Interview of Mathematical Experiences

After defining what math is to me, stating different problems that exist, and discussing gender issues I wanted to interview someone to see what it might confirm or deny for this particular person. I started out by asking questions that would give me an idea of what this female thought math was. Below is a narrative of the interview:

Me: Can you describe what mathematics is?

Her: Math is just problem solving using equations and numbers/

Me:Do you like or enjoy math?

Her: I am very good at math I just don’t like it.

Me: Could you tell me why you don’t like it?

Her: I don’t think it is useful in most situations. When am I ever going to use the quadratic equation in real life. I just had to memorize a bunch of equations that I’m going to forget at the end of the year anyways.

Me: So you say you are good at math, what make you say that?

Her: I get A’s in math.

Me: So you don’t really struggle?

Her: Nope

Me: Why do you think that is?

Her: My mom made me go to a learning center when I was little and memorize pointless facts that I used in my classes.

Me: Can you describe your worst mathematical experience(s)?

Her: In 3rd and 4th grade when we had to do the multiplication and division tables that were times. I could do them but not in a minute and it made me feel stupid.

Me: Can you describe your best mathematical experience(s)?

Her: (took a while to think) Nothing in particular. Maybe tanagrams, puzzles, anything that wasn’t just problems.

Me: How do you thin your math education could have been improved?

Her: If it was not based on memorization of equations and just using them in repeating practice problems with just different numbers.

Me: Do you think your enjoyment of mathematics has anything to do with being a female?

Her: NO

Me: Were most of your math teachers male or female?

Her: Mostly male.

After the interview I had a good grasp on her impression of what math is. I can take this interview into consideration for my classroom. Maybe not really force students to memorize equations all the time. Only emphasize the important ones that they should be using so often that it won’t really be memorization. I think that too many teachers reference the textbook as a guide for their curriculum. And this is great, but the textbooks emphasize everything equally. Teachers need to do a better job of transferring the importance of each concept to the students. Also, bringing meaning and purpose to everything they do! Looking forward to inspiring the world of those willing to absorb it.

 

Nature of Mathematics

Many people and commonly students believe that mathematics is the process of computing numbers in routine ways to obtain a specific desired result. They get frustrated when they can’t find a method in a textbook telling them a step by step procedure that fits the problem they are presented with. If that is what math is, then it should be easy for everyone. It would be like finding the map to a destination and following it. Many people try to do math this way and as a result can’t grasp the skills, beauty, or concepts of mathematics. This is because their idea of math is distorted. The idea of what mathematics is in society to do is so engrained that it is difficult to bring out the true mathematics. Football use to be the same way until society has shaped and molded it to be extremely accepted and now the majority of people participate in it.

Society needs to learn and experience what it really means to do mathematics. It is not solving problems on a worksheet or memorizing formulas. Mathematics is logically reasoning and thinking through a situation or question. For instance, I was flying on a plane and as we were landing I said to my friend, “I wonder what all has to go into landing a plane. The speed, the angle, the distance needed to stop.” This is an example of a mathematical question. After thinking the question, a mathematician would do mathematics by using his “tool box” to figure out the question either estimating an answer or finding exact ones. Now when I say “tool box” I mean an assortment of idea of where to start and mathematical knowledge that has been utilized previously that may help solve the problem. This may mean that the problem needs worked through several times before reaching a logical answer. Mathematics does just stop at an answer like commonly happens in schools. The next phase can vary but for this particular question a mathematician might investigate how the answer changes depending on the size or weight of the plain. Does the weather have anything to do with it? To answer some of these questions some research might need to be done to collect data, unless you have a bunch of planes in your backyard. With your data you can analyze it, look for patterns and reason through what the numbers mean. This part of the question involves some interpretation of the numbers. Depending on the mathematician this may look different. After answering some of these questions a mathematician might look at his work and ask, Could I find an answer in a different way? Why does this work?” Doing mathematics is playing with numbers and questions in an individually creative way.

Not everyone has the same mathematical questions or the same creative ways of finding answers, but everyone has these mathematical questions. Society just doesn’t recognize them as mathematics because they do not have a sense of “live” mathematics (that is, mathematics that isn’t staged in a textbook). If we can get society to just have conversations about these questions in a space where they would categorize their experience as mathematical (such as schools and workplaces), we could change the definition of mathematics.

Teaching Technology Lessons

So for my project, I have been working with Keegan, Danielle, and Paul Yu. We were creating a technology lesson on proportions and similarity. The lesson we had prepared involved Geogebra and other worksheets, so we decided to create a weebly (http://gvsujenison.weebly.com/) which worked easily for the students to access. The lesson had 4 main stages. The first was to review proportions using the computers and to actually have the students measure corresponding parts on the computer. This part was labeled as the Launch on the weebly page. The next step was to have students open the Geogebra link and have them play with it without moving the sliders. We had to give some basic instructions on how to use the tool first since this is the first time the students were seeing Geogebra. During this part the students were just playing with the tool and telling us what they observed (which was that the two figures were similar). The third step was to have the student move one particular slider (the top left) to 4.5 and see what they observe (the figures are not similar). The fourth step was to have each row move two different sliders and see what they can find. After each step we would have a discussion on the findings and construct mathematical arguments to support their conjectures. The last thing we were planning to have the students do was an exit slip which should only take about 5 minutes.

This past Tuesday, we actually visited Jenison high school and taught the lesson. Keegan taught first hour. Overall, the lesson did not go horrible. Being first hour and there was a strange person teaching their class, the students were not very talkative and it was difficult to get them engage in the activity. That is until Keegan and students move the sliders and play with the figure. You could see the students become instantly engaged and you could see the learning take place.

After the class had finished, Keegan, Paul, and I went to reflect on what had just happened. Basically, I said he should have used the board and reviewed some of the terminology with the students since they just learned proportions the day before. Jill (the original teacher) also said that she doesn’t use the term scale factor often and they use the term magnitude instead. I also suggested moving the 6 columns of desks together to create 3 columns of desk that are two desks wide so each student has a specific partner to discuss the activity with. When Keegan instructed the students to discuss certain things, they just sort of looked around and moved wherever. Paul mentioned that I should say to convince your partner using measurement that the figures are similar or not similar. Keegan had told the student to talk with their partner and convince them that the figures were similar. You could here the students just saying they were similar based on how they looked.

When I taught the lesson third hour, I thought it was very successful. First of all, the students were more awake and already talking as they came in the room. The review I added of terminology such as magnitude and proportions was a great help to the lesson. I left these up on the board the entire hour for students to reference (which they did). There was also a lot more conversation happening between partners about the figures being similar based on the measurements they took. I also think grouping the desks together helped entice conversation. I was not as happy with my conclusion as I would like to be. After I somewhat summed up the moving of the sliders I moved straight on to the exit slip we asked the students to fill out. I wish I would have reviewed the entire lesson with the students asking them what they learned as the lesson progressed. Keegan did this and it was great and I didn’t realize I did not until I had already instructed the students to do the exit slip.

Overall, I believe the students liked the change of their typical lesson and using the technology to learn. It was definitely a lot of work creating the technology lesson but I think it paid of with the student learning that came out of it. When I was observing Keegan during the first hour I paid attention to many things I would change and do differently wither because of my personal teaching style or the way the lesson just needed to be modified. However, I never paid much attention to the time management that Keegan used throughout his lesson. Then, when I taught third hour, I had to somewhat guess based on my instincts of how long each part would take.

Our next step is to write a formal review of the lessons and the experiences.

Accessible Mathematics, Book Review

For my MTH 495 course I read the book Accessible Mathematics which addressed important topics for many mathematics teacher. At the beginning of the book it stated, “The most important variable in determining the quality of education is the teacher.” I could not agree with this statement more and the students agree with it too. Generally, when a student dislikes a class or is engaged in a class they refer to the teacher and what they are doing that is either horrible or engaging. Going through the chapters of this book, any math educator can learn about different ways to be an engaging teacher and learn about the ways of boring non effective teacher.

One of the common acts of an ineffective math teacher is assigning tons of practice problems. Now I agree the students need to practice and implement what they learn. But if a student is doing something wrong and they do it 25 times the wrong way and then you correct it once, what do you think the students will remember. Also, there is never enough time to go over all the practice problems generally assigned and it can be counterproductive. Having students memorize formulas that they are not going to remember after your class anyways is a waste of time. This class time could be spent to further conceptual understanding.

The book notes that short warm-ups can be very effective because it gets the students focused on math and is a great way to make the curriculum on going. This leads to another point of vocabulary. The incorrect math language used is commonly the source of student misconceptions and thus teachers should always be aware of how they speak. A good teacher also keeps in mind the different ways students learn and thus should incorporate multiple representations and visuals. When choosing problems for students it is very beneficial to put them into real world contexts and situation that they can relate to. Doing so engages the students and grabs their attention. To keep students interests, the book also suggest to mix it up and not always stand at the board with notes , a textbook, or a worksheet. The book suggested bringing in the book of world records and making a math lesson out of that. Creative! An effective teacher additionally, should question everything. Ask the students why, or how they got their answer, how do they know, etc. I love to questions students because it can furthers there math abilities, it can uncover misconceptions, and eventually the students will be comfortable making mistakes.

All in all, this book was very insightful. I would recommend this book to someone just beginning to learn about math education. Many of the points the book made I had heard of before so it was a smooth read for me. There were a few new ideas I gained my reading this book and for that it was worth it. As a final point, like the book stated very well, “if we want better outcomes, we can’t keep teaching the same way!”

Roman Numerals to Hindu Arabic Number System

In 1170, Leonardo of Pisa, better known as Fibonacci, was born in Pisa Italy and is thought to be one of the most talented mathematicians of the middle ages. Most of his childhood was spent in North Africa where his dad worked as a merchant. This is how Fibonacci was exposed to the importance of numbers. The merchants had set prices for their goods and had to deal with set taxes on imports, etc. Once Fibonacci became a teenager, he began to travel the Mediterranean coast doing work for his father. While doing so, he learned many different systems for doing arithmetic and truly seen the many advantages of the Hind Arabic number system.

Contraire to what most might think, Fibonacci did not create the Hindu-Arabic number system. What he did do was make it accessible! In 1202, Fibonacci released the book Liber Abaci, meaning the book of calculations, which introduced the numbers system to Europe. The book sold the system itself by it simplicity of use. This new number system made expressing and doing math so much easier and all its greatness was expressed in the book. This book contains most of the methods for addition, subtraction, multiplying and dividing that is stilled learned in elementary schools. The book also contains his famous rabbit problem which also contains the famous Fibonacci sequence. Within this sequence also lies the golden ratio (1.31803…). The sequence can be found in many places such as nature, art, music, and is even used in computer science. This book significantly changed the way of the world of mathematics. It is crazy to think that over 800 years ago this number system was introduced and is still in prominent existence today. I think that we have used this number system for so long that it would be difficult to change again unless something as drastic as the change from Roman numbers to the new system was proposed or presented. Today, there is talk about how the number system is slightly flawed and sometimes difficult to learn. For instance, eleven and twelve really don’t even fit into the language of the number system very well and is sometimes difficult for children to learn. I can see the number system changing again one day but not to the extent that Fibonacci had presented.

Before the Hindu Arabic number system was introduced, Roman numerals were commonly used. Roman numerals were developed around 500 BC and consist of seven main symbols: I (1), V (5), X (10), L (50), C (100), D (500), and M (1000). Generally, the largest number was always on the left when writing with Roman Numerals. This is somewhat similar to how we think of our place value system today. The largest place value number is furthest left. In Roman Numerals, the numbers were added together left to right: LXVII → L + X + V + I + I = 67. With the new number system this was just done easier by adding 60 (the 6 hold the ten place and hence represents 60) with 7 (which hold the ones place). Roman Numerals further developed by using shorter notation. That is, instead of writing VIIII to represent 9, they would subtract whenever there was a smaller symbol before a larger symbol so that 9 could be expressed easier as IX (two symbols instead of five).

The Hindu Arabic number system further simplified the Roman Number system. However, we still see Roman Numerals often now days. For example, in chemistry compounds are written with roman numerals (Iron (II) Oxide, Iron (III) Oxide), clocks use roman numerals, book volumes, chapter numbers, and music markings (capital roman numerals to indicate major chords, lowercase to indicate minor). My question is why are we still using Roman Numerals today? Are we using them to show historical systems, the elegance, or do they still hold some significance over the Hindu Arabic system?

Making a Tessellation

 

Tessellation 1

 

Tessellation 2

To start making my tessellations, I began by making different shapes and seeing how they fit together in a nice way. My goal was to use curved lines in a tessellation because I think it gives them less of the standard pattern look. To make the tessellation I was going to make a design in one square grid and then reflect it multiple times. I wanted some of the shapes to connect and make new shapes when I reflected my original design. In my first tessellation, I used a lot of arc lines (semi circles) in corners of my original square and straight lines in the corners. I wasn’t really sure how it was going to look when I reflected it but in the end, hearts were the main new shape created by reflected my image. After getting a better feel of how an image might look after being reflected on Geogebra, I made a second tessellation. I really wanted to continue connecting the shapes so I again used arc lines in the corners this time which created a circle in the corner were 4 reflections met. After practice with the first tessellation, I feel that this one looks more complete.

Next, I researched what type of tessellation I had created. We had discussed in class that a regular tessellation was made up of congruent regular polygons. Also, we mentioned that a semi regular tessellation is made up of polygons so that every vertex is identical. Neither of these necessarily fit my images. I found that there are also demi regular tessellations as well. These tessellations can involve curved shapes not just polygons. However, I guess there is some debate in mathematics whether these are actually tessellations. I think both my images fit between a semi regular tessellation and a demi tessellation. This is because I did use some polygons but not all my vertexes are the same. I also used a bunch of curved lines, but not to the extent of the images I had seen when looking at demi regular tessellation.

How Did Greek Mathematics Develop?

When I think about the history of mathematics I wonder where numbers come from? Who decided that “one” would be written as “1”, and where did the symbols like pie and the equal sign come from? Well, just like the alphabet and English language has evolved, so has the way we use the mathematical system.

Although it is not certain, Greek mathematics is said to be built up from the Babylonian and Egyptian civilizations. The Greeks took what was known about mathematics and took a more advanced approach. That is, instead of studying mathematics using inductive reasoning, the Greeks used deductive reasoning. Greek mathematics was so different that the other mathematics at this time because of the reasoning and justifications that came with everything they did. A big part of math is asking questions and trying to find the answers and explanations that go with them. Greek mathematicians were some of the first to bring this part of mathematics to life. This particular style of studying mathematics lead to many discoveries for the Greeks. The Greeks helped formalize mathematics by introducing what we now call theorems and proofs. The Greeks created the foundation of math today. That is, everything we want to conjecture or claim to be true, mist be backed with reasoning and proof that it is so. This is primarily thanks to the Greeks. Overall, Greek mathematics can be broken down into three time periods: the Hellenic time, the Golden Age, and the Hellenistic time.

The Hellenic time (6th century B.C.) primarily involved four mathematicians and their associated schools. Plato studied mathematics in a way to understand reality and the world around us. He believed that geometry was the key to understanding the world. Pythagoras was credited with many discoveries and often related music to math. All of the most ancient mathematical texts include one of his bigger discoveries: The Pythagorean Theorem. Thales studied the geometry of lines and thought of abstract geometry which he  was credited with five theorems. Aristotle clearly distinguished axioms and posits.

In the Golden Age (5th century B.C.), Zeno of Elea proposed infinite and studied the relationship of points and numbers which lead to the discovery of irrational numbers.

The Hellenistic time (3rd century B.C.) was when Euclid presented his elements. He clearly defined point and line which lead to his axioms and postulates.

There is still much I do not know about how Greek mathematics developed, but then again there is still much I don’t know about how math came to be. I look forward to discovering more of the mysteries of mathematics!

What is math?

To describe math or my favorite thing about math it a challenge because math today is so enormous. That’s like asking me to choose my favorite ice cream and I can’t because there are so many good flavors and so many different combinations. Similar with math there are so many ways it could be described it makes it a difficult task. Generally, I would describe math as a resource for information. Math can be used to problem solve, find patterns, make predictions, provide reasoning, and much more which is all in the tool box of math – the resource for information.

I mean honestly what would we know or be able to do in this world without math. We use this resource in chemistry, physics, businesses, taxes, accounting, architecture … the list goes on! Say you’re playing Yahtzee and you roll all five dice to show all 3’s, and your friend says, “What is the likelihood of that actually happening?” Math is the resource you would have to utilize to figure that out. What if you’re at the hospital and you need a drug to remain in your system at about the same concentration for the next 10 hours? Your doctor would have to use math to figure out how often to give you the drug.

So obviously, math is important. My question is, “How did math come to be? Someone had to create the system or discover it at least. Then there are all the specific areas of study that people helped develop. I know Euclid was a big part of Geometry. Aristotle strongly believed in logic, mathematical reasoning and justification which likely lead to proofs. The most recent discoveries with math are in the application with technology. We have so many tools know that help us do math calculations such as calculators, math programs, search systems like Google. At the moment, I do not know a lot about the history of mathematics but I am interested to learn more in MTH 495.