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!”

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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

 

BLOG tes 1

Tessellation 1

 

BLOG tes 2

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.