Inspiring the Next Generation

I have some interesting news to share.

The Grade 10s in one of the schools where I sub began their poetry unit in English this week. I subbed for them yesterday.

One of their tasks yesterday was to write a poem in one of the forms they had already learned, then share these with the class. There were some very reluctant students; they had a low opinion about this sharing business, particularly their contributive involvement in it.



I decided to break the ice by sharing one of my poems. And I had access to two: those I published in my writing blog.

The poem I chose to share was Van Gogh and the Moon. It was a hit, particularly when I explained to the kids that the poem was an in promptu (five minute) response to a writing prompt in the local writing club.

So, yes, I got a chance to plug the Write Group as well; I told the kids that students from the school were part of the group, which peeked more interest.

But more importantly, it got each of the students to open up and share some of their poems, not just those they wrote in class yesterday, but those they had access to through their iPhones and other devices.

It was a perfect marriage of teacher and student sharing, technology (I used the Smart board; the students used their devices), and encouragement and modelling by example.

It never ceases to amaze me how well these teachable moments go when the teacher releases control and opens up to her or his students. (Of course, it also never ceases to amaze me how badly such moments go as well at times. There is a definite case for timing and thoughtful and responsive judgement here.)

These students have everything to be proud of. They have incredible imaginations, and a deep and active appreciation for written communication.

Moments like these remind me how much I love teaching, and learning with, these students.

This article is cross-posted in Digital Substitute and Stefras’ Bridge.


Inspiration Against a Lost Generation


This video serves two purposes. First, it is a creative way to communicate a profound and inspiring message for everyone, by juxtaposing opposing tones via a literary twist — a winding and unwinding technique common to folktales. Second, it is a reminder to me, and perhaps to many of you, that this is what my generation felt and talked about when we were in high school and graduating. I find the echo of my thoughts juxtaposed against this video nearly a quarter of a century later rather interesting.

There are so many parallels, some of which we recognize right away, some of which we forget until we are reminded.

For some reason, this video reminded me of my generation’s movement to curb pollution and yet the nearly simultaneous increase in vehicle turn-over and layers of packaging around otherwise small items. Today, our kids and students are still moving to curb pollution, though with a narrower scope — less, if any, emphasis on all the forms of pollution, including light, noise and odor pollution, and more emphasis on pollution that perpetuates and aggravates global warming. There is much happening in the World that they have little notion and control of, as was true for us. However, as we became more pollution, waste and recycling conscious and active, I wonder what their generation will accomplish.

What parallels do you see between the priorities and ideas of your students and those you had when you were their age?

Math: Let Students Earn Their Weight

mathhombre's Twitter avatar In a post he published yesterday, John Golden offered two geometric problems he had included in a math final for his preservice math teachers. Please visit his post and take a crack at his problems.





What I really liked about these problems was how open they were. The first problem, giving a circle geometry diagram, challenges John’s students to “Figure out some of the missing information in the diagram”. The second problem, also giving a diagram — this one of a series of lines, segmented only by the diagram edge, asks students to “Find more angles”.

At first, I thought this was a rather vague way to offer problems, particularly on a final exam where marks are summative and presumably weighted more. Then, I gave it some thought. I am not sure what John had in mind, but having questions which ask for “some” or “more” answers leaves the onus of how much work the student does and what weight the problem has on a test on the student. A way to assign a mark to such questions is to make “Each correct answer worth one mark”. Then all one has to do is make the test worth out of so-many maximum marks (less the total possible to get), order the questions so the “big” ones come first and voila … one has a test that the student weighs.

But what about students who answer only some questions thoroughly and skip others? One could make each question worth a minimum amount of marks, or make all questions equally difficult or multi-outcomed. There are likely other ways to ensure all outcomes are covered by the student.

It is an idea, one that just occurred to me as I read John’s post. What I really like about it is that it is truly open. The student cannot prejudge how much a question is worth and allocate effort accordingly. The grade the student gets is tied to the value the student, not the teacher, applies to each question.

What do you think? Do we herd our children with our judgement of worth? Should we be teaching them to do this intrinsically?

Oh, and by the way, my next post will show my solutions to John’s questions.

Engaging Kids: A Little Classroom Humour



There has been a recent rash of puns spreading around one of the schools I sub at. It has infected kids at all grade levels from 5 to 12. Of course, being called in to teach occasionally, I happened to walk into this contagious disease with no warning and no defence last Thursday and Friday.

My kids tried to infect me twice with puns on Thursday. Unfortunately, I was rather vaporous on that day, so I did not catch on to either attack and thwarted the jokes.

The Mistaken Challenge

The first attack came from my Science 9 students. I can not remember the pun and ruined the joke anyway. The students grudgingly revealed what they were trying to get me to say (without getting in trouble). I remember being glad I didn’t. My guess now is that the pun must have been inappropriate to school anyway.

(At this point, I should confess that I am a stickler when it comes to swearing or inappropriate topics from my kids. This deters my kids for about 15 seconds after I warn them not to engage in such behaviour. Then the fun begins: trying to find ways to tease Mr. Urban. This particular pun was their latest effort.)

Still, my kids were unaware that I hadn’t caught on. I am sure they have taken my sidestepping of the pun as a challenge, so I expect more cunning attempts to get me to break one of my own rules.

These kids just crack me up. They are so eager and clever. And for the most part, when I ask them to, they willingly engage in the learning activity at hand.

There is always room to play and enjoy class. My kids like joking with me; I am easy enough to let them bait me, yet usually wise enough to get out of their traps.



The Unintended Lesson

The second attack came from my Grade 12 math students. My Grade 12s were a little more cautious with their pun, choosing one that was barely offensive.

But, again, I did not catch on. And how spectacular the result.

I have watched these kids grow up from Grade 7 and am absolutely fascinated at how mature and confident they have become. I can’t tell you how awed and full of pride of them I am. So, yes, I was targeted again.

The pun was simple. My kids asked me “what is that under there?” and I was supposed to reply “under where?”

I did not.

Being obtuse


Really, it never occurred to me to even ask that. Over there were cabinets and shelves sitting without gap on the floor and a well raised table clearly with nothing under it.

I was supervising a probability quiz and wrote it myself along with them. (Probability, permutation and combination just confuse me. I can not make heads nor tails out of them. The quiz had a few sporting coin questions in it by the way.)

So I was thinking mathematically, systematically and about test question quality. I ended up pitching against the ambiguity of vague questions with my kids, particularly the one they were asking me, and they in turn kept trying rather desperately to get me to ask that magical pun-question. Dialogues of the obtuse are so amusing.

It all ended up in laughter and teacher-student bonding that would never have happened had I clued into the pun at any time.

One boy grinned that the joke turned out better than my kids had planned. A girl told me that I really got her thinking about clarity and definitions. Everyone, including me, ended the day with renewed energy and a smile.

Yeah, I was thick on Thursday. I normally take questions and comments at face value. I rarely look for ways to make this or that perverse by some lateral interpretation. I am eager to help.

And I love the way I am, and my kids. They can fool me any time they want, so long of course that doing so does not interfere with their learning activity.

I feel much closer today to these two classes of students, particularly the Grade 12s, as a result of this jocularity.

A little humour in the classroom is engaging and builds strong bonds. I am ecstatic that I subbed these kids on Thursday. A lot was won.



Pi Day: Part Tau

There has been increasing buzz about Pi and Pi Day lately, probably because July 22 — 22/7 — is approaching. And though any web search can find this buzz, I would like to add a comment about Tau Day (June 28), which is adding noise to the excitement.

“Ya! Do tip: Laud Tau, at dual Pi, today!” “I prefer Pi”



Tau, having been recently replaced by phi as the symbol of the Golden Ratio, is a proposed symbol for the value 2*Pi, lauded (to borrow from the palindrome above) by those who think 2*Pi is a better measure in all ways than Pi.

This claim, of course, has created a heated but sometimes hysterical discussion, and post and video “war”, between two extreme camps: the Pi’s and the Tau’s. Add to this a third camp, those who argue for Eta (Pi/2 or Tau/4) as the standard, and a fourth, who promote Pi/4 (Tau/8) instead, and we have a mathematical event to behold.




Of course, none of this really has serious implications, since we all know that Pi/3 or Tau/6 ¹, the sextant, with its relationship to the equilateral triangle, radian, Babylonian sexagesimal system, Earth year and rational cosine value, is the real fundamental unit of the circle.

It seems we are confused.

This of course is all in good fun. Yet tell that to Tau Beta Pi and Tau Alpha Pi, the Engineering and Engineering Technology Honor Societies, whose society names are at risk at both ends.



It’s happened before

Nor is this the first time a math convention has been questioned. In fact, it is not even the first time a circular measure has been questioned.

Before there was trigonometry (the study of measures of closed three-kneed figures or triangles), there was circle geometry (the study, which we still have — but lacking the now analytic trigonometric part, of regular closed no-kneed figures or circles). In circle geometry, the chord is king and emphasis is on the geometry and measurement of the circle, line segments and angles. Everything was working really well.

Then came sine!



Sine, of course, is half a chord, chord/2. It seems weird to us now, but at one time sine was the oddity trying to replace the convention. Yet, when it did become convention, a new field of math was born.

In a history shamefully oversimplified, circle geometry split into two fields and, for the sine portion, analysis, ratios, angles and triangles became the emphasis, so leading to the title of trigonometry. Circle geometry continues to deal with geometry and measurement. And the chord is the usurp outsider.

And history we witness

So we have Tau (whole), and Pi (half), and Eta (quadrant), and Pi/4 (octant), and Pi/3 (sextant). I wonder what history will come of this.

Importance: A rose in the classroom

Does it really matter what constant we use as the base unit for circle measurement? They are just names. Some formulae will work better in some situations than others, and this will change with situation.

How we choose to deliver the concepts to students is far more important than what we call them. Truth be told, all systems should be taught interconnectedly with no mention of which might be the opinionated best.

The key is engagement and problem solving. Students need to understand how to use the math, why it works, where it is used and how it is used. It would benefit them if they learn their own formulae, and we help them “conventionalize” these to fit what other mathematicians do and say.

Changing of the conventional Pi to Tau, or Eta, or either of the other measures, might change the nature of circle and trigonometric math in ways we can not predict at present, but that will come in the future. Today we have these five fundamental units, which are all arbitrary, math-founded and related. Who is to say which is best?

By any other name, is a rose not still a rose?



More resources

Want more? Visit Benjamin Vitale’s June 28th Pi is wrong! Here comes Tau Day, watch Vi Hart’s Pi is (still) wrong, and read Bob Palais’s original π is wrong! which started the Tau, then two-pi, movement.

Then read Mike’s response to The Pi Manifesto, from the creator of Spiked Math Comics, then the continued debate on On Pi Day we eat pie. On Tau Day we eat Taoists? and ‘Tau day’ marked by opponents of maths constant pi, including its comments.

Given that we are rather gossipy creatures, most of the posts, discussions and video have titles that attack poor Pi and its Day, or Days (March 14 or July 22). But a few out there attack Tau as well. The Eta’s, Quarter-Pi’s and Equilateral’s just cling where they might be heard. Sounds like a school yard, doesn’t it?

This post actually started as an update comment to my Math Challenge: Pi Day, but the comment morphed into a post of its own, so I decided to make it so. If you are interested, please visit my Pi Day math challenge post.


¹ Given its Babylonian pedigree, perhaps we should call Pi/3 Sedis, which is six in Assyrian.

Math Challenge: All-digits Arithmetic

Dennis Coble, @DennisCoble, just tweeted me this challenge an hour ago.

Here’s 1 that might interest you. Numbers 1-9 all used: 3 digit number, plus or minus 3 digit number, gives another 3 digit number.

As usual, the problem is deceptively simple, as is the solution. However, students could be engaged in their activity of this task for a full period. 😉 And the ordering of student ability and success could be shaken. 🙂