“Gifted” Is a Theological Word

Quite the juicy convo on Twitter:

Hard not to reply with every thought I have, but I want to keep the scope limited. One idea at a time.

In some sense, I work in “gifted education.” Big ups to BEAM, my favorite place to teach. This is a program that is addressing the intellectual hunger of students who are ready to go far beyond what they are doing in school. I have profound conviction that we are doing something worthwhile and important. (NB: to my knowledge, BEAM does not use the word “gifted” in any official materials, and most BEAM personnel do not use it with our kids out of growth mindset concerns.)

It is also true that I myself had a very different profile of needs from my peers at school as a young student of math. I taught myself basic calculus in 5th grade from an old textbook. I read math books voraciously through middle school, and in class just worked self-directedly on my own projects because I already knew what we were supposed to be learning. I am not mad that I didn’t have more mathematical mentorship back then — my teachers did their best to find challenges for me, I appreciated them both for that and for the latitude to follow my own interests, and in any case things have worked out perfectly — but looking back, at least from a strictly mathematical point of view, I definitely could have benefited from more tailored guidance in navigating my interests.

In this context I want to open an inquiry into the word “gifted” as it is used in education.

I hope the above makes clear that this inquiry is not about whether different students have different needs. That is a settled matter; a plain fact.

The subject of my inquiry is how we conceive of those differences. What images, narratives, stories, assumptions, etc., are implicit in how we describe them. In particular, what images, narratives, stories, and assumptions are carried by the word “gifted”?

This question is too big a topic for today. Today, I just want to make one mild offer to that inquiry, intended only to bring out that there is a real question here — that “gifted” is not a bare, aseptic descriptor of a material state of affairs, but something much more pregnant — containing multitudes. It is this:

“Gifted” is a theological word.

What do I mean?

A gift is something that is given; bestowed. My nephews recently bestowed on me a set of Hogwarts pajamas, fine, ok, but when we speak of “giftedness,” you know we are not discussing anything that was bestowed by any human.

By whom, then, is it supposed to have been bestowed?

You know the answer — by God. Or if not God, then by “Nature,” the Enlightenment’s way of saying God without saying God.

When we say a child is “gifted,” we are declaring them to have been selected as the recipient of a divine endowment. Each of these words carries a whole lot of meaning extrinsic to scientific description of the situation — selected; recipient; divine; endowment.

When we use this word in contemporary educational discourse, we usually aren’t consciously evoking any of this. Nothing stops a committed atheist from saying a kid is gifted. Nonetheless, I don’t think it can really be avoided.

Why I say this is how easily and quickly the full story — selected, recipient, divine endowment — becomes part of the logic of how people reason about what to do with a student so labeled. To illustrate with a contemporary slice of pop culture, the 2017 film Gifted, starring McKenna Grace, Chris Evans, Lindsay Duncan and Jenny Slate, hinges on the question of what is a family’s obligation to its child’s gift? How can a bare material state of affairs create a moral obligation? — but being chosen as the custodian of a divine spark on the other hand, it’s easy to see how to get from that to something somebody owes.

So, this is my initial offer. I’m not saying anything about what to do with this. For example, I am not evaluating Michael’s assertion that “giftedness is true.” I’m just trying to flesh out what that assertion means — to call attention to the sea of cultural worldview supporting the vessel of that little word.

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Teaching proof writing

I’m at BEAM 7 (formerly SPMPS) right now. I just taught a week-long, 18 hour course on number theory to 11 awesome middle schoolers. I’ve done this twice before, in 2013 and 2014. (Back then it was 20 hrs, and I totally sorely missed those last two!) The main objective of the course is some version of Euclid’s proof of the infinitude of the primes. In the past, what I’ve gotten them to do is to find the proof and convince themselves of its soundness in a classroom conversation. I actually wrote a post 4 years ago in which I recounted how (part of) the climactic conversation went.

This year, about halfway through, I found myself with an additional goal: I wanted them to write down proofs of the main result and the needed lemmas, in their own words, in a way a mathematician would recognize as complete and correct.

I think this happened halfway through the week because until then I had never allowed myself to fully acknowledge how separate a skill this is from constructing a proof and defending its soundness in a classroom conversation.

At any rate, this was my first exercise in teaching students how to workshop a written proof since the days before I really understood what I was about as an educator, and I found a structure that worked on this occasion, so I wanted to share it.

Let me begin with a sample of final product. This particular proof is for the critical lemma that natural numbers have prime factors.

Theorem: All natural numbers greater than 1 have at least one prime factor.

Proof: Let N be any natural number > 1. The factors of N will continue descending as you keep factoring non-trivially. Therefore, the factoring of the natural number will stop at some point, since the number is finite.

If the reader believes that the factoring will stop, it has to stop at a prime number since the factoring cannot stop at a composite because a composite will break into more factors.

Since the factors of N factorize down to prime numbers, that prime is also a factor of N because if N has factor Y and Y has a prime factor, that prime factor is also a factor of N. (If a\mid b and b\mid c then a\mid c.)

There was a lot of back and forth between them, and between me and them, to produce this, but all the language came from them, except for three suggestions I made, quite late in the game:

1) I suggested the “Let N be…” sentence.
2) I suggested the “Therefore” in the first paragraph.
3) I suggested the “because” in the last paragraph. (Priorly, it was 2 separate sentences.)

Here’s how this was done.

First, they had to have the conversation where the proof itself was developed. This post isn’t especially about that part, so I’ll be brief. I asked them if a number could be so big none of its factors were prime. They said, no, this can’t happen. I asked them how they knew. They took a few minutes to hash it out for themselves and their argument basically amounted to, “well, even if you factor it into composite numbers, these themselves will have prime factors, so QED.” I then expressed that because of my training, I was aware of some possibilities they might not have considered, so I planned on honoring my dissatisfaction until they had convinced me they were right. I proceeded to press them on how they knew they would eventually find prime factors. It took a long time but they eventually generated the substance of the proof above. (More on how I structure this kind of conversation in a future post.)

I asked them to write it down and they essentially produced only the following two sentences:

1. The factoring of the natural number will stop at a certain point, since the number is finite.
2. If X (natural) has a factor Y, and Y has a prime factor, that prime factor is also a factor of X.

This was the end product of a class period. Between this one and the next was when it clicked for me that I wanted proof writing to be a significant goal. It was clear that they had all the parts of the argument in mind, at least collectively if not individually. But many of the ideas and all of the connective tissue were missing from their class-wide written attempt. On the one hand, given how much work they had already put in, I felt I owed it to them to help them produce a complete, written proof that would stand up to time and be legible to people outside the class. On the other, I was wary to insert myself too much into the process lest I steal any of their sense of ownership over the finished product. How to scaffold the next steps in a way that gave them a way forward, and led to something that would pass muster outside the class, but left ownership in their hands?

Here’s what I tried, which at least on this occasion totally worked. (Quite slowly, fyi.)

I began with a little inspirational speech about proof writing:

Proof writing is the power to force somebody to believe you, who doesn’t want to.

The point of this speech was to introduce a character into the story: The Reader. The important facts about The Reader are:

(1) They are ornery and skeptical. They do not want to believe you. They will take any excuse you give them to stop listening to you and dismiss what you are saying.

(2) If you are writing something down that you talked about earlier, your reader was not in the room when you talked about it.

Having introduced this character, I reread their proof to them and exposed what The Reader would be thinking. I also wrote it down on the board for them to refer to:

1. The factoring of the natural number will stop at a certain point, since the number is finite.

(a) What does finiteness of the number have to do with the conclusion that the factoring will stop? (b) Why do you believe the numbers at which the factoring stops will be prime?

2. If X (natural) has a factor Y, and Y has a prime factor, that prime factor is also a factor of X.

What does this have to do with anything?

(I don’t have a photo of the board at this stage. I did do The Reader’s voice in a different color.)

Then I let them work as a whole class. I had the students run the conversation completely and decide when they were ready to present their work to The Reader again. In one or two more iterations of this, they came up with all of the sentences in the proof quoted above except for “Let N be…” and minus the “Therefore” and “because” mentioned before. They started to work on deciding an order for the sentences. At this point it seemed clear to me they knew the proof was theirs, so I told them I (not as The Reader but as myself) had a suggestion and asked if I could make it. They said yes, and I suggested which sentence to put first. I also suggested the connecting words and gave my thinking about them. They liked all the suggestions.

This is how it was done. From the first time I gave the reader’s feedback to the complete proof was about 2 hours of hard work.

Let me highlight what for me was the key innovation:

It’s that the feedback was not in the teacher’s (my) voice, but instead in the voice of a character we were all imagining, which acted according to well-defined rules. (Don’t believe the proof unless forced to; and don’t consider any information about what the students are trying to communicate that is not found in the written proof itself.) This meant that at some point I could start to ask, “what do you think The Reader is going to say?” I was trying to avoid the sense that I was lifting the work of writing the proof from them with my feedback, and this mode of feedback seemed to support making progress with the proof while avoiding this outcome.

Postscript:

As you may have guessed, the opening phrase of the sentence “If the reader believes…” in the final proof is an artifact of the framing in terms of The Reader. Actually, at the end, the kids had an impulse to remove this phrase in order to professionalize the sentence. I encouraged them to keep it because I think it frames the logical context of the sentence so beautifully. (I also think it is adorable.)

Think of a Brainy Black Woman in a Hollywood Film

So I’m psyched about Queen of Katwe (Disney), starring Lupita Nyong’o and David Oyelowo, based on the true story of young Ugandan chess champion Phiona Mutesi, which just came out. I’m definitely gonna see it this week.

I am also looking forward to the release this winter of Hidden Figures (20th Century Fox), starring Taraji P. Henson, Octavia Spencer, and Janelle Monae, based on the true story of Dorothy Vaughan, Mary Jackson, and Katherine Johnson, and their foundational mathematical contributions to the US space program. I have never ever ever seen a black female mathematician in a major film before.

This got me thinking: in my entire life up til now, have I ever seen a film released by a major Hollywood studio that centered on a brainy black woman and her brainy pursuits? I’ve been musing on this for about 24 hours now. I thought of exactly one: Akeelah and the Bee.

Can you think of any others?

Update 9/29: I thought of two more candidates. They don’t have that same “this woman is taking over the world with her mind alone” quality as all of the above, but they do have something:

Home (20th Century Fox, 2015): it’s not a major theme of the film, but the generally resourceful and awesome main character, voiced by Rihanna, does at a key point figure out the mechanism of a piece of alien technology while boasting of her “A in geometry”…

A Raisin in the Sun (Columbia Pictures, 1961): Beneatha’s intellectual pretensions don’t exactly drive the plot, but they are pretty central to her character. If you want to see what I mean and are up for being made a little upset, click here (the “in my mother’s house…” scene if you know it).

I want more! Please help!

Update 1/7/17: Having sat on this blog post for a few months now, I feel that the previous update dilutes the point a bit. Akeelah and the Bee, Queen of Katwe, and Hidden Figures, are the only movies of their kind I can think of. (Per the description above: produced by Hollywood, centered on a brainy black woman and her brainy pursuits.) I earnestly want to know if more exist. I am very excited there have been 2 inside of 6 months.

If I ask for “that kind of movie” only without the requirement that the lead be black and female, then we are swimming in them: Good Will Hunting, Theory of Everything, Imitation Game, Beautiful Mind, The Man Who Knew Infinity, Little Man Tate, Searching for Bobby Fischer, … shall I keep going?

For a quick and dirty numerical sample of the status quo: here is a list, compiled by a random IMDB user, of “movies about geniuses.” I found it among the first few hits upon googling “movies about smart people.” On this list I see 35 distinct titles. (The list says 42 but I see 7 repeats.) Of these, by my count the “geniuses” include 32 white boys/men, 1 black man, 1 East Asian man, and 1 white woman.

The fact that I managed, scraping my memory, to find a movie (Home) centered on a black girl who at some point in the film does something cool with her brain, is irrelevant to this stark picture. (This is not a knock on Home, which I loved.) If we want to bring it into the conversation, then we should put it in the context of every movie centered on a character that at some point does something cool with their brain. This is a lot of movies, way too many to make any kind of list.

If I allow the character in question not to be the main character (as in Raisin in the Sun; and if I allow us to leave Hollywood, 2012’s Brooklyn Castle and 2002’s Spellbound come to mind), then we are talking about every movie containing a character with plausible intellectual aspirations. Again, way too many to start listing.

The upshot: representations of brainy black women in (Hollywood) film have been exceedingly, shockingly rare. If you have taught in any place that has black people, you know that brainy black women are not rare in real life. Our national culture has had a very limited imagination in this regard. So let’s all effing go see Hidden Figures as soon as we possibly can. Independent of all this, I’ve heard it’s very good.

Pershan’s Essay on Cognitive Load Theory

Just a note to point you to Michael Pershan’s motherf*cking gorgeous essay on the history of cognitive load theory, centered on its trailblazer, John Sweller.

Read it now.

I’m serious.

I tend to think of Sweller as, like, “that *sshole who thinks he can prove that it’s bad for learning if you think hard.”

On the other hand, any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees, so you know that he’s talking about a real thing.

Michael has just tied it together for me, tracing how Sweller’s point of view was born and evolved, what imperatives it comes from, other researchers who take cognitive load theory in related and different directions, where their imperatives come from, and how Sweller’s relationship to these other directions has evolved as well. I have more empathy for him now, a better sense of his stance, and a better sense of why I see things so differently.

Probably the biggest surprise for me was seeing the connection between Sweller’s point of view on learning, and the imperatives he is beholden to as a scientist. I get so annoyed at the limited scope of his theory of learning, but apparently he defends this choice of scope on the grounds that it supports the scientific rigor of the work. I understand why he sees it that way.

The remaining confusion I have is why the Sweller of Michael’s account, ultimately so clear on the limited scope of his work (“not a theory of everything”) and the methodological reasons for this limited scope, nonetheless seems to feel so empowered to use it to talk about what is happening in schools and colleges. (See this for an example.) Relatedly, I’m having trouble reconciling this careful scientific-methodology-motivated scope limitation with Sweller’s stated goal (as quoted by Michael) to support the creation of new instructional techniques. The problem I’m having is this:

Is his real interest in supporting the work of the classroom or isn’t it?

If it is, well, then this squares both with the fact that he says it is, and that he’s so willing to jump into debates about instructional design as it is implemented in real classrooms. But it doesn’t square with rigorously limiting the scope of his theory, entirely avoiding conversations about obviously-relevant factors like motivation and productive difficulty, which he says he’s doing for reasons of scientific rigor, as in this quote:

Here is a brief history of germane cognitive load. The concept was introduced into CLT to indicate that we can devise instructional procedures that increase cognitive load by increasing what students learn. The problem was that the research literature immediately filled up with articles introducing new instructional procedures that worked and so were claimed to be due to germane cognitive load. That meant that all experimental results could be explained by CLT rendering the theory unfalsifiable. The simple solution that I use now is to never explain a result as being due to factors unrelated to working memory.

On the other hand, if his interest is purely in science, in mapping The Truth about the small part of the learning picture he’s chosen to focus on, then why does he claim he’s doing it all for the sake of instruction, and why does he feel he has something to say about the way instructional paradigms are playing out inside live classrooms?

Michael, help me out?

Reality Check: Does Anybody Care What Andrew Hacker Thinks?

So I’m just trying to figure out who, if anybody, cares what Andrew Hacker thinks about math education.

This is an earnest question.

At least since 2012, when he had an opinion piece in the NYT, he has been going on about how we should stop requiring “advanced” math, from algebra up, in schools. He was in the NYT twice again recently and has a new book out about it.

Now, of course both math educators and mathematicians are going to “care” in the sense that it annoys us. We are spending all this time trying to figure out how to improve students’ appreciation for and understanding of algebra etc., and out comes this dude talking about “scrap that whole project.” I remember there being at least a few responses in the MTBoS back in 2012, although Dan Meyer’s and Patrick Honner’s are the only ones I remember specifically. (Dan had some more fun with it a few years later.) And I was moved to write this reading mathematician Evelyn Lamb’s piece in Slate responding to Hacker’s book. (Dan’s responses succinctly summarize Hacker’s lack of imagination. Mr. Honner sees algebra in what Hacker wants to replace algebra with. And if you want to get more into the details, go read Lamb’s piece, it’s great.)

But annoying math teachers and mathematicians is definitely not the same thing as being remotely relevant. I mean, he is suggesting to do away with required algebra precisely at the point in history when, between the Common Core, the increasing quantity and stakes of standardized testing, and the incessant press handwringing about international competitiveness,[1] it seems to me that math, including advanced math, is more centrally ensconced in the curriculum than it’s ever been. (In my lifetime anyway. Possible historical exception of the immediate post-Sputnik era.) Is anybody taking him remotely seriously?

Yes, he has coverage in the New York Times and the Chronicle of Higher Ed. This doesn’t answer the question. He’s being intentionally provocative and succeeding in getting a rise. Is anybody taking his proposals seriously?

I eagerly await your thoughts.

*****

Postscript: Let me indulge myself to go ahead and give you my take, just for the record. Disclaimer that I haven’t read his book. I’m going on the 3 NYT pieces.

At the level of fundamental goals, he is upset about the fact that so many Americans have traumatic experiences with their math education, meanwhile graduating without basic numeracy needed for citizenship, and he wants to do something about it. I’m not mad at this, nor could I be.

I’m also sort of delighted that being a public intellectual counts for enough that this 86-year-old Queens College political scientist can mouth off in the NYT whenever he wants. I hope that when I’m 86, the NYT still exists and I can mouth off in it whenever I want.

But on to the merits themselves. I think he sees a real problem but I, like Dan, think he lacks imagination about both (a) what math education could be, and (b) what math is for – he doesn’t get it as a domain of human inquiry, or an intellectual inheritance, just as a tool, so he applies a utilitarian standard to it I’m sure he’d never apply to history, or art, say. But also, (c) I think he misdiagnoses the problem if he thinks removing required algebra (and up) will solve it. Algebra isn’t the first point in the curriculum where massive numbers of American children jump ship emotionally. This is already happening with fractions, and may begin much earlier. Hacker would never propose to take fractions, or even more fundamental stuff, out of the curriculum, since it’s obviously (even to him) part of the “citizen mathematics” he champions.[2] Doing a good job teaching math is a problem to be solved, not avoided. Finally, (d) I think he would probably puke if he really thought through the antidemocratic implications of a general public without advanced technical literacy while all the contemporary centers of power – finance, info tech, biotech, etc. – are technocratic and growing more so. He sometimes argues that you don’t need to know algebra to learn to code. Depends on what you are coding I suppose, but in any case this is beside the point. Sergey Brin does know algebra. Jamie Dimon does too.

Update 4/5: Sam Shah sends this graphic of number of people on feedly who “saved” or “favorited” Hacker’s Feb. 27 piece (which is about what he wants to replace algebra with):

benbs

Update 6/28: Patrick Honner wrote something relevant on the Math for America blog back in May – When It Comes to Math Teaching, Let’s Listen to Math Teachers.

[1] I’m only lumping these three things (CCSSM, high-stakes testing, and international-comparison handwringing) together from the point of view that all three seem to me to be moves in the direction of consolidating the consensus on the centrality of math in contemporary American education. I do not have them confused with each other and I don’t feel the same way about each of the three at all. For exmaple, I basically dig the CCSSM but (as any regular reader of this blog knows) I do not at all dig high-stakes testing.

[2]Patrick Honner’s post points out that Hacker’s notion of “citizen mathematics” almost surely involves algebra as well…

Lessons from Bowen and Darryl

At the JMM this year, I had the pleasure of attending a minicourse on “Designing and Implementing a Problem-Based Mathematics Course” taught by Bowen Kerins and Darryl Yong, the masterminds behind the legendary PCMI teachers’ program Developing Mathematics course, with a significant assist from Mary Pilgrim of Colorado State University.

I’ve been wanting to get a live taste of Bowen and Darryl’s work since at least 2010, when Jesse Johnson, Sam Shah, and Kate Nowak all came back from PCMI saying things like “that was the best math learning experience I’ve ever had,” and I started to have a look at those gorgeous problem sets. It was clear to me that they had done a lot of deep thinking about many of the central concerns of my own teaching. How to empower learners to get somewhere powerful and prespecified without cognitive theft. How to construct a learning experience that encourages learners to savor, to delectate. That simultaneously attends lovingly to the most and least empowered students in the room. &c.

I want to record here some new ideas I learned from Bowen and Darryl’s workshop. This is not exhaustive but I wanted to record them both for my own benefit and in the hopes that they’ll be useful to others. In the interest of keeping it short, I won’t talk about things I already knew about (such as their Important Stuff / Interesting Stuff / Tough Stuff distinction) even though they are awesome, and I’ll keep my own thoughts to a minimum. Here’s what I’ve got for you today:

1) The biggest takeaway for me was how exceedingly careful they are with people talking to the whole room. First of all, in classes that are 2 hours a day, full group discussions are always 10 minutes or less. Secondly, when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They never ask for hands, and they never cold-call. This means they already know more or less what the students are going to say. Thirdly, they have a distinction between students who try to burn through the work (“speed demons”) and students who work slowly enough to receive the gifts each question has to offer (“katamari,” because they pick things up as they roll along) – and the students who are asked to present an idea to the class are only katamari! Fourthly, a group discussion is only ever about a problem that everybody has already had a chance to think about – and even then, never about a problem for which everybody has come to the same conclusion the same way. Fifthly, in terms of selecting which ideas to have students present to the class, they concentrate on ideas that are nonstandard, or particularly visual, or both (rather than standard and/or algebraic).

This is for a number of reasons. First of all, the PCMI Developing Mathematics course has something like 70 participants. So part of it is the logistics of teaching such a large course. You lose control of the direction of ideas in the class very quickly if you let people start talking and don’t already know what they’re going to say. (Bowen: “you let them start just saying what’s on their mind, you die.”) But there are several other reasons as well, stemming (as I understood it anyway) from two fundamental questions: (a) for the people in the room who are listening, what purpose is being served / how well are their time and attention being used? and (b) what will the effect of listening to [whoever is addressing the room] be on participants’ sense of inclusion vs. exclusion, empowerment vs. disempowerment? Bowen and Darryl want somebody listening to a presentation to be able to engage it fluently (so it has to be about something they’ve already thought about) and to get something worthwhile out of it (so it can’t be about a problem everybody did the same way). And they want everybody listening to feel part of it, invited in, not excluded – which means that you can’t give anybody an opportunity to be too high-powered in front of everybody. (Bowen: “The students who want to share their super-powerful ideas need a place in the course to do that. We’ve found it’s best to have them do that individually, to you, when no one else can hear.”)

2) Closely related. Bowen talked at great length about the danger of people hearing somebody else say something they don’t understand or haven’t heard of and thinking, “I guess I can’t fully participate because I don’t know that idea or can’t follow that person.” It was clear that every aspect of the class was designed with this in mind. The control they exercise over what gets said to the whole room is one aspect of this. Another is the norm-setting they do. (Have a look at page 1 of this problem set for a sense of these norms.) Another is the way they structure the groups. (Never have a group that’s predominantly speed-demons with one or two katamari. If you have more speed-demons than katamari, you need some groups to be 100% speed demon.)

While this concern resonates with me (and I’m sure everybody who’s ever taught, esp. a highly heterogeneous group), I had not named it before, and I think I want to follow Bowen and Darryl’s lead in incorporating it more essentially into planning. In the past, I think my inclination has been to intervene after the fact when somebody says something that I think will make other people feel shut out of the knowledge. (“So-and-so is talking about such-and-such but you don’t need to know what they’re talking about in order to think about this.”) But then I’m only addressing the most obvious / loud instances of this dynamic, and even then, only once much of the damage has already been done. The point is that the damage is usually exceedingly quiet – only in the mind of somebody disempowering him or herself. You can’t count on yourself to spot this, you have to plan prophylactically.

3) Designing the problem sets specifically with groupwork in mind, Bowen and Darryl look for problems that encourage productive collaboration. For example, problems that are arduous to do by yourself but interesting to collaborate on. Or, problems that literally require collaboration in order to complete (such as the classic one of having students attempt to create fake coin-flip data, then generate real data, trade, and try to guess other students’ real vs. fake data).

4) And maybe my single favorite idea from the presentation was this: “If a student has a cool idea that you would like to have them present, consider instead incorporating that idea into the next day’s problem set.” I asked for an example, and Bowen mentioned the classic about summing the numbers from 1 to n. Many students solved the problem using the Gauss trick, but some students solved the problem with a more visual approach. Bowen and Darryl wanted everybody to see this and to have an opportunity to connect it to their own solution, but rather than have anybody present, they put a problem on the next day’s problem set asking for the area of a staircase diagram, using some of the same numbers that had been asked about the day before in the more traditional 1 + … + n form.

I hope some of these ideas are useful to you. I’d love to muse on how I might make use of them but I’m making myself stop. Discussion more than welcome in the comments though.