# The Learning Tree

A long time ago, a veteran colleague shared this analogy with me: Kids learn much in the same way that plants grow. Some plants can grow and mature quickly. Others are like the macadamia tree, which takes 7-10 years to produce those delicious nuts; after it produces the first batch of nuts, it can continue to produce for over 100 years. It doesn’t matter if it’s a tree, a flower, a bush or a plant; once it has fully matured, it is a beautiful sight. And once it shows its true potential, we wonder what took so long or why the growth process was such a challenge.

I recently encountered a young group of geometry students that certainly fit this analogy. During the geometry classes, the students used the investigative approach to learn about the distance formula. In the planning session with the teachers, I discussed the lesson and how the students could use their understanding of the Pythagorean Theorem to derive the Distance Formula, as well as the Equation of a Circle in standard form. We had high hopes for the students setting out, but we wanted to push them to a higher level since they had seen the Pythagorean Theorem a number of times in their previous classes.

As some plants grow and produce faster than others, not all the students caught on to the concepts at the same time. Some derived the formula quickly. Others took their time to get the formula—but they eventually got it. Still others didn’t get it at the end of the 90-minute period and would have to work at it during the next class meeting. But the students all shared one characteristic: they were able to apply their previous knowledge to a new situation.

I was impressed by how the students were able to use their previous knowledge to draw right triangles when given two points. I heard certain misconceptions, like “the square root of 13 is 6.5” and “the hypotenuse is always the diagonal side.” But I also observed their “a-ha” moments like when one student said, “the formula is like the Pythagorean Theorem!” Some students were able to see a formula after a few examples. These students were able to explain their findings and thinking to their peers. Other students needed more “think time” with the examples in order to derive the Distance Formula; they seemed to be bogged down in the calculations and couldn’t see the bigger picture of the generalization.

We want all students to be able to use their knowledge in new and unknown situations. They will need to do this when they take more math classes. They will need to do this when they go to college. The workplace will present them with numerous situations where they will need to apply their knowledge.

The Common Core State Standards call for students to see the how “geometric objects can be applied in diverse contexts.” Some students will be able to do this in 56 minutes, some in 90 minutes, and some in a whole lotta minutes. Like plants, students will eventually achieve the math standards set out in the Common Core. And as with the plants, we will wonder what took so long or why the growth process was such a challenge for the students.

In the spirit of the holiday season, I decided to remake a classic with a math twist. Enjoy and Happy Holidays.

On the n-th day of Math Class,
Geometry sent to me
Twelve computers computing,
Eleven points aren’t pointing,
Ten chords a-cutting,
Nine-ty is half pi,
Eight-sided stop signs,
Seven lines a-skewing,
Six arcs a-arcing,
Five golden means!
Formulas from Greeks,
Three point one-four,
Two-column proofs,
And a straightedge without a class fee!

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