Author Archives: Mike Poliquin

About Mike Poliquin

Math Teacher and Scholars' Bowl Sponsor at Conway Springs High School near Wichita, KS.

MOB: Teaching Prioritization

Term: MOB – Meat On Bones

I’ve committed myself to teaching students the Extreme Ownership “Laws of Combat” as our “Laws of Learning” this year in my secondary math courses.

In the first quarter of the year, my focus is to teach them two of the four: “Prioritize and Execute” and “Simple.” That’s not a plan, it’s just a new school-year’s resolution. It’s time to start figuratively “fleshing” things out. Thus the phrase I will use from now on: “Meat on Bones,” abbreviated MOB.

I’m not coming to this via my career as an educator — I found all of this by becoming a fan of Jocko Willink, one of the co-authors of Extreme Ownership: How U.S. Navy SEALs Lead and Win, and a secondary member of Eric Weinstein’s “Intellectual Dark Web.” This began as self-improving entertainment for me — learning about leadership from awesome people — but once I took a look at the search engine results for “teaching students to prioritize,” I realized that this is not new territory for teachers — or, at least, it shouldn’t be.

Teachers: have you ever heard of “executive functions”? If the answer is yes, then you are ahead of me on this path. I had not ever heard of executive functions. This is not good. Teachers should spend significant time learning about executive functions and how to teach them intentionally  — I have only been teaching them tangentially. This is not good.

Executive functions are:

  1. Inhibition — stopping behavior, actions, and thoughts at appropriate times
  2. Shift — moving freely between situations, thinking flexibly to respond correctly
  3. Emotional Control — modulating emotional responses to rationalize feelings
  4. Initiation — beginning a task, generating ideas, responses, and strategies
  5. Working Memory — holding information to complete a task
  6. Planning and Organization — managing current and future-oriented demands
  7. Organization of Materials — imposing order on work, play, and storage
  8. Self-Monitoring — monitoring one’s own performance and measuring it

“Prioritize and Execute” is Initiation.

This list came from an article at LD Online by Joyce Cooper-Kahn and Laurie Dietzel titled “What is Executive Functioning?”

I believe that, if we teach students to do these things well, they can learn anything they want to know. If I prioritize and execute in my own classroom, regardless of content, executive functions must come first: they are the key to learning how to learn. Most importantly, students who master these things are capable of imagining, selecting, planning, and executing plans that lead to achieving long-term goals.

If I’m a teacher at all, then this is a very high priority in teaching every subject: in fact, I’m not sure how we can get students to do challenging and worthwhile tasks without teaching executive functions. Perhaps this is why teaching students to do authentic projects well is such a challenge. Are we making these things priorities? Shouldn’t we?

I now realize that The Laws of Combat taught to U.S. Special Forces (or at least, by them) are militarized versions of executive functions. “Combat is reflective of life, only amplified and intensified,” according to Willink. When we strip down the basic combat doctrine of Navy SEALs, we find those executive functions waiting for us. They are not new, but the clarity created by combat experience illuminates them dramatically, which is why you should read Willink’s and Babin’s book.

Teaching Prioritize and Execute:

Here are some ideas for teaching this Law of Learning:

Discuss prioritization in the context of note-taking.

I’m planning to teach my students to take notes in the Cornell Notes format this year, so this will fit seamlessly into that lesson.

Discussion Prompt: “How do you select what to record as you take notes?”

Discussion Prompt: “When you did well on a unit, how did you decide what to study?”

Give a slide show of photos, paintings, or video clips.

Discussion Prompt: What do you think is the most important detail or big idea? (ask for each item in the show)

Select three motivational memes you think are “good.”

Rank them in order, from good to best.

Discussion Prompt: “What qualities did you use to decide how to rank them?”

Create an exemplary outline that designates prioritized main topics and lesser subtopics.

This complements note-taking when we discuss the purpose of keeping notes.

Create three different versions of this outline, blocking out some and less crucial content for struggling note-takers and more and more important content for advanced note-takers.

Have students use these outlines for both note-taking and study.

Give students opportunities to prioritize and experience consequences of failure.

Guide reflection and improvement through revision. Have students estimate progress, keep records of their progress, and revise schedules when they fall behind in achieving their benchmarks.

Give an assignment with clear expectations of the outcome and an analytic rubric of requirements, each with a point value. Have students make lists of tasks and put them into a sequence to follow from beginning to completion. Provide a template for scheduling the tasks.

Students assign a percentage value to each task according to information in the rubric and use these to aid in prioritization. Students record time spent on each element. Note that the weight of a task on the grade is not necessarily its priority: some lower-percentage tasks may have to be completed before high-percentage tasks can be attempted.

Set aside class time for reflection and revision of the plans during execution. Set aside time after completion for reflection and a final prioritization based on experience.

The ideas above came from an article at Edutopia by Judy Willis titled “Prioritizing: A Critical Executive Function.”

Time is an issue.

I can hear you think it. I understand. My own school year appears to have 158 days in it that could be devoted to instruction in a structured way, meaning that each is a full day of school with a full period of class assigned for the subject. At 52 minutes per period, we have 158 x 52 / 60 = 137 hours of instructional time in each of the three math courses I teach.

I think executive functions are the highest priority in my classroom and that they ought to be the highest priority in any classroom. If I give them the weight they deserve and teach them in parallel with the content I contracted to teach, then the right weight seems to me to be around 20-25% of our time, or 32 to 40 instructional days.

How will focusing so much on executive functions influence our progress through content?

First, this questions my prioritization. In my mind, there is no question that executive functions, which are skills everyone needs to succeed at life, are more important than math content. If priorities mean anything to us, then executive functions outweigh subject-specific content every time.

Second, I expect that, as students improve in executive functions, they will improve at assignments that are genuinely worth doing. I don’t know if they will do math exercises or worksheets better, but I really don’t care, as these things do not pop up regularly in adult or professional life.

I will, of course, teach my students the skills of mathematics and we will do exercises together to establish these skills — but I will also give them tasks to do that combine their developing executive functioning skills, which I stumbled into by way of my interest in leadership, with these mathematical concepts.

The challenge is to give students tasks to do that develop their math skills while also forcing them to use executive functions. Let’s give these a nice name: How about “worthwhile tasks?” Yes, that implies that assignments that do not force students to use executive functions might not be worthwhile. I’ll start putting up some fortifications on that ground. It feels like high ground to me.

How much content can we cover that way? Will time be an issue? My educated guess — at this point unburdened by research, though I will be building case studies and anecdotes all year — is that once we get executive functions ramped up, students who do well there will be able to move through content much faster while learning it more deeply. The executive functions will make the math I am teaching more accessible, even if we do not get to it this year, because, if these young people need that math, they will have the tools to acquire it independently. We may not get to all of it — but the students will have more and better tools for learning it or anything else they need to learn in the future.

Stay tuned to see how it works out.

Image Credits

Featured image from: CIO, address:


Book Report — History of Math, Part 2

IMG_1387We return to  Math Through the Ages: A Gentle History for Teachers and Others (Expanded Edition), by William Berlinghoff and Fernando Gouvêa, published in 2004 by Oxton House Publishers and the Mathematical Association of America. In this post I’ll continue my synopsis of its thumbnail sketch of the history of math. After that part, the book gives a series of twenty-five deeper sketches about topics from this narrative. I may leave that for the curious, but they may inspire me to post a few times as well.

We’ll take the subject’s development as described by our authors from the fall of Rome, at which original mathematics in Europe stalled, through Medieval times and up to the dawn of the Age of Enlightenment, setting the stage for the advent of calculus.

  1. The authors continue to emphasize that mathematics is guiding everyday life:

    “Of course, throughout this period [400 – 900 A.D.] people were still building, buying, selling, taxing, and surveying, so the subscientific tradition certainly persisted in all of these areas.”

  2. We now begin an account of Medieval mathematics in the subcontinent of India:

    “During this quiet period in Europe and North Africa, the mathematical tradition of India grew and flourished….It is likely that this tradition received some influence from the late Babylonian astronomers, and it is certain that Indian scholars knew some of the Greek astronomical texts. Astronomy was, in fact, one of the main reasons for the study of mathematics in India.”

  3. There are some names worth knowing here:

    “As in the case of Greek mathematics, there are only a small handful of mathematicians whose names we know and whose texts we can study. The earliest of these is Aryabhata, who did his mathematical work early in the 6th century A.D. In the 7th century, the most important mathematicians are Brahmagupta and Bhaskara [I], who were among the first people to recognize and work with negative quantities… Probably the most important mathematician of medieval India was another Bhaskara [II], who lived in the 12th century.”

  4. Of course, there is the Indian innovation in math that we take utterly for granted:

    “The most famous invention of the Indian mathematicians is their decimal numeration system….The history of this momentous step is obscure. It seems likely that there was some influence from China, where a decimal counting board was used. In any case, by the year 600, Indian mathematicians were using a place-value system based on powers of ten. They had also developed methods for doing arithmetic with such numbers….It quickly spread to other countries. A manuscript written in Syria in 662 mentions this new method of calculation. There is evidence that the system was used in Cambodia and in China soon after. By the 9th century, the new system of numeration was known in Baghdad, and from there it was transmitted to Europe.”

  5. The Indian tradition is also a big step toward our modern conception of the beloved subject of trigonometry:

    “Greek astronomers had invented trigonometry to help them describe the motion of planets and stars. The Indian astronomers probably learned this theory from Hipparchus, a predecessor of Ptolemy. Greek trigonometry revolved around the notion of the chord of an angle….It turns out, however, that in many cases the right segment to consider is not the chord, but rather half the chord of twice the angle. So the Indian mathematicians gave this segment its own name. they called it a “half-chord.” This name was mistranslated into Latin (via the Arabic) as “sinus,” giving rise to our modern sine of α [alpha].”

  6. Indian mathematics also put mankind on track to discover what we today call algebra:

    “Indian mathematicians were also interested in algebra and in some aspects of combinatorics. they had methods for computing square and cube root. They knew how to compute the sum of an arithmetic progression. They handled quadratic equations using essentially the same formula we use today except they expressed it in words….In addition to equations in one variable, the Indian mathematicians studied equations in several variables… Later, Bhaskara II … described a method that will find a solution of nx+ b = y2 in whole numbers whenever such a solution exists. Problems of this type are difficult and the Indian achievement in this area is quite impressive.”

Obviously, all of this knowledge made its way to Europe, but it was not a direct trip from central India to Western Europe.

  1. When the Islamic conquest period concluded with the advent of the Abbasid Empire, scholarship flourished for a while in Baghdad.t is important to note that Arabic mathematicians are those who did their work in Arabic. Plenty were not Arab ethnically nor Muslim in religion. Arabic, like Greek before and Latin later, was simply the common language of scholarship:

    “[The Abbasid caliphs] founded the House of Wisdom, a kind of academy of science, and began to gather together scholarly manuscripts in Greek and Sanskrit and scholars who could read and understand them….One of the first Greek texts to be translated was, of course, Euclid’s Elements. It had a huge impact. Once they had learned and absorbed the Euclidean approach, the Arabic mathematicians adopted it wholeheartedly.”

  2. The Arabic tradition produced some important names and words that we encounter in math and science every day, such as “algorithm” and “algebra”:

    “Muhammad Ibn Musa Al-Khwarizmi was one of the earliest Arabic mathematicians to make an enduring name for himself. his name indicates that he was from Khwarizm, a town (currently called Khiva) in what is now Uzbekistan…One [of his books] was an explanation of the decimal place value system for writing numbers and doing arithmetic, which he said came from India….Also by Al-Kwarizmi was the book of ‘al-jabr w’al-muqabala,’ which means something like ‘restoration and compensation.’…When this book was later translated into Latin, ‘al-jabr’ became ‘algebra.'”

  3. Many mathematicians are actually polymaths — people with multiple intellectual and academic talents — and one of the Arabic mathematicians is a fine example:

    “One of the most famous Arabic mathematicians was ‘Umar Al-Khayammi, known in the West as Omar Khayyam. He lived approximately from 1048 to 1131. Nowadays mostly known as a poet, in his day he was also famous as a mathematician, scientist, and philosopher. One of the goals of Al-Khayammi’s algebra book was to find a way to solve equations of degree 3….The challenge he laid down … was to be taken up by the Italian algebraists many centuries later.”

  4. Many of my students would agree with the Arabic mathematicians on certain points:

    “To the Arabic mathematicians, only positive numbers made sense.”

  5. The Arabic mathematicians were also obsessed with astronomy and trigonometry, but they investigated other areas more deeply:

    “Trigonometry was a major concern, mostly because of its applications to astronomy. The work on trigonometry led inevitably to work on approximate solutions of equations. A particularly notable instance of this is a method for approximating the nth root of a number, developed by Al-Kashi in the 14th century. Combinatorics also shows up in the Arabic tradition. They knew at least the first few rows of what we now call “Pascal’s Triangle,’ and they understood both the connection with (a + b)n andthe combinatorial interpretation of these numbers. Stimulated by Euclid and Diophantus, they also did some work in number theory.”

  6. As in other places in this thumbnail narrative, the authors emphasize that mathematics is developing in the work of more anonymous people, in this case, artists who were prevented by Islam from depicting the human body:

    “Finally, it is important to mention that practical mathematics was also advancing….Buildings were decorated by repetitions of a simple basic motif. This kind of decoration requires some level of forethought, because not all shapes can be repeated in such a way as to cove a plane surface. Deciding what sorts of shapes can be used in this way is really a mathematical question, linked both to the study of plane tilings and the mathematical theory of symmetry.”

At this point, the authors rewind the time machine a few centuries and discuss European scholarship and the transition from the intellectual doldrums of the post-Roman period to a more active Medieval period.

  1. Our first inklings of a Western mathematical Renaissance come from a decidedly curious French cleric:

    “Once [Medieval European] people became interested in mathematics, where could they go to learn more? The obvious thing was to go to places under Islamic control, of which the most accessible was Spain. Gerbert d’Aurillac (945-1003), later to be Pope Sylvester II, is an example.”

  2. Yes, yes he is: Gerbert was a proto-humanist, three hundred years before his time, not only learning advanced math and astronomy, but also reading Plato and Aristotle at a Spanish monastery.

    “Gerbert visited Spain to learn mathematics, then reorganized the cathedral school at Rheims, France. He reintroduced the study of arithmetic and geometry, taught students to use the counting board, and even used the Hindu-Arabic numerals (but not, it seems, the full place-value system).”

  3. We know that it was during these Medieval times that the Europeans began to establish universities, and these began with what they could resurrect of the Greeks:

    “Aristotle’s work did have a great impact, however. His work on the theory of motion led a few scholars at Oxford and paris to think about kinematics, the study of moving objects. Perhaps the greatest of these scholars was Nicole Oresme (1320 – 1382). Oresme worked on the theory of ratios and on several aspects of kinematics, but his most impressive contribution is a graphical method for representing changing quantities that anticipates the modern idea of graphing a function.”

  4. The Italians got a nice jump start on the Renaissance, in my opinion, because a young man learning his father’s mercantile business decided to publish what he learned from Arabic merchants:

    “Leonardo of Pisa (c. 1170-1240) was the sone of a trader. Traveling with his father, he learned quite a bit of Arabic mathematics. In his books, Leonardo explained and extended what he had learned. His first book was Liber Abbaci (‘Book of Calculation’). It started by explaining Hindu-Arabic numeration and went on to consider a wide array of problems…”

  5. The Italians kept at the math long after Leonardo of Pisa, whom we know as “Fibonacci,” was gone:

    “As the Italian merchants developed their businesses, they had more and more need of calculation. The Italian “abbacists” tried to meet this need by writing books on arithmetic and algebra….The culmination of this tradition was the work of Luca Pacioli (1445-1517), whose Summa de Aritmetica, Geometria, Proportione e Proportionalita was a huge compendium of practical mathematics, from everyday arithmetic to double-entry bookkeeping.”

    The tradition of the ten-pound math textbook obviously began here.

At this point the authors guide us into the Renaissance of math, in which a primary focus is marrying algebra to geometry to create the algebra content most American students endure (some even learn it, bless them) in secondary school and college, a process that takes us to the Age of Enlightenment (or Age of Calculus, as we call it in the math world).

  1. Navigation and astronomy are still driving math as we begin this phase, but when we complete it, mathematics will have a momentum separate from these, perhaps for the first time in its history:

    “Long-range navigation depends on astronomy and on a good understanding of the geometry of the sphere; this helped propel trigonometry to the center of attention….In parallel with the intense study of navigation, astronomy, and trigonometry, there was also growing interest in arithmetic and algebra. with the rise of the merchant class, more people found that they needed to be able to compute.”

  2. Math begins to take more familiar forms as this plays out:

    “A great many new ideas were introduced into trigonometry at this time. The list of trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) became standardized. New formulas and new applications were discovered.”

  3. New techniques in art demanded more mathematical prowess as well:

    “Somewhat related to all of this was the discovery of perspective by Italian artists. Figuring out how to draw a picture that gave the impression of three-dimensionality was quite difficult. the rules for how to do it have real mathematical content….Some [Renaissance artists], such as Albrecht Dürer, were quite sophisticated in their understanding of the geometry involved. In fact, Dürer wrote the first printed work dealing with higher plane curves, and his investigation of perspective and proportion is reflected both in his paintings and in the artistic work of his contemporaries.”

  4. And now, the Italians are ready to take on the challenge of Omar Khayyam: the general solution to the cubic equation:

    “The crucial breakthrough was made in Italy, first by Scipione del Ferro (1465-1526) and then by Tartaglia (1500-1557)….Both men kept their solutions secret, because at this time scholars were mostly supported by rich patrons and had to earn their jobs by defeating other scholars in public competitions.”

  5. No Internet? No problem. You just need someone who decides it’s okay to share information and put in in a printed book (the Internet of the Renaissance):

    “In the case of the cubic, this pattern was broken by Girolama Cardano (1501-1576)….Cardano was able to generalize [Tartaglia’s method] to a way of solving any cubic equation. Feeling that he had actually made a contribution of his own, Cardano decided that he was no longer bound by his promise of secrecy [to Tartaglia]. He wrote a book called Ars Magna, “The Great Art,” [my note: we mathematicians are not humble people]. This scholarsly treatise (written in Latin) gave a complete account, with elaborate geometric proofs, of how to solve cubic equations.”

  6. Remember, all this time mathematicians are managing the intellectual complexity of communicating math by writing it out using words. There were no variables, until:

    “Algebra began to look more like it does today towards the end of the 16th century, in the hands of Francois Viète (1540-1603). Among many other things, Viète worked for the French court as a cryptographer, a code-breaker who deciphered intercepted secret messages. This may be what led him to one of his most important innovations; the notion that one could use letters to stand for numbers in equations.”

  7. Two mathematicians capped this process of making algebra a mature mathematical science and setting the stage for the emergence of calculus, Rene Descartes and Pierre de Fermat:

    “Three innovations from this period were to be extremely important. First, the fact that no one could figure out how to solve the general quintic (fifth-degree) equation led algebraists to start asking deeper questions. Slowly, a theory about polynomials and their roots evolved. Second, Descartes and Pierre de Fermat (1601-1665) linked algebra and geometry, inventing what we now call “coordinate geometry.”… Third, Fermat introduced a whole new category of algebraic problems. These were related to the work of the Greek mathematician Diophantus, but went far beyond his work. Specifically, Fermat began asking “questions about numbers,” by which me meant whole numbers…. Unfortunately, for a long time, Fermat was alone in finding these questions interesting.”

    I feel your pain, Pete. I really do.

Well, that takes us from the Fall of Rome to the Dawn of Calculus, an intellectual journey of more than twelve centuries.

The next installment will wrap up my synopsis of the authors’ historical narrative. I will try to summarize how we got from high school algebra to an age of magic, where mathematics only a very curious elite understands makes our world work. The rest are at least dependent on that technology and, in too many cases, addicted to it. But that is another story in itself, I suppose.

Book Report — The History of Math, Part 1

In Extreme Ownership: How U.S. Navy SEALs Lead and WinJocko Willink and Leif Babin explain the importance of getting team members to believe in the mission. As a math teacher, this is one of my greatest challenges. In the words of Dan Meyer, Chief Academic Officer at Desmos, and a prominent pundit on math education, in his breakout TED Talk from 2010, as a math teacher

“I sell a product to a market that doesn’t want it but is forced by law to buy it.”

In other words, the team doesn’t believe in the mission. Dan (we have met) talked about his ways of getting them to believe in the mission. My favorite way, so far, is to engage students in the story of math and the people who discovered or invented it (that dichotomy is itself a fun or frustrating debate, depending on your perspective).

My reading program now consists of three different books: The Professional Chef (9th edition)the textbook for The Culinary Institute of America; Physics with Applications (6th edition), by Giancoli; and Math Through the Ages: A Gentle History for Teachers and Others (Expanded Edition), by William Berlinghoff and Fernando Gouvêa, published in 2004 by Oxton House Publishers and the Mathematical Association of America.

For this post, I’m quoting (rather extensively) the latter, as its ideas may help my fellow math teachers looking for hooks for lessons or even the year’s courses. I won’t be doing the one-and-done book report I’ve done for other books because I expect to take a slower pace through this book, and the other tomes I mentioned above are going to slow me down (and that’s okay with me, obviously).

Here are my highlights from the chapter titled “History of Mathematics in a Large Nutshell,” particularly the first two sections: “Beginnings” and “Greek Mathematics.”

  1. Mathematics developed with writing:

    “No one quite knows when and how mathematics began. What we do know is that in every civilization that developed writing we also find evidence for some level of mathematical knowledge.”

  2. We detect the development of mathematics by applications:

    “It became important to know the size of fields, the volume of baskets, the number of workers needed for a particular task. Units of measure, which had sprung up in a haphazard way, created many conversion problems that sometimes involved difficult arithmetic. Inheritance laws also created interesting mathematical problems. Dealing with all of these issues was the specialty of the ‘scribes.’ These were usually professional civil servants who could write and solve simple mathematical problems. Mathematics as a subject was born in the scribal traditions and the scribal schools.”

  3. We’ve found different evidence from different cultures, and then, as now, the hike from Egypt to Iraq was apparently not a popular trip for scribes:

    “… we have only a few documents that hint at what Egyptian mathematics was like….The situation with respect to the cultures of Mesopotamia is quite different….These two civilizations existed at about the same time, but there seems to be little evidence that either influenced the other’s mathematics.”

  4. Egyptian math (and learning it) was not terribly different from the way we see math today (which I think has more to say about us than the Egyptians):

    “The Egyptian mathematics of 4,000 years ago was already a fairly well-developed body of knowledge with content very similar to some of what we learn about calculation and geometry in elementary and high school today. It was recorded and taught by means of problems that were intended as examples to be imitated.”

  5. The Babylonians (remember, we know more about their math than about the math of Egypt) built math around the problems of government and management, and, after the work was done, the Babylonians went back for MORE:

    “The mathematical activity of the Babylonian scribes seems to have arisen from the everyday necessities of running a central government. Then, int he context of the scribal schools, people became interested in the subject for its own sake, pushing the problems and techniques beyond what was strictly practical. Like a musician who is not satisfied with playing at weddings and graduations, the well-trained mathematical scribe wanted to go beyond everyday problems to something more elaborate and sophisticated. The goal was to be a mathematical virtuoso, able to handle impressive and complex problems.”

  6. Despite the current preeminence of ethnic Chinese in mathematics, we have little to go on regarding the ancient development of the subject at the other end of the Silk Road:

    “We do not know a lot about very early Chinese mathematics….The mathematical texts we do have seem to reflect the rise of a class of civil servants who were expected to be able to solve simple mathematical problems. Like the texts from Egypt and Mesopotamia, they contain problems and solutions. In China, however, the solutions are often presented together with a general recipe for solving this type of problem.”

  7. What of the other Cradle of Civilization, you ask? Well you might:

    “We know even less about early Indian mathematics. There is evidence of a workable number system used for astronomical and other calculations and of a practical interest in elementary geometry.”

  8. The mathematics of China did not affect the Western development of the subject much, but:

    “The Indian mathematical tradition influenced Western mathematics quite directly.”

  9. Now we get to the meat of the early history of math, the culture that built the Mediterranean network of cultures in rivalry with the Phoenicians and, thanks to Roman victory over Carthage, wrote the history of math and many other things which dominates the story today:

    “Many ancient cultures developed various kinds of mathematics, but the Greek mathematicians were unique in putting logical reasoning and proof at the center of the subject. By doing so, they changed forever what it means to do mathematics. We do not know exactly when the Greeks began to think about mathematics. Their own histories say that the earliest mathematical arguments go back to 600 B.C.. The Greek mathematical tradition remained a living and growing endeavor until about 400 A.D..”

  10. It is the cultural hegemony created by Greek colonization and the military might of its successor civilization, Rome, that makes Greek mathematics what we consider it to be:

    “It is important to stress that when one speaks of “Greek mathematics” the main reference of the word “Greek” is the language in which it is written. Greek was one of the common languages of much of the Mediterranean world. It was the language of commerce and culture, spoken by all educated people. Similarly, the Greek mathematical tradition was the dominant form of theoretical mathematics.”

The rest of my reading at this point (the above and below quotes are from pages 6-24) makes a nice list which I have used for several years to build a historical reference framework of names and stories in my classes. The Greeks that make the cut are:

  1. Thales (circa 600 B.C.) —

    “… the first person to attempt to prove some geometrical theorems, including the statements that the sum of the angles in any triangle is equal to two right angles, the sides of similar triangles are proportional, and a circle is bisected by any of its diameters.”

  2. Pythagoras (circa 500 B.C.) —

    “Most scholars believe that Pythagoras himself was not an active mathematician … [Pythagoreans] seem to have been much concerned with the properties of whole numbers and the study of ratios (which they related to music). In geometry, they are, of course, credited with the Pythagorean Theorem. … It is likely, however, that the most important success often credited to the Pythagoreans is the discover of incommensurable ratios (my note: this led directly to the idea of irrational numbers).”

  3. Euclid (circa 300 B.C.) —

    “What we have are his writings, of which the most famous is a book called Elements. It is a collection of the most important mathematical results of the Greek tradition, organized in a systematic fashion and presented as a formal deductive science. The presentation is dry and efficient.”

  4. Archimedes (circa 250 B.C.) —

    “Archimedes wrote about areas and volumes of various curved figures …”

  5. Apollonius (circa 200 B.C.) —

    “…Apollonius wrote a treatise on conic sections that is still an impressive display of geometric prowess.”

The book then delves into a favorite topic of mine: the three great geometric problems of antiquity: the quadrature of the circle, the trisection of the angle, and the duplication of the cube. This is a favorite theme of mine when students are primed to discuss how and why people do mathematics, meaning discovering, proving, teaching, and publishing it.

I believe it is essential that we teach our students about the role of unanswered questions in driving human knowledge. The three great geometric problems of antiquity were not solvable with compass and straightedge — though the Greeks and many later investigators devised other tools and concepts and used them to solve the problems. The work they did under those limited conditions produced many prodigious mathematical achievements that, nevertheless, were not solutions to the original problems. A quick look at the careers of the great 20th century mathematicians shows that this is still what drives the science of mathematics forward today.

This section wraps up with another list of great mathematicians of the Roman imperial period, still called Greek, remember, because Greek was the language of intellectual and financial transactions:

  1. Ptolemy (120 A.D.) —

    “He wrote on many subjects, from astronomy and geography to astrology, but his most famous work is the Syntaxis, known today by the nickname given to it by Arabic scholars many centuries later: Almagest, meaning “the greatest.” Ptolemy’s book provides a workable practical description of astronomical phenomena. It was the basis of almost all positional astronomy until the work of Copernicus in the 15th century.”

  2. Diophantus (circa 220 A.D.) —

    “Diophantus … was probably one of the most original of the Greek mathematicians. His Arithmetica contains no geometry and no diagrams, focusing instead on solving algebraic problems; it is simply a list of problems and solutions. In the problems, Diophantus used a notation for the unknown and its powers that hints at algebraic notation developed a thousand years later in Europe…. Diophantus usually worked out the conditions under which his problems are solvable, thereby confirming that he was trying to find general solutions.”

  3. Pappus (circa 350 A.D.) —

    “Perhaps the most important part of Pappus’s work, from a historical point of view, was his discussion of ‘the method of analysis.’ …. Pappus’s discussion of analysis is not very specific. This vagueness ended up being very important, because the mathematicians of the Renaissance understood him to mean that there was a secret method behind much of Greek mathematics. Their attempts to figure out what this method was led to many new ideas and discoveries in the 16th and 17th centuries.”

    Note that, once again, the presence of an unsolved mystery drives mathematical discovery and innovation far more effectively than neat and clear presentations of methods and solutions. I think I’ll put that on a poster in my room this year, to answer all the complaints I will get along the lines of “Why don’t you just tell us how to do it?”, as though driving through difficulties to solutions and then discussing their merits has no intrinsic value.

  4. Heron (10 A.D. – 70 A.D.) is out of chronological order because he is noted for trying to bridge the chasm between the completely abstract endeavors of “scientific” mathematicians in the Greek tradition — the others named here — with the practical math that obviously was also developing throughout that time. Heron was more of an engineer, but his namesake formula for the area of a triangle in terms of its side lengths remains in the modern curriculum.

The authors note that both the “scientific” mathematical tradition of the great heroes of math and the “sub-scientific” mathematical tradition of merchants, government officials, mariners, and soldiers featured recreational problems, challenge problems, and puzzles. Like the Babylonians, those at all levels of the Greek mathematical tradition liked to do math “off the clock” as well as on. In fairness,we should note that we don’t have any indication of how common that was with the Egyptians, as we have spent much more time digging in their burial grounds than in the hearts of their cities (probably because those cities are under modern cities).

Well, that’s as far as I’ve gone, but stay tuned: this is just a first installment. There will be more of this, especially when I encounter historical information that I know I’ll use to help my troops believe in the mission in my classroom.

Book Report — “What Great Teachers Do Differently”

I admit that I several years longer than I should have to read anything other than a few articles by Todd Whitaker. That was certainly a mistake on my part.
I first encountered him as an author when I bought my textbooks for the last semester I was enrolled in a graduate program in school leadership. The book is What Great Principals Do Differently. It was at that time that I determined I really didn’t want to be a principal and abandoned the program. The book was never assigned, and it still sits on my bookshelf now.

I acquired Whitaker’s book What Great Teachers Do Differently from a pile of discarded books in the teacher workroom. The teacher who discarded it was leaving our school after a really rough year. I don’t think he had read any of it.

This is not the most recent edition of the book, by the way. The subtitle of the most recent edition heralds “17 Things That Matter Most,” so this book report will shortchange you by definition. I’ll buy and read the new edition next August, and I think I’ll reread it every year in early August for the rest of my career.

I’m not a great teacher — but I feel that I have learned over 21 years to try to do these things. I don’t do most of them terribly well yet, but it has made a huge difference in my teaching just to be trying to do all of these things. Nothing in this book is terribly surprising to a veteran teacher, but everything in this book needs to be part of any teacher’s preparation for a new school year — or part of the preparation of any person entering the profession.

Whitaker writes extremely well. His examples are excellent, and it is clear that he must have been a very reflective and, over time, effective teacher and leader of teachers. While I no longer aspire to lead a building as a principal, I want to be a more effective teacher and an effective leader of teachers from my role as a classroom teacher and professional educator. This book has too much good material in it and Whitaker summarizes it too well for anyone with my goals to leave it out of my annual routine (but next year, I’ll spring for the new edition and learn about the three “things that matter most” that don’t appear here).

In this brief paraphrasing of the “14 Things That Matter(ed) Most” when this first edition was published, I have also highlighted those areas that are most relevant to Teachership — the aspects of teaching that require the teacher to be a leader and to model good leadership for students (spoiler: the vast majority of the stuff is highlighted).

Here we go:

  1. Teachers must remember that education is about people, not programs.
  2. Teachers must express and uphold clear and consistent standards at all times.
  3. Teachers must react to misbehavior with a focus on preventing its recurrence.
  4. Standards for teachers must be higher than standards for students.
  5. The teacher is the only variable that the teachers control in the classroom.
  6. Teachers must keep classes positive, modeling respect and praising correctly.
  7. Teachers must ignore little slights and keep a positive and focused attitude.
  8. Teachers must maintain good relationships with students at all times.
  9. Teachers must focus on important concerns, seeking to preventing escalation.
  10. Teachers must plan and revise each lesson to focus on its essential goals.
  11. Teachers must plan communication for the best people they are trying to lead.
  12. Teachers must treat each person as if that person is the best person they lead.
  13. Teachers must focus on student learning, keeping standardized testing in its place.
  14. Teachers must build a nuanced understanding of the role of emotion in determining behaviors and beliefs, and use it to strengthen their practices.

Read the book. It’s an awesome way to charge up your preparation for the year.

Book Report — The Old Math

IMG_1379IMG_1380I just finished How to Become Quick at Figures, which was, as the graphic shows, published late in the 19th century. I found the attitude of the book fascinating. For instance, this, not a cover page, appeared immediately before the Table of Contents:


“It has been truly said that the great want of the age is men. Men of thought; men of action. Men who are not for sale. Men who are honest to the heart’s core. Men who will condemn wrong in friend or foe — in themselves as well as others. Men whose consciences are as steady as the needle to the pole. Men who will stand for the right if the heavens totter and the earth reels. Men who can tell the truth and look the world and the devil right in the eye. Men who neither swagger nor flinch. Men who are Quick at Figures. Men who can have courage without whistling for it, and joy without shouting to bring it. Men through whom the current of everlasting life runs still, and deep and strong. Men too large for certain limits, and too strong for sectarian bands. Men who know their message and tell it. Men who know their duty and do it. Men who know their place and fill it. Men who mind their own business. Men who will not lie. Men who are not too lazy to work, nor too proud to be poor. When in office, the workshop, the counting room, in the bank, in every place of trust and responsibility, we can have such men as these, we shall have a christian civilization — the highest and best the world ever saw.

And then this, right after the Table of Contents:


Do not spend your precious time in wishing, and watching, and waiting for something to turn up. If you do, you may wish and watch and wait forever. You can do it if you wish, but you must put forth the effort. Idleness and indifference never accomplished anything. It takes energy and push to make headway in the world, and an active, energetic, persevering man is sure to succeed. If he can not do one thing he will do something else. If he can not succeed in one direction he will in some other. He will do something. He will not waste his time in idleness. There is no lack of work, no lack of opportunities. Do what comes to your hand, and do it well. True progress is from the less to the greater. You must begin low if you would build high. Work is ordinarily the measure of success. Quit resolving and re-resolving and go and do something. —  School Supplement.

The author or sponsors of this book obviously felt that men worthy of leadership were in short supply. When aren’t they?

The author could be S. Stone, cited in the bibliography of Revelations of a Spirit Medium, a book to debunk charlatans published in 1922, or John Scott, listed author of a 1915 book that appears to be an updated edition of this one.

The book’s contents begin with many methods and special cases of arithmetic problems common in the mid- to late-19th century business world. There are then specific and brief chapters about business applications like simple and compound interest. The book includes references for the many different systems of measurement in use at the time, including avoirdupois, Troy, Imperial, and the nascent (at the time) Metric Systems. The last pages of the book feature parlor tricks and puzzles for entertainment purposes. It seems that this section was a resource for the book that cited it above in debunking numerological hocus-pocus.

As the tone of the block quotes above suggests, the book is designed for self-improvement at a time when the U.S. was growing rapidly in territory, population, and wealth. The 1880s were a period of recovery after the longest period of continuous economic contraction in U.S. history, 1873-1879, which, at 65 months, was actually longer than the period of contraction recorded for the depression between the 20th century’s two great wars. People of the time called this the “Great Depression.” It was not as severe in depth as the depression of the 1930s, but it was statistically longer. In 1883, a young man looking to make his fortune would have been encouraged by the continuing Second Industrial Revolution and the increase in settled territory where commerce would become more feasible, relatively continuously, for the next thirty years.

I also found it fascinating that the author or authors of the quotes above called up so many virtues that we might — or at least, I do — associate with honorable military service. The U.S. was still reeling culturally from the Civil War less than two decades earlier, and the leaders of businesses would have been people who lived through those difficult years and many of them would have been veterans. I’ve spent significant time around veterans of the U.S. armed forces, and they always seem to be wishing that the young people they encounter had more of these qualities. This may be something I hear more often because I am a teacher and like to have conversations with them about my work.

The arithmetic methods were fascinating to me. Many are very mechanical and specific. For instance: to square a two-digit number ending in 5 (useful for exactly 9 calculations), multiply the tens’ digit by one more than itself and append 25. So 75 squared is 5,625. I recognize in others basic content from my algebra courses: to multiply two mixed numbers, add the product of the whole numbers, the product of the first fraction and the second whole number, the product of the second fraction and the first whole number, and the product of the fractions. If we were to write the mixed numbers as the sums of whole numbers and fractions and apply the FOIL method, this is exactly the result we would get. The author is exhorting the student to do all of this mentally. My students would convert the mixed numbers to fractions, multiply the fractions, and the convert the product back to a mixed number, if that is appropriate. I can do that in my head, and I can do the method presented in the book in my head … but I wouldn’t challenge students in my classes to do it.

As I move on to my next book — I’m hoping to read one book per week and write about each one here — I am pondering how an updated version of this book would look in today’s context. Machines do all of this for us now, instantly. Most of us do not desire to know how the machines work or how the machines are programmed to do the work. This makes modern math seem like a kind of magic, perhaps even more than it seemed to be 140 years ago, at a time when an estimated one-sixth of the U.S. population, including 70% of its non-White population, could not read. I don’t like that math seems like magic. I’d like to do something about that. Hmmm…

Decentralize Command

Teachers are leaders. As leaders, they must reckon with the Laws of Combat: Prioritize and Execute, Cover and Move, Simple, and Decentralize Command. If a teacher is going to do these things in a formal leadership role, then the teacher has to be able to take a step back while students are learning and detach, observe, and be the strategic genius. That means that students have to be trained to self-direct and to make decisions, and that students must lead each other at times so that the teacher can gather data and manage relationships during class.


In Extreme Ownership: How U.S. Navy SEALs Lead and Win, Jocko Willink and Leif Babin give examples of Decentralize Command in both combat and business. In the combat examples, we see subordinates taking initiative and leading up the chain of command to prevent friendly fire incidents and make their teams more efficient and lethal. The business example explained that leaders cannot lead large numbers of subordinates effectively, and that leaders should plan the distribution of direct reports among subordinates leaders carefully.

If we do not apply Decentralize Command, then we have to have control of everything in the room at once. Control works in both directions: if we cannot step back from a group’s activities to observe and manage at a small distance, then that group controls us through its behavior and performance. This keeps us from stepping outside the activity to see how students fare when directing themselves and leading each other. This is not Teachership.

Having to control all action in the room also forces us to use identical activities to help each student learn. Each student is unique, so, while this might seem the most fair way to do things, it is actually very unjust, because it cannot optimize growth for each student. Students at either margin — those who struggle most and those who find the regular content of the course to be very easy — receive inferior service.

Decentralize Command puts all students into leadership positions by encouraging them to use initiative. In groups of any size that follow this Law of Combat, team members learn that good leaders will listen respectfully to their concerns and suggestions, and allow them latitude to make certain decisions according to the leader’s intent. Leaders who comply with Decentralize Command know that they do not have to carry the entire burden of planning and preparation: as subordinate leaders develop initiative, top leaders can delegate certain tasks.

Delegation frees a leader to manage the “big picture” and shape a clear “leader’s intent.” Then the leader can communicate “leader’s intent” to teammates so that they understand how to make decisions in leading within the team and up the chain of command. If we are practicing Teachership, then our students will grow in initiative throughout the year.

We are implementing Decentralize Command for two key reasons. First, students will learn content more deeply when they are leading and collaborating with each other. Second, students learn leadership when they must exercise their initiative and step up to help their teams succeed.

Decentralize Command Is a Better Way to Learn Content

Cornell University’s Center for Teaching Innovation provides a list of research-identified benefits for “collaborative learning” that opens with “development of higher-level thinking, oral communication, self-management, and leadership skills.” There are more on the list, but these are the benefits we want to achieve for our students in Teachership. The development of higher-level thinking — analysis, evaluation, and creativity — is a vital goal. Anyone can memorize, understand, and apply content, and we will teach those skills, but if students are to learn deeply and have that knowledge available to support more learning, then we must help students learn beyond the basics.

Possibly the most difficult thing for us to do is to get students to discuss or debate relevant academic content. By being present, we have a profound effect on how students participate in these activities. If a student leader can manage a team and if we can supervise the team at a small distance, then the teacher should arrange higher-level thinking activities that way. Students involved in this group will begin to take ownership of the team’s work, and they will engage in that work more completely than in a teacher-led full-class discussion or drill session.

Aren’t You Going to Teach?

We are not abdicating the teaching role to student leaders. Groups cannot begin to address problems and projects until the they can remember, understand, and apply the necessary concepts and skills. The teacher must present these to the whole class and guide practice until all students have access to the necessary knowledge by notes or other resources. Discussion, note-taking, and guided practice, led by the teacher, are still in the program — but their roles are to establish an arsenal of ideas, which students can explore and share in working on problems that challenge their analytic, evaluative, and creative skills.

We are not choosing the easy path in applying Decentralize Command or any of the other Laws of Combat. At its beginning, developing the correct conditions for collaborative learning under student leaders requires more work than teacher-centered instruction. Before implementing group activities under student leaders, we must train the class in effective note-taking and study methods, establish expectations for student self-discipline, and train all students in appropriate collaborative behaviors. We must also select and train the leaders. Falling short in any of these areas will prevent Teachership from happening at all.

What If Some Students Can’t Work Without the Teacher?

There is one exception to having students engage in collaborative learning under student leaders: some students will not acquire the necessary note-taking, social, and basic academic skills that support effective participation in these groups. These students will still be in a group and will still engage in collaborative learning, but this group will have the teacher as its leader, so that the teacher can continue to mentor these students and help them master content and skills while developing leadership skills.

There is an obvious supervision challenge here. When the room breaks to group work, we teachers take our seats in one corner of the room, with our backs to the corner. The groups we lead gather with us, and we use our location to supervise all students, as they should all be within 90˚ of our field of view. We plan our groups’ work so that, when students need a few minutes to work on a task, we may leave our seats for a few moments to observe and support the other groups.

Decentralize Command Helps Students Become Leaders

Students will have to make decisions and share ideas on their own initiative in these groups, and initiative is crucial to leadership at every level of the chain of command. Student leaders are not dictators, and we do not want their teammates to follow them blindly. We will train our leaders to check their egos, listen, and incorporate good ideas that come from teammates.

Leaders do have to make decisions — and teammates need to follow their leaders when that happens. But if leaders are serious about optimizing performance, then they will listen to others’ concerns and ideas, and incorporate them into their plans according to their judgment. Leaders will help their teammates take ownership of their work on problems and projects by listening to their ideas, adopting the best ones, and making counter-suggestions to flawed ideas. We will teach our students to do these things. When teams operate this way, our students grow in leadership, initiative, and responsibility.

When We Decentralize Command, Behavior Changes

We should begin to see more confidence and more efforts to lead up the chain of command, even when students are not working in groups. Kids push limits, and in this case, we hope they will. When Decentralize Command is working for us, our students will start to share ideas and concerns with us during whole-class activities. We must welcome and consider their ideas, listening respectfully, and then making the necessary call. If the students’ ideas are sound and helpful, then we should model the “ego check” and adopt them. If they are flawed ideas, we should make counter-suggestions respectfully. Good leaders are good listeners and amenable to good suggestions. We must model that.

Our students will take the new skills and habits they like best with them to other classes and to their activities beyond the classroom. As we observe them in their various public activities, such as sport, musical performance, or theater, we will watch for them to use initiative and to communicate effectively with their colleagues. When we see this sort of growth — in or out of the classroom — we need to recognize it. Praising a student for fulfilling specific tasks is often counterproductive, but we should look for opportunities to acknowledge broad personal growth and maturity.

Our students are not automatons and we do not want them to practice the bad habit of following bad leaders in lockstep. We also want them to learn how to be effective in helping leaders find the best solutions to team problems. We want students to become citizens who can lead and collaborate responsibly in a self-governing nation built on the blessings of liberty. To achieve that, they must have a share in directing their own work in groups of all sizes as they learn. We are the leaders of that enterprise, but we must practice good leadership to foster leadership qualities in our students.

Decentralize Command.

We Must Teach Initiative


  1. the ability to assess and initiate things independently.
    “use your initiative, imagination, and common sense”
    synonyms: resourcefulnessinventiveness

  2. the power or opportunity to act or take charge before others do.
    “we have lost the initiative and allowed our opponents to dictate the subject”

Here is an example of why we must teach initiative, taken from the powerful and thought-provoking film Most Likely to Succeed:

This scene reduced one of my favorite educators to tears when she saw it. It is a powerful example of what students are not learning in our classrooms.

We have one more Law of Combat from Extreme Ownership: How U.S. Navy SEALs Lead and Win to explore as we continue creating a foundation for Teachership. Before we explore that Law of Combat — Decentralize Command — we must state why it is important. Decentralize Command means granting trained subordinates the latitude to take and manage risks, fulfilling the leader’s intent without seeking permission. Educators are not used to cultivating initiative in the classroom, and parents and administrators may have objections to implementing Decentralize Command if the teacher fails to communicate what is happening and why.

Decentralize Command is different from the other Laws of Combat. Teachers have been trying with varying success to implement Cover and Move for a long time. It should surprise no one that we will teach students to collaborate in their learning or that we will collaborate with our colleagues and administrators. Everyone wants us to keep things Simple and to Prioritize and Execute. Decentralize Command will cause more concern because we will create opportunities for students to use initiative and to lead each other.

We must exercise our initiative to communicate with parents and administrators and address concerns that we should anticipate.

Why Decentralize Command?

It’s easy to justify Decentralize Command as a rule for combat or business, as the authors of Extreme Ownership do. Risk-taking and risk management are adult activities, and adults should know how to take responsibility for and mitigate risk. When do we teach this to our students? How can we help them develop initiative?

Soldiers fighting for their lives and their comrades’ lives must take risks and seize opportunities when that action fits the objective. There is no time to ask permission along the chain of command in the chaos of combat. Someone has to make a decision and go. Taking unplanned action to achieve objectives, save soldiers’ lives, and protect civilians is fundamental to the duty of any U.S. service member, and our best leaders train subordinates to do these things independently.

Business presents situations where managers or sales professionals must make decisions on the spot. If individuals don’t have the training and freedom to act in the company’s interests, then the organization is limited, if not paralyzed. Taking unplanned action to achieve objectives, limit liability, promote stewardship of resources, and meet legal obligations are fundamental to every role in a business, and our best leaders train their subordinates to do these things independently.

Do the best leaders in education do this? Do we give our students chances to make choices so that they can learn to optimize performance on their own initiative? How well will they do that in the future if we don’t teach them to do it now? They will do it — but shouldn’t they practice now when we can help them manage the risks?

Our students are not adults, and they are not in a chaotic environment like combat, nor are they in a dynamic environment like our economy. Educators control the classroom through curriculum, professional practice, and planning. This creates a chasm between the classroom and adult life that we must bridge.

Well-led participants in extracurricular activities have opportunities to take risks and see the consequences. Students who have jobs have chances to do this. Students who help their parents manage their households also have to practice initiative. Compared to these endeavors, classroom education is an inadequate learning experience most of the time.

Every student we teach will spend a lifetime needing to lead. Students need discipline and learning skills to succeed in school. They need to earn a high school diploma and prepare for the next mission, whether that is trade school, enlistment, college, apprenticeship, employment, or entrepreneurship. They must lead themselves.

Once they leave secondary school, they will have to learn and discipline themselves even more, and they will also need the character and confidence to act on initiative and manage risk. No one will plan and structure students’ adult lives the way we plan and structure their learning.

If a student doesn’t learn to do all of these things, then the student doesn’t learn leadership. If a teacher doesn’t do these things well, then that teacher’s students will miss a valuable model. Teachership has to include risk-taking and risk management opportunities for students and the modeling of initiative by the teacher. To learn to apply initiative in adult life, students must begin doing it while training for adult life.

We use Decentralize Command in the classroom to help students develop initiative, which is a quality we all hope to see in adults.

Own the Lines of Communication — or Fail

It is very important in implementing Teachership that we win people over to our way of thinking before we start the process of teaching students how to take and mitigate risk. Implementing Decentralize Command without communicating our plans to parents and administrators will derail the Teachership project before it has any time to work.

Imagine a parent who does not understand why some students might report to a student leader in class and not always directly to the teacher. Is the teacher even teaching those students? Imagine this parent pondering why a student is choosing among various learning activities. Is one choice better than the others? What happens if the first choice doesn’t work out? Will the consequences or delays be unacceptable? Parents will oppose what they do not understand.

Imagine a superintendent who does not understand what we are doing when parents are upset and express concerns at a personal appointment or at a school board meeting. The parent will discuss the matter with other parents who also do not understand. Imagine a principal who does not understand what we are doing when a cacophony of angry e-mails, phone calls, and surprise visits from parents begins. Administrators cannot support what they do not understand.

If we are going to implement every part of Teachership so that we can help our students become lifelong learners and leaders, then we must communicate those plans to parents and administrators before we begin. The first principle of Teachership is Ownership: if parents and administrators do not understand our plan, then we have not explained it yet. Our failure to communicate is not their fault.

The other adults in the chain of command want us to succeed in giving our students an excellent education. We are all on the same side, even when we disagree. We must explain why and how we plan to implement Decentralize Command to parents and administrators, and we have to build and maintain good relationships with them so that those lines of communication remain open and positive.

Two Excellent Questions to Answer Next

Now we can predict the two questions parents, principals, and superintendents will ask when we explain our plans to implement Teachership and Decentralize Command:

  1. How do we implement Decentralize Command in a classroom?
  2. How does implementing Decentralize Command help students to learn academic content?

We’ll address both questions in the next post.