### Anwar: Character background

Dec. 13th, 2010 09:01 pmAs you've probably all noticed, I haven't been around much lately. That's because I have a job that doesn't require ranting from school anymore and DOES require that I be on task pretty consistently, so I haven't had reason to pop over. I'll strive to be around more!

**( Cut for length; cross-posted to LJ )**

### Rebecca: Field Notes

Jul. 30th, 2010 09:37 amSent out with a rep from Min. of Heritage to look at an artifact. Supposedly Roman era, in Woodsleigh, in the Borderlands. Strange place for a Roman artifact; Romans didn't spend much time that far north, barely touched that valley. Woodsleigh is a throwback; was untouched by civ. until recently. Pensioners moving in en masse now.

Heritage person's odd. Emily Braithwaite by name. Doesn't suit her. Don't know why they sent her; doesn't know squat about Scottish/border history, or preservation, or what not to touch.

(Later that day)

HOLY SHIT WHAT THE FUCK JUST HAPPENED?????

/unprofessionalism. Let's try that again. Must not swear in notes for academic journal. Historical significance and all that.

Found artifact. Looks like a greek urn made of steel. Completely anachronistic. Was stored in a chest circa 1880. Very poor condition, but not valuable except as a middle-class attempt at an antique even if it were preserved. No great loss. Next to urn was a page torn out of a book; contained excerpt from "On the Morning of Christ's Nativity," w/ woodcut picture on back.

And sullen Moloch, fled,

Hath left in shadows dread

His burning idol all of blackest hue:

In vain with cymbals' ring

They call the grisly king,

In dismal dance about the furnace blue.

The brutish gods of Nile as fast,

Isis and Orus, and the dog Anubis, haste.

Barely had time to look at it before mechanical sound and the guard running away distracted me. Ran up to see what was going on. Very confused gentleman in 18th C banyan coat. Didn't know where he was. At all. Didn't know what country the Borderlands were bordering, didn't know the year. Barely had time to start to get a handle on him when Monster Robot came out of the ground. Braithwaite had stayed with artifact; Robot was near artifact. Something about the way she touched it caused him to - awaken? Sounds more like Star Trek every minute.

Braithwaite and I ran for lorry. Confused man followed; we took him along. Robot followed too, latched onto boot; lorry couldn't move with his weight. Confused man pointed out the robot wasn't trying to hurt us.

Don't know exactly what happened next. Something shifted, and the lorry crashed into a tree. GPS stopped working. All got out. Confused man took over, talked to Robot. Confused man says his name is Hero; not convinced, don't think he remembers himself. Robot is the Gesh. Robot looked at GPS, said it was useless b/c no satellites around this world. Around this time, saw a peasant. Either very authenticity-oriented SCAdian or a peasant b/w 1400-1700. Can't place it more than that; pre-Jacobean, pre-mechanization, all homespun, but not a noble so no fashion to speak of. Hand-made axe. Ran away before we could talk to him. Enough to see him: no satellites and clothes like that = time travel.

This just stopped being a resource notebook for an academic paper. Bloody hell.

Heritage person went and stole clothes for us. Peasant clothes, fifteenth or sixteenth century; undyed, plain, maybe poor Puritans? Need more info to figure that out. Peasant came by, told us Gesh shouldn't be out of the Grove. Reverend wouldn't like it. Reverend? So, post-Reformation, then. Getting more Puritan all the time. Peasant recognized Gesh. Gesh analyzed something about him; says he's not like us, lacks same chemistry. Suppose that makes sense if we're four centuries in the past.

Going into town shortly to see what we can figure out. Seriously - can handle Catholics and Anglicans, even Methodists most of the time, but Puritans? Why did it have to be Puritans?

### New Character: Rebecca

Jul. 29th, 2010 08:46 amGame system: Dr. Who, the new one

Players: Alex, Patrick, Kendra, and Me

GM: Piet

Character Background

I was born in Cambridge proper, in 1984, to two academics. My father was at that time a lecturer at King's College, Cambridge, where he was awarded a fellowship while I was still a small child; my mother, who had been a graduate student of his, had given up her academic pursuits in order to raise a family. This endeavour began with my brother, born eleven months and eight days before me. I believe my amiable nature was completely overshadowed by his boisterous egocentrism, which situation did not change for many years.

My early years were dominated by a strange hybrid of religious and academic fervour. My maternal grandmother was, I now believe, quite mad. She insisted on taking my brother and me to her church, an evangelical denomination brought to the UK by American ex-pats and extreme even for Americans, who in my experience tend to prefer extremism as a mode of living. My mother rarely attended, partly because her faith had dried up and partly because it gave her a few hours on Sunday morning to do exactly as she pleased. Nana took us to Sunday School and church, reinforced the fire-and-brimstone message, and succeeded in scaring me thoroughly out of my wits. After her death when I was eight, I ceased attending any church at all, and must steel myself to enter any establishment designated an evangelical denomination; I've pushed myself in recent years to enter other churches when there were no services happening, and I can generally manage to be there without my company noticing my discomfort, but I leave as quickly as I might without giving offense. I'm sure it says something profound about my relationship with my parents that they are completely unaware of this. It likely says something profound about me, too.

My father is the epitome of the absent-minded professor archetype. I believe he barely noticed my mother's attempts to ensure our family's position in the social life of the university; indeed she could often be heard to complain that she could not be certain he knew there were guests for dinner when she never saw him during the day and he refused to carry a phone or answer his desk phone while he was working. I believe he never heard them ring, and was saving face with my mother. In any case, his lack of attention was a factor in their divorce when I was sixteen. Only two people of our acquaintance were at all surprised by this: my father because he was oblivious to all but his work, and my brother, oblivious to all but himself.

My father's work was in history, where he focused on the Etruscans, forerunners of the Roman civilization and endlessly fascinating to him. He speaks, reads, and writes ancient Latin and Greek fluently, to the point where he can switch dialects between them and tell you precisely which century those dialects belong to; given half a chance he will expound upon the difficulty of ascertaining when certain words and phrases became most common. He insisted that my brother and I learn Latin and Greek, though my brother threw off all attempts to make him proficient at either. He has begun to read mathematics, somewhat late to the game, but I believe my father is simply content that he is studying anything at all. I am reasonably proficient in Latin and Greek, which skill has certainly helped in my chosen field; rather than pursuing antiquities and setting up a contest with my father, I chose Newnham College, there to read Scottish and Welsh history. The transition from a Gaelic culture to an English-speaking one is quite fascinating to me, in particular its connection to religion and by extension, to politics - the two cannot be disassociated during the medieval era. My Scots Gaelic is not as proficient as I would like it to be, but my Welsh is passable.

After my Master's degree, I decided not to pursue a doctorate immediately. Truthfully, I was desperate to remove myself from my father's house, and as my mother's flat in London was an even worse choice, it became obvious that I needed a job. The British Museum hired me as a junior curator on the strength of the recommendation of two of my professors, both of whom have interests in Father's more recent work. In the world of museum employment, education comes second only to connection as the deciding factor. I enjoy the job; I work primarily with the Welsh collection, though I'm involved in the Scottish collection peripherally and, due to my facility with dead languages, occasionally find myself aidng with Roman or Greek antiquities as well. I expect I'll return to Cambridge for a doctorate in a few years' time, and I'm hoping my credit with the Museum will be such that they will sponsor me in that endeavour.

### Altriona: Were we mistaken?

Jun. 10th, 2010 01:51 pmI would expect them to be bemused, for our vendetta is private, based in a supposition of crimes as yet unproved. But I would not expect them to remain baffled when we reveal the name of the beastman who travels with us.

The reason for their confusion? They say they have seen Mallion alive and well in Marienberg.

At first we believed this to be a simple hoax. We were certain of the identity of our own Mallion, despite his bestial appearance. But time went on, and it became clearer and clearer that people who should have known better had been taken in by the hoax.

We committed a horrible act of piracy and murder in the name of this vengeance. We treated with skaven to purchase warpstone with slaves - an act which will make us very rich - though the amount of warpstone we will send into the Empire to an unknown buyer could bring down the Emperor and permanently shift the balance of chaos in our homeland. We set a city to civil war. We leave mayhem in our wake wherever we sail.

And now, having committed ourselves to this course of action, having committed crimes to heinous to admit forgiveness, a letter comes from Marienberg - written in Mallion's hand and borne by Valadar's most trusted man.

The seal, of course, is not Malliion's. It is Father's. I wear Mallion's signet on a chain around my neck even now, knowing that I have no right to use it. But the wording, the writing, the sentiment - had I not left my brother in Sartosa to come to the meeting where I received the letter, I would have had no doubt but that my brother had written it.

Gallos, for it was he who was sent to talk to Lorandara and me, asked us to return with him to Marienberg. He had letters of passage allowing us to traverse the Tilean city-states where we are wanted as criminals. He would not compel us, but he asked. Everyone who knew of the request urged us to go. I refused. I informed Gallos that I would write a letter, the answer to which would prove if the person who had written to me was, in fact, my brother Mallion. The letter contained two lies. One was that I had tested the bestial Mallion using the same test I was about to give the pretender, and that he had passed. I had not yet tested him when I wrote that. The other related to the piece of poetry I used as my test. If he is my brother, he will recognize that lie and challenge me on it.

Lorandara did not come to the meeting with me. She waited behind for Mallion, and told him of my destination when he came out of the meeting which Lord Rackam had arranged to occupy his attention. I know not what transpired when Mallion and Lorandara returned to talk to Lord Rackam after she informed him of my departure, but whatever it was, he was tense and upset when he arrived at the meeting.

I told Mallion of the letter, and gave him a different test: a piece of poetry he had written many years ago, the theme of which was peculiarly appropriate to our situation. I hated my doubt and needed to lay it to rest. I needed my poet brother to be himself, to pick up the rhythm and rhyme and meter as of old, to duel with words as once he never would with a weapon.

He could not do it.

I think I hid my terror from him well. It struck me soul-deep. It helped that I went with the men in the rowboat while he and Lorandara took off flying for the ship. I had time to take hold of my emotions and conquer them, to convince myself that the loss of one poem was hardly proof of anything. I determined to set him another test, to give him another chance to show that he was my Mallion.

And yet my heart quaked. Was he ever my brother? Had he been my brother, but changed? How much could he be my brother if the poetry, that which so defined him, was lost to him?

He passed the second test, quoting verses that I helped him write, verses known to none but us two. We have decided to work with Mallion to keep his poetry alive in his soul.

So our Mallion is truly Mallion. What will we do if the pretender passes the test - if he knows that which only Mallion knows? How will we live with ourselves should our vengeance prove to be a hollow thing founded on a hellish misunderstanding? How so if Valadar is innocent of the patricide with which we have sullied his name?

### Altriona: If Mallion Ye Would Be

Jun. 10th, 2010 05:33 amIf you are indeed my brother Mallion, I owe you a lifetime of apologies and amends, and Valadar too - indeed our entire race. I am not sure yet if I hope that is the case.

It would seem there are two of you. One is with us, changed in form certainly, but I am convinced that he was once my beloved brother. Among other reasons, he passed the test I am about to give to you. However, Lorandara (who did not abandon you - she believes that she came on this voyage in order to stay with you as your loyal and loving wife, and her guilt over this possibility is great) wonders if it is possible for the magic that changed our Mallion into his present form to have split him in two, leaving you to wander the wood insensible for a time.

With this being the case, and knowing that this Mallion needs us much more than you, we could not come with Gallos at this time. If you are my brother, you will seek to understand and forgive that. It is certainly not the least of the things you will have to forgive us, should your identity be confirmed.

The following is an excerpt from one of your own poems. To the best of my knowledge, the working copy - the only one ever committed to parchment - is amongst Lorandara's books on the ship. I was there when it was written, there through much of the work that followed in an attempt to make it come right, and there when it was abandoned, nearly finished, but unsatisfactory. I suggested at least one simile that you adopted. I enjoyed it and regretted that you did not, so I remembered it - at least the stanzas you seemed to like.

I will give the first stanza. You will give as much of the rest as you remember. You will recount some of the changes made to it, the details of its composition, the purpose you had had for it. And if you convince me, we will all three of us return to Marienberg to face the Family and make such amends as we can.

*Letters in gold that twist and distort*

Music in notes none can hear

Pictures in colours that none here can see

Visions beyond all we know *

Music in notes none can hear

Pictures in colours that none here can see

Visions beyond all we know *

It is your turn. If ye be not my beloved brother, know that our vendetta as stated will continue. I think I hope that ye be he, whatever punishment my crimes may warrant. Know too that I acted in what good faith I could muster in a situation where no firm course presented itself, and would retain at least that small measure of honour, slim though it be.

With your own signet ring, I seal this letter. Please give my kind regards to my mother; would that I could spare her this heartache at the heart of her family!

One who may yet be again

Your loving sister,

Altriona

* Excerpted from a poem entitled

*Magical Moments,*by Graeme Montrose.

### This Week in Math: Area

May. 31st, 2010 09:03 amI realized at the beginning of last week that I didn't have enough marks for the report cards; I was short measurement and geometry marks, though I had taught all the other strands in third term. I decided measurement was the more crucial of the two, and also the one that I'd touched on at another time, so I figured I could get a mark for area quite easily.

Thursday was a simple review of area of a rectangle, which I was pretty sure they already knew. I was right; only two or three kids needed to discover the formula for area, and by the end of the lesson all of them had it. That was the grade five expectation met. So I looked ahead, because I like to give A's, and discovered that the grade six expectation was area of triangles and parallelograms, using the formula for area of a rectangle as the starting point.

My kids can do this, thought I.

So I started them on Friday with a sheet of grid paper and asked them to draw a 3cmx4cm rectangle. We figured out the area with no problem. Then I asked them to draw a diagonal line from one corner to an opposite corner, creating two triangles. With some demonstration of what I meant, they accomplished this.

I got them to tell me about the triangles. They were right-angled triangles. The other two angles were both acute. (We discussed why the other two angles in a right-angled triangle have to be acute at this point.) The length of one side of each triangle was 3 cm, and the length of another side was 4 cm. So far, so good, but nobody had yet come up with the word I was looking for, so I primed them: during the quilt unit, we talked about shapes like this. Compare these two triangles. Someone said they were the same, and then I managed to draw the word out of them: they're congruent, and they're rotated 180 degrees.

Then I asked them to look at the rectangle with its triangles and describe it using a fraction. I shaded one of the two triangles and asked for a fraction; without too much pulling, they said it was half.

Without belabouring the point, I asked: If the entire rectangle is 12cm^2, what is the area of each of the triangles? They got the answer - 6cm^2.

I asked them to draw a few more rectangles, bisect them the same way, and test this; when they counted squares and parts of squares, did they get that the triangles were in fact half the area of the rectangles? Next I asked them to draw a right-angled triangle and use what they knew about rectangles to come up with the area of the triangle. I was looking for the kids who could do it without completing the rectangle first; a few kids needed me to draw the congruent triangle for them so they could see what I meant by it. But within another ten minutes, kids were showing me their work and explaining it, so I knew they got it.

Then I started offering the challenge question. I took their grid paper and drew a triangle on it that wasn't a right-angled triangle, and asked them to use what they'd just learned to find the area of that triangle.

A couple of them started by finding the height, thereby creating two right-angled triangles, and then building rectangles on each of those. One or two drew a rectangle without creating the two right-angled triangles first; to those ones I asked how they knew that the triangle I'd drawn was exactly half of the rectangle they'd drawn, since the other triangles in their rectangle weren't congruent with the first one. Then they figured out that they could prove that the triangle was half of the rectangle if they drew in the height and made right-angled triangles.

At the end of the lesson, I gathered all the kids together who had completed the extension activity (I didn't force anyone to do it after they got the right-angled triangle, because that was already an A.) I explained the different terminology: when we're talking about rectangles, we use the terms length and width, but when talking about triangles, we use base and height instead; they mean the same thing in terms of their drawings but it's important to know the terminology so you know what other mathematicians are talking about. I also showed them a couple versions of the algebraic formula for area of a triangle: 1/2 bh, bh/2, and the formats they're more familiar with involving symbols for multiplication and division.

Next up: area of a parallelogram.

### Anyone interested?

May. 21st, 2010 01:29 pmI need some new French plays, which I'm prepared to write myself based on fables and fairy tales. I'd love to have them illustrated into a storybook that I could use to introduce the play, and some vocabulary cards with parts of the same illustrations for the major vocabulary. I'd pay to get them printed in colour and to get the storybooks bound. If I could drum up some interest in them, I'd then approach an educational publisher about publishing them as a series.

Anyone interested? I could use a francophone editor, too.

A couple of weeks ago, we started the book, "If the World Were a Village." Last Thursday, we got to the page about the most common languages in the global village. The page gave the breakdown of the number of people out of one hundred villagers who speak each of the eight most common languages in the world. The breakdown is as follows:

Chinese - 22, 18 of them Mandarin

English - 9

Hindi - 9

Spanish - 7

Bengali - 4

Arabic - 4

Portuguese - 3

Russian - 3

I was using this to teach the concept of the double bar graph. So I put these numbers up on chart paper and suggested to the class that we compare them to the native languages of our own class.

We came across a problem with Chinese. The only Chinese girl in the class has a Chinese father and a Vietnamese mother. She speaks both languages. So we put up an extra category for Vietnamese and went on. English wasn't too hard; 48% of my class speaks English as a their first (and in most cases, only) language.

Then we came across the biggest snag.

I have several Indian kids in my classroom. I also have several Pakistani kids. Between the nine kids who fall into one of these two categories, their are four languages with varying levels of mutual comprehension: Hindi, Urdu, Punjabi, and another form of Punjabi that is apparently incomprehensible to the speakers of the first form but which everyone agrees still goes by the same name. The Hindi people were willing to admit that the Pakistanis spoke a dialect of Hindi that they called Urdu, but the Pakistanis wouldn't give ground on the matter at all; yes, they understand the Hindi kids, but their language is a different language. All nine of them agreed as to why: India and Pakistan had been at war off and on for so long that no one in Pakistan wanted anything to do with India, while the Indians were always quick to point out that Pakistan USED TO BE part of India.

We finally separated out the concepts of race, religion, nationality, and language, pointing out that the four overlap quite a bit but they are not the same thing. We decided to discount race entirely (I admit to pushing that decision a little bit.) We came to the conclusion that the reason the different dialects of English are all called English is that there's no political reason to call them anything else, whereas the tensions between Pakistan and India make people of both countries want to separate their language from each other.

Then we brought it back around to math, pointing out that the author of the book probably made a decision to count Urdu as a dialect of Hindi, and count them all together, whereas in our classroom, we'd separated them for political reasons. The big understanding that came out of the discussion was that numbers can be used in different ways. Sometimes, to make a graph or some other representation of numbers, we have to simplify them, and sometimes when we simplify them, we lose some of their meaning. If the reader puts too much store in the simplified numbers, misunderstandings can happen as a result.

Next up: a bar graph comparing our class' level of access to electronics to the number in the global village. Part of that discussion will involve the word "privilege."

In other news, there's a correlation between the number of computers my students have access to and their success in school. Last year, only 60% of my kids had a computer at home, and several of those lacked internet access. Their scores on the standardized testing two years before were abysmal. This year, all but one of my kids has access to internet at home (if they're telling the truth, and I'm pretty sure they are.) Their scores on the testing were much, much higher, and they generally perform better.

### This Week In Math

Apr. 27th, 2010 11:30 amYou've been pretty lazy about laundry lately. You've been throwing socks into the drawer without matching them up first. You also haven't taken your laundry downstairs to wash it, so at the moment, there's only two unmatched pairs of socks in your drawer - one black, one red. Most of the time this isn't a problem because you've got lots of time to get dressed, but this morning, you're being picked up to go on a trip with a friend, and your alarm doesn't go off. Your friend knocks on the door expecting you to be ready, and you need to get dressed in a flash. You reach into the sock drawer and pull out two socks.

what is the probability that they match?

Now, I can't take credit for this problem. It's straight out of the Ministry of Education document entitled "Guide to Effective Instruction in Mathematics: Probability," for grade four to six. I do take some credit for how far we took it, however.

We started with paper bags, in which were two pairs of construction-paper socks. Kids did thirty trials and then predicted the actual probability based on their experiment. Then we figured out how many possible pairs there were, and how many of those pairs gave them a matching set. We got two matching pairs out of six possible pairs, or 2/6. Using what we knew of equivalent fractions, we reduced that to 1/3.

Then we extended it. What happens if we've got three pairs of socks? Is the probability of a matching pair better or worse or the same? I asked them to predict what they expected to find, and write down their prediction; then I asked them to prove it. Well, with three pairs, the probability is 3/15, which is 1/5. I reiterated, as I have many times, that it didn't matter too much if their prediction was wrong; what mattered was that, when they realized they were wrong, they went back to figure out why they'd made that prediction and checked to make sure they had it right now. I made sure we were using good scientific language for this process - hypothesis, experiment, proof.

Okay, so what about four pairs of socks? They figured out that the probability then was 4/28, and reduced that with help to 1/7. That was the end of day one.

On day two, I took the information we'd gathered the day before for two, three, and four pairs of socks, and organized it into a chart. Then I asked them to find the patterns, and use the patterns to predict the next term.

They came up with two patterns, only one of which I'd found myself. The first group noticed the pattern in the reduced fractions - 1/3, 1/5, 1/7 - and predicted that the next reduced fraction would be 1/9. Then they worked backwards to figure out the unreduced fraction of 5/45. The other group took the unreduced fractions - 2/6, 3/15, 4/28 - and figured out that the distance between 6 and 15 is nine, and the distance between 15 and 28 is 13 which is 9+4, and they postulated that the next term would be 13+4 more than 28, which is 45.

Anyone who got as far as seven or eight terms and gave a pattern rule that worked got a B. If they could use the phrase "theoretical probability" in their answer, that was bumped up to an A-, because the distinction between experimental and theoretical probability is a grade six topic according to the curriculum. Those who continued to develop the pattern for many more terms got an A.

Then I asked those who clearly understood that pattern to come to the carpet, and I introduced them to the concept of the nth term - when you don't know the term number, you can replace it with the variable n. If we could figure out how to get from the term number to the reduced fraction, consistently, then we could come up with any term even if they were out of order. So we looked at it, and realized that 1/3 is one less than two times two; 1/5 is one less than three times two; 1/7 is one less than four times two; and so on. So the denominator was two times the term number minus one. I showed them how to write this; 1/2n-1. To get an A+, all they had to do was show me that they could apply this to fill in two lines of the chart that were out of order, because the ability to solve an algebraic equation is a grade seven topic - two years above grade level.

In my class of twenty-five, I gave out exactly two B's. Everyone else got an A. My students on IEPs ALL got A's, and I didn't even have to adjust their expectations downwards; I just had to make sure they had access to support to clarify the patterns they saw.

The A's here are for both probability and patterning, so that's two A's on most report cards for my kids.

### Altriona: Intrigue

Apr. 22nd, 2010 06:23 amThe other sixty-nine elves aboard the Imholien vessel Trisk'a'levian are dead. Some, including Captain Amendil with whom I have ever been acquainted, fell to the magic bolts of Lorandara. Others, like the First Mate, made good target practice for me. Many - most - died at the hands of our crew.

I knew we could not risk taking them alive; that we may be able to afford to take slaves of them later, but not now. My heart weighed like a stone within me, but I did not question the necessity. It would seem, though, that my brother has not yet grasped that his gentle lady sister is gone, nevermore to return. He gave orders to crew of all three ships, unbeknownst to me, that no one was to be left alive.

I realized this as we boarded the vessel, having already relieved it of half the souls on board. Even as I was about to give the order, I became aware that the men were not looking to me to give one; they knew already what was expected of them and believed that I did, too. I insisted on one prisoner, who was removed to the Morehaig's Scythe upon my order after the rest of the battle was over.

I challenged Mallion over it. It seems he wanted to protect me and his wife from the worst we would have to do by not telling us of the orders until it was too late to change them. I insisted that in future, he inform me of all plans.

I do not believe he recognizes the difficult position in which he put me. It is I, ostensibly, who command the marines in battle. Had I balked, it would have been I who lost credibility and authority with our crew.

He agreed to discuss future plans with me in advance of the battle, and shared with me one of the goals. We are not going to kill Valadar; we will instead use the warpstone to transform him, and let him live. But first, we must practise with the stone on a prisoner.

Then he asked me to keep this information from Lorandara.

I do not pretend that Lorandara is as well able to handle the necessities of battle as I. She cried as she killed the captain, and I heard her whisper, "I'm sorry," before casting that first bolt. Her relative fragility is undeniable. Yet I cannot fathom how Mallion will go about keeping this plan from her. Nor can I understand why he would want to. Does he still not see us as equals? Must he still pull out the damn chivalry at every bloody turn?

Protecting her in this way is dangerous. It is condescending. And above all, it is impossible, as Lorandara has locked away the green stone.

Mallion made me promise I would not tell her, before he told me. I am getting very tired of keeping their secrets from each other. One day both will come out, and those two must needs deal fairly with each other on that day.

Even now, the old life of genteel and ladylike pursuits intrudes on the new, in the form of a respectful regard that must needs give way to true honour among us three. Perhaps I can still save Mallion if we can find that equality.

### This Week in Math, Week 2

Apr. 1st, 2010 10:59 pmFractions aren't about quantity so much as proportion.

The Problem:

Prove that 3/8 is greater than 1/4 but less than 1/2. Use at least two different representations to prove this.

Most kids used a number line first, bringing in their work from last week on equivalent fractions. A couple used simple arrays instead. A few got nowhere because they used three different representations for the three fractions; with their permission, we used that as a key teaching in the lesson: that comparing fractions is always easier when the wholes are the same size and shape.

The fun came when I changed the focus of the problem somewhat today. The problem became: is 7/9 greater than or less than 7/8?

First I instructed them to come up with a prediction. I made note of the kids who told me that 7/8 was greater because ninths mean smaller pieces than eighths; those are the kids who grasp the proportionality of fractions. But I didn't challenge the more common prediction that 7/9 was the greater number because nine was greater than eight. I simply told them to write down their prediction and their reason for it, reminded them of the lesson learned during the previous problem (that the whole must always be the same size and it helps if it's the same shape) and let them go at it.

I got a lot of number lines again. I suggested to a few kids that they use what they knew about multiplication and division; for example, could they draw a rectangle that was divisible by both eight and nine using graph paper? I guided a few kids through the process of figuring out how many squares were coloured in the 7/9 array versus the 7/8 array, and reaffirmed that I was proud of them for recognizing that their prediction had been wrong, and writing their revised answer. We briefly reviewed how a mathematical proof is the exact same process as the scientific method: develop a hypothesis, develop a method for testing that hypothesis, work through the method, compare your answer with your hypothesis, and conclude that you were either right or wrong.

I encouraged kids who clearly got it to use the best math language they could and to explain their pattern rule: as the denominator gets bigger, the pieces get smaller.

All of the kids who originally believed that 7/9 was bigger ended up revising their hypothesis and understanding the pattern rule by the end of the lesson. I have a few kids I need to see for guided math on Tuesday because they were helping with the raffle or otherwise unavailable for part of the lesson, though.

Next Week in This Week in Math: The Math of School Raffles. We did the raffle today, and I kept all the tickets for each item in a numbered brown paper bag for the express purpose of probability experiments next week. There were a few poor sports commenting on how someone (i.e. me - no, I didn't let them get away with it) must have cheated, since they put twenty tickets in one box and somebody else won anyway. I'm looking forward to examining the probabilities involved in that one.

The following week, we'll be doing coin tosses, which will lead directly into Pascal's Triangle.

### (no subject)

Mar. 28th, 2010 12:28 pmI also picked up two for school. One of the assignments this week is for a pair of kids to decide on the best way to cut a birthday cake so that the birthday child gets exactly twice as much as all the other attendees. Then they have to videotape one partner or the other explaining their method. They get to decide on the number of people at the birthday party, thus ensuring that each group will be representing different fractional amounts.

Downside: the camera has just enough memory for one video, so I may have to come up with an SD card to put into it temporarily until I can get the school to buy one. Also, I'll have to check on the battery situation.

Upside: they can record their presentations when they're ready, and I can show them when I'm ready, not to mention saving them for something like, say, a job interview about my math program in six weeks' time. I think that's worth using my own SD card.

### This Week in Math

Mar. 26th, 2010 07:54 amSo, this week in math, we started by reviewing what little we'd learned about equivalent fractions before the March break. Since exploratory learning is starting to become de rigeur for my students (as it should be) I took them to the computer lab and let them play around with the fractions manipulatives on the national library of virtual manipulatives website. The differentiation was easy. Don't get this at all? Try the basic fractions manipulative. You've already figured out the pattern rule? Great, do the grade six one.

On Monday, I gave them an assignment to do. They would get four or five pieces of regular paper in different colours, their choice. On each one, they'd write a different common fraction. The example, which I made up with my six-year-old on Sunday, was one half. Then they'd come up with three or four equivalents for it, and represent those equivalents in pictures. They had to explain the fractions and how those fractions were connected to the original fraction. There is a rubric - let me know if you want it.

The kids who were having trouble even accessing one half came for some guided math with me. We did fraction strips, and played around a bit with blocks, until they got to the point where they understood that if you divided all the pieces the same way, you had an equivalent fraction.

Then, because an abstract understanding is important to work towards, we took the equivalent fractions they'd come up with and analyzed them. How do you get from 1/2 to 4/8? If they expressed it as adding, I told them they were right, but they'd see the pattern faster if they thought of it as multiplication. So what would they multiply by? Some figured out the pattern independently; others needed to be shown several times, and finally got it with the help of more manipulatives. Then I encouraged them to use the pattern rule they'd found to come up with what they thought was an equivalent fraction. They had to come up with the proof that they were right by drawing a representation they could connect back to the original fraction.

As of this writing, most of my kids have finished two or three sheets of equivalent fractions. The rubric gives an A to kids who can connect their equivalents to decimals or percents, so I've explained the idea of repeating fractions to the kids trying that for 1/3. I suspect most will finish on time - the assignment is due on Monday.

Next Week: comparing and ordering fractions is so much more interesting when the denominators are different.

### Math Course Assignment

Mar. 22nd, 2010 08:46 pmFortunately,

**( I've started learning to play hand drum in that time period. )**

### Altriona: In business

Mar. 22nd, 2010 04:08 pmI can hear echoing through my consciousness the voices of my forebears, those great elves who built Imholien House into the enterprise over which my father presided with such consummate skill. From deep within my soul rises up a spring of inherited horror at my effrontery, that I, a lady of breeding and refinement, should ever throw off the bindings of polite society in order to keep company with brigands and thieves, leading them on a mission to intercept and make off with black powder and warpstone.

But I am no longer a scion of Imholien House. Here in my journal, I will forevermore thumb my nose at the lot of them. I am the instrument of my own revenge and my brother's, and I will allow no shades of guilt to colour my spirit.

Lord Rackham sent us on a first mission. With the city of Tobaro in such disarray, certain trade lines are not as well-policed as usual. He knew of a shipment of black powder out of Tobaro, to be exchanged with he knew not whom, on an outcropping of scree and rock just west of the Zombie Marsh. (We were assured that the zombies had rotted so thoroughly that it was truly more of a skeleton marsh now; I cannot say whether such news was more reassuring to my brother and sister-in-law than it was to me.) He suggested we arrange to take both the powder and the gold that would be paid for it.

Though we three discussed the possibility of sailing the captured ship into the port of Tobaro and blowing it up, thereby bringing down that cavernous pit of putrescence upon itself and finishing our retribution against the usurpers of the ancient elven port, we determined that such would not be wise, given that it would put us on the wrong side of our business partner's favour. Therefore we determined to do as we had been advised to do, taking the ship full of powder first, then treating with those who came to buy it long enough to o'erwhelm them. Accordingly we hid the Morehaig's Scythe on the wrong side of the outcropping of rock, the narrow beach on the other side of which was to be the site of the transfer. Our other ship, the fill-in-name-here, was to stand just slightly further off, to come up behind the mark and attack it. With Mallion flying messenger, this plan was easily accomplished as the sailors of the coast-hugging cargo ship were offloading the powder. He killed the captain of that vessel, though he was shot by someone off in the shadows just as the fight ended.

He flew back to our ship, where I was waiting with the men for his order to go ashore. Lorandara tried to heal him, and - I can barely find words for what happened to her. She convulsed, crying out continually in an agony of body I can barely comprehend. The sound tore at my sanity, for I believed her to be dying.

As Mallion took her to their cabin, I led the men landward, wading through the shallow water until we reached the beach full of kegs. The screaming behind us ceased ere we gained the beach, and I looked behind to see Mallion flying towards us and Lorandara, seemingly hale, skywalking.

My attention was diverted by a flaming arrow landing mere feet shy of the furthest barrel of black powder. I motioned the men back into the lee shadow of the rock, whence it would be difficult to get a sighting on us with a bow, and called out to the bowmen to show themselves, that we might treat with them.

"Go away, or we will blow it up!" they answered.

"Then neither of us will have it! Treat with us instead!" I answered.

This went back and forth until Mallion planted himself between the barrels and our hidden adversaries. Upon discussion, they revealed that it was not gold that they had to trade, but green stone.

Mallion went back to discuss this with Lorandara, who told him what it was. It seems it is the stuff of chaos, magic in brute form, and above all, worth several fortunes more than were on that ship to be traded for it. He offered to trade one vial of the green stone for the half of the black powder already sitting on the beach. They agreed, and the exchange was made.

Thus we got half of what we had come for, with little killing and continued access to more of that valuable and dangerous material. Lorandara took charge of it, and we are keeping a small portion for ourselves, in the hope that it may prove efficacious at some crucial moment.

I am learning more of the workings of the ship, for Mallion takes little interest in that aspect of captaincy and one of us must. Our second mate is good at interpreting orders, and I pay him close heed so that I might better turn my phrases as the men are used to hearing them. My hands are hardening in ways that handling a foil would never have achieved. Methinks I shall take quite well to this new life.

### Direct Patronage

Mar. 21st, 2010 07:45 amI hate the process of figuring out how a dozen different assignments, all relating to different expectations and all taught, supported, and assessed in different ways, go together to create one overarching letter that is supposed to sum up a kid's work for a term.

You know what? It doesn't. There's no way that it can. The same kid can be at three different levels in three different key expectations, and giving them the middle level doesn't recognize either their weaknesses or their strengths well enough to satisfy my professionalism, much less well enough to really represent the kid.

Furthermore, the parents don't look at that A and say, "Wow, you did really well on that brochure assignment! You put a lot of effort into it and used your information well, and your research came from lots of different sources! I'm impressed with your work!" No, the vast majority of parents look at the A and say, "You're really smart in English!"

Then they look at the C+ in number sense and numeration, and instead of saying, "It seems you were really struggling with multiplication. What can we work on together that will help you with that?" they say, "It's okay. Some people just aren't good at math." Which is a better message than the other most likely one: "You're stupid and lazy and that's why you got a low mark." But it's still not the truth. Neither of them are the truth. And since a parent's opinion is necessarily and properly more important to a kid than a teacher's, my repetition of the first message gets drowned out by their repetitions of the other messages.

For the parents out there, please, please, know this: no matter what your personal relationship with grades was in school, you need to put it aside. If there's one message I want to give you, as a teacher trying to improve your child's learning and give them hope for their future, it's this: marks are not a reflection of the child's abilities. They're a reflection of the child's achievement on a certain number of assessment tasks which may or may not accurately reflect the child's understanding of the material and almost certainly do not reflect the child's full potential. If you treat marks as indicators of work already done, and tie them directly to the learning that went into that work, then you'll probably avoid this trap. If you interpret marks as a reflection of your child's aptitudes, you are doing your child a significant disservice. Marks are only as good as the expectations they relate to and the tasks used to assess the child's achievement of that expectation. They do not reflect the child well at all. They're at best a necessary evil, at worst a horrible setback to kids who might otherwise be making great gains.

This rant brought to you by my second-term report cards and the letter C+.

### Philosophy of Mathematics Education

Mar. 19th, 2010 01:02 pmThe assignment has become a standard one for grade one students in Ontario: gather a collection of 100 items to bring in and explain to the class on the hundredth day of school, usually in mid-February. It filled my six-year-old with glee. She knew exactly what she wanted to bring in. I was less thrilled, because the collection she chose did not belong to her, but to me: my dice.

Nevertheless, we spent an hour one Sunday afternoon gamely dividing dice into baggies, categorizing them by number of sides, then by colours. Along the way, we explored concepts relating to the base-ten system and the absolute basics of multiplication.

My daughter has plenty of experience with dice. They’ve been a part of her life since birth. Plush dice, foam dice, and the vast array of dice used for roleplaying games by both her parents and all their friends, were her first introduction to numbers that weren’t on her fingers. For her, dice represent fun times with friends, groups of people laughing and telling stories around the dining room table, the adults who take an interest in her life even though they aren’t related to her – and math. So it was natural that when she needed a real-life collection to bring in for Hundreds Day, her first thought was dice. Dice are math as it is in her life.

There is a great deal of emphasis in mathematics education on making math real, on finding the ways to make the numbers concrete, tactile, visual. This emphasis is a vast improvement over numbers that never left the page, because it does facilitate a deeper understanding of mathematics, and that is the ultimate goal of mathematics education. (1) But my daughter’s experience with dice is evidence that it doesn’t go far enough. For her, dice are not something a teacher brings out to show how numbers work; dice are real life that we describe using numbers. The educational establishment has been getting it backwards. The goal is not to make math real. Math is already real. The goal is to teach how reality can be described using math.

Paul Lockhart, in his article, “A Mathematician’s Lament,” discusses how mathematics is the art of pure idea. (2) When we teach it procedurally, we strip from it the inherent creativity and beauty of it; but when we use it to describe our ideas, and engage students in describing progressively more complex ideas with mathematics, we find that everything is math. There’s no need to make it real because it already is. As teachers, our vision for our students should be to bring their mathematical understandings into the classroom. Where is the math in their lives? What forms of art exist in their cultures, and in the culture to which we’re introducing them, that can further their understanding of number and pattern and relationships? It is when we follow students’ mathematical understandings and extend them that we get the deep understanding of mathematical ideas that creates lifelong learners and problem-solvers.

Resources

1. Carpenter, T.P, Hiebert, J., Fennema, E., Fuson, K.C, Wearne, D., & Murray, H. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth. Heinemann.

2. Lockhart, Paul. (2008) “A Mathematician’s Lament.” Mathematical Association of America Online. http://www.maa.org/devlin/LockhartsLame