Has this ever happened to you?

*“My lecture went perfectly! I worked the sample problems. I could tell the students really understood the material.”*

The next day: *“I don’t know what went wrong! I thought all my students understood what I went over yesterday.”*

If we’ve been there, we don’t want to go back there. The teacher left something out of the equation when talking about the class – the students. There are a few key ideas we can take from this scenario to help us engage each and every student.

*“My lecture went perfectly!”*

*“My lecture went perfectly!”*

There is a place for direct instruction, but mathematics learning requires doing mathematics. *Your Mathematics Standards Companion, High School: What They Mean and How to Teach Them* has a set of suggestions about “What the teacher does” for each standard. Part of “What the teacher does” is having **good problems/tasks** prepared for the students to consider and having **good questions** related to the tasks that anticipate student thinking and that prod students along a productive path to deeper understanding of the mathematical learning goals. For example, instead of telling students what the definition of a function is, the teacher can provide samples of functions and nonfunctions and then challenge the students to determine what defines a function. When students have agency in their learning and are uncovering mathematics through exploring and discovering, they are engaged students.

*“I worked the sample problems.”*

*“I worked the sample problems.”*

Teachers need to use tasks that address the mathematics learning goals of the lesson. The tasks must have accessibility for each student. Sometimes, students need an entry into a problem. Stepping back and asking the students to state things they noticed about a problem is an open-ended, low-stress way to get students involved in thinking about a problem. Then students are invited to start reasoning about a task. Tasks can be of different types, but students should be encouraged to think about different solution paths and, as part of the whole class, about connections among and between the solution paths. All students are part of the discussion of solutions, and all students are learning from each other, so that each and every student is engaged in doing mathematics.

*“I could tell the students understood the material.”*

*“I could tell the students understood the material.”*

How are you gaining information about what the students understand? Having students do exit slips based on the learning goals for the day, writing what they learned from the lesson, asking one question about something they aren’t sure about are all ways to find out what students know and to engage the students in monitoring their own understanding. Eliciting information from students about their understanding engages them in the lesson.

Students can be engaged by making them a part of discovering and generalizing concepts, by being part of discourse about different solution paths based on problems and good tasks with accessibility for the diverse learners in the class, and by reflecting on their learning and sharing their understandings and questions. The focus of the classroom is moved from the teacher (lecture, doing examples, looking for cues) to the engaged students.