Yes, one thing I've always liked about the Desmos Classroom activities I've used (and that I try to do in the Desmos activities I write myself) is that you show consequences rather than green checkmarks and red x's. Here's a line, what's the equation? Student enters an equation, and it graphs the thing, so they can see for themselves whether their equation matches the line. If it doesn't it's not a BAD equation, it's just not the particular equation we're looking for at this moment. Set it aside, maybe we'll find a use for it later.
"Just try it" is the same energy I have in mind when I ask folks to just draft out their ideas. Make a rough draft. Just draft it out. So this resonated with me. Let's just see what happens, you know?
This will not be enough to prepare them for what we are facing. We need kids to be using all of their intelligence. The classroom is set up as a tool for somatic oppression. They need to be learning in 3D about the world they live in.
We need radical change to even "sort of" prepare them for the change that is already here. We are facing more than just societal collapse. We are facing something humanity has never before had to experience. There is nothing in 2D books, or constructed learning environments that will prepare them for the changes we are facing. We really, really, need to focus on reversing the oppression of their somatic intelligence. Because it costs them their life saving creative intelligence.
What this describes is a the equivalent of a band-aid being applied to a decapitation. This may seem insensitive but in a few years you will see I am understating the severity of the situation.
I also wonder about the premium that traditional math instruction places on efficiency, what's the most efficient path from problem to solution? Like we're all sitting at our desks solving equations all day, and the boss is going to write us up if we don't hit our quota. And if a student solves an equation through guess-n-check (or, heaven forbid, GRAPHING) do they understand less than the student who solves it algorithmically? If it's an application question, I'd think the student who graphs a rising function and finds where it crashes through the horizontal line might actually have a better understanding than the student who's correctly following procedure without know what makes it the correct procedure.
I don't think of traditional math being equated to algorithms only. A premium on efficiency means applying the most efficient method to the right question. Depending on the question, that can mean graphing, guess-and-check or an algorithmic method. It's a mathematical skill to recognise which method will most efficiently solve a particular question. For example, "5x<15" can be solved graphically or by guess-and-check but a good math student recognises that these methods are inefficient and not worth their time. The best way to convince a student of this is to just tell them. They can certainly be guided through one of the inefficient methods but the "valuable learning" from this (short) activity should be limited to not much more than "this is a waste of your time"
It's not really a question of how much the students know, but how much I know about what they know. To take your example, a student could solve 5x < 15 by dividing both sides by 5, but what do they really understand about this question, and are they just following a procedure because they've been told that's how you solve it? Compare that to a student who graphs y = 5x and then graphs y = 15 and then identifies the intersection point as significant, as a boundary of the solution set. Their thinking is much more visible to me. Does it matter that the second approach would take considerably more time? I don't see why, unless we think it's important to solve many of these in a short time.
Depends what you are trying to get out of the question. If the student genuinely thinks that the graphical way is the most efficient way to solve 5x<15 then I would argue that the student doesn't know as much about this question as you might think. If you want to say that the student is demonstrating some sort of ethereal superior understanding of the problem by visually demonstrating how to solve it graphically but not knowing that there is a far more efficient algebraic way to solve the problem, then I would say the burden of proof is on you to make that case.
Yes, one thing I've always liked about the Desmos Classroom activities I've used (and that I try to do in the Desmos activities I write myself) is that you show consequences rather than green checkmarks and red x's. Here's a line, what's the equation? Student enters an equation, and it graphs the thing, so they can see for themselves whether their equation matches the line. If it doesn't it's not a BAD equation, it's just not the particular equation we're looking for at this moment. Set it aside, maybe we'll find a use for it later.
"Just try it" is the same energy I have in mind when I ask folks to just draft out their ideas. Make a rough draft. Just draft it out. So this resonated with me. Let's just see what happens, you know?
The word "just" can be a bit problematic in math classrooms in some cases, but I don't think this case is that kind of problematic. See Wagner & Herbel-Eisenmann (2007): https://link.springer.com/article/10.1007/s10649-007-9097-x
This will not be enough to prepare them for what we are facing. We need kids to be using all of their intelligence. The classroom is set up as a tool for somatic oppression. They need to be learning in 3D about the world they live in.
We need radical change to even "sort of" prepare them for the change that is already here. We are facing more than just societal collapse. We are facing something humanity has never before had to experience. There is nothing in 2D books, or constructed learning environments that will prepare them for the changes we are facing. We really, really, need to focus on reversing the oppression of their somatic intelligence. Because it costs them their life saving creative intelligence.
What this describes is a the equivalent of a band-aid being applied to a decapitation. This may seem insensitive but in a few years you will see I am understating the severity of the situation.
I also wonder about the premium that traditional math instruction places on efficiency, what's the most efficient path from problem to solution? Like we're all sitting at our desks solving equations all day, and the boss is going to write us up if we don't hit our quota. And if a student solves an equation through guess-n-check (or, heaven forbid, GRAPHING) do they understand less than the student who solves it algorithmically? If it's an application question, I'd think the student who graphs a rising function and finds where it crashes through the horizontal line might actually have a better understanding than the student who's correctly following procedure without know what makes it the correct procedure.
I don't think of traditional math being equated to algorithms only. A premium on efficiency means applying the most efficient method to the right question. Depending on the question, that can mean graphing, guess-and-check or an algorithmic method. It's a mathematical skill to recognise which method will most efficiently solve a particular question. For example, "5x<15" can be solved graphically or by guess-and-check but a good math student recognises that these methods are inefficient and not worth their time. The best way to convince a student of this is to just tell them. They can certainly be guided through one of the inefficient methods but the "valuable learning" from this (short) activity should be limited to not much more than "this is a waste of your time"
It's not really a question of how much the students know, but how much I know about what they know. To take your example, a student could solve 5x < 15 by dividing both sides by 5, but what do they really understand about this question, and are they just following a procedure because they've been told that's how you solve it? Compare that to a student who graphs y = 5x and then graphs y = 15 and then identifies the intersection point as significant, as a boundary of the solution set. Their thinking is much more visible to me. Does it matter that the second approach would take considerably more time? I don't see why, unless we think it's important to solve many of these in a short time.
Depends what you are trying to get out of the question. If the student genuinely thinks that the graphical way is the most efficient way to solve 5x<15 then I would argue that the student doesn't know as much about this question as you might think. If you want to say that the student is demonstrating some sort of ethereal superior understanding of the problem by visually demonstrating how to solve it graphically but not knowing that there is a far more efficient algebraic way to solve the problem, then I would say the burden of proof is on you to make that case.