Organic Math: Part I

I’ll come clean: as a kid, I didn’t get math.  I was generally able to follow instructions and produce the answers my teachers were looking for, but on the whole it seemed totally arbitrary to me.  I was never quite sure how it impacted me and what it had to do with my day-to-day life even though I was constantly reminded of its importance with phrases like “math is everywhere” and “mathematics is the universal language.”

I don’t think I am alone.  Rote learning of mathematics was extremely common for my generation, and while I can only speak for myself, it solidified math as a subject that has to be learned instead of one that we should want to learn.  It is extremely difficult to encourage children to be curious and excited about learning something that their teachers feel negatively about.  Part of feeling good about learning math alongside preschool students involves reshaping the way we think about math as educators.  We can use our memories of top-down, standardized math curriculum as a counterexample to create a more inviting environment for our students to learn an incredibly important subject.

I feel that Loose Parts provocations are a fantastic place to start.  I’ve written before about them here on Without Windows in a more general context, but the purpose of this post is to provide specific ways in which the students of our school have learned to love mathematics through natural loose parts play.

Implementing nothing more than a few tools and natural surroundings, it is entirely possible to excite and stimulate the mathematical minds of young children.  What’s more, by learning math through relevant experiences, kinesthesis, and local flora and fauna, their discoveries will be more significant and rooted more deeply in their memories.

Leaves

Leaves are everywhere, but we tend to take them for granted.  They make fantastic learning materials in spring, summer, and fall.  If you choose to use them in the spring or summer, they don’t last very long once picked; some curl up and die in just a few hours.  Because they wilt so quickly, getting outside in the morning and picking them fresh can make for an excellent morning provocation.  Leaves also come with their own “dotted lines” to tear and cut along, and many have tissues that are so delicate even the tiniest of fingers can rip or snip them.  Regardless of age, they make an excellent material for the exploration of both simple and complex geometry.

At our school, we love to cut them.  Because they are so delicate, our students need to move slow and really pay close attention to how they are using their scissors to produce their desired results.  Many children start by trimming off the perimeter of the leaves to create familiar shapes, especially squares.  We love the leaves of honeysuckle because they are rounded at the base and pointed at the top, making it easy for younger students to make a triangle (by snipping off the tip) or a circle (by rounding out the base).

Over time, our older students discovered that instead of cutting shapes free-hand, they could use the natural structure of the leaves to facilitate their endeavors.  By folding the leaf in half along the stem, they found that could create a perfectly symmetrical square with only three cuts.  They were able to make an equilateral triangle in only two cuts. Eventually, they graduated to using familiar shapes to create more complex images, such as smiley faces (two eyes made from diamonds and a crescent-shaped mouth).  They even explored folding along other lines of the leaf to create symmetrical shapes away from the center.

On many occasions our students have even moved on to transforming their shapes into other component shapes, such as cutting a square into two right triangles, or cutting a trapezoid into a rectangle and two triangles.  A few of our older students have tried their hand at cutting out both capital and lowercase letter without breaking the perimeter of the leaf, resulting in both a letter made out of a leaf and a negative copy of the letter in the center of the leaf.

Logs

I have always been a huge proponent of woodworking with children.  Students need to be able to envision what they want to transform wood into, but the process of transforming a tree into wood they can use for their projects can sometimes be the most rewarding part of the activity.  While they work, students have countless opportunities for the exploration of standard and non-standard units, multiplication, and fractions.

To many of the boys at our school, the prospect of sanding, hammering, chopping, and sawing is enough to get them giddy with excitement.  Once we assemble a motley crew of lumberjacks, we venture out into the woods to find a suitable tree.  They like to select honeysuckle because on a rainy day (with enough bending and yanking) they can take down a large tree with the root cluster still intact.  Nothing makes a cadre of little boys feel mightier than pulling up a tree with their bare hands.

Next, they use a tape measure to decide how long they want the blocks to be.  Usually, we measure the entire length of the tree first.  We discuss potential unit length and use multiplication to determine how many blocks we can expect to produce when we are finished cutting.  A tree segment 24 inches long will make twelve 2 inch blocks, or 8 three inch blocks.  Our students tend to choose a length somewhere between 2 and 3 inches.

Processing the log requires teamwork.  They lock the tape measure at their desired length.  One student then moves along the log, while another uses a pencil to mark the agreed upon units.  Finally, we saw the lumber to produce the blocks.  When the cutting is done, we count the blocks to make sure our measurements were correct and evaluate why our yield may have been less or more than we expected (e.g. measuring a curved piece of wood with a straight measuring tool, or choosing a non-divisible block size).

Once the blocks are made, splitting becomes the activity of choice.  The students stand the blocks up, then hold a metal wedge along the diameter of the block.  When they hit it with a rubber mallet, the block splits in half.  They often compete with one another to see who can split the blocks with one stroke of the mallet, cheering and fist-pumping when they do.  One student has been known to refer to blocks that go flying off the table when split as “sploding blocks” or even just “sploders” for short.

Students continue this process to transform halves into quarters, quarters into eighths, and even eighths into sixteenths.  The boys at our school are in agreement that “quarters are cool.”  These fraction pieces made from a single blocks allows students to explore adding fractions by attempting to recombine them into a whole once again.  They can recreate a block by using two halves, one half and two quarters, four quarters, three quarters and two eighths, and so on.  The more they split, the more complex and variable their combinations become.

Holes and Water

Holes, puddles, and water are magical because, regardless of where you teach, they are abundant.  Whether on a playground or in the middle of the woods, water can create some incredible opportunities for good old fashioned fun.  Water is water; we often get ours from a rainstorm the night before, but water from a hose is just as effective in engaging students.  Children love to jump in puddles, splash each other, and frolic in the mud.  Gardens and patches of exposed dirt change form with the addition of large amounts of water.  Where muddy water pools, children tend to gather and investigate.

For our students, digging holes has been an integral part of our investigation of volume. A few months back, our class dug a deep hole (which we affectionately referred to as The Mud Pit) that became a gathering place for investigations of math and science.  When our students added water to the hole the first time, it disappeared, creating some frustration and prompting questions about where the water had gone.  After discussing saturation point, they recommitted to filling the hole with water.  As the continued to add water to the hole, it eventually started to accumulate and the water level began to visibly rise.  With the addition of a yardstick, they began to determine how many inches the water level rose with each bucketful of water they contributed.

When the hole was full, the time came to chuck stuff into the water.  As they threw in small and large items, they continued to monitor the water level.  The addition of too many large rocks caused the puddle to overflow even though no additional water had been contributed.  When one student added a favorite stuffed animal to give it a nice mud bath, the water level decreased much to their surprise.  They figured out that the bunny absorbed water instead of displacing it.

With their trusty yardstick, students were able to determine roughly how much water each item displaced when added to the hole.  Additionally, conversations about hole shape and size enticed students to explore the effect of different holes on their experiments.  Over the following months, they discovered that narrow holes have a water level that rises much faster than wider holes.  They were surprised to find out that it takes more buckets of water to fill a huge shallow hole a deeper one, even though the water level is sometimes only a few inches deep.

With so many lovely materials out in nature just waiting to be explored, I couldn’t fit everything into just one post.  Stay tuned for more Organic Math in the coming weeks!