The overall objective of this research is to better understand how students can learn to think and solve problems critically, and how dialogic-type mathematics pedagogies that deal with real-life or other meaningful everyday contexts can support them in meeting school mathematics' curricular needs as well as their needs in the real-world. That is, improving the learning outcomes of students in mathematics that address not only the mathematics curricular needs of schools, but also the students' quantitative competency needs beyond the school environment. This approach resulted from the inherent contradictions that the study identified in the use of mathematics in schools and in the context of everyday or real-life. The literature review suggests that these contradictions are related to the established perception that school mathematical knowledge involves 'abstract thinking', which has a 'different meaning' from that in real-life. I attribute this to the foundation of mathematical knowledge in schools, which is based on the belief that mathematical knowledge itself is a priori or mathematical apriorism, the 'Absolute or Absolutist' view, whereby mathematical knowledge is based solely on reason without recourse to real-world experiences or observations. Fundamentally, the study attempted to explore the use of everyday or real-life contextualised mathematical pedagogy in the development of student criticality or critical mathematical thinking skills. The criticality of students or critical mathematical thinking skills, the study's mathematical knowledge outcome, implies that students gain mathematical knowledge that is mediated by their own personal and other real-life experiences that help them to concretise their mathematical learning. I take this position because I believe that mathematical knowledge should be based on 'experience' or observations of the real world â that is, viewing mathematics as grounded in manipulations of physical reality or the real-world that is, knowledge is a posteriori. This is the 'Fallibilist' view of mathematical knowledge, which recognises that mathematical truths are 'fallible and correctable' and, as such, are subject to revision and correction as dictated by real-world events. The research employed a lesson study approach that encapsulates a mixed method of data collection for students aged 16 to 21 in Core and AS Level Mathematics classrooms in the UK. Quantitatively, the study developed measures for learners' metacognitive awareness of their critical mathematical reasoning processes (CMRP), relevant real-life pedagogy and critical mathematical thinking skills (CMTS) measures. The relationships between the measures were explored for two contrasting groups following the traditional AS level and the âCritical Core Maths' curricula. The key findings here are that (I) a validity study (following the Rasch Construct Development Approach) shows that these constructs are measurable, although some limits are identified, and (II) they can explain how differences in curricula and learner perceptions of pedagogy are associated with differences in these important learning outcomes. In particular, a programme designed to promote everyday real-life context mathematics, is associated with a significant increase in learners' CMTS, and this is partially mediated by the pedagogy as perceived by these learners as ânon-transmissionist' (or speculatively âconnectionist'). Furthermore, the investigation of observed lessons in the âadvanced' Core Maths curriculum suggests how the use of real everyday contexts can support learners in taking some control (power) over mathematics, making their own sense of the mathematics in ways that might support critical mathematical thinking. A Bakhtinian-Vygotskyan theoretical/analytical framework helps an analysis of how dialogue is entwined with context and mathematics in ways that empower learners to be critical. This qualitative work further offers a possible explanation of the statistical results showing the aforesaid association. In conclusion, I argued that the learners' criticality is encouraged by dialogic mathematics pedagogies posing contexts, where the learners can bring their own everyday real-life knowledge to bear on, in ways that they experience as being meaningful and communicable. Also, I further argued that, in mathematics lessons, spending more time on everyday real-life contextualised mathematics pedagogy may be more beneficial to students than time spent teaching a procedure, when the goal is to promote equivalent mathematical knowledge transfer to real-life or new problem domains.