Brief Wiki Update

While creating a new entry for factor in my wiki, I wrote this:

“In the arithmetic of known numbers, we focus on four activities: addition, subtraction, multiplication, and division.

In the arithmetic of variable expressions, particularly those known as polynomials, we also focus on three activities: addition (which at that level includes subtraction), multiplication (which sometimes includes division but not always), and factoring.

Every time we investigate new phenomena that we associate with new mathematical formulas, we will look at different ways to factor it in order to appreciate the usefulness of the formula and in order to gain insight: sometimes understand a different way to factor a formula suggests a view of the phenomenon it represents that we had not considered. We would never draw a scientific conclusion from such an activity, but we would certainly use it to create a new hypothesis for investigation.”

It struck me as something that belonged on the blog, too, so … now it is!

Lege, explora, cogita. Quaere verum.

Brother Francis #2: The Object of Mathematics

Brother Francis Maluf continues his objections, from this recounting of a lecture or article of his:

“Since in mathematics accuracy and clarity are achieved at the price of the reality and the goodness of the object, it is a danger of the mathematical mind to continue to sacrifice reality and goodness for the sake of clarity in every other field in which man must seek and find the truth.”

Brother Francis has decided that the purpose of mathematics is accuracy and clarity. I would say that accuracy and clarity are virtues of mathematics in the same way that temperance and justice are cardinal virtues; that hardly puts them first in the list of mathematical virtues.

The essence of faith is not found in the cardinal virtues — temperance, justice, prudence, and fortitude — but in the theological virtues — faith, hope, and love. Someone who said that the essence of faith should be justice and temperance would shoot well wide of the Christian mark, at any rate. We love justice and value temperance, to be sure, but they are not the bedrock of Christianity: Jesus chose his Apostles from men well acquainted with injustice and excess.

Just so, mathematicians — I feel like I’ve maneuvered myself into the apologist’s role for a religion rivaling Christianity, but I surely didn’t mean for that to happen — love accuracy and clarity but they are not the bedrock of math. Perhaps they were earlier in Brother Francis’ career: I recall a scene in Dead Poets Society where a stodgy old math teacher in an elite boarding school proudly announces to trigonometry students that their answers must be correct to four decimal places. If I had to treat math as a religion (it isn’t one) and had to defend it (which I am choosing to do for the sake of opposing Brother Francis’ assertions about math) then I would choose the creation, evaluation, and analysis of the correspondences between mathematical objects and real phenomena.

This analogical defense quickly dissolves at this point. Mathematics is not a religion; its precepts do not align with those of a religion for easy comparison. It is for this reason that Brother Francis’ choice of ground for leveling his assault on math’s elevated curricular status surprises me. For instance, if we solve an equation to discover two solutions, but only one fits the rules of the associated problem, we discard the meaningless solution: we prefer the other because it works; we do this out of utility, not for any moral reason. This kind of intellectual and ethical sloppiness pops up all over the landscape of mathematics. Why not object to this?

Philosophers object to a perceived lack of intellectual rigor in mathematical styles of proof and inductive reasoning. Surely a theologian who understands mathematics as well as Brother Francis did could craft a similar argument: Brother Francis doesn’t present any such argument here.

The most important failure of Brother Francis’ presentation to this point — and we’ve only scratched the surface at this point — is a misunderstanding of the purpose of mathematics. That purpose, again, is the creation, evaluation, and analysis of correspondences between mathematical objects and real phenomena. I have couched this purpose in the language of Bloom’s higher cognitive domains because I believe that’s where “real math” happens.

Students spend a great deal of elementary study building up to these levels of engagement, learning facts and rules, trying to understand how they interact to create paths from problems to solutions, and applying those solutions. This seems to me to be the extent of Brother Francis’ appreciation of math’s power; it is so incomplete as to be a distortion at best and utter ignorance at worst. In other words, Brother Francis ought to have known better. As a person pledged to defend the faith against the kind of relativistic utilitarianism we get from J.S. Mill and the  like, Brother Francis should know better than to value math only for its utility.

I might be tempted to speculate that Brother Francis composed this diatribe after witnessing a tendency of students who attain a certain level of facility with higher math and familiarity with the scientific writings of certain deistic or atheistic sources to question faith and perhaps even leave the Church for a time. If that’s right, I would join him in mourning the loss but I would also watch and wait: I thought I was too smart to be a Christian for a while, too, but here I am back in the arms of Mother Church and happily so. It’s speculation, of course, but I can’t resist wondering what motivated Brother Francis to speak in what seems to be such a despondent tone about a subject that, for many of us, reaffirms our faith in the religious truth of Creation and the integral inevitability of Salvation history.

If a systematic and appreciable set of rules, patterns, and strategies permeates the phenomena of the universe, then I find more encouragement and justification for believing as I do in its First Cause and Ultimate Purpose. What others see as a justification for abandoning faith I see buttressing my belief and renewing my hope. I’ll be teaching that this year; I won’t be warning my students against dabbling in the seductive accuracy and clarity of demon math.

Lege, explora, cogita. Quaere verum.

Brother Francis Maluf: My New Foil?

Click here to read the entire article if you like, or just savor the excerpts that will follow in the series of posts I’m going to write in response to it. This is what I found when I went looking for a Catholic patron saint for mathematicians (St. Hubert of Liege, but for no reason I can find) or Catholic saints who were also mathematicians (there are none).

I found Brother Francis Maluf, M.I.C.M, instead, and this article, apparently a digest or transcription of one of his lectures, is an inspiration for someone in my new situation as a teacher of mathematics at a Catholic school. You see, my students who hate math would love Brother Francis and his mistrust of the subject.

Lest you think that Brother Francis does not know about that of which he speaks or writes, here’s a link to his bio. I’m not tangling with an untutored troglodyte here. Brother Francis has all my respect, I assure you. Sadly, he won’t be directly responding to me: he passed away in 2009.

Let’s start here:

“When not overdone, and when counterbalanced by the proper correctives from the other types of knowledge, geometry and arithmetic, as they used to be taught, cultivated a few desirable virtues of the mind like clarity and precision, and sharpened the mind for the perception of harmony, rhythm, and pattern in the study of nature and of Holy Scripture. But even then, many saints and sages warned against the excessive preoccupation with such studies, and especially against the seductive clarity of mathematics; for it is not enough for the mind to be accurate and clear; we are bound to ask “accurate and clear about what?”

As I have come to understand them, learning and then teaching math as I have during the latter part of Brother Francis’ career, arithmetic and geometry are portals into an intellectual world where Man, created in the image of God, can create. Man cannot create in the physical world God has given: he can only sub-create, as God has already created the substance of that world in concord with His purposes, which are subtle and powerful. Man may only manipulate that substance imperfectly, but Man may approach perfection more closely by devising perfect plans and exact models. Apparently, the creative aspects of number theory and geometry were not features of the courses as teachers once taught them in that time that Brother Francis bemoans.

Creativity is apparently not a virtue of the mind that emerges from arithmetic and geometry. Of course, Brother Francis seems to be very wary of the idea that people construct knowledge. The notion that no one person understands something in exactly the same way that someone else does seems not to be familiar to him; I venture to guess that he would object to it. In my experience, my students can learn a concept well enough to execute the associated skills and to support the learning of other concepts and yet never see the original concept exactly as I do. When I present a lesson to my students, I am not imprinting it upon their minds but giving them a chance to fashion a unique understanding of it that fits their experience and God-given gifts. Learning, even learning something known long before, is a creative activity every time it happens. This difference in premises is a key to understanding the disagreement I have with Brother Francis.

Why would saints and sages warn against the “seductive clarity” of mathematics? I suppose we ought to begin with the Pythagoreans, an offshoot of the cult of Pythagoras that continued its mathematical studies and austerities in the Mediterranean region well into the first millennium of Church history. Pythagoreans had a host of beliefs that the Church would not tolerate, not least a belief in reincarnation, which was the hallmark of the teaching of the original Pythagoras. Associating the Pythagoreans’ extensive mathematical preoccupations with heretical teachings surely did not help the standing of the discipline of mathematics, a subject already difficult to grasp because such invaluable tools as printing presses, variables, base ten place value, and calculus had not yet entered the European intellectual pageant, and wouldn’t until the Renaissance and Enlightenment.

I also discern in Brother Francis’ sermon against modern mathematics a hint of the same kind of mistrust that greeted Galileo Galilei’s teachings, incomplete as they were at the time, on what came to be known as astrophysics. We all know that the Church made kindling of his writings and a prisoner of the great scientist because he stated that certain things moved when the Church did not, and even more controversially intimated that the movements were regular, predictable, and systematic. The Church much preferred that Earth be the center of the Universe and that the Hand of God regulate the constant movements of the celestial bodies. The notion that God could be so subtly powerful as to implant an elegant physical system into His Universe which fulfilled His Will regarding the relation of the Earth to its neighbors was beyond their comprehension, learned as they were about such things as the power of God.

So, too, it is with Brother Francis. Here, I bring in a favorite writer, J.R.R. Tolkien. In his cosmological fable, Ainulindale, Tolkien tells the fictional story of how Middle Earth came to be. In that story, a being — Melkor — made powerful by the One – Iluvatar (all-father) — has evil designs on creating things himself and becoming a powerful lord over other beings, and injects those thoughts into his participation in the creation of Middle Earth. The One explains the futility of such desires to Melkor this way:

“Then Iluvatar spoke,and he said: ‘Mighty are the Ainur, and mightiest among them is Melkor; but that he may know, and all the Ainur, that I am Iluvatar, those things that ye have sung, I will show them forth, that ye may see what ye have done. And thou, Melkor, shalt see that no theme may be played that hath not its uttermost source in me, nor can any alter the music in my despite. For he that attempteth this shall prove but mine instrument in the devising of things more wonderful, which he himself hath not imagined.’ “

The devices of mathematics can’t take us from God, because all that is — including mathematics — originates with God. We might become obsessed and distracted with math, but no more so than with any other human pursuit, intellectual or otherwise. God, the Being That defines existence by being (Exodus 3:14) and thus is the Source or First Cause of all that is, conceived the rules and patterns of the Universe. Our ability to create and discern such patterns and systems using our imaginations and mathematics is part of that conception: God surely has a Purpose for it, subtle and elusive though it is.

I suppose the big difference — other than his superior training and experience — between Brother Francis and myself is that I am free of this strange apprehension about studies that seem to him to distract us from or not to be in harmony with the study of Nature or Scripture. Frankly, I don’t see how that even can be the case: if a mathematician, full of Christian faith, desires to study nature and Scripture, as everyone should, surely that means that the Christian mathematician will see mathematical “harmony, rhythm, and pattern” in the propositions of theology as well as in the elegance of nature. That is what makes a mathematician a mathematician: Brother Francis, despite his own mathematical training and experience, apparently by intent, avoided becoming a mathematician or at least taking up that rubric of perception.

If I did that, I’d say I missed a chance to observe and think about the world in a different way, but I wouldn’t say I happily avoided a chance to devalue the role of faith in my life. I find myself wondering what specific experience caused Brother Francis to see math this way even as he taught it. There might be a valuable lesson in that for me.

Lege, explora, cogita. Quaere verum.

Course and Unit Structure for 2012-13

I want to use a very regular and stable unit and course progression for my new classes and students at Maur Hill-Mount Academy (MHMA) next year. I want the larger structure and the smaller structures within it to be similar to each other, and, when it is necessary to run through all or part of a sequence again, I don’t want that to violate the flow of the course. I want re-teaching to feel natural and not to be an obvious disruption.

I will identify four or five Essential Outcomes for each semester; these will each be the focus of a Unit. Each Unit will consist of several Instrumental Outcomes of Lessons. There will be a sequential and, if necessary, iterative process for learning the Instrumental Outcomes. The process for learning Instrumental Outcomes will be part of an iterative process for learning the Essential Outcomes. Finally, the process for completing a semester course will obviously have two iterations each year.

Note that there are conditions that help me decide whether to repeat a part of or all the sequence and that the plan allows students to take quizzes and Unit Assessments again if they review the content in a bona fide effort to perfect their knowledge.

Here’s my first take:

  1. Open the first semester with an assessment of prerequisite skills.
  2. Arrange interventions for students who need them based on the results of this assessment.
  3. Conduct an assessment of student knowledge of the next unit. 75% of the test should consist of Instrumental Outcomes and 25% should consist of executing and explaining the Essential Outcome of the Unit.
  4. Provide relevant and challenging enrichment for students who already know the content.
  5. Demonstrate the Essential Outcome of the Unit, and document the level of student participation. If the students are able to guide the entire demonstration themselves, then it is time for a unit assessment.
  6. Demonstrate the Instrumental Outcome of the Lesson, and have students take and annotate notes, delaying questions, suggestions, and observations until the demonstration is complete. This fulfills the first step in transferring control to students: the first time, I do it and they observe.
  7. Demonstrate the Instrumental Outcome of the Lesson, and have students take notes, but this time welcome questions, suggestions, and observations during the demonstration. This fulfills the second step in transferring control to students: the second time, I do it and  they help me.
  8. Have students attempt the Instrumental Outcome of the Lesson, and have students take notes but this time have students conduct an informal review of each other’s work and results, bringing the most vexing misunderstandings to my attention. This fulfills the third step in transferring control to students: the third time, they do it and I help them.
  9. Assign Practice in the form of Homework. Students attempt the homework individually or collaboratively, but each creates a homework document to submit. Students grade their homework themselves and then collaborate with classmates or me to perfect it; I expect to see perfect homework every time. This offers students the opportunity to experience the fourth step in transferring control to students if they are ready; they attempt the Instrumental Outcome of the Lesson without my help, at first. I simply delay my assistance until the next class — but I do not withdraw it yet.
  10. Assess the Instrumental Outcome of the Lesson by giving a brief quiz. I grade the quizzes myself and return them to students as soon as possible (often in the same class period) so that they may collaborate with classmates or me to perfect them. This fulfills the fourth step of the transfer of control to students: when we are ready to move on to new content, they do it and I observe.
  11.  If the brief quiz produces a median less than 80%, then re-teach by repeating steps 6-10. Assign top performers to serve classmates as peer tutors.
  12. If the brief quiz produces a median greater than or equal to 80%, then arrange interventions for those below 80%. Return to step 5. If, after repeating step 5, there are no more Instrumental Outcomes in the Unit, then prepare the class for a Unit Assessment. If, after repeating step 5, there is at least one remaining Instrumental Outcome, continue the iteration with step 6 and so on.
  13. Conduct the Unit Assessment by giving a test where the various skills and knowledge in the Unit takes up 75% of the test and the remaining 25% consists of executing and explaining an instance of the Essential Outcome of the Unit. This Assessment may be a written test but could take alternative forms — and should if students have particular needs or strengths. If the median for the class is 80% or more, then arrange interventions for those below 80% and move to the next unit. If the median is below 80% arrange enrichment for those whose scores are greater than or equal to 80% and intervene with those whose scores are less than 80%.
  14. If there are no more Essential Outcomes for the Semester, then prepare students for a Semester Examination covering all the Essential Outcomes explored in the class during the Semester. The test should weight the Instrumental Skills and knowledge from the units at 75% and executions and explanations of the Essential Outcomes at 25%.
  15. Administer the Semester Examination.

As always, feedback in the comments is very welcome!

Lege, explora, cogita. Quaere verum.

New Wiki Entry: Solutions of Inequalities

Click here to see it.

Since we’re officially into the summer, you can expect that I’ll be adding to the wiki daily. I’ll give a weekly listing of new entries: I don’t want to run up my number of blog posts just to report that I’ve continued to work on my wiki.

As always, I’d love to have collaborators — I’ve got one so far and would love to have more. Let me know here or at mikepoliquin@gmail.com or through the wiki if you’d like to jump in and become a part of it.

Lege, explora, cogita. Quaere verum.

Answer to Problem of the Week Ending May 25

19 May – 25 May 2012

A company packages its specialty beverage in a bottle that is a composite of a cylinder, a hemisphere, and another cylinder.

The vast majority of the volume of the bottle is cylindrical in shape with radius of 5 cm and height of 23 cm.

The hemispherical portion of the bottle above the cylinder has a radius of 5 cm (obviously) but is not a complete hemisphere, because the company affixes a cylinder with a radius of 1.25 cm to a matching opening at the “top” of the hemisphere.

The total height of the bottle is 30 cm.

What is the exact volume of this bottle and what is the height of the narrow cylinder at the top of the bottle?

Answer: I used the formula V = πr2 for the two cylinders and the disk application of integration to find the volume of the near-hemisphere to get the result:

(134,300 + 90√15)π/192 cm3

≈ 2,202.07 cm3

The height of the narrow cylinder is 2 + (√15)/4 cm ≈ 2.16 cm.

Lege, explora, cogita. Quaere verum.

My Ideal Education Paradigm #5: Reasoning Rubric

This turned out to be the easiest rubric to do, because someone has already done a terrific job of setting one up for us.

The educators among you will know the name of Dr. Robert Marzano, a giant in the development of newer and better classroom methods and curriculum. Anyone who wants to do anything significant to change the practice of education will do well to reflect on his work; as he is the only significant researcher I’ve encountered in those fields, he’s my guru when it comes to finding the right strategy for a new instructional challenge.

He or his minions have devised an excellent rubric for reasoning. It covers these ten thinking and reasoning tasks:

  1. Comparing and Contrasting: Do students identify the essential areas of sameness and difference among the things they analyze and evaluate?
  2. Analyzing Relationships: Do students make causal and correlative distinctions? Do students see all of the relationships they should? Have they identified interesting but unexplored relationships?
  3. Classifying: Do students correctly identify classes, intersections of classes, and complements of classes?
  4. Argumentation: Do students construct chains of statements that combine through syllogism, substitution, or classification to support a stated proposition? Do students define their terms clearly? Are their arguments valid? To what extent do they select debatable premises? Do students restate the opposing arguments correctly and refute them comprehensively?
  5. Induction: Do students generate reasonable hypotheses? Do students reason well from patterns? Do students appreciate the distinction between interpolation and extrapolation and the associated strengths and weaknesses?
  6. Deduction: Do students construct tightly molded sequences of reasons that lead to a promised conclusion with both a valid structure and well-established links to premises and given information?
  7. Experimental Inquiry: Do students devise experiments and generate results that give a clearer view of the hypothesis and initial observations than before? Are the students’ hypotheses both falsifiable and testable? Does the student draw a conclusion and chart a new direction based on the experiment?
  8. Investigation: Do students use diverse resources effectively to assemble a body of knowledge that supports more inquiry of a different type? Do students ask incisive questions? Do students make the effort necessary to explore all possible solutions to a problem?
  9. Problem-Solving: Do students have a broad arsenal of methods for dealing with the problems that arise in their chosen field of inquiry? Do students select and fulfill strategies in a repeatable and verifiable way? Do students understand the meanings of their solutions and know how to verify them?
  10. Decision-Making: Do students have a broad arsenal of methods for optimization of results given a situation that may call for several of those strategies in the student’s arsenal that apply? Do students understand the positive and negative consequences of their decisions? Do students weigh these against each other as objectively as they can in order to make a correct decision?

Here’s the Marzano rubric. I got it from the Utah Education Network. I might modify it as I use it, but for the moment, it’s the best thing I’ve seen for covering a very important and yet very broad region within the evaluation of any student’s achievement. Of all the rubrics we’ve looked at so far, this one will require the most annotation, because there is no way to be sufficiently specific on the form itself.

Next up: a rubric for professionalism.

Lege, explora, cogita. Quaere verum.

This Is What You Get When They Make Me Do “Field Day”

To my students, Field Day is a wasted day at the end of the year. In order not to waste it, let’s use it for its nominal purpose. Let’s talk about Fields!

I’m in the mood to review fields, for my own entertainment. Here we go! Hey, sit down back there! Calm down! I know this is exciting, but you have to control yourselves!

A group is a set, taken together with an operation, that fulfills four axioms:

  1. Closure: If a and b are elements of the set, performing the operation on a and b yields an element of the set.
  2. Associativity: If a, b, and c are elements of the set, then to perform the operation first on a and b and then on that result and c yields the same result as performing it first on b and c and then on that result and a.
  3. Identity: There is an element in the set, I, called the identity element, which, whenever we operate on it and any element a (including itself) from the set, or on the element a and the identity element I, the result is the element a.
  4. Invertibility: For every element in the set a, there is an element a* such that operating on a and a* or on a* and a yields the identity element I.

If it seems strange to see each operation rewritten in the opposite order, I apologize on behalf of the concept of a group. You see, groups do not necessarily have to have the property of commutativity. Commutativity means that the result of operating on two elements of the group is the same regardless of order. Lots of number sets, beginning with the integers, are groups under addition.

If a group DOES have the property of commutativity over its operation, then it is an Abelian group, which many people simply call a commutative group, for obvious reasons. Since addition is commutative, our familiar sets — integers, rationals, reals, complex numbers — are Abelian groups under addition.

A ring is a set, taken together with two binary operations (meaning that we perform the operation on two of the set’s elements at a time), is an Abelian group for one operation and a monoid for the other, with a distributive property. Regardless of their definition, we call these operations addition and multiplication, and use the symbols + and •, respectively. Using parentheses to indicate priority, we express the distributive property as a • (b + c) = (a • b) + (a • c). Remember that we haven’t stipulated any order for operations at this point.

A monoid is a set that, taken together with multiplication (in this context), has the properties of closure, associativity, and identity. Integers, rationals, and reals are monoids under multiplication.

A field is simply a ring whose elements, excluding the identity element for addition, also form a commutative group for multiplication. Obviously, zero is the object of that exclusion, and the reason for it is the impossibility of defining a result for dividing a given number by zero. The rational numbers, the real numbers, and the complex numbers are all fields — in other words, every day you have math class or have to do any genuine mathematics, it’s field day!

Isn’t field day fun? You got all this from me while I suffered through a scratched up DVD of Iron Man II in the school auditorium. Most days, I’m the happiest guy I know, but today, I assure you, you are having more fun than I.

Lege, explora, cogita. Quaere verum.

How My New Position Will Change My Blog

I have taken a new position for the 2012-2013 school year.

I shall leave my current post teaching Pre-Algebra and Algebra 1 to 7th, 8th, and 9th grade students at George S. Patton Junior High School at Fort Leavenworth, Kansas. I have also coached football and basketball at Patton. I have held this position for two very successful years. I will leave with happy memories of very special and dear students and wonderfully professional and supportive colleagues. They all have made me a better teacher and a better person.

I will teach Algebra 2, Trigonometry and Pre-Calculus, Calculus 1, and Calculus 2 to high school students (grades 9-12) at Maur Hill-Mount Academy in Atchison, Kansas beginning with the 2012-2013 school year. I will also coach the Scholars’ Bowl team; I may also coach other sports if needs and openings arise.

Some changes to this blog will coincide with this change in position — changes that I welcome joyfully.

I am returning to Catholic education after a two-year hiatus. I am a practicing Roman Catholic and I believe strongly that I am a better teacher when I infuse my teaching with my faith. My practice — in and out of Catholic schools — has improved steadily over the last seven years: this new position offers me a chance to put that practice together with my faith, which I see as a chance to be the best teacher I can possibly become.

Beginning Saturday, May 26, I will not restrain myself from expressing my faith in this blog. I do not know exactly how that will manifest itself; I have never maintained a blog while practicing as a Catholic educator. It may seem to some that very little changes; it may be that many things will change a great deal. All I can say with certainty is that I will no longer restrain expressions of  my faith here. I certainly have done so previously.

One might ask why I ever did restrain my expression of faith in this blog. Perhaps I should not have done so, but it seemed a prudent path given the nature of American public education. Did I sell out? I’ll have to reflect on that for a while.

I wanted this blog to attract attention from students, colleagues, and supervisors at my school, but I knew that my administrators were trying to limit overt expressions of political views and religious faith in my school. Public school teachers have a responsibility to avoid such entanglements in the view of many good administrators and school board members. In certain cases and situations, American courts of law have agreed. While no one ever explicitly said so, I believe I had to meet such standards if I wished to keep my position; it is a good position as such things go. I would have been very happy to stay and continue as before if this new opportunity had not presented itself. Once I knew exactly what this position was, I was pretty sure I’d take it if I could win the offer.

I willingly accepted restrictions on my religious freedom and freedom of speech as a condition of taking my position at Patton, and I will continue to abide by them until I have completed my service. Friday, 25 May, is the last day of the school year at Patton. I will return school property and vacate my classroom that afternoon and thus I shall be free to take up the mantle of a Catholic educator, a mantle I intend to appreciate and relish even more than I did in my previous stints as such.

Many things about this blog may not change at all when this happens. I will continue to work on my wiki. I will continue to seek collaborators. I will continue to put up at least one problem of the week every week. I will continue to try to work out fundamentals and ideals of teaching, learning, and education.

What will change? I will incorporate and infuse the values and ideals of Roman Catholic doctrine and tradition into my writing. I will also seek assistance and guidance from people who have taught these courses more than I have. One of these changes will certainly have a new impact, but surely discussions of pedagogy and math content are familiar fodder for us.

Please keep coming back to my blog. I will be asking for insights and support as I change gears for this exciting new challenge, and I continue to offer my support to anyone who wants help regarding education generally and math education particularly. I continue to believe in the peer-tutoring / collaborative-learning ideal: learners learn better from peers, and peers learn better by teaching. Let’s continue to leverage our interactions here to grow mutually, because, finally, we are all learning together.

From my point of view, I consider this change of venue and practice an opportunity to give you a more complete range of support and information. My faith is by far the most powerful part of who I am and what I do. That could never change, even in an environment managed to exclude it.

As I return to service in Catholic education, I believe I will be a better writer, blogger, sharer, teacher, and learner because of the freedom I will have to incorporate my faith into every aspect of my practice, including this blog.

I hope you’ll continue to join me here as I continue to explore this medium’s vast potential to help teachers teach and students learn.

Thanks for following me here or Twitter or wherever. I’m honored by your attention; it is my pleasure to follow many of you as well.

Lege, explora, cogita. Quaere verum.
(Look for this motto to change Saturday, too.)

Wiki Entry of the Week 19 May 2012

Just in case you’re new to the blog, I am building a wiki at http://mathwikibymrp.wikispaces.com.

This week I gained a collaborator, a fellow secondary teacher from Arkansas.

Whether or not you wish to be anonymous, please check in with a comment or message or e-mail (my e-mail is mikepoliquin@gmail.com) if you’d like to work with me on the blog or the wiki or in some other way. I’m always interested in working with others to improve this work.

This week’s wiki entry is “inequality.” Inequality is a much more complicated concept than equality, to be brutally honest about it. Mathematical equality is an inherently binary relation: two values either are or are not equal. There’s only one way to be equal — but there are five types of inequality: the broad “not equal to,” the more informative “less than” or “greater than,” and finally the more inclusive types of inequality, “greater than or equal to” and “less than or equal to.”

Read the entry here.

Lege, explora, cogita. Quaere verum.