In class (if I bring a class to the computer lab) I focus on activities that do push that focus on the math, as you write.

Out of class especially in a terminal mathematics class on the college level, I do admit that I allow assignments that may take considerable not-quite-math thinking: I let students do design projects that illustrate math or make math look nice, like knitting an abstract algebra sock (from Making Mathematics with Needlework edited by belcastro and Yackel) to writing an integration piece for clarinet to making an attractive computer program that does calculations relevant to the class content. I figure if they’re doing something pleasurable outside of class that they associate with mathematics and they *produce the appropriate mathematical parts correctly*, I am happy to have them do it! Teaching a terminal college math class, though, has some different pressures than teaching one math class out of many in a grades 5-12 context.

Excel is, largely invisible, once you learn the primary operations and don’t get into macros etc. In your example your students are thinking about how to get the results they want, and little else. As it should be.

Making little floating dock simulations is quite different. Then they are worrying about getting things to happen at the right speeds, with the right colours, whether the dock moves in tandem with the ramp. None of those are relevant to the learning of trigonometry, and positively distract from it. (There is a particular sort who will ace the simulation part but learn no Maths at all because they will brute force the answer.)

It is why I never use PowerPoint in class. If it is plain, then I may as well write it up on the board, because at least it is more easily adjusted. If it is fancy, then kids will spend at least some of their mental time focussing on the choice of colours, backgrounds and movements (and any errors in those will definitely distract them).

On the other hand, I was asking them to apply concepts they claimed they’d learned. It may be that the tech is distracting during the initial encounter with ideas, putting too much cognitive load on students. In the boat dock problem under discussion, I would not ask students to create a simulation until we’d gone through the problem on paper and discussed it thoroughly. Then creating a simulation is a different task, but it is one that allows students to test what they claimed they’ve learned.

None of those address the question of class time, though! One way I get around this is to allow students to do extra-credit projects like building a simulation (for my current class such activities can replace a low quiz grade, actually, but are not required).

It’s not about whether they are capable or not.

We need the technology to be as invisible as possible when we teach Maths — every time we direct them to working inside the technology, we risk losing their focus entirely.

Pythagorean Theorem is really only A LITTLE outside the narrow objective of using right triangle trig ratios, no? The big bonus in building this simulator is that it forces one to think about what the coordinates actually mean/represent and how they are related – applicable in many, many settings. But, I would admit, well outside your objective.

Dan, why did the maximum steepness of 34% of the original turn into 18° in this version?

I thought the use of % for slope, which is actually much more common in normal use (for example roof pitches, road signs warning of steep roads), was one more way they had to convert world into Maths.

That was my decision. I wanted the problem to focus on trigonometry rather than similar triangles & Pythagorus. I wouldn’t get worked up if someone wanted to change it back though.

I thought the use of % for slope, which is actually much more common in normal use (for example roof pitches, road signs warning of steep roads), was one more way they had to convert world into Maths.

True they can solve the problem using Pythagoras and never get to angles, but is that a problem in itself?

I have the same concerns that you expressed regarding simulations. In this case, I think many students are capable of creating their own simulation (like Jonathon’s) with a small amount of assistance. The beauty of doing this is that it is fairly easy to create a static diagram (low entry). And, there is instant feedback on what is working and not working as they build their models in desmos. Five lines for a crude desmos simulator.

But building the simulation often requires mathematics that’s way outside the objective I originally intended for the simulator. Sinusoidal modeling in my case and the Pythagorean theorem in yours. It’s a different objective.