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As the participants in this collaborative exercise who are mathematics education researchers espouse a cognitive perspective, it is not surprising that there were few genuine disagreements between them and the psychologists and cognitive neuroscientists during the process of generating a consensual research agenda. In contrast, the prototypical mathematics education researcher will mostly likely find the resulting list of priority open questions to be overly restrictive in its scope of topics to be studied, highly biased toward quantitative methods, and extremely narrow in its disciplinary perspectives. It is argued here that the fundamental disconnects between the epistemological foundations, theoretical perspectives, and methodological predilections of cognitive psychologists and mainstream mathematics education researchers preclude the prospect of future productive collaborative efforts between these fields. [Commentary on: Alcock, L., Ansari, D., Batchelor, S., Bisson, M.-J., De Smedt, B., Gilmore, C., . . . Weber, K. (2016). Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41. doi:10.5964/jnc.v2i1.10]

The authors of the Challenges article are to be commended for the time, energy, and thoughtfulness they obviously put into generating a collaboratively-derived, consensual research agenda in the form of a list of research questions framed as major challenges for the field of mathematical cognition. At the beginning of their background section, the authors fittingly describe some of the differences between researchers in mathematics education, psychology, and cognitive neuroscience in terms of their training, central questions and interests, research methods, venues for presenting their findings, and publication outlets. Furthermore, the authors rightly point out that “there has traditionally been little communication between researchers working in these areas; cross-citations (with education in particular) are comparatively rare.” And as they also correctly acknowledge, the type of exercise they carried out “is naturally limited by the experience and knowledge of its participants, and must trade off breadth of representation against depth of focus.”

Wisely, for their purposes, the authors chose depth over breadth by inviting only researchers for whom the study of mathematical cognition was a major component of their own work. Therefore, it is hardly surprising that: “genuine disagreements were few, which encourages us to believe that our different approaches are converging on similar conceptions of the key issues.” So while the advantage of including these kinds of participants was being able to collaboratively hammer out a research agenda with only a limited number of authentic disputes, the disadvantage is that the prototypical mathematics education researcher will probably find this list of “priority research questions” to be:

extremely narrow in its scope by failing to include situated, contextual, or ethnomathematical considerations, not to mention democratic access to important mathematical ideas or equity-based mathematics teaching;

notably restrictive in its implicit (and in one case explicit—Question 21) methodological bias toward quantitative, experimental methods, and

conspicuously limited to a cognitive psychological perspective to the exclusion of anthropological, sociological, linguistic, semiotic, historical, and political viewpoints.

Shifting perspectives in mathematics education research that took place during the late 20th century began to diverge from developments and advances in the cognitive psychology of mathematical thinking and learning. These changes have since led to genuine disconnects between not only

In the field of neuropsychology, a “developmental disconnection syndrome” (

Cognitive Psychology | Mathematics Education |
---|---|

1. Accessing magnitude from symbols | Access to important mathematics |

2. Numerical identity (encoded by the parallel individuation system) | Mathematical identity (one’s personal relationship with math) |

3. Translating between numerical formats (transcoding) | Translating research into practice |

4. Multi-voxel pattern analysis (of fMRI data) | Mathematical patterning activities and sequences |

5. Progressive alignment of numerical scales | Alignment between mathematics standards and assessments |

6. Connectionist modeling of numerical cognition | Making connections among mathematical ideas |

7. Object file system | Students’ understanding of mathematical objects |

8. Core knowledge | Common core |

9. Electrical brain stimulation can enhance numerical cognition | Effective teachers can stimulate students to learn mathematics |

10. Groupitizing (to facilitate enumeration) | Small group math instruction |

11. Emphasis on internal number representations | Emphasis on external numerical representations |

12. Parity judgments of Arabic numerals | Equity in school mathematics |

13. Experimental designs and quantitative methods are favored | Design experiments and qualitative methods are preferred |

14. Operational momentum effect | Teachable moments in mathematics learning |

15. Small number processing (e.g., subitizing) | Big (mathematical) ideas |

16. Benefits of finger-based numerical representations | Pitfalls of finger-based counting strategies |

17. Scalar variability—signature of the ANS | Perceptual variability principle (Dienes) |

18. Parental math talk | Classroom mathematical discourse |

19. Cross-cultural influences on number processing | Multicultural mathematics curricula |

20. Numerosity adaptation effect (visual sense of number) | Adaptive expertise (meaningful knowledge flexibly applied) |

21 .Centrality of working memory for mathematical processing | Superficiality of memory in mathematics learning |

22. Response production system | Productive disposition toward mathematics |

23. Speeded practice substantially fosters simple arithmetic skills | Repeated reasoning is required for internalizing what is learned |

24. Acuity of non-symbolic number sense | Mathematical sense-making |

25. Students with dyscalculia are impaired in learning basic number concepts and arithmetic |

Additional support for my claim of a disconnection syndrome can be found by comparing the topics covered in the recently published

Furthermore, and quite apart from judging the quality of the individual contributions to the mathematics education research handbook, terms such as

Finally, my examination of the 163 entries comprising the recently published

Engineering and clinical medicine are prominent examples of applied fields that build upon foundational scientific disciplines. In the case of the former, engineers draw on fundamental laws of physics, chemistry, and mathematics for designing, developing, testing, and manufacturing products and services used in everyday life (

In contrast to this account, the vast majority of present-day mathematics education studies and instructional practices do not appear to draw on the latest and best available empirical findings emanating from the basic science of cognition. This state of affairs is not entirely surprising, as mainstream mathematics education researchers consider laboratory-based, experimental quantitative studies of mathematical cognition (as well as learning and instruction) to be of only limited value to educational policymakers, administrators, and practitioners (

The participants/authors of the Challenges article have clearly demonstrated that cognitive psychologists, neuroscientists, and mathematics education researchers

In contrast to this rather pessimistic prognosis, it may prove more viable to try to increase mathematics educators’ knowledge of basic cognitive processing as it applies directly to pedagogy. Indeed, encouraging steps have already been taken in this direction.

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The author has declared that no competing interests exist.

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