Brilliant breakdown of when inquiry actually works vs when it becomes performative. The "airtight" criteria totally changed how I think about task design,especially the bit about chunking cognitive load into 5-15 minute blocks. Had a similar experence with my AP stats class where students tried multiple approaches to confidence intervals before we formalized it. That struggle phase seems key but only if the payoff lands cleanly afterward.
Great post! I agree that being confident students can accurately complete the "intermediate steps" is key for an inquiry lesson. I've taught lots of inquiry lessons that fell apart because of accuracy issues. I also agree that inquiry needs to be highly structured to work and that's a common misconception.
I think another element at play re: sense-making is trust. Whether you're using explicit teaching or inquiry, if the teacher regularly asks students to do things that are well beyond their abilities or end up feeling confusing, they lose trust. If lessons are well-structured and students see that teaching reliably leads to learning, they gain trust. That trust is key for students to make sense of challenging mathematical ideas.
Yes you’re totally right that trust is another factor here and one that is built over time. I think so many educators and admin know a high-trust classroom when they see it, but it’s less clear how that trust gets developed. Often it’s from repeated experiences with success that depend on the teacher making decisions that are related to how well we understand what our students understand plus the many nuanced ways we engage students in that learning. There are lots of efforts to shortcut that trust-building but I’m not convinced it’s possible to build it any other way.
I teach using many practices form the Building Thinking Classrooms framework, and when I think about the lessons that land perfectly, it is because they are, as you have described, “airtight”. I need to design my tasks so that there is only one possible interpretation- this, I have found to be, quite challenging. I have one class that has solid mathematical instincts. I can get away with a task that is less than airtight. However, I have another class that will find any and all “air holes”. If there is a possible way to misinterpret it- they all will go in that direction. This article has given me a framework for thinking about whether a task is worth using- particularly with those who will ultimately not walk away with the correct interpretation.
You bring up an important point that an activity that is airtight for one group of students might not be for another group of students. Knowing our students’ formative data well is an important part of determining which tasks are appropriate. This is difficult work but I really believe it’s the right work.
This is a great lesson using inquiry! I also think Trigonometry should be introduced after the Geometry unit on similar triangles, not in Algebra 1 or 2. Inquiry is the way most science should be taught, but it is slower as well as deeper. In math, I would have inquiry here and there, but not the majority of lessons, as the content requirements could be compromised.
Brilliant breakdown of when inquiry actually works vs when it becomes performative. The "airtight" criteria totally changed how I think about task design,especially the bit about chunking cognitive load into 5-15 minute blocks. Had a similar experence with my AP stats class where students tried multiple approaches to confidence intervals before we formalized it. That struggle phase seems key but only if the payoff lands cleanly afterward.
Great post! I agree that being confident students can accurately complete the "intermediate steps" is key for an inquiry lesson. I've taught lots of inquiry lessons that fell apart because of accuracy issues. I also agree that inquiry needs to be highly structured to work and that's a common misconception.
I think another element at play re: sense-making is trust. Whether you're using explicit teaching or inquiry, if the teacher regularly asks students to do things that are well beyond their abilities or end up feeling confusing, they lose trust. If lessons are well-structured and students see that teaching reliably leads to learning, they gain trust. That trust is key for students to make sense of challenging mathematical ideas.
Yes you’re totally right that trust is another factor here and one that is built over time. I think so many educators and admin know a high-trust classroom when they see it, but it’s less clear how that trust gets developed. Often it’s from repeated experiences with success that depend on the teacher making decisions that are related to how well we understand what our students understand plus the many nuanced ways we engage students in that learning. There are lots of efforts to shortcut that trust-building but I’m not convinced it’s possible to build it any other way.
Maybe someone who thinks a lot about confidence in math class should write a blog on this topic... :)
I teach using many practices form the Building Thinking Classrooms framework, and when I think about the lessons that land perfectly, it is because they are, as you have described, “airtight”. I need to design my tasks so that there is only one possible interpretation- this, I have found to be, quite challenging. I have one class that has solid mathematical instincts. I can get away with a task that is less than airtight. However, I have another class that will find any and all “air holes”. If there is a possible way to misinterpret it- they all will go in that direction. This article has given me a framework for thinking about whether a task is worth using- particularly with those who will ultimately not walk away with the correct interpretation.
You bring up an important point that an activity that is airtight for one group of students might not be for another group of students. Knowing our students’ formative data well is an important part of determining which tasks are appropriate. This is difficult work but I really believe it’s the right work.
This is a great lesson using inquiry! I also think Trigonometry should be introduced after the Geometry unit on similar triangles, not in Algebra 1 or 2. Inquiry is the way most science should be taught, but it is slower as well as deeper. In math, I would have inquiry here and there, but not the majority of lessons, as the content requirements could be compromised.