There are different communities of research that are currently active in investigating difficulties in mathematical learning, from cognitive psychology and neuroscience to mathematics education, but no solid common grounds from which to develop further studies have yet been reached. We seek to develop foundations for such common grounds.
Most of the literature from the field of cognitive psychology investigates typical development of basic number processing, introducing terms like “developmental dyscalculia”, “mathematical learning disability (or disorder)”, among many others for describing atypical development. However these terminologies as well as the corresponding definitions are still a topic of debate and there are still no consistent diagnostic criteria (the cut off scores vary from the 3rd to the 32nd percentile) or assessment tools applied across countries (in fact prevalence varies from 1,3% to 13,8%). Moreover, when taking into account the frequency of comorbidity and of heterogeneity, the situation becomes extremely complex.
On the other hand, the field of mathematics education highlights how the construct of “learning disability” or “disorder” does not afford to differentiate between difficulties that signal a stable disability in mathematics and those that are a result of deficient teaching experiences or lack of sufficient exposure. Therefore it is very difficult for the mathematics education community to build on the psychological literature when exploring specific teaching and learning processes in the presence of difficulties. Moreover, research in mathematics education has been going towards more sociological or anthropological views, even declaring to go beyond “deficit models”, and suggesting that mathematical failure more than an individual affair is a collective outcome of the student, the teacher, and the students’ social milieu.
 In general, a variety of teaching materials, strategies and theories has been developed within the various communities of research involved. However educators who work with “older” students (age 8 to 18), even when informed on their clinical condition, frequently find it hard to choose what to propose in each specific case.
Along these lines we use the acronym MLD in place of “mathematical learning difficulties”, as a broad category containing mathematics disabilities or difficulties regardless of whether they are innate (for example, developmental lag, related to specific or general cognitive deficits) or due to extraneous causes (for example, deficient teaching experiences, lack of sufficient mathematical exposure, social conditions, affective and affective factors). Our research assumes that basic cognitive processes are of great importance during the mathematical learning process, as well as the development of an appropriate mathematical interaction between the student with difficulties, the teacher, and his/her peers.
         We have developed a fourfold MLD model (Karagiannakis, Baccaglini-Frank & Papadatos, 2014) which revisits the main MLD hypotheses intertwining them in a holistic way. The domains of the model (core number, visual-spatial, memory, reasoning) can be used to outline the mathematical learning profile of a student, thanks to an appropriately designed computer-based assessment tool, the DeDiMa Battery (developed by Dr. Giannis Karagiannakis) that contains a range of mathematical tasks (Karagiannakis & Baccaglini-Frank, 2014). The DeDiMa Battery should be complemented with additional psychometric tests (measuring the IQ, the executive system, the possible presence ADHD, or of the autism spectrum…), to reach a more complete mathematical profile of each student. This approach is also in line with the DSM-V.
The fourfold MLD model and the DeDiMa Battery have proven to be valid and reliable (Karagiannakis, Baccaglini-Frank & Roussos, submitted). We hope these will be widely used internationally, leading to comparable data, on larger and larger samples, that can be used insightfully across countries and research communities. The individualized mathematical learning profile (at a certain moment in time) reveals the student’s weaknesses, but also his/her strengths, which can be used to design focused data-driven and personalized remedial interventions (see Karagiannakis & Coreman, 2014).
The design and study of such teaching interventions is a final main direction of our ongoing research. In particular, we are exploring the potential of the constructed framework for adapting existing and designing new didactical material and activities (Baccaglini-Frank & Scorza, 2013; Baccaglini-Frank, Antonini, Robotti, Santi, 2014; Santi & Baccaglini-Frank, in press) matching students’ profiles. These are directions that our research program is currently pursuing, with the explicit hope of crossing the boundaries of different fields of research, of informing all interested communities of research and fostering dialogues, and eventually of creating common scientific grounds.

• Baccaglini-Frank, A., Antonini, S., Robotti, E., & Santi, G. (2014). Juggling reference frames in the microworld Mak-Trace: the case of a student with MLD. Research Report in Nicol, C., Liljedahl, P., Oesterle, S., & Allan, D. (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36,Vol. 2 (pp. 81-88). Vancouver, Canada: PME.

• Baccaglini-Frank, A., & Scorza, M. (2013). Preventing Learning Difficulties in Early Arithmetic: The PerContare Project. In T. Ramiro-Sànchez & M.P. Bermùdez (Eds.), Libro de Actas I Congreso Internacional de Ciencias de la Educatiòn y des Desarrollo (p. 341). Granada: Universidad de Granada.

• Karagiannakis, G., Baccaglini-Frank, A., & Papadatos, Y. (2014). Mathematical learning difficulties subtypes classification. Frontiers in Human Neuroscience, 8:57, doi:10.3389/fnhum.2014.00057.

• Karagiannakis, G., & Coreman, A. (2014). Focused intervention based on a classification MLD model. In S. Chinn (Ed.), The Routledge International Handbook of Dyscalculia and Mathematical Learning Difficulties (pp. 265-276). London: Routledge

• Karagiannakis, G., & Baccaglini-Frank, A. (2014). The DeDiMa Battery: A Tool for Identifying Students’ Mathematical Learning Profiles. Health Psychology Review, 2(4), doi: 10.5114/hpr.2014.46329

• Karagiannagis, G., Baccaglini-Frank, A., & Roussos, P. (submitted). Detecting difficulties in Mathematics through a fourfold model (working title). Journal for Reasearch in Mathematics Education.

• Santi, G., & Baccaglini-Frank, A. (in press).Possible forms of generalization we can expect from students experiencing mathematical learning difficulties. PNA, Revista de Investigaciòn en Didàctica de la Matemàtica, Universidad de Granada, España.