These examples highlight that, for some students, their aspirations for mathematics learning relate to the avoidance of experiencing embarrassment in class. In terms of goal orientation they could be viewed as about any of the three aspects of *performance*-*avoidance*: avoiding looking incompetent or dumb (appearance), avoiding performing poorly in class (normative), or avoiding demonstration of a lack of ability relative to others and as judged by an authority figure (evaluative) (Elliot, 1999; Hulleman et al., 2010). Yet the last quote in the selection above highlighted that, for some students, being perceived by others as *smart* raised the fear of being teased about *that*, suggesting an influential role for peer expectations and social needs in students' mathematics learning.

Other emotional responses appeared to relate a mastery goal orientation to a motivational aspect of their learning. These are discussed in the next sub-section.

#### **<IP_ADDRESS> Students Relating Mastery Goals to Affect and Motivation**

The desire for understanding, expressed by many of the students, can be related to a mastery-approach goal orientation, of wanting to learn as much as possible to improve their knowledge, to understand the content as thoroughly as possible, and acquiring new skills (Elliot, 1999; Hulleman et al., 2010). Some responses in the *emotional response* code further linked this desire, to understand mathematics, to various motivational variables. The desire for being able to understand the mathematics and the experience of confusion leading to a subsequent lack of motivation are illustrated by the following responses:

My one wish is to understand math much more and to get it a lot more. My only weakness in learning is math. My wish is to stop getting frustrated when I do it all the time. If I understand math more then I would enjoy math a lot more. It would be my dream to get math and to understand it more. I have difficulty a lot in math and like I said I would really really REALLY LIKE MATH A LOT MORE if I UNDERSTAND! (Yr 5/6 boy #892)

To try a lot harder and focus more. I seem to doze off a bit when I get really confused. (Yr 7/8 boy #606)

Is to try and understand what I'm trying to learn and to always keep my head up and not to give up when I'm stuck on a question. (Yr 7/8 boy #167)

It would be that I could understand more that the teachers tell me and that I could understand fractions a bit better, I would also like to concentrate more in maths because sometimes I don't get what the teachers are saying. I think I learn differently to other people because sometimes I need to be told it in many different ways to understand. I think that I know how to do it but I just need the teachers to tell us in different ways. (Yr 5/6 girl #183)

These quotes illustrate that some students struggle to maintain their motivation when experiencing a negative emotion, such as confusion, in response to a task. As with the last quote above, some students also suggested a teaching approach for helping them to understand mathematics, such as being given multiple ways to solve a problem, suggesting their perception that a greater understanding will lead to increased motivation.

Some of the students' responses highlighted their desire for interesting and challenging tasks to engage them and to motivate them to learn:

To always be able to always do maths all day and for it to stay really fun and to get harder and harder so I can learn a lot about maths. (Yr 5/6 girl #23)

To be able to have more interactive, interesting and fun math activities for both me and the school. (Yr 5/6 boy #36)

My wish would be to make things more interesting in maths. It can get very boring sometimes and you don't really feel motivated to work. So maybe try and mix it up and try using the computers more or having breaks and letting us talk a bit more. (Yr 7/8 girl #312)

To be able to choose what I want to learn in class. I think this would help my learning a lot because I would have the motivation and energy (not that I don't) to do more work especially when I'm learning something I'm interested in. I would also like it if we could work in groups and partners more so we could work together in hard (or easy) problems. (Yr 7/8 girl #442)

These types of responses also evidence a mastery-approach goal orientation in terms of interest and curiosity—learning something interesting—as well as mastering a challenge (Elliot, 1999; Hulleman et al., 2010). In the third and fourth quotes above, the students explicitly referred to their self-motivation as being influenced by their level of interest. The first and fourth quotes also referred to hard or harder work, suggesting that some students do value challenge in learning mathematics. They can be related to Middleton's (2013) central motivational variable of *interest*, specifically situational interest from tackling a novel task that is interesting and challenging.

Overall, it can be seen that analysis of the large data set generated by an openresponse question provided a variety of insights into the expressed wishes of middle school students. A wide range of codes was created interpretively to capture the more subtle differences between the students' use of language in their responses. Qualitative analysis of the codes provided insight into how some middle school students related their experience in the mathematics classroom to their goals, affective responses, effort, and motivation for learning.

#### **8.5 Conclusion**

This study provided the opportunity to learn more about middle school students' motivation and engagement in mathematics from their perspectives, by inviting them to write their aspirations in their own words as a complementary methodology to predefined surveys. The high response rate and the low incidence of irrelevant responses suggest that when consulted, students similar to this cohort are likely to be willing to share their views sincerely. Some students referred to intrinsic aspects of their aspirations for learning such as their own interest, attitude, effort, and self-motivation. Some students described extrinsic aspects of their environments that are perceived by them to influence their engagement in learning, such as challenging or interesting work, learning multiple strategies, collaboration with peers, help from their teacher, or not being embarrassed by others in class. As previously discussed, the overall percentage of references to emotions was lower than expected. It is possible that an additional survey question requesting the *reasons* for their wish might have elicited more references to affective factors including emotions. Those students who did provide unsolicited explanations often included emotional references, which linked their goals explicitly to particular emotions. This resonated with Hannula's (2006) conceptualisation of motivation as "a potential to direct behaviour through the mechanisms that control emotion" and which is "structured through needs and goals" (p. 175).

Although there are conceivably many ways to analyse such a data set from over 3500 free-format responses, this study used a framework of wishes related to: learning or achievement, being taught, affect or motivation, specific task types, and working arrangement. It examined the students' responses using theoretical perspectives on intrinsic motivational variables and goal orientations, and extrinsic learning environments. Comparisons between year levels and genders at upper primary and lower secondary levels provided evidence that particular aspirations were more salient for particular groups of students. For example, more students from Year 7/8 expressed their wish to make more effort, compared to younger students. The Year 7/8 girls made more frequent references to the desire for fluency and for learning multiple strategies. Previous research found a mismatch between secondary teachers' negative predictions and students' positive attitudes about learning multiple strategies (Leikin et al., 2006). In Japanese secondary classrooms, a problem-solving oriented lesson involves students demonstrating and discussing a variety of solutions to one problem (Ohtani, 2014). There is more to learn about secondary students' views about learning this way (Lynch & Star, 2014), but it appears from this study that they may be more open to it than teachers predict.

The Year 5/6 students expressed more frequently their wish for more challenge in their learning. The Year 5/6 cohort also made more frequent reference to improving their knowledge or skills in a particular area of mathematics, such as fractions/decimals/percentage and times tables, yet both genders at Year 5/6 and at Year 7/8 referred to these areas. These topics also appeared in the list of five most frequent categories for girls and for boys. This suggests that for many students, targeted support to improve their understanding in a particular area of mathematics would be motivating. It would be worth investigating further why times tables is of particular importance to students, a finding Grootenboer and Marshman (2016) also highlighted with New Zealand middle school students.

The Year 7/8 cohort made more references to wanting to work in small groups, perhaps because they had experienced fewer opportunities in secondary school (Banilower et al., 2013). A recent study found that secondary teachers were able to broaden students' learning opportunities by using collaborative learning (DeJarnette & González, 2015); this study suggests that it might also be motivating for some students. More Year 7/8 students referred explicitly to teacher support (both socioemotional and instructional aspects) than those in Year 5/6, resonating with studies that found teacher support more strongly related to affective outcomes for older middle school students (Rolland, 2012). Overall both girls and boys and upper primary and lower secondary students made responses that were categorised across the full range of codes, suggesting that within one class of students similar to this cohort, a teacher is likely to find a diverse range of individual aspirations.

Evidence for all four goal orientations was found in the students' responses across several different categories, but with evidence that mastery goals are expressed more frequently than performance goals. This resonates with other studies on student goals (Brophy, 2005; Chouinard et al., 2007) but extends that work by suggesting that students do *spontaneously* generate evidence of performance goal orientations—that it is not survey items on this particular goal that suggest to students goals they don't actually hold. A mastery-avoidance goal orientation was evidenced in the small percentage of responses about *retaining: not forgetting, remembering, memorising, off by heart, revising*.More girls than boys evidenced this orientation, and this concern may relate to one possible reason why girls also have demonstrated higher maths anxiety in other studies (Grootenboer & Marshman, 2016; Plenty & Heubeck, 2013). A performance-avoidance goal orientation was evidenced in the small percentage of emotional responses that related to students' wishes not to be embarrassed.

It appears that a methodological approach giving students the opportunity to share their voice using a free-response format can provide worthwhile insights into the affective and motivational dimensions of a particular cohort of students' engagement and goals in learning mathematics. Individual and often detailed responses from the students provided further insight into the foundational desire for *understanding* mathematics, the influence of external features in their learning environments on their effort, and the key motivational factor of *interest* (Middleton, 2013), in particular *tasks* that are interesting and challenging.

There is more to understand and other conceptual lenses with which to view the students' aspirations for their learning. Because the data represent a snapshot in time, it was not possible to investigate the stability of students' expressions of wishes longitudinally. Perhaps a conclusion that can be drawn from the work to date is that seeking students' own voice on affective and motivational issues is valuable and may challenge some deficit theorising about middle school students' attitudes. Both the boys and girls expressed a wide range of hopes, which are overwhelmingly positive and often focussed on mastery-approach goals. Their language was often quite specific, suggesting that similar students to these are likely, when asked, to explain willingly and in detail what they desire in their learning and how teachers might support them. An implication for teachers is that rather than generalising these findings to other cohorts of middle school students, they might view their classes as comprised of many and varied individuals, to seek information about their specific wishes, and to find ways to incorporate students' own suggestions for promoting their engagement, learning, and achievement.

**Acknowledgements** The student participants who contributed to the study on which this paper is based are acknowledged with appreciation. Special thanks to Peter Sullivan for his oversight of the research and Daya Weerasinghe for his contribution to the data analysis. Gilah Leder is acknowledged with appreciation for feedback on earlier versions of this chapter.

#### **References**

*and modeling perspective on mathematics problem solving, learning and teaching* (pp. 405–431). Mahwah, NJ: Erlbaum.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Chapter 9 Implications of Training in Incremental Theories of Intelligence for Undergraduate Statistics Students**

#### **Valorie Zonnefeld**

**Abstract** This chapter documents the effects of training in incremental theories of intelligence on students in introductory statistics courses at a liberal arts university in the US. Incremental theories of intelligence examine the beliefs individuals hold of knowledge and how it is attained. An individual with an incremental theory of intelligence believes that intelligence can be developed. The research examined differences by gender in mastery of statistics and attitudes toward statistics for students who received growth mind-set training. A pre-test, post-test design utilised the Students' Attitudes Toward Statistics© instrument and the Comprehensive Assessment of Outcomes in a first Statistics course. An ANCOVA revealed that females gained more than males on their value of statistics (F(1, 63) - 9.40, MSE - 3.79, *p* - .003, η2 P - 0.134) and decreased less for effort expended to learn statistics (F(1, 63) - 4.41, MSE - 4.07, *p* - .040, η2 P - 0.067). Females also gained mastery of statistical concepts at a greater rate (F(1, 63) - 5.30, MSE - 0.06, *p* - .025, η2 P - 0.080) indicating a possible path to alleviate the under-representation of females in STEM.

**Keywords** Statistics · Mind-set · Beliefs · Affect · Gender

#### **9.1 Introduction**

At the heart of research in mathematics education is a desire that all students become competent and confident in mathematics. Unfortunately, this does not describe a multitude of students in educational settings around the world (Mullis, Martin, Foy, & Arora, 2012; Topper, 2011). In particular, females' success in mathematics is often hampered by negative gender stereotypes (Good, Rattan, & Dweck, 2012). This research focuses on differences by gender in a common undergraduate mathematics course, introductory statistics.

Students' beliefs about themselves as learners of mathematics play a key role in their mastery of mathematics (Nurmi, Hannula, Maijala, & Pehkonen, 2003; Schoen-

V. Zonnefeld (B)

© The Author(s) 2019

Dordt University, 498 4th Ave. NE, Sioux Center, IA, USA e-mail<EMAIL_ADDRESS>
M. S. Hannula et al. (eds.), *Affect and Mathematics Education*, ICME-13 Monographs, https://doi.org/10.1007/978-3-030-13761-8\_9

feld, 1992). Dweck's (1999) seminal research on the role of students' implicit theories of intelligence, also referred to as mind-sets, examines students' beliefs about knowledge acquisition. Dweck (2006) identified two implicit theories of knowledge: incremental and entity. Incremental theorists, also referred to as individuals with growth mind-sets, believe that intelligence is malleable and can be increased with effort (Bruning, Schraw, & Norby, 2011). Entity theorists view intelligence as unchangeable and are referred to as having fixed mind-sets (Dweck & Leggett, 1988). Effort cannot influence intelligence in an entity mind-set since intelligence is fixed. These mind-sets play an important role in academic persistence and achievement, both of which are important for college success (Good et al., 2012). Research has shown that mind-sets can be altered and that targets of stereotype threat decrease achievement gaps with mind-set training (Aronson, Fried, & Good, 2002).

This research is a continuation of a previous study on the role of mind-set training in introductory statistics college courses which noted that students who received mind-set training decreased in effort expended to learn statistics at a statistically significant greater rate while demonstrating less mastery of statistical content (Zonnefeld, 2015). One limitation of the previous research was a change in instructors and number of course credits and requirements between the control group and treatment group.

This follow-up research expands Dweck's important body of research by focusing on the differences within each treatment group by gender. All students completed both a pre-test and post-test which examined attitudes towards statistics and mastery of statistics. Students in the treatment group received training in incremental theories of intelligence throughout the semester during which they were enrolled in introductory statistics. Effects of incremental mind-set training were examined regarding student attitudes toward statistics and student mastery of statistical content by gender in both the control and treatment groups. Two research questions guide this research:

Research on implicit theories of knowledge has grown rapidly in recent decades. Unfortunately, only a handful of studies have examined creating a growth mind-set in the mathematics classroom (Kim & Kellert, 2010). This research answers the call from Dweck (2008) to "study ways in which the education environment can teach and support a growth mind-set over time" (p. 2). Shores and Smith (2010) have also called for future research, specifically in mathematics education, to focus on developing strategies to improve students' implicit theories of knowledge with the goal of helping students develop methods to cope with failure. This research heeds the call by examining the role of mind-set training by gender in introductory statistics courses.

According to the Organisation for Economic Co-operation and Development, college graduation rates vary from 22 to 60% worldwide (Weston, 2014). Pang (2010) suggested that improving students' experiences in mathematics courses will increase the perseverance of many aspiring college graduates. Statistics is a unique mathematics course that is required for many undergraduate students and the number of students who take statistics continues to grow (Onwuegbuzie & Wilson, 2003). Unfortunately, many of the students enrolled in statistics would not choose to take the course if it were not a program requirement (Cook, 2010). Subsequently, statistics becomes a 'problem subject' for many students (Onwuegbuzie & Wilson, 2003). An examination of possible methods to support students is valuable and timely.

Recent decades "have seen an extraordinary level of activity focused on how students learn statistics" (Aliaga et al., 2005, p. 8). Unfortunately, given the growth of research in both implicit theories of intelligence and statistics education, a paucity of research exists connecting mind-sets, attitudes, and achievement in statistics courses. Results of this study add to the knowledge base of implicit theories of knowledge and support continuous improvement in undergraduate statistics pedagogy.

One limitation to the generalisability of this study is the statistics curriculum implemented by the college. All course sections used a reform-based statistics curriculum which was under development. The curriculum utilised simulation-based methods which introduced inference earlier in the semester (Tintle et al., 2016). This curriculum was in its final stages of development and was not expected to be a factor in the study.

The quasi-experimental design of this study was a limitation since it only allowed for conclusions of association, not causation. A delimitation of the sample was that it was comprised of students from one college who enrolled in a specific statistics course. This also limits the generalisability of the findings.

An additional limitation of this research was the role of the environment. There is evidence that the educator affects the learning, attitude, and even mind-set of students in their class (Cutts, 2008; Dweck, 2008). The professors for the treatment classes were debriefed on the study, theories of intelligence, and the role of the teacher in fostering incremental mind-sets in an effort to create a supportive classroom environment. Unfortunately, different instructors were used for the control and treatment groups. Additionally, the number of credits increased from 3-credits to 4-credits from the control to the treatment semester.

The remainder of this paper is organised as follows. Section two reviews relevant research followed by section three which outlines the methodology utilised. Section four shares the results of the study and statistical analysis followed by conclusions in section five.

#### **9.2 Relevant Literature**

A review of relevant literature begins with current research on undergraduate statistics. This is followed by theoretical foundations, cognitive learning theory, and closes with gender stereotypes.

#### *9.2.1 Undergraduate Statistics Courses*

A common roadblock on many students' path to graduation is successfully completing a college-level mathematics course (Topper, 2011); this is a momentum point that is "very strongly correlated with postsecondary achievement" (Leinbach & Jenkins, 2008, p. 20). Introductory mathematics courses have earned the title 'gatekeeper to achieving a college degree' with Bryk and Treisman (2010) arguing that "math should be a **gateway**, not a gatekeeper, to a successful college education" (p. 1).

A commonly required mathematics course in many undergraduate programs is introductory statistics (Ruggeri, 2011). Statistics is a unique mathematics course that combines quantitative skills with analysis. As a result of the need for analytical skills in a diverse set of professions, the number of students required to take statistics has rapidly increased in the past half century making statistics the most common subject across disciplines (Ruggeri, 2011). Statistics has morphed from a course taught to "a narrow group of future scientists in agriculture and biology, to a family of courses, taught to students at many levels, from pre-high school to post-baccalaureate, with very diverse interests and goals" (Aliaga et al., 2005, p. 7).

As a mathematics course, statistics creates an obstacle for undergraduates, as many students experience anxiety and negative attitudes toward the subject (Evans, 2007; Ruggeri, 2011; Ruggeri, Dempster, Hanna, 2008a; Ruggeri et al., 2008b). As student diversity has increased in introductory statistics courses, so have the demands. "Today's teachers face challenges of motivation and exposition that are substantially greater than those of half a century ago" (Aliaga et al., 2005, p. 7). With the increase in diversity of undergraduate majors, students may not see the relevance of statistics in their lives, contributing to a struggle for motivation (Kim & Kellert, 2010).

One response to poor performance in introductory mathematics courses and low undergraduate retention rates is increasing the academic entrance requirements for students; however, ability is not the sole predictor of success (Lucio, Rapp-Paglicci, & Rowe, 2011). Educators recognise that students of similar abilities often achieve at vastly different levels (Boese, Stewart, Perry, & Hamm, 2013).

Cognitive learning theory may give insight into the disparity seen between students' achievements and their ability levels. While promising research has been conducted on mathematical mind-sets concerning students' belief in their ability to learn mathematics, no research is available on the impact of training in incremental theories of intelligence on their achievement and attitudes in introductory statistics courses. The next section will outline theoretical foundations for the research.

#### *9.2.2 Theoretical Foundations*

While there is not a clear consensus regarding the theoretical frameworks to conceptualise affect in mathematics education (Hannula, 2004), I chose to use DeBellis and Goldin's (1997) framework of four concepts in affect research: emotions, attitudes, beliefs, and values as it encompassed the aspects that best fit my research questions and the instruments chosen.

Beliefs are an important concept in affect. An examination of a student's epistemological beliefs regarding mathematical knowledge and its acquisition can reveal important keys to student affect and achievement since beliefs "act as a filter" influencing all thoughts and actions (Nurmi et al., 2003, p. 454). Schommer (1990) suggested that student's beliefs regarding knowledge are complex and vary on four dimensions: simple knowledge, certain knowledge, fixed ability, and quick learning. See Fig. 9.1 for descriptions of each.

Implicit theories of knowledge focuses on the third and fourth dimensions of fixed ability and quick learning. The next section discusses cognitive learning theory in detail.

#### *9.2.3 Cognitive Learning Theory*

Cognitive learning theory focuses on metacognition, the effects of underlying thought processes on the learner. Students do more than gain knowledge and master skills while learning; they simultaneously build beliefs about their ability to be successful learners (Boekaerts, Otten, & Voeten, 2003). This section starts with an examination of the role of attitudes in statistics education, followed by attribution theory and closes with mind-set theory.

#### **<IP_ADDRESS> Attitudes**

Student attitudes play an important factor in academic achievement creating a challenge for professors as anxiety and negative attitudes towards statistics are common (Chiesi & Primi, 2010; Evans, 2007; Kesici, Balo˘glu, & Deniz, 2011; McGrath, 2014; Onwuegbuzie &Wilson, 2003; Ruggeri et al., 2008a, 2008b). Chiesi and Primi (2010) examined undergraduate students enrolled in introductory statistics both before and after the semester using the Student Attitudes Towards Statistics—36© (SATS) instrument finding that attitudes at the beginning of the semester were related to achievement at the end of the course. An additional relationship indicated that students' attitudes at the beginning of the semester were directly related to their mathematical knowledge. This creates a challenge for many instructors since students enter introductory statistics with a wide variety of mathematical backgrounds. Evans (2007) found similar correlations between attitudes and achievement in statistics classes while Bandalos, Yates, and Thorndike-Christ (1995) observed a negative relationship between students' efficacy and their anxiety about statistics.

It is clear that attitudes impact students' achievement in statistics. Unfortunately, Evans (2007) found no methods in his research for instructors to help improve student attitudes. More research is necessary to understand and address the issues of statistics anxiety and negative attitudes toward statistics (Ruggeri et al., 2008a, b).

#### **<IP_ADDRESS> Attribution Theory**

Bernard Weiner contributed a key advance in metacognition through attribution theory. Attributions are an important aspect of cognitive learning theory that can have both positive and negative influences on students. Attribution theory was proposed by Weiner (1985) in the mid-eighties and since then, has been promoted by educators and educational psychologists as an effective method to improve achievement and motivation for students with academic difficulties (Banks & Woolfson, 2008; Robertson, 2000). Attributions are classified along three dimensions: locus, stability, and controllability (Banks & Woolfson, 2008; Boekaerts et al., 2003). Locus refers to the location of the cause as internal or external to the student (Shores & Smith, 2010). Stability is a belief about the possibility of change for the cause and controllability refers to the ability to affect future outcomes (Syed, 2013).

When students fall behind academically, it is important for teachers to understand what is causing their struggle (Shores & Smith, 2010). Of particular concern are failure attributions that are stable and beyond a student's control (Hall, Hladkyj, Perry, & Ruthig, 2004; Shores & Smith, 2010). Attributions occur frequently in mathematics. Boekaerts et al. (2003) found evidence in their work with 113 middle school students in the Netherlands that attribution patterns are different in mathematics than the subjects of native language and history. Students attribute success in mathematics to effort, but failure to ability (Boekaerts et al., 2003). Effort is an important causal attribution since it is closely tied to a growth mind-set. Students will use effort attributions only if they believe that their intelligence can be improved. This concurs with research from Harari and Covington (1981) who found that students who used effort attributions persisted longer in tasks following a failure.

Connections between attributions and self-efficacy have also been observed. Students who believed they were lower achievers, regardless of their teacher's perception of ability, displayed more maladaptive attributions (Banks & Woolfson, 2008). This finding points to the importance of attributions not only to low-achievers, but also for students who *perceive* themselves as low-achievers. Attribution theory is a foundation for mind-set theory and continues to play an important role in students' behaviours.

#### **<IP_ADDRESS> Mind-Set Theory**

Mind-set theory expands upon the foundation built by Weiner (1985) in attribution theory to a more comprehensive approach as it not only looks back explaining reasons for success or failure, but also looks forward to include goals (Dweck, 1999). An impetus for the introduction of mind-set theory is new research in cognitive neuroscience. Research in recent decades has revealed how the brain functions and that its ability to grow and increase in intelligence is much greater than was previously assumed (Cutts, 2008; Dweck, 2010; Good, Aronson, & Inzlicht, 2003). Knowledge about the increased neuroplasticity of the brain has powerful effects for individuals.

Mind-sets are metacognitive processes that an individual holds concerning beliefs about their cognitive abilities (Burns & Isbell, 2007; Mangels, Butterfield, Lamb, & Dweck, 2006). Anderson (1995) referred to mind-sets as knowledge structures that influence affective reactions and behaviours. It should be noted that mind-sets are domain specific (Dweck & Leggett, 1988) and affect motivation, goal setting, and persistence. In a typical population, approximately forty percent of individuals identify with the entity mind-set and forty percent relate to the incremental mindset with the remaining twenty percent of individuals not consistent enough in their responses to be classified as holding either theory (Dweck, 2008).

Entity theorists hold fixed mind-sets and see their environments and abilities as set with little that can be done to alter them (Dweck & Leggett, 1988). They believe that an examination of their current abilities predicts their future capacity (Dweck, 1999). Students with a fixed mind-set feel smart with easy, low-effort successes, outperforming other students (Dweck, 1999). The self-imposed rule to look smart prohibits entity theorists from seeking remediation when necessary. Students' theory of intelligence affects their behaviours with students holding entity mind-sets responding differently to failure (Burns & Isbell, 2007). Entity mind-sets are in contrast to the incremental mind-sets held by other students.

Incremental theorists hold a growth mind-set and believe that intelligence can be developed. They do not fear challenges or failure, but feel smart when they are fully engaged, stretching themselves, and putting their skills to use (Dweck, 1999). Incremental theorists thrive when they are growing and learning.

Evidence is mounting regarding the role that students' mind-sets play in learning mathematics. Research has shown that students with fixed mind-sets towards mathematics and science ability "are at a significant disadvantage compared to students who believe that their abilities can be developed" (Dweck, 2008, p. 1). In a study of seventh-grade students, teachers, who were blind to treatment, chose three times as many students in the growth treatment group as showing marked improvement in their motivation compared to those in the control group (Blackwell, Trzesniewski, & Dweck, 2007). One of the largest areas of difference between the entity and incremental mind-sets is in students' goal setting.

Theory of intelligence is a reliable predictor of student's goal orientation (Dweck & Leggett, 1988; Heyman & Dweck, 1998). Individuals that hold entity mind-sets tend toward performance goals, while individuals with incremental mind-sets favour mastery goals. Performance goals focus on demonstrating an individual's competence. Mastery goals focus on mastering a skill. An individual with this mastery goal will not see failing as a failure, but as a learning experience.

Goal orientation also affects individuals' views of effort. Dweck and Leggett (1988) found evidence that students with performance goals viewed effort and ability as inversely related. It is this irony that can be paralysing to students with performance goals. When they most need to exert extra effort, they draw back in a defensive response fearing that investing effort reveals their ability as deficient. In contrast to this, mastery goals provide an inoculation to failure in many students since failure is not alarming, but rather, an opportunity to learn (Dweck & Leggett, 1988). Mastery goals are associated with students who seek out challenging tasks, increase motivation, and strive under failure (Grant & Dweck, 2003).

Fortunately, there is evidence that mind-sets are dynamic and that individuals' tendencies change as they grow and develop with training in an incremental mindsets showing good potential in many studies (Anderson, 1995; Donohoe, Topping, & Hannah, 2012; Kim & Kellert, 2010). Training has also been linked to increased resiliency and academic performance in students (Donohoe et al., 2012).

While a majority of mind-set research discusses implicit theories of intelligence in a favourable light, there are also criticisms. Donohoe et al. (2012) questioned if it is possible to make a large difference in mind-set with only brief interventions. They, along with Blackwell et al. (2007), raised concerns about the longevity of the positive effects of implicit theory interventions. Some conflicting evidence has also arisen regarding the connection between mind-set and academic achievement (Blackwell et al., 2007). Furnham, Chamorro-Premuzic, and McDougall (2002) found evidence in a British study of college students that students' mind-sets were not related to achievement, but to their personality. The closing section of the literature review will examine research regarding gender stereotypes in mathematics.

#### *9.2.4 Gender Stereotypes*

The mathematics community has historically utilised a talent-driven approach to mathematics, emphasising that mathematics ability is innate (Clinkenbeard, 2015; Good et al., 2012; Leslie, Cimpian, Meyer, & Freeland, 2015). Alongside this emphasis is a persistent stereotype that males are more capable of mathematical thinking and quantitative reasoning than females (Dweck, 2008; Good et al., 2012; Steel & Aronson, 1995). At their heart, stereotypes represent entity mind-set beliefs. They are erroneous knowledge structures that apply fixed abilities to certain groups of people (Anderson, 1995; Dweck, 2008). Aronson et al. (2002) found that stereotype targets behave similar to entity theorists when their gender is made salient by choosing easier, success-assuring tasks.

The combination in mathematics of a talent-driven approach and gender stereotypes creates ideal conditions for stereotype threat. Stereotype threat is a burden that an individual feels to confirm cultural stereotypes which limit his or her academic abilities and achievement (Aronson et al., 2002; Grant & Dweck, 2003) and may provide important insight into why females have been historically under-represented in mathematics fields (Dweck, 2008). "A key factor driving students' intent to pursue math should be their personal sense that they belong in mathematics" (Good et al., 2012, p. 700). Women who identify with groups that have negative mathematical stereotypes may face obstacles. Aronson and colleagues (2002) claimed that a perceived stereotype could be strong enough to sway a student's implicit beliefs of intelligence. Students must first "be convinced that anyone can learn mathematics" or they have already crippled themselves by identifying with a 'have-not' group" (Clinkenbeard, 2015, p. 28).

Fortunately, there is evidence that training in incremental mind-sets can help females counter the negative effects of stereotypes. Good et al. (2003) conducted a study in which college students mentored seventh-grade students who were at-risk of stereotype threat. Students were divided into four groups. The first group received mentoring that encouraged a growth mind-set, the second group received mentoring concerning the transition to middle school, and the third group received both the growth mind-set and transition mentoring. The final group received an anti-drug message and served as a control for the experiment. At the end of the school year, females trained in the incremental mind-set received significantly higher scores on a standardised maths test (Good et al., 2003). Additionally, females showed greater gains than males and decreased the achievement gap. This finding supports Dweck's (2008) conclusion that females with growth mind-sets are less at risk of negative effects from stereotypes.

This review has examined literature regarding undergraduate statistics, theoretical foundations supporting the research, cognitive learning theory and the impact of gender stereotypes on female's mathematics achievement. The next section will examine the methods used to conduct the research.

#### **9.3 Method**

#### *9.3.1 Population and Sample*

The population for this study was undergraduate students at a small, faith-based, liberal arts university located in the United States. This study made use of a pretest/post-test design. The research was quasi-experimental since a treatment was applied, but the sample was not randomly selected. The sample consisted of students who enrolled in introductory statistics in the semesters under examination and completed all aspects of the research.

The treatment group consisted of 121 students enrolled in introductory statistics courses during the experiment with a gender breakdown of 47.8% female and 52.2% male. The response rate was 52.9% resulting in 64 students who completed pre-test and post-test assessments in the four introductory statistics classes between August 2014 and May 2015.

The control group consisted of 234 students who enrolled in introductory statistics during August 2013 to May 2014. The response rate was 47.4% resulting in 111 students who completed pre-test and post-test assessments. The gender distribution in the control group was similar to the treatment group with 52.3% female and 47.7% male.

#### *9.3.2 Instrumentation*

Two instruments were used to assess each student's attitude toward statistics and mastery of statistical knowledge: The Student Attitudes Towards Statistics—36© (SATS) and the Comprehensive Assessment of Outcomes in a first Statistics course (CAOS). Both instruments were implemented at the beginning and the end of the course. Since both instruments were pre-existing with established reliability and validity, a pilot study was not conducted.

The SATS© (Schau, 2003) instrument was selected to assess each student's attitude since it has been widely used in statistics education research (Bond, Perkins, & Ramirez, 2012; Chiesi & Primi, 2010; Coetzee & van der Merwe, 2010; Harpe, Phipps, & Alowayesh, 2012; Swanson, Vander Stoep, & Tintle, 2014). The 36-item online SATS© tool provided a valid representation of students' attitudes regarding statistics. Each item required a response on a seven-point Likert scale. Responses were grouped into six components of attitude: affect, cognitive competence, value, difficulty, interest, and effort. Internal consistency for all components of the SATS© (.66 < α < .85) was established (delMas, Garfield, Ooms, & Chance, 2007). See Fig. 9.2 for sample questions.

The CAOS (delMas et al., 2007) measures mastery of statistical concepts and has been implemented widely in statistical education research (Hannigan, Gill, & Leavy, 2013; Slauson, 2008; Tintle, Vander Stoep, Holmes, Quisenberry, & Swanson, 2011;

**Fig. 9.2** Examples of SATS*©* component questions. The asterisk (\*) represents a reversed scored item

Tintle, Rogers, Chance, Cobb, Rossman, Roy, ...&Vander Stoep 2014; Zieffler, 2007). The 40-item instrument was administered online to assess students' mastery of statistics at both the beginning and at the end of each semester. Internal consistency for all components of the CAOS (α < .77) was established by Nolan, Beran, and Hecker (2012).

#### *9.3.3 Growth Mind-Set Treatments*

The incremental mind-set treatment was given to all students in the four introductory statistics classes between August 2014 and May 2015 which comprised the treatment group. The training was designed using materials and findings from successful interventions in the review of literature. The treatment consisted of four incremental mind-set-training sessions throughout each semester. The goal of the sessions was to help students understand how the brain functions biologically with a focus on the malleability of intelligence. In this study, each of the four 15-minute training sessions was implemented approximately three weeks apart. Each session occurred during class time and was led by the researcher to ensure consistency. Verbal permission to conduct the trainings was granted by the professors responsible for teaching each of the introductory statistics courses.

A number of approaches have successfully been employed to alter students' mental constructs. Both attribution training and mind-set training have employed the use of videos (Boese et al., 2013), reading activities (Kim & Kellert, 2010), writing activities (Aronson et al., 2002; Hall et al., 2004), and computer-assisted instruction (Donohoe et al., 2012). In his study of attribution training, Robertson (2000, p. 118) found that no single medium was more effective than others. Given this finding, multiple approaches were implemented in an effort to differentiate the treatment delivery and connect with a variety of students in different methods.

The design of the sessions was adapted from the research of Berkeley, Mstropieri, and Scruggs (2011), Boese and colleagues (2013), Burk (2011), Cutts (2008), Hall

**Fig. 9.3** Treatment outline. An outline of each incremental mind-set training session

and colleagues (2004), and Sriram (2010) implementing combinations of a brief video, article, presentation, or activity which introduced concepts in incremental theory including mind-sets, the physiology of the brain, response to feedback, goals, and the role of effort. Following the success that Robertson (2000) found when combining strategy instruction with reattribution training, students were also given guidance in the sessions on successful techniques for mastering statistics. Dweck supported this approach with the rationale that telling a student to try harder is not enough without providing strategies to do so (Sparks, 2013). Pinxten et al. (2014) concur stating that "domain-specific interventions are more effective" than general interventions (p. 170). Students in the treatment were given guidance throughout each semester on how to respond to challenges and negative feedback in the course. They were also directed to resources to help improve their mastery of statistics including online material and tutoring that was available on campus, free of charge.

A consolidation activity closed each treatment to help students personalise the new information they received (Hall et al., 2004). The consolidation took the form of a small group discussion or an individual writing task. See Fig. 9.3 for the focus of each of the sessions.

#### *9.3.4 Data Collection*

The SATS© and CAOS instruments were administered online, outside of class during the first week of class and again during the last week of class for the semester. The CAOS assessments had been regular requirements of introductory statistics classes for the institution since the fall of 2011 in conjunction with a grant for the National Science Foundation. Similarly, the SATS© assessments had been regularly assessed since fall of 2013 in conjunction with the same grant.

Students who completed, at the minimum, the initial opt-out screen of all preassessments received credit for a daily assignment. Similarly, students who completed, at the minimum, the initial opt-out screen of all post-assessments received credit for a daily assignment.

The procedure for conducting the assessments ensured that students received the information and had adequate time to complete them. Students were introduced to the assessments during the first class period by the class instructor. After class on the first day, all students received an e-mail invitation with links to the assessments and a brief explanation of the purpose and procedures of the study. Each assessment was run through Survey Monkey™ and remained open for one week. A reminder e-mail was sent one day before the assessments closed. A similar process occurred with the post-test for each instrument.

To ensure that all individuals responsible for or affected by this research were informed and protected, appropriate permissions and reviews were completed prior to data collection including permission from the mathematics department and institutional research board of the institution. Additional permission was granted to use CAOS and SATS© data for the control group from an ongoing National Science Foundation study from the principal investigator of the project. Students were assured of the confidentiality of their responses in the invitation. Informed consent was given through an initial opt-out screen for each assessment. This ensured students that they had the opportunity to not participate if they were uncomfortable with an assessment.

#### *9.3.5 Data Analysis*

Robust data analysis procedures were used throughout this research. Prior to any analysis, thorough data screening was employed to ensure that no outliers distorted the data. The threshold to determine and remove univariate outliers was observations beyond five standard deviations of the sample mean. Observations that violated three or more diagnostic measures were removed as bivariate outliers.

Appropriate statistical analyses were employed to address both research questions. An alpha level of .05 was used for all tests. A measure of effect size was calculated for all analyses as it provides an index of the magnitude of the differences between means which is independent of sample size (Warner, 2013). Using Cohen's (1988) index guidelines, large effect sizes represented measures of .4 and greater, medium effects between .25 and .4 and small effect sizes between .1 and .25. The analysis for each research question follows. A post hoc power analysis indicated adequate power to detect large effects with ANCOVA for the treatment group (*N*Male - 32, *N*Female - 32, *d* - .40, 1 − β - .88) and medium to large effects for the control group (*d* - .3, 1 − β - .88, *N*Female - 58, *N*Male -53); (Faul, Erdfelder, Buchner, & Lang, 2009).

What differences exist by gender in the change in students' attitudes toward statistics for students who received training in an incremental mind-set and those who did not when controlling for initial attitudes?

The dependent variables for this research question were the post-test components of the SATS© score with covariates of the pre-test components of the SATS© score. The independent variable was the student's gender: male or female. ANCOVA was used to assess differences in post-test attitude scores while controlling for pre-test SATS© score by gender.

What differences exist by gender in mastery of statistics for students who received training in an incremental mind-set and for those who did not when controlling for initial mastery?

The dependent variable for this research question was the post-test CAOS score with the covariate pre-test CAOS score. The independent variable was the student's gender: male or female. An ANCOVA was used to assess differences in post-test mastery scores while controlling for pre-test mastery score by gender. The next section will share the results of this research.

#### **9.4 Results**

Section four examines the results of an intervention of incremental mind-set training in a college introductory statistics course. Areas examined included student attitudes toward statistics and student mastery of statistical content. The section opens with an examination of the response rate and sample demographics. The section then explores particular findings for each research question.

#### *9.4.1 Response Rate and Demographics*

The sample included all students who completed pre-test and post-test assessments in the introductory statistics classes between August 2013 and May 2015. The gender breakdown in the treatment group was evenly distributed with 32 females and 32 males. Adequate power was achieved to detect large effects (*d* - .4, 1 − β - .88, *N*Female - 32, *N*Male - 32) (Faul et al., 2009). The control group consisted of students enrolled in the introductory statistics course prior to the treatment between August 2013 and May 2014. The gender breakdown in this group was evenly distributed with 58 females and 53 males. Adequate power was achieved to detect medium to large effects (*d* - .3, 1 − β - .88, *N*Female - 58, *N*Male -53) (Faul et al., 2009).

**Table 9.1** Post-test SATS© scores by gender controlling for pre-test SATS© in treatment group

Numbers in parenthesis represent the change from pre to post test

\*Denotes significant difference at .05

aDenotes small or medium effect size

#### *9.4.2 Attitude Toward Statistics*

Research question one states: What differences exist by gender in the change in students' attitudes toward statistics for students who received training in an incremental mind-set and those who did not when controlling for initial attitudes? Table 9.1 depicts the results for the treatment group on each component using a seven-point Likert scale with higher scores indicating more favourable attitudes. An encouraging finding for females demonstrated greater growth in comparison to males regarding their attitudes towards statistics. Statistically significant results were present for the effort and value components.

Table 9.2 depicts the results for the control group on the change in attitudes for the students who did not receive growth mind-set treatments. No statistically significant differences between attitudes of males and females were present with the exception of the effort component.

The effort component measured students' perceptions regarding the amount of work they expend to learn statistics (Schau, 2003). While motivation is difficult to measure and can include unconscious aspects, effort is a description of behaviour which is "a dependable manifestation of motivation" (Hannula, 2006, p. 167). It is typical on the SATS© to observe a decrease in the effort component from the pre-test to post-test as students refer to intended effort on the pre-test prior to the semester and the post-test is a closer measure of their actual effort. The effort component decreased for both males and females during the treatment semesters, but males decreased at a statistically significantly greater rate than females (*F*(1, 63) - 4.41, *MSE* - 4.07, *p* - .040). Students in the control group that did not receive growth mind-set treatments showed similar, although weaker, findings by gender as females

**Table 9.2** Post-test SATS© scores by gender controlling for pre-test SATS© in control group

Numbers in parenthesis represent the change from pre to post test

decreased in the effort component at a statistically significant lesser rate than males (*F*(1, 110) - 3.98, *MSE* - 2.64, *p* -.049).

Value increased for both genders in the treatment group with females gaining at a statistically greater rate than males (*F*(1, 63) - 9.40, *MSE* - 3.78, *p* - .003). The value component measured students' views regarding the usefulness, relevance, and worth of statistics in their personal and professional life (Schau, 2003). Students in the control group did not demonstrate statistically significant differences by gender for the value component (*F*(1, 110) - 3.15, *MSE* - 1.40, *p* - .079). The next section will examine students' mastery of statistics.

#### *9.4.3 Mastery of Statistics*

Research question two states:What differences exist by gender in mastery of statistics for students who received training in an incremental mind-set and for those who did not when controlling for initial mastery?

Table 9.3 depicts the results including an encouraging finding that females in the treatment group demonstrated statistically significant growth in their mastery of statistics compared to males. No significant difference by gender was present for mastery in the control group. This provides strong evidence of a difference by gender in response to incremental mind-set training.

**Table 9.3** Post-test CAOS scores by gender controlling for pretest CAOS scores

Numbers in parenthesis represent the change from pre to post test

\*Denotes significant difference at .05

aDenotes small or medium effect size

#### **9.5 Summary, Conclusions, Discussions, and Recommendations**

The final section of this paper will summarise, make conclusions, discuss findings, and provide recommendations for future practice and research.

#### *9.5.1 Summary*

This research examined the effects by gender of an intervention of incremental mindset training on students' attitudes toward statistics and mastery of statistical content in an introductory statistics course.

A quasi-experimental, pre-test/post-test design was used in this quantitative research and robust statistical analysis procedures were used throughout. The population was undergraduate students at a small, faith-based, liberal arts university in the Midwest. The students were predominantly Caucasian and of traditional college age. While the sample was not randomly selected, it was representative of students who typically enrol in this course.

The results of this study add to the research of implicit theories of knowledge in undergraduate statistics courses and support continuous improvement in undergraduate statistics pedagogy. A noteworthy contribution of this research is the positive results for females with mind-set training regarding attitudes toward statistics and mastery of statistics.

#### *9.5.2 Conclusions*

Based on this sample the following conclusions emerged regarding the benefit to females of incremental mind-set treatments in this introductory statistics course.

Females in the mind-set treatment condition increased their value of statistics at a significantly higher rate than males did on the SATS© component of value. This is a significant finding given the historical underrepresentation of women in mathematics and STEM in general (Dweck, 2008; Good et al., 2003). The increase in female's value compared to males supports the notion that environments that communicate an incremental view of mathematics will improve the achievement of females in mathematics and science (Good et al., 2012).

Females in the mind-set treatment decreased in their effort towards statistics at a statistically significant lesser amount than males on the effort component of the SATS©. The effort component gives insight into motivation as it implies that an individual is "*moved* to do something" (Ryan & Deci, 2000, p. 54). This smaller decrease in effort for females is an indicator that females believed that they could "produce desired effects by their actions" allowing them to persevere when they faced challenges (Bandura, 1999, p. 28).

Females in the mind-set treatment demonstrated greater growth in their mastery of statistics compared to males as measured on the CAOS instrument. This concurs with Pinxten and colleagues' (2014) findings regarding the strong correlation between maths achievement and maths competence beliefs as growth mind-set training, at its core, addressed students' beliefs about their ability to learn mathematics.

#### *9.5.3 Discussion*

The purpose of the study was to determine the effects by gender of training of incremental mind-sets on students' attitudes towards statistics and their mastery of statistics. The first section examines effort, followed by value and mastery.

#### **<IP_ADDRESS> Effort**

Effort is an important behaviour to examine as time spent studying is the secondbest predictor of college readiness, following only socioeconomic status (Strayhorn, 2014). The statistically significant lesser decrease in effort reported by females than males in the treatment group is an important finding as "academic effort expenditure has been frequently investigated in relation to students' goal orientations and causal attributions of success and failure" because of its internal, controllable nature (Pinxten, Marsh, De Fraine, Van Den Noortgate, & Van Damme, 2013, p. 156). This provides evidence that training in a growth mind-set which includes goal setting and discussions of maladaptive attributions had a positive effect on females and the effort that they applied.

There is evidence that the valuation of effort and ability change with development as well. Harari and Covington (1981) found that early elementary students emphasised the role of effort in intelligence. This valuation gradually shifted throughout the educational experience to an emphasis on the role of ability in college students. As a result, many undergraduate students equate effort with lack of ability (Boekaerts, 1999). This is unfortunate since effort is required for students to work through struggle and is a normal part of learning mathematics (Su, 2016).

#### **<IP_ADDRESS> Value**

Similar to effort, value of mathematics also declines as students enter middle school and high school (Turner & Meyer, 2009). This research notes an encouraging finding, demonstrating a statistically significant improvement for females in the treatment group compared to males on the SATS© value component. This is important to motivation in light of expectancy-value theory which states that motivation is a product of both the expectation that an action will produce an outcome and the value placed on that outcome (Bandura, 1999). Additionally, if incremental mind-set training helps females value mathematics, it may also improve their performance to help diminish a persistent stereotype in the mathematics community that males are more capable of mathematical thinking and quantitative reasoning than females (Dweck, 2008; Good et al., 2003, 2012).

#### **<IP_ADDRESS> Mastery**

The statistically significant improvement in mastery of statistics, demonstrated by females in the treatment group when compared to males, is evidence of the importance of implicit theories of knowledge on student attitudes and behaviours. It provides evidence to support Urbina-Lilback's (2016) assertion that students should adopt a growth mind-set to gain an appreciation of effort and productive struggle in learning. The following section provides recommendations for future practice and research based on these encouraging findings.

#### *9.5.4 Recommendations*

Recommendations for practice include creating classroom environments that communicate an incremental view of mathematics. This is an important step toward increasing the representation and achievement of females in mathematics and science (Good et al., 2012). An understanding of the malleable role of intelligence should permeate all aspects of the classroom including discourse, assessment, and pedagogy (Urbina-Lilback, 2016). Good et al. (2012) conjectured that the disengagement of females with mathematics may result not from a disinclination, but from a decreased sense of belonging. In a large study of undergraduate students in calculus, they found that the perception of an entity-oriented environment and stereotyping were significant predictors of a sense of belonging for females. Fortunately, an incremental mind-set can help stereotyped individuals counteract the effects of stereotyping (Good et al., 2003).

Recommendations for future research include replication, as it is necessary to learn more about the effects of growth mind-set training on achievement and attitudes for both males and females. Research should replicate the current quantitative study to increase the sample size, but also add an aspect of qualitative research to understand why males and females respond in different ways. Additionally, more research on the role of mind-sets and incremental mind-set training should be conducted in various mathematics settings including elementary, secondary, and undergraduate classrooms to improve the generalisability of results.

#### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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## **Chapter 10 A Commentary: Extending the Research Discourse from Interests and Beliefs to Values**

**Wee Tiong Seah**

**Abstract** This commentary unpacks and interprets the various attributes that were examined in the chapters included in this section, that is, interest, motivation, personal meanings, and values. It looks at how they extend and deepen our understanding of the roles played by values and valuing in mathematics education research. It emphasises the opportunity for a re-focus on the purpose of learning mathematics to one of cultivating values and practising valuing, beyond acquisition of mathematical content knowledge and skills.

#### **10.1 Introduction**

This section features a collection of writings which investigate and deepen our understanding of mathematics learning and teaching from the perspectives of interest (Achmetli & Schukajlow), motivation (Bofah & Hannula; Middleton, Mangu, & Lee; Wilkie), personal meanings (Vollstedt & Duchhardt), values (Dobie), and multiple affective aspects including interest, motivation and value (Zonnefeld). Amongst these variables, values are often perceived to be the most fundamental, most internalised, and most stable (e.g. Bishop, 1988; Roccas, Sagiv, & Navon, 2017; Stern, 2000), thus guiding the nature, form and activities of one's motivation and attitudes. In this context, this commentary will adopt the values perspective to interpret and summarise the content, with the aim of exploring how the values approach to mathematics education (research) might be enriched further.

W. T. Seah (B)

Melbourne Graduate School of Education, The University of Melbourne, Melbourne, Australia e-mail<EMAIL_ADDRESS>
#### *10.1.1 Not Just Values, But Also, Values in Mathematics Education*

The 'values' construct is not new in the academic world (e.g. Kluckhohn, 1962), and this commentary will begin with a brief outline of its development. One line of research attention highlights the values that are inherent within the discipline, such as by UNESCO (1979) when it made reference to "the principal criteria which constitute the efficiency of this tool of investigation and communication [that is, mathematics]: intelligibility, brevity, accuracy, relevance, normality" (p. 10). Another direction of research tradition has been concerned with the purposeful and planned teaching of moral and civics values to students (see, for example, Thornberg & O˘guz, 2013). Yet another research direction has been focussing on the emergence and/or portrayal of some of these values in the process of teaching and learning school subjects such as history, languages, and science (e.g. Allchin, 1999; Hill, 1991).

The construct of values/valuing as it applies to mathematics teaching and learning explicitly has been a relatively recent one, however. It was only in 1996 when Alan Bishop made public his conceptualisation of the categories of values that are espoused in the Western school mathematics classroom, that is, the classroom as a site of intersection amongst mathematical values, mathematics educational values, and general educational values. That took place in a regional mathematics education research conference in Vietnam, however, and thus most members of the Western research communities were introduced to these categories even more recently. In Bishop (2008), he wrote that his "research approach to values and Mathematical thinking has to date been to focus on mathematical values, and on the actions and choices concerning them" (p. 84).

This development in mathematics education above was a follow up to Bishop's (1988) earlier assertion that Western mathematics as a scientific discipline is an expression of a set of three complementary pairs of mathematical values, namely, *rationalism* and *objectism*, *control* and *progress*, and *mystery* and *openness*. These mathematical values would later make up one of the three categories of values in the (Western) mathematics classroom mentioned above. While mathematical values express the nature of (Western) mathematics as a scientific discipline, the category of mathematics educational values would reflect the attributes of mathematics teaching and learning that are deemed important, and are thus expected to be contextualised differently in different cultures. The third category is made up of what Bishop (1996) called general educational values, which refers to the societal and cultural values that are represented and taught through the school education process.

Bishop (1999) sees "values in mathematics education … [as] the deep affective qualities which education fosters through the school subject of mathematics" (p. 2). DeBellis and Goldin (2006) extended this definition, when they wrote that "values, including ethics and morals, refer to the deep, 'personal truths' or commitments cherished by individuals" (p. 135). This reflects Halstead and Taylor's (2000) definition of values in the wider education scene, in which they regarded values as being the "principles and fundamental convictions which act as general guides to behaviour, enduring beliefs about what is worthwhile, ideals for which one strives, standards by which particular beliefs and actions are judged to be good or desirable" (p. 3), thereby differentiating values (that which is important) from beliefs (that which is true). Hill (1991) had proposed this similarly in the field of education in general, when he defined values as being "held by individuals to which they attach special priority or worth, and by which they tend to order their lives" (p. 4).

A consortium of some 21 research teams located in 18 different economies around the world has been collaborating on values-related mathematics education research since 2009. Called 'The Third Wave Project', it has been finetuning its operating definition of values. In the current view,

valuing refers to an individual's embrace of convictions which are considered to be of importance and worth. It provides the individual with the will and grit to maintain any 'I want to' mindset in the learning and teaching of mathematics. In the process, this conative variable shapes the manner in which the individual's reasoning, emotions and actions relating to mathematics pedagogy develop and establish. (Seah, 2018, p. 616)

These perspectives to values in mathematics education share a similar concern and objective, that is, examining what and how particular values would facilitate more effective mathematics learning and/or teaching. This is significant, given the many students around the world who are disengaged with school mathematics when they are as young as 12 years old, and many also do not choose to study the subject at the senior high school levels and beyond. The argument has been that when students, teachers, and other stakeholders value some particular enabling attributes of mathematics pedagogy, such as *open*-*endedness* and *application*, the subscription to these conative variables would provide the individual with the motivation, drive, and perseverance to see a purpose of studying mathematics, and to engage with the subject in school.

In this context, all the seven chapters in this section can be considered to be researching values/valuing to different degrees. In particular, both Dobie andWilkie's respective studies made use of the values variable in a more direct manner. Dobie's research question was: "What is the relationship between students' personal values and their ideas about the usefulness of mathematics?". The statements in her survey instrument begin with 'It is important to me that …', which is in line with the definition of values as being the attributes that are regarded as important. On the other hand, although Wilkie asked her participants to write down what wishes they had for mathematics learning, she had also explained that these were the things that mattered—and thus, were of importance and valued—to the students. Participantgenerated phrases such as 'being better at fractions' and 'wanting to make more effort' (see Table 8.1 in Chap. 8) could thus be interpreted as expressing convictions such as *fractions* and *effort* respectively.

The chapter written by Vollstedt and Duchhardt is in the same vein. They had used a questionnaire to survey 193 representative 15- to 16-year old students in Germany for their personal meanings towards mathematics. Although personal meanings may not necessarily represent personal valuing, some of the items in the questionnaire were asking the participants to indicate the extent to which they found particular attributes important—and thus, of value—as shown in Table 7.1 in Chap. 7. These would include stimulus statements such as "It is important to me that my family can be proud of my mathematics performance" and "It is important to me that at the beginning of mathematics class, the teacher summarizes the previous lesson".

Although Vollstedt and Duchhardt's chapter does not define values, it acknowledges values as a separate construct, while proposing that "a considerable amount of overlap can be assumed between the concepts and no sharp line of division can be drawn". The concepts referred to include personal meanings, and its conceptual overlap with the notion of values is understandable, given that personal meaning has been taken to denote relevance and personal significance, which thus relate to values' association with worth and importance. Vollstedt and Duchhardt also cited other researchers who regarded "meaning as a generating force for the development of values".

The research work of Bishop (1988), DeBellis and Goldin (1997), and Seah (2005) in the areas of values and valuing in mathematics education has been on identifying attributes of mathematics and its pedagogy which are regarded as being important by students or teachers, examining how these attributes promote a sense of engagement, will, and grit amongst learners, and investigating how teachers might subsequently promote the valuing of these enabling attributes to build engagement, will, and grit amongst the students. Yet, it is interesting that none of these researchers had been referred to in all the chapters here. Even Zonnefeld's reference to DeBellis and Goldin (1997) was related to their framework of affective concepts in general, rather than just on the characteristics of values themselves. In fact, the reference against which she had conceptualised values appears to reflect a view of values being a form of interest: "the value component measured students' views regarding the usefulness, relevance, and worth of statistics in their personal and professional life". Also, Dobie's research with personal values drew on Hill's (1991) definition of values in the general sense.

Instead, the expectancy-value theory which had been introduced by Eccles (1993) into educational research was used by Bofah and Hannula, by Achmetli and Schukajlow, as well as by Dobie. While this theory applies to education generally, its power here has been in facilitating a connection between the constructs of values and motivation, as can be seen in these chapters.

#### *10.1.2 Valuing Mathematics and Valuing Attributes of Mathematics*

There are two aspects to values and valuing in mathematics education. One is concerned with valuing mathematics, that is, an individual regarding the discipline as important. The other aspect operates independently of the first one, and is concerned with an individual having convictions with particular attributes of the discipline and/or its pedagogy.

Two studies in this section exemplify the first aspect. In one, Zonnefeld's teaching of growth mindset to undergraduates over a period of 12 months resulted in the female students valuing statistics much more than their male peers and also performing better than their male peers. This suggests a very significant relationship amongst the variables of values, gender, and performance. That this is in favour of female students further validates the importance of working with the notion of values, in terms of how this approach to mathematics education may empower female students' self-efficacy or confidence in being successful in understanding and performing in mathematics at school. In the other study, Bofah and Hannula's analysis of the 2011 Trends in International Mathematics and Science Study [TIMSS] data (Thomson, Hillman, & Wernert, 2012) for five African countries included an assessment of students' valuing of mathematics. Importantly, the study found that amongst the various factors analysed, "the association between perceived social support and achievement is accounted for entirely or partly by how the students value and like mathematics". It needs to be mentioned, though, that Bofah and Hannula's notion of mathematics valuing was as a motivational belief in the tradition of the expectancy-value theory, and thus it has some utility flavour.

Individuals' valuing of attributes of mathematics and/or its pedagogy is also evident in this section. In Middleton, Mangu, and Lee's analysis of the responses of some 24,000 United States students when they were in the 9th grade and then when they were in the 11th grade two years later, one of the motivation scales was 'effort'. Here, the extent to which a student emphasises the role of effort in studying mathematics may be regarded as an expression of his/her mathematics educational value, that is, a value that is concerned with an aspect of the process of learning mathematics (Bishop, 1996). In this study, "effort, interestingly decreased across all [four] categories of students, and was especially low for" the group of students who did not intend to pursue a STEM career, and for the group of students who had the intention in the 9th grade but no longer so two year later.

#### *10.1.3 The Relevance of Values to Contemporary Mathematics Curriculum Reforms*

An emerging trend of mathematics reforms in recent years has actually brought to the spotlight the significance of researching values. The publication of the 'Principles and Standards for School Mathematics' in 2000 by the National Council of Teachers of Mathematics [NCTM] in the United States had seen the introduction of five process standards alongside the usual content topics. These process standards represent the different ways through which content knowledge might be acquired and applied, and teachers are expected to foster these skills amongst their students. The five NCTM process standards are, namely, *problem solving, connections, communication, representations*, and *reasoning and proof* . This curriculum approach of a dual focus on content knowledge and process skills has since been a feature of many an education reforms across the world, such as in Australia, mainland China, Denmark, and Germany. In Australia, the national mathematics curriculum (Australian Curriculum, Assessment and Reporting Authority, 2016) requires schools to teach four identified 'proficiencies strands', namely *understanding, fluency, problem solving,* and *reasoning*. The point here is that whether they are called process standards, proficiency strands, or something else, these mathematics learning skills which are being emphasised in contemporary mathematics curricula are considered to be important in relation to learning mathematics well and effectively. That is to say, the Australian mathematics curriculum is requiring teachers to teach students to value *understanding*, *fluency*, *problem solving*, and *reasoning* at the same time that they are teaching the number line, decimals, symmetry, quadratic equations or probability.

This focus on teaching values at the intended curriculum and beyond is only too timely as we approach a time in human history when artificial intelligence and automation are redefining fundamental job skills in national economies around the world. Acknowledging that we will never be as skilful and efficient as robots and computers at computations and mathematical processing at the workplace an in daily life, (mathematics) education has the responsibility to prepare our next generations to stay relevant and to contribute to the emerging workforce and daily living needs. If soft skills such as teamwork and inclusivity are needed to complement the hard skills which machines and robots excel at, this is where mathematics education can be useful for 'robot-proofing' the human workforce. As should be evident now, these soft skills are also the values which can be nurtured through mathematics. To the extent that such attributes are already being introduced to students through mathematics lessons, we need to understand better how their teaching might be planned deliberately, and/or how it might be executed during class lessons.

#### **10.2 Comparing Results**

How do the findings represented in this section compare with those derived from related research conducted elsewhere? Wilkie's findings with regards to what upper primary and junior secondary school students in Australia wish for in mathematics learning resonates with the 'What I Find Important (in my mathematics learning)' [WIFI] study which was conducted in economies such as Australia, Hong Kong, Japan, Korea, Macau, Shanghai, and Taiwan in the first half of the 2010s (see Seah & Wong, 2012). For example, in both Wilkie's study and Kinone, Shinno, and Baba (2013), the valuing of students in the middle years of schooling was found to be different from what students in the senior secondary school years valued, suggesting that different values are associated with different mathematics topics and different cognitive demands while some other values stay the same. Of course, there is also the possibility that during these teenage years, the child's values system is still being consolidated in what Morris Massey (1979) calls the socialisation stage of human values formation.

In another WIFI report, Zhang, Barkatsas, Law, Leu, Seah, and Wong (2016) analysed data collected from 1386 Grades 5/6 students studying in Wuhan (a Chinese city), Hong Kong, and Taiwan. The analysed data suggested that these students were valuing *achievement*, *relevance*, *practice*, *communication*, *ICT*, and *feedback* in their respective mathematics learning experiences. Similar between this study and Wilkie's is the positive tone of the values expressed.What is different, however, is that while the Australian students' values were mainly reflective of mastery approaches to goals (as interpreted by Wilkie), there was a balanced mix of mastery and of performance approaches to goal orientations from the students in the three East Asian economies which top such international comparative tests as TIMSS and PISA. Admittedly, neither of these studies were designed to identify if the nature of a student's value is reflective of mastery or performance goal orientations. As such, the research methods used were not designed to determine if East Asian students' valuing of *achievement* reflect either mastery or performance goal orientation. Yet, the possibility of such a difference in student values between East Asia and Australia may provide another perspective to an understanding of the difference in mathematics performance between East Asia and elsewhere in the world.

Wilkie's data also echoes Grootenboer and Marshman's (2016) findings with New Zealand middle school students, in that fluency with multiplication facts—emphasising the valuing of *basic facts*—was highly valued amongst the Australian student participants.

One of the findings of Achmetli and Schukajlow's study has been that students' experience with constructing multiple solutions to mathematical problems did not bring about any increase in students' interest in mathematics. This is somewhat surprising, especially given that many researchers (e.g. McLeod, 1992) have conceptualised interest as being easier to manipulate when compared to other affective variables such as beliefs and attitudes.

#### *10.2.1 Values Alignment*

While both Dobie and Wilkie's studies are more directly related to values in mathematics education, it is also interesting to note that each of them advocates for teacher alignment of student values in their practice. Wilkie ended her chapter by calling for teachers "to find ways to incorporate students' own suggestions for promoting their engagement, learning, and achievement", which may not be easy to accomplish if what the students value individually represent a wide spectrum of attributes when considered as a class. Some of these values may be in conflict with one another too. On the other hand, Dobie's view was that "teachers can also work to increase alignment with students' values in the classroom". Indeed, values alignment by the classroom teacher is currently being investigated in a follow-up to the WIFI study, called WIFItoo, in Australia. The assumption has been that

values alignment should be regarded as already being part of day-to-day interactions. When individuals come together with their own value systems, they will always need to negotiate about different preferences and intentions to ensure that the interaction is successful. This calls for values alignment to take place, and it does not mean that one party needs necessarily to impose his/her/their values to the rest. There can always be middle-path compromises, for example. (Seah, 2018, p. 575)

After all, "all relationships … are claimed to be strengthened by aligned values" (Branson, 2008, p. 381). How this alignment might be achieved, however, is not yet known, and the WIFItoo study aims to document the various ways in which teachers and their students co-construct values alignment such that productive teaching and learning can take place.

#### *10.2.2 Values Intervention Programs for Mathematics Classes*

Perhaps reflecting the newness of this research area, the studies in this section—and indeed, in several instances elsewhere as well—are not directly associated with the implementation of values-related intervention programs or education initiatives. Zonnefeld's study, however, is worthy of mention here. In fostering undergraduate classroom environments that support the development of the incremental view of mathematics, a range of affective variables were assessed for changes. At the end of the experimental period, an increase of students' valuing was only one of two statistically significant changes that took place. That female students' valuing increased more than that of their male peers "is a significant finding given the historical underrepresentation of women in mathematics and STEM in general".

Values were also not intentionally examined in the longitudinal study conducted by Middleton, Mangu, and Lee, although student's valuing of effort was evaluated in the study. Over a two-year period, changes in motivation indicators such as identity, interest, and self-efficacy had contributed to volatile changes in students' intention to pursue STEM programs beyond Year 11: "Relatively short-term motivations appear to greatly impact career aspirations. An effort should be made to make STEM curricula continuously attractive in new ways throughout a student's high school education, given the volatility of STEM career intentions". The stability (over two years in this study) of the students' valuing of *effort* when compared to the other scales surveyed, reinforces the conception that values are very internalised within an individual (see, for examples, Bishop, 1996; Stern, 2000). There are certainly implications here for the mathematics education (research) communities to explore more deeply how the variable of values might be used as a more reliable indicator or predictor of individuals' intentions, decisions, and/or actions in the process of teaching or learning mathematics.

#### *10.2.3 Research Methodological Concerns*

As mentioned earlier, amongst the seven studies represented in this section, three might be considered to be researching values, while two other studies dealt with the valuing of mathematics and statistics. The remaining two focused on affective variables such as interest. All seven studies sourced their data from participants who were either in secondary schools or universities. There was no suggestion in the chapters why there was no primary school participant, though it may be argued that for variables such as values which are normally inferred from participant opinions rather than being presented explicitly through actions, a certain level of participant maturity and confidence is needed.

Indeed, the questionnaire method was evident in all the seven chapters, with five of these utilising the questionnaire as the only data source. Accordingly, five of the studies are purely quantitative, although this does not correspond to the five studies which had made use of the questionnaire method only. Wilkie's open-ended questionnaires were analysed qualitatively, for example. At the same time, Dobie had employed mixed methods to analyse her data which were collected through questionnaires, lesson observations, and interviews.

Amongst the studies represented in this section, conclusions about teacher and student valuing appear to be made through analyses of questionnaire responses, except in the case of Dobie's work in which lesson observations provided an additional source of information about the extent to which the students valued the usefulness of mathematics. The valuing of particular attributes, however, needs not necessarily be followed by actions which reflect this valuing. Competing values (see Seah, 2005) may lead to some values being suppressed by other values that are also subscribed to but are more dominant in an individual. Both Halstead and Taylor's (2000) view of values as being general guides to behaviour, and Hill's (1991) tentativeness when writing about values ordering our lives, allow for the competing values narrative.

Perhaps not surprisingly, amongst the chapters in this section, there is a dominance of research that were conducted in Western economies. Of the seven chapters, three were written by researchers based in the USA reporting on American research, two were from Germany, and one from Australia. Even though the remaining non-Western study, conducted by Bofah and Hannula, might be focussed on African nations, the researching was a secondary analysis of TIMSS data and conducted outside Africa. This form of documentation hides the fact that at the international congress whose presentations stimulated the production of this monograph, there were several other research presentations conducted by non-Western researchers and/or conducted in non-Western sites, such as Elizar (Indonesia); Atinuke and Juwe (Nigeria); Felix (South Africa); and Gun & Bulut and Haser, (Turkey) (Hannula, Morselli, Erktin, Vollstedt, & Zhang, 2017). Unfortunately, for various practical reasons, these studies are not represented here. This phenomenon is not unique to this event, of course. If we also consider the amount of related research which have been published in non-English language publications, it is easy to imagine that what we get to access from English-language books, reports, journals and magazines constitute only a small part of what we collectively know and understand. This is unfortunate, given that the constructs of values and beliefs are highly contextualised in the diverse cultures globally within which they are internalised, activated, and espoused. There are certainly implications as we grapple with the optimisation of the school mathematics learning experiences for refugees, migrants and expatriate children around the world. There is thus a rather urgent need for the international mathematics education community to come together, to collaborate on possible solutions to this issue. In this regard, the kind of cooperation we see in Bofah and Hannula's research and writing is worthy of further exploration by colleagues worldwide. Another potential solution strategy might be for Western researchers to base their academic activities in targeted non-Western cultures, so that there are opportunities for sustained understanding, discussion and reflections by all parties involved.

Thus, from the values perspective, this section has pushed the boundaries of academic knowledge in the field a great deal. The various chapters have reinforced the notion of values as something that is perceived to be of importance and worth (in facilitating mathematics pedagogy). In addition to Bishop's (1996) categorisation of values in the mathematics classroom into mathematical, mathematics educational, and general educational, two of the studies represented in this section suggest that these valuing might manifest as a general valuing of the subject. Values' role in regulating the stability of affective variables such as interest is also suggested by a study here.  It might be a case of different values being associated with different mathematics topics and different cognitive demands, while some other values stay the same across the grade levels. It may also be reflecting the experience of a teenager going through values formation. It is also probably unique that students in Australia and New Zealand value *basic facts* highly. The significant role of values alignment has also been reinforced through the studies represented in this section.

The deeper understanding that is gained through the findings of the studies represented in this section could not have come at a more appropriate time, as the world is witnessing a fast transformation to hybrid intelligence and automation at work and in daily life, which seems to threaten the relevance of learning mathematics at school. A re-focus of the purpose of learning mathematics to one of cultivating values and practising valuing to complement what robots do might just be what is needed to bring about engagement and interest in mathematics amongst students.

#### **References**

Allchin, D. (1999). Values in science: An educational perspective. *Science & Education, 8,* 1–12. Australian Curriculum Assessment and Reporting Authority (ACARA). (2016). *The Australian Curriculum: F-10 Mathematics*. Sydney, Australia: ACARA.

Bishop, A. J. (1988). *Mathematical enculturation: A cultural perspective on mathematics education*. Dordrecht, The Netherlands: Kluwer.

UNESCO. (1979). *New trends in mathematics teaching* (Vol. IV). Paris, France: UNESCO.

Zhang, Q., Barkatsas, T., Law, H. Y., Leu, Y.-C., Seah, W. T., & Wong, N.-Y. (2016). What primary students in the Chinese Mainland, Hong Kong and Taiwan value in mathematics learning: A comparative analysis. *International Journal of Science and Mathematics Education, 14*(5), 907–924.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## Part III Engagement and Flow

## **Chapter 11 Intertwinement of Rationality and Emotions in Mathematics Teaching: A Case Study**

**Marina De Simone**

**Abstract** In this paper, I will analyse a mathematics teacher's classroom actions using a theoretical framework that tries to bring together the cognitive and affective dimensions, often considered separately. In particular, I will concentrate on the relation between the actions of a teacher, Carla, in her grade nine classroom and her reasons for those actions, through a combined analysis of the language and the emotional elements that accompany her speech. To carry out this analysis, I will combine Habermas' theory of rationality and the concept of emotional orientation developed by Brown and Reid within the research in mathematics related affect. From a methodological point of view, I interviewed the teacher to detect some of her expectations for her teaching decision. To confirm that these expectations I identified a priori are those that actually guide Carla during her teaching practice, I also analysed her classroom activity focusing mainly on her "emotional indicators", namely gestures, facial expressions, emphasis of the words, repetition, and so on. Finally, I examined the lessons more carefully in order to identify intertwinement between the teacher's rationality and emotions.

**Keywords** Mathematics teacher · Decision-making · Rationality · Emotional orientation · Linear equations

#### **11.1 Introduction**

This paper draws on my earlier research, on how cognition and affect are intertwined in a mathematics teacher's decision-making processes (De Simone, 2015). Decisionmaking is a characteristic feature of teaching since teachers are constantly making decisions within the classroom. Indeed, many authors have recognized that decisionmaking has a crucial role in the work of mathematics teachers. For example, Bishop

M. De Simone (B)

Faculté de Psychologie et des Sciences de l'Education, Université de Genève, Bd du Pont d'Arve, 40, 1211 Geneva, Switzerland e-mail<EMAIL_ADDRESS>
(1976) considers decision-making as the activity that is "…at the heart of the teaching process" (p. 42).

The rationality and emotions of the mathematics teacher are considered in the research described in this chapter as two sides of the same coin. Thus I attempt to go beyond the Cartesian dualism between the *res extensa* and the *res cogitans* towards a holistic view of both body and mind. Indeed, the neuroscientist Damasio (1994) speaks in terms of "Descartes' error" in relation to "the abyssal separation between body and mind […]; the suggestion that reasoning, and moral judgment, and the suffering that comes from physical pain or emotional upheaval might exist separately from the body." (pp. 249–250) Damasio's work supports the view that accounting only for the rationality of the teacher would not allow one to make a complete—or even a valid—analysis of the process involved in teachers' decision-making.

In the first section of the paper, I will explain why and how the original research problem evolved. In particular, I will introduce the theoretical lens I chose for analysing the rationality of the teacher, which started with the work of Habermas. I will then continue to discuss some critical debates around Habermas' approach. This critique enabled me to formulate a kind of rationality in which the affective dimension of decision-making is emphasized, drawing on recent research in mathematics related affect. After describing the methodological choices I made in my research, I will move on to a case study analysis of one grade nine mathematics teacher, Carla whose decision-making will be shown to intertwine both rationality and emotions. In the light of this case study I will then discuss how and why the adopted theoretical framework allows an analysis of teacher behaviour in detail.

#### **11.2 Theoretical Framework**

Originally, my research aim was to study rationality in the teacher decision-making processes, in the context of discursive activity of explaining linear equations. I set my research in the framework of Habermas' (1998) philosophical theory of rationality, re-elaborated and adjusted to mathematics education (Boero, Douek, Morselli, & Pedemonte, 2010; Boero & Planas, 2014). In particular, Habermas speaks of discursive rationality of a rational being involved in a discursive activity. First of all, he defines a rational being as a human being who "can give account for his orientation toward validity claims" (Habermas, 1998, p. 310). Then, he explains that the discursive rationality has three different roots: the knowledge, the action, and the speech or, in a different manner, the knowing, the acting, and the speaking.

At this point, it already became evident that the discursive rationality of the teacher is deeply connected to her decision-making space, because a rational being involved in a discursive activity is a decisional agent. Each root of the discursive rationality is connected to a particular type of rationality: the knowledge is connected to the epistemic rationality, the action is connected to the teleological rationality, and the speech is connected to the communicative rationality. The structure of discourse determines an intertwinement of the knowledge, the action, and the speech "by bringing together" the three types of rationality (Habermas, 1998, p. 309). These three types of rationality are inseparable, since a rational being acts in a specific manner to achieve a goal, on the basis of a specific knowledge, and communicating in a precise way. Hence, the three types of rationality are always present in the discursive activity; the only thing an external observer can see is the prevalence of one of the three types over the others. For example, it could be that, in a specific moment of the discursive activity of the teacher, only the epistemic rationality is "visible" and there is no sign of the teleological one.

As Habermas affirms, the epistemic rationality is connected with a specific idea of knowledge that is not in a trivial sense a collection of facts. Habermas explains that, "in order to know something in an explicit sense, it is not, of course, sufficient merely to be familiar with facts that could be represented in true judgements.We know facts and have a knowledge of them at our disposal only when we simultaneously know why the corresponding judgments are true. Otherwise we speak of intuitive or implicit knowledge—of a 'practical' knowledge of how one does something" (Habermas, 1998, pp. 311–312). Epistemic rationality requires that we can also give an account of the justification for the knowledge at play. Conversely, someone is irrational when she puts forward her beliefs in a dogmatic way, "clinging to them although she sees that she cannot justify them" (Habermas, 1998, p. 312). By its very nature, the epistemic rationality is deeply entwined with the action and the use of language. For this reason, epistemic rationality is not a self-supporting structure, but it is always combined with action and discourse.

Furthermore, Habermas (1998) speaks of teleological rationality when "the actor has achieved this result on the basis of the deliberately selected and implemented means" (p. 313). Developing this perspective, Habermas affirms that a successful actor has acted rationally when, first, he knows and he is conscious of why he was successful, and then "this knowledge motivates the actor (at least in part) in such a way that he carries out his action for reasons that can at the same time explain its possible success" (p. 314).

Finally, he states that the communicative rationality "is expressed in the unifying force of speech oriented toward reaching understanding" (Habermas, 1998, p. 315). The communicative use of linguistic expressions not only gives a "life" to the intentions of the subject, but also represents the actually occurring communicative state and, furthermore, establishes interpersonal relations with a second person. Once again, the communicative rationality is strictly connected to the epistemic and the teleological ones: "what the speaker wants to say with an expression is connected both with what is literally said in it and with the action as which it should be understood" (p. 315). Hence, this philosophical lens seems to be suitable for analysing the decision-making processes of the teacher through the interplay among the different types of rationality.

However, several philosophers and social theorists have highlighted that, in the development of his theory, Habermas did not take into account the emotional side of human beings. For example, Rienstra and Hook (2006), quoting the philosopher Heller, asserted that "Habermas leaves no room for 'sensuous experiences of hope and despair, of venture and humiliation', accusing him of completely avoiding the 'creature-like' aspects of human beings" (p. 13). Moreover, Steinhoff stressed that not only will argumentation be used to obtain consensus from the interlocutor about the statements, but all sorts of non-verbal means also play a significant role in the discursive activity. In particular, Steinhoff (2009) affirms that when people conduct discourses "they do try to convince the listener (and for this purpose not all listeners have to be participants in the discourse), but not just with arguments; they also use emotion, rhetoric and all means of achieving strategic influence, and the conflict resolution that discourse might be aimed at does not have to be consensual either (why not, for example, through majority decision?)" (p. 147). Summarizing, emotions are connected to reasoning at every moment, in discerning what is true and what is not and providing justifications of what we are stating.

Putting together Habermas' ideas and the critical debates of them, I posited that affective factors are entwined with those of rationality in the decision-making processes of the mathematics teacher. Hence, the original research aim was enlarged, becoming a study of the intertwinement between the emotional and rational aspects in the decision-making processes of a mathematics teacher. There is a need for theoretical frameworks for articulating how these affective factors might be involved in decision-making.

Research in mathematics education has progressively extended to perceive the existence of a mutual interaction between the affective sphere and cognition in mathematics learning (Zan, Brown, Evans, & Hannula, 2006). But it was in the field of human neuropsychology that I found the most helpful ideas on the relationship between the affective and the rational sphere, starting with the work of Damasio (1994, 1999). Moreover, while educational research often treated decision-making, reasoning, and processes related to reading, language, and mathematics as detached from emotion and body, Immordino-Yang and Damasio (2007) have shown that "learning, in the complex sense in which it happens in schools or the real world, is not a rational or disembodied process; neither is it a lonely one" (p. 4). For them, emotion is "a basic form of decision making, a repertoire of know-how and actions that allows people to respond appropriately in different situations" (p. 7). This neurological research is also becoming applicable in the field of mathematics education. For example, Brown and Reid (2006) have developed and adapted the hypothesis of somatic markers (Damasio, 1994) for studying the decision-making processes.

Brown and Reid (2006) proposed a theoretical framework grounded on the notion of *emotional orientatio*n in order to study the decision-making processes of both the teacher and the students. Their study is significant for my research because of its interest in that particular emotional aspect of human behaviour, which Brown and Reid felt to be neglected, and which they saw as "related to the decision-making that happens before conscious awareness of the decision to be made occurs" (p. 179). Hence, the concept "emotional orientation" allows me to speak of the interconnection between rationality and emotion. In fact, as the words themselves suggest, the "orientation" in a teacher's decision-making processes is "emotional", that is, affected by emotions in a certain way. In fact, the authors refer to the notion of emotional orientation as "the criteria for acceptance of an explanation by members of a community" (p. 181) and they consider emotions to be the foundation of such criteria. So the criteria, namely the decisions that for example a teacher takes within the classroom are affected by emotions. To better explain this point, they also recall the somatic markers hypothesis of Damasio (1994), proposing to consider somatic markers proper as a basis for mathematical emotional orientations. Somatic markers are structures that inform our actions and decision-making, pushing us to decide something because "It feels right" in terms of its acceptance in a community. So, emotions related to being right are attached to somatic markers, sets of which constitute emotional orientations. In decision-making, "many possibilities are rejected because they are associated with negative somatic markers" (p. 180), while positive somatic markers imply possible behaviours, being revealed by the decisions of the teacher in the activity. If emotional orientation refers to criteria (decisions) to accept an explanation and if at the basis of the choice of such a criterion there are emotions, then we can see emotional orientation as an aspect linked to emotions when choosing certain criteria (decisions) and not others.

For operationalizing the notion of emotional orientation of a teacher, we need an adaptation in terms of her "set of expectations": the term "expectation" is connected to her "emotions of being right" when she uses specific criteria for accepting an explanation by the class rather than other ones (Ferrara & De Simone, 2014).

Summarizing, I will investigate the intertwinement of rationality and emotions in the teacher's decision-making processes, drawing on this adaptation of the notion of emotional orientation, analysing the decisions a teacher makes in the classroom through the interplay among the different types of rationality à la Habermas.

My research questions are twofold: how does the use of emotional orientation help us understand the decisions of teachers, and thus complement the Habermasian rationality framework and how might it help us understand why teachers make certain choices in their teaching and not others?

#### **11.3 Methodology**

This study utilizes data from my doctoral thesis (De Simone, 2015), which involved three teachers and their grade nine classrooms. The students were 14 years old and they studied in a scientifically oriented secondary school in Italy. In particular, I chose to focus on lessons on linear equations because this topic is a crucial component of the Italian national curriculum for high school. It is one of the first mathematical content areas in which the delicate "shift" is made from the arithmetical world to the algebraic one. In this chapter, I will concentrate on one of the teachers, Carla, who has completed a master degree for training other teachers.

Carla was first a priori interviewed about the topic of linear equations. The interview was twenty minutes long and was videotaped with the camera facing the interviewer and the subject. The interviews were transcribed for the analysis. Thereafter, five of her lessons were also videotaped. The teacher utterances and bodily actions were transcribed for data analysis.

**Fig. 11.1** The analytical framework

Concerning the structure of the analysis, I first considered the a priori interview. From what Carla explicitly described to me, I was able to identify some of her expectations with respect to her students. At this stage these expectations were just potential, because it could be that they would not actually drive the actions of the teacher in class. Hence, I also looked at what actually happened in the classroom in order to see if there was a correspondence between what the teacher stated *a* priori and how she actually behaved in the classroom. For determining whether the expectations were actually reflected in the classroom activity, I looked at the "emotional indicators". In other words, for determining the expectations which the teacher actually reflected in the classroom activities, I observed the gestures, the facial expressions, the emphasis of the words, the repetition, the rhetorical questions, the pauses, the tone of voice, and so on. For example, if a teacher had a very insistent rhythm of words in asking examples, then she actually had the expectation that students would be able to make examples. The emotional indicators informed me about the emotionality of the teachers, where the term emotionality comes from the Penguin Dictionary of Psychology in which it is defined "in terms of *behaviours* that are observable and theoretically *linked to the* (hypothetical) underlying *emotion*…" (Reber et al., 1995). Hence, the expectations of Carla were visible through her emotionality (see Fig. 11.1).

Then, I examined the lessons more carefully in order to identify the intertwinement between the rationality and the emotionality. In particular, I looked at Carla's decisions through the three components of rationality (epistemic, teleological, and communicative) and, looking simultaneously at the emotional indicators and expressions of their expectations, I was able to say something about *why* she made those decisions and not others.

The emotionality of Carla was revealed by her prosody, her gestures, her facial expressions, her postures, and so on. As her emotionality was found to be always intertwined with her rationality, the teacher emotionality can be interpreted in terms of the Habermasian rationality; as epistemic emotionality, teleological emotionality, and communicative emotionality.

For example, the *epistemic emotionality* surfaced when Carla decided to draw on the concept of identity she had already introduced previously and, remaining in a waiting posture, she nodded after confirming that students actually remembered it. Thus, it was not just the kind of knowledge she chose to consider (epistemic rationality), but why she decided to draw on that particular knowledge. The reason is connected to her expectation that the previous knowledge was valid for students and this expectation was made visible through the posture of waiting for something and her nodding after the confirming response by the students.

The *teleological emotionality* could be highlighted when Carla justified the meaning of solving a linear equation and, at the same time, her voice had the highest pitch corresponding to the word "sense". It was not just the action the teacher made to accomplish a goal, namely justifying the resolution of a linear equation for finding the solution, but also the fact that she had the expectation that the justifications serve to give meaning to what the students are doing. This expectation was made visible through the high pitch of her voice.

The *communicative emotionality* surfaced when Carla had an insistent rhythm of voice when asking examples. Then, it was not just a matter of the words oriented toward reaching understanding, but also why she decided to communicate with an insistent rhythm. The reason was connected to her expectation that students would be able to make examples and the expectation became clear through the insistent rhythm. Overall, I considered the epistemic emotionality as related to why the teacher used that specific justification of the knowledge at play, the teleological emotionality as related to why the teacher made that action to achieve a goal, and the communicative emotionality as related to why the teacher used that speech oriented towards reaching understanding within the classroom. For pragmatic necessities of analysis, these three types of emotionality could appear to be separated. Nevertheless, it is important to stress that they were always intertwined and present in the discursive activity of the teacher.

#### **11.4 Data Analysis**

For the topic of equations, Carla started with the definition of an open statement as a logic proposition in which the truth value is determined by the value of one or more independent variables. The open statement contains at least one variable, the value of which is selected within a referring universe set (e.g. *U*). The open statement is indicated by a capital letter of the alphabet (e.g. *P*) and the variable is indicated in brackets by a lower case letter of the alphabet (e.g. *x*). The open statement *P(x)* can be true or false with respect to the value assumed by the variable *x*, chosen within the universe set *U*. For example, given the universe set *U* - {*x* ∈ *N* : *x* ≤ 20} the open statement *(x)*: *x* is even, is true for {2*,* 4*,* 6*,* 8*,* 10*,* 12*,* 14*,* 16*,* 18*,* 20} and false for the others.

Carla explained what an open statement is in order to show linear equations as open statements in which the verb is "being equal to". This open statement can be true for one value ascribed to the variable (determined equation), it can be false for each value of the variable (impossible equation), or it can be true for all the values substituted to the variable (indeterminate equation).

The following passage comes from the a priori interview of Carla in which she spoke about her conception of the role of the previous knowledge of students:

The greatest problem I'm trying to solve - I realised and it's becoming dramatic in the last years - is the problem of the stability of knowledge, in that I feel that in many classrooms, except the good ones, students don't remember what we did and, for me, this is serious. That is, for example, in grade ten, I would like to refer to something that I did before, on which I have even insisted, without having to repeat it entirely (…) the big problem to solve, in that I persist a lot, is being able to find a way for constructing a core (miming a base with her hand), a base of knowledge (miming a list with her open hand), of abilities that stay. For me, aside from time economy -'cause, maybe, it's a bit annoying having always to recall it's really a matter that has to do with cognitive science, I don't know, I wouldn't know how to face it, but it's becoming a general problem, then we should look for (…) the problem is looking for meaningful activities that allow (…) fixing things.

As showed in this brief piece above, for Carla, classroom culture was very important for having knowledge stability. Hence, from what she declared, I identified her *expectation of constructing new knowledge from what has been already done in the classroom*.

The expectation determined choices, for example, as the next excerpt of Carla's lessons will show (Table 11.1). She recalled the definition of "open statement", for constructing that of equation, as a particular open statement with the verb "to be equal to":

#### **Table 11.1** First excerpt of Carla's classroom activity

#### **Table 11.1** (continued)

3. T: it is a phrase where a variable appears for which, first of all, we have to define [*pause and waiting,* Fig. 11.3] what is the universe set, right? Try to answer to the questions I make in order to [Fig. 11.4] summarize. For an open statement, can we say if it is true or false?

**Fig. 11.3** Carla pauses with her hands gathered together

**Fig. 11.4** Carla's gesture that conveys the meaning of summarizing

called a solution.

Carla focused students' attention on how they have already defined an "open statement". Indeed, she wanted to introduce the concept of equation as an open statement with the predicate "to be equal to". For this reason, she explicitly asked her students what an open statement is. She accompanied this request with the gesture in Fig. 11.2, through which she seemed to hope to have the classroom in her control. Then, she wanted to remind the students of the importance of saying what the referring universe set is. She was waiting for an answer from students as shown in Fig. 11.3 (#3). She expected someone to say something, because, in the previous part of the lesson, she stressed that for an open statement it is important to specify the universe set. In fact, the statement's truth value depends on the set in which it is defined. At the end, she answered herself with a rhetorical question ("what is the universe set, right?" #3). Then, she continued, because, probably, she felt confident that the students were aware of this fact. Indeed, the discussion continued and Carla with her students considered what the truth set of an open statement was. Later she was expecting, probably a little irritated (Fig. 11.5: #9), that students would remember its definition.

Carla's *epistemic emotionality* was constituted by the knowledge of the concept of an open statement, by the important role of the universe set (rational key), and by her hope that students had already learned the definition of an open statement (emotional key). In particular, she was waiting for a relevant feedback from her students because she had already explained open statements and, probably, she felt that she had been effective in doing that. This emotional key was disclosed, for example, by the gesture in Fig. 11.4; by the posture of waiting for students' response as shown in Fig. 11.5; by the rhetorical question ("what is the universe set, right?") through which it is clear that this knowledge had been already introduced. In fact, it was not necessary to explain it again, but just recalling it with a rhetorical intonation. Carla's *teleological emotionality* concerned both the recalling what they have already discussed about open statements in order to speak of equations (rational key) and her expectation that it would be sufficient to just review open statements, because she was expecting that students remember them (emotional key). This emotional key was disclosed by the verb "summarize" that comes along with the gesture in Fig. 11.4. This could be seen as a hint of her expectation about the validity of the previous knowledge: it was already known by the students, hence now it was important to answer her questions just to summarize the previous knowledge. As a human being, she could not detach her body from her speech, hence her discourse was full of emotional elements. In particular her speech was accompanied by postures, gestures, and rhetorical intonation which together contributed to her *communicative emotionality*.

Moreover, in the passage below (Table 11.2), Carla constructs the "properties of linear equations" by referring back to the "law of monotonies1" for equalities that were explained at the start of the year. As will be shown in the next section, students were able to construct this mathematical content themselves, by drawing on their previous knowledge.

1They are laws related to the substitution property, according to which adding/subtracting the same number to, or multiplying/dividing by it, both sides of an equality does not change the equality.

**Table 11.2** Second excerpt of Carla's classroom activity

In this brief passage, the teacher was speaking of the second monotony law (#1). When students constructed that knowledge, she remained—for the whole time—in the same attitude that was constituted by her closed fist, her nodding and tightening her lips (#1, #2, #3, #4, #5: Fig. 11.6). Hence, there was an *epistemic emotionality* from Carla that was not only the knowledge of the "laws of monotony" for equalities (rational key), but also the fact that Carla hoped for her students to remember what they had already done with equalities. This way they could see the analogy to equations, given that she was expecting students to consider equations as open statements of equalities (emotional key). This emotional key was shown, for example, by her emblematic attitude indicated, for example, by her pauses and her nodding (Fig. 11.6) when the students, one after the other, jointly constructed the principles of equivalence in #2, #3, #4, #5. The teacher was engaged in the discursive activity with the class not only from a rational point of view, but also from an emotional point of view: she has hopes, expectations, and also needs. Both these sides were reflected in her speech because a teacher cannot be purely rational only in her discourse. Hence, I speak of the communicative emotionality of Carla. It was disclosed by her posture, her gestures, her stopping in speaking, and the tightening of her lips. This information allows me to say something about why she decided to act in a certain way.

In the a priori interview, Carla spoke of the role of justification in her teaching activity, as can be seen in the following excerpts:

(gesture with joined fingers and she beats on the table) I insist very much on why the letters instead of numbers […] It is always a discourse to give sense, a meaning to what we are doing.

I always justify the use of letters, I anticipated it a lot, for example when we speak of problems I say that, through letters, we can solve a class of problems, instead of solving a single numerical problem […] When I treat the resolution of equations, I am very careful of the conscious use of the properties of equations. I'm worried that it is not a mechanical thing.

As shown above, justification was very significant for Carla and, for this reason, I infer that she had an *expectation that the justifications are necessary to give sense to what she and her students do*.

Moreover, Carla was not just concerned with giving justifications during her classroom activity, but, as can be seen below, she expected that justifying would become an ordinary practice also for the students, in order to give a sense to what they are doing:

(self-confident tone of voice) ah yes, sure, I ask always "let's justify your reasoning" (she begins to gesture), "let's justify your answer", yes, sure, both during the work within the classroom and in the exams in class" […] (Carla spoke about the national assessment for the middle school). For me, something is evolving in middle school. Many people say that this assessment "falls on the head", but it serves to give meaning to what is being done. Many tasks in this assessment have the question "Explain why", and then students have to expose and justify their reasoning […]. I make many tests of the type "know" and "know-how"; the "know" is the set of justifications that students give in relation to what they do. Concerning equations, for example, I gave a problem of the type "given an equation, write an equivalent one to it applying just the first principle of equivalence and explain why".

As anticipated above, the teacher not only gave reasons to what she does herself, but, at the same time, she also had *the expectation that also her students would feel the need to justify to give sense to what they do*.

I found these two expectations on the role of justification contributing to the shaping of Carla's emotional orientation, which was reflected many times during her classroom activity. For example, in the specific passage of her lessons that I'm going to show, Carla gave many equations to her students to solve and she explicitly declared that in order to solve them, they "have to give a meaning to what you read". In different moments, Carla posed her students questions like "How do you justify your answer?", or "Let's justify your answer" (Table 11.3).