European Journal of Education Studies
ISSN: 2501 - 1111
ISSN-L: 2501 - 1111
Available on-line at: www.oapub.org/edu
10.5281/zenodo.154128
Volume 2│Issue 4│2016
FOURTH GRADERS AND NON-ROUTINE PROBLEMS:
ARE STRATEGIES DECISIVE FOR SUCCESS?
Yeliz Yazgani
Assist. Prof. Dr., Elementary School Department,
Uludag University Faculty of Education, Turkey
Abstract:
This study aims to determine the explanatory and discriminative powers of non-routine
problem solving strategies used by fourth graders. Six problems were asked to 240
pupils. After scoring answers between 0 and 10, bottom and top segments of 27% were
determined based on total scores. Lastly, all scripts of students in these segments were
re-scored with regard to strategy use. Multiple regression and discriminant analysis
were utilized to evaluate data.
Results showed that strategies explain 84% of the problem solving success, and their
importance order is as follows: look for a pattern, work backward, make a systematic
list, make a drawing, guess and check and simplify the problem. Strategies which play a
significant role in distinguishing students are look for a pattern, make a systematic list,
work backward, simplify the problem, make a drawing, respectively. Further
researches using more problems, and examining students of different grades are
suggested.
Keywords: non-routine problems, problem solving, problem solving strategies, fourth
grade students, students’ success
Introduction
We are consistently confronted with problems on a daily basis such as deciding which
kind of shoe is more suitable for us, reaching an address we have ever been, planning a
birthday party, and deciding for whom to vote in an election etc. Whether they are
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
mathematical or not, these examples share a common core: A problem arises when a
living creature has a goal but does not know how this goal is to be reached (Novick and
Bassok, 2005; p. 321). Therefore, problem solving has been described as what you do
when you don’t know what to do ”ilstein, Libeskind & Lott,
.
Since it has so much importance in our daily live, it is inevitable to say that
problem solving is the cornerstone of school mathematics, and without the ability to
solve problems, the usefulness and power of mathematical ideas, knowledge, and skills
severely limited [National Council of Teachers of Mathematics (NCTM), 2000]. Besides,
mathematics education communities commonly agree that teaching problem solving
means teaching non-routine problems as well as routine problems. Actually, a large
body of literature about mathematical problem-solving show that non-routine problems
are the kind of problems which are most appropriate for developing mathematical
problem-solving and reasoning skills, and development of the ability to apply these
skills in real-life situations (e. g; Polya, 1957; Schoenfeld, 1992; Cai, 2003; London, 2007).
Because, routine problems can be solved using methods familiar to students by
replicating previously learned methods in a step-by-step fashion.
The followings are examples of one-step and two-step routine problems
respectively:
-
A bag contains 2 dozen cookies. Andy bought 3 bags. How many cookies did she get?
-
A book store purchases 3 packages of books, and each package includes 12 books. If price
of one book is 8 liras, how much do all books cost?
Non-routine problems are problems for which there is no predictable, wellrehearsed approach or pathway explicitly suggested by the task, task instructions, or
worked-out examples (Woodward, Beckmann, Driscoll, Franke, Herzig, Jitendra,
Koedinger, & Ogbuehi, 2012). Importantly, non-routine problems need reasoning and
higher-order thinking skills and often go beyond procedural skills (Kolovou, van den
Heuvel-Panhuizen & Bakker, 2009).
To have a more detailed opinion about what a non-routine problem is, following
problem and its solution can be revised:
“ male bee is born from an unfertilized egg, a female bee from a fertilized one. So, in
other words, a male bee only has a mother, while a female bee has a mother and a father. How
many total ancestors does a male bee have going ten generations back?
This problem can be solved with the help of the diagram in Figure 1. In this
diagram F means female and M means male.
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
Figure 1: Solution of the non-routine problem sample
Without completing this diagram until the tenth generation, solution can be reached by
using the pattern among numbers of bees in every generation: Add the first and second
number (1 + 1 = 2) to get the third number, add the second and third number to get the
fourth number (1 + 2 = 3) and so on. According to this pattern, solution of the problem is
55.
Non-routine problem solving strategies can be defined as procedures used to
explore, analyse, and probe aspects of non-routine problems in an attempt to formulate
pathways to a solution (Nancarrow, 2004). They play a very important role in the
mathematical process experienced by students while solving non-routine problems.
Results of recent studies have provided evidence for the use of strategies as a means to
enhance problem solving (Elia, Van den Heuvel-Panhuizen, & Kolovou, 2009). In the
literature, the most outstanding non-routine problem solving strategies are as follows:
Act it out, look for a pattern, make a systematic list, work backward, guess and check,
make a drawing or diagram, write an equation or open sentence, simplify the problem, make a
table, eliminate the possibilities, use logical reasoning, matrix logic, and estimation. (Altun,
Bintas, Yazgan & Arslan, 2007; Herr & Johnson, 2002; Leng, 2008; Posamentier & Krulik,
2008; Posamentier & Krulik, 2009).
However, most renowned ones that are in use at elementary school level will be
presented here:
a)
Make
a drawing or
diagram:
This
strategy
includes using
supportive
representations to solve problem. It provides a way to depict the information in the
problem and make the relationships apparent. In the following, there is a problem that
making a drawing strategy seriously facilitates to reach the answer in solving process,
and a fifth grader’s script about the solution of the problem (Figure 2):
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
Students in a class are standing in a circle; they evenly spaced and are numbered in
order. The student with number 7 is standing directly across from the student with number 17.
How many students are in the class? (Bilstein, Libeskind, & Lott, 1996; p.49)
Figure 2: A solution including make a drawing or diagram strategy
There are 20 students.
As understood from what the student did, drawings do not need to be detailed or exact.
The only things that need to be drawn are what is essential in solving the problem
(Reys, Suydam, Lindquist, & Smith, 1998)
b)
Look for a pattern: Finding patterns includes items or numbers that are repeated,
or a series of events that repeat. It enables persons to reduce a complex problem to a
pattern and then use the pattern to derive a solution. Often the key to finding a pattern
is to organize information. Following problem is a good example for which look for a
pattern strategy can be utilized solely:
An explorer found some strange markings on a cave wall. Can you find and complete the
pattern? (Reys, Suydam, Lindquist, & Smith, 1998; p.77).
In Figure , a fourth grade student’s answer to this problem is presented. The
student explained the pattern that he found by writing.
Figure 3: A solution including look for a pattern strategy
I multiplied the number by 2, and then
subtracted 1.
c)
Make a systematic list: This strategy is exactly what the name implies: making an
organized list that requires some method of counting things or making certain that
every possibility has been covered. Sometimes simple counting of the different ways or
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
situations would be impossible. As can be seen in the following answer of a sixth grade
student to How many three digit numbers can be formed from the digits 3, 5, and 7, if each
digit can only be used once? problem (Figure 4), an organizational scheme guarantees
that all thing required for counting have been included, and none were counted twice
(Van de Walle, 1994; p.50).
Figure 4: A solution including make a systematic list strategy
d)
Guess and check: When using this strategy, a person makes a reasonable guess,
checks the guess, and revises the guess if necessary. By repeating this process someone
can arrive at a correct answer that has been checked. Using this strategy does not
always yield a correct solution immediately but provides information that can be used
to better understand the problem and may suggest the use of another strategy.
In Figure , an eighth grade students’ solution to the question of Tolga’s team
entered a mathematics contest where teams of students compete by answering questions that are
worth either
points or
points. No partial credit is given. Tolga’s team scored
points on
questions. How many 5 point questions did the team answer correctly? (Bilstein, Libeskind, &
Lott, 1996; p. 49) is presented.
Figure 5: A solution including guess and check strategy
e)
Solve a simpler or similar problem: When problems that seem to be overwhelming
due to their complexity and/or large numbers are encountered, it may be useful to try
the same problem with much smaller numbers of fewer conditions. In this situation, one
of two things will happen: (1) solving the easier problem may help to see a way of
original problem; or (2) there is a chance that working a series of simpler problems,
starting with the easiest possible and systematically increasing the difficulty, may lead
to a pattern or generalization that will solve the original problem (Van de Walle, 1994;
p.52). For the question presented in the following, this approach may really work: Each
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
of following shapes consists of small triangles like the first one. How many triangles do you need
to make fifteenth shape?
“ fifth grade student’ solution represented in Figure
is completely convenient to the
essence of solve a simpler or similar problem strategy.
Figure 6: A solution including solve a simpler or similar problem strategy
Üçgen= triangle
f)
Work backward: Some problems are posed in such a way that people is given the
final condition of an action and are asked about something occurred earlier (Reys,
Suydam, Lindquist, & Smith, 1998; p.80). In another words, the work backward strategy
is useful when a series of events takes place and we know the result but need to
determine the condition at the beginning. If the events are arithmetic operations, the
task is to reverse these operations.
In other problems a series of unknown events results in some condition, and the
task is to find out what the sequence of events is. If not all steps are obvious,
occasionally the last one is relatively clear. From the last step you can then get to the
next to last, and so on (Van de Walle, 1994; p.51), like a fifth grader did to solve the
following question (Figure 7):
A bus driver started to drive from the terminal and dropped half of his passengers off at
every bus stop he stopped by. Meanwhile, no passenger got on the bus. After stopping by 3 bus
stops there were 8 remaining passengers on the bus. How many passengers got on the bus at the
terminal?
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
Figure 7: A solution including work backward strategy
Answer = 64
According to Tiong, Hedberg and Lioe (2005), the most recognizable and immediate
difference between non-routine problem solving strategies and algorithms is that nonroutine problem solving strategies do not guarantee solution, while algorithms (if the
procedures are followed exactly) guarantee a solution. Another and perhaps more
important difference between the two is that algorithms come with very specific
procedures, while non-routine problem solving strategies have non-rigorous
suggestions about what should be done. Also, algorithms are usually applicable to a
certain type of question in a specific topic area, while non-routine problem solving
strategies are generally applicable through types of question and topic areas. This raises
another important characteristic of non-routine problem solving strategies, that of
transferability. The most important reason to learn them is that because they can help
individuals solve problems in unfamiliar topic areas and expand their point of view.
In spite of the above mentioned importance of non-routine problems and nonroutine problem solving strategies, their place in math education in Turkey is limited.
In the general explanations at the beginning of elementary school mathematics
curriculum developed by Ministry of National Education (MoNE), guess and check, make
a systematic list, make a drawing, work backward, simplify the problem strategies are merely
mentioned, and just look for a pattern strategy were taught under the title of Patterns
and Tessellations
MoNE,
. Findings of nationwide research reveal that non-
routine problems, and strategies used to solve them, are not introduced to students
extensively in textbooks nor learning environments (Altun et al, 2007; Yazgan et al.,
2013).
Importance of the Study and Research Questions
Most research about non-routine problem solving is aimed at examining students’
currently used skills and attitudes on this subject without any intervention (e.g. HokWing & Bin, 2002; Wong & Tiong, 2006; Muir, Beswick, & Williamson, 2008; Wong,
2008; Elia, et al., 2009; Salleh & Zakaria, 2009; Mabilangan, Limjap & Belecina, 2012).
There are also some studies examining effects of an instruction on non-routine problem
solving skills of students (e.g. Follmer, 2000; Ishida, 2002; Nancarrow, 2004; Yazgan et
al., 2005; Johnson & Schmidt, 2006; Lee, Yeo, & Hong, 2014). Another group of studies is
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
focused on place of non-routine problems and strategies in mathematics textbooks and
syllabi (e. g. Tiong et al., 2005; Kolovou et al., 2009; Marchis, 2012;). Grade levels of
these studies vary from primary school to high school and their results can be
summarized in three points: (i) many students consider that non-routine problems are
more complicated and difficult than routine problems, and students may not initially
believe that the non-routine problems are mathematical since they are not familiar with
this kind of problem, (ii) providing students with a framework for the use an
application of strategies is beneficial and increased students’ level of confidence, iii
only a very small proportion of the problems included in the textbooks is non-routine.
In some textbooks series these problems are completely absent.
Still, studies investigating the impact of each of non-routine problem solving
strategies on the problem solving success of students are really rare. Moreover, none of
them are at fourth grade level (age 9-10). However, determining importance order and
distinctiveness of each strategy in terms of problem solving success may bring some
important outcomes out and they can be used to arrange textbooks and learning
environments. Therefore, this study first aims to draw a general picture about strategy
use of fourth graders, then (and more importantly) investigates the role of each strategy
in explaining problem solving success and in discriminating between successful and
unsuccessful students at fourth grade level.
Thus, research questions are as follows:
1. What is the overall situation about usage of non-routine problem solving strategies at
fourth grade level?
2. What is the role of each non-routine problem solving strategy in explaining the problem
solving success?
3. Which non-routine problem solving strategies are more effective in discriminating
between successful and unsuccessful students at fourth grade level?
Related Studies
Studies by Hok-Wing & Bin (2002) and Elia et al. (2009) will be mentioned more indepth here because of three reasons: (i) they were also at fourth grade level, (ii) they
aimed to seek student’s existing abilities on non-routine problem solving without any
instruction like this study; (iii) their methodologies were similar to the present study
with regard to allotted time for tests, and number of questions which tests included.
Distinct from this study, participants of Hok-Wing & ”in
’s study were
from four regions of China, and their findings showed that different and low
performances on the choice of problem-solving strategies of pupils from four regions
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
had been measured. Generally, the make a drawing strategy was successfully used by the
students, while usage of look for a pattern was unsuccessful.
Study carried out by Elia et al. (2009) is differentiated at the point of subject
being focused. Authors investigated strategy flexibility as well, and they proposed and
investigated two types of strategy flexibility: inter-task flexibility (changing strategies
across problems) and intra-task flexibility (changing strategies within problems).
Findings showed that these students rarely applied strategies in solving problems.
Among strategies, the trial-and-error (guess and check) strategy was found to have a
general potential to lead to success. The two types of flexibility were not displayed to a
large extent in students’ strategic behaviour. However, on the one hand, students who
showed inter-task strategy flexibility were more successful than students who
persevered with the same strategy. On the other hand, intra-task strategy flexibility did
not support the students in reaching the correct answer.
Although it was carried out with teacher trainees, study which is most related to
the present study was done by Altun and Sezgin-Memnun (2008). Unlike this study,
participants were given problem solving instruction. However, authors also tried to
find out which strategies are the leading indicators of problem solving skills. Multiple
regression analysis revealed that the relative importance order of the problem solving
strategies in terms of their effects on the problem solving success is as follows: simplify
the problem, look for a pattern, reasoning, make a drawing, make a systematic list, guess and
check, and work backwards. According to discriminant analysis results, the strategies of
reasoning, work backwards, make a drawing, make a table and simplify the problem,
respectively had a big impact in separation of successful and unsuccessful participants.
The analysis also confirmed that 80% of the problem solving success could be explained
by the problem solving strategies.
Method
Research Design
The research design of this study is of the descriptive type. This type of research
illustrates the situation at the time of the study. Namely, it involves gathering data that
describe events and then organizes, tabulates, depicts, and describes the data collection.
Descriptive research can be either quantitative or qualitative. As preferred in this study,
it involves collections of quantitative information that can be tabulated along a
continuum in numerical form, such as scores on a test or the number of times a person
chooses to use a-certain feature of a multimedia program, or it can describe categories
of information such as gender or patterns of interaction when using technology in a
group situation (Glass & Hopkins, 1984).
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
Participants
Two hundred and forty four students in fourth grade participated in the study from
eight different primary schools governed by MoNE (Ministry of National Education) in
Bursa/Turkey. Participating schools were chosen by using simple random sampling.
Participants did not have any experience with non-routine problem solving in their
school life before the current study, since this kind of problems have not been
emphasized in mathematics education in Turkey.
Information about Data Collection Instrument and Procedure
To measure the problem-solving success of students, a paper and pencil test (Problem
Solving Test – PST) comprised of six non-routine open-ended problems was
constructed by the author. Problems in the test were intended for using the strategies of
make a systematic list, look for a pattern, work backward, simplify the problem, make a drawing,
and guess and check strategies (see Appendix). Selection of these strategies were derived
from previous studies, projects and books (Follmer, 2000; Altun et al., 2007; Leng, 2008;
Posamentier & Krulik, 2009; Elia et al., 2009; Lee et al., 2014) dealing with strategies that
can be learned and used at fourth grade level.
PST was conducted by the author and students were given 45 minutes to
complete it. But, if a student needed, he/she could have extra time. Students were
encouraged about to write down all their thoughts while they were solving the
problems.
“ll students’ answers for each problem were scored between
and
by
categorizing them under the titles of correct, little mistakes stemming from calculation errors
or inattentiveness, insufficient answers despite understanding the problem and taking the right
action, wrong answers and no answer. Every student was scored between 0 and 60 points.
The author and another researcher evaluated answers independently and the given
points were compared to each other. If there was a big difference between points given
to a student solution, the solution was re-examined together by the scorers to determine
a common score. The focus was on the correctness of the students’ problem solving
processes. Strategy usage was not taken into consideration at this point.
Based on scores for each question in the PST instrument, Cronbach's alpha was
used to assess the reliability of the instrument. Generally, the lower limit for Cronbach's
Alpha should be .70 (Robinson, Shaver, and Wrightsman, 1991). The alpha coefficient
was computed as .713 for the PST indicating acceptable reliability for the instrument.
To assess validity of the PST, factor analysis with Varimax rotation was
performed based on the same scores as used for the Cronbach alpha reliability. Before
proceeding with factor analysis, it was necessary to assess the sampling adequacy. This
was done using the Kaiser-Meyer-Olkin (KMO) and Bartlett's test for Sphericity. The
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
level of significance of KMO and the Bartlett's test for Sphericity should be significant at
the .05 confidence level (Hair, Black, Tatham and Anderson, 1998). Table 1 below shows
that the results of the tests meet acceptable levels that permit proceeding with the factor
analysis.
Table 1: KMO and Bartlett's test results
Kaiser-Meyer-Olkin Measure of Sampling Adequacy
.74
Approx. Chi-Square
Bartlett's Test of Sphericity
Df
158.51
15
Sig.
.000
Factor analysis results suggested the existence of two factors. The eigenvalues showed
that the first and second factors explained 33.39% and 19.41% of the variance of PST
scores. Collectively these two factors explained over 52% of the variance. As seen from
the rotated component matrix (Table 2), factor loading of each item is greater than .45.
Thus, it can be said that problem solving success was properly measured by items of
PST. In other words, the administered PST test was validated.
Table 2: Rotated component matrix
Item
Factor 1
Factor 2
3
.72
.07
1
.68
-.26
2
.67
.34
5
.64
.24
6
-.04
.85
4
.40
.46
To answer the second and third research questions, (except the scoring system
explained above), all student scripts were evaluated with regard to strategy use again.
Instead of scoring each question separately, each student’s paper was considered as a
whole and usages of each strategies dealt with in this study were coded as 2 (correct
and effective usage of strategy), 1 (incomplete usage of strategy), and 0 (no usage of
strategy). So, every student had a point about usage of each strategy.
In Figure , one student’s answer to sixth question (In an exam, each right answer
was scored as 1 or 2. A student had 56 right answers and got 78 point. How many one-point
questions did he answer correctly?) In PST is represented to explain scoring processes of
problem solving success and strategy use thoroughly. This question could be solved by
using guess and check strategy, but the student did not prefer to use it. Actually, his
reasoning was quite different and elegant. He firstly supposed that all right answers
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
were scored as 1. Than he thought that he needs 22 extra points to get 78 points in total,
meaning that he has to convert points of 22 questions from 1 to 2. Thus, he found that
number of two-point questions was 22, and there were 34 one-point questions. Since his
reasoning and answer was completely right, this students’ problem solving success
score for this question was 10. But since he did not use guess and check strategy for any
questions in PST, his score about use of this strategy was 0.
Figure 8: “ participant’s answer to sixth question in PST
To account for inter-coder agreement (based on coding about strategy use),
Cohen kappa coefficient for inter rater agreement was calculated and it was found as
.81. Then, means and standard deviations about each strategy were computed. To
reveal how functional the strategies were in the problem solving success, multiple
regression analysis was executed by using strategy scores (independent variables) and
total problem solving success scores (dependent variable) for each student.
In order to see which strategies were more useful in discriminating between
successful and unsuccessful students, first bottom and top segments of 27% were
determined according to the total problem solving scores the students achieved in the
PST. The group at the bottom consisting of 27% of the students was composed of those
who had got a score of 9 or lower, and the group at the top consisting of another 27% of
the students achieving higher than 27. There were 63 students in each of bottom and top
groups. Later on, discriminant analysis was carried out based on the strategy and total
PST scores of students.
Results
Means and standard deviations about each strategy were used to have an overview
about strategy usage levels of participants (n=240). According to results shown in Table
3, it can be said that usage of look for a pattern and make a systematic list are quite close to
the average. However, low means for the use of other strategies may be an indicator of
the fact that students are not very familiar with these strategies.
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
Table 3: Some descriptive statistics on usage of strategies
Strategies
Mean
Std. Deviation
Make a systematic list
.82
.88
Look for a pattern
.95
.90
Work backward
.39
.72
Simplify the problem
.23
.61
Make a drawing
.36
.56
Guess and check
.16
.43
Multiple regression analysis was employed to determine each strategy’s contribution to
problem solving success with regard to the second research question. The results of the
regression analysis are given in Table 4.
Table 4: Multiple regression analysis results
Strategies
B
Std
β
t
p
7.77
.00
Dual correlation
error
Constant
4.14
.53
Make a systematic list
4.57
.35
.34
13.06
.00
.46
Look for a pattern
6.73
.38
.51
17.70
.00
.70
Work backward
5.31
.45
.33
11.74
.00
.57
Simplify the problem
2.73
.52
.14
5.22
.00
.36
Make a drawing
2.90
.59
.14
4.90
.00
.46
Guess and check
4.71
.71
.17
6.57
.00
.17
According to the dual correlations between the problem solving success and problem
solving strategies, it can be seen that the highest correlation coefficients belong to the
strategies of: look for a pattern (.70), work backward (.57), make a systematic list (.46) and
make a drawing (.46). Correlation coefficients which belong to guess and check and simplify
the problem strategies (.36 and .17) indicate that there is a positive but weak relationship
between each of these strategies and problem solving success. Generally speaking, there
is a significant relationship between strategies used and problem solving success (R =
.92, R2 = .84, p = .00). Strategies used in this study as independent variables, explain
almost 84% of problem solving success.
According to the standardized regression coefficients, the order of relative
importance of strategies in terms of their effects on the problem solving success is as
follows: look for a pattern, work backward, make a systematic list, make a drawing, guess and
check and simplify the problem. After analysing t-test results from the regression
coefficients, it appears that all strategies had a decisive role in explaining problem
solving success. The regression equation related to the success in problem solving is as
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
follows: problem solving = .
+ .
looking for pattern + 5.31 work backward + 4.57
make a systematic list + 4.41 guess and check + 2.90 make a drawing + 2.73 simplify the
problem.
Table 5: Wilks’ Lambda and F values of bottom and top groups
Wilks’ Lambda
F
Make a systematic list
.47
141.41*
Look for a pattern
.07
1638.52*
Work backward
.53
111.33*
Simplify the problem
.76
38.23*
Make a drawing
.91
12.15*
Guess and check
.99
1.00
Strategies
*Significant at .05 confidence level.
“ccording to the results of discriminant analysis considering the low values of Wilks’
Lambda and high value of F), strategies that play a significant role in distinguishing
successful and unsuccessful students were: look for a pattern, make a systematic list, work
backward, simplify the problem and make a drawing, respectively (See Table 5). The guess
and check strategy did not have any significant contribution. With the help of the
discriminant function, a classification with an accuracy rate of 98.4% was achieved.
Discussion
Non-routine problems which are usually not specific to any mathematical topic, have
no fixed procedure for solving, and require the use of one or more strategies to solve.
Moreover, they are especially challenging for many primary school students, since they
require an integration of several cognitive processes such as accounting for all
possibilities, visualizing relationships, etc. (Lee, Yeo, & Hong, 2014). Non-routine
problem solving skills and strategy use of fourth graders were scrutinized in this study.
In this context, findings suggest that fourth graders have difficulty in solving nonroutine problems without being exposed to specific training. This result is in line with
results found by Hok-Wing and Bin (2002) and Elia et al. (2009). The marginal place of
non-routine problems in Turkish textbooks and mathematics curriculum determined by
the MoNE could be an explanation for this situation. According to means computed for
each strategy, usage of two strategies was remarkable in the present study. These were
look for a pattern and make a systematic list. Relatively high application levels of look for a
pattern strategy can be explained by the fact that it is the only strategy which is
elaborated in primary schools in Turkey. In Hok-Wing and ”in
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’s study, use of
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make a drawing strategy was prominent, while guess and check strategy was the most
broadly successful strategy in Elia et al.
’s study. It seems that the most popular
strategies for students in fourth grade vary across countries depending on educational
systems or curriculum emphasis.
In regard to findings about the second and third research questions, the R 2 value
found through the multiple regression analysis demonstrates that in general,
knowledge of strategies explains 84% of problem solving performance. In other words,
students have to know about different problem solving strategies. Besides, 98.4%
accuracy in classification according to discriminant analysis results suggests that
strategies have a dominant and decisive role in determining novice and expert problem
solvers. Altun and Sezgin-Memnun (2008) achieved the similar results in this sense.
However, the most important contribution of this study is the detailed information on
the role of each strategy in explaining success, and in discriminating between successful
and unsuccessful students at fourth grade level. Even though almost all strategies had
statistically significant roles, three strategies were conspicuous: look for a pattern, make a
systematic list, and work backward.
Strategies of make a drawing, simplify the problem, and guess and check were not
used effectively by either low or high achieving students. Using the simplify the problem
strategy, students should first solve simple versions of the problem and then look for
patterns and make generalizations. It seems that the sophisticated structure of this
strategy makes it difficult for participants to use. This may be an explanation for the
ineffectiveness of the simplify the problem strategy in accounting for achievement and on
differentiating between good and poor problem solvers. Nevertheless, contrary to
expectations, most of the students did not prefer to make drawings which would help
them in visualizing and solving the problem. This situation can stem from the fact that
students are not encouraged by teachers to illustrate their thoughts by using pictures or
simple drawings while they are solving problems. Lastly, guess and check was the
weakest and least effective strategy. According to Elia et al. (2009), this strategy does
not entail high cognitive demand and it is widely used in a variety of mathematical and
everyday situations. However, results of this study exhibit an incongruity in this sense.
It is apparent that guess and check is often dismissed in math education in Turkey as an
unproductive or less sophisticated approach to solving problems.
Determining importance order and distinctiveness of each strategy in terms of
problem solving success may bring some important outcomes which can be used to
arrange textbooks and learning environments. First, since each of six strategies sought
in the current study has meaningful contribution to success, it can be concluded that all
these strategies are feasible to work at fourth grade level. High potentials of look for a
pattern, make a systematic list, and work backward might indicate that these strategies can
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Yeliz Yazgan FOURTH GRADERS AND NON-ROUTINE PROBLEMS: ARE STRATEGIES DECISIVE FOR SUCCESS?
be highlighted. However, other strategies should not be neglected. On the contrary,
impotent strategies such as guess a check should be given more time so that they can be
also be used effectively by all students.
Despite the fact that this study provides many meaningful conclusions, further
research is necessary. Firstly, the study included only six non-routine problems, and
using more different types of non-routine problems would provide more robust
evidence for the findings of this study. Maybe routine problems can also be involved to
provide more detailed information about the general problem solving success of
students.
Thinking processes of students were not analysed in detail in the present study.
”ecause this study was based on students’ written answers and students have
difficulties in writing down their thinking processes. Complementary qualitative
methods such as clinical interviews, observations and videotaping of pupils solving
problems could be used to collect data about students’ cognitive processes. In addition,
examining students of different grades age and school year level, mathematical abilities,
and educational systems would be more illuminating.
In summary, no matter how essential strategies may be as a component of
problem solving performance, it is evidence that they do not tell the whole story
(Schoenfeld, 1983). Nevertheless, if a primary goal of instruction is to develop students’
ability to think strategically so that they have a problem solving disposition including
the confidence and willingness to take on new and difficult tasks, non-routine problems
should be studied from the first grades of elementary school. Teachers may benefit from
non-routine problems that force students to apply what they have learned in a new
way. Furthermore, teachers should increase the variety of strategies which they present
to the students by considering grade level, and difficulty level of the strategy.
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Appendix
Problem Solving Test (PST)
1.
At a pizza restaurant, there are 4 toppings for pizza: Mushroom, sausage, onion,
and green pepper. How many different ways can you make a pizza with 2 toppings?
(Make a systematic list)
2.
While walking in a cave, a tourist saw the following code on the wall. He
recognized a relationship between numbers in each column. But there were missing
numbers in the code. Complete the code and explain the relationship. (Look for a pattern)
3.
A bus driver started to drive from the terminal and dropped half of his
passengers off at every bus stop he stopped by. Meanwhile, no passenger got on the
bus. After stopping by 3 bus stops there were 8 remaining passengers on the bus. How
many passengers got on the bus at the terminal? (Work backward)
4.
Each of following shapes consists of small triangles like the first one.
How many triangles do you need to make fifteenth shape? Explain what you thought.
(Simplify the problem, look for a pattern, make a drawing).
5.
At a birthday party, every child shakes hands with every other child. If there are
7 people at the party, how many handshakes take place? (Simplify the problem, look for a
pattern, make a drawing).
6.
In an exam, each right answer was scored as 1 or 2. A student had 56 right
answers and got 78 point. How many 1-point questions did he answer correctly? (Guess
and check)
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