Geometry Book

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Some of the geometry book I'm working on...

Eight pages of geometry

I forget which post Gordon said it in, but at one point he noted that nearly all books prior to the invention of printing were books of magic.  Sure, on the surface they might be called medical textbooks or scientific textbooks or books of geography or mythology or history. But at some level, all these books were books of magic — they were intended to change consciousness at some level.

Rufus Opus said something similar about making lamens. A lamen is usually a disk or a square that you wear on your chest during the conjuration of a spirit.  The act of writing one, of punching a hole in the parchment, and putting it on a string or a chain or a lanyard, is a creative act.  If the emblem you write on the lamen is the signature or symbol of a spirit, your hand is going through a kinesthetic meditation on the nature of the relationship between the conjurer and the spirit.

Something similar is happening as I create this book.  It’s a Moleskine Japanese Album, the larger size, so the pages fold out into this lengthy ‘wall’ or ‘screen’ of emblems — about 5 1/4″ x 8 1/4″ inches per panel, but about 115 1/2″ long — call it about 9′ 7 1/2″.

I think about this project from time to time — more lately, since I’ve been working on it the last few days — and every time I do, I’m somewhat more dismayed at the current state of geometry teaching in the United States.  By all the accounts I’ve found, and by the anecdotal evidence I’ve collected on my own, we’ve stopped teaching students to use rulers and compasses in the study of geometry.  It’s too hard to remember procedures, or students don’t know how to use those flimsy plastic compasses well and the good ones are too expensive, or Euclid isn’t widely available, or … or… or…

The excuses multiply like dandelions after a rainstorm.

I don’t know that this book “will become an heirloom of my house forever,” as one of the somewhat-more-fictional sagas would have it. But I do know that I learned more geometry from the construction of the book than I ever learned in a class.  And I wonder if there’s not a better way to teach geometry embedded in that discovery?

  • Each student gets a good compass, a good ruler, colored pens or pencils, and a blank notebook.
  • Each student learns the construction for a harmonious page layout
  • Each student learns a set of procedures for:
    • Perpendicular bisectors
    • duplication of angles
    • construction of parallel lines
    • construction of similar triangles
    • construction of polygons from given sides
    • construction of polygons within circles
    • transference of a given length or distance to another angle
    • construction of nets for 3-dimensional solids
    • construction of the root-2, root-3, root-4, and root-5 (phi/Φ) proportions
    • division of lines into thirds, fourths, fifths, eighths, ninths, and sixteenths
    • construction of grid and tile patterns
    • construction of simple polygonal combinations to find the sides of super-polygons.

This benefits future craftspeople, because they’re receiving an education in proportions and common mathematical relationships, and it’s not all algebraic notation.  It brings back the beauty of geometry to the mathematics classroom.  It gives all of society a common language for seeing mathematics in the natural world.  It trains future architects and engineers in precision diagramming, and gives future laypeople practice in reading such diagrams.

And it creates hundreds of unique copies of books of practical geometry that are themselves handbooks to a forgotten magic — a magic of beauty, of proportion, of color, of relationship, of graphic design. Students would get to learn ALL of that in the process of producing their own books over the course of a semester or a year. The quality of their book would gradually improve, as their understanding of the geometry improved, and as their love and care of the book improved. Think of all the other studies that could be folded into the creation of the book, too: handwriting, color theory, graphic design, book design, clear writing about mathematics, methodology.  The book is a grade — and students who kept their book up to date would find it useful while taking tests to remember what they had created in their own handwriting. The book itself would be a palace of memory for all the geometry they had learned, just as mine is.

All of the actual constructions are covered in Andrew Sutton’s book Ruler and Compass.  But actually implementing it is on the individual teacher.  And it’s likely the case that the teacher will need some substantial support from an administration that sees and cares about quality instruction.

But it can be done.

Geometry: back to work 

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It’s been a good long while since this particular project occupied my attention and focus.  However, I’m currently motivated to finish it — or at least finish the nine pages that I already have outlined and planned.  There are six more pages that are unplanned except for the margins, which means that I have a total of fifteen pages left to write, and maybe a card or panel to put in the pocket of the book, an afterword of sorts to explain the project a little better than I did at the beginning.

What project am I talking about? This one, the geometry book that I began a long time ago practically in a galaxy far, far away.  In fact, from the earlier entries from 2013, I can tell that I was already about sixteen pages into it.  Now, I’m thirty-seven pages into it, and I have fifteen left.  I’m almost the opposite point in this project as I was four years ago.  Funny how these things circle around, right?

The current pages, #36-37

Of course today is the day that I made a mistake.  I drew out the process of comparing 1:√2, and didn’t discover my error (on the right-hand page) until I had already inked the diagram and written the explanatory text.  Always check your work in geometry before you render it in pen!

The next pages laid out (and upside down for some reason)

No matter.  I had the room to be able to describe the process incorrectly, add in A WARNING IN CAPITALS AND RED, and then offer the correction. Typical medieval manuscript at this point, really — sometimes errors creep in, and the lowly scribe has to figure out how to offer the correction clearly and legibly in less space.  I managed.

As I said, I have nine pages remaining in this project that are already laid out.  A lot of this project is me working through Andrew Sutton’s book, Ruler and Compass from Wooden Books.

Why did I return to it, though? Well, first, I’m trying to clear my desk of unfinished projects. This one has been a big one, and it’s been on my mind to complete for a while.  But for another, I recently took up the opportunities and challenges of tutoring again.  And I’m tutoring a few young people in geometry.  So this project is serving to lubricate and rub the rust off of my geometry skills. Even so, I’m finding that the knowledge of actual geometric proofs isn’t quite as useful as one might imagine.

A lot of the work that students do in geometry class these days appears to be algebra. There will be one diagram (with a note beside it to say, not to scale or not rendered accurately), and then a lot of algebraic notation, and the student is expected to work without a ruler and compassed just their brain power and maybe a calculator, to solve the problem.

Say what??

I don’t understand.

Are we teaching geometry, or geometric algebra?  It looks like the latter, rather than the former.  And I understand that teaching actual geometry is challenging, and that it involves looking at a lot of diagrams and working out a lot of constructions by hand… but heck, that’s what we do as human beings. Isn’t it?

I said to someone on Twitter today that

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pardon, I can’t figure out the ’embed tweet’ system for my server.

But that’s (more or less) true — we use our hands to instruct our brains, and vice-versa.  How do we actually learn geometry if we’re not using the tools that geometry has used for thousands of years (or reasonable electronic replacements, though I’d argue that such tools are not as good as actually using hands to manipulate a compass)?

In any case, here’s a place where abstraction often gets the best of us.  I think it’s time to bring back some actual geometry to the classroom, and not simply ask students to do it algebraically.  This is a set of skills that belongs in our students’ hands, and not just in their heads.

Fidgeting and Hand Skill

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There’s a lot of outrage about fidget spinners right now. Some teachers are saying ban them! Other teachers are saying, Let students have them.

It’s a stupid argument.

Remember yo-yo’s? Finger skateboards? gear-powered spinning tops? String powered spinning tops? How about Rubik’s Cubes which made a comeback a couple of years back? Wind-up cars that did tricks?

Fidget Spinners have a place and a time in children’s hands.  And as some of you know, one of my mantras or principles is that What the Hands Do, the Mind Knows.  But here’s the thing.  If you don’t want the latest finger-toy-de-jour in your classroom, then you have to find other ways to put those hands to work, learning actual hand skills:

  • teach calligraphy
  • teach knitting
  • teach drawing
  • teach geometry with an actual ruler and compass
  • teach the use of a slide rule or abacus
  • teach the building of automata (cogs and gears)
  • teach carpentry and build yo-yos, finger skateboards, spinning tops, and fidget spinners.
  • teach contact juggling
  • teach juggling
  • teach beading
  • teach woodcarving
  • teach origami
  • teach flint-knapping
  • teach ceramics throwing on a wheel
  • teach students 3D geometry through the assembly of nets of the Platonic solids.
  • teach color theory and coloring at a more advanced level through color pencils.

The fidget spinner is an outward and visible sign of an inward need — a need for the hands to learn something.  Kids’ hands fidget because they’re of an age to want to do something, not just sit still.

(And I KNOW that we’re not making them sit still in schools — that they’re doing personal practice as well as listening, reading, writing, reflecting on their work and all that sort of stuff. That’s not what this is about).

Human beings need to use their hands. We learn things through manual dexterity, through touch, through manipulation of objects.  Our constant rejection of the toys-de-jour, be they yo-yos or balsa wood flyers or paper airplanes or fidget toys is part of the reason kids don’t learn as much in school as they could.

So if you want to fidget-spinner proof your classroom… figure out WHAT tool or hand-skill you want your students to have, learn HOW to teach it, and then TEACH THAT.

Knit: hat take2

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A while back, I tried making a hat. This Easter weekend, I got to finish it. In the process, I learned how to reduce and end a hat; and how to transfer knitting from working on ‘circular needles’ with a cable between the left and right needle, to four double-pointed needles.  Im sorry to say that my efforts to make a hat resembled something rather more like a floppy Frisbee cozy — more suited for covering a pie plate than a hat. The dome structure we associate with a beanie or slouch-stye knitted cap was almost entirely absent. As you can see, it was not an ideal construction.   It comes together nicely in the middle— but the outer edge, where one starts, is simply flat.

What went wrong?

The essence of the trouble is that I simply didn’t take the time to establish ribbing around the base of the hat. I should have begun with the end in mind — and started by establishing the defined edge.

The ‘defined edge’ that begins something like a hat is called 1×1 ribbing, and it’s done with a series of knit and purl stitches.

I did some investigation, and found several patterns from Tin Can Knits — not just hats, but also patterns for scarves and sweaters, shawls and socks. It seems to me that this is the core of a knitter’s repertoire, so I’ve printed out their patterns and I’ve been following along at home.

  1. Let’s knit a hat
  2. Knitting Socks
  3. Knitting Mittens and Handwarmers

So I’m starting again. This is actually take four — I put the ribbing on the  circular needles for a pattern and discovered that my needles were too long for the hat pattern I’m trying.  But the ribbing works. And in the process I’ve internalized the hand motions that need to happen when attempting to learn the purl stitch.

Which is not a minor accomplishment in itself — I don’t think I genuinely understood what the purl stitch did before today.  Yet now I do, sort of: it ‘digs a ditch’ in the yarn pattern, either resulting in cabling that stands out or recessed patterns that allow shadows to catch. This is Tin Can Knit’s language, sort of, not mine.   Yet now it has a purpose, a reason for being in my knitting tool-kit, so to speak: ribbing.

I’m kind of hoping this hat fits me.  I expanded it beyond the top TCK pattern size, in the hopes that it would fit my head… I like the idea that the first hat I make is for me.

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#edcampswct follow-up

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During the last session of yesteday’s #Edcampswct (see edcamp.org about what an Edcamp is), I led a discussion on MakerSpaces and Maker Programs.  I want to summarize what points I made there, and provide links to deeper insights on those subjects; and make a few further points that I don’t think I made in the time allowed, but were on my mind.

Here are the key points, which are further summarized below (@MrPerraultGES took a photo of my notes):

  1. Visual Thinking
  2. 2D makes 3D
  3. Tools Make Tools Make Things
  4. What Hands Make, Mind Knows
  5. Recycle and D.I.Y.
  6. Space Requirements
    1. Tool Storage
    2. Materials Storage
    3. Project Storage
    4. Workspace
    5. Input/Receiving
    6. Archive Process
    7. How-To Library
    8. Repair (and Sharpening)
    9. First Aid
  7. Best Practice vs. Liability
  8. (And to these 7 steps  I’m adding—
    1. Games and Game Playing
    2. Past vs. Future Orientation )

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Under-Education: Responding to @willrich45

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Will Richardson writes, over at Modern Leaners, about the problem of under-education:

 A phrase I find myself using more and more these days (and probably mispronouncing) is “raison d’être” or reason for being, as in asking school leaders “why does your school exist?” The easy answer is “to educate our children!” For centuries, that was probably good enough. Everyone knew, sorta, what it meant to be “educated.” You “learned” a lot of stuff about different subjects. You learned how to read and write and work with numbers to some acceptable degree. You were “prepared” for life, for a job or a college. You looked pretty much like everyone who was “educated” before you, passed the same tests, got over the same bars.

I think I get what Will is getting at, here.  Why does your school exist? is a really fundamental question. I mean, in general, “to educate our children!” is a great answer.  But one school I taught at, the “our children” part carried a caveat — and that caveat was “because the schools that are closer to where we live have failed them.”

I taught in a boarding school from 1996 to 2001, took a sabbatical year, and then taught in that same school from 2002-2010.  That school focused on student success by providing a one-on-one tutor for 45 minutes every day; by requiring all students to participate in athletics; and by providing a formal study hall for two hours every evening for completing homework.  In the early part of the 20th century, the idea of learning disabilities was not widespread or understood; my school helped develop a series of one-on-one tutoring processes, including championing the Orton-Gillingham methodology.  People from all over the country (and now from all over the world) sent their children long distances, because my old school could do what no other school could do.  So that sentence becomes, “My school exists to educate our children through time management and compensation techniques for their learning disabilities.”  I think that some of my former colleagues might dispute that baldly-phrased statement, but it’s a shorthand way of thinking about the school’s identity.

My old school’s day and pattern was adapted from a model similar to many other New England boarding schools. The most famous of these schools (and not one where I ever worked), is probably the Groton School, where the names of graduates are professionally carved into the oak-paneled walls of the study hall; and the names of its graduates who went on to the American presidency are gilded with 24-carat gold.  Students literally sit for study hall each evening with light and shadow playing out on the names of those who went before them; and each of them is impressed constantly with the reality, “No matter how great I become personally, I will never be as great as that other alumnus of this school, four-term U.S. President, who led the country out of the Great Depression and through World War II…”  Those students literally sit in the shadow of Franklin D. Roosevelt, and under the names of numbers of his deputies and aides — and it affects every one of Groton’s graduates deeply.  That school’s sentence becomes, “My school exists to educate our children to be problem-solving citizen-leaders at the highest levels.”

Sometimes the identifier is more regional.  I went to a junior high school for grades seven through nine; our athletic rivals were the other two junior high schools in town… not even the next town over.  That school’s sentence might be construed as “My school exists to educate our children who live within walking distance of this building.”  It was a funny, wonderful place: a good theater program, a good music program, Home Economics, shop classes and drafting, computer programming in the mid-1980s when that was still a weird thing.  Sure, all the regular classes in math and science, English and foreign language, history and civics (I had a civics class in there, how weird that seems today).  But it was the extras, I think, that shaped who I am today though it took twenty years to see it.

The Thinkers and the Makers

My mind, though, often returns to the schools in the shadows of Europe’s great cathedrals.  The so-called Cathedral Schools of the 9th through the 12th century focused on the seven liberal arts, which were the simplification and inheritance of the Latin educational system such as it was: the Trivium, consisting of Grammar, Rhetoric, and Logic; and the Quadrivium of Music, Arithmetic, Geometry, and Astronomy.  There wasn’t much time for more than that, really:  boys (and it was mostly boys) were taken in at 7 or 10, trained until they were 16 or 17, and then shipped off to be clerks to bishops and dukes, professional literates in a world lit only by fire (as William Manchester called it).

Those schools’ sentence ran like this: “My school exists to educate our children in Christian virtues, rational thought, and logical and mathematical relationships, for the purpose of tempering the warriors of the age with some professional advice-givers.”  They were pretty explicit about this in their writings; they knew that they were trying to bring about relative pacifism in an age of violence, warfare and feud, through education.

Down the street from here is a technical high school. Kids study regular academic subjects, but they also pick subjects like plumbing and electrical work, automotive repair and carpentry.  One of my friends is a graduate of that school;  he’s not a great writer, but he visualizes objects in three dimensions and builds them.  He’s a master builder, a contractor, a savvy businessman, and a leader both in his community and of his employees.  He’s no FDR, but I think he could say, “My school exists to educate our children in building and making and managing workers and materials.”

The Measurable and the Immeasurable

Will says,

So, why do we exist? What is our higher, more modern calling? How do we talk about that even? Haque says that one place to start is to put “intuition over computation,” or as I’m referred to it in the past, the “immeasurable ahead of the measurable.” Or, as Russel Ackoff says, again, “to do the right thing instead of trying to do the wrong thing right.” I think most of us get this, yet we seem unable to move from legacy thinking.

And I think this is one of those places where Will — as much as I love him — have to part ways.  Because I think about what it is that schools used to do, and what they so often appear to do now.  I think that we’ve probably drifted too far from legacy thinking, myself.

Because the essence of school is not necessarily to measure, but to teach measurement.  When I think to the Cathedral Schools, focused on the essentials that they could afford to save of Roman and Greek learning, they chose to save the abilities to think, speak, and write precisely; how to count, and account, for numbers; and to understand mathematical relationships in space and time and vibration.

My friend’s technical high school taught him how to cut 45°-angles into complex pieces of wood crown moldings for construction; how to keep an account book; how to price out a job; how to hire and pay workers; how to pay his own way in the world.  Measurement is at the core of his business and his success; and it is what his school taught him how to do.  He lovingly tells the story of one of his carpentry teachers inspecting a joint between two pieces of molding, holding the two wood scraps up to the window. “No. There is light shining through. The angle is wrong and the cut is not straight. Do it again.”  It is that love of accuracy and precision that makes him one of the most sought-after contractors in town.

The school where I used to work focused on teaching time management and attention management to students with ADD and ADHD; how to read effectively and to speak clearly; how to function with limited note-taking in high-information environments; how to move cleanly from one task to another.   It wasn’t exactly the exalted study of astronomy practiced in medieval times atop cathedral towers.  But it was the teaching of measurement and management.

Even Groton, with its oak-paneled study hall, was teaching the vitality of measurement and technical expertise: “one of your predecessors solved the greatest economic crisis in our country’s history, and you’re going to whine about a little algebra?” I can almost hear a teacher telling a recalcitrant student.    Except that my sense of things is that the environment of Groton does a good job of helping students understand that whining is not helpful. 🙂

In any case, a school is for educating our children in ways to measure the world.  We begin with counting and rulers, then teach angles, and a variety of formulae for converting one kind of measurement to another.  Temperature, humidity, wind speed.  Weight, velocity, angle of repose.  Supply, Inventory, Demand.  Current, resistance, voltage.  Light-years, Miles, Yards, Inches, Angstroms.  Pounds, Quarts, Grams, Micrograms. Syllogism, Fallacy, Validity, Conclusion.  Even in classes that focus on writing and reading, in storytelling, we are focused on methods that measure the world and make it return logical results: If this, then that; if not that, then not this.  The Cathedral Schools sent graduates into the world who knew the power of words, but also the structure of rational argument; they were not Christian fundamentalists, but deep and careful thinkers trained in Aristotlean and Platonic logic, Boethean grammar and Ciceronian rhetoric, with a skilled understanding of what to say, how to say it, and how to argue for and against different positions.  Aesop’s fables, annoying and boringly familiar as they are to us, taught medieval Frankish children Latin grammar, rhetorical style and logical thinking all at once; and prepared their minds to understand the formal relationships of geometry, the pitch relationships of music, and the  vast interplay of arithmetic, geometry and time that was astronomy.  My contractor friend builds houses from stacks of lumber — but stands in awe of another friend of his who knows to the board-foot and last chop-saw cut and nail-count how to build a Sonic drive-through restaurant.

Where we fail, I think, is that we spend far too much time taking the measure of the student.  In the pursuit of ever-more-accurate understandings of what is going on in the child’s mind, we fail to impart the critical lessons of how to apply differently-graded yardsticks, containers, and meters to various kinds of problems, or to analyze those results.  Nor do I simply mean physical tools like rulers and protractors — no, I mean the genuine tools of rational, evidence-based thought.  Instead, we look for evidence from the students themselves that they know what we’re talking about; and we give in probably too often to confirmation biases: a student says something, and we take their words at face value, hearing the measurements and evidence that we have laboriously collected for ourselves — instead of hearing all out their evidence, data, and interpretation.

But the child is not the center of the school.  

There will be some teachers and parents who will clutch their pearls  or adjust their ties at this.  And I think that we can agree that the best teachers that we know are the ones who care deeply about children and their welfare.  But true adults, capable and competent adults, are ones who know and understand the facts of the world — who are capable of measuring aspects of their lives, and interpreting those measurements against certain standards.  Doctors know the names of all the bones, muscles and ligaments in the body, all of the spaces and valves and passages; and they recognize when a blood-glucose level is too high or too low.  Lawyers know the names of the laws, how to look them up, and how to argue that one action either does or does not fit a particular interpretation of that law.  It’s a bit of great humor to me that Teachers measure and interpret children according to a variety of yardsticks, too.

It’s just that we forget sometimes that children are not in school to be measured; they’re in school to learn how to measure.

Where We Agree

I don’t think that Will and I are in any disagreement about one of the key challenges, which is finding ways to express what teachers want and need from the non-professionals: the politicians, the parents, and the other stakeholders.  We’re not medieval churchmen trying to save the scraps of a fallen empire from the dark ages warlords; we can’t claim God is on our side, and wow our opponents with secret knowledge of when an eclipse will occur.  We’re not all shop teachers, holding boards up to the window to look for the perfect angle.

But I wonder if a teacher focused on measurement for a few weeks in their lesson planning, if they did not see a marked improvement in the quality of their students’ work?  When we ask ourselves, before we begin each lesson, “what measurement process am I teaching today? What method for interpreting information am I offering to my students?” it may be that on some days we’re teaching fact-gathering.  On other days, we’re teaching the construction of logical thought.  On other days, we’re concerned with helping students string facts together into a story.  On other days, we’re teaching how to use a ruler or a thermometer, more basic tools that nonetheless reveal important truths.

But the centerpiece of our work as teachers, regardless of what subject we teach officially, is the work of helping students to measure the world.  And if a school isn’t doing that — then maybe that school should be closed.

Alchemy: vervain

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I am not by training or comprehension a chemist. But there is something about the chemical processes I quite enjoy.  The process from alchemical work that I love most, is of course calcinatio: the burning to white ash of a plant material (any material, really) soaked in alcohol.

Part of it is that I have an excuse to start up a fire in my athanor – excuse me I mean my Weber grill.  I put an iron pan of an herb soaked in grain alcohol on the grill, and wait while it gets hot enough to burn off the alcohol, and for the flames to turn blue.

But it’s only now — the twelfth or fifteenth time that I’ve done it — that I have any clear sense of what’s happening. The carbon and other flame-sensitive volatiles in the plant matter are burning off and being converted to vapor and smoke.  What is left is the residuum: the plant’s essentials that are not reducible by Fire to Air (smoke and vapor)  or Water (liquid converted to gas). This is the Earth of the plant, the remnant.

It’s interesting that even a century after the development of the periodic table of the elements, I can still see on a gross scale exactly what the medieval alchemists were getting at when they spoke about the four elements — and that it’s easier to understand what’s happening in an alchemical way than it is to understand in the terminology of modern chemistry.  There’s an opportunity for learning here, really: can we teach chemistry directly? Or should we help students understand the alchemical origins of modern chemistry, through direct observation of Fire (heat), Water (liquids and solvents), Air (the behavior of gasses), and Earth (irreducible solid components)?

Early in the process, of course, the visuals are stunning. Watching a fire burn at night, and gathering information about what’s happening, is deeply seductive.  This particular Vervain (Verbena officinalis) burns blue, partly from the alcohol used as a solvent to begin the breakdown of the plant matter, and partly from the vaporization of the burnable matter in the plant — cellulose, carbon, and volatiles.  What remains in the Earth of the plant — essential solid components that do not burn at these temperatures, and whiten (or at least gray-en) under the application of extreme heat until the ashes turn orange and sometimes even cherry-red in the flame.  When the flame cools and the charcoal dies down, you’re left with white ashes.

Eventually, this will be an alchemical salt — a spagyric tincture added to white ash of the same plant — which was one of the first efforts, historically, at making more potent medicines from herbs with known medicinal value.    I can’t say that I will ever want to take this salt.  But I do find it interesting and valuable to realize that an internal change has taken place in me from doing the work.

Namely, this: we are not complicated creatures, us humans.  We learn often learn things by doing, by copying, and by understanding a range of experiences with a range of words and language that matches what our senses tell us.  When the language drifts too far from what we can see, touch, taste, feel, smell, and experience as a whole — we tend to lose ourselves in abstraction.  But something burning on a fire is real and present in a way that engages all the senses, and teaches us things about the world that our ancestors knew intimately well.

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