Doing Math- Making Nets!

I’m really terrible at producing nets. It’s a huge challenge for me to be able to look at a 3-D image and visually imagine what it would look like in a 2-D net. Last week in class, we created three nets that together, made a cube. The shapes were a triangular prism, triangular pyramid, and square pyramid, making up  1/2, 1/6, and 1/3 of the cube respectively. Below is a photo of our nets we created in our group. Being in a group of three, we each constructed a net and placed them all together at the end. I wanted to revisit this activity to ensure I understood how each of the other two nets were created.

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When we created the nets, we decided to use a 1cm x 1cm cube, making the dimensions for our squares 1cm x 1cm, so I decided to stick with that when reviewing the mathematics for each net. I began with the triangular prism. With my  1cm x 1cm square, I knew the dimensions of the two triangles as well, because these were essentially just a square cut in half. I then had to find the hypotenuse of the triangles, where I used the Pythagorean Theorem to get 1.4cm. I knew my bottom most square was a 1cm x 1cm square, and my rectangle was 1cm x 1.4cm, because of the hypotenuse of the triangle.

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Next, I worked on the square pyramid. Since this is the one I made in class, I was pretty familiar with this one. I started with another 1cm x 1cm square since this too would be a face of the cube. The two smaller triangles were again, a 1cm x 1cm square cut in half, so I used my calculations from the triangular prism in order to label its hypotenuse. For the other two triangles, I knew that one side would have to be the hypotenuse from the smaller triangles, so from there I needed to find the hypotenuse. Using the Pythagorean Theorem, I found the hypotenuse to be 1.73cm.

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Last, I worked on the triangular pyramid. It made more sense to me to put the other two pieces together and then try and visualize the sides of this net and what their measurements then had to be, based on the measurements of the other nets. I realized that the two triangles that create a rectangle at the top is the same measurements as my rectangle in the triangular prism. I just needed to find the hypotenuse which again, is 1.73cm. Based on this, I knew the other two triangles had hypotenuses of 1.4 cm.

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It made sense with the numbers that one builds off the other. I also thought it was cool to see the other ways the nets could have been drawn. It may have been easier to figure out the math to each of them, depending on how you drew the nets. In a classroom, I think creating nets for 3-D objects could be extremely valuable. I hope to incorporate this as much as possible. It can really help students to learn and thoroughly understand geometry, while also giving them hands on activities where there isn’t just one right answer. Like we’ve seen, there’s many ways to do this and I think students can find much value in that. It was definitely an enjoyable activity that I think is important to have experience with.

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History of Math; Where the heck did this all come from?

As I was just discussing with a classmate, I knew little to nothing about the history of mathematics before this semester. I study it, teach it, but have no clue about many of the people behind it all. Sure, you hear of the big names throughout classes; Pythagoras, Euclid, but I had never realized how many mathematicians were involved, let alone heard of them.  The Greeks had such a huge influence on mathematics, and it’s so cool to hear about all of their awesome ideas!

A technique that I’ve learned through mathematics is being able to generalize. Learning about the Greeks, I discovered that they contributed this technique to the math world and without it, it’s crazy to think how much more difficult math would be to understand. We wouldn’t be able to say that the Pythagorean Theorem would always hold true or that any theorem in general would hold true. This makes our lives so much easier as math students. We use this everyday without thinking about where it came from. As a future teacher, generalization is a huge technique to be able to teach a class because this gives students a tool to prove statements, find universal patterns and make sense of mathematical concepts.

Many people can easily do mathematical processes, but the Greeks gave us a new way to approach and look at mathematics. In high school, students begin to more formally “prove” ideas, finding existing patterns, convincing themselves and an audience of a concept. This is a way to communicate in math, showing a convincing argument to prove a statement true. We can now look at math in a more abstract way that allows us to discover new patterns and make sense of them.

The Greeks had so much influence on the way we think about mathematics today. I continue to relate it back to teaching and think how I use these concepts everyday subconsciously to educate students.

What is Math?

This question seems so simple at first, but when you think of all the possibilities of responses from all of the many different types of people, it’s truly amazing how people portray it. When I think of this I immediately think of the common things that non-mathematicians might say. Personally, I know math is much more than the K-12 definitions that would normally be given. The world of math is so abstract and intriguing to think about. To me, math is a way of thinking about patterns throughout the universe. Math is interpreting and studying these patterns to find more patterns. What a concept.

Basic discoveries in math seem to be some of the most monumental for the simple fact that we had to start somewhere. I think that the concept of distance and space is one of the most important because without being able to measure or understand the world in that way, many math concepts would have not been found. This leads to geometry in general being a huge discovery, along with euclidean geometry. In order to perform any operations in math, addition/subtraction and multiplication/division are extremely important. With this, the discovery of infinity opens all sorts of new worlds in math that have become important when studying many topics. These concepts build on each other, creating new ones and without one, it’s difficult to understand another. The greatest discovery in mathematics in my opinion is the power of communication through symbols. In math there are so many symbols it seems crazy, but without them it would be difficult to communicate these concepts to one another.

Teacher Observations

It’s crazy how different teachers teach. Every time I get the opportunity to observe a new teacher, I think it’s awesome how they some how put their own voice into their teaching style. It amazes me because no one teacher is similar to another.

Of the math teachers I observed today, half were very investigation-oriented while the other half was more lecture based. The two that had a more investigative classroom both had their own styles within that. One had students doing everything in groups and circulated as they worked. It was clear that she believed that students work best in groups, interacting and building off each other. She would often stop at one group and work with them, questioning their conjectures, but allowed them to speak their opinions to each other while listening. She would then collect the class and have a group summarize their findings. I found it very encouraging that she facilitated such a discussion with such control.

The other teacher who had a similar classroom had students working in groups also, but led the class more so than the previous. She often brought the class back together and would explain the answer to the class without asking the students. It was clear she thought students worked well in groups also, and would help facilitate their group discussion. For both of these teachers, they used their observations and questioning as formative assessment, which I think is a great check in point to see where students are at.

The other two teachers I observed used worksheets that students followed along with. They didn’t sit in groups like the previous, but discussed as a class often. I thought it was interesting to compare the worksheet vs. non-worksheet methods and the whole class vs. small group discussions. Both seemed effective in each teachers way, and each teacher seemed confident in their style.

My personal opinion is that students learn mathematics better when working in groups and exploring as a team rather than the teacher facilitating most of the class. While this is my opinion, students can learn math in other ways as well so I liked observing other teacher’s inputs.

Literacy in Math

Today we taught a lesson on literacy in mathematics. The lesson had students completing different types of story problems and then continuing on to practice explaining their work, as well as analyzing student work (below is an example of a student’s work ofCan do a problem we did in class). It was fun and different, making it also challenging to know how best to address the topic with students. My goal throughout the lesson was to focus on both my questioning and answering; specifically whether or not I am clear and precise with my answers, explanations, and reasoning. I feel as though the lesson went well, but as always, as I look back there are things I would have liked to say or do differently.

As I was correcting the homework, one of my students brought up a question regarding division. They didn’t understand why when we divide 6(-3+5n)/6, that the 6’s cancel and we are just left with (-3+5n). Their thinking was that we should divide 6 by each individual term. I went on to explain that 6 was a factor of both the numerator and denominator. Although my explanation was decent for not expecting such a question at all, looking back I would definitely have explained this in a more thorough and concrete way.

When we moved onto the lesson, we first had students think-pair-share regarding the student work at hand and whether or not it was correct and if it was incorrect, they needed to find the mistake. The think-pair-share had students more engaged and we therefore had much more participation when we came back together as a class. The students were involved in the question and all were on the same page. I could easily walk around to groups and discuss the problem with them as other groups were discussing. It gave me good insight on where students were at and what ideas they had. Once we decided the student’s work was incorrect and found the error, we had the students write down an explanation they would give and we then analyzed some of them after they were finished. When we used student’s examples, their peers seemed more engaged because they could relate better. I then gave an example of a solid explanation and we critiqued the components that are necessary for the explanation to be “good”. If I were to do this lesson again I would have the students explain their own work because I think their peers would be even more engaged and participate more. Overall for teaching this lesson a first time, I felt it went well and I felt my explanations and question answering were precise. After looking at their homework tomorrow and analyzing it, I’m curious to see what they got out of the lesson and what they’ve improved on.

Here is an article that supports analytical writing throughout content areas.

Progress & What Is To Be

I had two main focuses when starting this semester; one being explaining in a way that students understand, giving clear and concise detail, and the other being classroom control/strategies for classroom management. I think that both of these will take time and experience, but it’s amazing what you can learn in such a short amount of time. It makes it more difficult to come into a classroom in the middle of the semester. Procedure is already set, so I obviously follow suite and it makes it a little difficult to find my own fit for my style of classroom management, but I also choose to take advantage of seeing other people’s style.

As for what I can do well as a math teacher, I feel as though when I began this semester I wasn’t very good at explaining things in a way that made sense to this age group (hence my goal), and I also wasn’t very good at answering students questions with a probing question. I have improved on these greatly and feel as though I do them well, for the most part, now; although I still have much to learn. I have recently learned many new techniques in order to aid my explanations, catering to my students.

Being in a classroom every day with the same group of kids and same procedures has really helped me to gain insight in the way they think. I’ve been able to anticipate misconceptions a little easier knowing the students, and also find ways to help guide them toward where they need to be, just from getting to know them and how they work. I think this is one of the biggest things I learned and I’m finding it crucial to not only engage your students in a way that we can all learn together, but get to know them in a way that allows you to know what’s best for them and what they need in order to be successful and how you can help with that as a teacher. I think this well help me with my student teaching and in my future as a teacher. I realize that many things will come with experience such as classroom management and knowing how to explain, catering to your audience in the most effective way. By learning about different styles and methods, I can try and find a best fit for me.

Goals

A teacher never stops truly learning. For now, I am targeting certain areas in which I hope to learn and grow most in order to improve as an educator.

Classroom management is one of my biggest areas of focus right now. I know it will take time for me to find my management style which fits best for me and my personality as a teacher, but my goal is for it to be as effective as possible in that all of my students are engaged and excited about math.

I’m realizing more and more how explicit and precise you need to be in a math classroom. It’s very easy for me to blow over little ideas that students may miss, simply because I’m so used to them or assume they know. This is going to be my biggest struggle, but I can see I’m already improving and realizing where I need to be more concrete in my explanations.