Academic journal article Australian Mathematics Teacher

# Unravelling Student Challenges with Quadratics: A Cognitive Approach

Academic journal article Australian Mathematics Teacher

# Unravelling Student Challenges with Quadratics: A Cognitive Approach

## Article excerpt

My secondary school mathematics students have often reported to me that quadratic relations are one of the most conceptually challenging aspects of the high school curriculum. From my own classroom experiences, there seemed to be several aspects to the students' challenges. Many students, even in their early secondary education, have difficulty with basic multiplication table fact retrieval. Difficulty retrieving multiplication facts directly influences students' ability to engage effectively in factorisation of quadratics, since factorisation is a process of finding products within the multiplication table. Finally, students also find it challenging to recognise and understand varied representations of the same quadratic relationship.

In my own classroom, I had explored various pedagogical strategies in order to mediate for the challenges that I have outlined--everything from rehearsal to real world applications. However, I felt that my pedagogical efforts lacked the necessary insight on how the brain creates memory and felt that my pedagogical directions might be enhanced with this knowledge. Therefore, in order to construct my own classroom solutions, I turned to cognitive science to assist me in better understanding the mechanisms of fact retrieval. I surmised that problems with quadratic relations might potentially be linked to the ways in which the brain constructs cognitive representations and this knowledge might in turn inform my pedagogical decision making as a classroom teacher. This article is a sharing of my investigation.

To better understand the problems students experience with quadratic relations, I draw from cognitive science researchers, Phenix and Campbell (2001), who suggest that order matters in the brain's ability to retrieve numeric facts. Their research is useful in understanding students' problems with factorisation and with identifying varied representations of the same quadratic relationship. Before I detail their research, I begin with an overview of the kinds of memory capabilities our brains have in order to situate why, as teachers of mathematics, we need to pay attention to Phenix and Campbell's claims.

Making the right kind of mathematical memories

Butterworth (1999) suggests that there are three types of memory our brains can create. Long-term autobiographical memory stores events with generalised timelines of when and where the events occurred. A student's memory of grade eight graduation is an example of long-term autobiographical memory.

Long-term semantic memory stores general knowledge not identified by a timeline for when the event occurred. Multiplication facts, for example, are stored in long-term semantic memory. Semantic, as in long-term semantic memory, implies associations to specific memories or meanings. In mathematics, semantic implies the ability to access certain knowledge over other knowledge based upon context. I refer to this process as "linguistic discrimination"--the ability to access one meaning over another meaning from long-term semantic memory of mathematical text; i.e., symbolic, numeric, visual, graphic, etc. (Kotsopoulos, 2006).

Short-term memory stores information temporarily. This information, or knowledge, may be lost if not eventually stored in long-term semantic memory. Multiplication facts can, alternatively to long-term semantic memory, be relinquished to short-term memory and thus lost to students during, for example, factorisation. Our goal, as educators, is to structure learning opportunities to ensure that mathematical facts and/or multiple representations of mathematical objects are stored to long-term semantic memory and are, thus, potentially accessible to students in the form of prior learning.

Therefore, how do we, as teachers, create learning opportunities that facilitate long-term semantic mathematical memories? Furthermore, how do we teach students to discriminate linguistically between meanings and access the appropriate long-term semantic memory? …

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