/he knew
Pascal’s Triangle. Since none of the students were familiar with it, the teacher-researcher
proceeded with the task. The directions were for students to find the pattern for which the given
portion of Pascal’s Triangle (the first five lines) was based on and upon identifying the pattern,
students were then asked to complete the next row of the triangle using the same pattern.
The teacher-researcher read the directions out loud to each participant and then asked
each child to think aloud as they tried to complete the task. Students were encouraged to try to
complete the task on their own but when they explicitly asked for assistance or the teacher-
researcher felt the student was “stuck”, she offered collaborative assistance, guiding the students
to find the answers themselves rather than giving them. Koro-Ljungberg, et al. (2012) suggests
the role of the researcher, in this case teacher-researcher, should be that of “observer and also to
prompt participants at points where they may not be giving adequate details on their thought
processes or during extended periods of silence” (p. 738).
Data Collection and analysis
The data collected included the transcribed audio tapes of each participant’s think aloud,
student activity sheets of Pascal’s Triangle, and teacher-researcher observation notes. The
students were audio taped, with their parents’ written consent, so as to ensure the teacher-
researcher captured each student’s think-aloud in its entirety.
Using a qualitative approach, participant audio tapes were transcribed and then coded for
emerging themes. The completion of the activity sheets, and how accurately each participant
solved the problem, was recorded for comparison. Teacher-researcher observation notes were
used to triangulate the other two methods of data.
Results
Participant #1, Anna (all student-participants have been given a pseudonym to protect
anonymity), approached the problem solving task by reading the numbers in the triangle out
loud. When no pattern was immediately apparent, she made the comment, “This is hard.” Since
there appeared to be no pattern in reading the numbers, Anna then proceeded to look for
multiplication patterns by noticing the 2 in row three and the 3 in row four multiply to get the 6
in row five. She then realized the multiplication pattern did not apply elsewhere so she went
back to reading the numbers out loud, first counting down the triangle and then up. After
another “this is hard” comment, she continued to look for a pattern within the numbers of the
triangle.
Once the teacher-researcher noticed Anna was “stuck” on reading the numbers, she
guided the participant by saying, “I’m thinking the word pattern might be a little deceiving.
Okay. So what I want you to think about is, think about it mathematically. For example, think of
a formula, like adding, subtracting, multiplying, or dividing. So see if you can see any
relationships between adding, subtracting, multiplying, dividing. Okay, but I want you to think it
out loud.”
As the participant continued, she reread the numbers out loud but also made the
comment, “I’m going diagonally.” This was a great example of her “thinking out loud” but Anna
continued to simply read the numbers. At this point, the teacher-researcher asked the participant
if she would like help. Anna said yes. Here the teacher-researcher took on the role as
collaborative partner by showing Anna the pattern of adding the two adjacent numbers above to
get the number in the next row. Once this was explained, Anna was able to identify other
patterns in the triangle, such as, “Okay, 1 and 4 is 5…Okay and then…4 plus 1 and then 1…5
plus 1 is 6”. At this point she got confused and said, “So…oh, okay…then 2, then 3 is…I’m
confused now.” This was another great example of the student-participant “thinking aloud” and
assessing her own ability to complete the task. The teacher-researcher, acting again as a
collaborative partner, suggested the student circle the numbers she was adding together to keep
her organized.
After more successful identification of the addition pattern within the triangle, Anna
again confused herself by adding numbers from different rows. When the teacher-researcher
asked if the pattern she used earlier jumped from one row to the other, Anna said no and was
then able to finish connecting all the given numbers, as well as complete the next row for the
triangle.
While participant #1 was unable to identify the pattern for Pascal’s Triangle on her own,
she was able to identify the additive pairs within the triangle once the pattern was explained.
Although she confused herself when she got farther down the triangle, she was able to continue
to apply the pattern with minimal redirection from the teacher-researcher. She was also able to
transfer what she learned about the pattern to successfully complete the next row in Pascal’s
Triangle.
Participant #2, Kevin, approached the task by reading the numbers in Pascal’s Triangle
row by row, which was different from the other participants who first read the numbers
diagonally. He then recognized that each row always begins and ends with ones. When looking
for a pattern, Kevin said, “Seems like it multiplies or something.” He applied this multiplicative
pattern to the 2 in the third row and the 3 in the fourth row, saying 2 x 3 = 6, but realized it didn’t
work in the following row when he said, “There’s not a second 2…there’s no 2 times 2 equals
4.”
Once Kevin realized the multiplication pattern would not work, he said the following,
“Let me think…Ummmm…Seems like 1 plus 2 equals 3…1 plus 3 equals 4…Don’t know where
that 6 came from…This is challenging.” Kevin was clearly on the right track by identifying an
addition pattern within Pascal’s Triangle. He began to second guess himself, however, when he
could not immediately make an addition connection for the 6 in the fifth row, so he went back to
multiplication and then again back to addition.
It was clear that Kevin was able to assess himself when he was thinking aloud. Once he
identified the addition pattern, he applied it to various numbers within the Pascal’s Triangle and
found that it worked. When he came to a number he could not find an additive relationship for,
he went back to his original thinking of multiplication, quickly dismissed it and went back to
what he knew for sure, “Okay, it always has to begin and end with 1…Ohhh….Ummm...always
begins and ends with 1…What’s the pattern?”
At this point, the teacher-researcher could sense some frustration from Kevin, so she
asked him if he would like some help. He said yes, so the teacher-researcher directed him back to
the addition pattern he identified earlier. He knew the pattern was addition but he did not see the
pattern for which two numbers should always be added. Once this was pointed out to him, Kevin
was able to complete the next row of the triangle with ease and confidence.
Participant #3, Bianca, approached the given task differently than any of the other
participants, “I’m gonna read it (the task) first so I can understand it.” Before she even tried to
read the triangle, Bianca’s first thought was, “Let’s see..ummm….This is hard…What is it like
suppose to mean?” At this point, the teacher-researcher explained that she needed to find a
pattern. After some silence, the teacher-researcher asked Bianca if she knew what a pattern was.
When she could not explain what a pattern was, the teacher-researcher replied, “A pattern is
something that occurs over and over, right? So you’re trying to find out how Pascal got these
numbers. What pattern did he use?”
Bianca continued to ask for help, “How did they get the 6?” until the teacher-researcher
said, “That’s what you need to figure out.” Bianca finally tried to identify any pattern within the
triangle herself by counting the numbers from 1, 2, 3, 4, 5…6, 7, 8, 9, even though not all these
numbers were represented in Pascal’s Triangle. After counting a few times, she expressed
extreme frustration when she again said, “It’s hard.” At this point, the teacher-researcher drew
the participant’s attention to the outside of the triangle, where all the numbers were ones. With
this observation, Bianca was able to make the connection that the rest of the outside of the
triangle will be ones also.
The teacher-researcher then tried to lead the participant to find the pattern within the
triangle by asking how she thought they got the number 2 in the third row. “Because I think they
went down counting it or by the sides.” Here, the teacher-researcher circled the two 1’s in the
row above the 2 and without having to say anything, Bianca exclaimed, “Oh!!! They added!” In
trying to find the additive pattern that got the 3 in row four, she added 3 plus 3. Bianca was
unable to make the connection from adding 1 plus 1 in row two to get the 2 in row three to
adding 1 plus 2 in row three to get the 3 in row four.
At this point the student-participant needed systematic modeling of how to connect the
two numbers that were being added to get the numbers in the following rows. The teacher used
lines and plus signs to do this. Bianca kept confusing herself with previous rows that we were
finished with by trying to add numbers from different rows, so the use of the lines connecting the
two numbers above and adjacent to any given number on the triangle proved to be helpful. Once
the five rows were explained, step by step, the student was able to complete the next row of
Pascal’s Triangle with little guidance from the teacher-researcher.
The fourth participant, George, began by stating, in more of a question-like format, “I
think you have to add...this…And then that goes in order?” He gave the teacher-researcher a look
that appeared to be searching for approval. The teacher-researcher replied with, “Okay, don’t ask
me, just try it first.” George then rambled on a bunch of numbers, some not even related to the
triangle. There appeared to be no rhyme or reason to his reading off of numbers, until his last set
of <6, 4, 10>. This set of numbers could be taken as an addition problem since 6 + 4 = 10, but
the student did not explicitly state it this way.
This student, when asked, could not explain what a pattern was so the teacher-researcher
explained, “A pattern, or think about this, a pattern is something that repeats itself, right? So, if
we didn’t have this row here, do you think that we got this row here by adding these? So here’s
1, 2, 3, 4…that would be a 4 there. If we were using the pattern that you were using, would that
work?” This statement may not make sense without the context it was made in. The numbers the
student-participant was rambling off when he first started were the numbers in numerical order
starting with the numbers in the last row: <4, 5, 6, 7, 8, 9, 10> and <6, 7, 8, 9, 10, 11>. The
teacher-researcher tried to help George see that his pattern, of simply counting in order, did not
apply to the rest of the triangle. For example, if we counted from the top of the triangle
beginning with 1, this pattern would only work on the diagonal row of <1, 2, 3, 4> but not <1, 3,
6>.
George did not appear to be following this line of logic, so the teacher-researcher directed
his attention to what she thought would be an easier line of logic for him to follow. “Do you
notice that the outside of the triangle is all 1’s? So what do you think will be the next number
here?” for which George responded, “Two.” The teacher-researcher repeated the question
slightly different, “If the outside of the triangle is all 1’s, what’s going to be on the outside of the
triangle here?” The student again answered, “Two.” At this point, the teacher-researcher asked
him why he chose two and he replied, “Because…the next number.”
Here the teacher-researcher had to re-explain what a pattern is to the student because he
continued to say two for this response. The teacher-researcher eventually had to tell the student
that the next number would have to be a one, since all the other numbers on the outside of the
triangle were ones. Once this was made explicit to the student-participant, he was able to figure
out that a one would have to go on the outside of the other side of the triangle.
Even after being directed to the possibility that the inside of the triangle might follow an
addition pattern, George was unable to identify the pattern for which Pascal’s Triangle is based.
After the teacher-researcher explained the additive pattern for every single number in the given
portion of the triangle, George was unable to complete the next row. His original answers were
<1, 6, 0, 0, 5, 1>. He was not even able to verbalize his thought process or explain how he got
these numbers, he just chose them. The teacher-researcher, wanting to give George one more
chance to show his understanding of the pattern within Pascal’s Triangle, showed him how to
add the 1 and 4 in row five to get 5 in the next row. The student-participant was still unable to
complete the sixth row of Pascal’s Triangle as his final answers were <1, 5, 9, 9, 5, 1>, getting
the nines incorrect.
Discussion
Individual differences between students’ think-aloud sessions
Kevin displayed high levels of mathematical problem solving and the ability to think
aloud while problem solving, much more so than the rest of the participants. He came closest
than any of the other participants in finding and applying the pattern within Pascal’s Triangle.
Kevin was able to verbalize his thought processes by saying, “It seems to get bigger…So it can’t
be smaller…Huh!!!!!” and “Oh, that’s it? So then the two above…That’s 10…That’s a 5 there
and a 5 there.”
Anna also displayed higher levels of mathematical problem solving and was able to
express her thinking more than Bianca and George. She tried multiple ways of identifying the
pattern within Pascal’s Triangle and, although unsuccessful on her own, was able to recognize
when something did not work. This self-reflection is important in the thought processes because
students will not know to try something else if they are not first able to identify that what they
have already tried does not work (Evans, 2008).
Bianca, although unable to identify or apply the pattern without assistance, was able to
verbalize her thought process, which was evident in statements such as, “I’m gonna read it first
so I can understand it” and “How did they get the 6?” Often times, asking questions when one is
trying to problem solve makes explicit what we do not know so that we are better able to solve
the problems.
George attempted the given problem solving task without much thought processing or
confidence. He looked to the teacher-researcher for guidance and approval from the very
beginning of the task. Unlike the other three participants, George was unable to complete the
task, even with guidance from the teacher-researcher. It is unclear from this study if he was
unable to complete the given task because he did not have the skills to problem solve or because
he did not have the prior knowledge and vocabulary skills to verbalize his thought process.
Differences between high-performing and low-performing students
In comparing the data between the two high-performing students and the two low-
performing students, we see clear distinctions. First, the high-performing students were able to
use their prior knowledge of patterns to actually look for patterns within Pascal’s Triangle and
recognize when certain patterns did not apply to the entire triangle, while the two low-
performing students were not. Kevin and Anna, the two high-performing students, made
comments and connections such as, “Okay, 1, 2, 6, 9 …That’s going by…oh, that doesn’t
work…I’m going diagonally” and “4 and 6, that’s 10…And then…that’s 10 and 5 is 15…10 and
10…6 and 15 that is..um 15…plus 6 equals…6 and 5 is eleven and ten is twenty one” (Anna).
Kevin was noted for saying, “Always ends and begins with ones..Seems like it multiplies or
something” and “2 times 3 equals 6…There’s not a second 2…There’s no 2 times 2 equals 4”.
The two low-performing students had to have the concept of a pattern explained to them,
and even after having it explained, were unable to transfer that knowledge to the present task of
finding the pattern within Pascal’s Triangle. There was evidence of this through comments made,
such as, “Oh!!! They added!” (Brenda), but immediately after she recognized the addition
pattern, she then tried to add 3 plus 3 to get 3, which was not the way she was shown. George
showed his inability to transfer the knowledge of patterns when he replied to the teacher-
researcher’s direction, “The thing I want you to notice. Do you notice that the outside of the
triangle is all 1’s? So what do you think will be the next number here?” He replied,“Two”, when
clearly the response should have been one.
Another distinction between the high-performing and low-performing students was their
use of mathematical language. Anna used phrases such as, “2 times 3 is six” and “I’m going
diagonally”, both examples of mathematical language; while Kevin was quoted as saying,
“Seems like it multiplies or something” and “Always begins and ends with 1…What’s the
pattern?”
Bianca, however, did not use any mathematical language until the teacher-researcher
prompted them. For example, Bianca only used the term “Oh!!! They added!” after it was
pointed out to her. George did use some mathematical terms, such as “I think you have to
add..this” and “And then that goes in order?” However, in both cases, the terminology was used
with no evidence of understanding why he was using them; he simply seemed to be saying them
with no context in mind at all.
Conclusions
As a beginning teacher in mathematics, the teacher-researcher was most concerned with
teaching her students a variety of formulas and equations, not necessarily how, when, or why
they should use said formulas and equations. She thought that giving her students a variety of
ways to problem solve would be sufficient in creating critical thinkers. What she did not realize,
however, was that problem solving is more than a list of strategies or skills; it is a way of
thinking (Jacques, 2005). Mathematics is a “sense-making activity” (Smith & Marvin, 2006) not
merely computational skills.
The purpose of this study was to “see” the thought process of 5th grade students during a
problem solving task in mathematics using a collaborative think-aloud protocol. This study did
not use collaborative learning in the traditional sense, where students were paired up with other
students, as the teacher-researcher acted as collaborative partner to the student-participants. The
results of this study support the research that says students are more likely to develop problem
solving skills by engaging in problem solving inquiries (Jacques, 2005).
From this study, the teacher-researcher found that students performing at higher levels of
mathematics outperformed those of lower abilities when using problem-solving strategies. A
correlation was also found between problem-solving and mathematical language as the use of
vocabulary knowledge better enabled the higher-performing students to verbalize their thought
processes. Students were also able to transfer what they knew about patterns, and the fact that
patterns repeat themselves, to the new task at hand. Although none of the participants were able
to identify the pattern used to create Pascal’s Triangle on their own, it was evident that the
higher-performing students were able to apply the pattern to the next line of the triangle once it
was explained to them, whereas the lower-performing students were not. This is an important
distinction because many researchers believe that the ability to transfer prior knowledge and
apply it to a new task is an indicator of a student’s overall level of cognitive development
(Karpov, 2003).
Following in the footsteps of neo-Piagetians’ ideas, students’ cognitive development,
which can be viewed using think-aloud strategies, such as were used in this study, is dependent
on what Karpov (2003) calls “information-processing capacity”. Students with higher levels of
information-processing capacity have more complex information-processing strategies from
which to draw from (Karpov, 2003). These levels are in turn affected by maturity and practice.
The earlier children are taught to think mathematically, rather than to simply memorize or
plug in formulas, the more they will develop the necessary information-processing strategies
needed to be successful in mathematical problem solving. It is not clear whether the higher-
performing students had earlier experiences with thinking mathematically but from the teacher-
researcher’s knowledge of the participants’ school history, one low-performing student was a
severe behavior problem and missed many days of school as a result, while the other low-
performing student was in bilingual classes until fourth grade, which was the time that he was
presented his first experiences with mathematical thinking in English.
There were no perceivable differences between genders. As a matter of fact, while the
higher-performing male participant outperformed the higher-performing female participant, the
opposite was true of the lower-performing participants. This makes it impossible to make any
claims as to whether gender plays a role in a student’s ability to verbalize his orher thought
process during a problem solving task.
Limitations and Implications for Future Study
There were two main limitations in this study that bear consideration for future research.
One limitation to this study was the fact that there were only four participants observed. The time
constraints the teacher-researcher faced restricted her ability to include more participants. In the
future, more participants would help strengthen the findings and make the results more
generalizable across a larger group.
Second, the use of a think-aloud strategy is contingent on the postulation that the student-
participants can effectively verbalize their thought process while actively engaged in a math
problem. Further inquiry into students’ exposure to and comfort using mathematical language is
needed.
A larger, more stringent version of this study would offer the education profession much
insight into students’ thought processes in mathematics. The importance of students’ prior
knowledge, ability to transfer this prior knowledge, and the ability to verbalize mathematical
thinking using appropriate mathematical language warrants future inquiries into this area of
research.
In summary, observing students using a collaborative think-aloud strategy during math
problem-solving uncovered important implications for math instruction. One was that students
need time to engage in think-alouds during math. An effective method for doing this would be to
incorporate more collaborative learning opportunities for students. Second, students’ use of
mathematical language is an important factor to consider when evaluating students in math
problem-solving. It may be that students understand how to solve a problem but lack the
vocabulary to properly verbalize it. Creating a vocabulary rich math environment may prove
helpful. In both cases, using collaborative think-aloud techniques would help teachers better
understand their students’ strengths and weaknesses in math problem-solving.
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Appendix A: Student Worksheet
Pascal’s Triangle: A mystical, mathematical pyramid
Please review Pascal’s Triangle. Identify the pattern. As you problem solve, think aloud your process so your partner can hear. You may use this space for your scratch work: Participant’s name: ____________________________________ Date: ____________________
1 1 1
2
1 1 3 3 1 1
6 4 1 4 1 1 1 5 5 10 10
1 1 6 6 15 15 20