Conceptual Understanding, Procedural Fluency, and Application

My colleague Liz created this graphic which nicely summarizes our project’s position on the relationship between conceptual understanding, procedural fluency, and application.

 

The balance between conceptual understanding, procedural fluency, and application

A typical week of teaching

 

The balance between conceptual understanding, procedural fluency, and application depends on your goals with your students and those goals should depend on what you know about your students. This is why I’m opposed to “Problem Solving Fridays” and “Practice Tuesdays” as these ways of deciding on goals over-simplify teaching to the detriment of student understanding.

 


Ambitious Mathematics Curriculum

In an age where we can provide instantaneous access to high-quality encyclopedias and generate customized user-generated-playlists for a billion people on the fly, we should be able to provide curriculum with more ambitious goals and more customizable content than what is currently provided to teachers. The typical curriculum resources teachers have access to today are […]


Instructional Routines as Formative Assessment

This article was rejected for publication so I decided to rework it and release it on my blog instead.   Across the United States, there is a continued focus on the use of formative assessment to improve the conditions for students’ learning. One common theme is that teachers want more support in implementing formative assessment […]


Expectations

In 1965 a pair of researchers, Robert Rosenthal and Lenore Jacobson, set out to study the Pygmalion Effect, which hypothesizes that if we hold high expectations for people’s performance, their performance will be better than if we hold low expectations.   What they found was startling, especially for younger kids. Students who had been randomly selected […]


The future of CLIME

CLIME (The Council for Technology in Math Education) is an affiliate of NCTM with the mission to: Empower math communities to improve the teaching and learning of math through the use of dynamic tools in a Web 2.0 world Last night members of CLIME and other interested people attended a meeting of CLIME to discuss […]


Making Mathematical Ideas Explicit

If you or your students are going to talk about mathematical ideas in your class, it is critical that everyone understands the idea being discussed otherwise they are less likely to either remember it or be able to participate in the discussion. Every time you or your students make logical leaps when explaining mathematical ideas, […]


Why Instructional Routines?

In our project, we organized our work this past year around the use of instructional routines (née instructional activities) with teachers. Our curriculum work has been largely focused on instructional routines, our professional development activities have been focused on instructional routines, our school-based work in some cases has shifted to focus on supporting teams using instructional […]


Teaching Problems or Teaching Mathematics

On the day before I first started teaching, the district coordinator came to me and handed me a piece of paper with twenty questions on it. “Here’s what you have to teach, David. If your students can answer all twenty of these questions by the end of the year, you will be fine.” Needless to […]


Planning Lessons

When I first started planning lessons, each lesson took ages to plan. I don’t really remember exactly what I wrote except that usually the lessons were based on choosing example problems to go over, producing a worksheet for students to work on, and assigning homework questions. Eventually I finally had some textbooks for students and […]


What is Ambitious Teaching?

A recent analysis by David Blazar has stirred up some interest about Ambitious Teaching. But what exactly is Ambitious Teaching?   According to Elham Kazemi, Megan Franke, and Magdalene Lampert: “Ambitious teaching requires that teachers teach in response to what students do as they engage in problem solving performances, all while holding students accountable to […]