go Deeper

Prior to the meeting:

The group selects a tape segment that offers rich possibilities for discussion about Cognitive Demand. The teacher(s) whose tapes are watched has veto power over framing and direction.

At the meeting:

The teacher (or a team member) explains context and what he or she was trying to achieve. We should hand out copies of the Dimension 2 observation & conversation guide pages as resources for reflection, and possibly point to other tools. The main handouts are the observation/discussion questions in slide 15. Stress that the main goal of this conversation is to think about the ways students are and can be supported to engage and persevere in productive struggle, individually and collectively.

Dimension 2 Cognitive Demand Strategies

Ask for patterns and outliers

To give students something more to think about while they practice similar problems or do problem sets, ask them to describe any mathematical patterns they noticed while working on similar problems. Or, ask them to find problems that seemed like outliers—ones for which the method didn’t work as expected, or with a solution that surprised or confused them. This can be given in addition to a regular homework assignment. Have a few students share their patterns and outliers in class, or have students discuss them in pairs.

Always, Sometimes, Never

Pose questions that require explanation and are not straight yes/no or one answer solutions. For example, instead of asking, “Is it true or false that when you multiply two numbers, the answer will always be bigger?” ask, “When you multiply two numbers, the answer will be bigger. Is this statement always, sometimes or never true. Explain your reasoning.” This gives students opportunities to explore more possibilities, such as positive / negative numbers and fractions.

Invent an argument

Come up with some fictional characters who disagree about a math idea you want to teach and ask students how they would resolve the argument. Make sure students also discuss why each side thinks what they do, even if the characters’ arguments aren’t correct.

Make categories

After a long problem set or homework assignment filled with similar problems, ask students to group the problems into categories. They can choose the categories they like, but encourage them to be mathematical. For example, after a homework assignment of solving systems of equations, students might choose to categorize the problems according to those that have solutions and those that do not; those that have solutions in different quadrants when graphed; those that were best solved with different methods; etc. Let students be creative! This gives students opportunities to see the bigger picture after doing detailed problems.

Open it up

Reframe a problem or problem set so that there’s more for students to discover. For example, when doing proofs with students, instead of asking them to “show” something, such as that the diagonals of a rectangle bisect each other (which assumes that it’s already known to be true), pose the thing to be shown as something to discover. Ask, “What do you notice about the diagonals of different quadrilaterals?” and give students time to draw, find patterns, and make claims about what they see.

Share Wrong Answers Like They’re Right Answers

Share an incorrect solution to a problem, saying “another student solved the problem this way. Identify similarities and differences between this solution and the solution you have. What makes sense to you? What questions do you have?” Encourage students to share their thinking in a whole class discussion or in small groups about their own solution to the problem as it relates to the incorrect solution the teacher provides. If possible, let the students come to the decision that the provided solution is incorrect and encourage them to justify their thinking.

Three Things

If students are stuck on a problem, ask them to state three things they know about the problem and three things they are wondering about. This can help students realize that they are making progress and have information that they can potentially draw on to make more progress.

Time alone

Give students some time (maybe 5 minutes) to work alone on a problem before working in a group, so that they have time to think and prepare their ideas. You may want to structure this alone time by setting expectations as to what students will share with their groups after 5 minutes. For example, tell students, “Be prepared to share one idea that you have for getting started on this problem and one question that you have about the problem.” or “Be prepared to share one thing that you know about this problem and one thing that you don’t know about this problem.”