# Thoughts on Teaching Math with technology

## Sunday, March 18, 2018

## Saturday, March 17, 2018

## Friday, March 16, 2018

### Mathematical Sculptures

I was trying to find information on the mathematics used by sculptors and instead of finding that, I found sculptures based on mathematics. I didn't know how many artists use mathematics to create their art or sculptures.

In today's world, mathematics is an important component in both animation and digital graphics. In addition, certain artists are using mathematics to create sculptures.

Ricardo Zalaya Baez wrote a proposal for classifying mathematical sculptures based on the type of materials and mathematical properties. He divides the art into geometrical sculpture, sculpture with concepts of calculus, sculptures with algebraic concepts, topological sculptures and sculptures with different mathematical concepts.

Through out the paper, he includes photographs of examples of each type of mathematical sculptures and their creators. Here are some who were chosen.

Helaman Ferguson who has a doctorate in mathematics but is an artist who uses mathematics to create his sculptures. One of his creations is Umbilic Torus SC is created using a computer program designed to direct a robot to care 144 different sandstone pieces which were later cast in bronze. Many of his other sculptures are variations on Torus.

Another artist is George Hart who creates geometric based sculptures. He has created a 3 dimensional print of the famous Sierpinski triangle. He is also known for his constructive geometric forms created from patterns and relationships.

Zachery Abel is a bit different in that he creates his works from everyday objects such as paperclips and those large clips you use with thick packets of paper. Zachery is a lecturer in mathematics for computer science at MIT. He does hold a doctorate in mathematics. If you check out his website, you can click on pictures of his works where he explains how he made the sculpture and the mathematical idea behind it.

Check out Bathsheba Grossman, an artist who uses computer programs to crate 3 dimensional printed steel. She uses pure math to create things like a Klein bottle opener, a gyroid, A Borromean rings Seifert Surface, etc. These sculptures are fantastic and fascinating.

If you read the paper by Baez, you can discover so many more sculptures who have produced mathematically based sculptures since the 1950's although there are some works from the early 1910's which could be classified in this group.

I am impressed with the huge number of artists out there who produced mathematically based sculptures out of so many different materials. I could see having students research various artists, their sculptures and contribute a page or two to a book created by the whole class on mathematical sculptures.

Let me know what you think. I'd love to hear. Have a great day.

In today's world, mathematics is an important component in both animation and digital graphics. In addition, certain artists are using mathematics to create sculptures.

Ricardo Zalaya Baez wrote a proposal for classifying mathematical sculptures based on the type of materials and mathematical properties. He divides the art into geometrical sculpture, sculpture with concepts of calculus, sculptures with algebraic concepts, topological sculptures and sculptures with different mathematical concepts.

Through out the paper, he includes photographs of examples of each type of mathematical sculptures and their creators. Here are some who were chosen.

Helaman Ferguson who has a doctorate in mathematics but is an artist who uses mathematics to create his sculptures. One of his creations is Umbilic Torus SC is created using a computer program designed to direct a robot to care 144 different sandstone pieces which were later cast in bronze. Many of his other sculptures are variations on Torus.

Another artist is George Hart who creates geometric based sculptures. He has created a 3 dimensional print of the famous Sierpinski triangle. He is also known for his constructive geometric forms created from patterns and relationships.

Zachery Abel is a bit different in that he creates his works from everyday objects such as paperclips and those large clips you use with thick packets of paper. Zachery is a lecturer in mathematics for computer science at MIT. He does hold a doctorate in mathematics. If you check out his website, you can click on pictures of his works where he explains how he made the sculpture and the mathematical idea behind it.

Check out Bathsheba Grossman, an artist who uses computer programs to crate 3 dimensional printed steel. She uses pure math to create things like a Klein bottle opener, a gyroid, A Borromean rings Seifert Surface, etc. These sculptures are fantastic and fascinating.

If you read the paper by Baez, you can discover so many more sculptures who have produced mathematically based sculptures since the 1950's although there are some works from the early 1910's which could be classified in this group.

I am impressed with the huge number of artists out there who produced mathematically based sculptures out of so many different materials. I could see having students research various artists, their sculptures and contribute a page or two to a book created by the whole class on mathematical sculptures.

Let me know what you think. I'd love to hear. Have a great day.

## Thursday, March 15, 2018

### Car Production

My curiosity in how math is used in real life leads me to explore a variety of topics. I'm still working on a couple more in regard to art but I'm pausing to look at the math involved in car production. We know if car companies make mistakes you end up with the Ford Edsel, a car known as one of the worst vehicles in history. Others include the Ford Pinto or the AMC Pacer.

So how do car companies go about deciding on new cars to market. One of the first things done is to create a market analysis by analyzing data to determine what sells, where it sales, and the price it sells at. In addition, they look at the horsepower needed, weight, fuel economy, and size before sending their detailed ideas to engineering. This mathematically based information that dictates the finished product.

In engineering, they use computer aided modeling to build the chassis inside the computer before virtually crash testing it again and again using calculus and physics based algorithms. Furthermore, they use geometry and trigonometry are used to design the suspension and anywhere there is a load bearing component fastened to another.

As for the exterior design, they use fluid flow programs to check on wind flow under, over, and around the vehicle. This particular computer program allows the engineers to fine tune the aerodynamics of the car design. Geometry and trig is used when designing the interior of the car. Its used to help make the interior appealing to the perspective buyer and the correct placement of pillars, dashboard, dials, steering wheel, radio, etc.

Once everything is set, its time to build the cars. In the old days, men did everything but now many cars are build by robots who are controlled by computer calculations so everything is accurate to the nano-meter so everything fits precisely.

Once the car is built, the math is not done. A certain number of cars are crash tested while others are taken out and road tested. Those that are road tested have precise measurements taken of its ride, handling, emissions, etc. These measurements are analyzed mathematically in addition to figuring out delivery schedules, delivery costs, and is sent to the dealership.

Math is used from start to finish and even beyond. When you take your car in to be fixed, you are often charged for parts and a per hour cost for labor. The car and its production has come a long way from the Model T build by Henry Ford.

Let me know what you think. I'd love to hear.

So how do car companies go about deciding on new cars to market. One of the first things done is to create a market analysis by analyzing data to determine what sells, where it sales, and the price it sells at. In addition, they look at the horsepower needed, weight, fuel economy, and size before sending their detailed ideas to engineering. This mathematically based information that dictates the finished product.

In engineering, they use computer aided modeling to build the chassis inside the computer before virtually crash testing it again and again using calculus and physics based algorithms. Furthermore, they use geometry and trigonometry are used to design the suspension and anywhere there is a load bearing component fastened to another.

As for the exterior design, they use fluid flow programs to check on wind flow under, over, and around the vehicle. This particular computer program allows the engineers to fine tune the aerodynamics of the car design. Geometry and trig is used when designing the interior of the car. Its used to help make the interior appealing to the perspective buyer and the correct placement of pillars, dashboard, dials, steering wheel, radio, etc.

Once everything is set, its time to build the cars. In the old days, men did everything but now many cars are build by robots who are controlled by computer calculations so everything is accurate to the nano-meter so everything fits precisely.

Once the car is built, the math is not done. A certain number of cars are crash tested while others are taken out and road tested. Those that are road tested have precise measurements taken of its ride, handling, emissions, etc. These measurements are analyzed mathematically in addition to figuring out delivery schedules, delivery costs, and is sent to the dealership.

Math is used from start to finish and even beyond. When you take your car in to be fixed, you are often charged for parts and a per hour cost for labor. The car and its production has come a long way from the Model T build by Henry Ford.

Let me know what you think. I'd love to hear.

## Wednesday, March 14, 2018

### Happy Pi Day

Pi is probably the most famous ratio in history. It is also one of the most famous irrational number known to man. Today is the day designated to celebrate the awesomeness of pi.

As you know, pi is the ratio of circumference to diameter which means that no matter the size of the circle, the ratio is always the same.

Although the concept of pi has been around for a long time, it was only in 1706 that William Jones first used the Greek symbol for pi. But The symbol did not come into popular use until Leonhard Euler used it in 1734. The letter p in pi represents the perimeter of a circle.

Interesting fact: Pi day became an officially recognized by the United States Government when House Bill 224 passed the first session of the 111th Congress of the United States in 2009. March 14th is the perfect day for pi since its set for the 14th day of the 3rd month or 3.14.

Now for some interesting facts about pi.

1. Pi day is also Albert Einstein's birthday along with several other famous people.

2. If you were to print the first 1 billion digits in regular font, it would cover the distance from New York City to Kansas City.

3. "I prefer pi" is a palindrome.

4. In "Wolf in the Fold" from the original Star Trek, Spock beats the evil computer by having it calculate pi to the last digit.

5. Pi as the secret code plays a part in two movies - The Net with Sandra Bulluck and Torn Curtain by Alfred Hitchcock.

6. There are no zeros in the first 31 digits of pi.

7. Givanchy marketed a cologne named Pi.

8. It is said that everybody's birthday appears somewhere in the digits in pi in order.

9. At position 763, there are 6 nines in a row. This is known as Feynman point.

10. The longest sequence of numbers to appear in order is 12345 and it appears eight separate times.

11. Pi has been calculated to the 1.24 trillion digits and calculations continue.

Have a wonderful Pi Day.

As you know, pi is the ratio of circumference to diameter which means that no matter the size of the circle, the ratio is always the same.

Although the concept of pi has been around for a long time, it was only in 1706 that William Jones first used the Greek symbol for pi. But The symbol did not come into popular use until Leonhard Euler used it in 1734. The letter p in pi represents the perimeter of a circle.

Interesting fact: Pi day became an officially recognized by the United States Government when House Bill 224 passed the first session of the 111th Congress of the United States in 2009. March 14th is the perfect day for pi since its set for the 14th day of the 3rd month or 3.14.

Now for some interesting facts about pi.

1. Pi day is also Albert Einstein's birthday along with several other famous people.

2. If you were to print the first 1 billion digits in regular font, it would cover the distance from New York City to Kansas City.

3. "I prefer pi" is a palindrome.

4. In "Wolf in the Fold" from the original Star Trek, Spock beats the evil computer by having it calculate pi to the last digit.

5. Pi as the secret code plays a part in two movies - The Net with Sandra Bulluck and Torn Curtain by Alfred Hitchcock.

6. There are no zeros in the first 31 digits of pi.

7. Givanchy marketed a cologne named Pi.

8. It is said that everybody's birthday appears somewhere in the digits in pi in order.

9. At position 763, there are 6 nines in a row. This is known as Feynman point.

10. The longest sequence of numbers to appear in order is 12345 and it appears eight separate times.

11. Pi has been calculated to the 1.24 trillion digits and calculations continue.

Have a wonderful Pi Day.

## Tuesday, March 13, 2018

### Remediation in Mathmatics.

The other night, I was reading a book on working with students who are well below grade level. The usual practice is to begin them where they are and work to get them caught up but the author of the book said this was not a good idea because they'd never get caught up.

This lead me to wonder what are some good ways to work with students who need remediation while allowing them to learn the new material just like their classmates.

Most students who spend time on remediation based computer programs know they are behind. Many have been working on the same computer program for a couple of years, yet they are not caught up yet. Perhaps they feel as if they may never catch up.

One article suggests increasing math rigor rather than slowing down. Teachers should intensify their instruction develop their abilities in math, develop better recall, improve learning behaviors, and help them move beyond solving problems using a memorized series of steps. In addition, it is suggested teachers help motivate students so they move past their belief that they cannot do math. Furthermore, instruction should include conceptual learning so a student has multiple ways to solve problems while providing opportunities for critical thinking and helping them connect to various concepts.

While looking at the topic, I came across the phrase "remediation through acceleration". Remediation is having students work on learning concepts from the past while acceleration is having students learn the material before the others in the class. This process gives struggling students a chance to stay up with their classmates. This is often done in a second class taught earlier than the regular math class.

Its well known students learn better when they have some prior knowledge of the concept and there is quite a lot of research supporting this. The idea is to expose students to the new concept so they have a chance to build prior knowledge before they are taught the topic in class. Acceleration does visit basic skills but only in the context of those skills to be immediately applied to newest concept.

Often times, the lack of prior knowledge is connected to vocabulary development. So it is important to include vocabulary development of critical terms. A student who has a rich understanding of a topic, when asked to write down what they know, will make a list using proper vocabulary to describe the topic.

For acceleration to succeed, the teacher needs to figure out exactly what skills and vocabulary the student needs in order to learn the new concept. The goal of accelerated learning is to have students:

1. Understand the purpose of the concept and real world connections,

2. Acquire critical vocabulary

3. Learned the basic skills needed.

4. Learned the new skills needed.

5. An idea of where instruction is headed.

When implementing this method, it is important to identify which students should be in it, deciding who will teach the class and when. Essentially, these students will be taking two math classes each day, one to help build the skills they need for the second class which will provide regular instruction as normal.

This is an interesting idea. In essence, it is providing scaffolding to help students keep up with their classmates rather than separating them into a slower "remedial" class where they feel as if they are not as smart as others. I'd love to hear what you think. Let me know.

This lead me to wonder what are some good ways to work with students who need remediation while allowing them to learn the new material just like their classmates.

Most students who spend time on remediation based computer programs know they are behind. Many have been working on the same computer program for a couple of years, yet they are not caught up yet. Perhaps they feel as if they may never catch up.

One article suggests increasing math rigor rather than slowing down. Teachers should intensify their instruction develop their abilities in math, develop better recall, improve learning behaviors, and help them move beyond solving problems using a memorized series of steps. In addition, it is suggested teachers help motivate students so they move past their belief that they cannot do math. Furthermore, instruction should include conceptual learning so a student has multiple ways to solve problems while providing opportunities for critical thinking and helping them connect to various concepts.

While looking at the topic, I came across the phrase "remediation through acceleration". Remediation is having students work on learning concepts from the past while acceleration is having students learn the material before the others in the class. This process gives struggling students a chance to stay up with their classmates. This is often done in a second class taught earlier than the regular math class.

Its well known students learn better when they have some prior knowledge of the concept and there is quite a lot of research supporting this. The idea is to expose students to the new concept so they have a chance to build prior knowledge before they are taught the topic in class. Acceleration does visit basic skills but only in the context of those skills to be immediately applied to newest concept.

Often times, the lack of prior knowledge is connected to vocabulary development. So it is important to include vocabulary development of critical terms. A student who has a rich understanding of a topic, when asked to write down what they know, will make a list using proper vocabulary to describe the topic.

For acceleration to succeed, the teacher needs to figure out exactly what skills and vocabulary the student needs in order to learn the new concept. The goal of accelerated learning is to have students:

1. Understand the purpose of the concept and real world connections,

2. Acquire critical vocabulary

3. Learned the basic skills needed.

4. Learned the new skills needed.

5. An idea of where instruction is headed.

When implementing this method, it is important to identify which students should be in it, deciding who will teach the class and when. Essentially, these students will be taking two math classes each day, one to help build the skills they need for the second class which will provide regular instruction as normal.

This is an interesting idea. In essence, it is providing scaffolding to help students keep up with their classmates rather than separating them into a slower "remedial" class where they feel as if they are not as smart as others. I'd love to hear what you think. Let me know.

## Monday, March 12, 2018

### Artists and Mathematics

I discovered another two or three ways art is connected to math but I'm not counting perspective, scales or anything like that. I'm referring to art work which has a particular mathematical slant but may not be created by mathematicians.

If you ever studied art history in high school or college, you might remember the cubism movement from the early 1900's.

The two most famous artists of that movement were Pablo Picasso and George Braque who began the movement. The name appears to have come from a comment on Braque who "reduced everything to geometric lines, cubes".

The artists broke everything down into planes so they could show different viewpoints at the same time in the same space using lines, angles, and shapes to create their distinctive style.

On the other hand, check out a more recent artist by the name of Frank Stella who created art through the use of irregular polygons that are bright and festive. He is an American born artist who spent several years in the 1960's creating art made up of lines, circles, etc.

His polygons not only have different length sides but they are also repeated patterns of broken circles or stripes that flow in geometric shapes. One is a square divided into four isosceles triangles using two diagonals. In each quadrant, there are stripes of two alternating colors going to the center.

Another one is horizontal stripes broken by rhombus divided into four triangles with the lines going outwards in an x shape so the lines meet the horizontal lines. Its in black and white and really really cool. Within that three year period, he created some wonderful pictures that used only geometric shapes and are awesome.

Other artists to look at are Simon Beck who creates art like Koch snowflake or Sierpinski triangle on snow using nothing more than a compass and his snow shoes. This art is large and covers a huge area. Its like he translates a small picture into a larger model. His art is fantastic and quite realistic. Then there is Hamid Naderi Yeganeh who uses computer programs based on mathematical formulas to produce computer generated art work which is intricate and three dimensional in appearance.

Check out Tom Beddard who creates Faberge Fractals. He generates them by using the output from one time as the input for the next run in a iterative formula. The art is quite detailed and absolutely breathtaking. Did you know there are different types of fractals? Each fractal produces a different type of picture. For instance, the L-systems produce a fern looking plant. Check this site out for more information on this.

Think about sharing these artists and their art with students to show them how mathematics can produce beautiful work worthy of being shown in galleries. I think its important to show students more than just the mathematics themselves. Sometimes you have to venture outside the box to give students an appreciation of the whole topic.

If you ever studied art history in high school or college, you might remember the cubism movement from the early 1900's.

The two most famous artists of that movement were Pablo Picasso and George Braque who began the movement. The name appears to have come from a comment on Braque who "reduced everything to geometric lines, cubes".

The artists broke everything down into planes so they could show different viewpoints at the same time in the same space using lines, angles, and shapes to create their distinctive style.

On the other hand, check out a more recent artist by the name of Frank Stella who created art through the use of irregular polygons that are bright and festive. He is an American born artist who spent several years in the 1960's creating art made up of lines, circles, etc.

His polygons not only have different length sides but they are also repeated patterns of broken circles or stripes that flow in geometric shapes. One is a square divided into four isosceles triangles using two diagonals. In each quadrant, there are stripes of two alternating colors going to the center.

Another one is horizontal stripes broken by rhombus divided into four triangles with the lines going outwards in an x shape so the lines meet the horizontal lines. Its in black and white and really really cool. Within that three year period, he created some wonderful pictures that used only geometric shapes and are awesome.

Other artists to look at are Simon Beck who creates art like Koch snowflake or Sierpinski triangle on snow using nothing more than a compass and his snow shoes. This art is large and covers a huge area. Its like he translates a small picture into a larger model. His art is fantastic and quite realistic. Then there is Hamid Naderi Yeganeh who uses computer programs based on mathematical formulas to produce computer generated art work which is intricate and three dimensional in appearance.

Check out Tom Beddard who creates Faberge Fractals. He generates them by using the output from one time as the input for the next run in a iterative formula. The art is quite detailed and absolutely breathtaking. Did you know there are different types of fractals? Each fractal produces a different type of picture. For instance, the L-systems produce a fern looking plant. Check this site out for more information on this.

Think about sharing these artists and their art with students to show them how mathematics can produce beautiful work worthy of being shown in galleries. I think its important to show students more than just the mathematics themselves. Sometimes you have to venture outside the box to give students an appreciation of the whole topic.

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