Errors in the mathematical learning process have traditionally been viewed as a critical issue, often considered an indicator of failure by both students and educators. However, in recent decades, a growing body of research in cognitive science, neuroscience, and pedagogy has led to a substantial reconsideration of the role of errors, transforming them from obstacles into potential catalysts for learning.
This blog post aims to examine the role of errors in mathematical learning, with a particular focus on students aged 14-18, a crucial period for cognitive development and the acquisition of advanced mathematical skills.
The main objectives of this blog post are as follows:
- Analyze the empirical evidence supporting the importance of errors in the mathematical learning process.
- Examine the cognitive and emotional mechanisms underlying error processing and their impact on learning.
- Five practical tactics for integrating errors into mathematics teaching.
In particular, this blog post is aimed at parents, mathematics teachers, educators, and school principals with the goal of disseminating the most important theories on the subject.
What is the Error-Friendly Pedagogy?
Introduction
The Error-Friendly Pedagogy is an educational school of thought that took shape in the mid-20th century, thanks to the significant contributions of two thinkers: Karl Popper and Henry Perkinson. This perspective introduced a substantial change in how learning is conceived, including in the field of mathematics.
In this educational approach, error assumes a central role in the construction of knowledge, where mistakes are no longer considered failures to be avoided, but become essential elements of the learning process.
Through the acceptance and analysis of errors, students can develop real, transversal, and transferable skills. These skills, which we might define as “antifragile”, not only withstand difficulties but are strengthened through them. In this new educational paradigm, errors become a starting point for a deeper understanding, in our case, of mathematics. They are no longer obstacles to overcome, but opportunities for growth and learning.
Cognitive Theories on Error
Cognitive theories have developed in opposition to behaviorist theories. The cognitive theories of learning developed by Tolman and the Gestalt psychologists posit that learning occurs through central brain processes, such as memory and expectations, which help build cognitive structures. This learning is not based on trial and error, but on a perceptual restructuring of the problem. Furthermore, there is latent learning, not immediately tied to reward.
Cognitive theories not only support error-friendly pedagogy, but consider it an essential element for deep and lasting learning. Errors are seen as learning opportunities, tools for metacognitive reflection, and means to develop cognitive resilience and flexibility. This approach is particularly relevant in the teaching of mathematics, where conceptual understanding and problem-solving are central.
Below we list some authors:
- Piaget’s Constructivist Theory: Jean Piaget (1976) considered error as a natural indicator of the learning process. According to this theory, errors are not simply mistakes, but represent active attempts by students to make sense of new information. In practice, encourage students to formulate and test their own hypotheses, even if they may lead to initial errors.
- Sweller’s Cognitive Load Theory: John Sweller (1988) proposed that learning is more effective when the load on working memory is optimally managed. Errors can be seen as indicators of cognitive overload. In practice, teachers are advised to structure lessons in a way that reduces extraneous cognitive load, allowing students to focus on essential aspects of learning.
- Ohlsson’s Error-Based Learning Theory: Stellan Ohlsson (1996) developed a theory that views error as a fundamental mechanism for acquiring procedural knowledge. According to this perspective, errors activate cognitive revision processes that lead to skill refinement. Suggestions for teaching: create dedicated spaces for students to reflect on their own errors and actively correct them.
The Importance of Making Mistakes
Making mistakes is inevitable in the learning journey and offers a valuable opportunity not only to learn and memorize information in the long term but also to develop persistence and resilience, allowing for a deeper understanding of concepts.
When students encounter an error, they are prompted to justify their thinking, seek examples, and construct arguments. This process is truly effective in developing logical mathematical thinking.
Showing students that there are multiple approaches to a problem and that failing with one method doesn’t mean failing altogether (Boaler, 2018). Errors often reveal gaps in understanding, and addressing them can lead to building stronger connections between different mathematical concepts.
Overcoming errors also contributes to the development of mathematical perseverance. Students learn that making mistakes is part of the learning process and that success often requires tenacity and persistence in the face of difficulties. Many of the most important mathematical innovations were born precisely from errors or apparent failures. If we encourage students to see errors as starting points for new ideas, we can cultivate a more creative and innovative approach to mathematics.
Finally, the process of making mistakes also encourages critical thinking by reconsidering one’s approach and exploring alternative solutions. This reflective and analytical thinking not only improves mathematical competence but also nurtures a mindset that values the problem-solving journey as much as the correct answer.
Cognitive and Emotional Mechanisms in Learning
But are all errors equal?
The relationship between emotions and learning has been at the center of numerous studies in neuroscience and psychology in recent decades. It clearly emerges how emotions play a primary role in cognitive processes related to learning.
Emotions, in fact, actively intervene in decision-making, idea formulation, and information memorization. When a concept is learned accompanied by strong emotional involvement, it is “cataloged” as important in our mind and will have a greater chance of being remembered in the future (Damasio, 1994; LeDoux, 1996). Conversely, if learning is associated with negative emotions such as fear, anxiety, or frustration, an “emotional short circuit” is created that can severely hinder the learning process (Immordino-Yang Damasio, 2007).
It has been demonstrated that effective learning implies strong emotional-cognitive activation. Moreover, academic success, as well as problems of self-esteem and insecurity, largely depend on early learning experiences and the emotions associated with them (Vygotsky, 1934; Piaget, 1967). Professor Daniela Lucangeli and her collaborators at the University of Padua have initiated warm cognition. This is a research study that takes into account the emotions underlying the learning process (Lucangeli, et al. 2015).
When students make mistakes, especially in a subject perceived as “exact” like mathematics, they might experience a range of negative emotions such as frustration, anxiety, shame, and a sense of inadequacy. The fear of making mistakes can lead students to avoid actively participating in lessons, thus limiting their learning and growth opportunities. Recognizing and addressing these negative emotions is fundamental to creating a positive and effective learning environment.
In light of this knowledge, it is essential that teachers are aware of the deep connection between emotions and learning and know how to promote a classroom climate where positive emotions, such as motivation, gratification, and a sense of self-efficacy, can foster student learning (Lucangeli Cornoldi, 2018).
Growth Mindset in Mathematics
The theory of Growth Mindset, developed by psychologist Carol Dweck in the 1980s and refined in subsequent decades, introduced a new paradigm in the field of education and psychology.
This theory, published in full form in her book “Mindset: The New Psychology of Success” in 2006, explores how people’s beliefs about their abilities significantly influence their learning and development.
The central concept of Growth Mindset is that intelligence and abilities are not fixed characteristics, but can be developed through effort, learning, and perseverance. This view contrasts with the “Fixed Mindset”, according to which abilities are innate and immutable.
Students with a fixed mindset tend to believe that mathematical ability is innate and immutable. This belief often leads them to avoid challenges for fear of failure, seeing errors as signs of lack of talent rather than as learning opportunities. Consequently, such individuals may easily give up in the face of difficulties and feel threatened by the success of their peers.
According to Carol Dweck’s theory (2006), students with a growth mindset believe that such ability can be developed through effort and practice because it is continuously evolving. This belief leads them to embrace mathematical challenges as growth opportunities, seeing errors as an integral part of the learning process. These students tend to persist in the face of difficulties and draw inspiration from the success of others, seeing it as a demonstration of what is possible through commitment and dedication.
A key element of this approach is the appreciation of errors. The feedback provided to students plays a crucial role in this process. Rather than valuing innate intelligence or talent, Dweck suggests that teachers (or business managers) reward the effort expended and reflect on the errors made because the class learns more from these. This type of feedback helps students develop a deeper understanding of their learning process and encourages them to persevere in the face of challenges.
Despite the popularity of the methodology in various fields and numerous scientific evidence, recent studies have instead demonstrated the poor correlation with academic results obtained in STEM subjects in classes of children between 7-9 years old (Dong, Jia, Fei, 2023). Here you can find a systematic review of the effectiveness of growth mindset in mathematics teaching (Bui, Pongsakdi, McMullen, Lehtinen, Hannula-Sormunen, 2023).
4 Strategies for Primary and Secondary School Mathematics Teachers
In this section, we will present four practical strategies for mathematics teachers in primary and secondary schools, aimed at integrating error as a resource in student learning. These approaches can facilitate a positive and creative educational environment.
1. Reward Better Conceptual Errors
If you are a mathematics teacher in middle or high school, you might find it useful to apply this technique. Organize your students into small groups of 3-5 and, at the end of the activity, ask them to share their work on the board. Establish the “Best Conceptual Error of the Week” award. This recognition should be given to a complex and conceptual error that can stimulate deep discussion and contribute to collective learning (Boaler, 2016).
2. Group Work with Self-Assessment
Working in small groups and implementing peer self-correction tests offers numerous benefits. It allows students to compare, share ideas, reduce their anxiety, and support each other, fostering a deeper understanding of concepts less focused on grades. Moreover, the practice of peer self-correction stimulates critical thinking and motivation. This approach also promotes a more inclusive and participatory learning environment.
3. Mathematical Journals
The use of mathematical journals encourages students to reflect on their learning processes and document their discoveries, difficulties encountered, and solutions. Regularly maintaining journals allows students to develop greater awareness, helping them better understand concepts. Teachers can use these journals to assess student progress and provide personalized feedback.
4. Learning Stations
Learning stations are positions within the classroom where students can work on different mathematical activities. Each station is dedicated to a particular topic or type of problem, allowing students to move between stations and work autonomously or in small groups. This approach promotes active learning and allows teachers to differentiate activities based on student needs, offering targeted support and stimulating curiosity.
Conclusions
Recent theories and research, such as Carol Dweck’s, demonstrate that an approach that values error can transform the study of mathematics from a feared subject into an exciting and engaging one. Embracing the concept of Growth Mindset and using errors as teaching tools allows the class to develop problem-solving skills and perseverance, essential for academic and personal success.
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