Guest Blogger: Scott Kim, IV

Well, this is the last installment of Scott Kim’s blog post on transforming mathematics education!  These are all important issues, and when you think about them all at once, they seem insurmountable.  It takes each of us working one at a time in our local communities, as well as groups of us working together in broader communities, to effect a change.  What is crucial is that we not only discuss these issues, but we do something about them.  Those of us who participated in the discussion a month ago at the Bay Area Mathematical Artists Seminar are definitely interested in both discussing and doing.

Scott suggests we need to move past our differences and find constructive ways to act.  No, this isn’t easy.  But we need to do this to solve any problem, not just those surrounding mathematics education.  It’s time for some of us to start working on these issues, and many others of us to continue working.  We can’t just sit and watch, passively, any more.  It’s time to act.  What are you waiting for?

Level 4. Resistance from SOCIETY (quarreling crew)

Sailing is a team sport. You can’t get where you want to go without a cooperative crew. Similarly, math education reform is a social issue. You can’t change how math is taught unless parents, teachers, administrators and policy makers are on board. Most adults cling to the way they were taught as if it were the only way to teach math, largely out of ignorance — they simply aren’t aware of other approaches.

Here are three ways society needs to change the way it thinks about math and math education in order for change to happen.

4a. Attitude. The United States has an attitude problem when it comes to math teachers. First, we underpay and under-respect teachers. And the situation is only getting worse as math graduates flock to lucrative high-tech jobs instead of the teaching profession. The book The Smartest Kids in the World and How They Got That Way describes how FInland turned their educational system around — they decided to pay teachers well, set high qualification standards, and give teachers considerable autonomy to teach however they think is best, with the remarkable result that student respect for teachers is extremely high.

Second, it is socially acceptable, even a badge of honor, to say that you were never good at math. You would never say the same thing about reading. Many people do not in fact read books, but no one would publicly brag that they were never good at reading. Our society supports the idea that parents should read to their kids at night, but perpetuates the idea that being no good at math is just fine.

Solution: respect teachers by paying them well, and value math literacy as much as we value reading literacy.

4b. Vision. The national conversation about math education in the United States is locked in a debate about whether we should teach the basics, or the concepts. As a result we see over the decades that the pendulum swings back and forth between No Child Left Behind and standardized testing on one extreme, and New Math and Common Core Math on the other extreme. As long as the pendulum keeps swinging, we will never settle on stable solution. The resolution, of course, is that we need both. In practice, schools that overemphasize rote math find that they must supplement with conceptual exercises, and schools that overemphasize conceptual understanding find that they must supplement with mechanical drill. We need both rote skills and conceptual understanding, just as kids learning to read need both the mechanical skills of grammar and vocabulary, and the conceptual skills of comprehension and argument construction.

Solution: We need a vision of math education that seamlessly integrates mechanical skills and conceptual understanding, in a way that works within the practical realities of teacher abilities and schoolday schedules. To form a vision, don’t just ask people what they want. A vision should go further than conventional wisdom. As Henry Ford is reported to have said (but probably didn’t), “If I had asked people what they wanted, they would have said faster horses.” Or as Steve Jobs did say, “It’s really hard to design products by focus groups. A lot of times, people don’t know what they want until you show it to them.”

4c. The will to act. As a child I grumbled about the educational system I found myself in. As a young adult I started attending math education conferences (regional meetings of the National Council of Teachers of Mathematics), and was astonished to find that all the thousands of teachers at the conference knew perfectly well what math education should look like — full of joyful constructive activities that challenged kids to play with ideas and think deeply. Yet they went back to their schools and largely continued business as usual. They knew what to do, but were unwilling or unable to act, except at a very small scale.

Solution: Yes, a journey of a thousand miles starts with a single step. And change is slow. But if we’re to get where we want to go, we need to think bigger. Assume that big long lasting change is possible, and in the long term, inevitable. As Margaret Mead said, “Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it’s the only thing that ever has.” I’m starting my small group. Others I know are starting theirs. What about you? 


Guest Blogger: Scott Kim, III

Today, I’ll post the third installment of Scott Kim’s blog on transforming mathematics education.  But before jumping into that, I want to share a little about Bridges 2018, which just took place in Stockholm, Sweden.  Because of my move and career shift, I decided not to go — at the time I would’ve needed to make travel arrangements, I didn’t even know whether I’d be living on the West coast or the East coast when I’d need to catch my flight!

In any case, my Twitter feed has been buzzing recently with tweets from Stockholm, and some have featured participants in the Bay Area Mathematics Artists Seminars.  Monica Munoz-Torres tweeted about Frank Farris’s talk on vibrating wallpaper patterns, which you may recall he gave at our March meeting of the BAMAS at Santa Clara University.

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And the Bridges Program Committee announced that Roger Antonsen won Best in Show for 2-dimensional Artwork for his piece, “Six Perfect In-Shuffles With 125 Cards and Five Piles.”

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Congratulations, Roger!

OK, now we’ll move on to Scott Kim’s commentary on transforming mathematics education.  His next point addresses a prevailing issue in mathematics education:  advances in technology relevant to teaching mathematics are moving along at a rate which outpaces curriculum development.

And it’s not just that.  Even if curriculum could be reimagined at a pace to keep up with technology, teachers would need to be retrained to use the new curriculum with the new technology.  Not just retrained on the job, but while students at university — meaning that institutions of higher education would need to have their faculty keep up as well.  This means resources of time and money, and the willingness and ability of mathematics and education faculty, as well as school districts, to embrace change.  A tall order, to say the least.

I could go on at length about this topic, but let’s give Scott a chance.  Again, if you just can’t wait for the fourth installment, feel free to go to Scott’s blog, where you see the post in its entirety.

Level 3. The wrong MATH (sailing in the wrong direction)

The mathematics we teach in school is embarrassingly out of date. The geometry we teach is still closely based on Euclid’s Elements, which is over 2000 years old. We continue to teach calculus even though in practice calculus problems are solved by computer programs. Don’t get me wrong: geometry and calculus are wonderful subjects, and it is important to understand the principles of both. But we need to re-evaluate what is important to teach in light of today’s priorities and technologies.

Here are three ways to update what we teach as mathematics.

3a. Re-evaluate topics. The Common Core State Standards take small but important steps toward rebalancing what topics are taught in math. Gone are arcane topics like factoring polynomials. Instead, real world mathematics like data collection and statistics are given more attention. As Arthur Benjamin argues in a brief TED talk, statistics is more important than calculus as a practical skill.

Solution: give kids an overview of mathematical topics and what they are for, long before they have to study them formally.

3b. Teach process. The widely used Writer’s Workshop program teaches the full process of writing to students as young as kindergarten. The process accurately mirrors what real writers do, including searching for a topic, and revising a story based on critique. We need a similar program for the process of doing mathematics. The full process of doing math starts with asking questions. Math teacher Dan Meyer argues passionately in his TED talk that we do students a terrible disservice when we hand them problems with ready-made templates for solution procedures, instead of letting them wrestle with the questions themselves. Here is my diagram for the four steps of doing math. Conrad Wolfram created a similar diagram for his Computer-Based Math initiative.

Solution: give kids an explicit process model for problem solving.

3c. Use computers. In an era where everyone has access 24/7 to digital devices, it is insane to teach math as if those devices didn’t exist. In his TED talk, Conrad Wolfram points out that traditional math teachers spends most of their time teaching calculating by hand — the one thing that computers do really well. By letting students use mathematical power tools like Mathematica and Wolfram Alpha, teachers can spend more time teaching kids how to ask good questions, build mathematical models, verify their answers, and debug their analysis — the real work of doing mathematics. And students can work on interesting real-world problems, like analyzing trends in census data, that are impractical to tackle by hand.

Solution: build and use better computer tools for doing math. Revamp the curriculum to assume the presence of such tools. Emphasize solving interesting problems, de-emphasize or delay learning about the mathematical mechanics for carrying out the computations. In other words, teach mechanics on a need-to-know basis.

Next week will feature the Level 4 of Scott’s remarks.  Until then!

Guest Blogger: Scott Kim, II

Today, I’ll continue with reblogging Scott Kim’s in-depth post about transforming mathematics education.  You might want to read last week’s post to get caught up.

I will say that the discussion generated quite a bit of interest.  Participants have been actively responding to each other in a very lively email thread.  The comments and discussions are still ongoing — I am having a hard time keeping up with them!  But in a later post, I’ll summarize some key ideas and observations made by members of the group.

But for now, I’d like to turn it over to Scott Kim.  Again, if you’re anxious to read the entire post, please feel free to go to his blog.  Or just be patient….  But you can see by looking at the heading that Scott is addressing a very important issue next.  I can still recall — when teaching gifted high school mathematics and science students — really understanding where the question “When am I going to ever use this?” comes from.

The answer is pretty simple.  Bright students want to know.  When I first started teaching at university, I thought it was the students’ job to find motivation for doing mathematics — after all, they were paying a lot of money for their education.

But I eventually realized that there are only about three months between the end of high school and the beginning of college.  Nothing magical happens to students to transform them into self-motivated human beings, hungering for knowledge for its own sake.

Actually, one of my goals is never to hear the question “When am I ever going to use this?” again.  If I do a good job teaching and motivation concepts, students will already be able to answer that question, and won’t need to ask it any more.

Yes, it’s a more challenging way to teach.  But I can tell you, for me, it has been worth it.

Now I’ll let Scott take over.  Enjoy!  We’ll look at the third level next week.

Level 2. Lack of MEANING (leaks)

The most common complaint in math class is “when are we ever going to use this?” And no wonder; the closest most kids get to using math meaningfully is word problems, which are typically dull mechanical problems, dressed up in dull mechanical narratives.

Traditional mathematics education focuses on teaching rote computational procedures — adding, dividing, solving quadratic equations, graphing formulas, and so on — without tying procedures to meaningful situations. Unfortunately most adults, including many teachers and administrators, think this is how it must be. But teaching only the rote procedures of math is like teaching only the grammar and spelling of English, without explaining what words mean, or letting kids read books. Mechanics without meaning is not just deathly boring, it is much harder to learn.

Here are three ways to plug the leaks of meaningless math.

2a. Use math. In our increasingly digital society, kids spend less and less time playing with actual physical stuff. All the more reason to get students out of their desks and into the world, where they can encounter math in its natural habitat, preferably integrated with other subject areas. My friend Warren Robinett told me “a middle-school teacher I knew would, after teaching the Pythagorean Theorem, take the kids out to the gym, and measure the length and width of the basketball  court with a tape measure. Then they would go back to the classroom and predict the length of the diagonal. Then they would go back to the gym, and measure the actual diagonal length. She said some of the kids would look at her, open-mouthed, like she was a sorceress.”

Solution: use problems that kids care about, and excite student interest.

2b. Read about math. Before we learn to speak, we listen to people speak. Before we learn to write, we read books. Before we play sports, we see athletes play sports. The same should apply to math. Before we do math ourselves, we should watch and read about other people doing math, so we can put math in a personal emotional context, and know what the experience of doing math is like. But wouldn’t reading about people doing math be deadly boring? Not if you are a good story teller. After all, mathematics has a mythic power that weaves itself into ancient tales like Theseus and the Minotaur. My favorite recent math movie is a retelling of the classic math fable Flatland, which appeals as much to my 7-year-old daughter as to my adult friends. Here’s a list of good children’s books that involve math.

Solution: read good stories about math in use.

2c. Ask your own questions. In math class (and much of school) we answer questions that someone else made up. In real life questions aren’t handed to us. We often need to spend much time identifying the right question. One way to have students ask their own questions is to have them make up their own test questions for each other. Students invariably invent much harder questions than the teacher would dare pose, and are far more motivated to answer questions invented by classmates than questions written by anonymous textbook committees. Mathfair.com goes further to propose that kids build and present their own physical puzzles in a science-fair-like setting. Kids can apply whatever level of creativity they want. Some focus on art. Some on story. Others add new variations to the puzzles or invent their own.

Solution: Give kids freedom to ask their own mathematical questions, and pursue their natural curiosity.

If we plug the leaks of meaningless math, we will grow a generation of resourceful mathematicians who understand how to solve problems. But are we teaching the right mathematics?  (To be continued….)

Transforming Mathematics Education: BAMAS, X

This past Saturday marked the tenth meeting of the Bay Area Mathematical Artists Seminars.  You might recall (see the post about Bay Area Mathematical Artists Seminars, VI) that at a recent meeting, we had a very stimulating dinner conversation about the future of mathematics education, with Scott Kim helping to guide the conversation.

Everyone was so engaged, it was unfortunate that the conversation had to come to an end.  So I invited Scott to lead a more formal discussion at a later meeting of the BAMAS.  We met at BAMAS member Stacy Speyer’s place — thanks for hosting, Stacy!

The discussion was quite animated.  Scott prepared a handout based on a lengthy blog post he wrote about various issues revolving around mathematics and mathematics education.  He graciously gave me permission to reblog his ideas.  The post is rather lengthy, so I’ll share it in installments.  You can go to Scott’s blog yourself if you can’t wait to read more.  So without further ado, I’ll let guest blogger Scott Kim take the wheel.  His original post was dated July 6, 2014.

Navigating Math Education

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Imagine that you are a sailor on a leaky boat that is on fire, sailing in the wrong direction, with a quarreling crew. Which problem would you fix first?

Well, that depends. If the leak is slow and the fire is raging, then you would put out the fire first. If the leak is gushing and the fire is small and contained, you would fix the leak first. It makes sense to fix the most urgent problem first.

What you would NOT do is fix one problem and declare victory. If your goal is to get to your destination safely, then you must fix ALL the problems, no matter how difficult. Anything less will not get you where you want to go.

Such is the situation with math education. The problems are so difficult and so numerous that it is tempting to fix one problem, and give up on the rest. And certainly we have to prioritize if we are to make progress. But if we are to get the ship of math education back on course, then we, collectively, must fix ALL the problems of math education. Nothing less will get us where we want to go.

Fixing all of math education may sound impossible or impractical. And indeed it is a formidable challenge. Well-meaning entrepreneurs who have launched successful businesses frequently grind to a halt when they try to start their own innovative schools. Resistance comes from all sides — standardized testing, textbook publishers, parents, administrators, government officials, and the students themselves trying to get into college.

But change is in the wind. America is losing its competitive edge, colleges are becoming impractically expensive, and the internet makes us dream of free education right now for everyone. I say we face the problem with eyes wide open, assess the full range of challenges we face, and look for the smartest moves that get us where we want to go.

With that in mind, here is my survey of the problems plaguing math education, and steps we can take to fix them. I’ve grouped the challenges into four levels that range from the tactical to the strategic: Mechanics, Meaning, Math, and Society.


Level 1. Faulty MECHANICS (fire)

The most obvious and urgent problem is that the mechanics of math are taught as a series of blink and you’ll miss it lessons, with little opportunity to catch up.

This one-size-fits-all conveyor belt approach to education guarantees that virtually everyone gradually accumulates holes in their knowledge — what Khan Academy founder Sal Khan calls Swiss cheese knowledge. And little holes in math knowledge cause big problems later on — problems in calculus are often caused by problems in algebra, which in turn are caused by even earlier problems with concepts like fractions and place value.

Here are three ways to fight the fire of poor pacing.

1a. Self-paced learning. The Khan Academy addresses the urgent problem of pacing by providing short video lectures that cover all of K-12 math. While the lectures themselves are fairly traditional, the online delivery mechanism allows students to work at their own pace — to view lectures when and where they want, and to pause and rewatch sections as much as they need. All lectures are freely available at all times, so kids can review earlier concepts, or zoom ahead to more advanced concepts. Short online quizzes make sure that kids understand what they are watching. And with an online dashboard that shows exactly how far each child has progressed, teachers can assign lectures as homework, and use class time to tutor kids one on one on exactly what they need.

Solution: the “flipped classroom.”

1b. Visual learning. I love the Khan Academy. My son hated it, because he, like many students, is a visual learner, and Sal Kahn’s lecture stick largely to traditional symbolic math notation. He would have done better with a visual experiential curriculum. Some kids are primarily audio or kinesthetic learners, some learn best socially. The bottom line is that different kids learn in different ways, and no one way is right for everyone. Education needs to address all learners, not just kids who learn in words.

Solution: teach every lesson three different ways.

1c. Testing for understanding. Nothing can change in education unless testing changes. Traditional standardized tests born of the No Child Left Behind era use multiple choice tests that assess only rote memorization of routine math facts and procedures. The new Common Core State Standards for mathematics, now entering schools across the nation, replaces standardized multiple choice tests with richer tests that include essay questions graded by human beings — a better way to assess mathematical understanding.

Solution: better assessment.

If we douse the fire of poor pacing in math education, we will increase test scores and student confidence. But there is more to mathematics than teaching the mechanics well.

I hope your interested is piqued!  Scott will continue next week….

Guest Blogger: Percival Q. Plumtwiddle, II

As promised, here is the second installment of my friend Percy’s essay on the significance of our Earthly existence being three-dimensional.  I hope these few words give food for much geometrical thought….

On Threeness (continued…)

Popping over a few dimensions in the meanwhile, the knowledgeable geometer is certainly aware that in dimension four and, indeed, in all dimensions exceeding this critical dimension, the situation is somewhat overwhelming, to say the least. So many thousands of uniform polyhedra abound that the prospect of their enumeration appears beyond the scope of even the most ambitious geometer. (I might recommend such a project, however, to an advanced practitioner of one of the Eastern religions who might, as a result of sufficient spiritual development, continue the undertaking in subsequent reincarnations.)

But in dimension three, how manageable our delight!  Seventy-odd uniform polyhedra clog the pores, multitudinous compounds and stellations frolic with the celestial spheres, etc.,  such as to render the average mortal awestruck.  Details are, of course, voluminous (though within reason), and more eminent geometers than yours truly have authored copious memoranda on said frolicking.  And there is yet much left unsaid, or perhaps, unmemorandized.  To presume that our friendly Supreme B. was unaware of such phantasmagoria would be heresy first class.  Fie!, I say, to such presumption!

As convincing as I have striven to be, I shall endeavor, with difficulty, not to be so narcissistic as to suppose that I have left no room for doubt in the reader’s mind. And so I entice with a few non-Euclidean morsels…

I shall not doubt the reader’s familiarity with the method for calculating the length of a circular arc, given the appropriate parameters. Perhaps less well-known is the method for calculating the area of a spherical triangle, with the attendant formulae relating the various parts of such a triangle only slightly more obscure. I purposefully omit their statements, secretly hoping that the reader will rediscover their simplicity and elegance afresh… In any case, the knowledgeable reader is well acquainted with the vast number of beautiful yet stable geodesic structures which may be constructed, the data for which constructions being directly calculated by way of the aforementioned simple formulae. Let not the skeptical reader be amused by such hyperbole — for one who has indeed fabricated such a structure or two with a few sheets of stiff paper and a bit of rubber cement will certainly view my descriptions as grave understatement.

In any event, as the informed reader is well aware, the formula for finding the area of the spherical triangles mentioned above may be successfully implemented by a child of ten, or perhaps a precocious child of eight, with a minimum of instruction, a pocket calculator, and, if necessary, the promise of an ice cream in the event of their successful completion. The informed reader is equally well aware that analogous calculations in four dimensions may be successfully implemented by no less that a ranking member of the cranially elite given the necessary data (and perhaps a few extraneous ones for good luck), a personal computer, and, if necessary, the promise of tenure in the event of their successful calculation. The superiority of the three-dimensional scenario, both fiscally and otherwise, is evident.

I pause, as a prelude to a conclusion, to ask the reader to reflect upon his or her stance vis a vis the proposed heightened status of Threeness. I imagine the reader to be of one of four minds. First, the reader may be insufferably bored at the current exegesis, in which case he or she will have undoubtedly ceased its perusal long ago. In this case, there is no reason for a continuation of discourse. Secondly, the reader may, to my ineffable delight, be wholly in agreement with the propositions contained herein. Should this be the case, I am not compelled to offer further justification for them. Thirdly, the reader may, to my profound disappointment, be immovably opposed to my thesis. (As an apology to those readers who fall in the second category, I must, rather that be labelled as a righteous fanatic, admit this frankly incomprehensible possibility.) I would not prolong the displeasure of such readers, those that there may be. Fourthly, the reader may be genuinely undecided, perhaps having never mused upon such matters previously, or perhaps still wrestling with deep foundational issues and at an impasse with regard to their resolution. In the former case, I urge an immediate excursion to that local library known for its excellent collection of volumes on the subject, followed by a thorough study of those treatises, and then a rereading of the current manuscript. In the latter case, I believe the matter rather more philosophical than mathematical, and I might suggest several worthwhile Buddhist sources (and caution the reader to avoid the twentieth-century European existentialists). In neither case would I find a need to argue further in this matter of Threeness.

And so, dear reader, I take my leave. It is my fondest hope that these few words have, at the very least, given cause for a leisurely intellectual frolic amidst sunny geometrical meadows, and at the very best, profoundly elevated the spirit. In either case, my work will not have been in vain.

Guest Blogger: Percival Q. Plumtwiddle, I

As I’m out of town for a few weeks, I thought I’d invite my good friend Percy Plumtwiddle to be a guest blogger.  He is an ardent enthusiast of geometry, and we have had many animated discussions on the geometry of three and higher dimensions.

But he has his stubborn side (don’t we all?).  And yes, his language is flowery and over the top, but well, we all have our eccentricities, too.  Nonetheless, he makes some interesting points — just how interesting, I’ll have to let you decide.

Also, his essay, while not too long, is a bit long for just one post — so I’ll break it into two installments.  I’ll return to my keyboard in a few weeks.  In the meantime, enjoy Percy’s unique perspective on geometry, and more broadly, life in general….

On Threeness

Were I God — and I assure you, dear reader, that such is not remotely an aspiration; and were it, even my dear friends and those who hold me in highest esteem (those that there may be) would chortle audibly at the consideration of such a prospect — and were it incumbent upon me to create a sentient being or two (admittedly, I would be somewhat embarrassed, were I God, to admit to the creation of many such beings which presently spoil an otherwise hospitable planet) — and had I the necessity of establishing a place of residence for the aforementioned sentient beings (not that it would be required, matter being at times inconvenient, especially when taking the form of a piece of chocolate cake hidden under one’s outer garments at the precise moment when a head-on collision with a rotund nephew is imminent) — I would not hesitate for a moment in making such a domicile, in dimension, three.

Lest the frivolous reader imagine such a dimensionality to be merely the whim of a somewhat eccentric, although omniscient, Supreme Being, or lest the pious reader imagine that such a dimensionality foreshadows the fairly recent (according to even the most conservative of cosmological estimates) doctrine of the Trinity popular in some Western religions, and lest the banal reader imagine that this third dimesion would be, as goes the cliche, a “charm,” and lest the reader of belles lettres be tempted to make a thoroughly trite reference to MacBeth — apply the cerebral brakes forthwith!  I intend to provide sound reason to dispel such imaginations, should the reader find it bearable to forage among slightly less green pastures.

I appeal not only to the reader’s intellect — being, unfortunately, acquainted much more intimately with self-proclaimed members of the cranially elite than I presently desire — but also to the reader’s aesthetic sensibilities, as potentially volatile as such an appeal may be.  But I have profound faith in this universal Threeness, so I proceed, fully cognizant of the perilous quagmire in which I may soon find myself immersed, should the fates allow.

Indeed, my argument is entirely geometrical, being of the school of thought that such argumentation is lacking in no essentials.  Should the reader be otherwise disposed, I urge an immediate cessation of the perusal of this document, the administration of a soothing tonic, and perhaps the leisurely reading of a light novella.

Several remarks of the Euclidean persuasion present themselves.  Oh, indeed, we are all familiar with the usual formulae for distance, angle, etc., etc.  These pose little difficulty, even in thirty-seven dimensions, once we are introduced to our good friend the Greek capital sigma.

But let us turn our eyes to the imaginitive world of polyhedra, a subject quite dear to my heart.  Another discussion entirely!  It is within this discussion that the force of my words comes to bear.  For indeed, I should be greatly surprised if when, upon strolling through the Pearly Gates and bidding good-day to St. Pete, I find that God did not have amongst his cabinet (a gaggle of geese, perhaps, but the precise nomenclature for an approbriation of Archangels momentarily eludes me) one for whom polyhedra are a consuming passion.

And what would give cause for raised eyebrows?  Follow along.  Examining your regular polyhedra of dimension two, we find squares, triangles, etc., etc., and, while these are no doubt pleasing to the eye, one might find oneself a trifle bored after pondering a polygon of forty-six sides.  For next comes the all-too-exciting forty-seven sider, and while one might appear to the casual observer nonplussed, the more astute attendant of human psychology would undoubtedly sense a consuming ennui.  Hardly worthy of God, or even a lesser member of his cabinet.  Of course, one may counter that I have neglected the likes of five-pointed stars, nineteen-pointed stars, and eighty-five-pointed stars, and one would be correct.  But these, too, soon become tiresome.

This ends the first installment of Percy Plumtwiddle’s essay, On Threeness.  Be sure to catch the second (and final) installment next week!

Guest Blogger: Geoffrey Owen Miller, II

Let’s hear more from Geoffrey about his use of color!  Without further ado….

Last week, I mentioned a way to combine the RGB and CMYK color wheels.

color wheels-02This lovely wheel is often called the Yurmby wheel because it’s somewhat more pronounceable then YRMBCG(Y). The benefit of the Yurmby is that the primary of one system is the secondary of the other. With the RGB system,

Red light + Blue light = Magenta light.

Red and Blue are the “primaries,” meaning they are used to mix all other colors, like Magenta, and they can’t be mixed by other colors within that system. The CMYK system is different because pigments absorb light and so it is the reverse of the RGB system:

Cyan pigment + Yellow pigment = Green pigment.

This spatial relationship on the wheel is important as it is an oversimplified representation of an important aspect of our vision. Our eyes only have three types of color sensitive cells—typically called Red, Blue, and Green cone cells. Each cell is sensitive to a different size of wavelength of light:  Long, Medium, and Short. When all three frequencies of light are seen together, we see white light. But more significantly, when we see a combination of Medium (green) and Long (red) wavelengths, our brain gets the same signal as if we saw yellow light, which has a medium longish wavelength!

color wheels-03

Going back to the Color Relationships, the first, and easiest to see is a Triadic relationship. Here three colors are chosen at similar distances from each other on the color circle. If we choose Red, Green and Blue, then you should feel a sense of familiarity. When light of all three colors is brought together we have white. But what if we move the triangle, you may ask? Well, let’s try Magenta, Yellow, and Cyan. To shorten the number of words written, I am going to increasingly use more abbreviations. C + Y + M = Black, right? Well yes, if mixed as pigment on paper it becomes a dark gray, as Cyan absorbs Red, Yellow absorbs Blue, and Magenta absorbs Green. But as light, they mix to White.

Cyan light is made up of both Green light and Blue light as that is how you make Cyan light: C = G + B. If we go back to our original equation

C + Y + M

and simplify further, we get

(G + B) + (G + R) + (R + B), or 2R + 2G + 2B.

color wheels-04

We can remove the 2’s as they are not important (think stoichometry in chemistry!) and we are left with RGB, or white. Now let’s try moving half a step clockwise:

 Magenta-Blue + Cyan-Green + Yellow-Red,

or

M + B + C + G + Y + R,

which simplifies to

(R + B) + B + (B + G) + G + (G + R) + R, or 3R + 3G + 3B,

which once again is RGB!

color wheels-05

The point of these “color relations” is to simplify the number of colors in use. Any three equally spaced colors would mix to white. A complementary color pair is the fewest number of colors necessary to engage all three cone cells in the eye; for example,

R + C = R + (G + B)= White.

A split complement is in between a complement-pair and a triad, as shown in the second figure below.

color wheels-06

So for the watercolors I started this discussion about, I had found an old tube of flesh color that I had bought while in Taiwan. I was oddly attracted to this totally artificial-looking hue as a representative for human skin, and it also seemed like a good challenge to get it to work harmoniously in a painting. To understand it better, I first went about trying to remix that flesh color with my other single pigment paints. It turned out to be a mixture of red, yellow, and lots of white — basically a pastel orange. The complementary color of this is a Cyan-Blue color, which I chose to be the pigment Indigo. I added to this pallet Yellow Ochre, which is a reddish-yellow, and Venetian Red which is a yellowish-red, which when mixed make an orange, though much earthier than the orange in that tube of Barbie-like flesh tone.

color wheels-07

Once I had the four colors I felt best about (which also took into consideration other characteristics of a pigment, such as how grainy they are), the painting became a process where intuition, chance, luck, skill, and the weather all played equal parts in creating a work of historical record — never to be repeated equally again.

Disciplines of Geography 10.28.11
Disciplines of Geography 10.28.11

–Geoffrey Owen Miller

Thanks, Geoffrey!  I’d like to remark that I asked Geoffrey to elaborate on the statement “Once I had the four colors I felt best about….”  How did he know what colors were “best”?  Geoffrey commented that it was “like Tiger Woods describing what goes through his mind when he swings a golf club.”

As I thought about his comment, I appreciated it more and more.  A color wheel is a tool to help you organize thoughts about colors — but a color wheel cannot choose the colors for you.  This is where artistry comes in.  Using your tools as a guide, you navigate your way through the color spectrum until — based on years of practice and experience — you’ve “found it.”

But this is exactly what happens when creating digital art!  It is easy to find the complement of a color from the RGB values — just subtract each RGB value from 1.  Sometimes it’s just what you want, but sometimes you need to tweak it a little.  While there are simple arithmetic rules relating colors, they are not “absolute.”

So it all comes back to the question, “What is art?”  I won’t attempt to answer this question here, but only say that having tools and techniques (and code!) at your disposal will allow you to create images, but that doesn’t necessarily mean that these images will have artistic value.

I hope you’ve enjoyed my first guest blogger — I’ll occasionally invite other bloggers as well in the future.  I’m off to Finland for the Bridges 2016 conference tomorrow, so next week I’ll be talking about all the wonderful art I encounter there!