Building Thinking Classrooms in Mathematics

Traditional note taking is essentially writing what the teacher writes. Many students can’t listen and write at the same time and most students never use their notes. This is clearly a non-thinking activity. Many students don’t ever write notes if they can get away with it.
To make notes more meaningful, tell students to write notes to their future forgetful selves. The notes are by them and for them. This will help them consolidate their learning. The teacher circles back in three weeks with a task the requires the use of the notes. Students will also use notes when doing their checking for understanding questions.
This may not be easy at first as students are use to writing what the teacher writes, which is easier and doesn’t require thinking. Notes should contain worked examples, annotations, and perhaps graphic organizers. Peter gives four examples. Students can work in their groups to make notes. Don’t check notes as it is likely to cause students write something just to get it done.

Evaluate what you value. Much research shows that the competencies valued most by teachers everywhere are perseverance, willingness to take risks, and ability to collaborate. Peter shows a typical wordy four-column rubric. He finds that for K-1 students a two-column rubric works better while students in grades 2-12 can handle three-column rubrics with as little text as possible.
To promote ownership, involve your students in rubric construction. Due to time constraints Peter only uses the rubric to evaluate three groups. Students are also encouraged to use the rubric to evaluate there own work.

The main idea is to help students see where they are and where they are going. To do this the teacher constructs a navigation chart. There are four columns. The column on the left contains one of the subconcepts from the current unit. The next three columns are labeled basic, intermediate, and advanced. The questions on a review test or checking for understanding questions are associated with the appropriate box on this chart.
Students can use this information to self-access. They can use different symbols to indicate if they answered each question correctly or incorrectly with or without help or with group collaboration. They can also indicate silly mistakes and questions not done yet.

For each unit set up a grid with four columns. The left column contains the individual learning outcomes such as “add and subtract proper fractions.” The other three columns are labeled Basic, Intermediate, and Advanced. As the unit progresses you put data in each cell that indicates how the student is showing or not showing understanding. You will score all three columns for some outcomes while others only have scores in one or two columns.
If only the basic column applies that row is worth a maximum of two points. If only the basic and intermediate columns are scored the maximum for the row is three. If, however, the advanced column is scored the maximum score is four. Each student will get a score for each row depending on the understanding they demonstrate. The sum of all student scores are then compared to the sum of the maximum scores to generate a grade, which you can scale. This lets students know where they are at for each outcome, which is more likely to promote thinking than just seeing the score at the top of a test.
You use observations, conversations, and tests to gather data. In some cases you may not need to give a test. If you do give a test, give labeled basic, intermediate, and advanced questions for each outcome. If students know the answer to the advanced question, they need not do the others.

We start with when to implement each chapter. Chapters 1 through 3 come first all at the same time. This is where you give thinking tasks to random groups of students who are all standing in front of erasable vertical surfaces. Chapters 4 through 8 can come in any order and can come more than one at a time. Chapters 9 through 11 should come in order one at a time. The chapters on assessment (12-14) come one at a time in any order.
Peter finds that teachers in their second or third year of this program often implement 1 through 9 altogether followed by 10 through 14. Above all keep the forrest in mind which is getting students to think rather getting lost in the fourteen trees that make up the forrest.

Peter is professor of mathematics education at Simon Fraser University in Vancouver, Canada. He is the current president of the Canadian Mathematics Education Study Group (CMESG) and the editor of the International Journal of Science and Mathematics Education (IJSME). He serves on the editorial board of five other international journals and is a member of the NCTM Research Committee. He has authored many books, chapters, and articles. He is a former high school math teacher and has received multiple major awards for his work. You can follow him on X(Twitter) @pgliljedahl. His email is liljedahl@sfu.ca. His website is peterliljedahl.com.