On Target?
Cognitive Demand
This set of targets concerns the cognitive demand associated with classroom activities.
We ask three big questions:
Grappling with the mathematics. What opportunities do students have to grapple with and make their own sense of mathematical ideas in this lesson?
Challenges and productive struggle. What challenges do students experience with the tasks and activities? What happens when students experience challenges? How does struggle with mathematical ideas support their engagement and understanding?
Supporting engagement. In what ways does the environment support active engagement and sense making?
Begin an interactive "target practice" by clicking on a target ABOVE.
If students are given work that is too easy, there is little for them to learn – and, they are likely to be bored or frustrated. If students are given work that is too distant from their current understandings and they can see no pathways to progress, then there is no pathway to learning; they are likely to be bored or frustrated as well. As Stein and Smith (1998) put it, “Tasks that ask students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that require students to think conceptually and that stimulate students to make connections lead to a different set of opportunities for student thinking.” The challenge is to find tasks and classroom activities that are framed in ways that provide students with meaningful opportunities for learning and that support their growth through active engagement with the content. Researchers refer to this as a productive level of cognitive demand. Once classroom activities have been set in motion, the challenge is to maintain a productive level of cognitive demand – not to “give the game away” when students find things too difficult, but to ask clarifying questions and to offer hints and suggestions that still leave the bulk of sense making to the students.
There are multiple ways to think about establishing a classroom climate and activities that support students in sense making. At the most general level, the mathematics students encounter has to be experienced as coherent and connected – and something students can make sense of (see, e.g., Schoenfeld, 1990). The same body of mathematics can be experienced as a set of rules and procedures (in which case the main challenge is to memorize them), as a body of work that is explained by that still requires memorizing, or as a collection of ideas that students can make increasing sense of by dint of their own work. This latter approach establishes a context within which students can expect to build their own understandings (with help, of course).
Then, there is the question of whether the classroom norms and activities support student efforts at sense making. Is there a climate of safety, so that students can take risks?
To what degree...
… do students feel safe taking risks and being wrong without it reflecting on their intelligence?
… are students invested in each other’s success and genuinely encouraging of each other when things get tough?
… is struggle is seen as a positive indication that students are learning and stretching themselves? (see Dweck, 2007)
… are students unhampered by issues such as stereotype threat, so they do not worry that a mistake will be seen as reflecting badly on a whole demographic group, (e.g. African Americans, women)? (See Steele, 1995)
and, what can we do to open things up even more?
Similarly, there is the question of how we handle what happens when students get stuck. As always, what we do depends on our perceptions of students and what we think they are ready for – but the overarching question is, can we help orient the students to productive directions without making things too easy, and turning challenges into exercises?
Can we, increasingly...
… be quick to clarify instructions but slow to close down the mathematical openness?
… pose questions that might lead students to find things themselves, rather than being told? (e.g., “why don’t you try out your conjecture by testing a few cases?” or “what would happen to the endpoint of this line segment if the person was moving faster?)
… induce students to compare their work with other students who have obtained different results, so that the resulting conversations iron out differences in ways that are at the students’ mathematical level?
… point to useful problem solving strategies such as looking at related phenomena, easier related problems, special cases, etc.?
… frame our interventions in ways that help make the students more independent (e.g. helping them develop and use tools and perspectives that will serve them in the future, so that they have more personal resources and need to depend less on the teacher)?
Cognitive Demand
The extent to which students have opportunities to grapple with and make sense of important mathematical ideas and their use. Students learn best when they are challenged in ways that provide room and support for growth, with task difficulty ranging from moderate to demanding. The level of challenge should be conducive to what has been called “productive struggle.”