Let's begin this discussion with a question. What is a task? I once asked a teacher what task students were going to do in class that day. The response was "No task. They are just going to complete a worksheet and then start on an investigation." Hmmmm..... aren't those tasks? My guess is that for some teachers a task is a rich math problem that stretches your mind. Like the one below:
For me a task is any activity given to students to do. That includes worksheets, quizzes, writings, discussions, etc. Suffice it to say -- tasks come in all shapes and sizes. That may be a little cliche but the emphasis is important. Teachers chose every day what tasks to give students. This choice is extremely important and one that bears deeper discussion.
"Effective mathematics teaching uses tasks as one way to motivate student learning and help build new mathematical knowledge through problem solving." (p.17)
What should a teacher consider when choosing a task? The decision is about the opportunities afforded students to make sense of problems and explore solution methods connected to the established goal of the lesson. (see practice 1) So what do I think about the task we used in the Leaping Lizard Lesson?
Using the taxonomy designed by Stein and Smith, I think that the Leaping Lizard task falls within the category of a higher-level demand. Students are encouraged "to engage in active inquiry and exploration" (PtA p. 19) I lean more towards procedures with connections as suggestions are provided to students for ways to look at the transformations (i.e. connect image and pre-image points). However, students are required to expend some cognitive effort to develop a deeper understanding of the definitions of the transformations.
While selection of the task is important so is its implementation. In an effort to help students that are struggling, a teacher can inadvertently GPS the task by "taking over the thinking" for students. This lowers the demand and effectiveness of the task. I've witnessed this happening with the Leaping Lizards task. The teacher told students upfront what they would discover and consequently students were no longer invested in doing the task. A teacher has to mindful to "support students without taking over their thinking" (p.24).
Knowing not to GPS and not doing it are two different things. It takes practice and intentional planning to change and prevent oneself from falling into that habit. I recommend the book 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay Stein. A number of years ago I was introduced to this book during a lesson study. It has been my guide ever since in implementing tasks in the classroom. We will revisit this again in part 4 of this series when we talk about student discourse.
Next time, the practice is use and connect mathematical representations and I will share one of my favorite tools when studying functions.