Monday, May 31, 2010

Web Roundup

Guillermo Bautista is starting a new blog carnival on Multimedia and Math, focusing on technology and connections. Check his announcement on Mathematics and Multimedia, and post to the carnival here.

Kate Nowak is writing about starting your own blog.   Given me some food for thought.  She links to Sam Shah's worthwhile article on it, too.

Knowing my love for games, Sue Van Hattum (Math Mama Writes) sent me the link to this neat Geometry Taboo game at  Teaching ninJa, which had some other neat posts, also.  Sue just posted/linked an awesome/awful video about the L-Curve distribution of wealth in the US.  Has me wondering what a reasonable or equitable wealth/income distribution would look like.  Is the L-Curve necessary in any way?

Michael Paul Goldenberg (no relation) reordered Dan Meyer's What Can You Do With It? series chronologically, instead of the usual blog reverse order.  Hope neither he nor Dan don't mind me sharing it here - he sent it out on the MathTalk mailing list.  MPG blogs at  It was some drudge work on Michael's part, but cool idea.  I had the list here, but he's since put it on his blog.  So go there.

There's kind of a standards based grading theme going.  I've got to figure out some stuff about that and how it applies to process goals.  Shawn at Think Thank Thunk and Kate at f(t) (where does she get the time?).

Comic from the often quite funny Chuck & Beans at the  Like the card company.

    Sunday, May 30, 2010

    Decimal Point Pickle

    I came up with a variation on what was already a decent game and got to pilot it with Mr. Schiller's 5th grade this week.  Esther Billings introduced me to the game she had found in the book, Nimble with Numbers by Leigh Childs and Laura Choate, Dale Seymour Pub., 1998

    The 5th grade came up with two names, Destination Elimination (which I like because it rhymes), and Decimal Pickle.  This suggested by a student who's answer for everything is pickle.  (I'm sure you know a student like that.)  But here, it reminded me of a childhood baseball game that none of the kids knew but kind of fits.  (The baseball game Pickle.)
     My Favorite Pickle

    Decimal Point Pickle

    Set Up:
    1.    2 or more teams or players.
    2.    Get a deck of cards and remove the Kings, Queens, 10s and Jokers.  Jacks stay in.
    3.    Each player or team makes a path with 10 spaces.  It can be straight and rectangles, or it can be curvy and circles, but it needs to have 10 spaces and a clear beginning and end.
    4.    Shuffle the cards.

    Playing:  Idea is that you’re going to fill in your path from small to big, flipping over cards to get possibilities.
    1.    On your turn, flip over a card.  If it’s red, flip over another card.  If it’s red, flip over another card.  But you never flip more than three.  If you run out of cards, shuffle up the used cards.
    2.    Arrange those cards to make a decimal number.  Jacks are the zeros. The smallest number you can make is .000, and the largest is .999.  Say your number.
    3.    Fill in your decimal number somewhere on the path.  But it can’t go before a smaller number or after a bigger number.  Your path has to start small and end big.  If there’s no place to fill in your number, you don’t. 
    4.    Winner is the first person to completely fill in their path, with all the numbers in order.

    1.    J ♥, 3 ♣.  You can make .03 or .30. 
    2.    5 ♥ hearts, so you flip 2 ♦, so you flip 7 ♥ hearts.  (You stop because you can’t have more than three.)  You can make one of .275, .275, .527, .572, .725 or .752.  Which you want depends on your path. 
    3.    Sample filled in path below.

    1.    Simpler:  Play where you always flip over 2 or 3 cards.
    2.    Play cooperatively.  Two players work together to fill in one path.
    3.    More complex:  Play with 10s, which fill in 2 places.  So 10 ♦, 5 ♠ can be .105 or .510.
    4.    More complex:  Play without the three card limit.  You could hit a 10 digit long decimal or longer!  (Pretty unlikely, but still…)
    5.    Make 12 space paths.
    6.    Play with Jokers as a wild digit.

    Teaching Notes:  As often with a new game I played me vs the class first.  It was clear that the blackjack-esque possibility of extra cards was exciting, and they quickly got the idea that it was a big advantage.  I didn't castigate anyone for saying "point two three" but often asked "so how do you say that number?"  I shared how I thought about getting numbers close together and they really ran with it.  In general hitting on lots of ideas about where to put numbers, how to divide up the path, etc.  In their 2 on 2 games, there was a lot of good discussion about strategy, how to leave space, and what they wanted to turn over.  There was a lot of excellent comparing of decimals of different length.  (One amazing discussion comparing .1 to .065)  Students got very creative with their paths and I was quite glad I hadn't brought any preprinted ones.  We actually wound up playing with everyday math cards, which thankfully came in black and blue.  Whew!

    If you give it a try, please let me know what you think.

    <--- 2nd="2nd" favorite="favorite" p="p" pickles="pickles">
    ---> PDF of the game.

    Thursday, May 27, 2010

    Glyphs to Data to Display

    Story of a 2 day lesson.
    In the preassessment for my 6 week math for elementary class, it turns out that the students were pretty strong on the basics of statistics measurements and displays.  So instead of spending a lot of time telling them what they know, we went right to data collection.  One of my favorite ways to collect data is a glyph.  The first teacher I saw use one was Char Beckmann, but I think there's an old Teaching Children Mathematics (er, Arithmetic Teacher) article about them.  [Found it:  Cartland, Patricia E., What's in a Glyph?, Feb 1996, 324-28] Definitely one of my main faults as a teacher is trying to get too much out of a lesson, so beware that here.

    Objectives:  TLW
    • collect statistical information
    • formulate questions
    • organize and analyze data
    • display data to address a question
    • consider what features of displays are effective
    • consider types of questions teachers ask
    • review how the next day's test will be evaluated
    Day 1: (40 min)
    Schema Activation:  what would you be interested in knowing about your classmates?
    They brought up music, sports, food, family background, and the like.  (I think of these as cultural identifiers.)

    Focus:  introduce the idea of glyphs.  The handout has this information on it:

    The circle on the other side of this page will be your face – but not the face you see in the mirror!  A face that tells much more about you…

    Hair:  Put a hair on your head for each person living in your residence.  Curly if they are 18 or younger, straight if they are 19 or older.

    Eyes: I purposely leave left and right ambiguous (like a mirror or a mask) because I want there to be issues in data collection, but
    • left eye - favorite pet :   circle-dog, oval-cat, triangle-bird, spiral-fish, X-exotic or other, crescent    don’t like pets       
    • right eye - favorite TV show:  rectangle-reality/game show, square-police/mystery, trapezoid-doctor/medical, rhombus-historical/documentary, kite-sports, Y-other (should be a chevron), crescent-don’t watch TV
    Nose:  Make a shape with as many sides as books you read for fun last month.

    Mouth  :If from outside the state, add a tooth for each year you’ve lived in Michigan.
    • Smooth line: from Grand Rapids area
    • Crooked line: from Michigan but not GR
    • Rectangle: from another state in US
    • Circle: from another country

    1)    What other characteristics of a face could we use to ‘store’ information?

    2)    What are some other categories of information that would be good or interesting to represent on a glyph?


    As a whole class they added categories for music (country, hip hop, chill, rock, classical, show tunes) as the left ear, food (italian/pizza, asian (incl. sushi), mexican, chocolate) as the right ear, and hobbies (sports, outdoors, shopping, games, etc.) as the eyebrows and came up with symbols -usually very pictographically, for each.  (It's okay for me that my starting questions are a bit boring,  as they ask about what they care about.  I don't have to do it for them.)  They completed their glyphs.

    Reflection:  look over the glyphs of the whole class and share their reactions.
    They remarked on how much they enjoyed making them, their interest in each others, and how cool it was to see them together.

    Day 2: (2 hours)
    Schema Activation: polish up your glyph (or make one if you were absent), move it to the back table, see what you notice about them as a group.  (Most people from Michigan, lots of pizza, music types remarked on, questions about living situation...)

    First we reviewed the types of elementary displays and covered any questions.  They asked about pictographs and boxplots.

    Teaching note:  I like talking about questioning with statistics anyway, but they were really interested in what Jo Boaler brought up about teachers' questions at the last book club, so the timing was perfect.

    Pick a column.  Pick questions in that column so their numbers add up to four or more.  Answer the questions. (Display) means that a display is required.  Make a poster of your answer that includes your justification.

    They collected data and dealt with interpretation issues, what to do with people who gave more than one answer, left/right, figured out pretty efficient ways to record their data, and started discussing display type.

    After the posters were mostly complete they were passed around, and each poster was evaluated by each other group using our communication rubric: 0, 1/2, or 1 each for clear, coherent, complete, consolidated and content.  (Created with Coffey so the consonants all alliterate appropriately.)  This is how their exams will be graded the next day, also.

    We came together as a group and discussed what makes for effective displays.  This is what they thought.

    We also discussed the question types and made connections with reading.
    • Literal.    Literal questions have answers that are found directly in the text or are answered by factual recall.  Examples:  How many people come from Michigan?  What was the main character's brother's name?
    • Application. Application questions require computation or processing to determine the answer from the information at hand.  This usually is considered more mechanical than involving conceptual reasoning.  Example:  What's the average number of books read this month?  How long was Bilbo's journey?  
    • Inferential.    Inferential questions require the answerer to create something unique, something that is implied by the information at hand.  Sometimes this is by combining prior knowledge and experience with literal information.  Inference may require students to imply, guess with support, or deduce.  Can be forward-looking.  Example:  How far would the average drive be from where people are from to Grand Rapids?  Why did the character do that?
    • Analysis.    Analytical questions may require synthesis of literal information with information from other sources.  They typically require justification.  Sometimes analysis is referred to as synthesis.  It revolves around examining the information in and from the problem and solution.  Reflective in nature.  Example: Is there a correlation between favortie food and favorite music?  Why did the character do that?

    Reflection:  Pick two of the following to address as a group.  Turn in your group response.
    • What issues came up in data collection?
    • How did you go from glyph to data?
    • What was a useful form for recording data?
    • How did you decide what graph to use?
    • Any decisions you would make differently if doing it again?
    • What do you see that was effective in other groups work?
    Typically people thought about the teaching implications of what we did, but also thought a bit about the effective display idea.

    Bonus:  better classroom decorations.

    He's Back!

    My favorite TED talk ever was Ken Robinson's message on How Schools Kill Creativity.  He's back with a call for learning revolution.  Bring it on!

    He connects education with finding what you love to do.  Crazy talk.  Awesome Lincoln quote.  Dry British wit.  Poetry.  What more do you want from 18 minutes? 

    Tuesday, May 25, 2010

    Teaching with Tech Links

    My colleague Paul Yu and I (with a lot of help from colleagues) are trying to put together a grant for equipping a classroom with current tech for math teacher preparation.  In the course of this we have connected with many resources.  Really I should have posted this earlier to get feedback from readers, but better late than never.  (Probably the inscription to be on my tombstone.  Actually... not bad.)

    Not to pander, but if you're a grant reviewer, welcome!

    I'll continue to update this post rather than add new posts.  If you have suggestions, I'd love to hear them.


    Just for Fun

    (The iPad Tango?!)

    Friday, May 21, 2010

    Math Teachers at Play 26

    Welcome to Math Teachers at Play 26!
    Coming to you from Michigan, the 26th state of the union, admitted on the 26th day of 1837.  (Did they do that on purpose?)  Of course this doubled the original 13 states, since mathematically, 26  is 2x6th Prime.  (Credit.)

    The US flag of 1837 is at left.  The 2 sets of 13 stars pick up the 13 stripes well, don't you think?  (Of course, things went south with the 27th state.)


    Case Ernsting wrote about Quines in the programming language Ruby at his MetaSpring Blog.  This is a quick bit of interesting logic about self-referential programs that he connects to other interesting mathematics, including Godel and Nerdsniping.

    jd2718 presents Puzzle: continued root (Part 1) and  Puzzle: continued root (Part 2) posted at JD2718, saying, "This might be a puzzle, this might be teaching, this might be advanced (advanced algebra)."  It's a good teaching tale of engaged students.  Jonathon writes a lot about many issues surrounding teaching.

    Hmm.  This will be the only Math Teachers at Play ever directly after a square and before a cube.  (Proof.)  Of course, in a 3x3x3 cube, only 26 cubes are visible, so you're really taking the 27th on faith.  I guess that makes 26 the third Rubik's Cube number?   If 8 is the previous and 56 is the next, what is the fifth Rubik's Cube number?  What is the closed form for the nth Rubik's Cube Number? 

    About teaching

    Iron sharpens iron.

    David Ginsburg presents An Important Point About Teaching Decimals posted at Coach G's Teaching Tips.  Just a quick note about saying your decimals correctly.  Have your students taught you any bad habits?

    Tracy Beach takes a swing at April showers bring Math Awareness Month 2010 and a new activity calendar posted at Dreambox Learning 

    Shana Donohue shares So, how much do I owe you? posted at The ZeroSum Ruler, saying, "My eleventh graders still struggle with adding positives to negatives. On the last quiz, the top student in the junior class solved a logarithmic equation incorrectly because she evaluated "x - 3x" to be "-4x". To combat this huge misconception, I created the ZeroSum ruler"  She'd love comments and suggestions.  Not sure where to put this - she is plugging a product, but seems to be really thinking about this tough content.

    Hmm.  The 26th US President was Teddy Roosevelt.  Robert Talbot, a math prof at a liberal arts college, wrote a post about what teachers can learn from Teddy's daily schedule.  Can you think of any other good Teddy math connections?


    NerdMom presents An enhanced Sieve of Eratosthenes posted at Nerd Family, saying, "Both our 8 and 6 year olds have been taking the long way around factoring (both fractions and multiplications) so NerdDad helped them make an enhanced Sieve of Eratosthenes."  (If you don't click you're bound to suffer some NerdGuilt.)

    Ian Byrd presents iPods, Apple Stock, &Authentic Math posted at Byrdseed Gifted Lessons.  He found some nice data and made it into an authentic problem (a la Dan Meyer) all while preparing students for a state assessment. 

    Cynthia presents Table TOP Math - make tops, learn math, have fun! posted at love2learn2day.  These look fun.  You have to write for the pdf, but she has video of the tops in action.

    Rivka presents Infinite evens and odds. posted at Tinderbox, based on some recent family investigations s'purred on by The Cat in Numberland.

    Ashley Allain presents Above Average Fun With the iPod Touch: Merging Gymnastics and Math posted at HyperHomeschool, saying, "This goal of this activity was to integrate math, technology, and our daughter's love of gymnastics creating an authentic learning experience."

    Denise Gaskins at Let's Play Math, and (hopefully) proud originator of this here carnival, shares Word Problems from Literature, noting "I’ve put the word problems from my pre-algebra problem solving series into printable worksheets. (The word problems are worked out and explained at their original posts.)She's been putting up some best-of lists for her blog that represent a nice overview of a lot of her writing.

    Hmm.  26 is an international organization promoting creative business communication.  So named because there are 26 letters in the English alphabet.  What would a display of the number of letters in different alphabets look like?  What would be the best type of display for it?


    Ryan O'Grady presents Wheel Spin posted at Maths at SBHS.  He took a disappointing result from a test and turned it into a lesson using video to gain understanding into the problem.  Very nice.

    Tom DeRosa presents Use a Dartboard to Review Geometry and Probability posted at I Want to Teach Forever.  Fun variation on the usual review games.  Reminds me that 26 is the score on darts that can incite mockery.  Because the dartboard has 20 between 5 and 1, a poor thrower will hit 5-20-1.  He also shares a do-it-yourself tetris set, that could easily be adapted to many of your own whiteboard objects.

    Guillermo P. Bautista Jr. presents Introduction to the Concept of Functions « Mathematics and Multimedia posted at Mathematics and Multimedia.  He's got some great exploded illustrations of cubes.  (This is the post to read if you're trying and at all stuck on the Rubik's Cube number problem I placed above.)

    Kitten presents Cops and Robbers *coughcough* posted at Kitten's Purring, saying, "Today for math, I did a graphing game on the computer called: Cops and Robbers."  This is a great example of self-extension.  One question leads to another leads to taxicab geometry ...
    Hmm.  The 26th amendment gave 18 year olds the right to vote in the United States.  How many people in the  country can vote today because of this?  I can't make Wolfram|Alpha tell me, although I'm pretty sure it must know.


    Advanced math

    John Chase presents Heat Conduction in a Rod posted at Random Walks, saying, "I built a little geogebra applet to graph the heat distribution throughout a rod. You can change the initial heat distribution function, ask for the distribution at any time, and change the temperatures applied at each end. The math requires an understanding of differential equations and fourier series, but the *results* can be understood by high school math students."  Nice support using technology to make content more accessible.  So hot it's cool.  (I just couldn't resist...)

    Fëanor presents Irrationality posted at JOST A MON, saying, "Another proof for the irrationality of sqrt(2)."  It's a historical proof by contradiction from early last century.  (Reducto ad absurdum just sounds like a Potter spell, though.)

    Edmund Harriss presents Hexayurts and African Villages posted at Maxwell's Demon, described as "Some thoughts on mathematical thought."  He finds math in an anthropology situation and then connects it to all sorts of other ideas, including fractals and lolcats.

    Host's Prerogative
    Some of my favorite blog posts from the recent past.
    Dan Meyer's TEDx talk has caused a furor quickly, even getting him on CNN.  I embedded it here because I was showing so many people.   Rare to see one item really move the conversation forward.  Thanks, Dan!

    I read everything Kate Nowak writes, because she's funny, honest and insightful.  Check out Fail for an example.

    The last host (MTAP 25) has a post about How to Teach Curiosity at Point of Inflection.

    Sue Van Hattum has been working more on her book than blog, but still found time to contribute a nice piece on reading a math book.

    Maria Anderson has her plenary address from the Michigan MAA conference up.  Tech to delight and excite.

    Hmm. Since there are 26 bones in the human foot, I tried my best to find something 26ish about the 2010 World Cup.  Best I could find was that it's the jersey number of deposed English captain John Terry.  Can a true futboler help me out with a soccer connection?

    By the way, I stumbled across a great source for number facts, including 26.  I had found or worked out many of this issue's factoids, and then found this resource - doh! (As a total aside, but on that page, Einstein had a great 26th year. "Albert Einstein (1879-1955), publishes 5 papers in Annalen der Physik (1905)  on the photoelectric effect, statistical mechanics, Brownian motion, special theory of relativity, and relationship between matter & energy: E=mc2" That might be the best year ever for an academic.)

    Thanks for visiting!
    That concludes this edition. Submit your blog article to the next edition using the carnival submission form.  Past posts and future hosts can be found on our blog carnival index page.

    Hope you enjoyed it!  Have a look around while you're here.  Besides Mr. Meyer's talk, I've been showing people this riff on the mac/pc ads for students vs. learners.  I'm asking for people to share their favorite math songs, and have been thinking about multiplying fractions too much.  So maybe you shouldn't look around...

    Best of the Spam
    If I included yours here and it is not spam, my deepest apologies.   Or if I somehow missed your submission, please let me know and I'll correct the situation.

    10 Essential Web Tools for Teachers
    Top 50 Special Education Blogs 
    Ways to Excite Young Readers

    Tuesday, May 18, 2010

    Book Club: What's Math Got to Do With It?

    In my math ed classes I've been trying to do some of our reading discussion in a book club style, as there's some good research support for teacher-run book clubs being one of the more effective forms of ongoing professional development.

    This semester my preservice elementary class is reading Jo Boaler's What's Math Got to Do With It?

    Today was the start, with them having read Chapters 1-3.  Through experimentation I'm currently wanting groups from 5-8.  (Haven't had a class of 9 students yet.)  After group discussions, wwe have a couple minutes for groups to decide what they want to share or what was important, and then we have a whole class summary.  Early on I randomly assign roles; later in the semester I let the groups choose.
    • Leader - responsible for overall discussion, uninvolved group members, checking on other assigned roles.
    • Questioner - if discussion stalls, posing a question
    • Recorder - Keep notes on what is said
    • Reporter - During whole class share responsible for representing their discussion.
    I also try to spur interactivity.  Otherwise groups just  give a report and then the next group goes.  One of the main student skills people learn is duck and cover.  If I ask a question then someone is going to ask me.  Since they will want their students to discuss, though, I try to connect coming up with questions or responses for each other to preparation for helping their students do the same.  I've also noticed that it helps make students into more active listeners.

    Today they noticed:
    • Get out of the rut of how you're thinking about teaching.  Don't have to teach like you were taught.
    • Old value of "If you don't know, then don't guess." vs how effective trial and error is as a problem solving method.
    • Examples vs definitions.  We know many more examples than definitions, but math can focus on definitions.
    • Math of a daisy:  never knew about that.  what else has mathematical qualities that we've never thought about?  Can math be integrated into other areas?  Connections are intriguing.  Engaging for students.
    • Tradition vs technology:  find a balance.  Need tech, because of its power and today's learners' expectations, but not always, because... (need skills, too soft, ... never really finished the thought.)
    • Math wars: parents going behind teachers' backs with little information.
    • Liked the book: good information.  Surprisingly light reading.
    • Balance between drill and math literacy.  Kids aren't comfortable with numbers/content; how do you get them comfortable with out drill and kill?  Activities vs. practice.
    • Why is math so awful in the US?  Employment statistics on how many more careers need math than the number of people we're graduating with math competencies.  How to interest students?

    It made them think about or wonder:
    • Communication with parents is crucial: newsletters.  Include all subjects, optional activities, involve parents.
    • Students teaching students in inquiry instruction.
    • Not right way vs wrong way, instead get studetns aware of the many ways.
    • Teach for understanding, not for practice.
    • What age level is this for?  
    • How do you assess to find out who's falling through the cracks?  Tests don't seem to always work.
    • Why couldn't we do conferring like Lucy Calkins?  Conferring.  Could elementary students use portfolios like we use.  Find their own problems.  (This started a discussion of MAPS testing and running records and DiBELs...)
    • When did we get scared of numbers?  When did math become scary?  How?
    • Kids need some practice.  But motivation matters.  Rewards?  Competition?
    • Our experiences.  If we had a question, the answer was often "look in your book."  The teacher needs in depth understanding of the content to lead a discussion.
    All in all, a lot of good thinking about math teaching in a short span.

    NEW 5/25:  2nd Edition of Book Club.  Students had their choice of Ch. 4, 5, or 6, and everyone read Ch. 7.

    Students noticed:
    • Ch 4 - America overtests compared to other countries.
    • Ch 5 - ability grouping negatively affects learning.  Low self-esteem.  So balanced teachig integrating students.  Leads to labeling. 
    • Nonhomogenous groups see and share more strategies.
    • Ch 6 - gender equality.  Female understanding vs. male memorizing.  Women can be more in depth learners.
    • Connect with reading differences by gender.
    • Questions.  As they get older, students ask less questions. Making sense of problems.  Incites thinking.
    • Ch 7 - Key strategies.  Lower achieving students focusing on procedures vs. concepts.  Math builds, so this is dangerous.  Connections make a knowledge cone.
    • Example: Four strategies: counting together, counting on, known facts, derived facts
    • Math can scar students.
    Students wondered about:
    • How to get parents involved?  Newsletters, etc.
    • How do they figure out who goes where?  Important to mix gender in groups.
    • Have to get students working with each other.
    • How to keep higher achieving students from doing all the work?  Stations, maybe, like in the Amazing Race.  Not everyday, but occasionally to teach how to learn together.  Everyone has to write.  Assigned roles, that switch each time.  (Science connection.)
    • How to create a question friendly atmosphere?  Maybe a question box.  Open time for students to come ask questions (before, lunch, after school)  Poker chips.  Each gets two chips and they have to ask at least two each day.  Emphasize questions are valuable.  Teachers ask questions that students might have. 
    • Why give struggling students more practice?  How is that supposed to help?
    • Awesome to do fun stuff everyday, but exhausting.  But on the other hand, you can do the same thing over and over until it's deadly boring.  Cross curriculum helps.  Make it memorable.
    • Beware the rut of teaching and repeating what's not working.

      Friday, May 14, 2010

      Quick Post: Wolfram Alpha

      From a gaming site:

      Background Research

      If you don’t care about where my information came from or how I compiled it, skip on over this part, but for the naysayers out there, this is what and how I did it. I’ll include links at the bottom of this article in case you kids at home want to try it out. In a rather complicated nut-shell… I referenced population data in Wolfram Alpha by state and county and compared that proportionally to the registered DCI users from the DCI player database online. I then cross referenced every county in the US via its latitude and longitude as recorded in the US Census from the year 2000 against the addresses for every National Qualifier event in the US and Canada as found in Wizards' online event locator. (The addresses found in the event locator were tossed through an online tool I found which converted them to latitude and longitude.) I then bounced all those coordinates though an Excel calculation to take into account the curvature of the earth and give me accurate distances (as the crow flies) to each event from every county. Once I had those HUGE data tables arranged and I had broken Google Docs spread sheet I reorganized the data into usable bits to tell me things like the minimum distance that someone in each county would have to travel to an event, and how that differs from the 2009 event schedule etc. I also weighted all the averaged distances by the affected population that would have to travel them.

      I am not teaching my students to use the available resources sufficiently well.  That is all.

      Thursday, May 13, 2010

      Multiplying Fractions, Times Three

      In a gig with Mr. Schiller's 5th grade last week, I was overcome with indecision.  They'd been working on fraction multiplication and I had three related activities, and asked the teacher to pick based on what he wanted for the class.
      The first was just skill practice.  (This is the option Mr. Schiller chose.)   I had never made it, but was confident that you could make a good fraction version of the Product Game.  This is almost what I made:
      The only change is that I originally left off 5/12, but the fifth graders convinced me that it should be on.  My thinking was that it would be nice if one unsimplified product was not available, creating a situation where players had to consider equivalent fractions.  But the game itself brought it up enough that I think it's unnecessary.  Click on the image for the full size image which should print properly.

      These students have been practicing fraction multiplication, and simplifying and 'unsimplifying' the result.  They have played the Product Game, which is the greatest math practice game ever.  (In the Connected Mathematics Project Prime Time module now, may be from the Middle Grades Mathematics Project before that.)

      I launched the game by playing me vs. the class.  Reemphasizing that you only get to change one factor at a time, the goal is to get four in a row, and the new idea that there are equivalent fractions.  If you multiply and get 6/12, you can cover 1/2, and vice versa.  Then the students played pair vs. pair.  At the end we summarized by discussing what they noticed about the game, and what they thought made for a good strategy.  I did point out to them that someone would tell them fraction division was hard, but they've already done it when they're figuring what to multiply 3/4 by to get 6/12.

      The second idea was to have the students develop the ability to make sense of their answers through a constructive representation.  Right now, I think the students are mechanically carrying out the multiplication, without much intuition to inform them if their answers are sensible.  These questions are adapted from an activity I do with my preservice elementary teachers.

      I like playing the video before the activity, but that is obviously optional.

      "Potatoes, mash em, boil em, stick em in a stew.” – Samwise Gamgee.

      Things are _______ (awful, bad, okay, good, great) because you have potatoes!  Draw a picture to justify each answer.  Write an equation or number sentence for each story, if you can.

      Find how many pounds of potatoes you’ve got if you search the cupboards and find…
      1)  1/2 a bag of potatoes, which started with 2 pounds of potatoes.

      2)  1/2 a bag of potatoes, which started with 2/3 pound of potatoes. 

      3)  3/4 a bag of potatoes, which started with 2/3 pound of potatoes. 

      4)  1 ½ bags of potatoes, which each started with 2/3 pound of potatoes.

      5)  1 ½ bags of potatoes, which each started with 3/4 pound of potatoes.

      6)  4 bags of potatoes, which each started with 1 1/3 pound of potatoes.

      7)  2 2/3  bags of potatoes, which each started with 3 ¼ pound of potatoes.

      8)  ____ bags of potatoes, which each started with ____ pounds of potatoes.
      (You make the problem!)

      The numbers are chosen pretty intentionally to allow for some connections and the possibility of relating the quantities to each other.  I like potatoes (of course) because they can be used for a discrete or an area model or a nice casserole.  My plan was to start with problem 2, demonstrating for the class a couple different models, and then have them start on number 1.

      The third option was to get at a new context for multiplication.  As anyone following the Keith Devlin multiplication fiasco knows, the prevalent contexts for multiplication involve repeated groups.  One of the other contexts that is often nice for rational numbers is the idea of stretching and shrinking.  That always puts me in mind of Alice, and how her terrific adventures began.

       Go Ask Alice
      “One pill makes you larger, And one pill makes you small
      And the ones that mother gives you, Don't do anything at all
      Go ask Alice, When she's ten feet tall” – Jefferson Airplane

      Sort of from “Using Alice in Wonderland to teach Multiplication of Fractions,” Susan Taber, MTMS, Dec 2006

      “There seemed to be no use in waiting by the little door, so she went back to the table, half hoping she might find another key on it, or at time she found a little bottle on it, ('which certainly was not here before,' said Alice,) and round the neck of the bottle was a paper label, with the words 'DRINK ME' beautifully printed on it in large letters.”  Alice in Wonderland, Lewis Carroll.

      It turns out that she drinks it, and shrinks to 1/6th her former size.  Now she later finds a cake…
      “She ate a little bit, and said anxiously to herself, 'Which way? Which way?', holding her hand on the top of her head to feel which way it was growing, and she was quite surprised to find that she remained the same size: to be sure, this generally happens when one eats cake, but Alice had got so much into the way of expecting nothing but out-of-the-way things to happen, that it seemed quite dull and stupid for life to go on in the common way.”  But soon, “Just then her head struck against the roof of the hall: in fact she was now more than nine feet high, and she at once took up the little golden key and hurried off to the garden door.”

      It made her grow almost 12 times larger.

      1)    What height would she be if at 10 ft tall she took a sip of on-sixth potion?

      2)    If she started at 5 ft tall, and then took a sip of one-sixth potion, how tall would she be?  In feet?  In inches?

      3)    If she took a bite of ten times cake and then a sip of potion, would she be the same height as what she did, which was take a sip and then take a bite?

      4)    Having had one sip and then one bite, how close can she get back to her original size?

      Let’s add to the story shall we?  Suppose she finds a times-three cookie, and a one-fourth soda.
      5)    Starting at 5 feet, what height does the one-fourth soda make her?

      6)    What effect would the times three cookie have, followed by the one sixth potion?

      7)    If she starts at 5 feet and wants to be 6 feet tall at the end of it, what should she eat?

      8)    In the story she wishes to pass through a 15” door.  If she had the choice of all four magic items, what should she do, starting off at 5 feet tall?

      9)    What other mixtures are possible with all four items?

      What does this have to do with fraction multiplying? 

      What did you learn from these problems?

      It's just such an amazing context, and that's without getting into the mushroom, which she uses for more controlled growing and shrinking later on.  Of course when my son is ready for these problems (soon, I think) we'll have to switch the context.

      He shrinks, too.  Perfect!

      Thursday, May 6, 2010

      Sing a Song for Making Sense

      I'm always on the lookout for a good math song.  Partly because I like the idea of math songs for immersion, but also because I'm just a geek and like them.  Thought I'd share what I've got, and hope that people might add their 2 cents.  Please?  Links to youtube where available, amazon or another source if it's out there.  Of course, this is music on the internet, so beware of piracy and anti-piracy.

      Best of the Best:
      What About 10?  - Dino 5, hippest counting song ever.  Even made our family's dance mix.

      Tom Lehrer:
      Could go in best of the best, but instead he gets his own category.  Just heard he stopped performing because he went back to being a math teacher.  There's a new collection of his work that includes a DVD.  Sounds like a must buy.

      New Math.  Greatest. Math. Song. Ever.

      That's Mathematics.
      What the heck, Werner Von Braun.  (Actual video of Lehrer playing.)

      They Might Be Giants have a whole album of math songs, one song each for important numbers, including the counting numbers, pi and infinity.  (Infinity's not a number, I hear you say.  But it is a cardinality... go ask Cantor.)

      Sesame Street:
      Obviously a great source of number songs.  Televison Tunes has many of these available - check out his Sesame Street songs.  My favorites are the Alligator King (7), King of 8, the Pinball Song, and, for the man himself, Song of the Count.

      Schoolhouse Rock:
      Again, at Television Tunes.  My favorites:  I Got Six (That's all there is), Elementary (Animals coming off the ark in pairs), and the phenomenal Three is Magic Number.

      Average Guy - Song of the Lakes (life is hard in the middle; way better than Joe Walsh's same named different song.)
      Statistician's Blues -Todd Snider (adult humor, but funny)

      Multiplication - Animaniacs, works through the standard algorithm for 47x83.  Which is somehow funny.
      Shape of the Sum - I Love Math (no, really)
      15 to 20 - Phenomenal Handclap Band (counting by 5s)
      1234 - Feist
      National Pi Day - TJ Zmina (to the tune of a Lehrer song; hence, better.)  Hat tip: Richard Zucker.
      One Was Johnny - Carole King

      Probably only math in my head:
      Multiplication - Bobby Darin
      Sdrawkcab Klat (Talk Backwards) - Steve Goodman, I use this to teach inverses and solving linear equations.
      1234567 - 1000 Robota, German punk counting. I don't speak German, so apologies if it's offensive.
      Seasons of Love - Rent.  (I had a student that knew this by heart and laughed when I asked them to find how many seconds in a year.)

      Math is a Wonderful Thing - Jack Black from the School of Rock
      Math Sucks - Jimmy Buffet

      Plea for help:  Denise at Let's Play Math! posted some calculus song videos, which combined with the new Tom Lehrer release that I heard about on Fresh Air  finally got me to do this post.  I haven't gotten too much into the homebrew stuff on YouTube.  If you have any suggestions for more math songs, please put it in the comments!  I especially love pop songs with accidental math.  I haven't used songs for memorizing ever, but I know some teachers love them.  Have you had a good experience with them?

      Someone needs to make this song.  Also, the song I'm alluding to in the title seems ripe for a math version.  "Sing a song of making sense, pocketful of why, four and twenty problems, one of them with pi..."

      Sunday, May 2, 2010

      Concentration Against Opposition

      Hosting the Play Date
      I am hosting the Math Teacher's at Play this month.  Please submit your own blog if you have something to share.  But also consider submitting something you've found valuable at someone else's blog.  It's the same form for either, and it just takes a second.  The link can always be found at the bottom of the right side column here.

      Complex Instruction
      I've shared before how NRICH is probably my favorite source for rich problems.  They also share bits of scholarly writing, host discussions and contribute in other ways to the math ed community.  This month, they've posted a wealth of resources on problem selection for group environments, leaning heavily on Jo Boaler's work with several of her articles or chapters; Dr. Boaler calls it complex instruction.  Take a look!

      Last and most oddly, I wanted to share some quotes from a Japanese gamer.  I play this game called Magic the Gathering, which is really the best strategy game ever invented.  (With the possible exception of global thermonuclear war, but you know the only way to win that...)  Players construct their own (or copy someone else's) deck of cards, and then face off against one or more other players.  It has a pro tour, and a small number of players who actually make their living playing.  The best analogy I've heard for it is a chess game with modifiable board, rules and pieces.  Each year the game expands and changes, yet remains simple enough for interested people to pick it up.  It has probability situations that will break your brain.  Frankly, it's amazing that I've gone this long without geeking out about it in this blog.

      The author, Tomaharu Saito, is one of the best players ever.  He is known for his concentration, and wrote an article on how to learn to concentrate.  I love when people share their understanding on how to learn, and really feel like that the best reason to teach math (for most students) is that it is such a good context to learn how to learn.  The full article is here; selected quotes below.

      I decided to write an article that would always be helpful to players, one whose ideas would not fade with time or format changes.
      One of the best reasons to write!  That's one of the reasons I like so much for my students to write.  But then we need to give them ways and opportunities to share.  I'm still working on that.

      In order to show 100% of your ability, concentration is crucial.  Frankly, as far as concentration is concerned, each person is different. And I feel like there are some people, although it may be only a handful, who can concentrate extremely well without training.
      However, this does not mean that most players are naturally sufficiently focused. I think that time spent training concentration can affect one’s ability to refocus when their concentration is broken. This will boost your deck’s power, and is itself a way of taking countermeasures against weaker decks.

      One of the things I am currently struggling with as a teacher is how to develop and support my students in becoming long thinkers, better able to apply themselves to significant problems, and to persevere through being stuck.  This feels relevant.

      When practicing concentration, I find it particularly effective to pretend I am playing in a real event and focus accordingly. If my only goal were to boost my concentration, it would be good to always play as though I were in a tournament, but I also like to watch deck development, tune my build, learn match-up odds, and work on other goals which can capture my attention and cause a distraction. Because this can also lower my efficiency, I have found it is not a good idea to always split my focus. It is for this reason that I make time to challenge myself to work on concentration.
      Remembering overall purpose, creating authentic conditions, assessing barriers to improvement.  Choosing to challenge yourself.  My teaching question is how to create purpose like that within the classroom.  You wouldn't do this for an exercise.  It needs to be relevant to your life.

      Also after reading a book or watching movies or television, make sure you are able to explain the subject to a third party. This is always effective in improving awareness. When it comes to explaining the subject to someone else after reading or watching, you need to have watched it carefully and tried to concentrate in order to pin down the main point.

      This is transfer!  He's talking about applying concentration in another circumstance.  The point about awareness in relation to concentration is new to me, but makes sense.

      There are various methods for training your concentration, but there are none that allow you to master the skill in a short period of time. There is a feeling that steadily bit by bit the skill increases, so persist in your attempts. 

      Also, one characteristic of concentration ability is that it is greatly affected by your every day life. In particular, people who stay up late should be attentive to this. If an individual does not spend enough time in sunlight various problems can ensue, and concentration ability is no exception.

      Clearly true.  He's talking about life long learning, and how lifestyle impinges on performance.

      Indeed, circumstances where concentration is broken and misplays are made are frequent, and naturally they make winning more difficult. I recommend self-confidence, but you can also learn my own method for recovering concentration.

      Have you heard of the Saito Slap?  If there is bothersome noise around me, I will slap my cheek hard to recover my concentration.

      He's not kidding.  He's not recommending everyone slap themselves, but instead talks about the usefulness of a routine that you can rely on, that by training triggers a response.  I think about that in my prayer life, but haven't thought about it for teaching.

      What are the times when it is easy to break concentration? When you recognize the times when it is easy for problems to arise, it becomes easier to cope with them.
      • The moment you think you’ve won
      • When you’ve lost a previous game due to a play error
      • When you’ve lost a previous round due to a play error
      • When you’ve lost a previous game due to a poor draw, mana troubles, etc.
      • When you’ve lost a previous round due to a poor draw, mana troubles, etc.
      • After some kind of trouble occurs
      • When something causes you to become irritated
      • When you think about things other than the match

      Perfect for an anchor chart!  I love the idea of thinking about ways you get stuck or when you give up on problems or...  What would this list look like in math class?

      Magic is a wonderful game. Right now, I am betting my livelihood on it, but even misplays and losses do not cost me my life. I don’t let other people get to me.

      If you relax a little, you can move forward.

      That deserves an amen for brother Tomaharu.

      Now to get more people to reflect metacognitively on their passions!