Subtraction isn't generally an operation but rather a shorthand notation. Really, x -1 = 8 is just x + (-1) = 8, and then to solve, we use the additive inverse and zero properties.
But the issue, which you are calling out, is that subtraction is treated as an operation, and students are supposed to apply these concepts, but really don't deal with them in a meaningful way leading up to 7th grade.
I really enjoy your posts as they are remind me of how I taught and thought about teaching math (now in admin). If you don't use "direct instruction" at times, students will not progress. A teacher is a mentor for learning. Direct instruction is one technique to show students how mathematicians think. It also helps guide students to see patterns that they will never discover on their own (mainly due to disinterest). In fact, it actually creates more engaging classrooms when students have tools to discover rather than a pure constructivist classroom.
I agree with you about subtraction from a strict mathematical perspective but I've never found that to be very helpful for teaching. We want students to understand a lot of things about inverse operations before they understand negative numbers so we need the concept of subtraction. Maybe there's a world where we redesign the curriculum to avoid that but I have trouble imagining what it would look like. Right now subtraction is a forgotten little sibling at times, which can make a mess of things for some kids who miss stuff and fall through the cracks.
Subtraction isn't generally an operation but rather a shorthand notation. Really, x -1 = 8 is just x + (-1) = 8, and then to solve, we use the additive inverse and zero properties.
But the issue, which you are calling out, is that subtraction is treated as an operation, and students are supposed to apply these concepts, but really don't deal with them in a meaningful way leading up to 7th grade.
I really enjoy your posts as they are remind me of how I taught and thought about teaching math (now in admin). If you don't use "direct instruction" at times, students will not progress. A teacher is a mentor for learning. Direct instruction is one technique to show students how mathematicians think. It also helps guide students to see patterns that they will never discover on their own (mainly due to disinterest). In fact, it actually creates more engaging classrooms when students have tools to discover rather than a pure constructivist classroom.
I agree with you about subtraction from a strict mathematical perspective but I've never found that to be very helpful for teaching. We want students to understand a lot of things about inverse operations before they understand negative numbers so we need the concept of subtraction. Maybe there's a world where we redesign the curriculum to avoid that but I have trouble imagining what it would look like. Right now subtraction is a forgotten little sibling at times, which can make a mess of things for some kids who miss stuff and fall through the cracks.