The instruction that you describe here seems very much like productive struggle to me! The problem seems like the equation of productive struggle with students figuring things out “on their own.” I see this a lot with science instruction (my subject) and inquiry, where there is the assumption that inquiry = students just figuring things out themselves. What you’ve described here is, in my view, a great example of (guided) inquiry.
You know it's funny, other people might say that this lesson is a great example of explicit teaching. I'm not very interested in attaching a label to my teaching because it often leads to assumptions about what that label means.
This post feels like a lot of bluster over very little substance.
Productive struggle means that you ask students to do some thinking, and to work through moments when they are stuck. When students are stuck on a problem, do you tell them what to do so they can just write down what you say? Or do you give them a hint that helps them continue to work independently? That is the difference between creating dependent students and allowing for productive struggle.
Productive struggle is NOT the same as inquiry, though inquiry also requires productive struggle. Who is telling you that they are one and the same?
Also, you are correct to realize that when struggle is not productive, it can be a bad experience for kids. You describe times when you have students try to figure out a rule or procedure for themselves, and some students are discouraged. That is real! You should not let those students just struggle unproductively. Knowing who, when and how much to help, that is the very challenging art of teaching. It sounds like you do let students explore sometimes, which is great. Some topics lend themselves much more to this than other topics. Multiplying negatives does NOT lend itself nicely to a discovery method. Other ideas like, say, the effect of a scale factor on the area of a shape, do lend themselves better. And it sounds like you do more explorations when it is appropriate.
I don't think I'd agree with Jacob that the lesson you describe qualifies as guided inquiry. Maybe I'd call it "Engaging discussion to develop new ideas." But your lesson is a far cry from the teacher who walks in at the start of the class and tells the students that today, we are going to learn 6 new rules, lists them on the board (neg x pos = neg, neg x neg = pos, etc) and then has the students practice. And if a student is stuck on a problem, the teacher repeats the appropriate rule and shows how it applies here, giving the student the correct answer. The truth is that many, many teachers will do just that, and little more. Those are the teachers who people like me are trying to reform.
In your lesson, you encourage all students to make connections between multiplying integers and what they just learned about adding integers, as well as what they know about the meaning of multiplication. You give room for some students to jump ahead of you with their logical reasoning if they can, by laying it out in such a thoughtful way. At the same time, other students will just wait for you to make it clear for them, but they will still be asked to figure out some small pieces on their own when they do the practice, because the practice is thoughtfully curated to keep them productively struggling. Beautiful.
That's fair, and it would be helpful for me to give an example of what I'm trying to contrast this with. Next year I'll be teaching the Bridges curriculum (not my choice). Bridges often designs a lesson by giving students one big problem and asking students to see what they can figure out. I agree with you that there is a much more explanation-based version of teaching out there that is not very effective and is different from what I'm doing. But there is also another version, often called productive struggle, that in my view asks students to figure out too much all at once. That perspective has become embedded in a number of elementary and middle school math curricula that are gaining market share in the US. Which makes all this very tricky to talk about without specific examples of what some of the terms can mean.
The instruction that you describe here seems very much like productive struggle to me! The problem seems like the equation of productive struggle with students figuring things out “on their own.” I see this a lot with science instruction (my subject) and inquiry, where there is the assumption that inquiry = students just figuring things out themselves. What you’ve described here is, in my view, a great example of (guided) inquiry.
You know it's funny, other people might say that this lesson is a great example of explicit teaching. I'm not very interested in attaching a label to my teaching because it often leads to assumptions about what that label means.
This post feels like a lot of bluster over very little substance.
Productive struggle means that you ask students to do some thinking, and to work through moments when they are stuck. When students are stuck on a problem, do you tell them what to do so they can just write down what you say? Or do you give them a hint that helps them continue to work independently? That is the difference between creating dependent students and allowing for productive struggle.
Productive struggle is NOT the same as inquiry, though inquiry also requires productive struggle. Who is telling you that they are one and the same?
Also, you are correct to realize that when struggle is not productive, it can be a bad experience for kids. You describe times when you have students try to figure out a rule or procedure for themselves, and some students are discouraged. That is real! You should not let those students just struggle unproductively. Knowing who, when and how much to help, that is the very challenging art of teaching. It sounds like you do let students explore sometimes, which is great. Some topics lend themselves much more to this than other topics. Multiplying negatives does NOT lend itself nicely to a discovery method. Other ideas like, say, the effect of a scale factor on the area of a shape, do lend themselves better. And it sounds like you do more explorations when it is appropriate.
I don't think I'd agree with Jacob that the lesson you describe qualifies as guided inquiry. Maybe I'd call it "Engaging discussion to develop new ideas." But your lesson is a far cry from the teacher who walks in at the start of the class and tells the students that today, we are going to learn 6 new rules, lists them on the board (neg x pos = neg, neg x neg = pos, etc) and then has the students practice. And if a student is stuck on a problem, the teacher repeats the appropriate rule and shows how it applies here, giving the student the correct answer. The truth is that many, many teachers will do just that, and little more. Those are the teachers who people like me are trying to reform.
In your lesson, you encourage all students to make connections between multiplying integers and what they just learned about adding integers, as well as what they know about the meaning of multiplication. You give room for some students to jump ahead of you with their logical reasoning if they can, by laying it out in such a thoughtful way. At the same time, other students will just wait for you to make it clear for them, but they will still be asked to figure out some small pieces on their own when they do the practice, because the practice is thoughtfully curated to keep them productively struggling. Beautiful.
That's fair, and it would be helpful for me to give an example of what I'm trying to contrast this with. Next year I'll be teaching the Bridges curriculum (not my choice). Bridges often designs a lesson by giving students one big problem and asking students to see what they can figure out. I agree with you that there is a much more explanation-based version of teaching out there that is not very effective and is different from what I'm doing. But there is also another version, often called productive struggle, that in my view asks students to figure out too much all at once. That perspective has become embedded in a number of elementary and middle school math curricula that are gaining market share in the US. Which makes all this very tricky to talk about without specific examples of what some of the terms can mean.