1. Evernote for reflection: this program was introduced to me by my CT during my student teaching semester. It is an app that can be downloaded to a smartphone or tablet, as well as an internet site where an account can be created. We had a shared folder for our geometry classes, and every day I would create a new entry with the agenda for the day. Then, at the end of the day, I would reflect on the lesson in the last column of our table. This could be something as simple as “this lesson worked perfectly today. The timing was great and the students seemed to enjoy what we did.” It could also be as detailed as “today was a disaster. Instead, this is what I would do in the future…” and then lay out a plan for next time I teach this concept. I really liked having this online journal to go back to so that I could see what techniques went over well with my students and which ones I could toss. It also helped me see how far I’ve come, as well as how far my students have come throughout the semester.

2. Avoid the book: we did a project called the tin man project to help students learn and understand surface area. They were given no formulas, told not to look in the book and set free in groups. Together, they brought in items (at least 1 of each of the following: cylinder, rectangular prism, sphere and cone) to construct their tin man or animal. Then, they had to figure out how to cover it using the least amount of tin foil possible, also taking into account that the length of one of the sides of the tin foil sheet they would receive for each body part was predetermined. The students were split into groups based on similar overall class grades so that I knew where to spend the majority of my time. The entire activity took about 5 days, and included a few practice problems on worksheets at the end of each day to make sure they could transfer the practical knowledge to mathematical problems. Overall, the students told me that they really enjoyed the project because it was a nice change of pace. Of course there were some students who just really prefer to work alone on worksheets and problems, but the point is that we were able to reach a wide variety of students.

3. Technology rules (usually)!: there is a website called infuselearning.com, and I’m convinced it’s magic. My students LOVE using this tool, even though they’re doing challenging word problems most of the time that we use it! Basically this app turns a tablet (in our case iPads) into a whiteboard. Students join your classroom then draw out their responses in various colors and submit them to the website. The website then allows you to display all the answers so that we can go over them, review whether we agree or disagree and discuss common misconceptions. Of course, students love making up their own display names (as long as it’s school appropriate) and drawing elaborate figures and detail into the representations of the story problems. I just think it’s great that they’re enjoying the work that we’re doing in class. Whenever they see it written on the board in the agenda, it’s all they can do to refrain from asking me every two minutes when we will start. That’s the type of enthusiasm I’m looking for from my students!

4. Give it purpose: this was feedback I received from both of my field coordinators, as well as something my CT and I discussed as a focus for this trimester. Instead of simply passing out worksheets, assigning problems and doing activities, tell the students *why *we’re doing them. They’re much more likely to buy into an activity if they understand that it has a deeper purpose or meaning. For example, I noticed students rolling their eyes during one exercise that was meant to show them a tool for studying–saying/explaining concepts out loud to ensure comprehension. It then dawned on me that I had never given them a reason to do this, I had simply asked them to get in groups and begin explaining. So, I then took the opportunity to let them know what a great study tool this was, because sometimes we think we understand a concept or how to do something, but then when we go to explain it to someone else, we realize we don’t understand it quite as well as we had thought. Just that quick prelude would have made a difference in how they viewed the activity (at least, I believe it would have), but so often we forget to give everything purpose and meaning. We know why it’s important, but students need to be told and reminded that there is a purpose for everything we do in the classroom. And if we as teachers can’t come up with the purpose, then that activity should probably be tossed!

The first method, inside/outside circle, was to show students the importance of saying their thoughts out loud. So often we think we know how to do something, but when it comes time to explain it to another person, we lack the ability to articulate our thoughts. This happens frequently in math, so I wanted to make sure my students understood the importance of this skill. It’s a great study tool for at home, and it was also nice to get up, stretch, move around and communicate with their classmates. It was different than the average classroom flow, so I think many appreciated the change. Of course, there were some nay sayers, and that’s okay! Not every student is going to love every kind of learning and teaching style–heck, some aren’t going to like any! But the point is to try to switch it up so that we reach every type of student.

Next, I broke them into groups based on their grades (there was a purpose to this!) and they worked on their “5-problem concept review.” They had a practice exam to complete during the previous days, and their task was to correct it using the answer key, noting which concepts them missed. Then, they would go to the post on their classroom webpage where I put 5 problems from each concept that were in their book for them to work on. The idea was for them to really focus in on what they *don’t* know and not waste time on what they *do* know. “Study smarter, not harder.” The reason I broke them into groups based on class grade, was so that I could spend more of my time working with the students who weren’t excelling in the class, and really focus on bringing them up to speed. For the “book study” type of kids, this was where they hit their “groove.” It allowed students to ask one-on-one questions on concepts with which they were still struggling.

Finally, we wrapped up with “space race,” which is on the Socrative website. There were just five questions with some basic calculations that they were to answer as quickly as possible on their iPads. They were automatically divided up into five teams, and the different colored rockets were shown on the screen. Depending on how many of students from each team got a questions correct, the rocket would move forward a little bit until one was the winner. This activity didn’t take long, and it wasn’t necessarily a study tool, but it did provide some enjoyment for them as they entered into their week of finals.

Overall, I really enjoyed the lesson today, because it had a combination of both teaching & learning styles. I didn’t have to be up at the front of the room talking at them, and they were able to learn from and teach each other.

]]>I had asked my coordinator to help me focus on how well I relayed the overall big picture of geometry to my students, as well as how engaged they were throughout the lesson. I want math to be taught to these students differently than it was taught to me. Rather than learning about each chapter or concept in the book as a separate idea, I really want them to understand how they all connect. Math isn’t meant to be studied in a detached manner, because it builds off of itself and relies on so many different components. We’ve been trying to ask them the “why” behind each thing we do in class. It’s so crucial that they critically think about each concept in geometry so that they’re able to apply what they know to all different situations. That is what I was aiming to teach and help them realize, not only in this lesson, but in every lesson. After discussing with my professor, I realized I could have done more to promote this idea of connectedness, but at the same time, we were using concepts and tools from the previous couple of lessons in order to accomplish the objectives and goals for the day. That being said, I think next time I could pause to have group discussions about what it is we are using/doing and how it relates to what we have done in the past.

As for the engagement piece, my coordinator was able to keep some data so that I could see where and when I need to refocus my energy to engage my students. In the beginning of the hour, the focus was fairly high, but as they began finishing the first activity (a parallel and perpendicular memory matching game) at various times, the first ones done became easily disengaged. I started to lose a few, so I tried to refocus their efforts on the next worksheet, which had an application to the parallel and perpendicular lines we had been learning about for the past day or two. It was a little challenging and I could tell that some students were going to need major motivation so that they wouldn’t simply give up. At some points, I felt like there wasn’t enough of me to go around and the data showed that these were the times that my students had a difficult time staying engaged. With them finishing at all different times, I was unsure of how to keep them all focused and on task. When I asked my coordinator for some recommendations, he talked about a strategy of “split-group-split-group.” That is, walk around and monitor/respond to questions from individuals or groups and then take time to come together as a group, especially when the same question continually pops up. This way I can ensure that everyone is on the same page and we’re all able to move forward together. I’ve actually had a chance to try this strategy since then and it’s been great!

I look forward to employing this tactic more often and trying more engaging and exciting math activities! It’s still my goal to show students that math is more than just memorizing steps and procedures–it’s making connections and applying what we know to figure out what we don’t.

]]>*Love and Math* by Edward Frenkel is the story of a Russian boy with dreams of becoming a mathematician, but due to his Jewish lineage, was unable to do so in the traditional route. This book is not only inspiring in the way in which Frenkel persevered through hardships and trials, but it’s interesting in how a young Jewish mathematician had to go about expanding his knowledge and horizons not all that long ago. He would sneak into other universities to attend math classes, meet with a tutor every couple of weeks and work on math himself in the confines of his home. Through this book, he hopes to “unlock the power and beauty of mathematics” for everyone—not just those who are mathematically minded.

He wants math to become part of culture—like science, DNA and cancer. It’s a vast, untraveled world that Frenkel wants to expose to society. His goal is to provide all with a general access to a seemingly inaccessible subject. From “skimming” this book, I believe that he does an amazing job of taking a difficult topic and making it readable for all types of people.

I highly recommend this book and hope to finish it in the near future myself. You can’t help but connect with this young man, because of all the adversity he faced and yet he still came out on top. This is what drew me into the book, and the math made me want to continue. It was so readable and extremely fascinating, that I wanted to know more and find out what he saw in mathematics. I know why I enjoy it, but I love hearing from others why they find it such an incredible area of study.

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There were, however, a few things I didn’t care so much for in the book. I am a very visual person, so just reading about a mathematical problem and its solution doesn’t really do it for me. I need to have it written out, in pictures, in steps, etc. to fully grasp what is happening. Although there were many instances where Maor did include pictures, there were many where he did not. Surely the book would have been much thicker with these pictures taking up more room, but it really would have helped me understand the math the first time around. Instead, I found myself reading pages over and over and over again. This next critique is going to sound odd, but it’s something that really bothered me while trying to read the book. The font is so dang small!! When I’m doing math and writing about math, you better believe I’m taking up pages upon pages because I want to be able to look at it without becoming overwhelmed. Trying to read through some of his explanations and problems literally made my head hurt at times.

So, after nitpicking a few items, I overall really enjoyed the book. It makes math history much more that tolerable–it makes it interesting! Reading about the mathematicians who invented the theorems we use and take for granted today gives me a whole new appreciation for what it means to invent and discover mathematics.

]]>From my research, the values in the first few rows of the rascal triangle support the claim that each element of the nth diagonal is congruent to 1 modulo (n-1). To explain this, in the fourth diagonal, each of 1, 4, 7, 10… is congruent to 1 modulo (4-1)=1 modulo 3. So, in these cases, n=1, 2, 3,… Moreover, we can see that the nth number on the rth row a∨(nr) is the nth number on the (r-n)th diagonal and n, r=1, 2, 3,… and the 0th row is excluded. This observation is fundamental in proving that the rascal triangle consists strictly of integers. Also, the sums of the rows of the rascal triangle are “cake numbers” (I had never heard of these before…). A cake number is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly *n* planes.

Another way to prove that all of the entries in the rascal triangle are integers, is by doing what the boys did…

They realized that the NE-SW diagonals exhibit a simple pattern: the m^{th} diagonal (starting the count with m = 0) is an arithmetic progression with m as a difference and 1 as the first term. The entry #n in the diagonal #m is simply *nm + 1*. The formula to generate the triangle is then

(m + 1)(n + 1) + 1 = [((m + 1)n + 1)(m(n + 1) + 1) + 1] / (mn + 1)

(CTK Insights).

I show this (or at least attempt to) in the picture I included, so that people could put a visual with the written work. I am so inspired by these students, and it makes me wonder how often we as math teachers shut down valid ideas. Not intentionally, of course, but without even realizing that our students could really be onto something. It just makes me think about how I’ll go about teaching in the future.

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“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.

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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.

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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.

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