Shooting Fish in a Barrel

     Earlier this month Graham Fletcher and Joshua Greene got together for a tag team take-down of a textbook lesson on finding the volume of rectangular prisms:

It never really had a chance.

    Their analysis and make over suggestions are spot-on.  And I hate to pile on, but what makes this especially egregious is that exploring volume of rectangular prisms, a major grade 5 content standard, can be, well, fun!  Rich and I have had success with activities described here and here, and I'll offer up another one that we tried out with the fifth graders last year.
     Andrew Stadel has written about the potential that starting arguments in math class has to  reinforce both practice standards (especially numbers 1 and 3) and engage kids in learning, and I agree.  For this project, I collected five boxes and asked the kids to guess their order from smallest to largest as measured by volume.  I didn't want the boxes to be too similar, just similar enough that it wouldn't be too obvious.

I chose these five...

...and gave the kids time to get their hands on them.

     

Argument started!

 With the intellectual need now built, it was time for the kids to resolve the dispute.  They first had to estimate the volume of each box...

...then find the necessary measurements.  We decided on nearest tenth of a centimeter to reinforce working with decimals.

We let them use a calculator because who wants to do all those calculations by hand?

The data was recorded.

    And, due to faulty measuring, improper rounding, and incorrect number crunching, the argument continued to rage!  In fact it took several days for the class to come to agreement on the volume of each box and the correct order.  But they were days filled with engagement, collaboration, discussion, and multiple content and practice standards.
    The activity isn't all that imaginative, and it's not that hard to prep for.  All you need are some boxes, rulers, and calculators.  And while it's not as easy as asking the kids to take out their books and do this...

.  ...I guarantee you'll have more fun.

   
   
 

A Post About Counting Circles

     "Children need repeated exposure to and practice with counting sequences in order to become fluent with counting.  Counting sequences help children understand relationships between numbers and further develop their abilities to to apply these understandings to problem solving situations."

Jessica Shumway

Number Sense Routines, pg. 56

     This year, our PLC is digging into Jessica Shumway's wonderful book:

Thanks to my principal for finding room in the budget to order multiple copies.

    Comprised of teachers grades 1,2, and 3, we are on a mission to both critically examine our preexisting math routines and explore new possibilities.  We have tackled the Calendar and Counting the Days in School routines, as well as the Quick Image/Dot Card routines that are now a part of our Everyday Math curriculum.  Shumway's book has been an invaluable resource, providing background into learning trajectories and pedagogy as well as practical, how-to advice and student work examples. 

     Shumway devotes an entire chapter to counting routines.  These routines have been a part of our curriculum for many years, commonly used as the warm-up to a lesson.  But direction on implementation and usage is pretty vague, as illustrated by the following example from the grade 2 Teachers Manual:

    As Everyday Math vets we are also used to seeing children engage in counting routines in their journal work:

Can you spot the error?

    We asked ourselves some tough questions, using Shumway's book as a lens through which to view these routines with fresh eyes.  Were we using the routine in a thoughtful way?  Or were we just doing it because it was there?  Were we in fact even doing it at all, or did the routine often end up on the cutting room floor due to lack of time?  Were there ways we could make the routine more engaging?  More meaningful?
   Shumway offers a host of different ways to engage students in a counting routine.  (See here for more detailed explanations as well as a video clip of a counting circle in action.  Scroll down to #2, Counting Routines.) Counting circles where each class member adds to the count individually, choral counting, start and stop, whole class, small group, kinesthetic; Shumway explores many variations. She provides questions that help facilitate discussions about patterns, place value, number sense, and strategy.
    After a lively discussion, and after engaging in a counting circle of our own, the teachers were ready to give it a try:

Shumway emphasizes the importance of planning.   Kristin, after determining her second graders needed practice counting by 10s in three-digit numbers, sketched out her idea for recording the class count.  Different patterns will emerge depending on how the count is recorded.  She  decided to record vertically, starting a new column when the digit in the hundreds place changed to highlight this particular trouble spot.

She recorded the count, and asked the class, "What do you notice?"  The kids responded, and Kristin had the opportunity to reinforce concepts of place value in the changing numbers.  There are also opportunities to explore the different strategies individual students employ to add 10.

Also from grade 2.  Jane decided to have her class count backwards.  Counting backwards is often neglected.  Notice how she has elected to record the count horizontally.

Maggie, another grade 2 teacher, knows that her students will be asked to count by 25s on an upcoming progress check.  She uses the routine to solidify the concept and also as a formative assessment.

 

Nicole uses an interactive hundreds grid with her first grade class.  She often has her students do jumping jacks as they count.

Larissa, grade 4, wants in on the action:

She plans to have the class count by a fractional amount, and has some questions ready for her class to think about.

What do you notice?  What are you wondering?

     "Counting sequences," Shumway asserts, "Help children understand relationships among numbers and further develop their abilities to apply these understandings to problem-solving situations."  Proof of that came last week:

Shannon's third graders were working on this task...

...and realized that skip counting would help them find their way to a solution.  I felt this was a good example of a counting sequence application to a problem-solving situation.

Larissa's class collected data on the circumference of their heads...

...and then organized the data in a line plot.

Counting by 1/2s from 49 to 56.  A perfect grade 4 counting circle task.

     It's been fun to watch teachers experiment with this routine.  The only way to learn how to do it is to, well, do it!  It won't be perfect and it might get messy, but that's OK.  Do what works best for you and your students.  Here are some reflections based on my observations around the school

• The teacher needs to be alert to student participation and engagement.  If the class is counting one at a time, once a child's turn is over, what will keep him paying attention to what's happening?  Conversely, if the class is counting chorally, the teacher needs to be aware of any students trying to hide behind more able classmates.  Those students would be good candidates for a small group counting circle.  
• When conducting a counting circle, it is not uncommon for a student to get stuck.  Shumway outlines moves a teacher can utilize in this instance, but unless a class culture of support and encouragement is in place (something Shumway also addresses in her book and Sadie Estrella emphasizes in her counting circle Ignite talk), the moment can turn into one of embarrassment and anxiety for the student in question.
• Planning is important.  What will you count by?  Where will you start?  How will you record the count?  What questions will you ask?  The more thoughtful you are, the more you will be able to get out of the routine.  And you will still be surprised by what students notice.  So be ready!
• It's important to keep your eye on the clock, especially during the discussion time.  Student interest can start to flag, and while there still may be ground to cover, if students are not engaged then nothing worthwhile can be accomplished.  In other words, keep things moving!
• Counting circle routines provide opportunities for students to employ practice standards.  Many students quickly realize that finding the emerging pattern that arises within the count will help them come up with the next number in the sequence, a good example of SMP 7 in action.  SMP 8 comes into play when the teacher asks questions like, "What number will the last person say?  How do you know?"

     As I've learned from hard experience (see here and here), there's more to counting than meets the eye. "Students who struggle with mathematics,"  writes Shumway, "Often lack counting skills." Shaky understandings about place value, underdeveloped abilities to recognize patterns, poor estimation skills, misguided notions about the way our number system is organized, the inability to think in additive and multiplicative ways; well planned and executed counting circle routines target all these areas, as well as promote critical thinking and problem solving skills.  What's not to like?

       

What I'm Looking For

     Last year Jo Boaler's multiplication task How Close to 100? caught my eye. I decided to bring it to our grade 3 teachers, and it quickly found its way into the rotation.

Roll two dice.  Use the numbers as factors and generate corresponding rectangular arrays.  Place them on a 10 x 10 grid, and see how close you can get to filling it up.  I liked color coordinating the equation with the array.  Colored pencils worked best.

     Although designed to build fluency with basic multiplication facts, we found the activity had other benefits.  In order to maximize your chances of filling up the grid, rectangles needed to be placed strategically.  Also, in order to calculate your final "score", you needed to do some computation.

Students experimented with different computation strategies, from adding up all their products to subtracting the total number of blank squares from 100.

The activity provided inspiration for a number talk:

     And when we experimented with a 20 by 20 grid, and dice that would generate higher factors, things got a bit more difficult...

And a little messier...

   

We also challenged the kids to find different ways fill up the grid using exactly 10 rolls:

     The rectangles came in quite handy when we got to studying area and perimeter:

There's really no escape from the scissors.

We had them cut out the rectangles, find their perimeters and areas...

...and sort them based on their relationship.

     I'm often asked, "What do you look for when choosing a game or activity to bring to a class?"  Once it passes through the first, most important test of might a kid who doesn't like math find  this engaging, I look for its potential to be extended or repurposed.  There's something about the idea of  becoming familiar with the way something works, then using that familiarity to build on or connect to something different, that appeals to my sense of how we grow and learn.  How Close to 100? is a good example of an activity that checked off those boxes.

"I Like This Game Because You Have to Think Hard."

    A Monday afternoon, working with one of my second graders, helping him fill in a blank hundreds grid.  Suddenly it hit me.  What if...

...using different colored pencils, players alternate capturing squares on a hundreds grid...

...while trying to get 4 squares in a row, column, or diagonal.

It would work  just like tic-tac-toe.

Each tic-tac-toe is worth a point, and you can keep track with tallies.  Play until the board is filled up.

   Theresa and I played a practice round:

Initially we tried 3 in a row, but things got very confusing and we decided that 4 in a row was better.  We also tried a game where a square could be used in multiple tic-tac-toes, but didn't like that either.  We also decided that the 0 and 100 would not count towards any player's tic-tac-toe.

And then it was time to take newly christened Number Grid Tic-Tac-Toe for a test drive.  (Apologies if someone holds the patent on this!)  Jane's class was first, and we decided to get crazy and use a grid that went from 1 to 120.  It took only about 5 minutes playing a demo game with a volunteer under the document camera and they were ready to rumble.

"Hmmm.  Where should I go?"

"I think I can figure this out in my head."

Playing it safe.

Using a number grid to help is so SMP 5.

A barn-burner.  You can see here that 34 was used twice, as was 46. 

   Over the next several weeks I tried it out with 3 more second grade classes and 1 third grade class.  I asked each to write down their reflections and reactions on the back of their game boards.  Here's a sample:

• I liked this game because it was fun to figure out where the numbers go.
• I like this game because you need strategy.
• I loved it so much but hard.  So hard!!!
• I think the game helps because to put the right number in the right place you have to count by numbers and learn to do that in the game.
• I liked it because I outsmarted (my opponent) when he tried to outsmart me.
• I like this game because you have to think hard.

The kids came up with some awesome suggestions for modifying the game, including:

• If someone gets stuck, the other player gets a point.
• If you get three in a row you get half a point.
• Add a few numbers scattered around the board.  (Love it!  This child wants it differentiated!)
• If you get five in a row you get 2 points.

Jane's class experimented with a 3 player game:

They found that 3 in a row was better with 3 players.

Maggie's class tried something a little more difficult:

"What goes here?"

Many possibilities...

If it's green's turn, where should he go?  Why?  Defend your answer.

     I'm happy to report that the game is wildly popular; for some reason it hits that elusive kid sweet spot.  I suspect it's because the cognitive demand is just right. (I like this game because you have to think hard.  Unspoken: But not too hard?)  And because the kids already know how to play tic-tac-toe, they can expend their mental energy on strategy and on figuring out which number goes in what square without worrying about a bunch of rules.   The teachers love it too because it's easy to explain and takes almost no effort to prepare.
   So grab a blank grid and some colored pencils and have some fun!
 
   
   
      

Standards for Mathematical Practice and the Cinema, Part 3

    In the first two installments of the series Standards for Mathematical Practice and the Cinema, I  explored how SMP 6 (Attend to Precision) played out in two classics: This Is Spinal Tap and March of the Wooden Soldiers.  I'd like to turn my attention to a different practice standard, SMP 5 (Use Appropriate Tools Strategically), and one of my favorite movies of all time, Hoosiers.
   The film, released in 1986, is loosely based on the true story of tiny Milan High School's Cinderella run to the 1954 Indiana state basketball championship. It stars Gene Hackman as Norman Dale, a disgraced, temperamental ex-college coach seeking redemption...

Barbara Hershey as Myra Fleener, a small-town teacher with unfulfilled dreams...

And the late, great Dennis Hopper as the alcoholic assistant coach "Shooter" Flatch.

   The Hickory High basketball team responds to Dale's unorthodox coaching methods.  They make an improbable march to the state basketball championship, and face increasingly more difficult competition along the way from schools much larger than their own.  The team is used to playing in their small, claustrophobic high school gym...

...and are visibly shaken upon entering cavernous Hinkle Field House, on the campus of Butler University in Indianapolis, where the championship final is to be held.  Afraid that his team will be intimidated by the size of the arena and become mentally psyched out before the game even begins, Coach Dale employs a wily teaching move:



So was the tape measure responsible for Hickory's last second 42-40 win over South Bend Central?  If so, it played a very small part.  Because we all know who the real hero was:

Jimmy Chitwood!