Here is a short video of some great teacher explanation.
In my view, the most effective teaching practice is one that strikes a balance between teacher control and student thinking, and adapts to the needs and strengths of individual students. This requires teachers to be responsive to student thinking, anticipate misconceptions, and provide targeted support and feedback. Ultimately, effective teaching is not just about imparting knowledge, but about facilitating learning and helping students develop the skills and confidence they need to succeed.
In Heibert, et al.'s book, "Making Sense: Teaching & Learning Mathematics with Understanding" (1997, Heinemann) there is a wonderful quote that has guided me about this tension in instruction. It's called 'The Dilemma.' "How to assist students in experiencing and acquiring mathematically powerful ideas but refrain from assisting so much that students abandon their own sense-making skills in favor of following the teacher's directions." (p 29) The authors then go on to say, "rather than resolving the dilemma by choosing one of the options, and rather than resisting or ignoring the dilemma, teachers should embrace it. By remaining open to the tension, teachers can remain sensitive to both the subject and the students... we believe it is possible for teachers to intervene in ways that stimulate and push students' thinking forward and, at the same time, promote students' autonomy... [by creating] an environment in which students reflect on and communicate about mathematics" (p.30)
This conversation reminds me of this distinction made by Parker James Palmer in his book, "The Courage to Teach":
This is a huge topic of discussion in math education right now, often with quite polarized views. I think this post is a very helpful perspective and addition to the conversation.
Love the video and agree that identifying students' strategies positions students as knowledge producers leading to positive math identities.
BTW, while Janet's observation about the T's lack of precision is true, I do not think the students would be confused and lose confidence. In a classroom like this one, students are not mimicing the teacher & are more likely to develop understanding. No one is perfect.
Thanks for sharing!
In the video the teacher did a great job listening and responding to the students' comments, but I couldn't get past the fact that she claimed 'dividing by a third is the same as multiplying by a third'. NO NO NO! I know she meant 'DIVIDING BY THREE is the same as multiplying by a third" but this kind of slip up is what makes students confused and lose their math confidence.