School: Redesign Homework


Around this time of year, I always think about how I’m going to re-design my teaching for the fall semester.  It doesn’t matter whether I’m teaching or not, I think about it.

A recent conversation with Dave Gray of XPLANE, Inc. got me thinking about his heuristic matrix from the book Gamestorming which he wrote with Sunni Brown. A heuristic matrix looks a lot like the grid from a spreadsheet, and which I used several years ago to redesign homework.

That grid looked something like this…

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I identified a bunch of broad categories that I wanted my students to learn about.  In this example, based on the broad theme of teaching about Ancient Greece, I have categories like religion, and aspects of art history, politics, literature, philosophy, and science and technology.

I then identified a variety of styles that I wanted my students to learn to write in. These formed the individual columns of the heuristic matrix.    These included paragraphs dealing with compare and contrast writing, where the same paragraph alternates between two different viewpoints or styles. There was also descriptive writing, involving a top-to-bottom explanation of a thing or a place.  Narrative writing, the description of a beginning-to-end process, was another category. Persuasive paragraphs offer reasons for holding an opinion, and attempt to persuade the reader to accept a particular viewpoint.  Exposition attempts to define or explain a person’s ideas or opinions without forcing them on the reader.  Reading comprehension, on the other hand, asks students to engage with an actual historical text.  Self-directed research is another category — independent projects of various kinds.

I haven’t filled in the heuristic matrix completely. Some of this is left as an exercise to the reader (which is to say, perhaps, that I’m lazy or that I don’t wish to think all of this through, or maybe that I don’t wish to share all of my thought process at once).  But the overall structure should be discernible.

I tried to do something similar with a mathematics heuristic grid for a lower grade, perhaps grade 2, grade 3, or grade 4.

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I’m not a mathematics teacher, so you’ll notice that the grid isn’t completely filled in.  But you’ll see what I’m trying to do… I’m trying to come up with a variety of mathematics exercises and activities that don’t revolve exclusively around the traditional “do these 20 problems to learn a type of procedure” worksheets or homework lists.  This is about inventing new forms of assignments and identifying how these can be used to teach or refresh skills that lie outside the usual curriculum norms.

And it’s important to note that YOU don’t have to fill in a grid completely, either. You may only generate one or two useful ideas from a heuristic matrix.  Yet if a few of those ideas have the chance to reinvigorate your teaching, that may be worth i.


“Greco-Roman” outfit 

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Be Hellenistic, not fatalistic. 

A friend of mine is going to the annual convention of the Society for Creative Anachronism, otherwise known as the Pennsic War. He needed some garb. The group he travels with are classicists, so a simple Roman-era tunic and a long rectangular himation or peplos — really a simplified toga — are all he needs. I figured out a way to cut two tunics, one sleeved and one sleeveless, from the fabric he brought me. This is the sleeved one. 

I’ll have to wait until his fitting this evening for photos of the pseudo/proto-toga. It’s simply impossible to photograph in a way that makes it look like something other than a long rectangle of cloth with stripes at the ends. Wrapped around a person it’ll look quite different, I believe. 

I have a new sewing machine. I did these projects with the old sewing machine because I’m at a critical stage in quilting two crib-sized quilts, and all my spare thread is wound onto bobbins of the old machine. But if I took down and store the old machine, I’d never finish those quilts. 

But… after this weekend, and this project for my friend, I got most of the quilting done. So I’m ready to store my old machine as of today. I have some time this afternoon, so I’m going to use the new machine to put some decorative stitching on the two tunics and the proto-toga. 

Quilt: triangles

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Had you told me at the start of this project what a terrible construction system triangles and hexagons were, I might not have believed you. I admit that. But I hadn’t expected them to be quite so much of a bear to construct as they actually were.

It seems simple enough, really. Cut triangles slightly larger than you want them to be. Clip the corners, or imagine them clipped, and imagine a 1/4″ seam allowance around each piece. As you sew them together, either in rows or in hexagon ‘blocks’, the pattern you’ve chosen begins to emerge. Alternate solid and patterned fabrics for a more elegant design with much more visual interest.  Seems simple, right?

Nobody explained what a bear those corners are in the middle. If you want a really elegant point on your star or on the fan of triangles at the center of your hexagon, that’s about 3x more work than “just sewing triangles together.”

Still, sooner or later you have to do it. It’s not possible to just keep building quilts out of squares forever. Sooner or later, you have to grit your teeth, stomp your feet, and assemble a quilt that uses triangles or hexagons.  You make your templates, slice up your fabric, and get to work sewing.
And things go wrong. You mis-marked a triangular piece. You didn’t mark a piece. You didn’t clip the corner of a triangle. You didn’t clip the correct corner at the right angle of a triangle. You removed too much of a corner. Your chalk wore off the piece of fabric. THere’s a dozen (a million) things that could go wrong. It doesn’t matter. “Build the whole prototype,” says a friend of mine in the engineering business. “That way, you know where the serious mistakes are.” There are a lot of serious mistakes in this quilt top.

Still, there are some successes.  Some of my center corners are pretty spot-on.  Some of my external centers look pretty good, too.  Some portions of this quilt look awesome.  And some percentage of those who see the finished product will never know there were any mistakes at all, once I’m done quilting it within an inch of its life.   The perfect is the enemy of the good, wrote Plato, as the words of Socrates.  And so it seems here, too — the more perfectly I try to make this quilt on the first try ever with this technique, the more likely the quilt will wind up unfinished in a drawer for months out of frustration.

And so it is that the quilt is here — pinned to its batting and backing, and ready for the quilting-sandwich: layers of stitching that will bind the upper layer to the bottom layer through the middle layer.  And then there will be bias tape to make, and edge-binding.

And then it will be done.

It’s certainly not the best quilt that I’ve ever made.  It’s certainly the best triangle-based quilt I’ve ever made, given that it’s the only triangle-based quilt I’ve made (though not the only hexagon-based quilt I’ve made — see English Paper Piecing and some of my further insight.)  But most of what it is, is a learning experience.  I’ve made this quilt, and I now know enough of the process that a range of similar patterns and workings are now open to me.  I can do this again and again, as needed and as desired.

Just don’t ask me to make a triangle-based quilt for free. Ever.

Quilts: cut and sew

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I want to learn the core angles of quilt sewing. Since most quilts are simply tiling patterns writ in fabric, a large percentage of this work is done in squares or rectangles on the one hand; or triangles, diamonds, and hexagons on the other.  These shapes and their variants are pretty much the only ones that tile easily. 

I started by making some six-pointed stars using a flat triangle pattern. When three flat triangles are grouped, one gets an equilateral triangle. when one groups six equilateral triangles, a hexagon results. Half-hexagons can be used to form an edge to a field of hexagons, turning a hexagonal tile patttern into a rectangular ore square field. Many hexagons and half-hexagons together form a quilt… Who knew?? 

Much of the early work consists of lining up sheets of fabric and then putting a template on them to slice out triangles.  I felt like I cut out hundreds if not thousands of triangular fabric tiles yesterday. You can see the piles of them in the first photograph here, a lot of grays and blacks and very dark blues, with some Celtic knot work fabric, too. 
Once that fabric gets sorted by color and type, it begins to feel like not enough, though.  

Nonetheless, one has to keep going. Breaking up big pieces of fabric into smaller ones just results in a mess. Fabric has a warp and weft that holds it together.  Once you start cutting into it, you break up its internal integrity and it will start to unravel.  You’ve dissolved the bonds that hold it together.  Now you need to begin to recombine it.  

The key things to consider about that recombination are color, texture and weave.  People like complementary colors rather than clashing colors.  They like patterns, but they don’t want too many patterns next to one another.  There need to be places where they can rest their eyes on relatively neutral hues, so that a patterned fabric can then grab their attention. That’s a lot to hold together in mental clarity. 

And so we begin with somethiing relatively neutral, and matching the stars in certain particulars.  
Hexagons have a particular logic to them when you start with triangles, as I have. Quite naturally the three stars are going to draw the eye first and foremost.  So the quilt has to be built out around them. I have enough fabric to make twelve or thirteen more of these two-grays hexagons, but not enough to make a whole quilt this way. So my next step is going to be to construct another pleasing hexagon design, and interleaved the two-tone gray hexagons with that new design, while trying not to distract from the stars.  My partner also recommended making more stars but in radically different colors. That could work too. 

Yesterday at the fabric store, I met a woman who was making fabric furnishings for a Russian Orthodox congregation: linens for the altar and stoles and robes for the clergy.  She was working on the stiff linen and brocade chalice cover, and had come in to find some more gold braid for the cover. It was beautiful. I have a long way to go yet, but it was a reminder that all kinds of people need all kinds of custom sewing work. Increasingly I’m prepared to handle it. 

Commonplace book

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I’ve been experimenting with commonplacing. In the 1600s through the early 1900s, the commonplace book was a system of gathering texts and quotations in one place, usually a blank notebook, for the purpose of recollecting information and remembering key ideas about virtue, truth, health, leadership or what have you.

Doctors used them for recording “pearls”, key ideas about a pair or triad of symptoms and a specific diagnosis. Politicians used them to note useful quotations for speeches, and historians used them to categorize events and trends in the age before statistical analysis made more nuanced discussions possible.

I’m using a Moleskine/Evernote-branded softcover notebook to record poetry that I’m trying to memorize; pieces go into the book in the order that I’ve memorized them or intend to commit them to memory.  I attended a Burns Night supper in January last year; and I made an effort to memorize Robert Burns’ Epigram on Bad Roads, which is the first poem in the book, as you can see.

“I’ve now arrived —
thank all the gods!
Through pathways both rough and muddy;
a certain sign that makin’ roads
is no’ this people’s study.
Though I’m not with Scripture crammed
I know the Bible says
that heedless sinners shall be damn’d —
unless they mend their ways.”

It was nice and useful to memorize a funny poem for a change, instead of a serious one.  Most of my poetry tends to be pretty serious; and I tend to memorize serious poetry.  It’s a useful reminder that I should from time to time work on funny poetry as a form — both to memorize, and to write.  Something to practice!img_5468

Further on in the book, in the last three pages or so, is an index page listing the poetry and other elements I’ve put in the book.  Here’s part of that index, listing on page 1 the Epigram on Bad Roads, and Langston Hughes, and John Keats, and so on.   William Blake’s Auguries of Innocence takes up pages 7-11. You can see that I’m working on memorizing quite a lot of Thomas Taylor’s translations of the Orphic Hymns, as well, and the Aleister Crowley hymn for Coffee (not Covfefe).  The index continues; I’ve listed all of the pages, even if I haven’t filled them yet.  It’s rather more similar to the Digital Ambler’s Vademecum, really, or an Enchiridion, than a true commonplace book. A true commonplace book should not only have a table of contents at the beginning, but also an index by subject, such as hope or valor or kindness or coffee. Such an index would help one find appropriate material within the book more rapidly and easily.

img_5469Not everything in the book is poetic. Two pages include a list of all of the U.S. Presidents in order, which I’m working on memorizing, not just with their names but also their years.  It’s occurred to me frequently that this list serves a useful purpose as a time-counter; it’s much easier to remember when something occurred in time if you remember who was president at the same time.  That’s part of the reason why I also have the similar list of the Kings and Queens of England a few pages on from this — The English royal list extends back in time to 1066, and it creates a useful parallel list for European affairs.  Maybe I should also work on the list of the Emperors of Japan…


Geometry Book


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 


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.

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