The Rookery

The Many Ways We Math

A recent exploration of factors

There is a lot out there about literacy and the importance of a literature rich classroom and home.  There is not nearly the same amount of cultural focus on numeracy - and yet number underlies every aspect of our lives -- even as I type my computer has translated my keystrokes into hexadecimal numbers that correspond to pixels on an xy axis. (Woah!)

 

We do math every day in the Herons. We do math in a lot of different ways.  We often do math that the Herons don't realize is math.  I hope that students are able to use number to explore and understand their world.  I hope that they like to play with number.  But I also hope that they develop mathematical habits of mind so that they see math as a process of observing, making conjectures, testing and re-thinking.  Communication is key.  "Failure" is important because it is only through testing a conjecture that we can make it stronger. 

One of the challenges of teaching math is that, in order to discover patterns, make conjectures and talk about math, students need a certain amount of facility with computation and vocabulary.  It's not enough to just explore.  Students need to solidify those explorations and be able to use that knowledge efficiently to push their learning ceiling higher.

To help facilitate that process, our math program has many parts.  I realized that we sometimes don't talk explicitly about the purpose of each element and wanted to take the opportunity to do that:

Exploration Math (often called "Illustrative Math") - This is when we work with children to use numbers, algebra, data, graphing and geometry to make se

A record of our class conversation about the factor exploration.

nse of the world.  We focus on a math concept such as fractions for 2-3 weeks.  The Illustrative Math curriculum guides our work and we supplement it with problems and teaching strategies from many other sources including Marilyn Burns, YouCubed, Nrich, Math Recovery and Think Math.  During Exploration Math time, we use games, open ended problems, and other math experiences to build students' understanding of a mathematical concept.  We do a lot of work that emphasizes talking and writing about math to solidify our own understanding of concepts and master a shared math vocabulary.   Once the concept is solidified, we often introduce algorithms so that students can use their understanding effectively when solving more complex problems.

Foundation Math - This is an almost daily practice session when students are gaining ease with computation.  We focus on understanding algorithms and using them accurately to get answers to computational problems.  While calculators are an invaluable tool, we also want students to be able to solve problems accurately and easily using paper and pencil.  In addition, working problems regularly helps information "stick," solidifying students' memory of a skill.

Fluency Math - It is really important that students know some things in math automatically (Here is a document that explains the "why").  By fourth and fifth grade, we are focussing on multiplication fluency.  We use a variety of tools to track our ability to recall math facts automatically.  Initially, students work on multiplication facts in this order:0s, 1s, 2s, 5s, 10s, 11s, 9s, squares, and then the "last 10 facts" which have not been memorized in the preceding sets:  3x4, 3x6, 3x7, 3x8, 4x6, 4x7, 4x8, 6x7, 6x8, 7x8.  Students then "flip" their multiplication knowledge to master division and finally, common percent/fraction/decimal equivalencies (e.g. 1/5=20%)

Number Talks - We often write a problem on the board and have students solve it mentally then share their pathway to a solution.  This encourages math communication and flexible thinking. It also shores up number sense.

Math Problem of the Week - a multi-step problem in which students are asked not just to give an answer but to explain how they arrived at their answer.  The focus is on explaining the process of solving and it develops students' ability to communicate about math succinctly and clearly.

"Recreational Math" - A lot of elementary math focuses on computation but that is only a small fraction of the world of math.  We take the time to introduce students to some of the biggest questions in math in a way they can understand.  The Game of Sprouts is an example of this kind of math.  Students love exploring these new ideas and, even though some of the math used in explaining will not be taught to them for years, it's valuable for them to be excited by these big math ideas.  (Click on the link to find out just how big an idea our little math challenge was.)

Serendipitous Math - Part of my job as a teacher is to look for ways to pull math into our daily classroom life.  Math conversations can pop up in meeting.  Impromptu class surveys help us crunch data efficiently.  A current event might give us a chance to noodle on the numbers involved (just how much snow does it take to make a giant snow shark?)  Math is everywhere and it helps us understand things.

Whew.  It's a lot of math and, when we do it right, it's exciting and engaging and real.  I very rarely have a student ask, "When are we going to use this?" because they are using it right now.

I will write more soon about the ways families can support numeracy at home.  But I wanted to get the conversation started.

Becoming a Big Bird

The Herons at work during guided recess.

When students travel upstairs and become fourth graders in Prairie Creek, we call them "Big Birds." Their role in the school changes significantly and they are held to very high expectations -- especially when we are with younger classes.  These first six weeks we are very directly teaching what it means to be a big kid at Prairie Creek.  For the most part, the rules around games (gentle tagging, call yourself out) and the woods (stay 10 feet from the creek) were learned long ago.  But now they are the keepers of the Prairie Creek way -- they are the ones who show the younger students what it means to include and care for each other.

During guided recess, we focus on how to connect to younger children. We even have a "Bird Watchers Guide" that's like a field guide to the kids they'll find at recess. Our older students learn to get down to eye level and be gentle in their voices.  They learn how to play tag gently enough to be safe with smaller children. They learn how to reach out to shy kids or kids who have been left out.  They learn how to constructively guide bossy kids and how to manage arguments about missing goldens in the woods (it's always the squirrels who took it).

They learn many kids' names and learn the importance of using those names so that kids feel seen and known in our school.  They also learn the power of their kindness.  By the very fact that they are a "big kid" they can bestow "coolness" on a child who might be left out of a game.  They can make that child feel like the center of the universe just by paying attention and being kind.  Conversely, a harsh word is amplified far more than they might realize so they must manage their impulsivity and think twice before saying something unkind.

Big birds are expected to care for younger students throughout the school day.  If they see a younger child in the hall, they say "Hi" with a smile.  If someone is fooling around in the bathroom, the older students are expected to remind them of the correct behavior and guide them back to their classroom.  If we are at All School Gathering and a younger child is having trouble listening, their Big Bird buddy reminds them gently with our silent signal.  For some kids, it wasn't so long ago they were the ones receiving the gentle reminder!  It's a big step to be the one doing the guiding.

Big Birds have additional leadership opportunities at Prairie Creek.  They often organize a student council and lead school-wide activities, service and celebrations.  Last year, the Big Birds founded an environmental club and with the help of Ryan and some St. Olaf students they build a hoop house. That group will continue their work this year.

The transformation these first weeks is astonishing.  The hesitant almost third graders who walk through the door on the first day become fully fledged Herons (sorry for the bird pun).  They are ready to lead and they are also ready to have more freedoms - the other benefit of taking on more responsibility.  It is truly an exciting time.

The Herons led a gentle game of "Mirror, Mirror"

Connecting one on one build the community

Serendipity and Learning

It is very difficult to write or speak succinctly about progressive education but, when I'm asked to, I talk about authenticity.  Teaching progressively means that the students feel that what they are learning is real and means something.  There are many ways to create authentic moments in the classroom, but one of the most joyful is serendipitous learning.

Serendipity is defined by Oxford as "the occurrence and development of events by chance in a happy or beneficial way."  Serendipitous learning is learning that happens seemingly by chance (more on that "seemingly" in paragraph 7).  We discover and explore something together as a class.  The unexpected has happened and we feel a need to figure it out.  An example from this week:

As we settled in to read aloud in the meeting area on the first day, a drop of something fell on a student as she came into the meeting area.  "Something dripped on me!"  Suddenly all eyes went up.  "That tube has lots of drips on it!"  "Is it leaking?"  "Is it water?!?" I asked the students to share what they noticed.  The drops were in a line at the bottom of the tube.  The tube had an air vent in it so where was the water coming from?  I asked them if the tube was going to be hot or cold - almost everyone thought hot.  We sent the tallest Heron up on a chair to test, "Cold!" She reported.

Hmmmm. The windows were open (even with the air conditioning on, I like to encourage fresh air circulation) and it was a warm humid day.  I asked the student to hypothesize where the drips were coming from if it wasn't leaking out of the tube. "The air?"  What do you mean? "Like rain comes out of the air." There was a lot of excited chatter as the Herons considered this.  Several students volunteered how the air has water in gas form in it -- I drew up a diagram on the board to illustrate what they were talking about.  "So how does the water get from gas form to drips?"  Eventually we got to thinking about temperature - it's very cold high in the sky where clouds form and the tube was cold to the touch.  I shared that warm air can hold a lot more water than cold air in gas form and the Herons has an epiphany.  The tube was cooling off the air and the water was coming out of it. "CONDENSATION!" one student yelled.  Another followed with evaporation.  We drew the water cycle on the board.  "We have our own mini weather system in the Herons."  So cool.  Now on to the read aloud.

What is this?!

One of the reasons we began forest school was because of the way it engenders serendipitous learning.  We never know what we are going to find outside.  Just this Wednesday, we discovered three different kinds of bumps on the underside of oak leaves.  What were they?  iNaturalist said they were galls.  I gave a quick mini-lesson on what galls were (fifth graders were already on the lookout for goldenrod galls and the tasty grubs they hold).  These galls were too small for us to find a grub.  What could be in there?  Students build a bar graph out of the leaves they were finding separated into the three different kinds - there were a lot more white fuzzies.  Then we saw a small solitary wasp on one of the leaves.  Was it laying eggs? How could we tell?  (It flew up too high to observe, sadly.)

Another group called us over the the magnolia tree.  It has strange globby growths at the ends of the branches.  Were these galls?  One of them seemed to be splitting open and had a bright red thing popping out.  A seed? We dissected it to learn more.  Students remembered that the flowers were at the ends of the branches and that flowers often turn into fruits...this seemed to be a very strange seed.  "Can we plant it?"

Ironically, serendipitous learning doesn't actually just "happen" even though it feels that way to the class.  As a teacher, I am constantly watching for the opportunity to bring something unexpected forward.  I try to create an environment in which we are noticing things all the time and I'm ready to ask the questions that bring us from noticing to understanding.  The class, too, has to learn to go with the unexpected.  Sometimes we're even conspiratorial about it, "I know we're supposed to be doing this but this other thing is amazing.  Let's go that way for a while and then get back to our "supposed to."  (A secret? A surprising number of our serendipitous journeys actually are standards based (one has to really know the standards to teach progressively because you are always watching for the best moment to bring something forward.))

I, too, have to be learning along side the class to make the learning truly authentic.  I'm modeling curiosity and, importantly, an openness to not knowing (yet).  In a very short time, curiosity becomes the capital of the classroom - that's who we are. The more practice we get as a class finding the fascinating, the easier it is to discover things to be interested in.  Serendipity is what has kept me in the classroom all of these years - I never know for sure exactly what the day will hold and I wouldn't have it any other way.