Describe your blogging experience in this course. Do you think you will continue using your blog? Why or why not?

Before this course, I didn’t have much experience with blogging. I have played around some with the program; however, I still have a lot more to learn. I’m curious to see how my students will react to blogging. If used correctly, I think it can be very beneficial in the classroom. As I become more comfortable using it, I plan to start small and have students use it as an electronic journal.

What did you learn about yourself and your abilities or interests in Math or Algebra?

Going into this class, I already knew that I probably wouldn’t have as much experience as others taking it. I’ve only actually had 2 full years teaching in a Math classroom, so I don’t feel I had as much to give in regards to ideas or past successes/failures on certain topics. I do feel comfortable with the material, but I feel that I still have a lot to learn.

Did you learn or discover anything you found particularly interesting through your course activities or your own internet research? Describe one interesting discovery and why you found it fascinating.

I’ve definitely learned many interesting and helpful activities from this course. Being a new teacher, I really wanted to bring from this course a way to make my students want to learn Math. I have learned many ways to bring our everyday lives into the Math classroom. From allowing the students to make math definitions their own, or allowing them to do investigations on the Math Trail, these are just a few ways that will hopefully peak the students interest in Math.

Do you think you will use journals with your students? Do you think you will use blogs? Why or why not?

I will definitely encourage writing activities to my students. I’ve taught several subjects and I’ve always had my students keep a journal. As stated before, I would definitely like to try an electronic journal using blogging. With teenagers today being so technology savvy, I think it will be a great resource in the classroom.

After reviewing your classmates post, would you alter your definition?

After reviewing my classmate’s posts, I wouldn’t do much to change my definitions. At this level, I believe that a formal definition is too abstract for middle school students. I liked the idea of having the students create their own definitions and then comparing it to the formal one, a lot like we’re doing right now. When looking at variables, I do need to give more examples of different types of equations. It’s important that they know that there are one-step, two-step equations, etc. When looking at functions, I need to first give my students experiences that introduce them to the idea that a function takes a specified input and returns a unique output. They need lots of hands on practice with blocks, tiles, beads and to develop the vocabulary to describe those more concrete patterns. Then I can move on with more abstract patterns in functions.

Student Understanding

To make sure that students grasp the difference, they will have to give an example of an equation, and then give an example of a function. In the past, I’ve had difficulties with functions. It is important with functions that they know there is a relationship with the 2 quantities and that for every input there is only one output. I would show the students examples that show the world is full of relationships. For example, how far you run depends on how fast you run. Getting rid of a headache is a function of taking aspirin. I would then have the students list 5 relationships that exist in the real world. They will have to explain what effect one has on the other; how does one depend on the other?

Pascal’s Triangle is a geometric arrangement of the binomial coefficients in a triangle. The rows of Pascal’s triangle are conventionally enumerated starting with row zero, and the numbers in each row are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On the zeroth row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. For example, row two contains the numbers 1 3 3 1, where each element is generated from the sum of the two numbers directly above it. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

The outer left and right side of the triangle consists of all 1’s.  This pattern of adding the two numbers above to get the number below is continued throughout the triangle.  When you dissect the triangle row by row you start to see that every odd row is symmetrical and creates a palindrome.  For instance, row 3 is: 1, 3, 3, 1 and row 5 is: 1, 5, 10, 10, 5, 1. Both rows can be read forwards and backwards.

There is a “math mind” -some people have it and some  people don’t.

This is a view of many people out there. There are two types of math people, you either like the subject or you don’t. I believe that some of those people that do hate math, it has nothing to do with their abilities. I think people get discouraged at a young age with math and then just “shut off” from the subject. Some people assume they can’t do it so they choose to hate it instead. And then others get it stuck in their mind their never going to use it so then they also choose to hate the subject. I’ve always been good at math and I really looked forward to going to class, yet most of my classmates dreaded it. I think there are a lot of people who might appreciate math if it was presented in an applicable problem-solving tool. I had some good teachers who really helped make math interesting, which is why I love it today. This is the image of math we students. I also believe that some people just aren’t logical thinkers, and are much better at arts and languages. The brain works in two ways, creatively and logically. Math is hard for some people because they aren’t logical thinkers. If math people were forced to do art throughout the whole high school curriculum, they would hate it just as much because they would have a much harder time than someone who is naturally artistic.

There is always one best way to do a math problem.

I don’t agree with this myth. This is another reason why I love this subject. As a math student, I loved that there was more than one way to get to a solution. I feel that what works best for one student might not work for another. Think about the many different ways people figure out a tip for a restaurant bill, or how much they will save on a sale item. Of course there might be a “best way” to arrive at a solution, but students should be encouraged to find more than one solution to a problem and to be able to explain each. The focus sometimes needs to be on how students arrived at an answer, rather than the answer itself.

“Fractals” and “Nature” and “Patterns”

Were there ideas or concepts you were not familiar with? What were they?

“Fractals are to chaos what geometry is to algebra.”

I have heard about fractals, but I didn’t realize they could be found with such diversity in nature. They turn up in fruits and vegetables, germs, snowflakes, plants, animals, mountain ranges, shorelines, and many more places. The delicate Queen Anne’s Lace (shown below left), which is really just a wild carrot, is a beautiful example of a floral fractal. Each blossom produces smaller iterative blooms. This particular image was shot from underneath to demonstrate the fractal nature of the plant. The nautilus is one of the most famous examples of a fractal in nature. The perfect pattern is called a Fibonacci spiral.

What images did you find particularly striking?

I have always been mesmerized by the beauty and power of lightning. The pictures shown below are great examples of fractals found in nature.

Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?

Fractals are unique patterns left behind by the unpredictable movements – the chaos – of the world at work, so of course they can be found both in the workplace and at home. Look out the window and you can find fractals in the branch of the trees or vegetables in your garden. Look down and see them in the veins of your hand or legs. The crackle of paint on your walls can create a fractal pattern.

How can you adapt this webquest activity for your classroom?

I think that student’s would be interested and also amazed on how mathematics and nature tie together. I would approach this topic the same way and allow students to look at different images of fractals found in nature. I would then have them identify where they find fractals at home and in the classroom.

“Fibonacci” and “Phyllotaxis” and “Prime Numbers”

Were there ideas or concepts you were not familiar with? What were they?

I was vaguely familiar with Fibonacci, but knew that he discovered a pattern for the population of rabbits. I found it interesting that Fibonacci was born with the name of Leonardo Pisano and that he wasn’t even recognized until much later on. I never really thought that his finding has anything to do with items we find in nature. In the nineteenth century Fibonacci numbers were discovered in many natural forms. For example, many types of flower have a Fibonacci number of petals: daisies tend to have 34 or 55 petals, while sunflowers have 89 or, in some cases, 144. The seeds of sunflowers spiral outward both to the left and the right in a Fibonacci number of spirals.

What images did you find particularly striking?

I found the picture below of a daisy core both intriguing and striking. Daisies have always been my favorite flower; however, I’ve never looked at them in terms of mathematics. In the close-packed arrangement of tiny florets in the core of a daisy blossom, we can see the phenomenon in almost two-dimensional form. The eye sees twenty-one counterclockwise and thirty-four logarithmic or equiangular spirals. In any daisy, the combination of counterclockwise and clockwise spirals generally consists of successive terms of the Fibonacci sequence. Wow, that’s amazing!

Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?

The Fibonacci number patterns occur frequently in nature. We’ve already observed that you can find patterns in the arrangement of flowers such as calla lilies and daisies. You can look out your window and see the whorls on a pine cone, the numbers of rings on the trunks of palm trees, and the patterns of snail shells. Go into the kitchen and get a pineapple from the refrigerator. Pineapples have rows of diamond-shaped markings, or scales, which spiral around both clockwise and counterclockwise. If you count the number of scales in one of these spirals, you are likely to find 8, 13, or 21.

How can you adapt this webquest activity for your classroom?

I would list several Web sites for my students that had examples and images showing Fibonacci’s number patterns. Students can then print their findings and then on a separate sheet of paper write down what patterns they see or anything interesting. They can then brainstorm other places that these patterns are found.

Non-Traditional Pattern

A pattern that doesn’t follow a repetitive format.

Linear Pattern “Kid Language”

A pattern that repeats in either direction on a line.

Linear Pattern (formal definition)

If the plotted points make a pattern, then the coordinates of each point may have the same relationship between x and y values. In such a case, the x and y values are connected by a certain rule.

Linear Patterns exist when the points examined form a straight line.

One of the easiest ways to remember a linear pattern is that the word “linear” begins with the word “line.” The definitions are the same in that they both mention that there is a line with a pattern. The formal definition is more detailed and states the way the line is formed by mentioning x and y coordinates.

It is important for them to understand that patterns exist in their everyday life, and that these patterns can be organized and extended. I believe for students to fully understand linear patterns they must investigate the definition using models and graphs. I would draw my students into a discussion by asking them to predict and explain the growing pattern of a certain item. They can then use pattern blocks and draw graphs, which will allow them to make predictions and see the patterns of their data. Working from the concrete to the abstract is especially important for students who have difficulty with mathematics, and using pattern blocks or algebra tiles help students make this transition.