A one-degree angle would certainly be difficult to draw accurately, especially without the aid of other tools like a protractor. However, mentally making a one-degree angle in visualized imagination is not only possible, but it is important for students' emergent understandings of angular measure. When students begin learning about length,…

Every educator knows the sinking feeling of a lesson gone wrong. As teachers look around the room and realize that many of their students are just not getting it, they often feel like failures. However, the struggle students experience as they persevere through high-quality challenging tasks is not a sign of failure, but rather a key aspect of…

Even before the release of the National Council of Teachers of Mathematics (NCTM)'s Principles to Actions (2014), teachers paid significant attention to mathematics tasks based on their positive influence on student learning (Doyle, 1988; Kazemi, 1998; NCTM, 1989, 2000). Over the last almost quarter century, we have learned a lot about high-level…

Providing students the autonomy and choice to learn when productive struggle becomes unproductive is a core teaching belief in the author's classroom. The author teaches in a multiage classroom where he has 16 first- and second-grade students, ages 6, 7, and 8. Like other educators, he believes in the importance of productive struggle. The author…

Probabilistic reasoning underpins much of middle school students' future work in data analysis and inferential statistics. Unfortunately for many middle school students, probabilistic reasoning is not intuitive. One specific area in which students seem to struggle is determining the probability of compound events (Moritz and Watson 2000). Research…

We share ideas for preparing for and enacting high-cognitive demand tasks in ways that support students in articulating and justifying their ideas. We offer strategies for developing and posing several types of purposeful questions: (1) eliciting thinking, (2) generating ideas, (3) clarifying explanations, and (4) justifying claims.

The authors who are mathematics teacher educators, have found that classifying, composing, and transforming shapes (in particular, rotations and reflections) are areas of difficulty for adults as well as for children. However, these are also some of the most important geometric ideas. They are fundamental topics in the K-8 Geometry and Measurement…