Finding Ways to Nguyen Students Over
Fraction Division via Rectangles
I appreciate Christopher Danielson's post on common numerator fraction division because it's important to examine how various algorithms work and how we can help our students become more flexible with their thinking. It's not surprising that I teach fraction division using rectangles, and I really believe the kids seem to grasp it better because it's visual.
Someone says, There's one left over.
I'll start with this problem: 3/4 ÷ 2/3.
But before we do fraction division, I ask kids about whole number division. What is 8 ÷ 2? What is 15 ÷ 5? Eventually we settle on something like: asking what is 8 divided by 2 is the same as asking how many groups of 2 are in 8.
Then we apply the same question to 3/4 ÷ 2/3 as "how many groups of 2/3 are in 3/4?"
I guide them through this process:
- Let's draw out 3/4 and 2/3 on paper.
- Let's use grid paper instead to draw our rectangles. I think you can show 3/4 much more accurately on grid paper than on a circle.
- Please draw 2 rectangles of the same size. (By doing this, we are really dividing two fractions using the common denominator strategy. Christopher writes about it here.)
They ask, Any size?
- What size do you think? Does it matter?
- Shade the first one to show 3/4 and the second one to show 2/3.
- So maybe we should think about the size of the rectangle more carefully. Look at the problem again. Three-fourths divided by two-thirds. Hmmm... What dimensions should our rectangles have so it's easy to divide into fourths and thirds.
- Bingo! I'm drawing these with you. Okay, so two rectangles of 4 by 3 — or 3 by 4 — doesn't matter.
- I'm shading in 3/4 on the first one and 2/3 on the second one.
- So our question is: How many groups of 2/3 are in 3/4?
- Because I colored mine in, can you help me ask the question again using colors instead?
Someone responds, How many pinks are in the greens?
- Yeah. And how many little squares are pink? Okay, eight.
- So, I'm going over to the green here and round up 8 pink squares.
- I'm able to round up one group of 2/3 (pink) in the 3/4 (green).
Someone says, There's one left over.
- How much is this one little green square left over worth?
- Right! 1/8 because we called 8 little boxes as one, so 1 little box must be 1/8.
- Our answer then is 1 and 1/8.
A few students say, I get it.
- How do we know that our answer of 1 1/8 is correct?
- Okay, we'll use a calculator.
I purposely use an online calculator where I'm entering the fractions as they appear. I don't need to distract them right now with decimals or talk about parentheses. This is from CalculatorSoup.
- Let's do this again. Now with a mixed number just for fun. Let's do 1 1/2 ÷ 2/5.
- How many rectangles are we drawing?
- What dimensions should they be?
- Oh, but we have more than 1 whole here, so...?
- We should have something like this then.
They say, How many groups of orange are in blue?
- So let me round up the groups of orange that are in the blue.
- I got three. And the leftover is? Right, three. Three out of...?
More students say, Three-fourths! Three and three-fourths. I get this!
This online calculator from Calcul allows for entries of mixed numbers.
- Okay, your turn to do one all by yourself.
- Please do 2/3 ÷ 1/2. (Same one Christopher used.)
I think these kids' papers show understanding.
While these are not there yet.
I don't know. But it seems that drawing pictures and doing more visual stuff start to disappear in middle school. Below is our textbook's treatment of "dividing fractions and mixed numbers" — Chapter 5, Section 7 — the full 3 pages before the Exercises. Notice the two circles at the start of the section — that's pretty much it. And circles are great if you have denominators of 2, 4, and 8.
I think if I can get my kids to first see the answer, then I can sell them the other algorithms — like multiply the reciprocal — and not come across as a fraud.
I also want to point out that I normally see this visual below for division of fractions. My way is different than this — I deliberately ask kids to draw 2 rectangles whose dimensions are the denominators.
Venturing Into the Sequel of Penny Pyramid
I didn't get around to doing Penny Pyramid when I first saw it last year. But Dan's 3-post series and Nathan's recent mention of it were the reminders I needed to make it happen.
Act 1
- how many pennies
- how much money is that
- how long did it take
- who in their right mind would do this/who has that patience
- how much does it weigh/is the table gonna collapse
- what is the volume/surface area/height
- what is the ratio of pennies from one level to the level above it
(Student who gave the highest high guess did correctly say her written number as
"one hundred quadrillion." It made me happy that she knew this.)
"one hundred quadrillion." It made me happy that she knew this.)
Acts 2 and 3
Lauren F: Is there a way to multiply consecutive numbers quickly? You showed us the addition one...Maddie: Isn't that the exclamation point operation?Gabe: But we're not multiplying consecutive numbers!Mia: Doing 40 by 40 then by 13 gives all huge numbers, so we're doing a simpler problem, then find an equation.Lauren P: Our group is finding a pattern and making a table.Gwen: We're doing layer by layer. There are more of us (4 instead of 3), so it's pretty quick to divide up the work.Gabe: I already have the answer because I was too eager to do the math, but I didn't say anything to the group. (He got the answer about 2 minutes after we formed groups.)Julia: And I got it 3 or 4 minutes after Gabe.Angela: And I got the answer after Julia. Without her help.Me [to Gabe, Julia, and Angela who were in same group]: Aren't you guys special. You seriously just sat there and did nothing then while I walked around?Julia: Well, yeah, we're kinda admiring our work.Me: Geez Louise. What do you think I'd have asked you if I knew you'd found the answer to this pyramid?Gabe: If it was 100 high?Me: No. A million high. A billion high.Gabe: Hehe. That's why we didn't want to say that we're done.
- Two students figured out why each stack had 13 pennies.
- Their other questions were answered to their satisfaction, except we didn't know exactly how long it took Mr. Bezos to build it, but we talked about how we might be able to estimate this.
- Kids remembered from last week's lesson that a square pyramid has 1/3 the volume of a cube with same dimensions, but that our penny pyramid had jagged lateral edges.
While everything up to this point had gone as well as I'd expected. Kids immediately responded to the video with WOAHs and WOWs. They asked solid questions in both Acts 1 and 2. They worked well in groups. I'm proud to say that my kids do these 3-Acts like Matt Vaudrey does mullets. If there were a 3-Act lesson throwdown, my kids would kill it. :)
However, the kids and I knew that no one really struggled with the task of just finding the number of pennies. The math was pretty basic and with a calculator, 40 layers of pennies didn't make anyone break a sweat.
What was meant as an "extension" or "sequel" really needed to now become the focus of our lesson — at least for this group of students who valued a good struggle. We needed to try to figure out the equation for this penny pyramid.
But I also realized that it would be very unlikely for my 8th graders to come up with the equation because it involved summation of a sequence. (You're right, Nathan, it is unlikely, even for Gabe.) But the process of getting there might be worth it. I wouldn't be their teacher if I didn't ask them to explore the patterns that they might see along the way.
I gave them small interlocking cubes and colored chips so they could build smaller models of the pyramid.
Their collective frustration arose from how "simple" the pyramid was built — nothing more than a sum of layers whose square dimensions were consecutive.
Incomplete Cube
We started with a smaller problem. We did a 5 x 5 square pyramid with a height of 5. We didn't like the "jagged" lateral edges of the pyramid either, hence we pushed the cubes into one corner like this so at least the cubes stacked squarely.
One way would be to imagine that we had a whole 5 x 5 x 5 cube, then subtract from this the small cubes that were missing. We noticed the missing pieces were these L-shapes.
We see a pattern in these missing L-shapes:
- 4 pieces of (2n-1) or (n-1)(2n-1) or 2n2-3n+1
- 3 pieces of [2(n-1)-1] or (n-2)(2n-3) or 2n2-7n+6
- 2 pieces of [2(n-2)-1] or (n-3)(2n-5) or 2n2-11n+15
- 1 piece of [2(n-3)-1] or (n-4)(2n-7) or 2n2-15n+28
Incomplete Rectangle
How else can we see this pyramid? Because my mind has a tendency to reshape things into rectangles, I flattened the pyramid into an incomplete rectangle like this:
The dimensions of the rectangle were straightforward enough, and unlike the missing L-shapes of the incomplete cube, the missing pieces here were rectangular and came in pairs. For example, in the above right sketch, the missing pieces were two 1 x 4 and two 2 x 3 rectangles. But if n were even, then the number of missing pieces would be pairs of rectangles plus 1 lone square piece.
I talked with them about the sigma notation, and since they knew how to add {1 + 2 +... + n} quickly — we refer to this as "Gauss addition" in class — they thought it was fun to learn the new symbol.
Then we went into WolframAlpha and typed in what we wanted. BAM! Equation came up with the "newly" learned summation notation.
The kids saw patterns. They learned a fancy new sign. They knew that the right math could help solve for any penny pyramid. But I really think they look forward to learning more math in high school.
I have 6^2 students. How many do you have?
When my daughter Sabrina was 23 years old, her grandfather would have been 34 if he were still alive. Unfortunately my father had already passed away on his birthday when I was 62 years old.
While the above paragraph is all true, I don't know of anyone who writes or talks like this. No one writes or speaks of people's ages in exponent form.
No one except in this 7th grade Pre-Algebra textbook that we'd adopted for our school five years ago. I was a big part of that "we" even though I don't teach Pre-Algebra. I liked all the bells and whistles, like online textbooks and resources, that Glencoe had to offer over the other publishers. But bells and whistles might serve the teaching of physical education better.
These are just some examples in two sections on exponent rules.
On page 541:
Also on page 541:
On page 559:
On page 561:
To be fair, the textbook does offer questions where numbers are appropriately written in exponent form, including:
- area of continents
- mass of a molecule
- distance between planets
- number of cells in culture
- computer processing speed
But this beauty on page 521 leaves me breathless:
Ask the School Budget Gods?
As the school year winds down, I'm wondering if you would like to ask your school budget gods to consider the following — and none of these people asked me to say any of this:
1. Mathalicious
Please sign up or renew your membership because you simply cannot get a team of Karim Ani, Matt Lane, Chris Lusto, and Kate Nowak in one place. That's crazy brain power. (EA Jackson just joined the team. She must be awesome too.)
I want these folks to eat and sleep well to continue their amazing work. Right now I think they're just drinking a lot of pureed kale.
2. Yummy Math
Brian Marks and Leslie Lewis work tirelessly to crank out relevant lessons on current topics. For just $15 a year, you can get solutions to nearly 200 lessons.
I don't know of another teacher who has a family and works as hard as Andrew does. Besides me. Did you know that you can honor his work by "tipping" his videos? This is the same great teacher who's behind the amazing Estimation180.com. Andrew puts out all these high-quality videos on his own time and dime.
I recommended Nico's wonderful book in a post back in March. So here it is again.
By far the best $25 our school could spend for an annual membership to anything is access to The Math Forum's great PoWs. The $25 is for one class of up to 36 students to submit their solutions to the Current PoW online (each additional class is just $10 more). Max Ray, Annie Fetter, Suzanne Alejandre are now familiar names to my students. You can't put a price on personal feedback.
All of us have spent countless hours outside of school to do this tough job of teaching. Add to that we also spend a lot of our own money to make our lessons come alive for our kids. But we need to specifically seek school funds for these higher-quality stuff. (I need to remind myself this.) Our administrators mean well, but they often waste money on shitty stuff because they don't know it's shitty.
Thank you.
(Comments are closed.)
When I Let Them Own the Problem
From our textbook:
Stuff like this makes my heart sink. (I actually wrote that it makes me fart — but that's very unladylike. And I'm trying to write better.)
There is essentially nothing left in this problem for students to explore and figure out on their own. If anything, all those labels with numbers and variables conspire to turn kids off to math. Ironically even when the problem tells kids what to do (use similar triangles), the first thing kids say when they see a problem like this is, "I don't get it."
They say they don't get it because they never got to own the problem.
I wiped out the entire question and gave each student this mostly blank piece of paper and the following verbal instructions:
- Make sure you have a sharpened pencil. Write your name and date.
- Inside this large rectangular border, draw a blob — yes, blob — with an area that's approximately 1/5 of the rectangle's area. No one will die if it's not quite 1/5.
- Next, draw a dot anywhere inside the rectangle but outside the blob. Label this dot H.
- Now, draw another dot — but listen carefully! — so that there's no direct path from this dot to the first dot H. Label this second dot B.
I asked the class if they knew what they just drew. After a few silly guesses, I told them it was a miniature golf course: blob = water, point B = golf ball, point H = hole location.
The challenge then was to get the ball into the hole. Since you can't putt the ball directly into the hole due to the water hazard, you need to make a bank shot.
(Some students may have drawn the blob and points in such a way that this was not really possible, at least not in one-bank shot. I let them just randomly pull from the stack of copies to pick a different one. I made a copy of their sketches first before they started their work.)
The discussions began as they started drawing in the paths. One student drew hers in quickly and asked, "Is this right?" I replied, "I'm not sure, but that's my challenge to you. You need to convince me and your classmates that the ball hitting the edge right there will bounce out and travel straight into the hole. Does it? What can you draw? What calculations are involved?"
What I heard:
- The angle that the ball hits the border and bounces back out must be the same.
- Because we're talking about angles, something about triangles.
- This is like shooting pool.
- Right triangles.
- Similar right triangles.
- Do we need to consider the velocity of the ball?
- This is hard.
- I can't figure out how to use the right triangles.
- Similar right triangles because that'll make things easier.
- Even though it's more than one bounce off the edges, I'm still just hitting the ball one time.
- I think I got this.
- I have an idea.
- Wish my golfer is Happy Gilmore.
BIG struggles, so I was happy and tried not to be too helpful. (I struggled big time too on some of their papers! And I think this made them happy.)
Lauren explained in this 55-second video how she found the paths for the ball to travel. I also had her explain to the whole class later at the document camera.
Jack took a different approach. Instead of measuring the sides and finding proportions to find more sides to create similar triangles like Lauren did, he started with an angle that he thought might work [via eyeballing] and kept having the ball bounce off the borders at paired angle until it went into the hole. (His calculation was off — or his protractor use was inaccurate — as he had angles of 90, 33, and 63. Or maybe if he had a better teacher, he'd know the sum of the interior angles of a triangle was 180.)
Gabe was quieter than usual today. When he finally shared, his classmates realized he was the only one to solve the entire problem using just constructions with a straightedge and compass. He walked us through his series of constructions until he found point C on the bottom border where the ball needed to bank off and end up in hole H.
Imagine none of this thinking and sharing would have occurred if I had given them problem #24 in the book.
Half of my kids were still struggling and working to find the correct bank shot(s), but they were given the chance to struggle. And none of them said, "I don't get it."
**********
The cutest thing also happened while we were doing all this math. Yesterday (Monday) I bragged to the kids — and I'm doing it again right now — about the Rolling Stones concert that we went to on Friday. I am still over the moon ecstatic that we got escorted into the Pit from our way-in-back-floor-seats!!!!
Anyway, a kid today started humming to the tune of (I Can't Get No) Satisfaction and quickly others joined in with THESE LYRICS:
I can't get no similar triangles
I can't get no similar triangles
'Cause I try and I try and I try and I try
I can't get no, I can't get no
When I'm drawing in my lines
...
This lesson leaves me so full and proud. Their singing to the Stones while struggling in math makes me crazy in love with them.
Just so you know, I swooned shamelessly in front of my students over a 70-year-old rock star's butt.
[Added 05/08/13]
Today I had the kids work on someone else's paper (remember I made copies of their papers before they worked on them) and find similar triangles to make the bank shots. Because I purposely told the kids to draw in the blobs and the 2 points without any mention of where exactly to place them, it was then by chance that these papers below allowed for one-bank shots to get the ball into the hole.
The ones below, however, are some of the ones that would not work with just one-bank shots, but I had the kids create similar triangles on them anyway because that was the learning goal of the lesson.
Do These Two Things
Here's a bold claim that I can make here and now: I have made more teaching mistakes than any other teacher I know. I have the years to back me up.
You've asked your students to work out of the textbook for an entire period? I've done that more than once. You've snapped at a kid and made him cry? I've done that. Or you've cried in front of the class because you're so fed up with their ungrateful and spoiled behaviors? I've done that. You've doled out a factory-made test without checking through all the questions on it? I've done that. You've brought ungraded papers home and shredded them? I've done that. (Although this may not be a mistake at all; we know crappy assignments must be burned.)
You've punished the whole class for something one or two kids had done? I've done that.
You've denied your kids' opportunities to think deeply because you gave them all the answers? I've done that. You've given them timed tests? I've done that. Worse, you've given them timed tests preceded by this lie: just relax and do your best. I've done that. You've shown a video without previewing it? I've done that. You've made a promise that you couldn't keep? I've done that.
This list drones on and it's already exasperatingly dull.
If I may shift my attention then to the two main things that I've learned to do over the years so that when I do make my mistakes, the kids are incredibly quick to forgive me. Just two things.
Teachers are not in it to climb some corporate ladder to reach the thick-carpet land. The ladders we know are the ones we climb on to proudly hang our students' work in our classroom. Or we scaffold a lesson to get students to play around on that ladder of abstraction. We are here to learn right along with the kids. Teachers are hypersensitive to that metacognition thingumabob.
Maybe it was during my second year of teaching when a veteran teacher dropped in to give us new teachers a short presentation about what to do and what not do as a teacher. He told us to not talk about ourselves to our students, that the kids are not interested.
I disagree. I think you should tell stories.
1. Do talk to your students about yourself. We are adults, they are children, we are teachers, they are our students, so of course the topics have to be appropriate. But it's disingenuous and selfish to say that we want to know more about our students yet not reciprocate in this endeavor. How else may we create that magical "rapport" that everyone talks about?
I tell sporadic stories about what's going on in my life — past, present, future — dispersed between solving equations and talking about math. These are not planned conversations, they just come out naturally and haphazardly. These light moments came up within last week.
2. Tell them about things you're not good at. To balance out my heard-all-too-often outbursts of I'm brilliant!, I tell kids about the many things I cannot do. I tend to tell the class this when I sense they are struggling with a math concept. And when I'm no good at something that they are really good at, I shower them with genuine admiration.We were playing Nim in class and discussing how many chips to remove on one's turn. This made me remember when someone at a department store was trying on a shoe of mine that I'd removed in order to try another shoe myself. I guess it looked new enough that she thought it belonged to the store and wanted to ask the salesman how much it was until I told her it was my shoe! Give me back my shoe, lady! (Two kids couldn't stop laughing about this.)I told them briefly about my trip to D.C. and the soft sheets in the hotel room. I told them about my visit to the Holocaust Museum. One student asked, "Did you cry?" Another student replied before I could, "She cried when we did bad on a test, so what do you think?" We eased right back into simplifying the next rational expression.I was hungry and told them about the so-so enchiladas I'd made for dinner the night before. The kids shared their preference or indifference about red and green enchilada sauce. Quickly the conversation was centered around "mystery meat" coming from our school cafeteria. We agreed that our favorite school lunch is the teriyaki chicken. Then we were all quiet again thinking if these two triangles shared a height or base and what the ratio of their areas might be.
I can't tread water. I think I can swim. But that's just it. I have to constantly swim or I'll drown. I can't do the eggbeater routine with my legs. My teacher took me to the deep end and said, "It's really easy. Just watch me..." Ten minutes later, she said, "You're right. You can't tread water. Oh, look, our time is up." I tell this story knowing that the kid who's struggling with what we're doing right now is on a swim team. He says, "I'll teach you. I'm a good teacher."Once I skied straight into a big pole while taking ski lessons. I know I'm not supposed to look at the pole because looking at it turns it into a giant magnet and me into an iron rod. I didn't give up though. Not after I saw a guy take a giant tumble and had taken forever to get up. I was way cooler when I fell.I bring in a math problem that I cannot solve. Then I share a different one on another day.I can't sing. My husband, bless his heart, showers me with affection and compliments ad nauseam. But even he can't lie about my tone deafness. What I'm about to reveal next has only been known to a handful of people outside my family. Here goes: I was in 2nd or 3rd grade, standing on a small platform at the front of the class with another classmate, we have a song to sing together. I remembered how cute I must have looked because I wore a pretty dress. We did not get far into the song when my duet partner turned... and...(are you ready for this?)... slapped me in the face! She fucking slapped me. I was that bad.
Monya and Kishi
Dear Mr. or Mrs. Textbook Writer,
My name is Joey and I'm in Mrs. Win's algebra class. She asked me to write to you because she's afraid if she wrote it nobody in your department would read it. She said stuff that comes from kids normally gets better attention. I don't know about that. I've been writing to this girl for 4 months now and she doesn't know me from the carpet.
My teacher, Mrs. Win, lost it [again] today when she saw this problem in our textbook.
Page 484, Exercise 33:
While hiking in the San Bernardino Mountains, Monya and Kishi stop for lunch on a ledge 1000 feet above a valley. Kishi decides to climb to another ledge 20 feet above Monya. Monya throws an apple up to Kishi, but Kishi misses. The equation h = -16t^2 + 30t + 1000 represents the height in feet above the valley of the apple t seconds after it was thrown. How long did it take for the apple to reach the ground?
Well, our teacher — actually our whole class — wants to know why on earth do you always use the "strangest" names in all them math problems. I don't mean disrespect to call them strange, but Mrs. Win was saying that in her 23 years of teaching (she's old, huh? but she gives us candies when we tell her she doesn't look that old), so that's like 3000 students she's taught, and she's never had a kid named Monya or Kishi. There was a Shaniqua in another word problem. On page 402, there's a LaTanya. Olinda lives on page 440, and Ofelia on page 39.
Some of us think you do that to be politically correct or culturally sensitive. (I don't quite know what culturally sensitive means but I heard someone say that in class so I'm just passing it along.) Mrs. Win thinks you use these particularly unique names just in case there is a 1/3000 chance you could raise the self esteem of a kid named Monya when she sees her name in a math problem. Monya is a girl, right? See, we aren't so sure ourselves.
Our teacher normally calls on one of us to read the problem out loud to the class. And every single time. I mean every single time. The reader butchers the name and no one in the room can help us correct it. By the time we're done reading the problem — all worked up and bent over on the pronunciation of the name — we don't know what the heck the problem is about. Then teacher makes us read the problem again, this time quietly to ourselves. But the names don't get any easier just because you say them in your head.
Look, Mrs. Win's real name is Phuong Nguyen. She had to change her first name to Fawn because she said she got tired of hearing Phuong mispronounced and having to spell it every time. She'd change her last name too if she could. And she said she still might do that some day since there are so many Nguyens around that nobody would miss or care if one went missing from the census.
My point is our teacher doesn't mind if she ever saw her name Phuong in a math problem for as long as she lives. Why all her siblings but one changed their names: Hien is Henry, Tuong is Tom, Loan is Lucy, Nga is Kim, and Chau is Monique. So, if there are real people who change their names to make it easier for the rest of us, then why do you have to come up with fancy names for the imaginary people in your math problems?
Well, I'm done talking about the names. And if your company continues to use these names, then could you please at least give us a pronunciation guide like they do in the dictionary? (I can't remember ever having to use a dictionary, but that's what our teacher said.)
Oh, aside from the names, Mrs. Win wasn't too happy about the problem either. She was pissed because the question was about apple throwing and that's just lame. She didn't care how long it took for the apple to drop to the ground, she was more mad that either one person couldn't throw or one person couldn't catch a lousy apple. Hikers who couldn't catch food normally end up dead. While I don't mind you giving us the equation right there in the problem, but our teacher hates that shit. Sorry to use a bad word, but if Mrs. Win wrote this letter, I'm afraid she'd say the same thing.
We love Mrs. Win because she's like the coolest math teacher ever. She just wants us to learn and we learn better when she's not all pissed off.
Thank you and have a great day!
Sincerely,
Joey
Sleep On It
This lesson is the result of my spending 4 nights on a really nice bed wrapped under thousand-thread-count sheets.
Part 1: The Mattresses
- I asked the kids to guess how the size of the little rectangles on paper compare to the real mattresses. (It's 2% of actual.) We ignored the thickness.
- Then I asked them what the scale factor would be. Many didn't know.
- Next, with a partner, I had them cut out a set of 5 mattresses from construction papers, a different color for each size. Each paper mattress needed to be 8% of the original.
I almost threw out this cutting part because I didn't really have a follow-up for these pieces, but seeing how some struggled with measuring and cutting that I'm glad I kept it in. Here's a set of the 8% that we used as a key to check for precision.
- Based on the given price of a certain size mattress, how much should the other sizes [of the same type] cost? And they had to explain how they arrived at the prices, based on what criteria.
Part 2: The Flat Sheets
- The handout. To give them more practice with "finding the percent of," I told them that flat sheets are made so that the width is 160% wider and the length is 130% longer than the width and the length of each sized mattress, respectively. Knowing this, find the dimensions of each sheet.
- Then I gave them the price per square foot of each brand of sheets set, and they needed to figure out the rest.
**********
I thought it went well. (When I spend too much time creating a lesson, I convince myself and the kids that it'll be awesome. It kinda works.)
A very simple rates/proportions lesson, but I'm reminded of some key things:
- The kids weren't cutting for the sake of cutting — we're not here to make stuff pretty, we're here to make things accurate, and in doing so things look pretty. They had to use a ruler to mark two end points before drawing a segment. They had to understand the significant digits allowed by a simple ruler. Decimals are still tricky for some. Junior high boys are more clumsy with scissors. It's smart to draw the rectangle in the corner of the paper instead of smack dab in the middle of the paper.
- Student to his partner, "Finding the perimeter doesn't make sense. We have to find the area because that's the material you sleep on."
- Eight percent is not point-eight.
- Their proportional reasoning is still developing.
- Dividing by 12 is not how you change square inches to square feet.
- "But that's what the calculator says" is a poor excuse on any day of the week.
[Added 05/01: The idea of unit rate needed a lot of attention and review. Kids weren't sure to divide price by area, or divide area by price. What I learn regularly is kids will take two numbers and operate on them, but they don't know what the answer means and what unit the answer carries.]
Darren Kuropatwa and Intel Visionary Conference 2013
Back in February I received an email from Judy Salpeter of Tech & Learning with the subject line: An invitation to participate in Intel's Visionary conference (April 24-25).
Here's a condensed version (and Judy herself bolded the last couple sentences):
I had to read the email again, and again, and Googled Darren Kuropatwa (@dkuropatwa). Very quickly I learned that Darren is a guru in education technology. I also came upon a post from Dan Meyer with Darren's name in the title. Darren lives in Winnipeg and "knows" me, yet I didn't know him at all. This was so completely backward.Dear Fawn,I got your name from Darren Kuropatwa, who is leading a workshop at Intel's Visionary Conference in Washington D.C. on Thursday, April 25 (with an opening reception on the 24th). We (Tech & Learning magazine) are helping Intel with conference planning and organizing, which is why I'm reaching out to you.The theme for this year's conference, which is geared to approximately 100 school, district, state and government education leaders, is "Walking the Walk."...Darren will be leading the math workshop (with Diana Laufenberg, David Jakes, and Sara Martin leading the three others). Each workshop leader was asked to name two visionary educators in the field to invite as participants (to share examples and contribute in other ways) and Darren named YOU. Are you available and interested?
I forwarded the email to my superintendent who immediately gave me the three days off to go. I couldn't have replied back to Judy any faster with a YES!
After Darren learned that I'd be delighted and honored to attend, he sent me a very kind email, in part:
Mike Thayer (@gfrblxt) was the other invited educator. Mike writes thoughtful and provocative posts about the tough and rewarding teaching profession. It's the kind of big-heart writing that leaves you full of gratitude when you finish reading one of his pieces. I love this from Mike:I thought of you because you do good stuff. You share it. And you're honest about stuff that is or isn't effective in teaching math. You're someone I know I wanted in the room with me.
... that mathematics departments stop acting the wallflower and get out there in the middle of the floor. We need to grab some new partners and learn to dance.
Darren's work — and "work" here is an inadequate word to describe what I think is more his "humanity" — is prolific, gorgeous, generous, honest. He always has the student — the learner — in sharp dead-center focus with every lens that he uses to engage us in the important work that we do with children. Darren walks the walk — I mean literally — as he shares his thoughts on "learning about learning" in a series of #WhileWalking videos. His latest, number 90, about Public vs Private or Personal vs Collective is important.
The Conference began Wednesday evening with dinner at Smith & Wollensky near Dupont Circle. I got to sit between Darren and Mike at the "educators' table." We had to stand up, one by one, and introduce ourselves. It wasn't long before I realized the room was packed full of people with 3-letter acronyms after their names: CEO, CSO, SVP, CIO, DEA, DOT, CDO, etc. And because there wasn't much to my 3 letters of MRS, when it was my turn to talk, I told a roomful of 100+ strangers about my fondness of wearing padded bras. (You had to be there. Darren and Mike wished they weren't.)
On Thursday, after the General Session, attendees would go to one of four pre-assigned curriculum workshops (Language Arts, Math, Science, Social Sciences), then break for lunch, and then go to another pre-assigned workshop in the afternoon. All this happened at The Fairfax at Embassy Row.
Darren lead both our AM and PM math workshops. Mike and I circulated through the room to facilitate the activity and answer any questions. A few days prior, Darren let us know that he'd be having the participants work through the "Des-man" activity — this made me giddy with delight. Clearly Darren wasn't exaggerating when he wrote that he was burning the candle at both ends when we saw how much thought and planning he'd put into this presentation. Darren did an awesome job of course — with exemplary cadence and patience of a great teacher! I also thought the three of us made a great team.
We ended the Conference with a Closing Reception in the Lounge. We shared more wines and delicious hors d'oeuvres. I really didn't want the evening to end, but both gentlemen had to catch the next flight and train home.
This space that I call my blog. My dumb blog. My insignificant blog. Maybe it's not that dumb or that insignificant. It was how Darren found me.
How can I not be blown away that someone from Winnipeg trusts me enough from just reading the contents of this space to invite me to this fancy schmancy conference?
Thank you to Darren.
Thank you to Mike.
Thank you to Intel and Tech & Learning.
I need more time to digest what I'd learned from the Conference and from talking with Darren. I'm already stealing his ideas on student blogging and student sharing. SHARING is oh-so at the very heart of what Darren does.
It's fitting then that I wrap up with Darren's #WhileWalking 87: Learning is sharing — in 1 minute and 41 seconds, he sums up what my blog wishes it could do.
Oh, but wait, there's more. The next evening, Friday, I got to meet Andrew Carle (@tieandjeans) and Rachel Kernodle (@rdkpickle) in Georgetown for drinks!!!!!
And, and. Last Saturday, my daughter and I had lunch with Annie (@sophgermain) in Troy, NY!!!!!
Admit that you're turning green.
Louis Zamperini
Louis Zamperini.
My youngest kid, Sabrina, and I make our way to our assigned Table 33. We're here at an awards dinner honoring the county's top seniors. Sabrina is one of 3 seniors from her high school. I'm a proud parent sitting among other proud parents, eating too many bites of the chocolate ganache cake.
An older gentleman sits across from me at our large round table. He has no kid next to him. He says he's here to listen to Mr. Zamperini.
Louis Zamperini.
The evening's agenda moves at a good pace. The room is a little colder than I'd like but then I'm always cold. I'm grateful for a tall glass of hot green tea to wrap my hands around. I smile at my baby girl who's no longer a baby. She might very well end up at RPI this September. Kate's alma mater. I can't ask Kate to convince Sabrina not to go there. It's just New York is so far away. We are flying there this weekend for a campus tour, then Sabrina will make her decision by the end of next week. I touch my daughter's cheek and wish she were ten again.
Sabrina crosses the stage to receive congratulatory handshakes, several certificates, a trophy, a scholarship letter that we'll need to fill out and send in to receive the check in the mail, and a hardbound book, Unbroken, by Laura Hillenbrand who also wrote Seabiscuit.
Oh, and a bookmark, autographed by Louis Zamperini.
The speaker on stage says that Angelina Jolie will direct a feature movie based on Unbroken. The Coen brothers will add their magic to it.
The lights quickly dim in the room, and the same voice from the stage promptly announces, "And here's a short clip from that CBS Special."
Two big screens in the room project the 60 Minutes' Documentary.
[4/17/2013
I found the same clips from awesomestories.com
My life is just made smaller and bigger by this short video. My face still damp with tears when the lights go on again.
His World War II. My Vietnam. His 47 days in a life raft surrendered to the edgeless sea. My 4 days in a fishing boat domed by the edgeless sky. In his post-traumatic nightmares he strangles his tormentor; mine are similar. Oh, but how measly is my survival compared to his. His story of redemption renders mine trivial and petty.
Louis Zamperini. His name means nothing to me at the start of the evening. I vaguely remember seeing his name in an email as tonight's keynote speaker. I vaguely remember dreading all speakers at award ceremonies.
Louis Zamperini, at age 96, is helped onto the stage by his son. I'm in the same room with Mr. Zamperini. We are on our feet clapping. We don't want to stop clapping, but we have to stop because we need to hear more of his story, from him, right now, right here in this room. He sports a red USC Trojans cap and immediately makes us laugh as he throws a few jabs at UCLA. We laugh even harder when he tells us that the brutality of the Japanese POW camp prepared him for his 54-year marriage. The gentleman at our table who's only here to see Zamperini blows a loud finger-whistle at this.
His words nimble, gracious, full of life. I come to this evening selfishly thinking only of my daughter, grateful and proud of the recognition she's earning. But the man in our presence tonight is too big for anyone of us. The moment is too generous. I feel blessed and rub my daughter's hand.
It is not the shiny trophy or the embossed certificate that Sabrina will carry with her. It will be Louis Zamperini's story — his survival, his redemption — that I hope she'll carry with her forever. Just as I will.
