Let's Go To the Movies!

     If you haven't checked out the lessons and activities at Robert Kaplinsky's site, you should.  That's where we ripped off the highway sign activity, which I described here.  And don't miss his blog,  which is very thought-provoking and deserves a read.  Earlier this year Rich and I adapted his movie theater project and turned it into a week long problem solving exploration for our fifth graders.
     We started by giving the kids a folders with the following information from the local multiplex:

The first page included pricing information, movie times, ratings, and summaries, and the capacity of each theater, which Rich found out from the manager.

The second page had the rest of the movie listings.

The last page had the food prices, which Rich copied down from the theater.

We asked the kids to take a look.  They started reading each synopsis, talking excitedly about which movies they had seen and which ones they wanted to see.  They laughed about "Jackass Presents: Bad Grandpa".  They whispered about the "R" rated movies.  They debated the merits of seeing "Thor: The Dark World" in 3D.  They drooled over the snacks.  Rich and I exchanged a look.  The hook was baited, now all we had to do was reel them in.
   I decided to use the strategy I had tried last year with Shannon's fourth graders.  They would create the problems themselves, and then solve them.  We borrowed a phrase from their ILA teacher: "thin questions vs. thick questions".  We encouraged them to write some questions that were straightforward and could be solved in a relatively easy manner (thin questions) as well as more challenging, complex problems (thick questions).  They started on their own, then met in small groups to discuss and refine their questions.

Here's an example of two questions from a student notebook.

I had to include this one, which provoked a very interesting "discussion" between the question's creator and another student that I was fortunate to capture on video.  The student insisted that the problem made no sense.  No one would pay to see a movie, leave halfway through, and then go to see the second half of another movie because that would be a "waste of money".  The student who created the question tried to explain that the scenario was not supposed to be taken as anything that might realistically be expected to happen; he just thought it would be a challenging problem to solve.  The other student would have none of it, and this back and forth went on for a good 5-7 minutes.

Groups got together and put their most interesting questions on chart paper.  Rich and I looked them over and selected about 15 for the class to work on.  They varied by topic and skill level.  I encouraged Rich to post them around the room.  The kids had several days to look them over before they were asked to dig in. This time, we assigned each question a letter and asked the kids to list their top 3 choices in their notebooks.  We gave them time during class to tour the room and take a careful look at each one before they made their choices.

The following day they entered the room and saw large, blank pieces of construction paper and markers along with the questions arrayed on desks and in corners all over the room.  They were told to get working on their first choice, but that if more than three people were already working on a problem, they were to try an alternate choice and come back later.  We also provided post-its in case kids wanted to provide comments.  Some worked alone, but most solutions were the result of collaborative efforts.

This was problem C.

These kids tried to find the difference by subtracting and got the wrong answer.  This led to an interesting discussion about why adding and subtracting time is different then adding and subtracting whole numbers.

This group used a number line.  Their work was correct but the answer is wrong.  In the middle post-it the group explains that their answer was a "typo".

This group also used a number line, which they termed "useless".  One of the students who answered the question plotted it out and revised their comment on the number line to "not useless".

This was question F.  It involved calculating with decimals. 

This group was close.  When they went to find the total, they copied $87.75 as $87.00 and were off by $0.75.

This group added incorrectly and was off by $10.00.

A correct answer!

My favorite: question A.  The reason: great example of a problem with a big vertical scale.

This student added up the costs of each individual item in each meal or combo package, found the total cost, and then calculated the difference between that and the price of each package.  It took him two days.  Talk about perseverance!  He claimed that for one of the choices you'd be better off buying each item individually.

A very basic response.

Also basic, but this student has a notion that perhaps better value may be found in another option due to the relative sizes of the drinks and foods that are offered.

The student who wrote the question also provided an answer.  I'll let his work speak for itself. 

  As the experience progressed from beginning to end, Rich and I engaged in a continuing conversation about what was happening.  Our reflections:

• We felt that this was an effective model for setting up conditions where students can be engaged in both traditional and non-traditional problem-solving scenarios. 
• Motivation and engagement came from: student-designed questions that originated from a high-interest topic, and student choice regarding which problem(s) to solve.
• Mathematics arose from a need, not the other way around.
• Low barrier to entry and plenty of room to scale up.
• Tremendous amount of opportunity for communication; both verbal (in the discussions students had during the question formation phase and also during the problem-solving phase) and written (in the way students express their solutions on paper.)

We liked this "movie theater" project so much that we started contemplating another, similar experience for later in the year.  Great Adventure anyone?

Lots of questions

I've been carrying around a copy of the Common Core math standards for several years now.  It's all annotated and dog-eared from use doing summer curriculum revisions, and from all the times I've been in classrooms, seen something taught (or taught something myself), and asked: was that really necessary?  I thought I knew them well, but you learn something new every day.
  Monday we had a math specialists meeting and one of our tasks was to check and revise our grades 1-5 end- of- year tests.  While revising and aligning the grade 2 assessment I came across this:

• CCSS.Math.Content.2.G.A.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

I did a double-take, and checked the rest of the grade 2 standards.  This was the only mention of anything related to fractions, it was appearing in the geometry domain, and there was no formal symbolism.  No numbers, no numerator and denominator, no fraction bar, only descriptive language.  Words.
I tracked back to grade 1 and found this, again in the geometry domain.

• CCSS.Math.Content.1.G.A.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of,fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Again, describe using words.  Then I looked at grade 3 and found this:

• CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
• CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
• CCSS.Math.Content.3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
• CCSS.Math.Content.3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
• CCSS.Math.Content.3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
• CCSS.Math.Content.3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
• CCSS.Math.Content.3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
• CCSS.Math.Content.3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
• CCSS.Math.Content.3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

¹ Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, 8.

So in third grade fractions has its own domain, with 7 separate standards, that treat fractions in a very formal way.

So this has got me wondering:

When we teach "fraction readiness" to first and second graders, should we stay away from formal symbolism altogether?  Was that the intention of the common core writers?  Would focusing exclusively on the words improve a child's understanding of fractions, and how might this be related to what Dan is talking about here ? (read the quotes he pulled out from Pimm's book)

Are exercises like this:

From our grade 1 math journal

and this:

Grade 2

and this:

More from grade 2

helpful or counter-productive?
And even if children are able to complete these pages (and others like them) correctly, does that indicate real understanding, or is it an example of Gardner's "correct answer compromise"? 
And for very practical purposes: What has to happen in grade 3 to help children make what seems to be an enormous leap towards a very formal understanding of fractions?
Lots of questions.  Any thoughts?

What Would Happen If We Took a Problem Apart and Put It Back Together?

Ben Blum-Smith's challenge to, "describe and (jargonlessly) name concepts for thinking about task design" got me thinking about an activity my colleague Shannon and I tried last year with our fourth graders.
  After watching Dan Meyer's TED Talk  Math Class Needs a Make-Over, I couldn't get out of my mind the part where he shows a problem from a text book and then visually strips away all the elements until he's down to the graphic.
  What would happen, we wondered, if we took apart a problem like that and then asked the kids to put it back together?  We decided to take a grade 4 PARCC prototype, remove the question and some of the information, and present it to the kids as, well, just what it was: an incomplete question.

     We asked them to imagine what the question might be.  They brainstormed some ideas in their notebooks, and then got together in groups to put down some of their questions on chart paper.  Some were basic and pretty much what we expected.  But many were complex and needed additional information, supplied by the kids, in order to solve.  This was something we did not anticipate, but once it started the floodgates opened and all this math just started pouring out!

Some of the original questions. They  were first posted in the room; later they went up on a bulletin board.

Next, we gave them the same problem, but with the rest of the information.  Still no question.

Again, they were asked to take a guess as to what the question might be.  They started getting closer.

Here's an example from the second go-around.  

And another.  

Again, we were surprised at the complexity of the questions.  There was a high level of engagement; as long as they didn't have to figure anything out they thought they were getting away with something!
     Finally we gave them the whole thing:

They went to work immediately.  They had spent so much time with the task they had internalized its parameters.

  Then we realized we had this whole pile of really interesting questions, and that it would be a shame for them all to go to waste.   And even better, they were kid questions!  We did some culling, made sure there was variation in the problem content, and that there were some easy, medium, and hard ones.  We taped them all around the room, and told the kids to spend a few days looking them over and thinking about which ones most interested them.  We then put big sheets of construction paper and some pencils and markers under each question and let them have at it.  Some kids wanted to solve the problems they had written, others were interested in solving ones their friends had written.  Interestingly, some questions could not be solved because the authors had not included enough information.  Those had to be revised.
     When all was said and done, we agreed that it was a very productive problem-solving experience.  It had a low barrier to entry, it scaled both horizontally and vertically, and had a high engagement level.  Shannon and I agreed that one of the reasons for that was because of the student-centered nature of the project.  The questions were created by the kids, and they had freedom to decide which ones they wanted to tackle.  Kudos to Shannon for turning her class into a "math lab" and for using the lesson again this year with a new group of fourth graders.  Any help with a name to describe this activity?

62

62 is my new favorite number.  Ironically, if you add its digits and double that sum, you get my OLD favorite number. What can we say about this most amazing number?

• 62 is between 61 and 63.
• 62 is even.
• 62 is a composite number.  Its factors are 1, 62, 2, and 31.
• If you reverse its digits (26) and add, you get a palindrome: 88.  If you add those digits you get my old favorite number.
• 62 has 6 tens and 2 ones.
• 62 is a repdigit (222) in base 5.
• 62 is the only number whose cube (238,328) consists of 3 digits each occurring two times.
• 62 is the code for international calls to Indonesia.
• Sigmund Freud had an irrational fear of the number 62, but I don't.  It's my new favorite number.

Why is 62 my new favorite number?  Sit back and watch.

Posting and Toasting

     A few weeks ago Michael Pershan asked us to blog about our favorite non-education related internet site.  So here's mine: Posting and Toasting.  It's a blog for the suffering New York Knicks fan community.  Seth Rosenthal, who posts game summaries, other Knicks news, and pictures of interesting lizards, fish, and birds, is spot-on hilarious.  And I even found something I could use in school. This from his recap of last night's train wreck vs. the Sixers describing the complete lack of Knick defense:

"The rotations are like some sort of fifth-grade math word problem. If Iman thinks Carmelo should be guarding Evan and Carmelo thinks Tyson should be guarding Evan and Tyson thinks Iman should be guarding Evan, then who is actually guarding Evan? NOBODY. NOBODY EVER. "

     And I can't sign off without acknowledging the originator of the phrase "posting and toasting"(along with spinning and winning, moving and grooving, slicing and dicing, bounding and astounding, wheeling and dealing,stumbling and bumbling, and shaking and baking, as well as other memorable turns of phrase), the former Knick great and current color analyst:

Walt "Clyde" Frazier

     And my daughter and I got to meet him at his restaurant, Clyde's Wine and Dine, after a Knicks game last year!  He's just as cool in person as he is on TV.

Big Inning

Sometimes very subtle shifts in what we do as teachers lead to profound changes in our classrooms.  I was reminded of this again earlier this week.
 Monday morning I sat down with Rich to talk about the next (grade 5) unit, which would involve lots of adding and subtracting of fractions with unlike denominators.  We knew that being able to find common denominators was going to be a key skill.  I shared something I had used with students in the past: a multiplication/division facts table in a plastic sheet protector with an expo marker and an eraser.  He had some tables already laminated.

Kids can find common denominators by highlighting multiples...

...uncovering the ones they have in common...

... and identifying the lowest.  Simple.  

In the past we would have handed out the tables and told the kids what to do.  We would have modeled the procedure and practiced examples together.  But we decided to do something different.  What would happen, we wondered, if we just gave them the tables and asked them to figure out how to use them?  We decided to find out.

Here's a copy of their task:

As soon as we saw what was happening to those tables, we knew something important was going on.  Some groups started checking off all the multiples.  They got confused about which ones went together.

One student, I'll call him Gabe, came up with something that looked like this:

We looked at Gabe's chart under the document camera and argued its merits.  Students agreed that Gabe had a good idea; it was easy to understand and use.

This is what Alan's table looked like.  He created a system that involved marking multiples with a "shape code".  We looked at that under the document camera.  The kids had fun trying to figure out what Alan had in mind.  Alan had a big smile on his face as he stood by the smart board while his classmates analyzed what he had done.  They agreed that Alan's system was pretty cool!


 After a quick demonstration of how to use the common multiples to make fractions with common denominators, the class went off to work with their tables.
    As we wrapped up the lesson, we had the kids reflect on the usefulness of the tables.  Maya remarked, "I used Gabe's special technique to find common denominators."
 To understand what this means, you would need to know that both Gabe and Alan are two of our basic skills students, and Maya is one of the brightest students in class.  Had we just explained to them how to use the tables, they would have had more time to practice the skill, but no time to engage in a problem solving activity that required them to come up with a logical way to organize their thinking.  Sam's and Alan's approaches to the problem would not have seen the light of day.
 As the class filed out for lunch, Rich whispered, "Did you see how Gabe got all puffed up when Maya told the class about how she used his special technique?"
  Not every lesson can be a home run.  But a whole bunch of little singles can add up to a big inning.  Ask Gabe and Alan.

 

That's Gold, Jerry! Gold!!

   Readers know what a big fan I am of Andrew Stadel's estimation180.  I've blogged about it here and here.  But the more we dig into the site, the more amazed I am at how rich the activities really are.  Here's an example from last Thursday.

Class opened up with the activity from Day 55.

What's the capacity of the cylindrical vase?

First of all...

• What's a cylinder?
• What's capacity?  How do we measure it?

     OK, now it's time for the number line.  Kids have gotten used to this routine.  They've already got their notebooks out and are working on their boundaries (too low, too high) and their "just right".  They still think it's funny that the teacher is actually asking them for wrong answers!  We have a general discussion of strategies.  Where are your numbers coming from?  It's now automatic:  they use the can as a referent.  They use the other vases and glasses and containers they've seen on previous days to inform their estimates.  They describe the attributes of the vase.  They compare.  They imagine.  They agree with each other.  They disagree, too.

     Next we start the video.  We pause it at the 14 second mark.  Time to evaluate estimates and revise.  This is important: the original estimate is not crossed out or erased.  Your response was not "wrong".   The new estimate must be placed on the number line. 

The video is started again.  Another stop at the 21 second mark.  (Mr. Stadel has disappeared!  Where did he go?  Let's make an inference!)  Time to reevaluate and make a final estimate.

Now we play it out to the end.  Cheers, applause, groans.  The answer is 41 ounces and that is also placed on the number line.  General discussion about strategies.

The original estimate was actually the closest!

     We're ready to move on to the day's lesson (something to do with polygons), when a normally quiet girl in the back of the room pipes up, "So how many cans of soda would fit in the vase?"
Great question!  What do you guys think?  Turn and talk and figure it out!  The vase holds about 3 1/2 times as much liquid as the can?  How do you know?

     15 minutes has gone by since the beginning of class.  I'm breathless.  Instinctively I know that a lot has been covered in that short period of time, and none of it was contrived.  It was all embedded in an engaging activity that all students could access.  (Later I counted 4 separate content standards and all 8 standards of mathematical practice.)  As the kids put their notebooks away, Jeff turns to me to me and says, "That was more worthwhile than anything else we could have done in those 15 minutes."
 I turn and respond with what has become our signature line this year:

Fun and Games in Fifth Grade

Taking some type of content and asking kids to use it to create a board game is not a revolutionary idea. But organizing an entire unit around it was something we had never tried, until this year when Rich came to me with the idea. He wanted to try it out with his fifth graders.  We decided to use the project not as a culminating activity, but as an ongoing part of instruction in a unit on fractions and decimals.
  Here was the plan: after teaching a lesson on, say, converting mixed numbers into improper fractions, the kids would practice the skill by creating question cards to be used for their board game instead of doing pages in a journal or worksheets.  The answers would be put on the back of the cards so the game would be self-checking.  They could then work on constructing their game boards.

                                                                                                                       

A typical card.

     We liked the idea that the kids would be learning and practicing the skills not because "you need to know this for the test", but because they needed them to be able to create questions and answers in order to make their game, as well as to be able to play the games that their classmates would create.
    Some of the game boards that got constructed were very plain and simple; others were much more elaborate and required lots of thought, planning, trial and error, and compromise (and yes, there were some heated arguments).  But every group managed to come up with something.

This game was called "Monopoly Crazy Cackle".

This game had a "Box of Doom".

This game was called "Toilet Twister" and featured a toilet bowl (lower left corner).

This group decided to make their own game tokens.

It was easy to land in the "Devil's Snare" but difficult to get out, as I learned the hard way.

This board had a zoo theme.  It was very colorful. 

They were given a day or two to "test drive" their games: fix any inconsistencies or correct wrong answers on the backs of their cards.  Some groups realized that their games were confusing and did not really work, so they had to revise them as best they could.  Tuesday and Wednesday they got to play each other's games.  We decided to leave one group member behind to help explain the rules and generally referee, but as an active participant. 

As we reflected on the experience, we realized there were some problems that we would need to address for next time.  Among the most significant:

• Groups should have been required to submit their questions on paper to be checked before being committed to a card.  There were some confusing and off-topic questions and wrong answers.
• Game boards needed to be checked more carefully to make sure that a minimum amount of questions had to be answered. Some of the games, while quite thoughtful and creative, did not include enough question-answering in their play.
• Kids needed to be held accountable for their work answering questions (Thanks Theresa!)  After the first day we did require the students to record their questions and answers on a separate sheet, but by then it was a little too late.
• Give the groups big manila envelopes to store their cards and boards instead of folders.  This was a suggestion that came from one of the kids; every time his group went to get their folder to work on their game, all the cards fell on the floor!

With all that being said, we felt that the project was a net win.  There was a tremendous social benefit, lots of collaborative problem solving, space for creativity and imagination...and all during math class!  We asked the kids to reflect, and the response was overwhelmingly positive.  One student described it as, "Tons of fun!"  Another one called it, "Imaginative."  And my favorite was the student who gave it a score of "499.99/500".  What math teacher wouldn't like that rating!