Nature of Mathematics: Is Math a Science?

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I had honestly never really given this a second thought. Math is always associated with the sciences, so why wouldn’t it be one? However, turns out that this is much more of a controversial issue than I ever realized. I mean, sure I think of math as its own language, but I’ve never identified it as literature or speech–it’s just always been a science. But why? Here is the dictionary’s definition of the word science: the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment. I’m going to dissect it and see what conclusions I can draw.

“The intellectual and practical activity encompassing the systematic study” If I look at this part first, it is clear to me that the science label fits well with mathematics. It takes active participation and involvement to truly do math. Plugging and chugging numbers into formulas is, I suppose, math, but it’s not the true heart of exploring numbers and patterns and theories. This takes deep thought and actual “doing” to achieve. And as for the systematic study part of this definition, the first thing that comes to my mind is proofs. Sure, proofs give us mathematicians the opportunity to be creative and spread our literary wings, but there is still a systematic approach of sorts we must follow. That is, we can’t just jump from assumption to assumption in order to prove a theorem–we must use only the axioms we know and build upon them to reach the end product. This system is one that is common across all mathematics.

“Study of the structure and behavior of the physical and natural world” Though math may not focus on explaining why polar bears only live in cold climates, it can tell us why 2 polar bears plus 2 more polar bears is 4 polar bears; why 6 polar bears minus 1 polar bear is 5 polar bears, and so on–mathematical abstractions arises naturally from the physical and natural world. Math helps explain how objects exist in the natural world, and the majority of math has numerous real-life application. Areas such as Algebra and Calculus help us understand things such as the basic rules of motion. Math is an integral part of everyday life, so I find it difficult to think of an argument that supports the idea that math doesn’t deal with the structure and behavior of the physical and natural world.

“Through observation and experiment” In my opinion, this is the most obvious part of math. It’s learning by doing and observing. Noticing patterns and seeking out new ones is what mathematicians love to do. It takes numerous rounds of trial and error, critique and communication to solidify a mathematical conjecture as a theorem, but that’s part of what makes mathematics a science. Observing what people have done in the past and building on these ideas is very similar to what scientists do. Though they aren’t an exact mirror of each other, math possesses all the characteristics of the science definition.

 

Communicating Math: Amicable Numbers

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Amicable (which means friendly) numbers were discovered by Pythagoreans a really, really, really long time ago (see, even old guys liked finding “cutesy” math things). It is said that their properties stem from Pythagoras’ response to the question, “What is a friend?” He responded, “A friend is one who is the other I.” Now most of us here in the 21st century hear that response and say, “Huh?,” but when we dissect two amicable numbers, it becomes much clearer. There are many ways to define or describe Amicable numbers, but the clearest definition I’ve found so far is this: Two numbers are called Amicable (or friendly) if each equals to the sum of the proper divisors of the other (proper divisors are all the divisors excluding the number itself).   

The first pair of amicable numbers ever discovered are 220 and 284. Let’s take a look:

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220

284: 1, 2, 4, 71, 142, 284

I put the final factor of each number in italics to show that these are not proper factors of each number, so when summing the factors we will not be adding these. If you plug the factors 1-110 into your calculator and add, you should get 284 as the answer. Similarly, if you plug the factors 1-142 into your calculator and add, you should get 220 as the answer. And there we have it, Amicable numbers.

Thabit ibn Qurra, braniac he was, decided to take this a step further–he proved that numbers of the form T(n)=3*(2^n)-1 (now called 321 numbers) are also very special. If the 321 number for n and n-1 is prime and 9*2^(2n-1)-1, then the numbers (2^n)*T(n)*T(n-1) and (2^n)*(9*2^(2n-1)-1) are amicable. Don’t ask me how he managed to figure this all out–he must have had a lot (and I mean a lot) of time on his hands! And, even after all this work that Thabit did, the only known examples of this are for n=2, 4 and 7. That’s it. Check ’em out and see for yourself! (Hint: n=2 is our 220 and 284 example :))

For many years, this was the only known pair (220 and 284) of amicable numbers. It wasn’t until 1636 that Pierre de Fermat discovered another: 17296 and 18416. Descartes contributed the third pair: 9363584 and 9437056, and from there Euler took over, finding 63 pairs. Today, roughly 11994387 pairs of amicable numbers are known. 

 

Weekly 2: History of Math, The House of Wisdom

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For my weekly work I did some research into the House of Wisdom and its influence on mathematics. To begin, the House of Wisdom was a library, translation institute and school established in Baghdad, Iraq. Works on agriculture, mathematics, philosophy and medicine were translated into Arabic here.  The House of Wisdom helped transform Baghdad into a hub for the study of humanities and for sciences, including astronomy, chemistry, zoology and geography, as well as alchemy and astrology. It is said that the House of Wisdom was the key institution in the Translation Movement and was considered to have been a major intellectual center of the Islamic Golden Age (Institute Mohamed Ali). This is simply a brief background on the House of Wisdom–next I’ll get into the role it played in mathematics.

The Abbasid Caliph, al-Ma’mun, established the House of Wisdom and invited Al-Khawarizmi (who will later come to be known as the Father of Algebra) to come and help him with something. He wanted to prove Allah’s existence through the “complexity and beauty of” mathematics.  Al-Khawarizmi began by translating ancient Greek and Indian texts. From the great Indian book on math, The Opening of the Universe, al-Khawarizmi adopts the idea of the zero as a number. This opened up a whole new world of mathematical possibilities and complexities unknown to the world at this time. The old Roman numeral system made more complex math almost impossible, but with a number system that includes 0, al-Khawarizmi introduces new ideas. Areas of mathematics such as algebra and geometry of the Greeks are developed, which eventually lead to math such as trigonometry and calculus. 

However, he still has a problem. Zero cannot be proven to exist using math, because even though the Indian texts he’s translated insist that zero divided by zero equals zero, al-Khawarizmi knows that dividing anything by zero is impossible. So, he decides that zero must simply be accepted without being proven–much like the existence in Allah. al-Khawarizmi is later cited by European mathematicians by a name that much more closely resembles ‘algorithm’ than it does his actual name. The mathematical word ‘algorithm’ is derived from his name, and, as many know, means complex, mathematical formula. Not only is he remembered by this, but also for his book titled Kitab al-Jabr wa-l-Muqabala, in which he lays out the principles of Algebra.

The House of Wisdom hosted scholars and researchers of all fields and studies, not just mathematics. However, the discoveries that were made by al-Khawarizmi truly provided the foundation of mathematics as we know it today. His discovery of the number zero and the concept of algebra laid the basis for almost every other type of mathematics that is taught in schools across the world. Therefore, the role of the House of Wisdom in the history of mathematics is more fundamental and crucial than I (and I’m guessing most others) ever realized. 

 

 

 

Daily 2: LCM & GCF

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For my daily work I took a look at the LCM (least common multiple) and GCF (greatest common factor) to see if I could come up with any connections or observations. Here are some of the notes I took while working on this task…Image

So, to begin with, I just started choosing random integers (ones that I knew were not prime, because then their greatest common factor would be one and their least common multiple would be their product) and determined their GCF and LCM by simply listing out all of the factors of each integer and some of the multiples and seeing where they overlapped. I was then thinking about it and thought there had to be a way that made more sense, or a way that would better illustrate a connection between the two. I decided to try prime factorization on the two integers to see what those factors presented.

First I determined how to find the GCF using the prime factorization results. I noticed that it had to do with the factors that both integers had in common. For example, 10 and 12 have 2 in common (10 has 2 as a prime factor once, while 12 has 2 as a prime factor twice). Since they only overlap once, 2 is the GCF. However, for the integers 12 and 18, they both have at least one 2 and one 3, so the GCF is 2*3=6. I tried this for the remainder of my integers and found it to be true, so I moved onto LCM. Since I had some idea of how to approach this at this point, I began by trying to just multiply the common factors (from my prime factorization), but realized this wasn’t quite right. So, after a bit of tinkering, I finally came up with this: you multiply all of the prime factors together, unless there are any overlapping ones. Then you just use that number once. For example: 9, 12 (9: 3, 3; 12: 2, 2, 3) 2*2*3*3=36. All of the prime factors are multiplied except for one of the overlapping threes.

I found this website, which does a great job (in my opinion) representing this relationship visually.

http://www.lamath.org/journal/vol5no1/Venn_Diagrams.pdf

So how does this tie in with axioms? Let me tell you…

An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true. So, they are guiding lights to doing mathematics. Euclid proved entire postulates using only the few axioms he had predetermined. Likewise, using the prime factorization in the LCM/GCF exploration, I used basic rules (axioms) to determine what constituted as a prime number and what I could therefore do when dividing the integers further and further. Basic statements about what are prime numbers and division allow us to expand our explorations and practice to new things, like prime factorization, GCF and LCM, for example. Then, once I had finished coming up with a relationship, I wrote down the basic rules I had created for doing so. Thus, creating more axioms.

 

What is Math?

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When I was younger I used to think that math was all about formulas, equations and numbers–and that it always had a “right” answer. I had a hard time coming up with those “right” answers, and often felt discouraged and dumb based on the accusations and actions of my teachers. So, I decided to write math classes off as “Cs get degrees” courses (just kidding, I was never someone who could settle with a C, so I just cried about the fact that I would never understand :)). Who needed a subject that just made you memorize equations and plug-and-chug data with no real goal in sight?

This attitude carried through my elementary and middle school years, so I saw no reason why it shouldn’t continue right through high school, too. That was until my freshman year geometry class. I was expecting day after day of boring notes and homework problems, but what we got instead were group projects, collaborative assignments and lessons in which we were supposed to prove theorems. My teacher did the impossible–he made math fun. I loved the challenge of being expected to work through difficult problems with classmates and see what we could discover on our own. It was so much cooler to take what I knew and shuffle it around until something new emerged. And working with classmates made everything possible. There were times when I had no idea what to do next (these would have been the times in the past where I would have given up and called it good enough), but thankfully had two or three other minds to bounce ideas off of and gain new insight from. I learned that math was more than I had ever thought. Math literally helped explain the world around me, and applied to nearly every facet of my life. Sure, it included things like numbers, variables, operations and solutions like I mentioned before, but it was more about thinking critically, problem solving and working collaboratively with others than I had ever thought before. And so began my love affair with mathematics.

I recently had another “aha” moment during my semester of teacher assisting in a sixth grade math classroom. I realized the power of justifying one’s response or reasoning when it comes to math. My favorite questions (and likely my students’ least favorite question) quickly became “why?” Why does that work? Why is that the solution? Why are we able to do that in mathematics? It soon became habit for them to have their explanation prepared to share with myself and the rest of the class. It was wonderful to see them explain their thoughts to their peers using pictures, words and gestures–it was like they were now the teachers!

So, I guess I would conclude that “math” is a slightly challenging word to define. It’s almost just as much data and numbers as it is teamwork and thought processes.

Beating the Blahs

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This long stretch between Christmas break and Spring Break is taking its toll on the staff and students at school. We’re all ready for a break, and it’s beginning to show. Nevertheless, I’m pushing through and making the best of each day–what else can you do, right?

My partner and I have been experimenting with a variety of teaching methods to see which ones work best for, and continually engage our students. We’ve found that lessons involving physical movement tend to be best at keeping the kids interested in math. We’ve also had some great discussions about  major topics, like negative numbers, the relationship between addition and multiplication, etc, which I love seeing from middle schoolers. They’re working on cooperating with partners and sharing their thoughts in front of the class. This is something that has been extremely effective–having them come up to the board and justify their solutions and thought processes with everyone else. Watching their excitement about being able to assume the role of the teacher is inspiring, because it shows their understanding of a concept and confidence in justifying their reasoning. I try to call on them as often as possible to give them this opportunity. It can often be difficult to ensure that I am getting all of the students to volunteer equally, because some volunteer more than others. However, I think it’s important to get everyone up to the board because it gives me a better sense of the entire class’s understanding, rather than just those whose voices tend to dominate the classroom. I try to regularly reinforce that making mistakes is no big deal–in fact, it can actually help us learn more as a class. This has gotten a few more volunteers to come up and share their thinking with the rest of us.

In order to “beat the blahs,” our school is hosting a slew of events and activities to motivate the students to keep pushing through. We’re currently in the midst of spirit week (which the teachers may be enjoying more than the kids!) as well as activities like student vs. staff basketball games, mini-olypics and more. It has seemed to make the school days a bit more enjoyable for the students, but a break is definitely much needed.

I’ve been teaching at least one hour a day most days, and it is amazing to me how much more comfortable the students are around me compared to when I first arrived. I feel like I’ve built up a good relationship with them, and I really take the time to get to know them–their likes, dislikes–and try to incorporate them, even if only briefly, into the daily lessons. It’s the little things that count. 🙂

That’s about all there is to report as of now. I’m still loving every moment of being in the classroom and teaching the students. In fact, I’m learning quite a few new words myself–urban dictionary has become my new favorite tool! Ha-ha

What? So What? Now What? (Round 2)

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Today was my first day being observed by my content field coordinator, so this post will serve as a reflection of both my lesson and the debriefing I had with him. It will follow the what? so what? now what? format, just like my third blog post.

WHAT?: Today I taught a lesson on evaluating algebraic expressions to first hour. I have to say that I went into it quite nervous, because first hour tends to be a bit challenging when it comes to classroom management. However, I made sure to set goals for myself so that I knew what I was working toward. My goals were to make connections back to the main goals/objectives and vocabulary for the day and to try to keep the students engaged constantly so that I wouldn’t waste time asking for a zero (the system my school uses to get students’ attention). I don’t like to be that teacher who is constantly telling the students to be quiet, or that I need them to stop doing such and such–I think it should be an entire class effort, because it’s their classroom, too.

I taught my lesson and worked through the activity with the students, mostly circulating through the room while they worked to answer questions and monitor their work. The activity had them working in pairs with manipulatives (little cubes of different colors in this case) in which they counted the number of each color cubes they had and let these represent the variables in the algebraic expressions. Their job was to substitute the values they found into the expressions and evaluate them. The two vocabulary words which went with this lesson were substitute and evaluate.

As I reflected with my field coordinator, I realized that I hadn’t done as well with the fulfillment of my “making connections” goal as I would have liked. Although the lesson went well overall, there were opportunities I missed to connect to previous lessons, or even concepts from earlier in the day. I was, however, able to make connections with the vocabulary from the day, as well as spend a relatively small amount of time on getting the students’ attention. You win some and you lose some, I guess.

SO WHAT?: Being able to lead this lesson completely on my own was an amazing experience and showed me that I can do this whole teaching thing! There are definitely aspects I would like to change and improve for future use, but that’s why this observation was so great. I was able to discuss what worked and what didn’t so that I can make the necessary adjustments for next time. The discussions with my coordinator showed me things I forgot (such as referring back to previous concepts which connect) and things that could go more smoothly next time (such as giving the students explicit instructions and tasks so that there is no question, and no reason for them to be off task) as to what they are supposed to be doing. Sometimes we become so absorbed in the teaching that we don’t pick up on the little things in the classroom that may be causing a distraction or may not be clear to students. It’s nice to have the opportunity to be observed so that I know what is going on in all areas of the room when my eyes and ears are with other students.

NOW WHAT?: Two areas I really want to focus on are classroom management and explicit instructions. I have a feeling that the explicit instructions will help address some of the classroom management, so I’ll start there. Students seemed to be a little confused when I was going through the worksheet, so next time I would be sure to pass that out first before going through it, but without the manipulatives so they aren’t tempted to start before hearing the instructions. If I can go through an example with them, then they will have a model to at least base their work upon, even though our answers won’t be exactly the same due to the variation in colored blocks. Also, when students were at the board writing their answers, I need to give the other students a task so that chatting doesn’t become a temptation. As far as classroom management, I’d really like to try out some different methods of getting (and keeping) the students’ attention. That way, they’re less likely to engage in side conversations. I’m not sure what these methods are yet, but I will be doing my research and trying them out so that I can report back!

This experience was more than helpful and I feel so much more prepared to teach even from that one observation. Having a second set of eyes to monitor your lesson and classroom skills is a lifesaver!!