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EURASIA Journal of Mathematics, Science and Technology Education, 2022, 18(12), em2187
ISSN:1305-8223 (online)
OPEN ACCESS Research Paper https://doi.org/10.29333/ejmste/12579
© 2022 by the authors; licensee Modestum. This article is an open access article distributed under the terms and conditions of
the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/).
chium@nccu.edu.tw (*Correspondence) linf@math.ntnu.edu.tw kailin@ntnu.edu.tw toshi@mail.mcu.edu.tw
TsungjuPanda@gmail.com 107102009@g.nccu.edu.tw
The interplay of affect and cognition in the mathematics grounding activity:
Forming an affective teaching model
Mei-Shiu Chiu 1* , Fou-Lai Lin 2 , Kai-Lin Yang 2 , Toshiyuki Hasumi 1,3 , Tsung-Ju Wu 2 ,
Pin-Syuan Lin 1
1 National Chengchi University, Taipei City, TAIWAN
2 National Taiwan Normal University, Taipei City, TAIWAN
3 Ming Chuan University, Taipei City, TAIWAN
Received 10 September 2022 ▪ Accepted 18 October 2022
Abstract
This study aims to build a framework for affect-focused (or affective) mathematical teaching
(AMT), while promoting higher-order mathematical learning (e.g., pattern finding and deep
understanding). The data sources were the class mathematics grounding activity designed by
Taiwan’s mathematics educators, aiming to enhance students’ affective performances in learning
mathematics with a theoretical base on the enactivist perspective. Qualitative methodology
identified features of affective mathematics teaching and formed a framework for AMT, which
defines AMT as transforming natural languages to mathematical languages, highlighting student
agenda of upward learning (interest, sense, utter, and present), met by teacher agenda of caring
(cultivate, amuse, reflect, and explain). Finally, the enactivist embodiment activities are embedded
in the pedagogical structure of 4E phases: entry, entertainment, enlightenment, and enrichment.
Affect and cognition interplay in each phase.
Keywords: affect, cognition, mathematics education, qualitative methods
INTRODUCTION
Successful acquisition of mathematical knowledge is
essential for academic development and all aspects of
human cognition for everyday life (Menon & Chang,
2021¸ Sun et al., 2018). As a subject of “the study of
patterns and relationships” (Burton, 1994, p. 12),
mathematics is often set within the cognitive domain.
However, there has been a call for greater emphasis on
affective variables (e.g., Goldin, 2000). This is also in line
with the recent emphasis on fulfilling both character (i.e.,
curiosity, compassion, and courage to take actions) and
cognition for success in today’s education (OECD, 2021).
Affect plays an important role in mathematics
activities (Hannula, 2019; Zan et al., 2006). Earlier work
showed that affect is especially vital in solving non-
routine creative mathematical problems (Chiu, 2009).
Affect is particularly evident in mathematics teaching for
higher-order (cognitive) learning. Students as
mathematicians experience affective challenges when
learning mathematics (Burton, 1994). Projects, games,
and puzzles are challenging tasks for children to exercise
their creativity and control over their mathematics
learning.
Feeling anxiety, exerting control, and exercising
creativity are affective issues in their striving for
progression and new ideas. For example, in solving
discrete mathematics, five representational systems are
involved: verbal-syntactic (natural language), imagistic,
internalized formal notational, executive control, and
affective representations (Goldin, 2004). In learning
geometry, students’ incomprehension mainly arises
from multiple representations and representation
transformations (Duval, 2006). The incomprehension
raises affective responses and can be addressed by
natural languages. Affective responses with natural
languages, therefore, may assist learners in
representation registrations and transformations, which
appear to be missing in literature and would provide
opportunities to fill the gap by research.
Chiu et al. / The interplay of affect and cognition in the mathematics-grounding activities
2 / 15
Recognizing the need to expand beyond the cognitive
domain, the current national curriculum in Taiwan, the
12-year basic education (Ministry of Education in
Taiwan, 2014), began incorporating affective aspects in
learning and teaching to nurture lifelong learners
capable of facing the challenges of a fast-changing
lifestyle and information overload. In line with these
guidelines, the mathematics curriculum envisions the
ideal of mathematics as a language, practical pattern
science, cultural literacy, sense-making learning
opportunity, and avenue to use diverse tools (e.g.,
computer) (Ministry of Education in Taiwan, 2018).
In accordance with this vision, the current study aims
to build an affect-centered theory on mathematics
learning and teaching from both bottom-up (context)
and top-down (content) methodologies. For the context
aspect, the in-class mathematics grounding activity
(MGA) developed by Shi-Da Institute for Mathematics
Education (SDiME, 2022; Yang et al., 2021) in Taiwan can
serve as a source for building an in-depth mechanism
framework for mathematics learning and teaching. The
rationales are that the MGA aims to enhance students’
affective outcomes by building fundamental
mathematical knowledge, manipulating concrete
representations, and engaging in gamified activities
(Wang et al., 2021). Challenging, engaging, and
motivating mathematical tasks in the real classroom
context will provide insights into high-quality
mathematics teaching that balance or harmoniously
integrate both affect and cognition from a bottom-up
perspective.
For the content aspect, a top-down perspective
utilizing past literature on affect in mathematics learning
and teaching will add to the insights from the
aforementioned empirical research. As a result, a
qualitative methodology will be used for theory
building, applying empirical cases, literature, and
researchers of this paper as the participants to co-build
the theories. Concretely speaking, this study aims to
answer the following research questions.
1. What are features of instructional design that
support affective mathematics teaching?
2. What is an abstract framework that can address
the features identified?
Answers to RQ1 are presented in the results section.
RQ2 is presented in the discussion section, which
synthesizes the answers to RQ1 by drawing from the
current literature and authors’ insights. The following
literature review first draws upon empirical and
theoretical research on affect in mathematics learning
and teaching. The next review goes to an enactivist
perspective on mathematics education, which is the
theoretical basis for the empirical data used in this study,
the MGA.
The Interplay of Affect and Cognition in Mathematics
Learning and Teaching
McLeod’s (1992, p. 578) seminal work conceptualized
affect into the dimensions of emotions (affective states
such as joy or anxiety), attitudes (positive or negative
predisposition toward an activity), and beliefs (learned
perspective toward an object), each differing in intensity,
stability, and cognitive involvement. Affect can include
two kinds of systems: global (trait) vs local (state) affects
(Hannula, 2012) and positive (e.g., confidence and
interest) vs negative (e.g., anxiety and frustration) affect
(Goldin, 2000) in mathematical learning. Subsequent
scholars added other dimensions such as values
(DeBellis & Goldin, 2006) and motivations (Hannula,
2012). All these constructs appear to place affect in all the
processes of mathematics learning.
Early mathematical learning or problem-solving
theories or frameworks, however, mostly use a cognitive
perspective (Polya, 1945), including sociocultural
concerns (Francisco, 2013). Later studies focusing on
learners appear to trigger the addition of an affective
perspective (Voskoglou, 2011). Despite the intention to
distinguish between cognitive and affective issues in
mathematics education in order to delve into them in
depth, a line of research considers that affect and
cognition are indispensable or interweaving in
mathematics learning and teaching.
Mason et al.’s (1996) work appears to be the first to
formally address the issue of the interweaving of affect
and cognition in mathematics learning. The three phases
of mathematical problem-solving (entry, attack, and
review) formally incorporate both cognitive and
affective concerns in mathematical thinking. For
example, being stuck is viewed as an inevitable part of
mathematical processes. The key to conquering being
stuck is to reflect on prior experiences and emotional
moments. This line of research continues. Gomez-
Chacon’s (2000) study highlights six emotional
responses (calmness, confidence, cheerfulness, being
great, being blocked, and frustration) in affective and
cognitive contexts during mathematics learning.
Contribution to the literature
• Build a 4E Affective (Mathematics) Teaching (4EAT) Model, with a four-phase pedagogical structure:
entry, entertainment, enlightenment, and enrichment (i.e., 4Es) based on the enactivist’s perspective.
• Cross boundaries between affect and cognition by defining affective mathematics teaching as
transforming natural languages to mathematical languages.
• Overcome educational systematic constraints or tensions by aligning teacher agenda with student agenda.
EURASIA J Math Sci Tech Ed, 2022, 18(12), em2187
3 / 15
Cognitively, students experience a flash of intuition,
explore the correctness of understanding, and seek
strategies.
Mathematical teaching also manifests the issue of the
interplay of affect and cognition. At the beginning and
midway of solving problems, teachers’ affective support
is needed for students’ calmness and active participation
as problem-solvers, claim-makers, and solution-
reporters (Empson, 2003). A recent study by Marmur
and Koichu (2021) uses students’ key memorable events
(KMEs) to identify essential discursive events in
undergraduate lessons. Students’ affect or emotions are
highly related to key mathematical teaching events (e.g.,
lack of understanding), highlighting students’ needs for
heuristic-didactic discourse (meta-level learning), which
requires instructors’ investment, in order to increase the
opportunity of student learning affordance.
In summary, although the cognitive context appears
to directly fit most goals of mathematics curricula, the
affective context appears to be associated with
alternative, higher-order, and broader scopes of
mathematical learning. This line of research provides
evidence of the interplay of affect and cognition. Using
affect in mathematics may also initiate a new avenue of
mathematics research to advance mathematical learning
and teaching.
The MGA’s Theoretical Basis: An Enactivist
Perspective to Mathematics Education
The MGAs, the empirical data used in this study, base
their theoretical basis on the enactivist perspective of
mathematics education and are designed through the
process of metaphorizing, scaffolding, and gamification
(Yang et al., 2021). Enactivist perspectives insist learners
learn by situating and engaging themselves in the
context. What and how learners learn are co-determined
by themselves as human beings and their broader
ecological systems or “from cells to culture” (Hannula,
2012, p. 146). Desirable learning occurs only when
learners actively engage in and/or are triggered by
suitable teaching activities. This dynamic exemplifies the
enactivist perspectives to learning, as encapsulated by
Hannula’s (2012) three major propositions, as follows:
1. Emergence and co-emergence in fuzzy
boundaries: Learners spontaneously learn
cognitively or affectively; locally or globally; and
individually or socially. Learning may occur
within and beyond mathematics classrooms. This
is especially true with the advance of information
and communication technology (Chiu, 2020).
2. Structural affordance constraints: Learners’
action is constrained by the system. Instructors’
pedagogical designs offer affordances or
opportunities to learn mathematics. A salient
example is that young learners’ approaches to
mathematics are largely influenced by school
mathematics, with their confidence and interest in
mathematical problem-solving decreasing over
time (Hannula, 2019).
3. Embodiment: Mathematical learning is activated
by bodily experiences including gestures (body
movements), thinking (computational thinking),
and linguistic expressions (Kopcha et al., 2021).
Affect is a natural function of everyday human
activity, including mathematics learning. Affect is
essential for human survival, innovation, and
interaction, although affect’s role is often
recognized as weaker than that of cognition.
This enactivist perspective provides a fertile, flexible
ground for this study to build knowledge from authentic
mathematics learning experiences. In order to find the
content of ‘affect’, it is especially important to analyze a
context where affect is the aim of the mathematics
teaching design, such as the MGAs.
METHOD
Data Source and Sample
The major data source comprised all 42 MGA class
videos developed through the project “Just Do Math”,
implemented by SDiME (2022), starting in 2014. The
project is a response to a special phenomenon:
Taiwanese students have high achievement but low
affect (e.g., interest) in mathematics, as indicated by the
Program for International Student Assessment (PISA)
(OECD, 2014). The aims of the project, therefore, are to
develop mathematics activities to raise students’ affect
and ability to learn mathematics. At the time of writing
this study, the “Just Do Math” project has successfully
expanded through the professional development of local
school mathematics teachers (Chang et al., 2021). The
original activities have also been gradually adjusted to
fit the context of certain mathematics teachers.
The 42 MGA videos analyzed in this study are
available on YouTube (playlist on YouTube:
https://www.youtube.com/channel/UCj--Hy76_ZKsy
Gw_cP5HgLw/playlists?view_as=subscriber). Some
related sources included relevant open data shared
online (e.g., teachers’ Facebook posts). The datasets
generated during this study are presented in the two
supplementary materials (Supplementary Material 1 &
Supplementary Material 2). This study was part of a
larger project, which obtained the approval of the
institutional review board of National Chengchi
University (NCCU-REC-202105-I030).
Measures
Qualitative data analysis methods are used to
analyze the videos of the MGA in class (SDiME, 2022).
The initial coding scheme comprised three parts: lesson
structure, teacher contexts, and students’ issues. The
teacher contexts and student issues with both cognitive
Chiu et al. / The interplay of affect and cognition in the mathematics-grounding activities
4 / 15
and affective aspects were a combination of coding
schemes used by Mason et al. (1996) (including the three
phases of mathematical thinking, key moments, and
affective issues), Gomez-Chacon (2000) (including local
affective responses, affective contexts, and cognitive
contexts), and key moments (Marmur & Koichu, 2021).
While the affective aspect focuses on psychosocial
behaviors and teaching materials, the cognitive aspect
emphasizes students’ acquisition of the (declarative and
procedural) knowledge of mathematics or activities. The
analysis framework is presented in Table 1
(Supplementary Material 1 presents the detailed
analysis results of four videos).
Lesson structure
A lesson encompasses multiple teaching events in a
linear progression. The lesson structure identified key
phases that an excellent MGA would follow. Mason et
al.’s (1996) three phases of mathematical thinking (entry,
attack, and review) served as a starting framework.
During the process of coding, the coders aimed to
answer the question: “What is the structure (phases
developing over time) of the lesson?”
Teacher affective and cognitive contexts
Teacher contexts aimed to identify key moments
along the development (phases) of the lesson. The coders
asked in the coding process: What are the teacher’s
affective and cognitive contexts (e.g., instructional
vocabularies, behaviors, and material uses) that may
raise students’ affective and cognitive issues?
Student affective and cognitive issues
The affective and cognitive issues were students’
responses to the teaching context. During coding, the
coders kept in mind: What are students’ affective and
cognitive issues in relation to the teacher’s affective and
cognitive contexts, respectively?
Data Analysis
The content of the videos were qualitatively analyzed
using a combination of phenomenography (Marton,
1981), grounded theory (Charmaz, 2000; Strauss &
Corbin, 1990, 1998), and general qualitative data analysis
methods (Miles & Huberman, 1994). The qualitative data
analysis procedure included an iterative process of open
coding, theme finding, and theory building. The data
analysis process also included techniques of constant
comparison and dialogue with literature. The video-
narrative methodologies were also applied, starting with
transcribing verbal and non-verbal behaviors by clips,
followed by identifying critical events, coding,
constructing storyline, composing narrative, and
presenting results in different grain sizes to support the
discussion of issues (Derry et al., 2010; Powell et al., 2003;
Wilkinson et al., 2018). Detailed data analysis steps for
the RQs are presented, as follows.
RQ 1 data analysis steps
1. The first two authors discussed and identified the
most desirable affect-focused MGA (MGA1)
among all the 42 MGAs. The MGA1 had salient
pedagogies linking mathematics learning content
with student affects, using student affects as
teaching materials, and transforming student
affects into higher-order mathematical learning.
2. Analyzed MGA1: The key moments or critical
events of teachers and students were transcribed,
photocopied, and analyzed as narratives, a
procedure similar to a multimodal interaction
analysis (Wilmes & Siry, 2021) and video studies
(Derry et al., 2010; Powell et al., 2003). An initial
coding scheme was formed.
3. Used the initial coding scheme to compare the
analysis results between MGA1 and the next
MGAs until the features of MGA1 merged clearly
and solidly, or reached saturation, using a term of
qualitative methodology (Fusch & Ness, 2015).
This initial analysis was conducted using the
videos’ original language (Chinese) and used four
videos to reach saturation (Supplementary
Material 1). For example, challenge, curiosity, and
fantasy activities (Middleton, 1995) induced
learners’ situational interest (Hidi & Renninger,
2006; Rodríguez‐Aflecht et al., 2018).
4. Narrated MGA1 to identify features, as presented
in the results section (Supplementary Material 2).
RQ 2 data analysis steps
5. Went beyond the identified features and
generated a framework (theory or model) for
affective mathematics teaching for higher-order
learning.
6. Made associations between the identified features
and literature.
RESULTS (RQ1): THE STORY OF
‘RECTANGULAR NUMBERS’
The four steps of data analysis for RQ1 successfully
identified the most excellent affective teaching,
‘Rectangular Numbers’ (MGA1 in this study, video
available on YouTube: https://www.youtube.com/
watch?v=fImiotpvqBo&list=PLylUUAvq6ZNUSGm0I1
Table 1. Analysis framework
Coding schemes
Lesson structure
Affective (teaching) context
Affective (learning) issues
Cognitive (teaching) context
Cognitive (learning)
EURASIA J Math Sci Tech Ed, 2022, 18(12), em2187
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b5OfJ71Ys4GisCI&index=23) (SDiME, 2022). MGA1’s
key moments emerged vividly by further continual
comparisons with the other MGAs. This constant
comparison between MGA1 and the other MGAs further
revealed distinct features of affective mathematics
teaching (or reached theoretical saturation). This process
engendered a picture of an affective (affect-focused)
mathematics teaching: A clear lesson structure of four
phases, and the distinct features in each phase.
Lesson Structure: Entry, Entertainment,
Enlightenment, and Enrichment
The ‘Rectangular Numbers’ was a grade-5
mathematics class on point, prime, and composite
numbers (Yang et al., 2021). The teacher guided students
on the graphical meanings of prime numbers and
composite numbers and completed an exhaustive list of
factors for a given number through a game called ‘Go
Pieces’. The video lasted for 12.52 minutes.
The analysis started with Mason et al.’s (1996) three
phases of mathematical thinking (entry, attack, and
review). However, the teaching was found to follow four
phases: entry, entertainment, enlightenment, and
enrichment (including formative assessment after class)
(“4Es”). The reason may be the added ‘entertainment’
phase, in which students are immersed in mathematical
games.
The 4Es capture the major characteristics of
constructivist approaches to teaching and learning for
conceptual changes (Driver & Oldham, 1986). The
teacher started his teaching by reducing students’
barriers to entry into deep understanding of basic
meanings of a “rectangle” and explaining the rules of the
two-player game: One posed a number, and one formed
a ‘rectangle’. Students won if a rectangle was formed;
otherwise, they lost (Phase A). Then, the students
entertained themselves by participating in the game
(Phase B). Next, the teacher enlightened students with
correct answers, game results, and winning strategies
(Phase C). Finally, the teacher enriched students’
understanding by linking the previous hands-on,
embodied experiences (including feelings toward the
numbers in the game) to formal mathematics knowledge
(Phase D).
Features: Fully addressing each phase of the 4Es in the
right order
Excellent affective mathematics teaching has a
simple, clear structure of 4Es.
1. The entire teaching followed the phases of 4Es
once only.
2. Enough time was allocated to dig deep into each
task in each phase of the 4Es.
3. Except for the entertainment phase, where
mathematical games engaged students through
peer interaction, the other three phases (entry,
enlightenment, and enrichment) focused on
student-teacher interaction.
4. The teacher’s language, material, and activity use
gradually changed from natural/physical to
mathematical/symbolic languages. This fits
Piaget’s theory from concrete to abstract
representations or multimodal uses and from
natural (informal) to mathematical (formal)
language uses (Nunes, 1997).
This feature echoes a coherent, deep pedagogical
approach (Stigler & Perry, 1990). Teaching of each phase
adequately prepares students for the teaching of the next
phase based on cognitive development principles.
Phase A. Entry
(Video time from 0:24 to 4:10; three affective and
three cognitive key moments)
The lesson starts with the teacher (Mr. Chu)
reviewing prior knowledge using many scaffolding
questions (teacher cognitive context). Students
experienced a review scaffolding from easy to hard
(student cognitive issues), bridging past and new
learning (coded AC1, where A=Phase A, C=cognitive
aspect, and 1=the first code of ‘AC’).
Teacher (T): What did I draw on the blackboard?
Students (Ss): One dot.
T: What can two dots link together to form?
Ss: A line.
Then Mr. Chu gradually added dots (until 4 dots).
T: Aha! Can I use no matter how many dots to
form a line?
Ss: Yes.
T: Now I move the places of the dots. (Move four
dots to form a square.) What’s this called?
Ss: Square.
T: Can eight dots form a square?
Ss: Yes.
Mr. Chu invited Jeff to demonstrate how he formed
his graph on the blackboard (1:05) (Figure 1).
Figure 1. Jeff’s method (Source: Authors’ image based on
the MGA1 video)
Chiu et al. / The interplay of affect and cognition in the mathematics-grounding activities
6 / 15
After viewing Jeff’s method, Mr. Chu elicited other
students’ opinions rather than indicating Jeff’s mistakes
(teacher affective context). Jeff and/or all students
would feel less egocentric and feel a sense of a learning
community because they were not criticized directly but
reached out to the community. Engaging in the
community might reduce their negative emotions
towards learning mathematics (student affective issue)
(coded AA1, where first A=Phase A, second A=affective
aspect, and 1=first code of ‘AA’).
S: There is a space in the center.
T: Today we must form a full spaced square … or
a rectangle (1:24).
Mr. Chu invited another student to demonstrate the
formula in forming a solid square using eight dots.
Students underwent natural (‘in a row’ and ‘dots’ visual
aids) to mathematical languages (‘mathematical
calculation expressions’ (AC2) (Figure 2).
T: How many dots in a row? … This can be
recorded as 4×2.
The teacher asked a student (David) to play the game
with him. Students felt involved and on equal footing
with the teacher and each other (AA2).
T: David gave me a number, “13”, wanting me to
make a square or rectangle. If I can form, I can
obtain one point.
The teacher intentionally made mistakes, invited
students to judge, and asked for reasons. Students
clarified game rules by learning from mistakes of the
teacher (AC3). The teacher formed the shape shown in
Figure 3.
T: Is it a square or a rectangle?
Ss: Neither.
T: Why not? (Mr. Chu invited Mary to reply)
Mary: Because there is a hole.
T: There is a hole, so it is not … (Mr. Chu waited
for Mary to reply).
Mary: A rectangle.
T: Very good! Thank you! Please sit down (2:05).
The teacher gave compliments for correct responses
and contributions to the learning community. Students
would thereafter feel social recognition or confidence
(AA3).
Feature: Invite to enter by reminding prior knowledge
and introducing interesting games
Phase A introduced the game and reviewed prior
knowledge through the teacher’s interactions with
students on equal footings. Students’ situational interest
in the game appeared to open the avenue for students’
entry into learning mathematics.
The affective key moments in this phase were
teachers inviting alternative opinions to supplement
incomplete or incorrect student answers (rather than
directly indicating students mistakes), playing with
students to demonstrate how to play games, and
providing positive feedback for correct responses.
Through this context, students experienced the learning
issues of feeling diversity, involvement with equal
footing with the teacher, and social recognition in the
learning community.
The three cognitive key moments include teachers’
reminding the basics for the new learning content
through Q & As (rather than direct teaching), connecting
daily languages with mathematical languages that
students had already learned, and making intentional
mistakes to clarify game rules. It is inferred that students
thereby could review prior knowledge, check their
learning, and obtain further competency to play the
games.
Phase B. Entertainment
(Video time from 4:11 to 5:57; 2 affective and 1
cognitive key moments)
Students began playing the game in pairs (4:11). Mr.
Chu was not present in the video; it can be inferred that
he enacted a competitive game through Phase A
activities and students’ behaviors. A scoring system was
activated and might trigger students’ desire to win
(BA1). Given the playful essence of the activity, students
looked happy and focused during the game (BA2).
Visual aids (times tables for 11 to 19×1-10) served as
hints to support students in playing the game (5:36)
(BC1).
Feature: Entertain by playing games
In Phase B, even without formal teaching
(interventions), it can be inferred that students
Figure 2. Forming a solid square using eight dots (Source:
Authors’ image based on the MGA1 video)
Figure 3. The teacher’s formation (Source: Authors’ image
based on the MGA1 video)
EURASIA J Math Sci Tech Ed, 2022, 18(12), em2187
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continuously practiced the key lesson content during the
game and intuitively gained mathematical knowledge.
Phase B had two affective key moments (BA1-2).
Teachers enacted competitive games, which used
scoring systems to rank players as winners or losers.
Teachers could also enact playful activities (games)
without any scoring system. Students would experience
focus and excitement, with strong, diverse emotions in
playing competitive games and positive emotions in
playing games.
Cognitive key moments were limited in Phase B. The
reason may be that this phase was students’ play time;
the teachers limited their interventions to a minimum.
This MGA1 teacher provided conventional
mathematical calculation support. Answering students’
questions was a variation of this kind of support often
observed in other MGAs.
Phase C. Enlightenment
(Video time from 5:58 to 8:44; two affective and three
cognitive key moments)
After the game (Phase B: Entertain), Phase C
(Enlighten) began (5:58). Mr. Chu asked a series of
positive scaffolding questions about the game results,
repeating student responses and inviting them to say
‘yes’ before checking answer correctness. Students
would feel positive acknowledgement before being
assessed as winner or losers (CA1).
T: Some numbers may allow more than one
approach to form (a rectangle), right?
Ss: Yes.
T: Perhaps two approaches?
Ss: And three approaches. And four approaches.
T: Yes, there may be three approaches. Is there any
number which allows three approaches, but you
only wrote out two?
Ss: Yes.
Mr. Chu presented the correct answer table on the
blackboard (Table 2). Based on Table 2, Mr. Chu
designed the key question, ‘Is there any miss?’ and
enacted group discussion. Students learned
collaboratively to answer the question (CC1).
T: OK, shall we find it? Let’s check your
approaches. Two in a group collaborate to check
whether there is any mistake (on your (guided
inquiry) worksheet).
Then, Mr. Chu invited and interacted with students
to demonstrate their results, rationales, and teaching.
Students became teachers, active learners, and
contributors in the learning community (CA2). Mr. Chu
invited two girls working in a group (Alice and Betty)
onto the podium with the teacher to share their answers,
where the distance between the students and teacher
was shortened (CA3).
T: In which number did you omit an approach?
(6:18).
Alice: 24.
T: 24 has a miss. Which did you overlook?
Alice: 2×12.
In the process, Mr. Chu reconstructed inaccurate
answers. Students gradually used more mathematical
languages (CC2).
T: 2×12 is missing. Why?
Betty: Because it is not included in the
multiplication table of nine …
T: How did you find (it)?
Alice: We should not use multiplication, but
division.
T: Oh! Later we find that in addition to using
multiplication, division is another choice to find
more answers … However, division may also miss
some answers. How can we solve this issue?
Betty: Divide one by one ... like 1, 2, 3, ..., 10 (7:00).
T: When can 24 no longer be divided?
John: Start from two.
T: Oh, so 24 divided by two equals 12, and then
three, four, …
Adam: Divided by two is 12. Divided by three is
eight. Divided by four is six. Divided by five is
Table 2. Correct answer table
Numbers and records
Number
1
2
3
4
5
6
7
8
9
10
…
21
22
23
24
25
Record
X
X
X
2×2
X
2×3
X
2×4
3×3
2×5
…
3×7
2×11
X
2×12
3×8
4×6
5×5
Chiu et al. / The interplay of affect and cognition in the mathematics-grounding activities
8 / 15
none. Divided by six returns to four. [It] cannot
repeat. So, divide until six.
Then, Mr. Chu asked students their winning
strategies. Students described their intuitions about the
game (CC3).
T: Raise hands, the winners … What’re your
secrets to winning? How did you win the game?
(7:54).
S: Give others the number that cannot be divided
…
A meta-knowledge about the mathematics of prime
and composite numbers was gradually generated
through students sharing winning strategies. In the
process, self-monitoring arose.
Feature: Enlighten by inviting mathematical intuitions
from playing the game, including answers, results, and
strategies
Phase C appeared to be a stage bridging the game
(hands-on experiences) in Phase B and the lesson
objective of mathematics teaching in Phase D. Students’
natural languages gradually transformed to
mathematical languages by Mr. Chu’s raising key
questions (e.g., checking answer correctness and
identifying winning strategies), rephrasing students’
responses, and discreetly adding more formal,
conventional mathematical terms and expressions.
There were two affective key moments. First, teachers
repeated student utterances and generated ‘yes’ teacher-
student dialogues before providing answers or game
results that identified winners or losers. This ‘yes’
atmosphere would increase students’ sense of
acceptance before revealing the game results. Second,
teachers invited and interacted with students to
demonstrate their results, rationales, and findings,
during which students became teachers, contributing to
the learning community.
The cognitive aspect included three key moments.
The teachers designed key questions for small group
discussion, reconstructed students’ inaccurate answers,
and asked students their winning strategies during the
game. By identifying key concepts, linking mathematical
concepts, and employing higher-order thinking, it is
inferred that students would move towards
collaborative wisdom, initial mathematical patterns, and
mathematical intuitions in relation to their experience of
playing the games.
Phase D. Enrichment
(Video time from 8:45 to12:36; three affective and
three cognitive key moments)
Based on the emergence of self-monitoring in Phase
C, Phase D formally introduced conventional
mathematics knowledge by using student emotions as
teaching materials. Perhaps the most distinctive event of
this teaching is that the teacher asked students’ feelings
about playing the game. Thus, students divulged their
emotions during the competitive game in Phase B (DA1).
T: If I give you ‘23’, would you try your best to
form it?
Greg: Yes.
T: Did you finally form it?
Greg: No.
T: How did you feel then? (Use a dramatic
emotional voice.)
Greg: It is a bit frustrating.
T: This number makes you frustrated because you
cannot form it for many times. If s/he gave you a
number other than 23, like 22.
Greg: Yay (a cheerful sound)! (use two hands to
show two Vs--sign for victory, with laughter on
his face.) (Other students also laugh.) ...
T: ‘13’ (9:22).
John: Very worried and stressful. I want to beat
“13” then (with a sad sigh and humorous voice,
while gesturing as if beating something) ...
T: May I ask if s/he gives you ‘22’?
John: So happy (with a cheerful smile on his face,
both hands forming V (victory), and dancing) …
Mr. Chu reflected the process like telling a story, from
playing the game, expressing feelings, and associating
their feelings with the number. Students revisited the
whole process through personal stories and stayed
focused (DA2).
T: So happy, so happy. Look at this (correct
answer) table now. There are many numbers in
this table. These numbers are just numbers, but
because you have played the game, … you will
have a feeling for these numbers. It may be joy, it
may be anger, it may be sadness, and it may be
despair. Those are your moods. Let’s first divide it
into two categories, according to your mood, or
the pattern you see, and give it a name.
After that, Mr. Chu asked students to name the
mathematical phenomenon they perceived. They wrote
on their worksheets and created their own terms (DC1).
T: What do you call those numbers which cannot
form (a rectangular or square)?
EURASIA J Math Sci Tech Ed, 2022, 18(12), em2187
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S: Bad eggs.
T: Bad eggs. How about the numbers that can?
S: Good eggs.
Mr. Chu asked several more students and wrote their
answers on the blackboard (Table 3).
Mr. Chu then summarized the discussion by
associating students’ creations and mathematics as an
aid for pattern recognition, which was then used to
introduce formal mathematical terms. Students were
exposed to both their own natural languages and formal
mathematical languages, thereby building mathematical
knowledge in cognitive aspects (DC2).
T: Mathematicians divide these numbers into two
categories. One is that they can be arranged in
rectangles or squares. Mathematicians think that
this category, no matter which number, can be
decomposed into two other numbers and
multiplied together. In other words, it is
synthesized from two numbers ... Therefore,
mathematicians call these numbers of composite
numbers ... Line point numbers only have lines
and points.
T: Which number is special? (Mr. Chu ask
questions for deeper understanding, which
strengthen student learning [DC3]).
S1: ‘1’.
T: Why?
S2: Because it cannot form a line.
T: Although you feel the invalid numbers are
surprising or weak, mathematicians do not. They
think they are numbers that cannot be
decomposed any further. What are they called?
(Mr. Chu showed “prime number” on the .ppt).
Ss: Prime numbers (12:06) (Only until now was
conventional mathematics introduced, gradually
from students’ natural languages to mathematical
languages).
To deepen student understanding, Mr. Chu
pretended to be oblivious and asked them to teach him.
Students rephrased what they had learned gradually
from unclear to clearer mathematical concepts by
becoming a tutor (DA3).
T: I’m now a student who does not understand
mathematics. Please tell me what is a prime
number? How will you explain it?
S1: A number with only one and itself.
S2: No multiplication (on the correct answer table)
(12:20).
More students stated their experienced mathematical
phenomenon. Interaction between Mr. Chu and students
continued until the meaning of prime numbers could be
clearly addressed by the students using their own
words. Here, students experienced a sense of presence in
a mathematics learning community.
Feature: Enrich by linking natural (emotional) terms to
abstract mathematical terms
Phase D addresses formal, conventional knowledge
in mathematicians’ world. A unique pedagogy is the
process of arousing students’ emotions by revisiting the
game, reporting, and classifying their emotions into two
categories based on experiences of playing the games in
Phase B, and connecting students’ and mathematicians’
lexicons with mathematical rationales, which were also
experienced by the students during the game.
The final phase features presence, where students feel
gifted through higher-order mathematical knowledge
and skills. The affective key moments manifested by
directly asking students’ feelings about the numbers
used in the game, using story-telling throughout the
whole process to link students’ emotions to the numbers,
and students’ teaching. Thus, students divulged their
strong emotions about the game (results or objects
[numbers]), riveted in personal stories, and became
helpers (teachers) in the learning community.
The three cognitive key moments were the teacher’s
initiating activities for students to create terms for a
mathematical phenomenon they experienced while
playing the game, linking student creations to
conventional mathematical knowledge, and asking
questions (e.g., “why”). It is inferred that these activities
would deepen the understanding of the newly acquired
mathematical knowledge.
Table 3. Students’ answers
Invalid numbers
Valid numbers
‘Bad eggs’
‘Good eggs’
Line and dot numbers (because the numbers can only form line or dot)
General numbers
Weak
Potent
Surprise numbers (because we both want to give it to each other in the game)
Square-rectangle number
Explosion numbers
Very good
Chiu et al. / The interplay of affect and cognition in the mathematics-grounding activities
10 / 15
Student Outcomes and Teacher Reflections
Mr. Chu shared a student’s work after the lesson on
an open Facebook post (Chu, 2020):
I named the numbers that could not form a
rectangular (or square) as ‘dedication numbers’.
Reason: They must be ‘prime numbers’ because
only 1 and itself can become them (e.g., 1×19=19),
not others. It’s like our fathers love our mothers …
exclusively forever …
(On the other hand,) the numbers that can form a
rectangle (or square) are the ‘companion
numbers’. Reason: Even numbers must be (the
‘companion number’). There are many partners
multiplied together, and they will not feel lonely
(e.g., 8=1×8, or 2×4) ...
Mr. Chu also shared his own reflection.
(Students) learned more than just mathematics;
more importantly, thinking like a mathematician
… The student said, ‘prime number’ and
‘composite number’ but couldn’t explain the
reasons why mathematicians used these names.
‘Names given and explained by yourself’ gives
name essence of the humanities of mathematics.
(Students provided the following names for prime
numbers and composite numbers)
‘Hate number’ and ‘like numbers’.
‘Oh my God number’ and ‘Oh yes number’.
‘Dedication number’ and ‘companion number’.
‘Introvert number’ and ‘extrovert number’ …
(This is a) real experience, like a mathematician
forming mathematical concepts.
In summary, Mr. Chu’s teaching of ‘rectangular
numbers’ was selected by the research team as excellent
affective mathematics teaching. The major reasons are
that students’ affective or emotional states are strongly
stimulated through playing games and directly
transformed to connect with the targeted mathematical
topics. The teacher’s interactions with students created a
desirable atmosphere of mathematical learning in both
affective and cognitive aspects.
DISCUSSION (RQ2)
MGA1 appears to feature the merits of the best
mathematics teaching practices such as engaging,
listening, using questions, preparing, assimilating with
rich representations, learning community, discovering,
and uplifting students’ roles (Maher et al., 2014;
Schulman, 2013). A 4E affective (mathematics) teaching
(4EAT) model (Figure 4) is posited on the basis of the
dialogue between the answers to RQ 1 and the literature.
Beyond Boundaries (Enactivist Proposition 1) by
Defining Affect as Natural Languages to Approach
Mathematical Languages
This addresses the proposition of the enactivist
perspective in going beyond fuzzy boundaries. Given
the focus of this study, the following definitions aim to
define affective and cognitive mathematics separately
and transcend their boundaries.
Affective mathematics=Natural languages for (learning
and teaching) mathematics
Affect as natural languages. This pedagogical design
is consistent with the notion that affect represented by
natural languages are more automatic and precedent,
which should be resolved before cognitive, intentional
issues (Duval, 2000). One salient example is that an
affective mathematics teacher would invite students’
features of the enactivist MGA and contribution in the
learning community. This includes unlocking students’
hidden emotions from entertaining games. The teacher
accepted students’ languages by repeating them.
Affective mathematics emerges here and now, co-
emerging in fuzzy boundaries with cognitive
mathematics and all entities in and beyond mathematics
classrooms (Chiu, 2020; Hannula, 2019). Affective
mathematics materials are stories, daily languages (e.g.,
emotional responses of frustration and joy), and daily
activities (e.g., games, home design, and financial
simulations). Affective mathematics events are
psychosocially-related activities, involving psycho-
somatic behaviors, interaction, and perception, which
prompts a learning community.
Figure 4. 4E affective (mathematics) teaching (4EAT) model
(Source: Authors’ own elaboration)
EURASIA J Math Sci Tech Ed, 2022, 18(12), em2187
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Cognitive mathematics=Professional languages in
mathematics
Mathematics is a study of abstract patterns and
relationships (Burton, 1994), which moves mathematics
toward a pure, context-free, and cognitive domain of
knowledge. The national mathematics curriculum was
designed by mathematics experts and educators who
have been learners in the system and endorses the
mission to transmit the cultural heritage to the next
generations. This forms instructional or structural
affordance constraints.
Affective mathematics teaching=Transforming natural
languages to mathematical languages
Affect can serve the cognition essence of
mathematics. Affect or emotions naturally emerge from
students’ daily lives and playing the games (in Phase B),
while cognition is the main goal of mathematics with
abstractive, conventional knowledge rooted in teachers’
minds and the mathematics curriculum. Perhaps one
naturally generated affective response is the anxiety of
self-identification and social recognition as being
winners or losers if a competitive game is activated.
Using affect naturally emerging from playing games as
teaching materials is to use emotional arousal (e.g.,
anxiety and pride) and reflections to engender a deep,
experiential understanding of mathematics.
Affective mathematics teaching will fully transform
students’ informal daily activities, events, or languages
to formal mathematical activities or languages as
practiced in the natural curriculum and the professional
world (Nunes, 1997; Stipek et al., 1998). This will reduce
the difficulty of learning mathematics with affective
responses represented by natural languages as
automatic, precedent issues coming before
mathematical, cognitive, and intentional issues (Duval,
2000). Further, these key moments (features) delves into
the essence of mathematics-thinking like a
mathematician and being consistent with a discovery-
oriented teaching (Askew et al., 1997).
A salient practice of affective mathematics teaching is
to use students’ affective responses to a mathematical
game as teaching material, as in the enrichment phase of
MGA1, where the teacher invited students to map the
names of negative and positive emotions (mainly toward
the numbers and the results in the game) to prime and
composite numbers. Subsequently, students’ affective
responses (personal, natural languages) from playing
the game (natural activities) are used as teaching
materials and fully transformed to cognitive
mathematician-like thinking (mathematical languages
and activities).
Beyond Constraints (Enactivist Proposition 2) by
Fitting Teacher Agenda to Student Agenda
According to the enactivist perspective to
mathematical learning, learning opportunities are
entrenched in the teaching context or system. Learning
opportunities can be manifested in students’ agenda and
teaching context in teachers’ agenda.
Student agenda: Upward learning=isUP (interest,
sense, utter, and present)
The theme of students’ learning agenda is the issue of
upward learning or growth mindset (Yeager et al., 2019).
Students are born potentially curious about the world
and mathematics. However, despite striving to learn
mathematics (Burton, 1994), most students gradually
lose their confidence and interest throughout
educational stages (Hannula, 2019).
The results of RQs 1 and 2 highlight four phases of
student learning issues in experiencing the MGAs:
interest, sense, utterance, and presence. Students should
be motivated by situational interest and reminded of
prior learning, experience hands-on playful activities,
utter the facts of the playful experiences, and celebrate
the rewards of obtaining abstract mathematical
knowledge and skills.
Teacher agenda: Cultural teaching=CARE (cultivate,
amuse, reflect, and explain)
The theme of teachers’ agenda is acknowledging
mathematics as a cultural product and implementing the
mathematics curriculum by the cultural educational
system. This theme, however, forms both opportunities
and constraints.
To avoid the structural affordance constraints,
teachers need to care about students’ agenda. Teachers’
agenda needs to begin by cultivating student interest
and foundations, followed by enacting playful activities
to enrich sense-making, inviting students to reflect on
the activities, and finally explaining the newly learned
conventional mathematics knowledge and skills as
addressed in the curriculum by eliciting students’
previous experiences in the lesson.
Tension or harmony?
Both teachers and students may be constrained by
their themes and agendas. With teachers’ superior
status, they assume the role of building a mathematics
classroom with tension or harmony. As a saying by
‘Zhuangzi (莊子)’, ‘It’s hard to tell a worm that lives only
until summer about ice (夏蟲不可語冰)’. Teachers or
mathematicians are the survivors in learning
mathematics; they are capable enough to live until the
winter and know what ice is (the cold, abstract, and
cognitive knowledge of mathematics).
Chiu et al. / The interplay of affect and cognition in the mathematics-grounding activities
12 / 15
Tensions occur if teachers fail to care about students,
who strive to learn professional (cognitive) mathematics.
Students will potentially lose interest and reject learning
mathematics, like a summer worm with warm, affective
mathematics competencies (e.g., natural languages with
playful tendencies) dying before winter’s (cold,
cognitive mathematics) arrival. Harmony occurs if the
proposed teachers’ agenda fits students’ agenda.
Beyond Embodiment (Enactivist Proposition 3) by
Enacting Affective Mathematics Teaching Through
4Es Phases
Affective experiences naturally arise from
embodiment or daily activities, including mathematics
learning, though it is typically perceived as cognitive
experiences. Affective mathematics teaching can
successfully link natural embodiment activities with
mathematical learning through four phases: entry,
entertainment, enlightenment, and enrichment (‘4Es ‘).
The 4Es capture the major characteristics of enactivist
approaches (Hannula, 2012; Yang et al., 2021) and
constructivist approaches to teaching and learning for
conceptual changes (Driver & Oldham, 1986).
The reasons for the appropriateness of 4Es may be
that they align with the basic structure of the traditional
three-phase lesson design (motivating, main, and
synthesis activities), but adds an ‘entertaining’ element
which specifically tackles the issue of affect-focused and
enactivist design of the MGA (SDiME, 2022). Further, the
4Es fit the four steps of typical Chinese writing:
introduction, elucidation, transition, and conclusion. It
also matches our natural, physical experiences of the
four seasons, making the lesson structure easily
acceptable and possibly automatically adopted, though
with some adjustments by teachers.
4Es’ affective and cognitive mathematics
While enactivist 4Es develop (or move) along the four
teaching phases, affective and cognitive mathematics are
interwoven along the 4Es phases. However, relative or
sequential roles of affective and cognitive mathematics
teaching in each phase can be derived from the answers
to RQs 1 and 2. The interplay between cognitive and
affective mathematics would build a positive
atmosphere for learning mathematics.
Phase A: Cognitive to affective mathematics: Entry
starts with reminding prior mathematical knowledge
and ends in game preparation. Situational interest is the
key through inviting students to play the games.
Phase B: Affective with cognitive mathematics:
Entertainment involves students actively experiencing
playful activities and intuitively sensing mathematics.
Students’ natural language use dominates this phase of
game playing, while the cognitive mathematical learning
is implicit or embedded.
Phase C: Cognitive with affective mathematics:
Enlightenment of mathematical minds confronts
students with facts (game results), mistakes, and
patterns, while building a safe affective atmosphere.
When performance is the concern, potential affective
issues (e.g., frustration, pride, and confidence) arising
from social comparison and recognition may deserve
notice by educators.
Phase D: Affective to cognitive mathematics:
Enrichment builds upon students’ creation of terms
(starting with affective/emotional
languages/representations) for the mathematical
phenomenon, experiencing the lesson like a story, and
linking formal mathematics knowledge, skills, and
terms. The underlying mechanism may be that
emotional languages may not completely satisfy the
experienced (mathematical) phenomenon. Pattern
recognition gradually emerges and leads to the creation
of concise, abstract mathematical languages.
CONCLUSION
Contribution
This study used a qualitative methodology to analyze
mathematics teaching lessons focusing on promoting
student positive affective responses to learning
mathematics using an enactivist perspective. Dialogues
between the lesson analysis results and literature
identify features of affective mathematics teaching. A
framework for affective mathematics teaching (4EAT
model) is further built to add theoretical interests and
pedagogical insights to enactivist’s perspectives in
mathematics education.
1. Cross boundaries between affect and cognition by
defining affective mathematics teaching as
transforming natural languages to mathematical
languages.
2. Overcome educational systematic constraints or
tensions by aligning teacher agenda (with care
through the pedagogical phases of cultivating,
amusing, reflecting, and explaining) with student
agenda (with upward learning tendency through
the phases of interest, sense, utterance, and
presence).
3. Extend embodiment activities to a four-phase
pedagogical structure: entry, entertainment,
enlightenment, and enrichment with relative
emphasis and sequence between affective and
cognitive mathematics teaching in each phase.
Limitations of This Study and Suggestions for Future
Research
This study conducted in-depth case studies, which
are widely used by studies on student local affect during
mathematical teaching (e.g., Marmur & Koichu, 2021).
While qualitative methodologies rely on contextual cases
EURASIA J Math Sci Tech Ed, 2022, 18(12), em2187
13 / 15
to infer theories, this inductive nature suggests using
quantitative methodologies to validate the findings.
Experimental studies can also be conducted to validate
the theory and models. Large-scale surveys can validate
the model further.
Author contributions: M-SC: methodology, visualization,
validation, data curation, project administration, & funding
acquisition; M-SC & F-LL: conceptualization; M-SC & TH:
writing-original draft & editing; M-SC, T-JW, & P-SL: formal
analysis & investigation; M-SC & P-SL: supplementary materials;
& F-LL & K-LY: resources & writing-review. All authors have
agreed with the results and conclusions.
Funding: This work was supported by the Ministry of Science and
Technology, Taiwan (MOST 110-2511-H-004-001-MY3).
Ethical statement: The authors stated that the study was approved
by the Research Ethics Committee of National Chengchi
University on July 23, 2021 (Reference number: NCCU-REC-
202105-I030). Informed consents were obtained from the
participants.
Declaration of interest: No conflict of interest is declared by
authors.
Data sharing statement: Data supporting the findings and
conclusions are available upon request from the corresponding
author.
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