From Our SEL Field Notebook: Thoughts, Emotions and Mathematical Reasoning

How has your own work as an educator impacted your research?

I have been involved in mathematics education for many years, initially as a primary (elementary) teacher and mathematics leader, then as a regional mathematics consultant, and currently as a mathematics education university tutor. A recurring theme in these roles—arising from dialogue with students, parents, teaching assistants, and teachers—is the relationship between learning mathematics and an individual’s attitudinal and emotional responses, sometimes called the cognitive-affective interplay.

I have observed the impact that positive emotions can have on enhancing mathematical thinking (for adults and children) and also the impact that mathematical thinking can have on stimulating feelings of satisfaction and pride. However, I have also observed the ways in which mathematical thinking can result in feelings of frustration and even anxiety, and how these more negative emotions can create barriers to engaging with mathematical thinking.

These experiences inspired me to investigate and research the relationship between thinking and emotions during mathematics lessons. I am interested in exploring the interactions between thoughts and emotions during mathematical reasoning activities—in particular, researching how teachers can make a difference within an activity in a lesson rather than over a longer term.

What drew you specifically to mathematics education research?

Throughout my career, I have been interested in the learning and teaching of mathematics. I am passionate about students having high-quality, rich experiences in learning mathematics, and these include opportunities to behave mathematically through conjectural thinking. However, creating a conjectural learning environment is pedagogically demanding for teachers for a wide range of reasons. Through my own engagement in and creation of mathematics education research, I wanted to contribute to our understanding of the learning and teaching of mathematics, particularly that involving the cognitive-affective interplay.

We know that perseverance is a vital social-emotional skill for young mathematicians. In your research, how do you gauge perseverance in students? What have you learned about how students develop perseverance?

Mathematical reasoning—particularly during problem-solving activities—involves a suite of well-researched mathematical behaviors and processes: having a go and making trials;   spotting patterns and relationships; forming and testing conjectures; generalizing and forming convincing arguments about why the generalizations are true.

Whilst these are not necessarily approached in a linear way during mathematical reasoning, the learning from one process needs to be applied in others. For example, testing a conjecture may lead to the realization that the conjecture is not true, which then instigates further trials with the aim of revealing other patterns or relationships that might support revisions to the original conjecture. This means that reasoning through a line of inquiry requires ongoing reflection self-regulation, decision-making, and overcoming being stuck. This is not always easy to do and requires perseverance. Successful perseverance in reasoning through a line of inquiry involves movement between these reasoning processes and culminates in forming generalizations with convincing justifications as to why they are true.

In general terms, perseverance can be interpreted as “keep going when things get difficult”   or   “not giving up”—and in many aspects of life and learning, these interpretations may work well for individuals. However, when we engage in activities involving mathematical reasoning, being persistent, telling ourselves to “keep going” and “don’t give up” is not necessarily sufficient to help overcome the barriers or difficulties encountered. This is because a persistent, “keep going” approach can lead to repetitious, reactive behaviors that override reflection and self-regulation, such as re-creating trials using the same system or approach despite already having noticed a pattern or relationship. Students with limited perseverance in mathematical reasoning can become stuck by predominantly making trials and spotting patterns. This can lead to limited perseverance in mathematical reasoning.

Teachers can assess students’ perseverance in mathematical reasoning by observing their reasoning processes and their movement between, or stasis in, these processes. To support students to improve their perseverance in mathematical reasoning, teachers can emphasize the central activity aim of forming justifications as to why generalizations are true. This can support students to adopt this important goal and the reasoning process to help them to persevere toward it. It may be that this requires additional lesson time, but it is well worth spending more time on fewer problem-solving activities to support students to construct convincing justifications because this not only enhances their mathematical thinking and deepens their understanding, but may also enable them to experience successful perseverance in mathematical reasoning and pride and satisfaction in their work!

What has your research shown about the enjoyment children express in learning math compared to their levels of persever- ance with difficult mathematical tasks?

Enjoyment in learning is a welcome emotion for teachers and students, and it is often prized by teachers in mathematics lessons because in this subject we can sometimes see more negative emotions or attitudes becoming a barrier to engagement and learning.

However, in my research, I found that for some children, their in-the-moment enjoyment acted as a barrier to their successful perseverance in mathematical reasoning. In these instances, the children tended to act persistently by engaging in repetitive processes, particularly by making further trials when they had already spotted and generalized a mathematical pattern. They gained great pleasure from the repeated application of the pattern to make more solutions, and this led to them forming the in-the-moment goal of making as many solutions as possible rather than forming justifications as to why the generalized pattern worked. As their enjoyment of making trials shifted their activity goal, this acted to prevent their ongoing reflection, decision-making, and self-regulation. Furthermore, because they were focusing on making more solutions rather than forming justifications, and they did this successfully, they did not experience being stuck and consequently had no emotional feedback to alert them to reflect on their actions. This then prevented them from examining the generalization they had formed and constructing the mathematical reasoning for why this was true, thus limiting their capacity to persevere in mathematical reasoning to form justifications of their generalization. So whilst enjoyment can be a highly desirable emotion in mathematics lessons, for some children—and this is counterintuitive—it can act to create a repetitious cycle that impedes their perseverance in mathematical reasoning.

From an educator’s perspective, what can we do in our classrooms to create the conditions for students to persevere in mathematical reasoning?

I am really pleased to be asked this question. These are some practical steps that teachers can take to create the conditions for students to successfully persevere in mathematical reasoning:

• Provide students with opportunities to develop reasoning behaviors, particularly activities that involve applying reasoning processes to develop a line of mathe- matical inquiry to produce assertions, develop a mathematical argument, and justify conclusions.
• Focus students’ attention on forming generalizations with explanations and justifications by establishing this as the goal of the activity.

• Assess students’ perseverance in mathematical reasoning by observing the mathematical processes that they’re engaged in. Look for:
  • Successful perseverance in mathematical reasoning by observing students’ movement between reasoning processes, that is, a progression from making trials and pattern/relationship spotting to conjecturing, generalizing, and forming convincing justifications.
  • Students who engage in only one or two reasoning processes, particularly those who repeatedly make trials and apply patterns. These students may have established an in-the-moment goal that does not relate to forming justifications about generalization (and they may not have realized this), thus limiting their success in persevering in mathematical reasoning.
  • Expressions of pride and satisfaction rather than enjoyment as indicators of successful perseverance in mathematical reasoning

Have any recent findings in your work been unexpected or surprising to you?

Yes, definitely. I had expected to find that becoming and overcoming being stuck, and the emotions related to this, such as frustration, bewilderment, and even anxiety, were a central theme in developing students’ perseverance in mathematical reasoning. However, when students experience these more difficult emotions, they have cues to help them to recognize that they are stuck during mathematical reasoning. This recognition is an important step toward trying to overcome the difficulties encountered. Here we can see why it is important for students to experience struggle and difficulty in learning mathematics.

I had not expected to find that, for some students, enjoyment could be a barrier to successful perseverance in mathematical reasoning— and that enjoyment could impact on the in-the-moment goals that students form, creating cycles of repetitious behaviors that deflected their attention away from constructing justifications about why their generalizations were true. The idea that enjoyment can act as a barrier in this way is counterintuitive and does not seem to align with our desire, as teachers, to want students to enjoy learning.

What is your favorite aspect of your work?

It was such a privilege to work with teachers to develop pedagogies that supported students to break out of their repetitious use of processes during mathematical reasoning. My favorite and the most heart-warming aspect of the work was observing this group of students experiencing intense pride and satisfaction—and not just enjoyment—in their mathematical reasoning following teachers’ interventions.

What are one or two projects you’re working on right now?

I’m currently developing and applying this work in a couple of ways. I’m supporting student teachers and teachers to apply and further develop these findings in their practice. And, importantly, I’m working with student teachers and teachers to understand the interplay between emotions and thinking, and how this can impact on their own experiences and perseverance during mathematical reasoning as well as those of their students.

What do you hope to achieve with your research?

Reasoning is central to learning mathematics and being mathematical—and persevering through a line of inquiry is an important aspect of this. However, it’s not easy, and overcoming struggles and difficulties, whether these relate to thinking or emotions, is a necessary part of this perseverance. I hope that this research supports educators involved in the learning of mathematics to deepen their understanding of the widely known interplay between emotions and thinking in mathematics, so that they can support their students to be able to successfully persevere in mathematical reasoning—and experience the related satisfaction and pride!

Dr. Alison Barnes is a principal lecturer in education, specializing in teaching primary mathematics education and postgraduate education research at the University of Brighton in England. She teaches the MA education, professional doctorate in education, and primary mathematics education BA (Hons) with QTS courses, and is course leader for the university’s MA education, run in partnership with the Mauritius Institute of Education. Dr. Barnes’s research focuses on supporting professional learning in education in order to enable practitioners to further develop their professional understanding and knowledge and by doing so enhance their personal learning and that of their students.