While the moon walks are the first things to come to mind when recalling the Apollo missions, their impact extended beyond the moon. The Apollo missions created a need for technologies that revolutionized our world, which eventually led to the device you are using to read this article. The missions also produced stunning images of our planet, leading to environmental movements and conservation efforts to protect our only home. Perhaps most importantly, the Apollo missions were a springboard for our other explorations, including landing multiple rovers on Mars, landing a probe on a comet, sending a probe to Pluto, and even sending the Voyagers into interstellar space.

As we reflect on the historic achievements of the Apollo program, we must remember that it all started with a vision and broad support from public and private entities. In order to tackle the problems of our age, we must adopt a similar strategy. Just as reversing climate change may seem impossible today, landing a man on the moon was once seen as impossible. And yet, 50 years ago today, we did just that.

]]>Unknowingly, the instructor has communicated their own fixed mindset belief about the students. The belief is that intelligence is a fixed quantity as opposed to growth mindset, where intelligence is thought to be changeable. Previous studies have found that instructor expectations for students can affect their performance, but what about the instructor’s beliefs about their students' intelligences?

To answer this question, researchers from Indiana University looked at grades from over 600 science, technology, engineering, and mathematics (STEM) courses, representing over 15,000 students. To understand whether the instructor had fixed or growth mindset beliefs, the researchers posed two questions to the instructors that tested how much they agreed with the idea that intelligence is not something that can be changed.

Perhaps unsurprisingly, the researchers found that students in the fixed mindset instructors’ courses earned lower grades than students in the growth mindset instructors’ courses. The effect was even more pronounced for students who identified as Black, Latino, and/or Native American. This likely is yet another contributor to the race gap in STEM (and college completion in general).

To ensure a more equitable classroom, instructors must be aware of their beliefs and how they may promote or hinder student achievement.

]]>Start by drawing a square. Let’s make it 8.5 inches long on each side so it fills up the width of a standard sheet of office paper. Now, draw a circle with a diameter of 8.5 inches within that square.

Now we need something to drop onto the drawing, like grains of rice or M&Ms. Begin by covering the square and circle with the objects and then count the number that are within the circle and the total number of objects. From this information, we can estimate the value of π. We can calculate the area of a circle with π r^{2}, where r is the radius (the distance from the center of the circle to the outside). Based on how we drew the circle and square, the square will have a side length of 2r, so the area will be (2r)^{2 }or 4r^{2}.

Back to the M&Ms: to guess how many of the dropped candies will land inside the circle, we can calculate the ratio of the circle’s area to the square’s area: (π r^{2})/4r^{2} = π/4. When I tried this with 100 M&Ms, I found that 78 of them were inside the circle and 22 of them were outside the circle but inside the square. This means 78% of the total M&Ms landed in the circle. From our ratio calculations, we estimated that π/4 candies would land within the circle. Now, we can compare the fraction of the M&Ms that landed in the circle to the theoretical number to estimate π. Doing so, we find π/4 = 0.78 so π is about 3.12. The actual value of π is 3.14159265…. so our result isn’t a bad approximation. If we wanted to get a better estimate, we could use a lot more M&Ms and a much bigger drawing.

Of course, counting all the M&Ms takes a long time, so using more than a few hundred M&Ms isn’t practical. Instead of actually dropping M&Ms onto a square and circle, we could write a computer program to simulate dropping thousands or even millions of candies onto a circle and square and counting them up. The computer can “drop” and count a million M&Ms in a matter of seconds. When I tried this, I found that 785,389 of the 1 million simulated M&Ms landed within the circle, leading to an estimated value of 3.141556, which is even closer to the true value of pi than our estimate using only 100 M&Ms.

This is only one of many ways to approximate π. Nevertheless, this is a simple way to estimate π with just regular household materials.

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