Organic Math: Part II

Rote learning works for some students, but it isn’t for everybody. More traditional methods of teaching mathematics can sometimes be too abstract for some students, which leads to the ubiquitous belief that they are “bad at math.” Because each student engages with mathematics a little bit differently, it makes sense to investigate mathematics in a suitably dynamic learning environment. The natural world is diverse and constantly changing, making it the perfect place to engage students with sometimes incompatible interests and abilities.

The everyday objects students come in contact with form the connection between them and the natural world. Whether a pine cone, a big stick, or a bed of flowers, natural objects have to potential to make all the difference.

Flowers

Because they are so diverse and visually appealing, flowers can easily captivate the curiosity of even the youngest students. In our classroom, the children love to pick and arrange them, but they also have an extreme interest in drying, shucking, and replanting them. In the spring and summer, especially with access to open fields or large expanses of grass, flowers can become a powerful tool for estimation, counting large quantities, and multiplication.

The Clover Flower Game came about one morning when the students and I were sitting in a yard with a stack of hula hoops. After spinning and rolling the hoops for a while, I decided to throw one into the yard and then count the clover flowers it encircled. The children became excited and took turns tossing their hoops as well. Soon, they formalized the rules of the game. Once all the hoops were thrown, the player with the most clover flowers was declared the winner.

But over time, the students began to change the rules, requiring them to exercise more than just their ability to count. One of our younger boys likes to include a flip rule, which allows each player to flip the hoop from where it lands exactly one time to attempt a higher score (similar to a mulligan in golf). The caveat to this rule is that there is no counting allowed before a player chooses to flip or not. They can only look at where their hoop landed, estimate how many flowers are within, and visually assess whether or not the adjacent areas have the potential to increase or decrease their score. When he first introduced this rule, it actually did him a disservice because he always elected to flip. His assumption was that no matter where he tossed the hoop, he could always gain a better score by adjusting his position. After a few tries (and admittedly a little frustration), he determined that accurate estimation and careful thought held the key to using his singular flip to successfully increase his score.

One of our girls chose to take the game in an entirely different direction. Based on her rules, the students threw their hoops, estimated the number of clover flowers, then counted them after. The winner was determined by whose estimation was the most accurate instead of who had the most flowers.

By turning the game on its head, she repeatedly beat her classmates. Instead of tossing her hoop as far as she could and trying to get as many flowers as possible, she chose to look around at her feet for a patch of ground with only one or two flowers, then drop her hoop. With only one flower, her estimation and the actual trefoil count within her hoop always matched, guaranteeing her victory over other students who managed to painstakingly encircle fifteen or twenty flowers.

Recently, the game became even more complex when one of our older students noticed that other flowers were present between the trefoils. “The clover flowers are worth one each, but the little yellow buttercups are super special. They are worth two each,” he explained. On that particular round, I had tossed into a very sunny patch of ground and snagged almost twenty clover flowers. He diligently added up his score, starting with the buttercups. “Yes… Three buttercups makes… six points!” After tallying his remaining sixteen flowers, he added the numbers together. “Six points from the yellows, sixteen from the whites…” Thinking for a while, he came up with the sum and pumped his fist in the air, celebrating his narrow victory over me.

Games, whether competitive or non-competitive, can be powerful vehicles for students to hone their gross and fine motor abilities, but also to sharpen their minds and flex their mathematical muscles in the moment. To many of our students, the Clover Flower Game is just another field game, similar to tag or soccer. But as they shift and manipulate the rules of the game, they organically explore more and more mathematical concepts including everything from estimation to arithmetic to multiplication and order of operations. Mental math, accurate estimation, and the memorization of integers are necessary skills, but they are also practical and exceedingly useful in the everyday lives of both children and adults. Calculating tips, creating schedules, and managing personal finances are just a few tasks that have a direct correlation to these formative skills.

Seeds

Seeds are a powerful tool for mathematical investigation because they are one of a few objects in nature that have a tendency to appear in a seemingly infinite quantity. In our program, we often intentionally encourage our students to practice counting seeds when we are relatively certain that a final count would be nearly impossible to acquire.

Sunflowers are a particular obsession of our current class. Last year, our students grew giant sunflowers in our lower gardens. My co-teacher, an experienced gardener, made sure to harvest a few heads from the sunflowers as they began to wilt so she could dry them and bring them back to school at a later date. When our students discovered that they could remove the seeds one at a time with their fingers, they of course naturally progressed to removing them en masse by smacking the sunflower head against the ground.

But once a huge quantity of the seeds were removed, tallying them become incredibly difficult. After having to start over a few times, one of our older girls mentioned that, “they all look the same, and I keep getting mixed up.” On that particular occasion, while the students did count to the upper limits of the numbers they knew, they eventually gave up on such a daunting task.

Pine cones present a similar opportunity for students to practice numeration. They often shock students in their sudden ubiquity; one moment, there is nothing beneath the pine trees, while just a week later, there are so many pine cones that they seem to fully cover the ground. Last year, a massive pine tree fell and blocked the road to our school, and I managed to run over and negotiate salvaging part of the fresh tree for the students to explore that day. I filled my backpack with pine cones (even I had no idea how many I harvested), brought two large trash bags filled with fresh pine branches, and one massive section of the sap-covered exterior bark of the tree.

When I unzipped my backpack and showed the students the surprise within, they immediately wanted to find out how many were inside. One at a time, they took turns removing the pine cones and counting in sequence. One of our students narrated the process as it occurred. “Okay, that is 21, now you get to do 22. There, that makes 22.” Like the sunflower seeds, the larger the pile of tallied pine cones became, the harder it became from them to determine which pine cones they had counted and which they hadn’t. Sometimes they lost count and had to start all over.

But unlike the episode with the sunflower seeds, these particular students had the idea to separate the pine cones into easily countable groups. Instead of counting to higher numbers and invariably losing their place along the way, they determined a way to calculate the total number of pine cones while only needing to count to five over and over again. When they were finished separating the pine cones into groups, they proceeded with their total tally, but made sure to keep the groups distinct to minimize the possibility for error.

Tallying seemingly infinite quantities of objects adds an entirely new dimension to mathematical exploration that is lost when learning by rote. It embodies a synthesis of both critical thinking and numeration. Practical math differs from rote mathematics because it acknowledges that numbers are living, breathing concepts that sometimes need unfolding. Before math can happen, there is often a logic problem that needs to be solved, such as which methods to use, how to maintain accuracy, and what to do with the solution once the math problem is solved.