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Metacognition Application: The Use of Chess as a Strategy to Improve the Teaching and Learning of Mathematics

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Metacognition Application: The Use of Chess as a Strategy to Improve the Teaching and Learning of Mathematics

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The aim of this study was to explore chess as an alternative strategy to improve the teaching and learning of mathematics. Purposive sampling was used to identify 25 experimental group learners and 26 control group learners. Experimental learners were drawn from two schools that offered chess, while control group learners were drawn from five non-chess-training schools. As such, 51 learners provided the data. The researcher administered a group test in an attempt to triangulate the findings of the study. Using chess as an alternative strategy to improve the teaching and learning of mathematics, it emerged from the main findings that the control group learners made far more contextual errors, compared to the experimental group, on the various problems that were presented. It also emerged that non-experimental learners were poorer in applying required basic steps to arrive at answers to the activities given to them compared to the experimental learners. The study further found that while both groups committed blended contextual and procedural errors, control group learners were dominant as compared to the experimental learners. This implies that the experimental group applied more metacognition in various problems and therefore outperformed the control group in the group test. Based on the above findings, the study concludes that chess training improves both teachers’ and learners’ application of metacognition in supporting learners’ performance in mathematics. The study recommends that since the use of chess serves as an alternative strategy to positively improve the teaching and learning of mathematics, chess training can be introduced to schools to enhance the mathematics performance of learners.
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Research Article
Metacognition Application: The Use of Chess as a Strategy to
Improve the Teaching and Learning of Mathematics
Simon Adjei Tachie
1
and Johnson Motingoe Ramathe
2
1
School of Mathematics, Science and Technology Education, North-West University, South Africa
2
Post Graduate Student, School of Mathematics, Science and Education, North-West University, South Africa
Correspondence should be addressed to Simon Adjei Tachie; simon.tachie@gmail.com
Received 11 October 2021; Accepted 31 March 2022; Published 11 May 2022
Academic Editor: Enrique Palou
Copyright © 2022 Simon Adjei Tachie and Johnson Motingoe Ramathe. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
The aim of this study was to explore chess as an alternative strategy to improve the teaching and learning of mathematics.
Purposive sampling was used to identify 25 experimental group learners and 26 control group learners. Experimental learners
were drawn from two schools that oered chess, while control group learners were drawn from ve non-chess-training
schools. As such, 51 learners provided the data. The researcher administered a group test in an attempt to triangulate the
ndings of the study. Using chess as an alternative strategy to improve the teaching and learning of mathematics, it emerged
from the main ndings that the control group learners made far more contextual errors, compared to the experimental group,
on the various problems that were presented. It also emerged that non-experimental learners were poorer in applying required
basic steps to arrive at answers to the activities given to them compared to the experimental learners. The study further found
that while both groups committed blended contextual and procedural errors, control group learners were dominant as
compared to the experimental learners. This implies that the experimental group applied more metacognition in various
problems and therefore outperformed the control group in the group test. Based on the above ndings, the study concludes
that chess training improves both teachersand learnersapplication of metacognition in supporting learnersperformance in
mathematics. The study recommends that since the use of chess serves as an alternative strategy to positively improve the
teaching and learning of mathematics, chess training can be introduced to schools to enhance the mathematics performance
of learners.
1. Introduction
For the past decades in the 21st century, scholars have
engaged themselves in related research activities to investi-
gate and suggest eective strategies that may help educators
to prepare learners on how to navigate through the increas-
ingly globalized world and inter-connected problems associ-
ated with learnerspoor performance in mathematics and
how these strategies, applied by teachers in teaching and
learning situations, can help learners overcome their poor
performance in mathematics globally in the 21st century
[1]. It is believed that learners need skillful strategies in order
to deal with the competitive global market changes and
employment through the application of the concept of meta-
cognition; this being abstract in nature and which we focus
on learnersthinking about their own thinking when faced
with problem-solving activities in todays sophisticated
world so as to further prepare themselves for graduating
from colleges of learning [2].
The concept of metacognition as proposed by Flavell [3]
in his research work, focused merely on developmental psy-
chology, which particularly targeted childrens thinking of
their own thinking processes in problem-solving and which
was also inuenced by Jean Piagets important work in
developmental psychology [4, 5]. Due to the abstract nature
of metacognition, dierent terms such as self-regulation,
meta-memory, and executive control are used interchange-
ably in describing/dening the same basic phenomenon for
improving the thinking abilities of learners in problem-
solving situations.
Hindawi
Education Research International
Volume 2022, Article ID 6257414, 13 pages
https://doi.org/10.1155/2022/6257414
The concepts of cognition and metacognition are widely
used interchangeably in our daily lives to mean the same
thing and it is important to explain the dierence between
these two concepts for the betterment of our research
environment. Flavell [3] explains cognition as the state of
learning and understanding the outside worldwhereas
metacognition refers to the process of engagement with
higher order thinking about how to understand and how
to create a better learning experience [4]. Related studies
have demonstrated that learners need skills such as prob-
lem-solving, metacognition, and critical thinking [610] to
enable them to think critically when faced with problem-
solving situations in order to nd appropriate solutions to
their problems in teaching and learning. In line with this,
Greenstein [11] also states that critical thinking includes
the concept of analyzing information, dissecting the problem
faced, applying relevant strategies, ideas, logical inquiry,
drawing conclusions, evaluating evidence, making accurate
judgments, and analyzing assumptions in order to nd a
solution to a problem which is very important in the game
of chess; likewise in the teaching and learning situation [1].
1.1. Background. In chess, there are various pieces, namely,
Queen, King, Rook, Bishop, Knight, and Pawn, and all have
their distinct way of moving across the board. Queens move
diagonally, horizontally, and vertically; Kings just move one
step in every direction, The Rooks move vertically and hor-
izontally, Bishops move diagonally, the Knights move two
squares vertically and then one square horizontally, while
the Pawns move on the les and capture diagonally.
Giménez et al. engaged in a test on several games to
ascertain the skills correlation that learners had in those
games such as Dots and Boxes, Wari, and Trac Lights in
respect to mathematics factors such as numerical and geo-
metric, counting, and rotation. This test was done to investi-
gate exactly how and with what every game correlated in
relation to each mathematical factor. The current studys
main focus explores chess as a strategy to improve the teach-
ing and learning of mathematics. Drawing from Todor
et al.s [12] study, resolving the problem-solving of tasks
compatibly, can be modelled by childrens practice in chess
using Costa and Kallicks [13] concept of habit of the
mind.A game of chess is more than a game: it relates to
mathematics as it involves calculations, basic arithmetic,
and geometry applied in enhancing problem-solving abilities
of children. The above can be interpreted as a solid connec-
tion between chess and the mathematics domain.
Djakov [14], and Chase and Simon, cited in Trinchero
and Sala [15] indicates that chess players can memorize
chess positions which arose from previous games because
of their good memory. Gobet [16] and Severin et al. [17]
further conrm that professional chess playersknowledge
and acquisition of games in the database assist them in
evaluating and applying the best combinations, thus ensur-
ing that they play or choose best variations and make the
best decisions.
Gobet and Simon [18] concur in that chess players usu-
ally have excellent memories, stating that chess players are
able to recall positions and random legal positions on a
chessboard. Gobet and Simon [18] further claim that chess
players have specic knowledge of structures which are
embedded in the long-term memory resulting from pro-
longed chess practice. This means that if chess players have
a good memory of positions and classic games, and they
use the latter skill to their advantage in becoming expert
chess players, then they are able to apply that skill to math-
ematics learning, an aspect that this study seeks to explore.
1.1.1. Knowledge Gained as Learners Learn Chess. There are
dierent forms of knowledge that learners may acquire and
develop as they play chess. Ormrod [19] describes metacog-
nition as onesrealization and reliance on ones own sub-
conscious thought processes and vigorous ventures which
are utilized to modulate ones processes to optimize learning
and memory; in other words, comprehension and control
of our subconscious thought processes.
Flavell [3] further explains metacognition as thinking
about thinking, while Martinez [20] says metacognition is
the monitoring and control of the thought algorithm,
which Bechtel [21] describes cognition as a dynamic men-
tal mechanism process that can be a chain of information-
processing steps.
From Figure 1, for achess player to make the rst move
on a chessboard, he or she has to apply his/her metacogni-
tive abilities to think of the possibilities of the rst move,
of which there are about 20 move possibilities. Bart [23],
Jerrim et al. [24], and Kazemi et al. [25] argue that chess is
a game that can be helpful to learnersstudies by developing
their metacognitive abilities. According to Bart [23], chess is
a cognitively demanding activity; subsequent to that, it ame-
liorates learnersintelligence, attention, and reasoning abili-
ties, thus developing many skills unrelated to chess.
Jerrim et al. [24] and Garner [26] share the same senti-
ments: that is, that chess is a cognitive enhancer. A general
population that is involved in intellectual activities more
often than not present with superior cognitive abilities com-
pared to others [27], and this is found in an intellectual
activity such as a game of chess.
1.2. Problem Statement. The problem of poor performance
in mathematics and the causes and solutions thereof have
been a subject of concern for many years. South Africa cur-
rently faces a crisis in mathematics education which has
been identied in ve cycles of the Trends in International
Mathematics and Science Study (TIMSS) conducted since
1995 with Grade 9 learners. Grade 9 South African learners
were placed last in the Third International Mathematics and
Science Study [2831] and in 2015, South African learners
were some of the lowest performing candidates of the 39
participating countries [32]. A report by Evans [29] indicates
that South Africa is ranked second last in the world in terms
of mathematics and science prociency.
For eight years, as a researcher, and as a Head of the
Department of Mathematics and Science in high school, I
have been working and organizing workshops at my school.
During this time, I have come to realize that most mathe-
matics teachers in the circuit use traditional approaches in
the teaching and learning of mathematics even though the
2 Education Research International
Department of Basic Education (DBE) has developed a vari-
ety of policies aimed at improving teachersmethods of
teaching mathematics in the country. Some of the schools
in the country have been equipped with computers and soft-
ware programs to teach mathematics; however, it has not
been established whether these have had a positive eect
on mathematics performance to date. To the best of my
knowledge, little research has been conducted on the use of
chess as an alternative strategy to improve the teaching
and learning of mathematics, particularly in the Motheo
Education District of the Free State Province, hence the
current study.
1.3. The Research Questions. Based on the above back-
ground, the main research question for this study was: In
what ways does chess serve as an alternative strategy to
improve the teaching and learning of mathematics in schools?
The current study further sought to answer the following
sub-research questions:
(i) To what extent do schools provide learners with
chess training?
(ii) To what extent do learners understand chess?
(iii) How does learnerspractice in chess inuence their
performance in mathematics?
1.3.1. Theoretical Framework. Pragmatism as a research par-
adigm nds its philosophical foundation in the historical
contributions of the philosophy of pragmatism [32] and, as
such, embraces a plurality of methods. Tashakkori and
Teddlie [33] state that as a research paradigm, pragmatism
is based on the inclination that researchers should utilize
the philosophical and methodological approach that func-
tions well for a certain research problem that is being inves-
tigated. It is often related to mixed methods [3436] where
the focus is on the results of the research and on the research
questions rather than on the techniques and strategies.
Duy et al. [35] suggest that pragmatism accepts that
there can be single or multiple realities that are open to
empirical inquiry. A signicant set of ideas of pragmatist
philosophy is that knowledge and reality align with beliefs
and habits that are socially constructed [37]. Pragmatists
doubt that reality can ever be certain for one and all [38].
They view reality as a standard idea and assert that reality
is what functions; thus, they maintain that knowledge claims
cannot be completely taken from contingent beliefs, habits,
and experiences [39]. For pragmatists, reality is absolute as
far as it helps us to get into satisfactory relations with other
parts of our experiences [40]. According to Baker and Schal-
tegger [41] and Ray [42], the truth is whatever proves itself
good or what has stood the scrutiny of individual use over
time. But it is also of utmost importance to keep in mind
that it is not necessarily guaranteed that if pragmatism
works, it is now an absolute truth [43]. In pragmatism, an
empirical approach takes preference over idealistic or ratio-
nalistic approaches [44].
A signicant set of ideas of pragmatist epistemology is
that knowledge comes from the foundation of experience as
in chess training. The understanding of the world is largely
impacted by social experiences. Thus, the socialization of
individuals or group of people in certain activities were to
understand a situation; in this case, the learners practicing
of chess to develop themselves mentally to improve their per-
formance in mathematics. Individuals have specic knowl-
edge that emanates from unique experiences. Nonetheless,
the bulk of knowledge is socially shared as it comes from
socially-shared experiences; therefore, all knowledge is social
knowledge [36], hence the use of pragmatic theory in this
study. Pragmatist epistemology does not view knowledge as
reality [45]. Instead, it is created with the aim of eectively
managing peoples existence and operations in the world
[46]. According to Berkman [47], playing chess requires con-
centration, visualization, analytical thinking skills, abstract
thinking, creativity, critical thinking skills, and bring about
cultural enrichment and early intellectual maturity. As a
result, learners who train in chess will be able to espouse such
skills in their mathematics class and perform better, hence
the theory. Many pragmatists in the forefront looked at the
work of Dewey and maintained that the epistemology draws
from Deweys concept of inquiry which amalgamates beliefs
and actions through a process of inquiry [34, 36, 46].
1.4. Literature Review. In our previous article (Tachie and
Ramathe in press), we highlighted the game of chess and
the way learners learn chess to inuence their performance
in mathematics. The study revealed that in non-chess-
playing schools, only 4% (N=2) indicated that they could
algebraically notate the game whereas all 20 learners
(100%) in chess-playing schools conrmed that they could
notate the game. The game of chess comprises a board with
32 chess pieces, 16 of which are white and 16 black.
Figure 2 shows that in chess there are various pieces,
namely, a Queen, King, Rook, Bishop, Knight, and Pawn,
and all have their distinct way of moving across the board.
Queens move diagonally, horizontally, and vertically;
Kings just move one step in every direction. The Rooks
move vertically and horizontally, the Bishops move diago-
nally, the Knights move two squares vertically and then
one square horizontally, while the Pawns move on the les
and capture diagonally.
Figure 1: The chessboard setup [22].
3Education Research International
1.4.1. Time Spent Learning Chess. Time spent on learning
and practicing chess is vital in ne-tuning the relevant skills
and abilities. According to Gobet and Simon [18], as already
stated, mastery of chess needs specialized knowledge, as well
as titanic brain capacityto learn previous games and cur-
rent games patterns, ability to recall, fathom, synthesize,
apply, evaluate, and problem solve certain move orders so
as to win games. This applies to learners from the beginner
stage to grand master levels.
Mastery in chess requires deep training and study. It
comes only from having more, better, or more eciently
organized knowledge in a domain. Gobet and Campitelli
[48] assert that mastery of ones chess skills requires max-
imum focus and determination as well as investment in
time. They further indicate that the best time to commence
chess practice or play in local chess clubs is 12 years old.
On the other hand, Blair et al. [49] claim that training in
chess and development of chess skills play a signicant role
in social and academic interactions of individuals. Chess
rehearsal extrapolates the abilities which learners can use
to comprehend mathematics [50]. Amount of time spent
on training in chess is of great signicance to mathematics
performance of learners, giving an indication of chess
being a strategy to improve the teaching of mathematics
in schools.
Various studies have suggested a range of times for chess
training. Sala et al. [51] indicate that some studies suggest
that learners participate in a chess lesson rst and then allo-
cate 30 minutes for learners to play against each other, while
some studies give learners two-hour lessons daily spread
over 12 days resulting in a total of 24 hours to introduce
new skills. Sala and Gobet [52] report having given learners
30 hours of chess instruction. Trinchero [53] blended 10 to
15 hours for a chess course, supported by computer-
assisted training (CAT) to enhance learnersmathematical-
solving ability. In the current study, I took note of the
research and worked with learners that were introduced to
chess training over a whole year while they were studying
mathematics at the same time. In the subsequent year, the
study was conducted with the hope of seeing the inuence
of chess training on mathematics performance.
Studies conducted by Trinchero [54] and Kazemi et al.
[25], looked at the eects of chess courses on Grades 3, 5,
8, and 9 learnersmathematical problem-solving ability.
The current study focused on Grade 9 learners who were
participating in chess training as well as those who did
not participate in chess training but all took mathematics
as a subject.
1.4.2. Experimental Groups vs Control Groups. Various
studies have been conducted to compare a normal groups
performance with a dyscalculia group with a signicant dif-
ference being found [54]. In another study, Trinchero [53]
reported that three types of groups were compared: the
group that was actively involved in chess, the group actively
involved in checkers and was treated as a control group, the
third group that was passive. The group that played chess
together with the passive group indicated a slightly better
performance in comparison to the active control group
(playing checkers) in mathematics problem-solving. The
three groups showed no statistically signicant dierence
in mathematical problem-solving. However, the results
showed that in counting skills, problem-solving, and compu-
tational tasks, the students who received chess training had
higher mathematical scores [55].
In the evaluation of the role of chess heuristics in
promoting childrens mathematical problem-solving skills,
Trinchero [53] compared the eectiveness of two dierent
types of chess training in enhancing mathematical
problem-solving abilities. A meta-analysis reported that
chess players outperformed non-chess players in several
cognitive skills [56]. In a study conducted by Aciego et al.
[57] in Northern Italy, the comparison group, which played
soccer or basketball in comparison to the chess group, had
better cognitive abilities, better coping and problem-solving
capacities, and better socio-aective development. In the
study conducted by Hong and Bart [54] in Korea, the
experimental group (students at risk for academic failure)
received chess lessons for three months (12 lessons) and
the last six lessons took place in a computer lab with chess
software in pursuit of establishing if there was an impact of
chess on cognitive abilities and minimal impact was found.
The study presents the paradigm adopted. Regarding prag-
matism, this study explains the history of pragmatism and
pragmatism as a research paradigm.
1.5. Research Approach. In this study, a qualitative research
approach was used. The research design is the case study
design used for the structural framework that guided the
researchers in the planning and implementation of the
research study. It also helped the researchers to achieve opti-
mal control over the factors that could inuence the study
[58]. Polit and Beck [59] contend that a research design
maps out a detailed plan which speaks to the research ques-
tions and helps attain the research objectives, hence the use
of a case study in this research.
Figure 2: Chess piece movement on a chessboard.
4 Education Research International
1.6. Population of the Participants. The population of the
participants consisted of Grade 9 learners from seven high
schools in the Thaba Nchu region of the Free State in South
Africa, the researchers having established which schools
oered chess training. Grade 9 learners normally have a
solid mathematical foundation hence being selected. Grade
9 learners who were doing mathematics at the seven (7)
chess-playing schools were subsequently sampled for the
data collection.
1.7. Sample of the Participants. Various studies have yielded
dierent results and the researcher has found loopholes.
This study, conducted in a rural area of South Africa, focuses
on secondary school learners who were aged 15-16. All par-
ticipants were normal chess-playing and normal non-chess-
playing participants and the training was done over a year by
chess instructors who are mathematics educators as well.
Purposive sampling was used to identify an experimental
group and a control group at Schools X and Y. Purposive
sampling is a non-probability sampling or purposeful sam-
pling procedure, which involves selecting certain partici-
pants based on a specic purpose rather than randomly
[60]. Purposive sampling was used in this study to select par-
ticipants for the second phase of data collection. The
researchers made use of the control group and experimental
group in chess training from the seven (7) schools; ve (5)
schools were found to have no chess training while two (2)
schools provided chess training. The schools were given
pseudonyms, they being Schools A, B, C, D, and E and were
the ones which did not oer chess training which Schools X
and Y oered chess training to their learners.
From Schools A, B, C, D, E, X, and Y, 26 learners indi-
cated that they did not have chess training and were hand-
picked in accordance with Table 1 as the control group.
Twenty-six learners of the non-experimental group were
given a test totalling 50 marks (Annexure H) and the proce-
dural errors and the contextual errors were examined.
Learner performance results during the year recorded on
SA-SAMs as well as classwork and homework were used
for document analysis. In addition, lessons were observed.
In Table 2, from School X and Y, 25 learners formed the
experimental group of 15 and 10, respectively, as indicated
by Table 1 and the same scope of work that was adminis-
tered to the non-experimental group was also administered
to them.
1.8. Instruments for Data Collection. Table 3 summarizes
the alignment of the research questions to the instrument
utilized.
The test was administered to the groups and these results
were compared to learner performance recorded in SA-
SAMS. Prior to the commencement of the main study, a
pilot study with three (3) learners who were not part of the
nal study was conducted using the test. The test was scru-
tinized by the supervisor and revisions made according to
initial ndings. In the control group, the group test was
administered with the assistance of educators of the seven
various schools. The learners concerned wrote the test dur-
ing their study periods and the test was out of 50. The test
was written on the same day across the seven schools.
1.9. Group Test and SA-SAMS as Additional Research
Instruments. The researcher also administered a group test
in an attempt to triangulate the ndings of the study. This
instrument was used in order to improve validity of the nd-
ings as it was used to check learnersperformance under a
controlled environment during which they were closely
supervised on common items. These results were an inde-
pendent source of evidence to gauge learnersgeneral math-
ematics performance which helped to validate the ndings.
2. Data Analysis, Presentation, and Discussion
Document analysis was conducted in an attempt to establish
the inuence of chess training on mathematics performance
on Grade 9. Qualitative data gathered from two of the seven
schools is presented, interpreted, and compared with the
information obtained from the literature study.
2.1. Inuence of LearnersPractice in Chess on Their
Learning of Mathematics. In this section, comparison of test
results is done and errors are identied.
2.2. Comparison of Test Result Errors. All learners wrote a
test and after scoring the test, the researchers were able to
identify the procedural and contextual errors. Procedural
errors are those mistakes that learners make but have some
understanding of how to solve the problem. Contextual
errors are those mistakes which learners make when they
have no understanding of the problem. We sampled several
procedural errors and several contextual errors from the test
to identify how learners performed in general. At this stage,
the idea was to establish whether there were any signicant
dierences in learnersperformance in the selected tasks.
The test was marked out of 50. Marks were recorded and
the type of errors which learners made were classied into
three themes, namely, contextual errors, procedural errors,
and a combination of both contextual and procedural errors,
namely, blended errors. Analysis of the qualitative data in
this section involved comparing similarities and dierences
in ndings and this was executed in the format of themes
as follows:
(i) Theme 1: Contextual errors.
Table 1: Distribution of control group learners.
Schools Number of control
group learners
Control group learners
pseudonyms
School A 4 A, B, C, D
School B 4 E, F, G, H
School C 4 I, J, K, L
School D 4 M, N, O, P
School E 4 Q, R, S, T
School X 3 U, V, W
School Y 3 X, Y, Z
5Education Research International
(ii) Theme 2: Procedural errors.
(iii) Theme 3: Blended contextual and procedural errors.
2.3. Theme 1: Contextual Errors. Participants were assessed
on the manner in which they simplied expressions. The
expected rst step was to expand the expression by removing
the brackets, and the second step was to apply the FOIL
method which is to multiply the FIRST, OUTER, INNER,
and LAST terms to simplify the expression.
In Figure 3, a control group participant from School A
brought down the rst term as it was, while the second term
was simplied correctly. However, in the second step, the
learner did not apply the FOIL method. This was supposed
to expand the rst term by multiplying the brackets but,
instead, the learner squared what was inside the bracket.
This was wrong. The learner had applied metacognition in
solving the problem but fell short of giving him/her an
opportunity to expand the rst term by multiplying the
bracket in order to arrive at the correct answer even though
we were not really interested in the correct answer but rather
how the procedure was applied so as to arrive at the right (or
wrong) answer. Clearly, the learner could not apply skillful
strategies through the application of the concept of metacog-
nition in solving the problem, which of course focuses on the
learners thinking about his/her own thinking and that could
have helped him/her to arrive at the correct answer using all
the necessary steps involved [4, 5].
In Figure 4, the experimental learner AX from School X
made a number of incorrect algebraic manipulations to ren-
der the whole exercise futile. The rst mistake was the failure
to apply the FOIL method to correctly expand the rst
bracket. This was followed by the incorrect interpretation
of the negative sign which was converted into a multiplica-
tion sign by simply multiplying the terms in the second
bracket next to the negative sign. Failure to cautiously treat
the subtraction sign as an algebraic operation that separates
terms was one of the sources of the problem for this exper-
imental learner in comparison to the control group learners.
Though the learnersnal answer was correct as per the pre-
vious step, the contextual mistakes that were committed ear-
lier had an eect on the nal answer as a result of the
incorrect application of metacognitive abilities in solving
the problem [4, 5]. This was similar to the previous learners
errors. According to Brown [61] and Flavell [3], metacogni-
tive knowledge consists of awareness and understanding that
can help a learner learn eectively [5]. The above learners
simply failed to engage in higher order thinking about how
to understand and present the problem step-by-step, in
order to create a better learning experience that could lead
to a permanent change in learning in life [4]. In this regard,
the teachers should take their time to guide learners in their
eort to teach mathematics for eective understanding of
their learners. Teachers should also take their time to care-
fully explain the mathematical operations or signs, where
they should be used or not used, and what happens if they
use in the wrong place when attempting to build or improve
their metacognitive skills of thinking.
To arrive at the correct answer, the following steps were
to be followed:
x3
ðÞ
2
xx+4
ðÞ
=x2
6x+9x2
4x=10x+9✓✓
ð1Þ
As such, it is clear that despite the fact that learners were
drawn from two dierent groups, it emerged that they both
made the same error but merely diered in the ways used
and steps taken when making the mistakes.
In another question, a control group learner X drawn
from chess-training school Y was also observed to have com-
mitted the same error (Figure 5).
The control group learner was supposed to apply the dis-
tribution law which was to multiply each term inside the
bracket by the outside term. However, learner X instead
applied factorizing by grouping which did not apply in this
Table 2: Distribution of experimental group learners.
Schools Number of experimental group learners Experimental group learnerspseudonyms
School X 15 AA, AB, AC, AD, AE, AF, AG, AH, AI, AJ, AK, AL, AM, AN, AO
School Y 10 AP, AQ, AR, AS, AT, AU, AV, AW, AX, AY
Table 3: Alignment of data collection instruments to the research questions.
Questions Data collection
Main question
How does training in chess inuence the performance of learners in mathematics? Questionnaire, document analysis,
and observation
Sub-questions
To what extent do schools provide learners with chess training? Interviews
To what extent do learners understand chess? Interviews
To what extent does learnerspractice in chess inuence their performance in mathematics? Interviews, document analysis,
and observation
6 Education Research International
instance. This is because there must be two common
brackets and outside each bracket a plus or minus sign; only
then can factorizing by grouping apply. This means that the
learner got the rst step wrong and, in addition, failed to
note that 3a
2
bc
2
is a single term which thus cannot be split
as manipulated. The learner even proceeded to poorly sim-
plify the incorrect previous step.
While the nal answer was not entirely correct, an
experimental learner in Figure 4 generally performed better
in the sense that 3a
2
bc
2
was correctly treated as a single
term (see Figure 6).
In addition, while the learner failed to uphold all expo-
nential laws correctly on variables, there were correct alge-
braic manipulations. As such, there were not entirely the
same challenges that the control group learner encountered
but were experienced by the experimental group learner. In
fact, the control group learners also failed to note that
3a
2
bc
2
is a single term which thus cannot be split as was
manipulated by one learner. The learner even proceeded to
poorly simplify the incorrect previous step. The question
one may ask is: Do teachers take their time to explain what
is meant by a term, expression, etc. to learners when they
are executing their duties as mathematics educators? In most
cases, educators minimize or ignore this simple information
learners require and the steps learners should take when
confronting any problem which leads them to make simple
mistakes in their attempts to solve mathematical problems
thus resulting in poor performance in the subject and which
further derails their metacognitive thinking abilities. In
short, learners often fail to apply certain skills such as prob-
lem-solving, metacognition, and critical thinking [610] that
enable them to think critically when faced with problem-
solving situations.
The question on simplifying fractions using exponential
laws was generally poorly done especially by the control
group learners. The ability to follow the correct steps was
critical.
In the case of Figure 7, the rst step taken by the control
group learner K from School C was correct in distributing
the exponent to each term inside the bracket in the denom-
inator. In the second step, the learner was supposed to apply
the Laws of Exponents and specically the second law of
exponents. This says when you divide, the bass is the same
as the one that should subtract the exponents. The learner
should have identied the fact that there were two bases, 4
and x, then applied the Laws of Exponents correctly. How-
ever, the control group learner was completely incorrect as
he/she dropped the negative exponents and introduced a
negative sign on the terms.
Figure 8 shows the attempt made by the experimental
group learner and the mistakes made.
The very rst step (identifying the Lowest Common
Denominator (LCD)) was correct. However, the learner
failed to correctly apply the required steps to arrive at the
correct answer. For instance, the learner was supposed to
say 4 on the rst term goes so many times into the LCD
and then multiply by the numerator on the rst term; in
the second term, the learner should say 2 goes so many times
into the LCD and then multiply by the numerator of the sec-
ond term. Lastly, he/she should say 4 goes so many times
into the LCD and then multiply by the numerator of the
third term. The learner did not apply the correct steps and
thus arrived at the incorrect expression.
Figure 3: Control group learner B, School A.
Figure 4: Experimental learner AX, School X.
Figure 5: Control group learner X, School Y.
Figure 6: Experimental learner AR, School Y.
Figure 7: Learner K, School C.
7Education Research International
2.4. Theme 2: Procedural Errors. In this example, learners
were required to identify and apply the correct concept. In
this case, the learner was expected to apply the BODMAS
Rule, namely, Bracket of Division, Multiplication, Addition,
and Subtraction.
Learner N from Figure 9 in a non-chess-training school
was a control group learner who omitted the required steps;
this gives an indication that he/she used the calculator.
Learners were instructed not to use a calculator. The lack
of a clear step-by-step approach which does not show all
workings does not guarantee a learner getting full marks,
even if the learner is correct.
In this question, learners were simply required to use
substitution to arrive at an answer.
The experimental group learner AS of School Y in
Figure 10 missed marks when he/she failed to do the actual
substitution before any simplication. In order to calculate
by substitution, the learner should have substituted -2 where
he/she saw xin the whole expression. The learner failed to
show the full correct substitution in the rst step except on
the third term. The learner correctly presented the second
step but did not add and subtract correctly in the last step;
hence, he/she arrived at the incorrect answer.
In comparison to the control group, learner L from
School C in Figure 11 scored lower marks than the experi-
mental group learner due to a complete failure to do the
actual substitution and subsequently arriving at 36 instead
of 32. Though there was consistence in getting the nal
answer wrong, on average the control group Learner L
scored less than the experimental Learner, AS. As such, the
experimental and the control group learners made proce-
dural errors but they were not entirely similar.
2.5. Theme 3: Blended Contextual and Procedural Errors. We
also established that learners from both groups committed a
blend of contextual and procedural errors.
In Figure 12, the control group learner W was supposed
to solve the expression within three steps. On the rst term,
2 on the denominator of the mixed fraction inside the
bracket should have been multiplied by 2 whole plus 1 on
the numerator to get (5/2)2 and then he/she solved the sec-
ond term by converting the decimal fraction (0,5) to fraction
(1/2)2 since it was not possible to simplify the whole expres-
sion with fractions and decimals mixed without the use of a
calculator; thus, the use of a calculator contributed to him/
her making a contextual error. The expected procedure was:
21
2

2
+0,5
ðÞ
2=5
2

2
+1
4=25
4+1
4
26
4=13
2
:✓✓ ð2Þ
Figure 13 shows how a learner from the control group
committed a procedural error by doing the rst step
incorrectly.
It is unclear how the participant arrived at the second
step without using a calculator. Furthermore, in the second
step, the learner missed the addition and was also unclear
about how -2 in the previous step became 2; thus, all the
workings were incorrect.
In the same question, experimental group learner AQ
made a contextual error.
The learner AQ from Figure 14 got the rst step incor-
rect because he/she calculated the square root of 144 and
25 in the same root instead of adding 144 and 25 rst in
the rst term thus everything is incorrect. The learner was
also unclear about how became 5. This meant that the
learner failed to apply the correct manipulation in step 1
which aected the rest of the workings.
Learner D in Figure 15 was a control group learner who
committed a procedural error.
The learner correctly changed the division sign to a mul-
tiplication sign by reciprocal, which means that the numera-
tor becomes the denominator and the denominator becomes
the numerator on the second term. The learner was then to
take out the common factor on the rst term which was 2x.
He/she should then have simplied the expression to the
simplest form. The learner completed the rst term correctly
by changing the division sign and bringing down the rst
term as it was; however, when coming to the second step,
the learner did not take out the common factor. The learner
rather decided to add unlike terms, and as a result, the whole
solution was incorrect.
Figure 8: Learner AN, School X.
Figure 9: Learner N, School D.
Figure 10: Learner AS, School Y.
8 Education Research International
2.6. Quantitative Representations. Figure 16 shows results for
control group test results emerging from the written exer-
cises and tests.
The test results from the control group from seven non-
chess-playing schools indicated that out of 26 learners who
wrote the test, 10 made both contextual and procedural
errors, while 11 made contextual errors and 5 made proce-
dural errors.
Figure 17 shows the results of the experimental group
selected from the two chess-playing schools who wrote the
same test as the control group.
The results from the 25 learners in the two schools,
Schools X and Y, were as follows: One learner was found
to have both contextual errors and procedural errors, four
learners were found to have made only contextual errors,
while 10 learners were found to have procedural errors,
and 10 of the learners were found to have made none of
the errors.
For a chess player to make the rst move on a chess-
board, he or she has to think of the possibilities of the rst
move, there being about 20 move possibilities. Bart [23], Jer-
rim et al. [24], and Kazemi et al. [25] argue that chess is a
sport that can be helpful to learnersstudies. According to
Bart [23], chess is a cognitively demanding activity which
subsequently ameliorates learnersintelligence, attention,
and reasoning abilities thus beneting many skills unrelated
to chess. Jerrim et al. [24] and Garner [26] share the senti-
ments that chess is a cognitive enhancer. It is argued that
learners involved in intellectual activities such as chess, more
often than not, demonstrate glimpses of superior cognitive
abilities compared to others who are not involved in such
activities [27].
2.7. Performance in Classwork and Homework. Frank and
Hondt [62] and Scholz et al. [55] state that many studies
reported aligned factors in chess instruction with mathemat-
ical problem-solving; however, only a few of those studies
utilized sound methodological methods; therefore, the con-
clusions that were reached are questionable. Interestingly,
Isabella as cited in McDonald [22] reviewed various studies
which indicated aligning the eects of chess on mathematics
problem-solving in classrooms. She alludes to the fact that
benets of chess in mathematics might arise from the fact
that chess uses notations which might assist learners in
understanding mathematics. Isabella further indicates that
notation is associated with visio-spatial patterns on a chess-
board, while mathematics is associated with pure symbolic
manipulation.
2.8. Comparison of Overall Group Test and SA-SAMS
Results. The overall performance of the two groups in com-
parison was also analyzed and is presented in Figure 18.
Figure 11: Learner L, School C.
Figure 12: Learner W, School X.
Figure 13: Learner H, School.
Figure 14: Learner AQ, School Y.
Figure 15: Learner D, School A.
9Education Research International
This gure shows interesting results pertaining to the
overall achievement of participants in the two groups on
the dierent tasks. It emerged that in the group test admin-
istered by the researchers, experimental learners had an
average performance of 78%, which was double the mean
mark obtained by the control group. It further emerged that
in Term 1, the experimental group had a 30% mean mark
compared to 17% of the control group, while in Term 3,
the control group obtained a mean of 28% compared to
39% which was obtained by the experimental group. Overall,
it was clear that the experimental group outperformed the
control group in the group test and the SA-SAMS results
from Terms 1 and 3.
2.9. Five-Number Summary Comparison of Group Test and
SA-SAMS Marks. The overall performance of the two groups
is reected using a ve-number summary, which is a set of
descriptive statistics that provides information about a data
set. It consists of the ve most important sample percenti-
lesthe sample minimum, the lower quartile or rst quar-
tile, the median, the second quartile, and the maximum.
The ve-number summary is presented in Table 4.
Table 4 indicates that the learners from the experimental
group outperformed the control group in all aspects. For
instance, in the test, there was a dierence of 7% between the
minimum marks, with the best learner in the control group
attaining 60% compared to the experimental learner who
scored 100%. The SA-SAMS marks also showed a similar
trend despite variations in the range of marks for each compo-
nent. There were also reduced dierences in SA-SAMS Term 3
performance compared to the other two assessments. Despite
that, experimental learnersperformance was far better than
the control group learnersperformance throughout.
In various studies done, with some comparing a normal
groups performance with a dyscalculia group, a signicant
dierence was found [54]. In one study, the group playing
chess was compared to an active control group (playing
checkers) and a passive control group. While the chess-
treated group and the passive control group slightly outper-
formed the active chess-playing control group in mathemat-
ical problem-solving, the dierences were not statistically
signicant. The three other groups showed no statistically
signicant dierence in mathematical problem-solving. The
results showed that in the counting skills, problem-solving,
and computational tasks, the learners who received chess
training had higher mathematics scores [55].
3. Summary of the Findings
3.1. Inuence of LearnersPractice in Chess on Their
Learning of Mathematics. The following were observed by
the researchers during the study.
3.1.1. Contextual Errors
(i) The study found that control group learners made far
more contextual errors compared to the experimental
group on various problems that were presented. For
39%
42%
19%
Control group learner test result errors
Both procedural and contextual errors
Contextual errors
Procedural errors
Figure 16: Control group test result errors.
4%
16%
40%
40%
Experimental learner test result errors
Both procedural and
contextual errors
Contextual errors
Procedural errors Non-procedural and
contextual errors
Figure 17: Experimental group test result errors.
0
10
20
30
40
50
60
70
80
90
Mean group
test
Mean SA-
SAMS term 1
Mean SA-
SAMS term 3
Experimental group
Control group
Figure 18: Comparison of group test and SA-SAMS results.
10 Education Research International
instance, in one of the problems in the second step,
the learner did not apply the FOIL method which
was supposed to expand the rst term by multiplying
the brackets; the learner squared what was inside the
bracket thus going totally wrong
(ii) Failure to cautiously treat the subtraction sign as
an algebraic operation that separates terms was
one of the sources of the problem for one experi-
mental learner in comparison to the control group
learners
(iii) Control group learners also failed to note that
3a
2
bc
2
is a single term which thus cannot be split
as was manipulated by one learner. The learner
even proceeded to poorly simplify the incorrect
previous step
3.1.2. Procedural Errors
(i) The study also found out that control group learners
were poorer in applying the BODMAS Rule com-
pared to the experimental learners. As such, some
ended up omitting the required steps and used the
calculator, even though the instruction was for
learners not to use calculators
(ii) The use of calculators was also possibly evident in
the experimental group when learners missed marks
for failing to do the actual substitution before any
simplication
3.1.3. Blended Contextual and Procedural Errors
(i) A control group learner failed to properly rst
convert a mixed fraction into an improper fraction
before rst-stage simplication. Furthermore, the
learner did not convert the decimal number to a
fraction since it was not possible to simplify the
whole expression with fractions and decimals
mixed without the use of a calculator, thus the
use of a calculator contributed to committing a
contextual error
(ii) Addition of unlike terms was prevalent among
learners from the non-experimental group and more
so than learners from the experimental group
4. Contribution of the Study to the Field
The present study is signicant to the extent that it will oer
concrete evidence to link chess training and good perfor-
mance in mathematics, thereby adding to the existing body
of knowledge regarding tools that can be used to facilitate
the teaching and learning of mathematics. If chess training
is found to have an inuence on mathematics performance,
it means that learners would do well in mathematics and
could result in the production of a generation of engineers,
architects, and many other mathematics-related professions.
Chess can then be incorporated into the teaching of mathe-
matics, and teachers and the Department of Basic Education
would benet immensely as teaching and learning of math-
ematics would be more eective and fun. Furthermore, the
ability to develop a more focused approach through
increased concentration and developing emotional intelli-
gence are characteristics which indirectly are associated
with success in life yet they are exhibited by most chess
players as well as mathematicians. The study is based in
the rural areas, with communities previously disadvantaged,
so learners performing in mathematics would bring eco-
nomic emancipation to the society as whole. Binev et al.
[63] indicated that the European parliament and Spanish
parliament have agreed to the implementation of a chess
course in schools where it functions as an educational tool
which takes place during school hours.
5. Conclusion
In conclusion, based on overall performances in the assess-
ments analyzed, experimental group learners outperformed
control group learners in the group test as well as the SA-
SAMS marks from Terms 1 and 3. The experimental groups
teachers applied metacognitive abilities in their executed
activities more than the controlled groups teachers.
6. Recommendations for Practice
From the conclusion in line with the ndings of the study,
the following recommendations are made:
(i) Since the inuence of chess training has been
proven in the literature, it is recommended that
chess training be introduced to schools in order to
enhance the mathematics performance of learners
Table 4: Five-number summary of group test and SA-SAMS results.
Assessment Group Five-number summary
Min Q
1
Median Q
2
Max
Group test Control 20 30 40 47 60
Experimental 27 67 83 97 100
SA-SAMS Term 1 Control 3 10 15 28 28
Experimental 14 24 32 34 40
SA-SAMS Term 3 Control 16 21 26 32 51
Experimental 27 26 41 42 53
11Education Research International
(ii) Chess training should be encouraged by administra-
tors and embraced by teachers in order to amelio-
rate mathematics performance
(iii) While marking mathematics activities, it is advis-
able to use positive marking in procedural errors if
all steps are shown as this assist in the building of
condence
6.1. Recommendations for Further Study. Based on the nd-
ings of this study, the following recommendations are sug-
gested for further study:
(i) A large-scale study on the inuence of chess among
mathematics learners in high schools in South
Africa could be conducted
(ii) A compilation of a well-structured program on how
to go about introducing chess as a learning area/
subject
(iii) Chess as an extra-mural activity by way of clubs and
competitors/championships be formed or encour-
aged in schools to support the development of
learnersmetacognitive abilities in mathematics
learning
Data Availability
The data is captured in the article.
Conflicts of Interest
The authors declare that they have no conicts of interest.
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WITHDRAWN: Teddlie, C., & Yu, F. (2007). Mixed methods sampling: A typology with examples. Journal of Mixed Methods Research, 1(1), 77-100. DOI 10.1177/1558689806292430 Article withdrawn by publisher. Due to an administrative error, this article was accidentally published OnlineFirst and in Volume 1 Issue 1 of publishing year 2007 with different DOIs and different page numbers. The incorrect version of the article with DOI: 10.1177/2345678906292430 has been replaced with this correction notice. The correct and citable version of the article remains: Teddlie, C., & Yu, F. (2007). Mixed methods sampling: A typology with examples. Journal of Mixed Methods Research, 1(1), 77-100. DOI 10.1177/1558689806292430