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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 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.

1. Introduction

For the past decades in the 21st century, scholars have

engaged themselves in related research activities to investi-

gate and suggest eﬀective 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 learners’poor 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 learners’thinking about their own thinking when faced

with problem-solving activities in today’s 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 children’s thinking of

their own thinking processes in problem-solving and which

was also inﬂuenced by Jean Piaget’s important work in

developmental psychology [4, 5]. Due to the abstract nature

of metacognition, diﬀerent terms such as self-regulation,

meta-memory, and executive control are used interchange-

ably in describing/deﬁning 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 diﬀerence between

these two concepts for the betterment of our research

environment. Flavell [3] explains cognition as “the state of

learning and understanding the outside world”whereas

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 [6–10] to

enable them to think critically when faced with problem-

solving situations in order to ﬁnd 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 ﬁnd 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 ﬁles 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 Traﬃc 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 study’s

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 children’s practice in chess

using Costa and Kallick’s [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 conﬁrm that professional chess players’knowledge

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 speciﬁc 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

diﬀerent forms of knowledge that learners may acquire and

develop as they play chess. Ormrod [19] describes metacog-

nition as one’s“realization and reliance on one’s own sub-

conscious thought processes and vigorous ventures which

are utilized to modulate one’s 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 ﬁrst move

on a chessboard, he or she has to apply his/her metacogni-

tive abilities to think of the possibilities of the ﬁrst 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 learners’studies by developing

their metacognitive abilities. According to Bart [23], chess is

a cognitively demanding activity; subsequent to that, it ame-

liorates learners’intelligence, 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 identiﬁed in ﬁve 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 [28–31] 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 proﬁciency.

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 teachers’methods 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 eﬀect

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 learners’practice in chess inﬂuence their

performance in mathematics?

1.3.1. Theoretical Framework. Pragmatism as a research par-

adigm ﬁnds 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 [34–36] where

the focus is on the results of the research and on the research

questions rather than on the techniques and strategies.

Duﬀy et al. [35] suggest that pragmatism accepts that

there can be single or multiple realities that are open to

empirical inquiry. A signiﬁcant 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 signiﬁcant 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 speciﬁc 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 eﬀectively

managing people’s 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 Dewey’s 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 inﬂuence 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 conﬁrmed 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 ﬁles

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 ﬁne-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 capacity”to 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 eﬃciently

organized knowledge in a domain. Gobet and Campitelli

[48] assert that mastery of one’s 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 signiﬁcant 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 signiﬁcance 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 ﬁrst 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 learners’mathematical-

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 inﬂuence

of chess training on mathematics performance.

Studies conducted by Trinchero [54] and Kazemi et al.

[25], looked at the eﬀects of chess courses on Grades 3, 5,

8, and 9 learners’mathematical 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 group’s

performance with a dyscalculia group with a signiﬁcant 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 signiﬁcant diﬀerence

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 children’s mathematical problem-solving skills,

Trinchero [53] compared the eﬀectiveness of two diﬀerent

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-aﬀective 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 inﬂuence 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

oﬀered 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

diﬀerent 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 speciﬁc 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; ﬁve (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 oﬀer chess training which Schools X

and Y oﬀered 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

ﬁnal study was conducted using the test. The test was scru-

tinized by the supervisor and revisions made according to

initial ﬁndings. 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 ﬁndings of the study. This

instrument was used in order to improve validity of the ﬁnd-

ings as it was used to check learners’performance under a

controlled environment during which they were closely

supervised on common items. These results were an inde-

pendent source of evidence to gauge learners’general math-

ematics performance which helped to validate the ﬁndings.

2. Data Analysis, Presentation, and Discussion

Document analysis was conducted in an attempt to establish

the inﬂuence 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. Inﬂuence of Learners’Practice in Chess on Their

Learning of Mathematics. In this section, comparison of test

results is done and errors are identiﬁed.

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 signiﬁcant

diﬀerences in learners’performance in the selected tasks.

The test was marked out of 50. Marks were recorded and

the type of errors which learners made were classiﬁed 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 diﬀerences

in ﬁndings 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 simpliﬁed expressions. The

expected ﬁrst 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 ﬁrst term as it was, while the second term

was simpliﬁed correctly. However, in the second step, the

learner did not apply the FOIL method. This was supposed

to expand the ﬁrst 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 ﬁrst 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

learner’s 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 ﬁrst mistake was the failure

to apply the FOIL method to correctly expand the ﬁrst

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 learner’sﬁnal answer was correct as per the pre-

vious step, the contextual mistakes that were committed ear-

lier had an eﬀect on the ﬁnal answer as a result of the

incorrect application of metacognitive abilities in solving

the problem [4, 5]. This was similar to the previous learner’s

errors. According to Brown [61] and Flavell [3], metacogni-

tive knowledge consists of awareness and understanding that

can help a learner learn eﬀectively [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

eﬀort to teach mathematics for eﬀective 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:

x−3

ðÞ

2

−xx+4

ðÞ

=x2

−6x+9−x2

−4x=−10x+9✓✓

ð1Þ

As such, it is clear that despite the fact that learners were

drawn from two diﬀerent groups, it emerged that they both

made the same error but merely diﬀered 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 learners’pseudonyms

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 inﬂuence 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 learners’practice in chess inﬂuence 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 ﬁrst 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 ﬁnal 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 [6–10] 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 ﬁrst 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 speciﬁcally 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 identiﬁed 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 ﬁrst 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 ﬁrst term goes so many times into the LCD

and then multiply by the numerator on the ﬁrst 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 simpliﬁcation. In order to calculate

by substitution, the learner should have substituted -2 where

he/she saw “x”in the whole expression. The learner failed to

show the full correct substitution in the ﬁrst 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 ﬁnal

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 ﬁrst 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 ﬁrst 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 ﬁrst step incor-

rect because he/she calculated the square root of 144 and

25 in the same root instead of adding 144 and 25 ﬁrst in

the ﬁrst 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 aﬀected 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 ﬁrst term which was 2x.

He/she should then have simpliﬁed the expression to the

simplest form. The learner completed the ﬁrst term correctly

by changing the division sign and bringing down the ﬁrst

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 ﬁrst move on a chess-

board, he or she has to think of the possibilities of the ﬁrst

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 learners’studies. According to

Bart [23], chess is a cognitively demanding activity which

subsequently ameliorates learners’intelligence, attention,

and reasoning abilities thus beneﬁting 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 eﬀects of chess on mathematics

problem-solving in classrooms. She alludes to the fact that

beneﬁts 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 ﬁgure shows interesting results pertaining to the

overall achievement of participants in the two groups on

the diﬀerent 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 reﬂected using a ﬁve-number summary, which is a set of

descriptive statistics that provides information about a data

set. It consists of the ﬁve most important sample percenti-

les—the sample minimum, the lower quartile or ﬁrst quar-

tile, the median, the second quartile, and the maximum.

The ﬁve-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 diﬀerence 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 diﬀerences in SA-SAMS Term 3

performance compared to the other two assessments. Despite

that, experimental learners’performance was far better than

the control group learners’performance throughout.

In various studies done, with some comparing a normal

group’s performance with a dyscalculia group, a signiﬁcant

diﬀerence 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 diﬀerences were not statistically

signiﬁcant. The three other groups showed no statistically

signiﬁcant diﬀerence 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. Inﬂuence of Learners’Practice 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 ﬁrst 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

simpliﬁcation

3.1.3. Blended Contextual and Procedural Errors

(i) A control group learner failed to properly ﬁrst

convert a mixed fraction into an improper fraction

before ﬁrst-stage simpliﬁcation. 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 signiﬁcant to the extent that it will oﬀer

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 inﬂuence 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 beneﬁt immensely as teaching and learning of math-

ematics would be more eﬀective 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 group’s

teachers applied metacognitive abilities in their executed

activities more than the controlled group’s teachers.

6. Recommendations for Practice

From the conclusion in line with the ﬁndings of the study,

the following recommendations are made:

(i) Since the inﬂuence 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

conﬁdence

6.1. Recommendations for Further Study. Based on the ﬁnd-

ings of this study, the following recommendations are sug-

gested for further study:

(i) A large-scale study on the inﬂuence 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

learners’metacognitive abilities in mathematics

learning

Data Availability

The data is captured in the article.

Conflicts of Interest

The authors declare that they have no conﬂicts of interest.

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