Low Floor, High Ceiling, Wide Walls#

Resnick and Silverman (2005) offer

The Logo programming language is often described as having a low floor and high ceiling: it is easy for novices to get started (low floor) and possible for experts to work on increasingly sophisticated projects (high ceiling). In our own work (especially in recent years), we have put less emphasis on high ceilings and more emphasis on what might be called “wide walls.” That is, we have tried to design technologies that support and suggest a wide range of different explorations. (p. 118)

Mathematics#

D'Ambrosio & Katsberg (2008) offer

In considering high expectations and worthwhile opportunities for all, Hiebert and his colleagues (1997) describe the importance of the tasks chosen for instruction in supporting all students. They describe tasks with multiple access points as those that allow students to engage in meaningful inquiry by drawing on whatever mathematical knowledge and experiences they know are useful in solving the task at hand. Tasks with multiple access points create opportunities for all students to engage in the construction of mathematical understanding. Tasks that limit students’ accessibility can result in a meaningless experience for many. Still, tasks must be problematic and deal with important mathematical ideas in order for the student to garner something meaningful from the assignment.

Low-threshold, high ceiling tasks#

Boaler and others reinforce the importance of including all. From De Geest & Lee (2019)

Other characteristics of ‘low-threshold, high-ceiling tasks’ are: they offer multiple methods, pathways and representations; inquiry opportunities are included; they involve asking the problem before teaching the method; they add a visual component; and they ask pupils to convince and reason and to be sceptical (Boaler, 2016, p. 90).

Low-floor, open-middle, high-ceiling tasks#

Liljedahl (2020, p. 23) offers the following definitions

Task type	Description
Low-floor task	a threshold that allows any and all learners to find a point of entry, or access, and then engage within their level of comfort
High-ceiling task	have ambiguity and/or room for extensions such that students can engage with the evolving complexity of the task
Open-middle	A problem structure where a task has a single final correct answer, but in which there are multiple possible correct ways to approach and solve the problem

References#

D'Ambrosio, B. S., & Kastberg, S. E. (2008). Strategies to Promote Equity in Mathematics Education. In Educating Everybody's Children: Diverse teaching strategies for diverse learners.

De Geest, E., & Lee, C. (2019). Promoting a positive learning environment. In C. Lee & R. Ward-Penny (Eds.), A Practical Guide to Teaching Mathematics in Secondary School. Taylor & Francis Group.

Making sense : teaching and learning mathematics with understanding, James Hiebert(Author)Mary Montgomery Lindquist(Writer of foreword.)

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

Resnick, M., & Silverman, B. (2005). Some reflections on designing construction kits for kids. Proceedings of the 2005 Conference on Interaction Design and Children, 117--122. https://doi.org/10.1145/1109540.1109556