The Man Who Counted

The Man Who Counted is an enchanting math story book.  Originally published in 1949 in Portuguese (O Homem que Calculava), it was presented as a translation of a 13th century (Islamic calendar?, 1942 CE, maybe?) work by Malba Tahan, who is typically listed as the author, translated by Breno de Alencar Bianco.  Both of whom are fictitious.  The real author seems to be Júlio César de Mello e Souza from Brazil.

A student in Calgary gave me first copy, correctly deducing that I would LOVE it.  You can find a pdf of the entire thing online, and the copyright is complicated enough that I can't figure out if it's legal.  There's a current publication of the book, too, and it's a nice one to have.

All this is by way of introduction, as one of our promising preservice teachers, Cassie Becker, wrote up some very nice problems from the book, and was willing to share them here.  She did this for a choice workshop, where students have freedom to follow up or pursue an item of interest.  In general, I find that the students make amazing choices.

The Man Who Counted

(Chapters 1-9)

Beasts of Burden

Close to an old half abandoned inn, we saw three men arguing heatedly beside herd of
camel. Amid the shouts and insults the men gestured wildly in fierce debate and we could
hear their angry cries:
“It cannot be!”
“That is robbery!”
“But I do not agree!”
The intelligent Beremiz asked them why they were quarreling.
“We are brothers,” the oldest explained, “And we received thirty-five camels as our
inheritance. According to the express wishes of my father half of them belong to me, one-
third to my brother Hamed, and one-ninth to Harim, the youngest. Nevertheless we do
not know how to make the division, and whatever one of us suggests the other two
disputes. Of the solutions tried so far, none have been acceptable. If half of 35 is 17.5 if
neither one-third nor one-ninth of this amount is a precise-number, then how can we
make the division?"
“Very simple,” said the Man Who Counted. “I promise to make the division fairly, but
let me add to the inheritance of 35 camels this splendid beast that brought us here at such
an opportune moment.”

Can you explain why this would be a good idea?

Food for Thought

Three days later, we were approaching the ruins of a small village called Sippar when
we found sprawled on the ground a poor traveler, his clothes in rags and he apparently
badly hurt. His condition was pitiful. We went to the aid of the unfortunate man, and he
later told us the story of his misfortune.

His name was Salem Nasair and he was one of the richest merchants in Baghdad. On
the way back from Basra a few days before bound for el-Hillah, his large caravan had
been attacked and looted by a band of Persian desert nomads, and almost everyone had
perished at their hands. He, the head, managed to escape miraculously hiding in the sand
among the bodies of his slaves.

When he had finished his tale of woe, he asked us in a trembling voice, “Do you by
some chance have anything to eat? I am dying of hunger.”
“I have three loaves of bread.” I answered.
“I have five,” said the Man Who Counted.
“Very well,” answered the sheik. “I beg you to share those loaves with me. Let me
make an equitable arrangement. I promise to pay for the bread with eight pieces of gold,
when I get to Baghdad.”
…
Then Salem Nazair said to us, “I take leave of you my friends. I wish however to
thank you once more for your help and, as promised, to repay your generosity.” Turning
to the Man Who Counted, he said, “Here are rive gold pieces for your.
To my great surprise, the Man Who Counted made a respectful objection. “Forgive
me, O Sheik! Such a division, although apparently simple, is not mathematically correct.
Since I gave five loaves, I should receive seven coins. My friend, who supplied three
loaves, should receive only one.”
“In the name of Muhammad!” exclaimed the vizier, showing a lively interest. “How
can this stranger justify such an absurd division?”

Each piece of bread was divided into three portions and each man ate an equal portion of bread. Can you justify this division?

The Four Fours

“Did you notice that this shop is called The Four Fours. This is a coincidence of unusual importance.”
“A coincidence? Why?”
“The name of this business recalls one of the wonders of calculus: using four fours, we
can get any number whatsoever.”

Can you make 2,5,17,26,34,91,135, etc using only four fours?

dweekly @ Flickr

Going to Market

And the shopkeeper told the following “Once I lent 100 dinars, 50 to a Sheikh from
Medina and another 50 to a merchant from Cairo.
“The sheik paid the debt in tour installments, in the following amounts: 20, 15, 10 and 5 that is

Paid 20 and still owed 30
Paid 15 and still owed 15
Paid 10 and still owed 5
Paid 5 and still owed 0
Total 50              Total 50

“Notice, my friend, that the total of the payments and the total of his debt balance
were both 50.”
“The merchant from Cairo also paid the debt of 50 dinars in four installments, in the
following amounts:

Paid 20 and still owed 30
Paid 18 and still owed 12
Paid 3 and still owed 9
Paid 9 and still owed 0
Total 50            Total 51

“Note that the first total is 50—as in the previous case—while the other total is 51.
Apparently this should not have occurred. I do not know how to explain the difference of
1 in the second manner of repayment; I know that I was not cheated, as I was paid all of
the debt, but how to explain the difference between the total of 51 in the second case and
50 in the first?”

Can you explain to the shopkeeper why this happened?

Pennington @ Flickr

Seventh Heaven

The sheik addressed the three of them: “Here is the esteemed master calculator.” And.
to Beremiz he added, “Here are my three friends. They are sheep rearers from Damascus.
They are facing one of the strangest problems I have come across. It is this as payment
for a small flock of sheep they received here in Baghdad, a quantity of excellent wine, in

21 identical casks:
7 full
7 half-full
7 empty

They want to divide so that each receives the same number of casks and the same
quantity of wine. Dividing up the casks is easy—each would receive 7. The difficulty, as
I understand it is in dividing the wine without opening them, leaving them just as they
are. Now, calculator, is it possible to find a satisfactory answer to this problem?”

Can you find a solution to the problem?


Three and Thirty

“Your total bill, with your food, is 30 dinars,” was the reply. Sheik Nasair wished to
pay the bill, but the men of Damascus refused, which led to a small discussion and an
exchange of compliments, with everyone speaking at once. At last it was agreed that
Sheik Nasair, a guest, should pay nothing and that each of the others should pay 10
dinars; so 30 dinars were handed to a Sudanese slave for his master. A few moments
later, the slave returned and said, “My master says he made an error. The bill is 25 dinars,
and he has asked me to return 5 to you.”
“That man of Tripoli is most honorable,” remarked Sheik Nasair. And taking the five
coins, he handed one to each of the three men, so that two remained. After exchanging a
“lance with the men from Damascus, the sheik handed them as a reward to the Sudanese
slave who had served them food.
At that moment, the young man with the emerald rose and, looking gravely at his
friends, said. “This business of paying over the 30 dinars has left us with a serious
problem.”
“Problem? I see no problem.” replied the sheik, astonished.
“Oh yes.” said the man from Damascus. “A serious and seeming ridiculous problem.
A dinar has disappeared. Think now. Each one of us paid 9 dinars. Three times nine is 27.
Adding to these 27 the 2 that the sheik gave to the slave, we have 29 dinars. Of the 30 we
handed over to the man from Tripoli, only 29 are accounted for. Where, then, is the other
dinar? Where has it disappeared to?”

Can you explain to the men where the 30th dinar went?

Engagement with a Purpose

Trying to document our senior student teacher seminar. Lesson by Dave Coffey, (@delta_dc, Deltascape)

Dave’s quote for the day:

You must be the change you wish to see in the world. – Mohandas Gandhi

We watched part of Alan November’s TEDxNYED talk (picking up after the barbershop, about 6 min in):






What do the novice teachers notice?

Overheard snippets:
I’ll do extra work all the time. Write a letter…
Empowering students…
Present the idea of what they are learning…
Could give them a topic, send them to research, and what do you get.

Whole class discussion:
A student shares: I’m at a progressive school, and it’s hard to think about going back. The book isn’t example, example, exercise. They’re an idea, a goal, an objective, and then it’s an investigation. The problems guide you through finding the information about the topic. The parent would have to do the whole investigation to help. I did a demonstration of communicating what you’re doing, and then the students were responsible for being able to do that. Putting the work on the students is what we’re doing, and we’re there to guide them.  On the test had a check question and 19/20 had the quadratic formula right.

Q:  What’s a way for this to work in another classroom?
Dave wonders about:

Platform	Audience	Purpose
class blogs	Parents	here’s what we’ve been doing
class blogs	Peers	here’s what you missed
letter	...	...

Student response

• Home or class work? Up to you.
• Thinking of students that don’t have access to a computer… we had a designated note-taker. If someone missed they could copy.
• Saw a student teacher have learners come up to be a scribe.

Not about can do or can’t do. About can do and not yet.

One student opines:  I wish they’d say this in their videos. They come off as you have to change everything. He mentioned that students say they’ll do things for their fiends, and I will, too. But is that what you want from me as a student? But if it was just me, I wouldn’t do this portfolio. (No offense.) To totally allow students to do what they please. “It’s so nice, it’s a great metaphor…” But in reality it’s crazy. Are you kidding?

The teacher of the JK Rowling fanwriter said – she’s not a good student. Could the teacher meet her halfway? There are probably things desired for her that are not met by writing like Harry Potter.

Teachers are more important than ever, to provide that structure.

by Priki @ Flikr

A student shares:  I covered triangles and they constructed definitions, and classifications. Homework was a brochure or a puzzle. Had to have name and properties and a picture for each kind. Non-homework doers was cut in half. Next day quadrilaterals. They could make a bumper sticker or a questionairre for interviewing a quadrilateral. Ask questions, but you can’t ask the type. Again the majority of kids turned that in.

What is working? What isn’t working? What will increase engagement? This example had choice, a framework, a purpose…

We then offered a choice to work on portfolios, presentations, or cooperatively planning engaging lessons.  No one chose the lessons, with so much hanging over there heads.  (A mini-lesson for us.)  Still, it was a lively discussion and helped us process Dr. November's TED talk.

Learning Math Anchor Charts

Anchor charts are a way to summarize learning that you wish to preserve, build from, or be able to reference.  While they started out in my classroom as lists or concept maps, students building from previous students' work have started edging into metaphor territory as well.  I had the camera with me to capture the charts, and the students wanted to present them, so I taped it.  I think it would have been more valuable to capture them working on it.  The worked very intently, quickly going from "what does he mean?" and "what did we do about this?" to debating relative importance and discussing key features.  Very cool.

The research they're referencing includes:

• Conditions of Learning by Brian Cambourne 
• Relational/Instrumental Understanding by Richard Skemp
• The Learning Principle by the NCTM
• "Active Learning of Mathematics" by Jim Smith, in the March 1996 Mathematics Teaching or the October 1999 Mathematics Teaching in the Middle School
• Vygotsky's Zone of Proximal Development. (My favorite article about this.)

The video with the students explaining their posters is below the poster images.













This is a solar system model - a little hard to see. They had fun developing the metaphor and the extending it. Used the idea of earth, other planets, the sun, constellations to all symbolize different roles.

















(Students knew they were being taped for publication, and had a later chance to withdraw.)

Product Game... again!

It is no secret to my students how much I love the Product Game.  It is fun, not just fun for, you know, a math class.  The strategy required is at least as good as Connect Four, which is a surprisingly deep game.  The practice value is huge, as students have to compute many, many more products than would ever be done on worksheets.  The mathematics has connections, as the products lead to factorization, which enhances the strategies available.  But even the pedagogical structure is nice, as you consider moving one factor leads to considering families of multiples that are good for learning multiplication facts in a way that promotes fluency and efficiency.  The first I saw of it was from the Middle Grades Mathematics Project, the precursor of the excellent Connected Mathematics Project middle school curriculum.  (Which still has the game.)

So I love to adapt it.  It's never quite as good as the original, but often the great structure of the original allows new features to come to light.  Here's a previous handout that has the original Product Game and and Integer variation.

The fifth grade class I'm working with is beginning multiplication of decimals, by considering whole number times decimals that include tenths.  They're starting the whole counting up the decimal places routine, without much though of unitization.  If you have 5 bags with 4 apples each, you've got 20 apples.  If you have 5 groups with 4 tenths each, you've got 20 tenths... it's just that we don't often look at 2-point-OH as 20 tenths.  With my class I'd be looking for a context to start at this - probably money.

This class expects games from me, though.  I thought we had played the product game already (that was last year, Mr. Golden!) - oops.  In this version of the game, there's markers to make for your team.  I didn't have my usual two color counters available, but I've also learned that making markers or game pieces is a point of engagement and pride for some of the students.  Others are content with a quickly scrawled initial, and that's okay, too.  Mr. Schiller set a time limit of 3 minutes for making your markers, which was a good idea and the right amount of time.

We were to start by playing me vs. the class, so I could model some of the multiplying strategies I wanted to share.  But my son made me some excellent University of Michigan and Michigan State markers, and, being a proud alum, I had to be State.  There were students who couldn't bring themselves to being on U of M's team, and how could I argue?  So we played Spartans vs. Wolverines.



Product Game Decimal

The group play got us through all the rules and allowed us to model a lot of the whole x tenths.  But we never got up to the hundredths.  So we discussed how to get those.  I think they knew on some level it was tenths times tenths, but had a bit of the 'we haven't been taught this yet' syndrome.  I used the analogy of a dime being a tenth of a dollar, so what's a tenth of a dime? "A penny!" And what part of a dollar is a penny?  How many does it take to make one dollar?  It is terrific that they are used to seeing one cent written as .01

It was interesting seeing them play.  They started out almost entirely in whole x whole, and then were forced to the whole x tenths by the game play.  And if the game went on long enough, to the hundredths.  Several people got calculators to explore this, and a couple got to the calculators and then beyond them in the space of the hour.  Some students were done with the game after one session, but others were definitely up for more.  Hope you get a chance to try it and get a little bit addicted.

Photo credits: From Flickr, jeff_golden, 24oranges.nl and fireflythegreat.

LaTeX in Blogger

EDIT AGAIN (August 21, 2011): I seem to have this working again, using the following code:


Which I got from watchmath's author in this blogpost.

(OLD EDIT:  You may notice that you see $$ instead of math formulas.  WatchMath's server used to go down periodically, so they've moved the service to Google's, but I haven't had a chance to change the code.  That's annoying to have to go back to fix old posts and disinterests me in their service.  Here's the post explaining their changes. )

ORIGINAL POST:
Guillermo has a quick post about using LaTeX in blogger.  That's great, as it is frustrating to show nice math.  My question was if you needed a widget, but it turns out you can do it using the Edit HTML tab in the composition window.  For example,

$2x + 3 = \frac{1}{2}$

$\displaystyle \frac{x+3}{(x+2)(2x-1)}$

$$ \frac{x+2}{2x-1} $$




Quick and easy!  It uses the WatchMath service.







Install LaTeX?

I Support Teachers

Eugene Victor Debs, 1922
by Moses Wainer Dykaar

“Ten thousand times has the labor movement stumbled and bruised itself. We have been enjoined by the courts, assaulted by thugs, charged by the militia, traduced by the press, frowned upon in public opinion, and deceived by politicians. 'But notwithstanding all this and all these, labor is today the most vital and potential power this planet has ever known, and its historic mission is as certain of ultimate realization as is the setting of the sun.”

 - Eugene Debs

I count myself blessed to be a part of this vocation.  To teach is such amazing work, and so difficult.  Part of why I was drawn into the education side of math rather than the research side of math is the social relevance, but it also appealed to the math me because it is such a hard problem.  We know so much and yet it is a tiny part of what we need to know to support real learning.  Every class from 6 to 60 new people with different experiences and different needs that you want to lead to new learning in a subject that people consider the toughest... alright!  That's a problem worth solving.

This work has led me to dozens of schools, and given me the opportunity to work with scores of teachers, and to educate hundreds more.  Never have I encountered a group more likely to be intent to seek the good of others, and be willing to sacrifice themselves to do it.  OK, maybe the monks I lived with... maybe.

It is a continual mystery to me why we don't have more of a voice in education policy.  Bill Gates has more of a voice than we do.  The only hope we have of gaining a voice is to stand together.  As any group, we have made and will make mistakes.  Some unions have made gains that are not to their own benefit, let alone the students.  But if we are going to be a profession, we need to stand together. 

http://edusolidarity.us/
Full list of blogposts supporting edusolidarity. (Google Spreadsheet.)

Unity does not require uniformity.  Groups of people that agree completely about everything usually have one member.  We are going to work alongside teachers who believe in 20 minutes of lecture and 50 repetitive homework problems with no assessment other than the unit test.  We are going to work alongside geniuses who see math throughout their life and lead students to forge their own understanding - while mastering the skills needed to excel on the state test.  All these teachers are likely to be doing what they're doing because they think it is the best thing for their students.

Given the choice of these people giving their heart and lives to students and... well, it's not actually clear who the alternative is.  Youtube videos?  Faceless corporations?  Tax cuts for businesses?  

That's not a problem.  I support teachers.

“Solidarity is not a matter of sentiment but a fact, cold and impassive as the granite foundations of a skyscraper. If the basic elements, identity of interest, clarity of vision, honesty of intent, and oneness of purpose, or any of these is lacking, all sentimental pleas for solidarity, and all other efforts to achieve it will be barren of results.” 

- Eugene Debs

Photo credit: cliff1066@Flickr

Math Teachers at Play 36

Welcome to the 36th edition of

Math Teachers at Play!

How many rhombs can be traced on the edges of this figure?

36 has long been one of my favorite numbers, but faced with this carnival, it was hard to figure out why.  It's a square number that's a product of two squares, but that's not too rare.  (Why?)  It's the 6th perfect square and the sum of the first six odds, but that's not too remarkable.  (Why?)  It's the 8th triangular number, but not a Sierpinski step or anything... wait!  It's a square triangular number?  How common is that?  1, 36, then...?

8+9+9+10... will numbers like
that have a special property?

Number
Denise, the founder of this here carnival here, has an activity on Times Tac Toe at Let's Play Math.

Erlina at Mathematics for Teaching (formerly keeping Math Simple) has a nice introduction to integers based on sorting.

Can you see the four 9-squares and
nine 4-squares in this brick?

Guillermo has posts on addition and subtraction of integers.  He updated recently from his old wordpress domain to mathandmultimedia.com.

Marilyn at Mad Kane's Humor Blog had a Pi Day limerick.  Given Pi Day's proximity to St. Patrick's Day, that's a double score. Bonus: Math Jokes 4 Mathy Folks had mathy St. Patrick's jokes.

Pat blogged at Pat's Blog, hmmm, about Pi from a classic Venn illustration in 1888 to an internet meme cartoon from 2011. 

John has fascinating info on a class of prime numbers called the Limerick Primes at the Endeavor.

How can you quickly determine if 1926
is a multiple of 36?  Is it close?

Algebra
Mimi at I Hope This Old Train Breaks Down... has a complete activity on (beyond) composition of functions, adapting and extending an NCTM activity, and also an idea for a neat telephone function composition activity.  She's prolific, so poke around the blog while you're there.

Alexander, author of the amazing Cut-the-Knot, has a great discussion of inverse functions, relating three different Mathematics Teacher articles at the CTK Insights blog.

How big is this 36 in real life?
Why do you think so?

Geometry
Becky at Wide Open Campus has a quick little activity called Nature's Pattern Blocks.

Chris at M.O.B. has an activity on Mirror Imaging Monsters.  Arrrgh, reflectional symmetry.

Maria at Homeschool Math Blog has an excellent problem with a chord. Good challenge with a different perspective.




 
Connections
Katie has a travel site, Tripbase, with the nine most mathematically interesting buildings.  I might make it ten for either the Plaza de España or the Alhambra in Spain, both of which have some great tessellations.  What would you add?

Cheong ponders Are You a Number? at Singapore Math.

Representation
Caroline at Maths Insider has a post on strategies for your visual, auditory or tactile learner.

Sue, writing at Math Mama Writes...,  use graphs to make connections to increase her understanding of trig identities.

I never got a chance to clean up and make pretty this 36 knot that I designed for this carnival.  Why on earth would I think that this somehow related to 36?

Why would 36 degrees be
newsworthy?

Teaching and Learning
Rachel at Quirky Mama has monthly math activities for preschoolers.

Maria at The Math Mom recommends a short post with 5 tips for not transferring your math anxiety or a long and serious post on if it's okay to be smart.

Peter has a collection of links to enthuse about math at Travels in a Mathematical World.  Peter is half of the Math/Maths podcast.

Jacob tackles memory vs. mastery learning at License to Teach.


What about this?
Carson shared a nice graphic relating math to careers at When Am I Going to Use This?  But it's at an online education site, which Dan Meyer wrote quite forcefully against, in terms of predatory recruitment, at dy/dan.  He was writing specifically about the Top N lists.  What do you think?

 The other issue to be on the look out for is the EduSolidarity online rally, March 22nd.  If you don't stand for something, you'll fall for anything.

So Long and No Longer
Sorry this was a wee bit late.  I had a busy couple of days between Knot Fun (Celtic knot activity on axioms) and a Tech Symposium presentation, for which I assembled this glog (interactive poster) about the Tech I Use.  See you next month inside Maths Insider hosted by Caroline.  Submit an article!

Mathy pictures made in Geogebra.  Other pix from Flickr - the photographer's name for each is in the file name.  And a special MTAP shout out to Leo Reynolds, who has got to be the most prolific number photographer in history.

Knot Fun

afborchert

In class today I gave the preservice middle school teachers a choice: finish up our work on decimals or do a St. Patrick's Day activity.  They all yelled, "Decimals!"  End of story.

No, wait, they yelled for the St. Patrick's Day activity.  (Though I go by mathhombre, I'm more fear na hÉireann than boricua.  Not to look at me.)  I take St. Paddy's pretty seriously.

To introduce the activity below, we considered something I saw recently.  A teacher told her students to multiply decimals by lining up the decimals.  They were having trouble remembering when you have to line up decimals, so she told them to just always line them up.  The students then wanted to bring the decimal straight down.  Some use the place holding zero, some even put an extra zero after 3.7, and a row of 000 in the products.

Several of the preservice teachers knew this was wrong, though, like some of the middle school students.  You count up the decimal places.  We noticed that this works, but why?  Nobody knew.  One student thought that it must be connected to counting the zeroes when multiplying by powers of ten.  Good connection!


So we considered how all of math is often presented as rules to students.  You must do this. You have to, and then you.  When really to mathematicians, there are axioms, conjectures and theorems.  Axioms are our starting assumptions.  Euclid discovered (or recorded) that to logically proceed, we had to have a place to start.  Like, 'two pints determine a line.'  (Cheers.)  The students were also familiar with the Parallel Postulate.  What you can prove from those axioms are theorems.  What you think might be true are conjectures.

Design 36

In the activity below, the rules for making knots are the axioms.  I charged the preservice teachers with trying to decide if they were all axioms.  Mathematicians, after all, do not like making additional assumptions.  Were some of the rules consequences of other rules?  Could they make conjectures about the patterns they saw?  This activity was adapted from Symmetry Shape and Space, an amazing math for general education text by Kinsey and Moore.


As they did the activity, several people struggled to make sense of the rules.  "Give it a go!"  Compare with your table mates.  After everyone was trying, I shared a couple students' work, and a knot I was working on for the Math Teachers at Play Carnival.  Then, they were off to the races.  Some students worked on making their knots beautiful.  Some worked on technique.  Some worked on mastering the rules.  There was a lot of natural cooperation.  One table specialized, with a student who drew the setup and another who traced the strands.  They quickly became interested in how many strings were in each knot.  We played some Irish music via Pandora and I did show them my daughter's favorite song from The Secret of Kells. (This beautiful animated movie has some great images of illuminated text.)




Knot Rules

Ank0ku

It's important to have some Knot Dot paper for their experimentation. (The link is to some 2-sided paper on Scribd.  Yes I know most paper has 2 sides.)  I bring about 1.5 sheets per person.  Some will take some sheets to go.

They formed several conjectures through their work. They decided Rules 5 and 6 were non-mathematical, but rather stylistic.  They noticed that seemingly symmetric knots involved flipping the crossings on corresponding pieces.  They made a conjecture that on knots with dimensions bigger than two the greatest common divisor of half the side lengths was the number of strings.  Maybe.  But if the GCD was one, there was one string.  And they conjectured the converse was true, too.
It was a very engaging activity.  I was interested in the math value, though, and a quick 0 to 5 finger survey found they thought it was about a 3.5, with a range of 2 to 5.

Éire agus mata go Brách!

Screwtape for Teachers

From The Number Devil

One of my favorite books of all time is The Screwtape Letters by C. S. Lewis.  Very practical and grippingly clever, I can read and reread it to great benefit.  On a recent family trip we were listening to a passable dramatization of it, not as good as the excellent audiobook read by John Cleese, and I resolved to make a bible study of it.  As I was writing that, a small bit really struck me as a teacher.  In letter 2, Screwtape, a senior devil, writes the following to Wormwood, a junior tempter whose patient has just horribly become religous.  The book is brilliant, and fascinating for putting the whole life of faith thing on its head.

"Work hard, then, on the disappointment or anticlimax which is certainly coming to the patient during his first few weeks as a churchman. The Enemy allows this disappointment to occur on the threshold of every human endeavour. It occurs when the boy who has been enchanted in the nursery by Stories from the Odyssey buckles down to really learning Greek. It occurs when lovers have got married and begin the real task of learning to live together. In every department of life it marks the transition from dreaming aspiration to laborious doing. The Enemy takes this risk because He has a curious fantasy of making all these disgusting little human vermin into what He calls His "free" lovers and servants—"sons" is the word He uses, with His inveterate love of degrading the whole spiritual world by unnatural liaisons with the two-legged animals. Desiring their freedom, He therefore refuses to carry them, by their mere affections and habits, to any of the goals which He sets before them: He leaves them to "do it on their own". And there lies our opportunity. But also, remember, there lies our danger. If once they get through this initial dryness successfully, they become much less dependent on emotion and therefore much harder to tempt."

-CS Lewis, The Screwtape Letters

I know that's pretty religious language, but it reminded me so strongly of what I want for my students.  I want their freedom, and for them to be able to learn to do the mathematics on their own.  But I'm afraid that it is so easy to try to tempt them into this life.  "Hey, look - this stuff is fun!"  When that's truthful, I'm okay with it.  But it has led me sometimes to cutting out something because it's not fun, or describing something that is not fun as fun.  When I am willing to carry my students, I am not teaching well.

The other aspect of this quote that connected for me to teaching is the anticlimax, the laborious doing.  There is dryness to picking up math as a discipline.  We have a lot of rules, and some skills to practice.  My son is taking a Tae Kwon Do belt test tomorrow, and we had quite a talk today about the nature of work.  It is most often NOT fun.  It's a blessing when it's satisfying.  But when his jumping back kick got working, he was proud of his accomplishment.  And it was his accomplishment.  And if he passes the test tomorrow (which they are only cleared for if the teacher thinks they are ready; TKD-as-math post later?) that will be his accomplishment, too.  When he got through the dryness, he was engaged in the task; the work resulted in increased ability.  I also think he learned something about if he works, then he improves.  I'm not naive enough to think that once is all that takes, but it's a step.

I also had an experience recently of trying to have a discussion on an anti-reform blog comment thread, and that feels like the mystified devils here.  (Not to equate devils and anti-reform people.)  When I wrote about understanding and process as a goal, it felt like they just didn't get it.  They want the students to get the right answer as quickly as possible.  How can we hope to get them to that point if we are not willing to reward and cajole them to the end result?

PS. The bible study that inspired this post is here.  Not very bible-ey this time, as I'm hoping the participants can provide the scripture.

PPS.  "Do remember you are there to fuddle him. From the way some of you young fiends talk, anyone would suppose it was our job to teach!" - Screwtape in Letter 1.  The most explicit teaching reference in the book.

PPPS.  I tweeted this quote earlier: "Experience is the mother of illusion" - Screwtape paraphrasing Kant. Our students experience of math (and learning) has created a powerful illusion that they know what it is because of what it has been.  I think we need to shatter that illusion by sharing the truth of our vision of mathematics with them.  Can I get an Amen?

Battle of the Century

Dorky little skit I wrote a while ago. Should we film it?


Fractions vs. Decimals

The Battle of the Century

Ringside Announcer (RA): Welcome ladies and gentlemen to the Battle of the Century: Fractions vs. Decimals!

Old Man Fractions has been king of the hill for so long he can remember the pharaohs. But relative new-comer decimals has been rocketing through the ranks past previous contenders like Mixed Numbers and Percents, buoyed by the rise of science and handheld technology. Tonight they settle the issue once and for all, mano a mano.

Color Commentator (CC): That’s right, Jim. And they have both clearly prepared. Fractions has developed his upper body so much he looks positively improper. Decimals has emphasized speed work, and is awfully quick to the point. Hey, looks like they’re ready to start.

RA: They come out swinging! Fractions looks like his strategy is to corner decimals and work his weaker visual representations. Oh there’s a pie model and a fraction strip combo! Decimals finally lands a 100 grid haymaker and gets back out to the center of the ring.

CC: Looks like that speed work is paying off, Jim. Decimals is coldly calculating without having to hit any special menu buttons on the calc, if you know what I mean.

RA: Not really, Howard, but I’m used to it. Oh! Decimal made a rounding error and Fractions lands an uppercut.

CC: That’s exactly the answer, Kid Decimals!

RA: The traditionalists are out of their seats, cheering on Fractions. Even the French are into it!

CC: He’s certainly got that je ne sais quoi, eh, Jim?

RA: Huh? Back to the action, Fractions is pressing his advantage. But decimals sees an opportunity and – oh! The referee calls time!

CC: I don’t think it was intentional, but that was definitely below the vinculum.

RA: The referee gives Decimals a warning and they’re back in. Fractions still looks a little wobbly, and Decimals presses the advantage, really working over Fraction’s arcane and misunderstood algorithms.

CC: Invert and multiply that! Whew!

RA: Fractions gives a nice example of unit fraction multiplication and is back in the fight. Oh, and lands a nice left hand on a complicated long-division problem.

CC: Decimals looks like he doesn’t know if his point is going left or right, Jim.

RA: It’s back and forth at this point folks. Fractions simplifies nicely, and catches Decimals a good one. Decimals lands a nice easy comparison, but Fractions hits a unit confusion counter-punch.

CC: That’s half of something, alright.

RA: Then Decimals comes right back with a repeating combination! Oh, and a non-terminating, non-repeating wallop! Fractions has no answer for that.

CC: Right in the Pi hole! Practically transcendental ring work, Jim.

RA: They’re really taking a beating out there. Howard, I think the crowd’s getting confused about what’s important here.

CC: I think you’re right, Jim, there’s kind of a baffled silence. Not that unusual at a rational battle like this one, though!

RA: That’s time. The fighters move to their corners. The judges communicate their decision to the ref. It’s pretty close on my scorecard, Howard. What do you think?

CC: Did you double check your answer, Jim? Nothing would surprise me –

RA: The ref is ready and brings both fighters to the center of the ring.… he pulls up both fighter’s hands! It’s a draw!

CC: The judges have called them equivalent! Oh, man! Looks like we’re in for a rematch.

Photo credit: tanita1 @ Flickr