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Explicit versus inquiry#

See also: inquiry-learning, explicit-teaching

When applied to learning mathematics - Boaler & Selling (2017) - describe the two approaches as active engagement (inquiry) and passive engagement (explicit) in mathematics.

Findings/perspectives#

Boaler (1998) reports on case studies from two schools (which I've chosen to equate to the two extremes)

  • Traditional, textbook-based (explicit?) teaching - "students developed a procedural knowledge that was of limited used to them in unfamiliar situation" - with a significant literature available to support this.
  • Open-ended activities (inquiry) - "developed a conceptual understanding that provided them with advantages in a range of assessments and situations"

Boaler & Selling (2017) found that students from the open-ended activities school - as adultes - "move into significantly more professional jobs, despite living in one of the lowest income areas of the country". That the two different approaches led to very different relationships with mathematical knowledge as adults.

Fry and Hilman (2018) describe the growing prevalence of discourse around the effectiveness of explicit instructions from the mid-2000s and briefly summarises literature citing some of its limitations. The Boaler articles include mention of this with additional literature.

  • procedural lessons increase student disengagement
  • explicit teaching limits students' opportunities to exercise conceptual agency

A different approach: Non-curricular, scripted curricular, as-is curricular#

Liljedah (2020) cites research exploring the value of the following three types of lessons. Findings included:

  • Type 1 tasks got more students to think than Type 2 (both types designed to encourage thinking)
  • Type 3 tasks encouraged more successful completion, than type 2 (both designed around the same curriculum goals)
  • But even more students successfully completed when three (or sometimes more) lessons using type 1 tasks preceded a Type 2 lesson.
Lesson type Description
1. Non-curricular Highly engaging thinking tasks, card tricks, and numeracy tasks are used without concern for curriculum. Echoing: inquiry-learning, rich-mathematical-tasks etc
2. Scripted curricular tasks Curricular tasks are turned into problems, which are posed to students as building on previous work, but requiring additional learning. Echoing productive-failure
3. As-is curricular tasks Standard direct or explicit-teaching of routines directly linked to curriculum goals. An approach that tends to promote "mimicking, not thinking" (p. 26)

To follow up

  • Kerkhoff (2018) - 18 rich tasks, 18 classes covered almost all curricular outcomes

References#

Boaler, J. (1998). Open and Closed Mathematics: Student Experiences and Understandings. Journal for Research in Mathematics Education, 29(1), 41--62. https://doi.org/10.2307/749717

Boaler, J., & Selling, S. K. (2017). Psychological Imprisonment or Intellectual Freedom? A Longitudinal Study of Contrasting School Mathematics Approaches and Their Impact on Adults' Lives. Journal for Research in Mathematics Education, 48(1), 78--105. https://doi.org/10.5951/jresematheduc.48.1.0078

Fry, K., & Hillman, J. (2018). The Explicitness of Teaching in Guided Inquiry. In Mathematics Education Research Group of Australasia. Mathematics Education Research Group of Australasia. https://eric.ed.gov/?id=ED592484

Liljedahl, P. (2020). Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning. Corwin Press.