The fact that Desmos is listed as a "study tool" doesn't do it justice. Desmos is the "conjecture-testing, judgement-free, learning-math-by-playing" tool.

Very interesting article Dan. I loved playing sports because it was fun to be with friends and have a common goal. I think that this could also be applied to math by having students work in groups as they explore math concepts. When I was a student in high school, I mostly did math by myself. In college, I've had classes where I work in my classmates and it's been fun to see how people approach problems in different ways.

Hi Jack, I had the same thought! I always loved playing soccer because of the comradery of my team. I enjoyed working socializing, and my teammates motivated me to work hard. I am currently a college student, and throughout my time being a math major in college, I have found it difficult to get this sense of fun/common goal back. It took multiple classes of being on my own to finally seeing familiar faces and getting a study group together. I think If math classes were seen as more of a collaboration then many students would feel more comfortable having math discussions with peers and forming these "teams".

This also emphasizes to me the importance of physical intuition/visualization when first approaching or creating a problem/challenge. Even if the physical limitations are not “real world”, like in games like Portal, Fortnite, Civilization, etc., the player/problem-solver usually has a general idea of what to try and how to explore - quite clearly gaining information about if their attempts and explorations led to their intended outcome. The same goes for sports and esports - even when the rules and physics are arbitrary. For me, this freedom of exploration is what sets apart a good (re: fun) tutorial stage of a game from a bad one. Did they just tell you to “hit this button” and then “run over here”, or was it presented as some sort of challenge, where you learn how to play the game through exploration and naturally interacting with the game environment.

Contrast this with a typical traditional lesson, where many players often have no idea how to even begin to explore, what exploration would even look like, or even if they reached their goal/solution/equation. These lessons typically exist only in the realm of a weird written language and rule-following.

This is one thing that I think the best Desmos lessons do really well, and plays a main role in making these lessons effective (mainly thinking of Function Carnival, Marcellus the Giant, Transformation Golf, etc.).

This is so true of how math has been presented traditionally, and many online platforms are not changing much. I made this video where I use RPA software to automate completion of an IXL to show that there's not much thinking involved in some of these activities: https://youtu.be/fsdo5SmX26I

We should strive to have the vast majority of our curriculum be unable to be automated. We should let students be human in math.

I think what makes the definition debate interesting (although I agree definitions are not the most interesting part) is that changing your definitions in math change what kind of math you can do. (Like with spherical/hyperbolic geometry.) I don't know that a debate over a definition is valuable, as by-definiton (pun intended) it cannot be resolved, but it is useful to analyze why a definition is chosen and what purpose it solves, what it allows and what it doesn't.

100% in favor of any pedagogy or curriculum that asks students to poke at definitions and develop new ways of thinking mathematically. (Spherical geometry! Polygons with non-integer side lengths! Etc!)

I just don't really see this particular debate contributing to that intellectual adventurousness. It just seems to elevate personal preference to the level of religious dogma in an area where the core religious texts are ambiguous.

The fact that Desmos is listed as a "study tool" doesn't do it justice. Desmos is the "conjecture-testing, judgement-free, learning-math-by-playing" tool.

Very interesting article Dan. I loved playing sports because it was fun to be with friends and have a common goal. I think that this could also be applied to math by having students work in groups as they explore math concepts. When I was a student in high school, I mostly did math by myself. In college, I've had classes where I work in my classmates and it's been fun to see how people approach problems in different ways.

Hi Jack, I had the same thought! I always loved playing soccer because of the comradery of my team. I enjoyed working socializing, and my teammates motivated me to work hard. I am currently a college student, and throughout my time being a math major in college, I have found it difficult to get this sense of fun/common goal back. It took multiple classes of being on my own to finally seeing familiar faces and getting a study group together. I think If math classes were seen as more of a collaboration then many students would feel more comfortable having math discussions with peers and forming these "teams".

This also emphasizes to me the importance of physical intuition/visualization when first approaching or creating a problem/challenge. Even if the physical limitations are not “real world”, like in games like Portal, Fortnite, Civilization, etc., the player/problem-solver usually has a general idea of what to try and how to explore - quite clearly gaining information about if their attempts and explorations led to their intended outcome. The same goes for sports and esports - even when the rules and physics are arbitrary. For me, this freedom of exploration is what sets apart a good (re: fun) tutorial stage of a game from a bad one. Did they just tell you to “hit this button” and then “run over here”, or was it presented as some sort of challenge, where you learn how to play the game through exploration and naturally interacting with the game environment.

Contrast this with a typical traditional lesson, where many players often have no idea how to even begin to explore, what exploration would even look like, or even if they reached their goal/solution/equation. These lessons typically exist only in the realm of a weird written language and rule-following.

This is one thing that I think the best Desmos lessons do really well, and plays a main role in making these lessons effective (mainly thinking of Function Carnival, Marcellus the Giant, Transformation Golf, etc.).

This is so true of how math has been presented traditionally, and many online platforms are not changing much. I made this video where I use RPA software to automate completion of an IXL to show that there's not much thinking involved in some of these activities: https://youtu.be/fsdo5SmX26I

We should strive to have the vast majority of our curriculum be unable to be automated. We should let students be human in math.

Great video. Just added to the main post.

Wow, thanks! That means a lot coming from you.

I think what makes the definition debate interesting (although I agree definitions are not the most interesting part) is that changing your definitions in math change what kind of math you can do. (Like with spherical/hyperbolic geometry.) I don't know that a debate over a definition is valuable, as by-definiton (pun intended) it cannot be resolved, but it is useful to analyze why a definition is chosen and what purpose it solves, what it allows and what it doesn't.

100% in favor of any pedagogy or curriculum that asks students to poke at definitions and develop new ways of thinking mathematically. (Spherical geometry! Polygons with non-integer side lengths! Etc!)

https://blog.mrmeyer.com/2013/discrete-functions-gone-wild/

I just don't really see this particular debate contributing to that intellectual adventurousness. It just seems to elevate personal preference to the level of religious dogma in an area where the core religious texts are ambiguous.