Dr Peter Jenkins, Head of Mathematics Curriculum Development
From 2020, Year 12 students in Queensland will sit external examinations that assess content learned over almost two years. Evidently, the ability of students to effectively remember a large volume of content, skills and understanding will be crucial for success. In Mathematics, there is often a reluctance to talk about the role of memory, possibly due to a fear that focusing on the act of remembering somehow minimises the importance of conceptual understanding. But not only is memory crucial in the process of developing conceptual understanding (Byers and Erlwanger, 1985), conversely it is the development of conceptual understanding that makes mathematical ideas memorable.
As noted by McInerney (2014), the most important characteristic that makes information memorable is meaningfulness. Meaningful information is information that is already related to networks of ideas, or schemas, in long-term memory, and thus gives rise to the feeling of making sense. The richer the connections made with existing schema, the more potent and robust the memory of the new concept.
Helping students see mathematical information as meaningful can be challenging for teachers, yet forms a large part of what happens in the mathematics classroom. Our school-wide pedagogy, based on the Harvard Project Zero Cultures of Thinking, is an ideal framework to accomplish this challenge. Encouraging students to engage in specific thinking routines that require them to call on prior knowledge, their expectations and intuition means that this new information can then be integrated into their existing schemas. Of course, this process is intellectually taxing, and can be demanding on the information-processing abilities of students.
The cognitive system responsible for the temporary holding of information available for processing is known as working memory. There is a well-established direct relationship between the capacity of working memory and mathematical performance (Alloway and Passolunghi, 2011). A larger working memory essentially allows one to keep more incoming information and information drawn from long-term memory at the front of their mind simultaneously, thus allowing them to make sense of new information more easily. Because working memory is limited in capacity and duration, it can easily be overloaded. This phenomenon, known as cognitive overload, results in minimal change to long-term memory (McInerney, 2014), and thus represents a severe impediment to learning.
When cognitive overload occurs in mathematics learning, students fail to make sense of new concepts, and instead must resort to remembering rules and procedures that are often meaningless in isolation. While this may allow them to experience limited success in simple, familiar situations, they may struggle when presented with a problem in any unfamiliar context. Further, their memory of such rules and procedures will be fragile, and could be easily confused.
Furthermore, the high level of connectedness and dependence between mathematical concepts means that in order to make sense of new concepts, students typically must be able to readily access a variety of other concepts from long-term memory. Yet, if these prerequisite concepts cannot easily be retrieved, it makes the process of connecting future concepts to existing schemas much more difficult.
This all underscores the importance of the mathematics classroom experience, where teacher expertise and student attention must combine to make learning possible. When students are absent from mathematics class, they can often learn important procedures by studying worked examples or reading over notes, but they miss out on the activities and discussions that help them think about the new ideas in particular ways that make them meaningful. Skilled teachers minimise the risk of cognitive overload by carefully sequencing new information, and prioritising manageable key principles that serve as powerful keystones of subsequent learning. As an example, a key principle in solving algebraic equations is that ‘performing any operation to both sides of an equation does not change the solution set’. Such a principle can readily be made meaningful by an effective teacher, and subsequent techniques and procedures for solving specific types of equations may then be made meaningful by linking them to this key idea.
Given the deleterious effect that overloading working memory has on learning, much research has been focused on techniques for reducing working memory load in order to facilitate the changes in long-term memory associated with schema acquisition. For example, Chandler and Sweller (1992) showed that when information contains both a diagram and statements (neither of which can be understood in isolation), then physically integrating the statements within the diagram significantly reduces cognitive load. More significantly, cognitive overload can be reduced by a process known as overlearning. Overlearning refers to the continuing repetition or practice of a skill past the point of first mastery, so that the skill becomes automatic. Once automatic, the skill contributes almost nothing to cognitive load (McInerney, 2014). This explains why students who have overlearned multiplication tables seem find it so much easier to make sense of factorising algebraic expressions: no effort is required in determining factors of numbers in such expressions and so all their working memory can be dedicated to making sense of the actual concept.
The amount of practice required to reach automaticity with certain mathematical skills is significant, but it is not impossible to achieve. Further, the type of practice is important. Stobart (2014) points out the importance of practice being purposeful and deliberate. This seems especially relevant in mathematics, where there is always a danger of repetitive practice reinforcing the wrong things if it is done without thinking. For example, a student who practices solving 20 equations, all having the exact same structure (for example, having the form ), can very quickly forget the necessity of the key principle described previously, which reinforces meaningless manipulations of symbols. This can be averted by teachers who ensure that the set of problems students are encouraged to practice have enough variation to eliminate the possibility of false patterns emerging, and require repeated application of the key principles in order to solve them.
Clearly, memory is important in the learning and doing of mathematics. Yet, mindless memorisation of facts and formulas will not take students far. Instead, engaging in classroom activities and discussions that help make new ideas meaningful together with purposeful practice are important key skills. These skills enable students to think more effectively, remember more and be successful in their learning.
Alloway, Tracy & Passolunghi, Maria Chiara. (2011). The relationship between working memory, IQ, and mathematical skills in children. Learning and Individual Differences. 21. 133-137
Byers, V., & Erlwanger, S. (1985). Memory in Mathematical Understanding. Educational Studies in Mathematics, 16(3), 259-281
Chandler, P. & Sweller, J. (1992). The split-attention effect as a factor in the design of instruction. British Journal of Educational Psychology. 62 (2): 233–246
McInerney, D. ed., (2014). Educational Psychology: Constructing Learning. 6th ed. Pearson Education Australia
Stobart, G (2014) The Expert Learner. Challenging the Myth of Ability. L Stoll & L Earl (series eds.) Expanding Educational Horizons. McGraw-Hill, Open University Press