The cognitive leap: enhancing algebraic thinking in STEM education
By KJ Ndweni
Welcome to a world where operations and numbers no longer align with the familiar rules of arithmetic. Instead, this world dances to the abstract rhythm of algebra. Have you ever wondered if the key to unlocking the mysteries of advanced mathematics lay not in the memorisation of formulas, but in comprehending the relationships between quantities from foundational years?
How can teachers bridge the intimidating gap from arithmetic to algebra, where abstract concepts become intuitive, and learners’ mathematical fascination flourishes?
My exploration into these posed questions reveals a revolutionary approach to algebraic thinking and algebra education that could contribute to the reshaping of the future of STEM learning.
When looking at the concept of algebraic thinking, we note how it represents a critical juncture in the cognitive development of mathematics learners. However, significant challenges are presented by the complex transition from arithmetic to algebra, and this often hinders learners’ progress.
Through drawing insights from my recent research, this article aims to highlight the nuances of these challenges and explores innovative pedagogical strategies, explicitly focusing on the application of Davydov’s Vygotskian-based theory.
The transition challenge
Transitioning from arithmetic to algebra can create a cognitive gap in learners, as highlighted by Pratiwi, Herman and Suryadi (2019).
The reason behind the cognitive gap is the late introduction of formal algebra education, typically during the middle years of primary school or early secondary, leaving learners with limited foundational experience in algebraic thinking and reasoning. This late introduction to formal algebraic education is a sudden shift from concrete arithmetic operations to abstract concepts. It can be disorientating and thus lead to frustration and difficulties among learners.
Understanding the difficulties
My research aimed to identify and comprehend these challenges through semi-structured interviews with first-year students at the University of Bristol in the United Kingdom. The findings of my research consistently alluded to the sudden pedagogical shift as a primary obstacle. Students from my study recalled struggling to grasp the new algebraic methods introduced typically during the middle years of primary school or early secondary after years of arithmetic-focused learning. This sentiment is alluded to by Silver and Kennedy (2001), who suggest that the late introduction of algebraic concepts can cause a significant learning barrier.
Davydov’s Vygotskian-based theory: a solution
In my attempt to address the challenges, I turned to Davydov’s Vygotskian-based theory. The argument that was made by Vygotsky is that algebra, characterised by relational analysis of quantities, should precede arithmetic, which focuses on the development of number concepts (Schimattau, 2004). This approach asserts that an early introduction to algebraic reasoning has the potential to build a robust cognitive potential, facilitating smoother transitions and deeper comprehension.
Research Methodology
This study employed an interpretivist epistemological approach and used semi-structured interviews to gather qualitative data. The definition of the epistemological approach started emerging as I pondered on the research questions for this study. Thinking about how I would ask participants questions related to the study, I considered that the questions had to be related to my suggested proposal of the Vygotsky method.
As a mathematics education researcher, I sought to deeply understand the difficulties first-year students face when learning algebra. In order to achieve this, I adopted an interpretive epistemological paradigm from a constructivist ontological approach (Guba & Lincoln, 1998).
This approach was particularly relevant as it granted me the opportunity to explore the students’ subjective experiences and construct meaning from their perspectives. Through the utilisation of semi-structured interviews, I was able to gather rich qualitative data that shed light on the nuanced challenges these students encounter and their learning processes. The thematic analysis of these interviews revealed three main themes: the nature of the transition challenge; effective teaching strategies; and attitudes towards the Vygotskian approach.
Key findings and strategies
- Challenges in transition
The primary difficulty I identified from the interviews was the sudden shift from arithmetic to algebra. Students found it challenging to adapt to new symbolic representations and abstract thinking required in algebra.
- Effective teaching strategies
Strategies that were suggested in the interview included the use of real-life examples, group discussions, and meta-commenting, where teachers provide ongoing feedback and suggestions to facilitate understanding. Using meta-commenting during algebraic activity is supported by Coles and Ahn (2022), who argue that meta-commenting can facilitate students’ understanding of algebra. These methods can help bridge the cognitive gap and make algebra more relatable and less intimidating.
- Attitudes towards Vygotskian theory
My findings highlighted the participants’ attitudes towards the proposed Vygotskian theory, which suggests that algebra should precede arithmetic. Initially unfamiliar with the theory, the students found it helpful once introduced.
To introduce students to Vygotsky’s approach, I designed an engaging activity (see below) demonstrating the principles of early algebraic reasoning. One method involved drawing two lines of different lengths, prompting participants to explore and comprehend relational concepts before delving into numerical operations.
This hands-on experience below helped illustrate the benefits of introducing algebraic thinking at an earlier stage in the learning process:
Algebraic activity
Drawing two lines
Questions:
- What do you think about the relationship between those two lines? Which line is longer than the other one?
- Can you help me to make a comparison between those three lines?
Introducing Vygotsian Theory
- Can you relate this to some of your teachers’ strategies in teaching algebra?
- Do you think it is helpful for children to understand this topic? Why?
Future Directions in STEM Education
This study asserts the notion that early and sustained engagement with algebraic reasoning is crucial. Integrating Vygotsky’s theory into the primary school curriculum could revolutionise how algebra is taught, providing learners with a more intuitive and solid grasp of mathematical concepts. This foundation comprehension is vital not only for success in STEM fields but also for fostering critical thinking and problem-solving skills that are applicable across various disciplines.
Conclusion
The transition to algebraic thinking represents a pivotal moment in STEM education, one that requires careful pedagogical consideration. By adopting innovative approaches, like Davydov’s Vygotskian-based theory, teachers can better provide support to learners through this transition, ensuring the development of skills and confidence required for future academic and professional success. My findings underscore the significance of early algebraic reasoning and highlight effective strategies that can make a lasting impact on learners’ mathematical journeys.
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