We are charged with providing opportunities for our students to become self-regulated learners. We guide them to take control of their own learning by instilling certain habits of mind, including:
- thinking flexibly
- striving for greater accuracy and precision
- questioning and problem solving
- applying past knowledge to new situations
- thinking and communicating with clarity and precision
- accessing prior knowledge, transferring that knowledge
This description above (from the CISD Learning Framework) provides a challenge we choose to accept in our mathematics classrooms.
How do we do this in the mathematics classroom? What learning experiences do we design so that our students have opportunities to establish these habits of mind?
The best resource I can recommend for any work in building educator content knowledge of mathematics is Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, and Bay-Williams.
I am currently leading a book study with my Elementary Math Content Specialists over this book with monthly reflections that require evidence of understanding. For example, this month the tasks associated with Chapter 3 (Teaching Through Problem Solving) are:
- Based on what you have learned about problem solving in Chapter 3, identify a problem with multiple entry and exit points [educators post their problems on a group board so their peers can access them and provide feedback]. If you have an opportunity to utilize the problem task or activity with learners, capture a sample of student work and post it as well.
- Do you curate (gather from other sources) great problems? Where are your favorite places to go to find problems for your learners? Feel free to talk with other math educators on your campus to identify great resources for problems. Post one resource you utilize for great problems.
This book describes multiple entry and exit points.
Because your students will likely have a big range in “where they are” mathematically, it is important to use problems that have multiple entry points, meaning that the task has varying degrees of challenge within it or it can be approached in a variety of ways…Tasks should have multiple exit points, or various ways that students can demonstrate understanding of hte learning goals. (page 37)
A problem is defined here as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific “correct” solution method. (page 34)
Related to this is the description of low floor, high ceiling tasks for our learners. In his blog post, Ten Design Principles for Engaging Math Tasks, Dan Meyer describes these problems:
Set a low floor for entry, a high ceiling for exit. Write problems that require a simple first step but which stretch for miles. Consider asking students to evaluate a model for a simple case before generalizing. Once they’ve generalized, considered reversing the question and answer of the problem.
When exploring problems, check out youcubed.org.
At the top of the website, choose Ideas & Tasks, then Tasks. Then, under Grades, select Low Floor High Ceiling.
How powerful are intentionally designed/chosen low floor, high ceiling tasks for learners at any level of understanding? They are written to be accessible to all learners (even those not yet exhibiting understanding of certain, specific skills) and enriching to those ready to make connections and demonstrate understanding in more abstract, or elaborate ways.
Do you have any other resources you use to identify tasks with a low floor and high ceiling? I’d love to hear about it!