Monday, March 18, 2024

A New Verification For Robots

In the field of robotics, ensuring safety is paramount, especially when it comes to robot motion in shared spaces with humans. A new safety-check technique has emerged, promising to prove with 100 percent accuracy that a planned robot motion will not result in a collision. This groundbreaking development has the potential to revolutionize the field of robotics and make human-robot interactions safer than ever before.

Traditional safety-check techniques rely on probabilistic methods, which can sometimes lead to false positives or negatives, posing a risk to safety. However, the new technique, known as "formal verification," takes a different approach. By using mathematical algorithms and logic, formal verification can rigorously prove that a planned robot motion will not result in a collision, eliminating the possibility of errors.

One of the key advantages of formal verification is its ability to handle complex robotic systems with multiple moving parts. Traditional methods struggle to cope with the complexity of these systems, often leading to incomplete or inaccurate safety checks. Formal verification, on the other hand, can analyze intricate robotic motions and provide a definite answer regarding safety.

The implications of this new safety-check technique are far-reaching. In industries such as manufacturing, healthcare, and logistics, where robots often work alongside humans, it improves safety which is crucial. Using this technique, companies can have confidence that their robotic systems will operate safely, reducing the risk of accidents and injuries.

Moreover, formal verification can also accelerate the development and deployment of robotic systems. By providing a fast and reliable method for safety checking, developers can streamline the testing process and bring their robots to market faster.

While this improved technique represents a significant advancement in robotics safety, it is not without its challenges. Implementing formal verification requires specialized knowledge and expertise, and the technique may not be suitable for all robotic applications. However, with further research and development, formal verification has the potential to become a standard practice in ensuring the safety of robotic systems.

This new safety-check technique of formal verification promises to revolutionize the field of robotics by providing a rigorous and accurate method for ensuring collision-free robot motion. With its potential to enhance safety and accelerate development, formal verification represents a major step forward in the advancement of robotics technology. Let me know what you think, I'd love to hear. Have a great week.

Friday, March 15, 2024

How To Establish Mathematical Goals.

One of the seven effective mathematical teaching practices is to establish mathematical goals as a way of focusing learning. It is important to set clear and achievable mathematical goals since it is essential for guiding learning and ensuring academic success for students in grades K-12. These goals not only provide direction but also help educators tailor instruction to meet the diverse needs of students. Today, we'll explore the importance of establishing mathematical goals while outlining strategies that work across different grade levels.

Why is it important to establish mathematical goals? Setting mathematical goals help students and educators stay focused on what needs to be achieved, providing a clear path for learning progression. In addition goals provide a basis for assessing student progress and evaluating the effectiveness of instructional strategies. By establishing goals, educators can differentiate instruction to meet the individual needs of students, ensuring that all learners are appropriately challenged. Furthermore, clear goals can motivate students by providing a sense of achievement and progress as they work towards mastering mathematical concepts.

Setting goals will be different in the early grades from those used in high school. In the kindergarten to second grades, goals should focus on building foundational skills such as number recognition, counting, and basic operations. Goals may include mastering addition and subtraction within 20, understanding place value, and developing spatial reasoning skills.

Whereas the goals for grades 3 to 5 should expand to include more complex operations, such as multiplication and division, fractions, and basic geometry. Students should also develop problem-solving skills and the ability to apply mathematical concepts to real-world situations.

  1. Middle school goals should focus on deepening understanding of mathematical concepts, including algebra, geometry, and statistics. Students should also develop critical thinking skills and the ability to analyze and interpret data.


    By the time we set goals for high school students, we need to set goals to prepare students for college and career readiness, focusing on advanced topics such as calculus, trigonometry, and advanced algebra. Students should also develop the ability to use mathematical models to solve real-world problems.  


    In regard to actually writing goals, the goals should align with state and national standards to ensure that students are meeting grade-level expectations. One way is to use student data, such as assessment results and observations, to help with goal-setting and tracking student progress over time. Don't forget to involve students, parents, and other stakeholders in the goal-setting process to ensure that goals are meaningful and achievable. Finally, monitor student progress towards goals and adjust instruction as needed to ensure that all students are on track to meet their goals.

Remember that establishing clear and achievable mathematical goals is crucial for focusing learning and ensuring academic success for students in grades K-12. By setting appropriate goals and using data to monitor progress, educators can help all students achieve mathematical proficiency and develop the skills they need for future success. Let me know what you think, I'd love to hear.

Wednesday, March 13, 2024

Finding Solutions Through Visualization

Mathematics is often seen as a subject of numbers and equations, but it can also be a visually creative endeavor. One of the ways to solve problems is to create some sort of drawing or visualization since drawing representations of mathematical problems not only helps in understanding complex concepts but also in predicting their resolutions. Today, we explore the art of drawing mathematical problems and how it can lead to insights into their solutions.

Drawing mathematical problems involves creating diagrams, graphs, or illustrations that represent the problem at hand and help visualize each problem. This visualization can provide valuable insights into the problem's structure, relationships between variables, and potential solutions. For example, drawing a graph of a function can help in understanding its behavior and identifying key points such as intercepts, maxima, and minima.

In addition drawing mathematical problems can help in predicting possible solutions by allowing us to see patterns, relationships, and symmetries that may not be apparent from the equation alone. For instance, drawing a geometric figure can reveal hidden congruence or similarity relationships that can be used to solve a problem. Similarly, drawing a diagram of a trigonometric function can help in visualizing its periodic nature and predicting its behavior over a certain interval since you "see" all aspects of it.

One famous example of how drawing can find solutions in mathematics is the Four Color Theorem. This theorem states that any map can be colored using only four colors in such a way that no two adjacent regions have the same color. While the proof of this theorem is complex, it was initially conjectured based on the observation that maps could be drawn in such a way that only four colors were needed, leading mathematicians to search for a proof of this conjecture.

There are numerous benefits by creating visual representations. By drawing mathematical problems, people are not only able to find solutions but also has several other benefits. Drawing can aid people in understanding complex concepts, exploring mathematical ideas, and communicating solutions to others. Furthermore, drawing can enhance creativity, critical thinking, and problem-solving skills, thus making it a valuable tool in mathematical education.

So creating drawings of mathematical problems is an art that can lead to insights into possible solutions. By visualizing problems, we can see patterns and relationships that may not be apparent from equations alone, consequently helping us find solutions and deepen our understanding of mathematical concepts. So, it is important to teach students that the next time they encounter a mathematical problem, try picking up a pencil and sketching it out to see possible answers. Let me know what you think, I'd love to hear.

Monday, March 11, 2024

Pi Day Is Coming Up On Thursday.

 

On Thursday, we celebrate pi day which is a beautiful look at a wonderfully helpful irrational number. Every year on March 14th, mathematicians, scientists, and enthusiasts around the world celebrate Pi Day, a day dedicated to the mathematical constant π (pi). Pi, often approximated as 3.14, represents the ratio of a circle's circumference to its diameter and is a fundamental constant in mathematics, with an infinite number of decimal places that never repeat.

Pi Day was first celebrated in 1988 by physicist Larry Shaw at the San Francisco Exploratorium. Shaw, known as the "Prince of Pi," organized a march around the museum's circular spaces and concluded the event with a pie-eating celebration, honoring both the mathematical constant and the delicious dessert.

Today, Pi Day is celebrated worldwide with various activities, including pi recitation contests, baking and eating pies, and exploring the significance of pi in mathematics and science. Many educational institutions and organizations host events to raise awareness about the importance of mathematics and inspire interest in STEM (science, technology, engineering, and mathematics) fields.

Pi Day can be celebrated in so many different ways. One of the most popular ways to celebrate this spectacular day is to bake pies with the pi symbol on top. One place I worked provided pieces of pie for everyone. At that same place, I had a pi trivia search through the building. One of the facts included a British air squadron that used it for their symbol. Other possibilities include holding a contest to see who can recite the most digits in pi, or pi based art or music, host a pi run, or learn more about pi.

In recent years, NASA has joined the celebration by hosting the NASA Pi Day Challenge, an educational activity that encourages students and the public to solve a series of math problems related to space exploration. The challenges are designed to showcase how pi is used in real-world scientific calculations, such as calculating the size of craters on Mars or the volume of propellant needed for a rocket launch. Do a quick check on the internet to find out more about these activities. In addition, NASA has previous years activities available should you want to look at some of those.

Remember Pi Day is not only a celebration of a fascinating mathematical constant but also a reminder of the importance of mathematics in understanding the world around us. Whether you're solving complex equations or simply enjoying a slice of pie, Pi Day is a time to appreciate the beauty and significance of mathematics in our lives.

Friday, March 8, 2024

I've Learned To Use Manipulatives To Clarify Misunderstandings.

 


Over this past year, I've discovered how useful manipulative are in helping to clarify missing information in a students knowledge base. As you know, I am currently teaching grades 6 to 12 in a two room school house (yes they still exist in Alaska) and I've resorted to manipulative to help clarify student misunderstanding.

I had study hall this past weekend and one of my Algebra I students was having difficulty distinguishing between tens and tenths.  Apparently, she thought they were the same so I used a place value chart (borrowed from the elementary classroom) to help clarify this topic.  I showed her the ones place and then pointed to the column on either side after which I identified the tens and tenths, emphasizing the whole numbers end only in s while the decimal value ends in ths.  I then went to the hundreds and hundredths followed by thousands and thousandths.  She said this is the first time she understood the difference.

Since the elementary teacher is out on medical leave,  the sub in there just turned 21 and doesn't have a strong math background so she sends students to me.  I had one who pretty much understood when fractions were different as to which was larger but equivalent fractions he struggled with so I pulled out those fraction strips and used them to show him.  He could see using fractions strips better on how certain equivalents were the same rather than just coloring in sections of printed squares or rectangles.

In addition, he could see why you would need the same denominator to compare different fractions.  He was also able to connect to why you multiply the denominator and numerator by the same number from "playing" with these strips.

For my 7th graders, we hit multiplying and dividing decimal numbers, so out come the base 10 manipulative so that we could use as we worked through questions.  I found that using these for division was easier for them then for multiplication. They did understand that if the number was 3.4 divided by 6, they had to change some of the larger pieces into equivalent smaller ones.  Out of my 3 students students in that class, one cannot multiply anything but one digit by one digit numbers so he is struggling to show his work.  

Then there is the Algebra I group who had trouble adding like terms so again, I pulled out the base 10 pieces because I can use the ones as ones, the tens as x, the 100's as x^2 and the 1000 block as x^3.  I used these pieces to represent the terms in the equation so they could see what could be combined.  It worked so well and helped them see by x^3 doesn't get combined  with x^2.

In the meantime, I look for other possibilities of using the limited supply of manipulative to clarify mathematical concepts and understanding.  Let me know what you think, I'd love to hear.  Have a great weekend.