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[following up from here]

Appeal To Their Intuition

"How much cash is this?" Take guesses. The student risks nothing with a guess but that investment pays off huge for the teacher over the life of the exercise because the student wants to know who guessed the closest.

Build Slowly

Again, ask "how much cash?" but also ask "how heavy?" Show them the weight. (I zeroed out the jar from every weight measurement you'll see here. Don't worry about it.) Spitball some ideas for determining the value of those coins. You're trying to motivate the idea that the weight of the coins ties directly to how much the coins are worth. Pull up the relevant Treasury website.

Then mix in some nickels. Scoop out a small sample. Play with that. Set up a proportion between value and weight.

Iterate

Now you have pennies, dimes, nickels, and quarters. I took nine sample scoops, everything from small to big.

I formatted these at 4×6 so I could print them out at our local one-hour shop for a few bucks and put one in front of every student.

Throw A Curve Ball

Some will finish quickly. You tell them you have a jar of coins that weighs 5,500 grams. You reach in and pull out 14 nickels. How much is the jar of coins worth?

They'll run these calculations and come up with an estimate of $55. You tell them it was really $34, which is huge error. Ask for sources of error. Then toss this up and talk about it.

Confirm The Answer

$84.00, if you were curious.

It's essential to give some kind of visual confirmation of the answer, both so we can give credit to good initial guesses and so we can talk about sources of error. (ie. "who was off by the most? did sample size matter at all?")

Miscellaneous

  1. Show them CoinCalc, the backend of which does exactly what we've done here.
  2. This activity follows-up nicely on the goldfish activity, where we used a small sample of fish to determine the total population of a lake.
  3. We yield the floor to Jason Dyer and anybody else who would like to debate the question, "why are we doing this digitally?"

Download

Here's the entire learning packet [62MB].

Michael Caratenuto:

Personally, I think that this particular image lacks opportunities for inquiry. Perhaps if it was presented with other kinds of door locks leading students to come up with and answer the question, “which is the most secure lock?” [emph. added]

This is exactly right. The latest WCYDWT? installment has provoked the usual litany of Really Interesting Bite-Sized Questions, the sort of prompts that will play great in the Applications & Extensions & Assorted Mindblowers section of your lesson plan but which, on their own, aren't a lesson plan. Those questions don't provoke the kind of iterated, increasingly difficult practice that students need for skill development.

Again, this image on its own is insufficient. With some creative modifications, however, it will carry you through permutations. Here is that lesson plan in its broadest strokes.

Start with the image.

Tell them the code is 1 digit long. Tell them the code is 2 digits long. Tell them it's as long you want it to be. I respected the rule of least power here, which meant that when I took this photo I tried to stay out of the way of your lesson planning. Have them write down all the possible codes for n=1, n=2, n=3, etc. The increasing obnoxiousness of the task will motivate a formula for the general case. That's arrangements.

Tell them the lock is a 4-digit lock. Now turn on the blue light.

Ask them to list the possible codes. You can iterate this a bunch of times until they have discovered on their own this tool that mathematicians call a factorial.

Remind them it's a 4-digit lock. Then put up this image. It will be confusing, but only for a second. Ask them to list every possible code.

Iterate this with two and three buttons until they have generalized permutations. Then maybe you iterate the entire thing with another keypad lock.

Then maybe you dip into the comments of the original WCYDWT? post and help yourself to some very-interesting follow-up questions. I recommend Alex's.

Let me close by saying how shocked I am at how little all of this costs.

[Update: Bruce Schneier has a good follow-up on information leakage. Two photos.]

[Update II: due to the peculiarities of many car door locks punching in "123456" tests both "12345" and "23456." Consequently, there is a number string 3129 digits long that will test every five-number comination.]

[Update III: more information leakage.]

[Update IV: more information leakage.]

I like this. The iPhone application RulerPhone will measure anything, in any photo, so long as the photo includes a credit card. It's a great use of proportional reasoning, which, if pressed to name one, would be The Mathematical Skill I'd Most Like My Students To Retain After High School.

I added it to the What Can You Do With This? segment featuring The Bone Collector, which seemed like an obvious pair to me. In trying to find the best classroom entry point for this program, I can only think of the question, "How can we break this thing — trick it into giving an incorrect measurement?" I imagine someone can do better.

Session Title

Games And Puzzles That Develop Sequential Reasoning

Better Title

OMG MICHAEL SERRA!!1!

Presenter

MICHAEL SERRA!!1!

Narrative

A structure not dissimilar to Megan Taylor's yesterday, where Serra debuted games and puzzles and gave us time to tease them out.

I sat with two former colleagues in the back — all of us now at different schools. One teacher enthused over Sudoku puzzles. They challenge kids. Kids like them. It gets them comfortable with numbers. The other enjoys Serra's games and puzzles, like Lunar Lockout. Both cite improved student disposition toward math and improved deductive reasoning.

I disagreed with them. In general, I find it dangerous to put too much distance between "fun time" and "math time" preferring, instead, to have that cake and eat it too, creating as many challenges as I can that are both fun and mathematically rigorous. (Which Sudoko, to put it plainly, isn't.) My task is harder, I think, and I know I fail at it more, but I'm more satisfied on balance.

It was a good conversation. Feel free to interrupt us.

Serra's best offering for my money was Racetrack Math:

It's like this:

  1. Draw a racetrack on graph paper, however crude.
  2. You and your opponent start anywhere on the starting line.
  3. You travel along vectors. You may increase or decrease either the x-value, the y-value, or both, but only by one unit per turn.
  4. First person to the finish line wins.
  5. (P.S. No crashing.)

This gets very interesting very quickly. You start out with tiny vectors which lengthen by one unit every turn. If you fail to notice the side of the track off in the distance, though, and fail to slow down in time, you crash. (Which I did in the example above.)

I hereby toss all of my battleship exercises in the recycling bin. This is a much more straightforward introduction to positive/negative coordinates since each new turn is relative to the last turn rather than relative to this strange coordinate axis thing.

Plus, your students can create racetracks of their own, of infinite complexity, within seconds. Serra cited some kids who created a pit lane, which you had to enter on your second lap, and oil slicks, on which you could not adjust your vector at all. I'm impressed.

Visuals

PowerPoint. Which is tough when you're asking people to solve a puzzle. If someone suggests an alternative route to the one you have programmed into your slide, you have to dodge their answer a bit.

Handouts

Blank puzzles and games to draw on. Again, paper is not dead. How do you do this digitally? Load each picture one at a time into Skitch and pass a stylus back and forth? Moderation, please.

Homeless

  • "There is no research that demonstrates these games improve outcomes in other mathematical procedures like two-column proofs," Serra admitted reluctantly. "It has to be there. I know it is.

a/k/a My Qualified Disaster
a/k/a The Trouble With Tech

previously on dy/dan

We started with four variables (text messages, beers per day, etc.) which we tracked for 2.5 months in quad-ruled notebooks attempting to transform the quotidian details of our lives into extraordinary infodesigns a lá Nicholas Felton.

This was a departure for me. A tech-driven, student-led, design-infused mathematical project. Things went wrong.

This is a comprehensive autopsy of our Feltron Project. I post it here, in its entirety, a) for my own review next year, b) for your criticism. If you aren't in the mood for the full, bone-by-bone dissection, please scan down to the section headed What Really Happened. These are problems I don't know how to solve.

The Lesson Plan

a/k/a What Was Supposed To Happen

  1. We selected variables.
  2. We discussed them, making them more interesting (disaggregating "hugs per day" into "boy hugs" and "girl hugs") and more manageable (tracking "fast food I eat" instead of "what I eat").
  3. We tracked them for ten weeks, checking ourselves for consistency every two weeks, and then we stopped.
  4. We spent one hour marveling over Nicholas Felton's annual report, dissecting it for meaning, identifying the mathematical operations (average, maximum, minimum, sum) and the mathematical forms (pie chart, line graph, histogram, stacked bar graph, map) he used.
  5. We spent six hours entering our data into Excel sheets.
  6. We spent two hours teaching and deriving ten facts of our lives using average, maximum, minimum, and sum functions in Excel.
  7. We spent two hours teaching and deriving four graphs of our lives using pies, lines, and bars.
  8. Raw facts and graphs in hand, we spent thirty minutes discussing and distilling Felton's graphic design savvy into the two principles I thought my freshmen could reproduce with crayons and paper if they had nothing else:
    1. colors, Felton uses a two-color design (shades of black, shades of blue) which, apart from distinguishing his hierarchy (titles in black, data in dark blue, accents in light blue, etc.) keeps down costs when designing for a large print run.
    2. grids, the kind your eyes can't see but which your brain loves, the kind which imposes order on what would otherwise be a completely disordered data set, so while Felton jumps from music to movies to drinks you know where to find everything.
  9. We spent another two hours in class tying up loose ends in Excel and then a week designing our Feltron Projects.

What Really Happened

a/k/a Help.

  1. Only 55% of my students submitted the final Feltron Project1.
  2. Many of the other 45% stopped tracking early in the project, which meant assigning them review work, new work, or busy work while everyone else worked in Excel.
  3. Those who kept up with the project quickly staggered their progress (based on pre-existing computer ability, typing speed, and attendance) which saw me dashing between desks, explaining and re-explaining the same procedures over and over again.
  4. Our mobile computer lab a) comprised just fifteen laptops, and b) was available for check-out only once a week, c) if that.
  5. Kids lost work. I had them send their Excel files to themselves and then download the attachment the next day. Trouble was kids sent old files to themselves or they named files computer arsenic like "<<xxxx….davidsfeltronz!!!….xxxx.xls>>" which put both Excel and Gmail into simultaneous cardiac arrest2.
  6. I overestimated my students' computer fluency. Name it: locating saved files, opening programs, using a trackpad, using modifier keys, sending e-mail. These tasks all required constant, patient re-explanation. Missed that mark by a country mile3.
  7. None of them had used Excel before. Ever. Many didn't have it at home. One triumph of this project — recognized by a lot of students — is that my kids are now somewhere in the top quintile of Excel users. This will doubtlessly prove useful again in their lives — not in the when-will-we-ever-use-this-in-real-life? sense, like they won't be able to find food or shelter without Excel, just that it will open up a lot of interesting opportunities.

What Mattered

a/k/a Grading

  1. Faithful Tracking
  2. Interesting Findings
  3. Clear Design

Students ranked themselves on a ten-point scale across each index. Given how deeply we had immersed ourselves in exemplary work over two-and-a-half months, with only a few exceptions, I gave them exactly the grades they felt they deserved.

What I'll Do Next Time

a/k/a If There Is A Next Time, Obviously

  1. Host screencasts online demonstrating essential Excel procedures4.
  2. Strengthen our analysis. A student's text message graph plunged for a week when her parents confiscated her phone and spiked when she pulled a boyfriend in May. Students positively thrilled to see those connections but we didn't build any of that analysis into the project grading. Should've.
  3. Employ a Kuropatwa-esque rubric to better inform kids what constitutes "clear design" or "faithful tracking."
  4. Discuss design in greater depth, incl.
    1. showing them what my own Feltron would look like with rangy, mean grids or spasmodic colors;
    2. showing off the good and bad from this year's class;
    3. comparing/constrating Khoi Vinh's approach to grids and David Carson's insane anti-grids;
    4. showing them Aesthetic Apparatus' beautiful work in just three-or-fewer colors;
    5. compare 3D graphs alongside 2D hoping a lot of students will reconsider the choices they've made in life.
  5. Make a more obvious point of my own Feltron Project. Playing along with your students isn't even optional here. I made sure I ran through the collection process with my students (for empathy, if nothing else) but I should've made a larger point of my own struggle and process.
  6. Find collaborators. This was insane. I should not have gone at this alone5.

Students On Feltron

Just do a month.

JG, smart; we'll multiply a month by 12 to extrapolate for a year.

Everyone should track the same thing because it'd be really cool to see which people are like you.

BP, also smart; resolved, then, that we'll select three variables independently of the class and then select a common classroom variable for the fourth.

I like the chalang. It feels like I acopolished something hard and it made me feel good.

BS, sic sic sic; whose mother, in an IEP meeting, said of his Feltron notebook, "He carries it everywhere."

Felton On Feltron

Nicholas Felton consented to an e-mail interview on his process which will appear in this space tomorrow.

Gallery

I have installed student work — everything from awful to exemplary, but mostly exemplary — into a Flickr set.

Handouts

  1. Feltron Project Outline
  2. Nicholas Felton Analysis Sheet
  3. Excel Chart Illustrations
  4. Excel Formula Sheet
  5. Map Infograph Template
  6. Final Review Sheet

To Conclude

This was a different, necessary kind of insanity for me to finish my fourth year teaching even a little eager for a fifth. The price tag was steep. To accommodate this time-sucking project-based learning, we skipped a third of our logic unit in Geometry and fully jettisoned last year's Platonic Solids project.

If I weren't already guzzling away at this barrel of standards-based Kool-Aid, I'd write something agitated and truly inexcusable here about curriculum narrowing or the time cost of NCLB, but I remain convinced we need to settle on a list of necessary skills and then decide horse-in-front-of-cart-style on the best tools and projects to teach them6. I do not know if this was that.

There are twenty-four hours. No exceptions. I'm uncertain Feltron was the best use of our time.

I put Feltron to rest now, surely the weirdest assignment I've concocted in a four-year career. I post this here to solicit the usual gallery of critique and construction but also because, at some point in this whole blogging thing, I forgot how else to end a project if not with rigorous and public self-critique.


  1. Controlling for age: 48% of freshmen and 63% of upperclassmen completed the project.
  2. For the record, I originally sought GoogleDocs out for this project but they maxed out at something like fifty rows where we needed hundreds.
  3. There were exceptions, naturally, but Digital Immigrants™ outnumbered Natives™ at 15:1, many of which Natives one day, I have little doubt, will grow up to be edubloggers.
  4. incl: sorting columns, using formulas (avg, min, max, sum, countif), saving/sending work, creating new sheets, filling down the date.
  5. Any takers?
  6. Noted here: Jay Greene's j'accuse directed at teachers who complain that NCLB exigencies leave them with no time for fun project but who also wile away the last month of school with parties, assorted time wasters, etc. We didn't start computer lab work with Feltron until after our round of state assessment.

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