In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission again comes from my former student Daniel Adkins. His topic, from Pre-Algebra: rational and irrational numbers.
What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)
The largest hurdle to overcome in mathematics, is the introduction to foreign, and new concepts. Quite often, individuals are stuck with their “old math”. When a new object appears before them, they won’t play with it, or recognize that even though it’s new, it still works with what we already know. This is especially true when it comes to introducing concepts of new sets of numbers, such as the imaginary numbers, or irrational numbers. More often than not, when it comes to irrational numbers, students freeze up. I believe that the best way to prevent this, is to show that students already know a lot about this set. By taking it step by step and reminding students what they already know about rational numbers, you can show them they have known about irrational numbers in some form or fashion for quite some time.
A simple two step project would be to first introduce the concept of an irrational number, then the instructor can draw a circle with a marked radius, and say, this is my pizza pie. Now I want a piece of pizza pie, but when it comes to pieces of pizza pie, I’m particular. I want to proficiently partake of my pizza pie by partitioning it perfectly, to where each piece is equally cut. If all I know is the radius though, how can I know where to cut it? Eventually students will point out that by finding the circumference, and then dividing the circumference by how many pieces you want, you can make sure they’re all equal. At this point, point out that you have a ratio of pieces to circumference, but how did the students get to the circumference in the first place? 2*pi*r so that means the radius of a circle is in a ratio to its circumference right? So we can right pi as some sort of fraction correct? If the students are aware that this isn’t possible, then the digging isn’t necessary, but if they aren’t ask them to try and write it as a fraction.
The second part of this exercise would be to emphasize nested sets. Divide the students up into 2-4 groups, and have a several Natural, whole, integer, rational, and irrational numbers written on pieces of colored paper (with each team having 1 color). Students will line up in front of “nestable” baskets spread out in front of them labeled by the different sets of numbers as listed above, and will one at a time aim for the smallest set that their number can fall in. After all the papers have been thrown, the papers will be collected and compared as a class, and each paper made in the correct basket will count as a point for that team. At the end of it all, put the numbers back in their baskets and show how the baskets can all fit inside of each other, except that the irrational and rational baskets are the same size, and so they can’t nest inside of each other. This can be emphasized by drawing it on the board. This exercise reminds students of what it means for sets to share qualities, and that irrational numbers don’t have the same qualities that rational numbers do.
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
Throughout my high school career, it was never brought to my attention that there were conflicts within the history of math, or in fact that there even was a history of math. In fact, it wasn’t till my collegiate years that my classmates and I came to learn such things were at one point a problem, that math could have different viewpoints.
The individual who is credited with discovering irrational numbers is Hipassus of Metapontum. He was a philosopher who studied Pythagorean based concepts, and while trying to use the Pythagorean Theorem to solve for a ratio between a unit square’s side length and its diagonal, he learned that there wasn’t such a thing. At the time, the other Pythagorean philosophers believed that only positive rational numbers existed. So when Hipassus introduced his discovery to them, they weren’t exactly happy. The story varies, and no one may ever know what truly happened to him, but some of the more versed stories range from the other Pythagoreans simply killing him, to Pythagoras himself ostracized him, and upon the Gods discovering the abomination he discovered, they had him drowned by the sea to hide it away.
Regardless of the validity of these stories, it shows how discoveries like these can often cause turmoil in time periods.
How was this topic adopted by the mathematical community?
Hipassus’s discovery caused such a drastic response because of two reasons; first off, it contradicted the core belief of Pythagoreans that Mathematics and geometry were indefinitely correlated, as in they were completely inseparable. But it also raised another problem that would eventually be brought up by another philosopher named Zeno. The problem was in the discrete vs. continuous argument, and how geometry couldn’t solve it. All in all, when Hipassus introduced this concept, it was met with malice. Many individuals would write this off as simply how things were “back then”, but a closer examination at something like imaginary numbers will reveal a similar pattern. It wasn’t until the Middle Ages when Middle Eastern mathematicians introduced concepts of algebra that irrational numbers became fully accepted within the mathematical community.
All in all, the stories behind things like irrational and imaginary numbers should be shared within schools much more often. Not only is it extremely interesting, and can convince students to do their own research, but it also shows that people were afraid to learn new thing, that these foreign concepts that are terrifying now, were terrifying to the people who discovered them too. It teaches students that instead of ostracizing others for bizarre concept, but instead to analyze them themselves. Because those bizarre concepts, may become commonplace. It shows students that Hipassus was on the right side of history, even though he was alone for quite a while.