Dedemonizing frustration in mathematics

A few years ago, after an interview with a family, the school's principal called me to his office. As it turned out, the family went to complain because I told them that I wanted their son to get frustrated and the principal wanted to hear my side of the story. The family wasn't too far off, but those weren't my words. During the interview, we talked about their son's mathematics frustration, I told them that mathematics can be ungrateful because hard work doesn't always pay off, but this only happens in the short term, with perseverance, sustained effort eventually bears fruit. What seemed to bother that family was what I told them about that frustration not being something negative, it's in a controlled environment after all, and that we could take advantage of that frustration because it could help him learn more and better.

The truth is that I was only based on my intuition, which might be wrong. I have done some research and feel like sharing what I've been reading with bibliographic citations and references at the end.

Traditionally, frustration in mathematics has been considered something negative [5, 10, 11, 15 & 21], it might be so because research has mostly been focused on preventing it and not on its possible benefits [17]. Frustration doesn't have to be a negative consequence of solving problems, it might even be what fosters heuristic strategies; the desperation triggered by frustration could facilitate trying new ways, even taking non-logical steps, that might help solving the problem [9].

Frustration appears because of a lack of progress when solving a problem expected to be solved with less effort and whose value is low. If solving the problem is of high value but there are no means to solve it, anxiety appears, and if it is of no value, boredom comes with eventual disengagement [14 & 15].

Observed emotion to emotion transitions and its hypothezised causes. Adapted from [8].

John Dewey, one of the greatest philosophers of the first half of the twentieth century and one of the pioneers of the philosophy of education claimed that "the origin of thinking is some perplexity, confusion, or doubt" [3]. In fact, for deep learning there needs to be challenges [7]. This perfectly fits with the new theory of executive intelligence which considers that in order for a student to engage in a complex learning process there ought to be a "cognitive dissonance" [13].

"the origin of thinking is some perplexity, confusion, or doubt"

Adapted from @nerdbugs.

Mathematicians enjoy thinking and learning, we are used to challenges. As a matter of fact, we develop "a certain attitude toward doing mathematics, accepting that problem solving will require some potentially frustrating, non-goal-oriented behavior. There will be times of play and many detours that must be taken for progress to be made" [18]. To struggle is a natural part of "doing mathematics" [20], and so, frustration is one of the emotions most often experienced while solving math problems [10 & 12].

Instead of aiming to eliminate frustration in mathematics, meta-affect (becoming aware of one's emotions and addressing them) should be developed so that the feelings about emotions associated with difficulty are productive in terms of learning. With an appropriate development of meta-affect, frustration would indicate that the mathematical problem is interesting and non-routine, meta-affect would confer the problem the intrinsic value of learning something, making it worth solving [2]. Affect, together with cognitive and conative processes, fundamentally determine students’ mathematics-related activities [1]. There already are cognitive-emotional training strategies developed for solving mathematics problems [4].

Frustration would indicate that the mathematical problem is interesting

When students face an impasse (jam, dead end or conflict), they realize that they need to learn something and for a moment they care about learning [19]. Efforts must be made to ensure that the unavoidable confusion experienced by students during problem solving is productive rather than hopeless [5].

So, meaningful learning requires facilitating that students encounter obstacles, challenges with contradictions or difficult decisions that promote critical thinking and deep inquiry [6, 7 y 19]. There ought to be support and guidance through that naturalization of frustration during problem solving, though, to prevent confusion from being too frustrating and make the struggle productive [5, 7 y 15]. "When students persevere through frustration, they develop a greater sense of accomplishment, resilience, and a deeper understanding of the content" [21].

It seems that I wasn't misguided, but it is not about getting students frustrated for the sake of frustration, as we just read, frustration is a reaction to unsolved confusion and that confusion must be deliberated, almost surgical, to provoke thinking and foster comprehension. Additionally, this process must be monitored to avoid negative consequences and benefit from the positive ones.

Wondering about the fact that thinking requires some teasing and making an effort, it reminded me of the catchy gym phrase "No pain, no gain" and I came up with this:

References

[1] De Corte, E., Depaepe, F., Eynde, P. O. t. y Verschaffel, L. (2011). Students’ self-regulation of emotions in mathematics: An analysis of meta-emotional knowledge and skills. ZDM - International Journal on Mathematics Education, 43(4), 483-495. https://doi.org/10.1007/s11858-011-0333-6

[2] Debellis, V. A. y Goldin, G. A. (2006). Affect and meta-affect in mathematical problem solving: A representational perspective. Educational Studies in Mathematics, 63(2), 131-147. https://doi.org/10.1007/s10649-006-9026-4

[3] Dewey, J. (1910). How we think. D C Heath. https://doi.org/10.1037/10903-000

[4] Di Leo, I. y Muis, K. R. (2020). Confused, now what? A Cognitive-Emotional Strategy Training (CEST) intervention for elementary students during mathematics problem solving. Contemporary Educational Psychology, 62. https://doi.org/10.1016/j.cedpsych.2020.101879

[5] Di Leo, I., Muis, K. R., Singh, C. A. y Psaradellis, C. (2019). Curiosity… Confusion? Frustration! The role and sequencing of emotions during mathematics problem solving. Contemporary Educational Psychology, 58, 121-137. https://doi.org/10.1016/j.cedpsych.2019.03.001

[6] D’Mello, S. y Graesser, A. (2011). The half-life of cognitive-affective states during complex learning. Cognition and Emotion, 25(7), 1299-1308. https://doi.org/10.1080/02699931.2011.613668

[7] D’Mello, S. y Graesser, A. (2012). Dynamics of affective states during complex learning. Learning and Instruction, 22(2), 145-157. https://doi.org/10.1016/j.learninstruc.2011.10.001

[8] D’Mello, S., Lehman, B., Pekrun, R. y Graesser, A. (2014). Confusion can be beneficial for learning. Learning and Instruction, 29, 153-170. https://doi.org/10.1016/J.LEARNINSTRUC.2012.05.003

[9] Goldin, G. A. (2000). Affective Pathways and Representation in Mathematical Problem Solving. Mathematical Thinking and Learning, 2(3), 209-219. https://doi.org/10.1207/s15327833mtl0203_3

[10] Hannula, M. S. (2015). Emotions in Problem Solving. Selected Regular Lectures from the 12th International Congress on Mathematical Education ISBN: 9783319171869, 269-288. https://doi.org/10.1007/978-3-319-17187-6_16

[11] Lehman, B., Matthews, M., D’mello, S. y Person, N. (2008). What Are You Feeling? Investigating Student Affective States During Expert Human Tutoring Sessions. En E. A. R. N. S. L. Beverley P. Woolf (Ed.), LNCS (Vol. 5091, pp. 50-59). Springer, Berlin, Heidelberg.

[12] Lehman, B., Mello, S. D. ’, D’mello, S. y Person, N. (2008, junio). All Alone with your Emotions: an analysis of student emotions during effortful problem solving activities. Workshop on Emotional and Cognitive issues in ITS at the Ninth International Conference on Intelligent Tutoring Systems. https://www.researchgate.net/publication/228630627

[13] Marina, J. A. y Pellicer, C. (2015). LA INTELIGENCIA QUE APRENDE: La inteligencia ejecutiva explicada a los docentes. (The Intelligence that Learns: Executive Functions for teachers). Santillana Educación.

[14] Muis, K. R., Psaradellis, C., Lajoie, S. P., Di Leo, I. y Chevrier, M. (2015). The role of epistemic emotions in mathematics problem solving. Contemporary Educational Psychology, 42, 172-185. https://doi.org/10.1016/j.cedpsych.2015.06.003

[15] Pekrun, R., y Stephens, E. J. (2010). Achievement Emotions in Higher Education. En J. C. Smart (Ed.), Higher Education: Handbook of Theory and Research (Vol. 25, pp. 257-306). Springer. https://doi.org/10.1007/978-90-481-8598-6_7

[16] Quintanilla, V. A. y Gallardo, J. (2020). Identificar experiencias emocionales para mejorar la comprensión en matemáticas. UNO, 88, 24-33.

[17] Riegel, K. (2021). Frustration in mathematical problem-solving: A systematic review of research. STEM Education, 1(3), 157-169. https://doi.org/10.3934/steme.2021012

[18] Sinclair, N. (2004). The Roles of the Aesthetic in Mathematical Inquiry. Mathematical Thinking and Learning, 6(3), 261-284. https://doi.org/10.1207/s15327833mtl0603_1

[19] VanLehn, K., Siler, S., Murray, C., Yamauchi, T. y Baggett, W. B. (2003). Why Do Only Some Events Cause Learning During Human Tutoring? Cognition and Instruction, 21(3), 209-249. https://doi.org/10.1207/S1532690XCI2103_01

[20] Warshauer, H. K. (2015). Productive struggle in middle school mathematics classrooms. Journal of Mathematics Teacher Education, 18(4), 375-400. https://doi.org/10.1007/s10857-014-9286-3

[21] Young, J. R., Bevan, D. y Sanders, M. (2024). How Productive Is the Productive Struggle? Lessons Learned from a Scoping Review. International Journal of Education in Mathematics, Science and Technology, 12(2), 470-495. https://doi.org/10.46328/ijemst.3364

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