Life Among the Elms

Problem of the Week

One of the most important parts about working in math is being able to explain one's thinking to others.  Whether you're convincing colleagues of a budget line item, or working with your partner to buy enough food for the week, we need to be able to share our strategy.

Today's homework, the problem of the week, is designed to hone students' skills in this area.  We write about our math thinking often during math workshop but it is a skill that demands a lot of practice.  The problems are not necessarily meant to push a child's computational abilities, but to challenge them to explain their problem solving and math thinking.

For example, today's problem states:  Today I found 6 coffee cups in the Elms (true story!).  If I found the same amount for a week, how many would I find all together?

Now, you might find yourself asking right away, "Wait a second, does she mean a 7 day week, a 5 day week or just the remainder of this week?"  Any of those assumptions is "correct" as long as you explain what you chose in your solution.

We talked about using a rubric to help us categorize the detail in explanations.  Using a scale of 1 to 5, students could see very concretely what a "complete" solution was.  Here is what we came up with:

Level 1 - Simply the answer, or an attempt that shows no understanding of the math situation. 

e.g. (for this week's prooblem) "30"

Level 2 - A label or two.  Numbers that show computation.  An attempt that shows a weak understanding of the math situation.

e.g. "6x5=30"

Level 3 - Labels.  Computation for multiple steps.  Answer is labled clearly.

e.g. "6x7days=42   You'd find 42 cups"

Level 4 - Complete explanation but a computational error.  Solid computation and labels but more limited explanation.

e.g.  "6 cups x 5 days in a school week = 30 cups  Anwer:  30 cups by Friday."

Level 5 - Complete explanation and correct computation.  Pictures often add to explanation.

e.g.  "Step 1:  I decided to think about a school week, Monday to Friday that has 5 days.

Step 2:  Each day had 6 cups.  That's 5 groups of 6.  I used multiplication (6 cups x 5 days) and found there were 30 cups total

Answer: There were 30 cups at the end of the week (and one empty cover.)

Some students had great questions about the rubric, including asking if it was a grade.  We explained how using a rubric can help show how they can grow in an area.  They can find out in a very specific way how they can deepen their skills as a math thinker and communicator.

Candy Consultants, Wrestlemania and Greek Sieves

We are wrapping up a short unit on multiplication.  It began with an unexpected debate.  We had been doing a lesson on how to add, subtract and multiply with multiples of ten when the commutative property of addition came up.  Students knew that they could "flip flop" the numbers in an addition problem and, after reflection, they agreed that they could not do the same for subtraction.  Could you flip multiplication?  About a third of the class wasn't sure.  "Three groups of two is different than two groups of three."  The pro-commutative group tried to convince us but struggled until we looked at multiplication as an array, specifically, a rectangular candy bar.  Was the candy bar with three rows of two squares the same as the candy bar with two rows of three squares.  Everyone agreed that it was.

Sensing a rich vein, I became the owner of a candy company who wanted students to design the best box to hold 24 cube chocolates.  This gave students the opportunity to experiment with arrays and multiples.  Many chose to build their boxes out of graph paper - a great spacial exercise.

When everyone had a chance to present their ideas to the shareholders.  I explained that we now were facing a new challenge - we needed to find the most flexible number of candies so that we could change our box often.  Consumers like things to be "new," afterall, so we wanted to know the number of chocolates that would give us the most boxing flexibility.  The table the students created allowed us to explore primes and square numbers as well as see which numbers had the most factors.

Our multiplication minds warmed up, I introduced multiplication wrestling (which you may recognize from your algebra days as the F.O.I.L. method).  This is a multiplcation algorithm that emphasizes the "groups of" nature of multiplication.  Students can see how quantities can be broken done into smaller quantities that are easier to multiply and then reassembled to find the final product.  We used base ten blocks to create a concrete representation of what we were doing with the numbers on the board.  I was excited to see several students setting up their own problems with the blocks when we had open work later in the day.

Today we explored the Sieve of Eratosthenes, an ancient tool for finding prime numbers.  By crossing out the multiples of a prime, you begin to winnow down the remaining integers, slowly revealing only prime numbers.  (The link above is a slick interactive web version.)  Don't worry, we only went to 100 or 200, not the two million plus digit number that is the largest prime number discovered yet.

I hope this overview gives you an idea of how we are attempting to weave more concrete, computational math into authentic, engaging experiences.  By following students' ideas and needs, we help them construct the understanding that "math makes sense."

The Gourd Election Heats Up

The gourd candidates, Ovifera Peponapsis and Crescentia Alata, had their first debate on Wednesday.  The students prepared questions that probed the candidates views on everything from domestic "fertilizer" production (the student reminded us that "fertilizer" really meant "oil") to affirmative action (should warty gourds receive special consideration for jobs?) to health care (several of the gourds are getting a bit mushy in the stem).  Cathy, Elizabeth and I were thrilled with the sophistication of the questions and the rapt interest of the class during the debate.  The last question was, I kid you not, "How do we know we can trust you?"

Many of the students were surprised at how the debate affected them.  "Both of the candidates made a lot of sense.  I don't know what to think now," was a common response.  I felt it mirrored many voter's frustration when trying to tease out the positions of candidates.

I am playing Senator Ovi, the Free Gourd candidate, and was a bit nervous going into the debate.  During our party caucus, only two students joined my table.  I asked them which of the platform planks they found most convincing.  They actually disagreed with much of the platform but had joined the Free Gourds because "the other tables were too crowded" and "well, I'm married to her so I figured I better vote the same way."  Things weren't looking too good.  Luckily, I was able to slip in a few zingers during the debate and the polls (taken just this afternoon) are showing a statistical dead heat with 30% of voters still undecided.

The other teachers were interested in joining in the gourd election and I proposed this to the class today.  They were very skeptical.  Most felt there was no way that the younger kids were ready to understand the issues we had been studying like health care and the taxes.  Others brought up that some kids couldn't read the ballot.  The concerns gave us a great opportunity to talk about ways Americans have been kept from voting in the past (poll taxes, literacy tests.)  We decided to do a test poll of the 2/3s to see if they are aware enough of the issues to be ready to learn about the Poligourd and Free Gourd platforms ( Download Gourd_parties.pdf.)  Others started work on a TV special about the elections.  A third group began an aggressive poster campaign.  Tomorrow, we'll work on presentations about the candidates and issues that we'll present to the Tamaracks (and perhaps the Maples and Pines if they're deemed ready by the polling committee).

I hope that your dinner table has been filled with some good gourd politics conversations.  I'm going to be writing a press release this weekend, and welcome any comments or insights that you have.

Taking a Closer Look

I spent Wednesday afternoon in my son James's classroom.  I wasn't volunteering; I was observing.  James attends Montessori Children's House, and part of the Montessori philosophy is to observe children as they work.  Both parents and teachers are encouraged to watch children carefully in order to learn about them.  When I was there, I tried to disappear from the children's awareness.  I sat in one place and avoided making eye contact.  If a child approached me, I would nod to acknowledge them and then shift my focus.  They soon forgot about me and I had the opportunity to truly see my son and the environment in which he spends most of his day.  It was wonderful.

I hope that you'll take the same opportunity in the Elms.  I invite you to come in and just watch.  How does your child interact with our classroom world?  How do they settle into their work?  How do they overcome difficulty?  You may not have any epiphanies, but you'll certainly come away with a little more understanding -- even if it's why those sushi erasers are so important.

And I'm sure I'll learn from you, too.  Elizabeth Treat, the uber-volunteer who has been spending three days a week with us, often "just" watches during activities.  Our conversations have brought many things to light for me -- I don't get a chance to be invisible in the Elms as much as I'd like.

So...please come.  I'll put together a little sheet with things to look for and tips on how to "disappear."  Let me know when you're coming so I can make sure we don't have an unusual schedule that day.

See you soon.

Dinosaur Math

I've been doing a lot of thinking about math recently and you can expect to read about it fairly often on this page.  Teaching math can feel like a balancing act.  On the one hand, computational skills are vital to math literacy.  They must be taught and practiced.  On the other hand, the beauty of the mathematics and its connection to the natural world is also crucial.  Without exploring the wonder of math, it is dull and lifeless.  Without knowing some foundational pieces, one is blind to the wonder.

The lessons of the past two days illustrate how we try to nurture both elements in our students.  Yesterday, we did a pretty straight forward lesson on place value.  We looked at how our place value system works and why it is an efficient system that had advantages over abandoned systems like Roman numerals and Egyptian glyphs.  We briefly hypothesized why Base10 might be so popular with humans (10 fingers?).  We practiced reading big numbers.  We learned how to place commas in numbers.  We used place value to easily add large numbers together.

Today, based on a brief conversation yesterday with G.S., I asked the class how a creature, say a T-rex would count.  After all, it has only 3 fingers.  I'll admit to a lot of roaring and stomping as I, the T-rex, became more and more frustrated trying to write a note to my mother in which I wanted to tell her that I had eaten four velociraptors.  The only digits I had were 0,1,2,3.  How could I write the quantity four?  Suggestions abounded and eventually, we hit upon a place value system but, instead of a group of ten being the next place value, four would be.  At this point, about 10% of the class was really understanding what was going on.  I added 13 more velociraptors to the original 4.  Now how could I tell my mom?  A new place value was needed, four groups of four.  We practiced with a few more values and more heads started to nod.  The dinosaur was feeling pretty good about things.

Another student then pondered, "What if you had a different number of fingers?"  Instantly, I announced that we had landed in Hexadeciland -- home of a species with 16 fingers.  They, of course, had a system with 16 digits -- 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.  Kids giggled as they announced, "I'm going to be A this year!"  "My brother just turned F!!!"  Someone asked, "How old would my Dad be in Hexadeciland -- he's 54"  We figured it out and then figured out his age in Dinosaur Base4.  Students were howling to discover that I would be 203 in Dinosaur.  More hands shot up and we figured out ages for siblings, grand parents, great grandparents.  At one point, a student was jumping up and down saying, "I get it!  I get it!"

We ended by imagining what Gourdland would count in.  After all, they just have one stalk -- so it must be Base2!  (That makes me 100011, by the way.)

At different points, I re-assured students that they should feel very confused.  Everything they had learned since pre-school was based on their ten fingers, after all.  I also assured them that we were just playing and that they weren't responsible for being able to do this by themselves.  But they left the room laughing and imagining different people they'd want to figure out in dinosaur.  It was all just a little bit crazy -- and wonderful.  Our goal for the day was accomplished.

The United States of Gourd

As you may have heard, the Elms recently adopted some gourds.  They've been describing their gourds, measuring them, and creating personas for them.

Predictably, some gourd weddings are already planned.  Gourd clothing has been made.  Cars have been designed.

When I saw the gourds at the farmers' market, I knew right away the stir they would cause in the Elms.  Fourth and fifth graders still love imaginative play -- but they sometimes need permission to engage in it.  A class wide project grants that freedom and it's a wonderful thing to watch.

The gourds also called out to me for another reason.  I've been searching for a way to approach this November's elections in a way which avoids the shallow partisanship that often mars elementary units on elections.  Passions run high but students aren't prepared for informed politcal discourse -- things devolve quickly, especially when I'm not there to referee.

But a presidential election is too important an opportunity to ignore.  I became a teacher in large part because I believe in the importance of an informed, critical electorate.  What to do?!  GOURDS.

On November 4, there will be an election in the United States of Gourd.  We'll be reviewing the structure of the government - the Sentate, House of Gourds, Supreme Gourd and the executive branch.  I'm planning on re-writing with the some students the platforms of the Repbulican and Democratic parties with gourd-centric planks.  There will be debates and history lessons as the gourds get ready to vote.  I'm inviting you to join the fun -- perhaps you'll be a guest lecturer on the Gourd Bill of Rights.  Maybe you'll share something on gourd civil rights (some are smooth...some are bumpy...).  Maybe you'll want to create the Statue of Gourdity for the classroom.  We'll be working out a lot of the details together as the election nears.

The Internet and Your Child

On Thursday, the Elms and Tamaracks signed an Internet contract (Click on link for text).  This is a very important document for us and we take a lot of time with the students explaining our expectations for computer and Internet use. If students purposefully break the contract, they are not allowed to use computers while they are at Prairie Creek.

We believe that the Internet is an invaluable tool in a progressive classroom.  It allows us instant access to amazing information, stoking the fires of curiosity.  There is so much power in being able to celebrate a child's question by stopping everything and finding an answer.  The Internet gives us that ability.  But, sadly, the Internet is also a place filled with commercials, bad information and false identities.

We speak to students directly about what to do if an inappropriate site appears on a screen.  We have filters on our computers but we teach the children how to do a search as though there were not filters.  This prepares them to use the Internet safely regardless of the setting.  We insist that they use a "kid friendly" search engine (all of which are linked from the Prairie Creek site) and that they go only "one step" from these search engines.

We insist that children protect their personal information and never give out their name, age, city, address or phone number.  E-mailing is only allowed with an adult, using the adult's account.  Children are not allowed to check or write personal e-mail while at school.  We are frank about the number of people pretending to be somebody else on the Internet.  Students have a surprising number of stories to share about this in their own lives.

Finally, we ask them to promise not to use the Internet at home or at school to hurt people.  It is too easy to write hurtful things in an e-mail and we urge students never to write anything they wouldn't say in person.  We also insist they never write anything anonymously.  We talk to them about cyber-bullying and how students use social networking sites to gang up on individuals.  I hope by talking about these things early, we can help students recognize unhealthy patterns when they arise.  Perhaps if they are armed with knowledge, they will have the courage to stop it from happening.

We hope you will use this opportunity to have a long conversation with your child about computer use.  Set your own expectations for their computer use at home and at friends.  We use a written contract to emphasize the size of the responsibility -- you can do the same thing at home.  Many parents have come to me after their child has left PCCS, wishing they had set limits earlier, before the technology use came to be seen as a right by their child.

I have a lot of resources about Internet and technology use and children.  Please don't hesitate to ask questions if you have them.

Our Prairie Home

As Sarah and I folded up the tents on Monday, I reflected on how many people chipped in to make the campout go smoothly.  Every time I turned around it seemed someone had seen a way to help out and had just jumped in.  Tables got set up, cereal was doled out, stings were doctored, four-square games policed, the fire was built, kids were ferried to activities, and tables were taken back to school.  Thank you.  Your generous and willing spirit is the same one we hope to encourage in the Elms and Tamaracks.  Many hands do indeed make lighter work.

Special thanks to Chris and Dan Kelley who opened their home and prairie to us.  Their unique location allowed us to explore very different aspects of the prairie during our day together.  I asked the students what they learned about the prairie and their responses were deep and varied.  Gary Wagenback also gave generously of his time -- an exciting wagon ride was followed by a lesson on prairie soil composition (did you know it's like playdough when it's wet?)

Thank you to all drivers, chaperones, campers, snack providers, and paper good provisioners.  I was truly touched by your eagerness to help out.  I really enjoyed the day.