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, too, grow concerned around the "efficiency" side of things. Most thorough, long-term learning typically involves trial-and-error by a fairly self-motivated learner. A process most would call time intensive and inefficient...but is it really? I think we make a mistake when we don't allow students to explore a solution space and really understand what is going on. I would also add that many of today's best "computational solvers" working in multiple dimensions do a great deal more "guess and check" and exploration of geometric spaces over set algorithm application. These are skills we should foster in today's students so our future can be expanded by our creative math thinkers the contemporary classroom has fostered. AI improved the matrix multiplication algorithm......why didn't it just learn our "efficient one" and use that?
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?
I didn't see your post until I submitted a comment suggesting "Let's try it" instead of "Just try it." I was purely going off the sentiment, without knowing that Beth had published anything about the use of that word in classroom discourse! I'm going to check that out.
This might be dumb, but can I suggest a "Let's try it" instead of a "Just try it" classroom? I appreciate the contrast to the 'Just Do It' phrase, but there feels like there is something significant to "Let's" vs. "Just" that might be more than pedantic. "Let's try it" gives the sensation that we are a team figuring this out together, where "Just try it" feels like "I know already but go ahead and play around you silly mortals." The connotation distinction between them makes the latter feel more like what you are describing in the post (and in your other work).
In the world of "necessary and sufficient" we overlook the need to provide "levels of learning" so that students can build on the scaffolding they currently have. I have 3 boys, all are good at math, all in different ways. I do not think most children could start understanding 3D space before they start with a number line but I have one that sees the world this way. By offering an opportunity that is guided and grounded in appropriate leveling through scaffolded assignments, we can start on a pathway to reach every learner. These Desmos activities allow a constrained level of trial and error that can help many learners and for those that choose to stay with math, an opportunity to understand many dimensions-either now or in the future. I say we prepare them so they can "be the change" instead of "prepare..for the changes". Our children are the future not passengers in the future.
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, too, grow concerned around the "efficiency" side of things. Most thorough, long-term learning typically involves trial-and-error by a fairly self-motivated learner. A process most would call time intensive and inefficient...but is it really? I think we make a mistake when we don't allow students to explore a solution space and really understand what is going on. I would also add that many of today's best "computational solvers" working in multiple dimensions do a great deal more "guess and check" and exploration of geometric spaces over set algorithm application. These are skills we should foster in today's students so our future can be expanded by our creative math thinkers the contemporary classroom has fostered. AI improved the matrix multiplication algorithm......why didn't it just learn our "efficient one" and use that?
https://www.quantamagazine.org/ai-reveals-new-possibilities-in-matrix-multiplication-20221123/
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
I didn't see your post until I submitted a comment suggesting "Let's try it" instead of "Just try it." I was purely going off the sentiment, without knowing that Beth had published anything about the use of that word in classroom discourse! I'm going to check that out.
This might be dumb, but can I suggest a "Let's try it" instead of a "Just try it" classroom? I appreciate the contrast to the 'Just Do It' phrase, but there feels like there is something significant to "Let's" vs. "Just" that might be more than pedantic. "Let's try it" gives the sensation that we are a team figuring this out together, where "Just try it" feels like "I know already but go ahead and play around you silly mortals." The connotation distinction between them makes the latter feel more like what you are describing in the post (and in your other work).
You and Dan Finkel are totally tracking!
https://mathforlove.com/2024/10/creating-lets-try-it-classrooms/
That's crazy how similar that was- thank you for sharing!
In the world of "necessary and sufficient" we overlook the need to provide "levels of learning" so that students can build on the scaffolding they currently have. I have 3 boys, all are good at math, all in different ways. I do not think most children could start understanding 3D space before they start with a number line but I have one that sees the world this way. By offering an opportunity that is guided and grounded in appropriate leveling through scaffolded assignments, we can start on a pathway to reach every learner. These Desmos activities allow a constrained level of trial and error that can help many learners and for those that choose to stay with math, an opportunity to understand many dimensions-either now or in the future. I say we prepare them so they can "be the change" instead of "prepare..for the changes". Our children are the future not passengers in the future.