What is mathematics? How should we make sense of mathematical cognition research
Written by Camilla Gilmore who is a Professor of Mathematical Cognition and co-director of the CMC. Edited by Bethany Woollacott.
This post summarises Camilla’s EPS Prize paper that was recently published by the Quarterly Journal of Experimental Psychology. The paper is open access and linked at the end of this blogpost.
Finding the bigger picture in mathematical cognition research
As academics we typically look forward, focusing on the new studies we want to run, the datasets we are analysing or the papers we need to write. We rarely spend time looking backwards and thinking about the bigger picture that guides the work that we and others do. However, last year I had the opportunity to write a review paper and spent some time reflecting on the huge growth in mathematical cognition research over the past two decades and the progress that has been made.
In doing so, one of the things that struck me is that we do not have a shared viewpoint on what mathematics is. As a result, it is difficult to bring together findings from different studies and we don’t know how a set of varied skills, processes and knowledge combine to allow individuals to be mathematical.
“We do not have a shared viewpoint on what mathematics is“
I felt that a framework that does this might be helpful so we can:
- understand how different research findings fit together;
- identify outstanding questions for future research;
- inform the choice of mathematical measures;
- and provide a shared language to discuss these issues.
I hope that the resulting multi-level framework for mathematical cognition provide impetus for researchers in the field to have these bigger picture discussions.
Mathematics as a multi-componential domain
What is mathematics? It’s well-established that mathematics is not a single construct. It encompasses a wide range of domains (e.g., arithmetic, geometry, algebra) and involves a combination of skills, knowledge and processes.
This creates two challenges for researchers:
- How do we identify and understand the mechanisms underlying mathematics learning?
- How do we decide what to measure when studying mathematical cognition?
Does it matter if one group of researchers studying, for example, the relationship between inhibitory control and mathematics choose to use a comprehensive mathematical achievement measure and another group of researchers interested in the same topic choose a timed measure of arithmetic fluency?
Levels of mathematics
I propose that it might be helpful to think about mathematical cognition as involving three (or more) levels, depicted in the diagram here.
At the highest level we have overall mathematical achievement. This is typically measured by broad curriculum measures or composite standardised measures that incorporate a variety of mathematical domains and may include reasoning and problem solving. Measures of mathematics achievement typically require individuals to identify the mathematics required in contextually based problems in order to select appropriate strategies and to combine different skills and knowledge to answer a given question.
Overall mathematics achievement emerges from proficiency with specific components of mathematics. This level of the framework captures an individual’s performance in coherent sub-components of mathematics for which it may be anticipated that they will use a more-or-less consistent set of mathematical knowledge and skills. For example, specific components of mathematics may include number fact retrieval, algebraic reasoning, understanding of arithmetical relationships, and adaptive strategy selection.
These specific components of mathematics in turn recruit basic mathematical processes. These are lower-level processes that underpin the specific components described above. Here, it is helpful to consider the lowest levels of mathematical processes that cannot be easily subdivided and measured in a meaningful (mathematical) fashion. This might include, for example, magnitude comparison, order processing, spatial-numerical associations, intuitive geometrical knowledge, and place-value understanding.
The nature and content of the specific components of mathematics and basic mathematical processes are likely to change over development and learning.
The framework and existing evidence
We can think of mathematical cognition as comprising these three levels of increasingly specific processes and the links between them. However, these mathematics-specific elements of the framework do not operate in isolation. General cognitive skills may be independently related to each of these levels. Informal and formal learning experiences may influence the development of each level, as well as the links between them.
If we examine the existing, and rapidly-growing, mathematical cognition literature, we see there is some evidence for each of the links a-e, in the framework (see the paper for examples and references). Much of this evidence is currently correlational, for example showing associations between specific mathematical (or cognitive) processes and proficiency with specific components of mathematics or overall mathematics achievement.
What do we need to know?
The framework draws attention to several outstanding questions that need answering for us to have a comprehensive understanding of mathematical cognition.
- What are the most important basic mathematical processes and specific components of mathematics that are necessary to understand the mechanisms of mathematical cognition?
A helpful task for the field is to investigate which basic mathematical processes are essential for higher-level mathematics performance and which specific components of mathematics form coherent elements of knowledge, skills, and understanding.
- How specifically mathematical are the basic mathematical processes and how general are the general cognitive skills?
It may be that the basic cognitive processes involved in mathematics are better conceived of as a continuum between more general and more specific, rather than a dichotomy between domain-general and domain-specific.
- What is the role of affective factors (anxiety, motivation, enjoyment etc.)?
We know that these factors are associated with overall mathematics achievement but it is less clear how they relate to basic mathematical processes, the involvement of general cognitive skills in different components of mathematics or the way that individuals interact with different learning experiences.
The links in the model represent crucial mechanisms that we need to understand. Paying more attention to the mechanisms between different elements of mathematics, rather than just correlational evidence of associations, would help us to further develop theory.
The framework is not intended to be a model of mathematical processing or to capture everything that is involved in learning mathematics. I hope instead that it will trigger conversations amongst researchers, provide a structure to think about the commonalities and differences between studies and help researchers to think more precisely about the measures of mathematics that they use in their studies. Different measures of mathematics are not inter-changeable and should be selected carefully in light of the specific research questions of interest. What should not drive decisions about the use of measures is the simple ease of the measure involved.
“What should not drive decisions about the use of measures is the simple ease of the measure involved“
Thinking of mathematics in terms of this framework may also contribute to debates about pedagogical decisions by identifying the range of processes, skills and understanding that children’s learning experience should seek to target.
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