Yes, that Tenison paper is a great example of arithmetic modelling in ACT-R, and especially connecting it to the modern fMRI approach for validation!   For an example of the other sorts of math modelling that's more psychology-experiment-based, this paper gives some of the low-level detail about how such a model would work, and maps it onto human errors:
  - "Toward a Dynamic Model of Early Algebra Acquisition" https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.53.5754&rep=rep1&type=pdf

(that work was expanded on a few times, and led to things like "Instructional experiments with ACT-R “SimStudents”" http://act-r.psy.cmu.edu/?post_type=publications&p=13890 where they made a bunch of simulated students and ran them through different teaching regimes)

As for other cool tasks, the stuff about playing some simple video games is pretty compelling to me, especially in as much as it talks about what sort of learning is necessary for the precise timing that develops.  http://act-r.psy.cmu.edu/wordpress/wp-content/uploads/2019/03/paper46a.pdf  Of course, this is not as good in terms of getting a high score as modern deep learning game-playing approaches, but it is very good in terms of matching human performance and learning trajectories.   Another model I find rather cool a model of driving a car, which then got combined with a model of sleep deprivation to generate a model of sleep-deprived driving: http://act-r.psy.cmu.edu/wordpress/wp-content/uploads/2012/12/9822011-gunzelmann_moore_salvucci_gluck.pdf  

One other very cool application, I think is the "SlimStampen" flashcard learning tool developed out of Hedderik van Rijn's lab at the University of Groningen, in the Netherlands: http://rugsofteng.github.io/Team-5/  The basic idea is to optimize learning facts from flashcards by presenting a flashcard fact just before the ACT-R declarative memory model predicts that a person is going to forget a fact.  This seems to improve learning considerably http://act-r.psy.cmu.edu/wordpress/wp-content/uploads/2012/12/867paper200.pdf and seems to be pretty reliable https://onlinelibrary.wiley.com/doi/epdf/10.1111/tops.12183  

I think that sort of task might be modellable with ACT-R -- the hardest part might be getting or gathering the animal data to compare to!  Most of the time ACT-R models are validated by comparing to human data gathered by taking a room full of undergraduates and making them do some task 100 times each.  It's a bit trickier to do that with animals.  But that does seem like something that would be interesting research for someone to do!  

Ooo, very good questions!  :)  I think there are a few different reasons why....  one small clarification, though, I don't think ACT-R shrunk to a small group -- I'd say more that it gradually grew from a small group (starting out of John Anderson's lab at CMU) up to about 100 active researchers around the world, and then sort of stabilized at that level for the last decade or two.

But, as for why it didn't take over everything or at least get more widely known, I'd say one big reason is that the tasks it historically focused on were very specific -- usually things involving looking at letters and numbers on a screen and pressing keys on a keyboard or moving a mouse.  So lots of the early examples were that sort of specific experimental psychology task.  It's been expanded a lot since then (car driving, for example), but that's where its history was, and so for people who are interested in different sorts of tasks, I can see them maybe initially feeling like it's not relevant.  And even now, lots of the tasks in the paper you provided are so far away from even modern ACT-R that I can see people believing that they can just ignore ACT-R and try to develop completely new theories instead.

Another more practical reason, however, is that there's a pretty high barrier to entry to getting into ACT-R, partly due to the fact that the reference implementation is in Lisp.  Lisp made tons of sense as being the language to use when ACT-R was first developed, but it's very hard to find students with experience in Lisp now.  There's been a big movement in the last decade to make alternate implementations of ACT-R (Python ACT-R, ACT-Up, jACT-R), and the latest version of ACT-R has interfaces to other languages which I think will help to make it more accessible.  But, even with a more common programming language, there's still a lot of teaching/training/learning required to get new people used to the core ideas.  And even to get people used to the idea of sticking with the constraints of ACT-R.  For example, I can remember a student building a model that needed to do mental arithmetic, and it took a while to explain why they couldn't just say "c = a + b" and have the computer do the math (after all, you're implementing these models on a computer, and computers are good at math, so why not just do the math that way?).  Forcing yourself to break that addition down into steps (e.g. trying to recall the result from memory, or trying to do a memory recall of some similar addition fact and then doing counting to adjust to this particular question, or just doing counting right from the beginning, or doing the manual column-wise addition method in your head) gets pretty complicated, and it can be hard to adjust to that sort of mind-set.

I will note that this high-barrier-to-entry problem is probably true of all other cognitive architectures (e.g. Sigma, Clarion, Soar, Dynamic Field Theory, Semantic Pointer Architecture, etc: https://en.wikipedia.org/wiki/Comparison_of_cognitive_architectures ).  But one thing ACT-R did really well is to address this by regularly running a 2-week summer-school (since 1994 http://act-r.psy.cmu.edu/workshops/ ).  That seems to me to be a big reason why ACT-R got much more widely used (and thus much more widely tested and evaluated) than the other architectures that are out there.  There was an active effort to teach the system and to spread it into new domains, and to combat the common approach in computational cognitive modelling of people sticking with the one model that they (or their supervisor) invented.  It's much more fun to build my own model from scratch and to evaluate it on the one particular task that I had in mind when I was inventing the model.  But that just leads to a giant proliferation of under-tested models.  :(  To really test these theories, we need a community, and ACT-R is the biggest and most stable cognitive architecture community so far.  It'd be great to have more such communities, but they're hard to grow.  

I agree that there isn't an overarching theory at the level of specificity of ACT-R that covers all the different aspects of the mind that cognitive science researchers wish it would cover.  And so yes, I can see cognitive scientists saying that there is no such theory, or (more accurately) saying that even though ACT-R is the best-validated one, it's not validated on the particular types of tasks that they're interested in, so therefore they can ignore it.  

However, I do think that there's enough of a consensus about some aspects of ACT-R (and other theories) that there are some broader generalizations that all cognitive scientists should be aware of.  That's the point of the two papers listed in the original post on the "Common Model of Cognition".  They dig through a whole bunch of different cognitive architectures and ideas over the decades and point out that there are some pretty striking commonalities and similarities across these models.  (ACT-R is just one of the theories that they look at, and they point out that there are a set of commonalities across all the theories, and that's what they call the Common Model of Cognition).  The Common Model of Cognition is much more loosely specified and is much more about structural organization rather than being about the particular equations used, though, so I'd still say that ACT-R is the best-validated model.  But CMC is surprisingly consistent with a lot of models, and that's why the community is getting together to write papers like that.  The whole point is to try to show that there are some things that we can say right now about an overarching theory of the mind, even if people don't want to buy into the particular details of ACT-R.   And if people are trying to build overarching theories, they should at least be aware of what there is already.

(Full disclosure:  I was at the 2017 meeting where this community came together on this topic and started the whole CMC thing. The papers from that meeting are at  https://www.aaai.org/Library/Symposia/Fall/fs17-05.php and that's a great collection of short papers of people talking about the various challenges of expanding the CMC. The general consensus from that meeting is that it was useful to at least have an explicit CMC to help frame that conversation, and it's been great to see that conversation grow over the last few years.  Note: at the time we were calling it the Standard Model of the Mind, but that got changed to Common Model of Cognition).

I think maybe our disagreement is about how good/useful of an overarching model ACT-R is? It's definitely not like in physics, where some overarching theories are widely accepted (e.g. the standard model) even by people working on much more narrow topics -- and many of the ones that aren't (e.g. string theory) are still widely known about and commonly taught. The situation in cog sci (in my view, and I think in many people's views?) is much more that we don't have an overarching model of the mind in anywhere close to the level of detail/mechanistic specificity that ACT-R posits, and that any such attempt would be premature/foolish/not useful right now.

Makes some sense to me! This is part of why my post's conclusion said stuff like this doesn't mean you should believe in ACT-R. But yeah, I also think we have a disagreement somewhere around here.

I was trained in the cognitive architecture tradition, which tends to find this situation unfortunate. I have heard strong opinions, which I respect and generally believe, of the "we just don't know enough" variety which you also espouse. However, I also buy Allen Newell's famous argument in "you can't play 20 questions with nature and win", where he argues that we may never get there without focusing on that goal. From this perspective, it makes (some) sense to try to track a big picture anyway. 

In some sense the grand goal of cognitive architecture is that it should eventually be seen as standard (almost required) for individual works of experimental psychology to contribute to a big picture in some way. Imagine for a moment if every paper had a section relating to ACT-R (or some other overarching model), either pointing out how it fits in (agreeing with and extending the overarching model) or pointing out how it doesn't (revising the overarching model). 

With the current state of things, it's very unclear (as you highlighted in your original comment) what the status of overarching models like ACT-R even is. Is it an artifact from the 90s which is long-irrelevant? Is it the state of the art big-picture? Nobody knows and few care? Wouldn't it be better if it were otherwise?

On the other hand, working with cognitive architectures like ACT-R can be frustrating and time consuming. In theory, they could be a time-saving tool (you start with all the power of ACT-R and can move forward from that!). In practice, my personal observation at least is that they add time and reduce other kinds of progress you can make. To caricaturize, a cog arch phd student spends their first 2 years learning the cognitive architecture they'll work with, while a non-cog-arch cogsci student can hit the ground running instead. (This isn't totally true of course; I've heard people say that most phd students are not really productive for their first year or two of grad school.) So I do not want to gloss over the downsides to a cog arch focus. 

One big problem is what I'll call the "task integration problem". Let's say you have 100 research psychologists who each spend a chunk of time doing "X in ACT-R" for many different values of X. Now you have lots of ACT-R models of lots of different cognitive phenomena. Can you mash them all together into one big model which does all 100 things?

I'm not totally sure about ACT-R, but I've heard that for most cognitive architectures, the answer is "no". Despite existing in one cognitive architecture, the individual "X" models are sorta like standalone programs which don't know how to talk to each other. 

This undermines the premise of cog arch as helping us fit everything into one coherent picture. So, this is a hurdle which cog arch would have to get past in order to play the kind of role it wants to play.

That sounds right to me.  It gives what types of information are processed in each area, and it gives a very explicit statement about exactly what processing each module performs.  

So I look at ACT-R as sort of a minimal set of modules, where if I could figure out how to get neurons to implement the calculations ACT-R specifies in those modules (or something close to them), then I'd have a neural system that could do a very wide variety of psychology-experiment-type-tasks.  As far as current progress goes, I'd say we have a pretty decent way to get neurons to implement the core Production system, and the Buffers surrounding it, but much less of a clear story for the other modules.