Session Information
WERA SES 02 C, International Lessons Learnt on Student Preparedness and Transition
Paper Session
Contribution
Theoretical Framework/Research Questions
The perspective that primary and secondary schooling needs to prepare students not only for post-secondary schooling but also for careers that may not require such, has been public policy in most countries. In that context one question that arises is to what extent exposure to rigorous formal mathematics is associated with strong mathematics literacy? We address this question making use of the PISA mathematics literacy assessments and the Opportunity to Learn (OTL) questions that premiered in PISA 2012.
Specifically, we used the questions that measure the content coverage of formal and applied mathematics (OTL) that are cumulative over the student’s schooling up to the age of 15, taking into account the international grade placement of mathematics topics as developed in the TIMSS curriculum study (Schmidt et al. 2001). These measures were then employed to examine the relative importance of exposure to Formal Mathematics, Applied Mathematics (AM), and Word Problems (WP) to mathematics literacy (ML). Research based on the TIMSS provides strong evidence that exposure to formal mathematics is associated with student performance on a curriculum-based test. The outstanding question is whether a student’s ability to apply these concepts as measured in PISA (ML) is mostly related to exposure to topics explicitly oriented around mathematics applications and word problems, or whether exposure to formal mathematics concepts also has a strong relationship.
The Opportunity to Learn (OTL) construct specifies what occurs in schooling, the process of instruction in the classroom. Caroll’s (1963) model was among the first to specifically address classroom learning. Bloom and his colleagues made use of Caroll’s theory in the OTL measures developed in the First International Mathematics Study. Most of the research that has investigated the relationship of OTL to what students know has been conducted with measures of students’ learning of the mathematics taught in school. The PISA emphasis on ML raises the question as to what occurs in schools that might support the development of ML as defined and measured in PISA (De Lange, 2003). This uncertainty about what it is in schooling that may support the development of ML is related to at least two issues. One has to do with the nature of learning and the other has to do with the nature of instruction.
How Students Learn focuses on three “fundamental principles” (NRC, 2005). The development of ‘mathematics proficiency’ requires students to have “a foundation of factual knowledge (procedural fluency), tied to a conceptual framework (conceptual understanding), and organized in a way to facilitate retrieval and problem solving (strategic competence)” (Fuson et al., 2005). Although all three of these competencies should be outcomes of schooling, the PISA ML focus would appear to be most closely associated with the last of these.
As soon as one specifies something to be learned, the question “how shall this be accomplished” immediately follows. This becomes an instructional or pedagogical issue. Responses to this question have historically contrasted a focus on “pure” versus “applied” approaches to mathematics teaching (de Lange, 1996). Instruction focused on “pure” mathematics centers on the study of the academic aspect of mathematics: its definitions, relations, and structure. Upon mastery of these, students may then be asked to apply their knowledge to real world situations. In contrast, the applied approach focuses on connecting mathematics to the real world at all times. The meaning and value of mathematical rules and definitions must be illustrated through practical, real world situations.
Method
Expected Outcomes
References
References Carroll, J. B. (1963). A model of school learning. Teachers College Record, 64(8), 723-733. de Lange, J. (2003). Mathematics for literacy. In B. L. Madison & L. A. Steen (Eds.), Quantitative literacy: Why numeracy matters for schools and colleges (pp. 75-89). Princeton, NJ: The National Council on Education and the Disciplines. de Lange, J. (1996). Using and Applying Mathematics in Education. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International Handbook of Mathematics Education, Part 1 (pp. 49-97). Dordrecht: Kluwer. Fuson, K. C., Kalchman, M., & Bransford, J. D. (2005). Mathematical Understanding: An Introduction. In M. S. Donovan & J. D. Bransford (Eds.), How students learn: history, mathematics, and science in the classroom (pp. 217-256). Washington, DC: Committee on How People Learn, A Targeted Report for Teachers, Division of Behavioral and Social Sciences and Education, National Academy Press. National Research Council. (2005). How students learn: history, mathematics, and science in the classroom. Washington, DC: Committee on How People Learn, A Targeted Rport for Teachers, Division of Behavioral and Social Sciences and Education, National Academy Press. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H. A., Wiley, D. E., Cogan, L. S., & Wolfe, R. G. (2001). Why Schools Matter: A Cross-National Comparison of Curriculum and Learning. San Francisco: Jossey-Bass.
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