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.
There is a note in the Progressions for the CCSS (EE 6-8, Revised 2023) on page 177 that states "Because students aren’t expected to do arithmetic with negative numbers until Grade 7, equations in standard 6_EE_7 are restricted to positive p and q. However, students in Grade 6 might solve equations of the form x – p = q where p and q are positive."
I found myself thinking about how many elementary students (and certainly 6th graders) think about subtraction as removal by default. So the problems you posed (8 – 3, 10 – 1) really don't activate anything useful for solving x – 1 = 9. I don't mean that as a critique, just as an observation. It's probably the students that have a more dense network of number relations under their belt that have more immediate success here. If I know all the partitions of 10, then I'd immediately recognize 1 and 9 one of those partitions. So for students with that "structuring" knowledge it doesn't depend on their knowledge of addition and subtraction.
I wonder if something like "list all the equations you can that use 3, 5, and 8" would be a helpful setup. I want to activate the idea that addition and subtraction are two sides of the same coin and I can use one to solve the other.
More to the point of your post though, it's easy to talk at students for long periods of time and it's also easy to provide students tasks to somehow "figure out". It's much harder to plan exactly what needs to be made explicit, visible, and directly explained and what aspects of a lesson benefit from inquiry and student exploration.
Interesting that's in the progressions, though I find the vagueness of the phrase "students...might solve" pretty frustrating. Should they do it or not? I would argue yes but it's ambiguous.
Your point about partitions of 10 is really interesting to me. I lean on those often when I'm teaching a tricky skill for the first time. For instance, when I'm introducing a problem like -4 - 6 where students are subtracting from a negative I lean on a lot of partitions of 10, or very small numbers like -1 - 3 or easy numbers like -10 - 5. But there might be something that trips students up where they can do those problems because of their comfort with those specific numbers but it doesn't transfer when the numbers get trickier. Good food for thought.
I like the way you say "dense network of number relations." There's a big difference between a student who knows 5 + 3 = 8 and a student who knows 5 + 3 = 8 and 3 + 5 = 8 and 8 - 5 = 3 and __ + 3 = 8 and ___ - 3 = 5 and sees the connections between all of those. That "dense network" is a huge part of how students understand the inverse operations side of equation solving, it's not obvious to many students that to solve x + 3 = 8 one can subtract 3 from 8, but if you have that dense network it makes sense and fits in nicely.
The most unfortunate mixture of approaches to teaching vs. readiness to learn happens when teachers of students with uneven prior knowledge/foundational skills think that their students aren't paying attention because the class isn't "interesting" enough... and they completely abandon all "lectures" and go full throttle on having students "discover" math.
As you mentioned, students who are learning a new concept or who have a shaky understanding of a particular concept need clarity and to be guided to a place of firm understanding before they can explore on their own. Otherwise, in the best of circumstances, they're just doing random "hands-on" activities and having a good time... and then the teacher has to re-explain everything again anyways when the problem becomes more abstract. In the worst of circumstances, the teacher is frustrated because they feel that their students always just "want the answer" without "putting in the effort" to learn on their own... and conversely their students feel frustrated that their teacher "doesn't teach anything." (Real conflict I faced as a HOD.)
Yea it's a tricky kind of second-order thinking. Reasonable people think "students don't seem interested in this, so I should try to make it more interesting." But the cause of disinterest isn't how interesting the topic is, it's deeper in student skills and feelings of success that are under the surface.
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.
There is a note in the Progressions for the CCSS (EE 6-8, Revised 2023) on page 177 that states "Because students aren’t expected to do arithmetic with negative numbers until Grade 7, equations in standard 6_EE_7 are restricted to positive p and q. However, students in Grade 6 might solve equations of the form x – p = q where p and q are positive."
I found myself thinking about how many elementary students (and certainly 6th graders) think about subtraction as removal by default. So the problems you posed (8 – 3, 10 – 1) really don't activate anything useful for solving x – 1 = 9. I don't mean that as a critique, just as an observation. It's probably the students that have a more dense network of number relations under their belt that have more immediate success here. If I know all the partitions of 10, then I'd immediately recognize 1 and 9 one of those partitions. So for students with that "structuring" knowledge it doesn't depend on their knowledge of addition and subtraction.
I wonder if something like "list all the equations you can that use 3, 5, and 8" would be a helpful setup. I want to activate the idea that addition and subtraction are two sides of the same coin and I can use one to solve the other.
More to the point of your post though, it's easy to talk at students for long periods of time and it's also easy to provide students tasks to somehow "figure out". It's much harder to plan exactly what needs to be made explicit, visible, and directly explained and what aspects of a lesson benefit from inquiry and student exploration.
Interesting that's in the progressions, though I find the vagueness of the phrase "students...might solve" pretty frustrating. Should they do it or not? I would argue yes but it's ambiguous.
Your point about partitions of 10 is really interesting to me. I lean on those often when I'm teaching a tricky skill for the first time. For instance, when I'm introducing a problem like -4 - 6 where students are subtracting from a negative I lean on a lot of partitions of 10, or very small numbers like -1 - 3 or easy numbers like -10 - 5. But there might be something that trips students up where they can do those problems because of their comfort with those specific numbers but it doesn't transfer when the numbers get trickier. Good food for thought.
I like the way you say "dense network of number relations." There's a big difference between a student who knows 5 + 3 = 8 and a student who knows 5 + 3 = 8 and 3 + 5 = 8 and 8 - 5 = 3 and __ + 3 = 8 and ___ - 3 = 5 and sees the connections between all of those. That "dense network" is a huge part of how students understand the inverse operations side of equation solving, it's not obvious to many students that to solve x + 3 = 8 one can subtract 3 from 8, but if you have that dense network it makes sense and fits in nicely.
The most unfortunate mixture of approaches to teaching vs. readiness to learn happens when teachers of students with uneven prior knowledge/foundational skills think that their students aren't paying attention because the class isn't "interesting" enough... and they completely abandon all "lectures" and go full throttle on having students "discover" math.
As you mentioned, students who are learning a new concept or who have a shaky understanding of a particular concept need clarity and to be guided to a place of firm understanding before they can explore on their own. Otherwise, in the best of circumstances, they're just doing random "hands-on" activities and having a good time... and then the teacher has to re-explain everything again anyways when the problem becomes more abstract. In the worst of circumstances, the teacher is frustrated because they feel that their students always just "want the answer" without "putting in the effort" to learn on their own... and conversely their students feel frustrated that their teacher "doesn't teach anything." (Real conflict I faced as a HOD.)
Yea it's a tricky kind of second-order thinking. Reasonable people think "students don't seem interested in this, so I should try to make it more interesting." But the cause of disinterest isn't how interesting the topic is, it's deeper in student skills and feelings of success that are under the surface.