In my last post I wrote about some materials we can use to engage students in puzzling through problems, the Contexts for Learning Mathematics units.  These materials present a wonderful model for giving students the opportunity to think through the meaning of a problem.  Then they engage students in applying their prior knowledge to find a variety of solution paths.

We can really help our students grow as mathematicians if we provide similar opportunities throughout the curriculum.  By presenting students with scenarios, problems, or puzzles that are unfamiliar, but for which they have sufficient prior knowledge to figure out a solution path, we give them the opportunity to think mathematically in constructing their understanding.  By engaging students in discussions about these scenarios, problems, or puzzles, we encourage them to think together about what makes sense and how they can apply what they know.  We are asking them to do math rather than just follow a procedure that has been explained to them.

So how do we do this?  Here are examples using the variety of types of problems in the table of common addition and subtraction situations (click here) and the table of common multiplication and division situations (click here).   At some point in the year, primary students might be comfortable with “add to” and “take from” situations where the result is unknown, but unfamiliar with “take from” situations where the change or start is unknown, or “put together” situations where one addend or both addends are unknown.  We can engage students in making sense of these unfamiliar situations by asking them what they notice about the new situations and what they wonder.  Rather then showing them what procedure to use, we can empower them to think mathematically by asking them to discuss the situation.  We can help them learn to use strategies that help all of us make sense of problems and situations: retelling, acting out (physically, with manipulatives, or with pictures), and visualizing.

Similarly, in the middle grades, when students are comfortable with “unknown product” situations with equal groups, we can have them discuss and apply sense making strategies to figure out “unknown product” situations where we are comparing (ex. “A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?”).  Rather than showing them what procedure to use, by discussing, acting out, and visualizing comparison situations,  students can work together to think about the meaning of these unfamiliar situations.  That will help them apply their knowledge about multiplication and make sense of the connections between these types of problems.

Talking about what they are doing and thinking, helps students learn to use  sense making strategies.  As they agree and disagree with each other, they learn to think through what makes sense and try out different ideas on a regular basis.  This helps them develop a “growth mindset”.

We need to provide regular opportunities for students to puzzle through math problems and ideas.  These ideas are right there in our curriculum.  When we work on new content standards, we can provide problems or situations for which students  have prior applicable knowledge.  We can use questions and prompts that engage them in modeling ideas and thinking about relevant concepts and skills.   Then our students can do the thinking and talking that will help them figure out how to apply that prior knowledge.  Let’s engage our students in puzzling and wondering throughout the content standards!