Author: imoc seo

Sudoku, the classic number puzzle, has captured the minds of puzzle enthusiasts around the globe. Originating from Japan, the name Sudoku translates to “single number,” reflecting the puzzle’s core principle: each number should appear only once in each row, column, and grid. This article will introduce you to the basics of Sudoku, walk you through some examples, and offer tips to enhance your solving skills.

What is Sudoku?

Sudoku is a logic-based puzzle game typically played on a 9×9 grid divided into nine 3×3 subgrids. The objective is to fill the grid with numbers from 1 to 9, ensuring that each row, each column, and each 3×3 subgrid contains all the numbers from 1 to 9 without repetition.

Basic Rules of Sudoku:

1. Each row must contain the numbers 1 to 9, without repetition.
2. Each column must contain the numbers 1 to 9, without repetition.
3. Each 3×3 subgrid must contain the numbers 1 to 9, without repetition.

How to Solve a Sudoku Puzzle

Step-by-Step Example

Let’s walk through a simple Sudoku puzzle to understand the solving process.

Initial Puzzle:

5 3 _ | _ 7 _ | _ _ _

6 _ _ | 1 9 5 | _ _ _

_ 9 8 | _ _ _ | _ 6 _

——+——-+——

8 _ _ | _ 6 _ | _ _ 3

4 _ _ | 8 _ 3 | _ _ 1

7 _ _ | _ 2 _ | _ _ 6

——+——-+——

_ 6 _ | _ _ _ | 2 8 _

_ _ _ | 4 1 9 | _ _ 5

_ _ _ | _ 8 _ | _ 7 9

Step 1: Start with the Easy Ones

Begin by looking for rows, columns, or subgrids where only one number is missing. For instance, in the first row, the missing numbers are 1, 2, 4, 6, 8, and 9. However, given the other numbers in the row and subgrid, you can often narrow down the possibilities.

Step 2: Use the Process of Elimination

In the first 3×3 subgrid (top left), we are missing the numbers 1, 2, 4, 6, 7. By checking the rows and columns intersecting the empty cells, we can often deduce which numbers go where.

Example Fill-In:

• For the empty cell in the first row, third column, the possible numbers are 1, 2, 4, 6, 8, 9. However, since 6 and 9 are in the same column and 8 is in the same row, the number for this cell must be 2.

Step 3: Repeat the Process

Continue using the process of elimination and logical deduction for the remaining cells. Let’s fill in a few more:

• For the second row, second column, the missing numbers are 2, 3, 4, 7, 8. Since 3 and 8 are already in the same subgrid, we need to see which other numbers fit based on the column and row.

Intermediate Puzzle:

5 3 2 | _ 7 _ | _ _ _

6 _ _ | 1 9 5 | _ _ _

_ 9 8 | _ _ _ | _ 6 _

——+——-+——

8 _ _ | _ 6 _ | _ _ 3

4 _ _ | 8 _ 3 | _ _ 1

7 _ _ | _ 2 _ | _ _ 6

——+——-+——

_ 6 _ | _ _ _ | 2 8 _

_ _ _ | 4 1 9 | _ _ 5

_ _ _ | _ 8 _ | _ 7 9

Step 4: Solve the Puzzle

By continuing to apply these methods, you will gradually fill in the entire grid. Here’s the completed puzzle for reference:

5 3 4 | 6 7 8 | 9 1 2

6 7 2 | 1 9 5 | 3 4 8

1 9 8 | 3 4 2 | 5 6 7

——+——-+——

8 5 9 | 7 6 1 | 4 2 3

4 2 6 | 8 5 3 | 7 9 1

7 1 3 | 9 2 4 | 8 5 6

——+——-+——

9 6 1 | 5 3 7 | 2 8 4

2 8 7 | 4 1 9 | 6 3 5

3 4 5 | 2 8 6 | 1 7 9

Tips for Solving Sudoku

1. Start with the obvious: Fill in the easy cells first to gain momentum.
2. Use pencil marks: Write possible numbers in cells to keep track of your thoughts.
3. Look for patterns: Familiarize yourself with common patterns and techniques, such as naked pairs, hidden pairs, and X-Wing.
4. Stay organized: Work methodically through rows, columns, and subgrids.
5. Practice regularly: The more you practice, the better you’ll get at spotting solutions quickly.

Conclusion

Sudoku is a fantastic way to challenge your brain, improve your logical thinking, and enjoy a bit of quiet time. Whether you’re a beginner or an experienced solver, the satisfaction of completing a Sudoku puzzle is a reward in itself. So, grab a puzzle, follow these steps, and immerse yourself in the fascinating world of Sudoku!

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. In this three-part series, we briefly discuss these three factors and paint a different picture: mathematics is a subject that is ever-changing.

In this third and final part, we discuss the-

Needs of the Society

The usefulness of mathematics in everyday life and in many vocations has also affected what is taught and when it is taught. In early America, mathematics was considered necessary primarily for clerks and bookkeepers. The curriculum was limited to counting, the simpler procedures for addition, subtraction, and multiplication, and some facts about measures and fractions. By the late nineteenth century, business and commerce had advanced to the point. that mathematics was considered important for everyone. The arithmetic curriculum expanded to include such topics as percentages, ratios and proportions, powers, roots, and series.

This emphasis on social utility, on teaching what was needed for use in occupations, continued into the twentieth century. One of the most vocal advocates of social utility was Guy Wilson. He and his students conducted numerous surveys to determine what arithmetic was actually used by carpenters, shopkeepers, and other workers. He believed that the dominating aim of the school mathematics program should be to teach those skills and only those skills.

In the 1950s, the outburst of public concern over the “space race” resulted in a wave of research and development in mathematics curricula. Much of this effort was focused on teaching the mathematically talented student. By the mid-1960s, however, concern was also being expressed for the disadvantaged student as U.S. society renewed its commitment to equality of opportunity. With each of these changes, more and better mathematical achievement was promised.

In the 1970s, when it became apparent that the promise of greater achievement had not fully materialised, another swing in curriculum development occurred. Emphasis was again placed on the skills needed for success in the real world. The minimal competency movement stressed the basics. As embodied in sets of objectives and in tests, the basics were considered to be primarily addition, subtraction, multiplication, and division with whole numbers and fractions. Thus, the skills needed in colonial times were again being considered by many to be the sole necessities, even though children were now living in a world with calculators, computers, and other features of a much more technological society.

By the 1980s, it was acknowledged that no one knew exactly what skills were needed for the future but that everyone needed to be able to solve problems. The emphasis on problem-solving matured through the last 20 years of the century to the point where problem-solving was not seen as a separate topic but as a way to learn and use mathematics.

Today, one need of our society is for a workforce that is competitive in the world. There is a call for school mathematics to ensure that students are ready for workforce training programs or college.

Conclusion

International Mathematics Olympiad play a vital role in shaping the intellectual and analytical landscape of society. They not only foster critical thinking, problem-solving skills, and creativity among students but also prepare them to tackle complex real-world issues. By encouraging young minds to engage with challenging mathematical concepts, Olympiads help cultivate a future generation of scientists, engineers, economists, and leaders who can drive innovation and progress. Moreover, the collaborative and competitive nature of these competitions promotes a culture of academic excellence and perseverance.

As we face increasingly complex global challenges, the importance of nurturing a strong foundation in mathematics through Olympiads cannot be overstated. They are not just competitions; they are essential platforms for equipping society with the tools and mindset needed to build a better, more informed, and innovative world.

 

 

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. This three-part series briefly discusses these three factors and paints a different picture: mathematics is an ever-changing subject.

We have already discussed the Needs of the Subject in the previous blog. In this second part, we discuss-

Needs of the Child

The mathematics curriculum has been influenced by beliefs and knowledge about how children learn and, ultimately, about how they should be taught. Before the early years of the twentieth century, mathematics was taught to train “mental faculties” or provide “mental discipline.” Struggling with mathematical procedures was thought to exercise the mind (like muscles are exercised), helping children’s brains work more effectively. Around the turn of the twentieth century, “mental discipline” was replaced by connectionism, the belief that learning established bonds, or connections, between a stimulus and responses. This led teachers to the endless use of drills aimed at establishing important mathematical connections.

In the 1920s, the Progressive movement advocated incidental learning, reflecting the belief that children would learn as much arithmetic as they needed and would learn it better if it was not systematically taught. The teacher’s role was to take advantage of situations when they occurred naturally as well as to create situations in which arithmetic would arise.

During the late 1920s, the Committee of Seven, a committee of school superintendents and principals from midwestern cities, surveyed pupils to find out when they mastered various topics. Based on that survey, the committee recommended teaching mathematics topics according to students’ mental age. For example, subtraction facts under 10 were to be taught to children with a mental age of 6 years 7 months and facts over 10 at 7 years 8 months; subtraction with borrowing or carrying was to be taught at 8 years 9 months. The recommendations of the Committee of Seven had a strong impact on the sequencing of the curriculum for years afterward.

Another change in thinking occurred in the mid-1930s, under the influence of field theory, or Gestalt theory. A 1954 article by William A. Brownell (2006), a prominent mathematics education researcher, showed the benefits of encouraging insight and the understanding of relationships, structures, patterns, interpretations, and principles. His research contributed to an increased focus on learning as a process that led to meaning and understanding. The value of drill was acknowledged, but it was given less importance than understanding; drill was no longer the major means of providing instruction.

The relative importance of drill and understanding is still debated today. In this debate, people often treat understanding and learning skills as if they are opposites, but this is not the case. The drill is necessary to build speed and accuracy and to make skills automatic. But equally clearly, you need to know why as well as how. Both skills and understanding must be developed, and they can be developed together with the help of International Maths Challenge sample questions.

Changes in the field of psychology have continued to affect education. During the second half of the twentieth century, educators came to understand that the developmental level of the child is a major factor in determining the sequence of the curriculum. Topics cannot be taught until children are developmentally ready to learn them. Or, from another point of view, topics must be taught in such a way that children at a given developmental level are ready to learn them.

Research has provided increasing evidence that children construct their own knowledge. In so doing, they make sense of the mathematics and feel that they can tackle new problems. Thus, helping children learn mathematics means being aware of how children have constructed mathematics from their experiences both in and out of school.

End Note

As we have explored, a child’s journey through mathematics is deeply intertwined with their cognitive development, critical thinking skills, and overall academic success. By addressing their individual needs, providing appropriate support, and fostering a positive learning environment, we lay the foundation for a lifelong appreciation and understanding of mathematics. But what about the broader context? How does mathematics serve society at large, and what influences has it made in history? In our next blog, we will delve into these questions, examining the societal needs in mathematics and its profound impact on the course of human history.

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. This three-part series briefly discusses these three factors and paints a different picture: mathematics is an ever-changing subject.

In the first part, we discuss-

Needs of the Subject

The nature of mathematics helps determine what is taught and when it is taught in elementary grades. For example, number work begins with whole numbers, then fractions and decimals. Length is studied before area. Such seemingly natural sequences are the result of long years of curricular evolution. This process has involved much analysis of what constitutes a progression from easy to difficult, based in part on what is deemed necessary at one level to develop ideas at later levels. Once a curriculum is in place for a long time, however, people tend to consider it the only proper sequence. Thus, omitting a topic or changing the sequence of issues often involves a struggle for acceptance. However, research shows that all students do not always learn in the sequence that has been ingrained in our curriculum.

Sometimes, the process of change is the result of an event, such as when the Soviet Union sent the first Sputnik into orbit. The shock of this evidence of another country’s technological superiority sped curriculum change in the United States. The “new math” of the 1950s and 1960s was the result, and millions of dollars were channeled into mathematics and science education to strengthen school programs. Mathematicians became integrally involved. Because of their interests and the perceived weaknesses of previous curricula, they developed curricula based on the needs of the subject. The emphasis shifted from social usefulness to such unifying themes as the structure of mathematics, operations and their inverses, systems of notation, properties of numbers, and set language. New content was added at the elementary school level, and other topics were introduced at earlier grade levels.

Mathematics continues to change; new mathematics is created, and new uses of mathematics are discovered. As part of this change, technology has made some mathematics obsolete and has opened the door for other mathematics to be accessible to students. Think about all the mathematics you learned in elementary school. How much of this can be done on a simple calculator? What mathematics is now important because of the technology available today?

As mathematical research progresses and new theories and applications emerge, the curriculum must adapt to incorporate these advancements. For example, the development of computer science has introduced concepts such as algorithms and computational thinking into mathematics education. These topics were not traditionally part of the elementary curriculum but have become essential due to their relevance in today’s technology-driven world. Additionally, as interdisciplinary fields like data science and quantitative biology grow, there is a pressing need to equip students with skills in statistics, probability, and data analysis from an early age, and here, the International Maths Challenge is playing a crucial role. This integration ensures that students are prepared for future academic and career opportunities that increasingly rely on mathematical literacy. Furthermore, globalization and the interconnected nature of modern societies require students to understand complex systems and patterns, necessitating the introduction of topics such as systems theory and network analysis. Consequently, the curriculum evolves not only to preserve the integrity and progression of mathematical concepts but also to reflect the dynamic and ever-expanding landscape of mathematical applications in the real world.

End Note

In the next blog, we will move towards the second factor: understanding the need for mathematics in a child’s education can set a foundation for problem-solving, logical thinking, and even everyday decision-making. In the following blog, we will delve into why mathematics is not just a subject but a vital tool for a child’s overall development and future success. Stay tuned for further updates!

Feedback plays a vital role in mathematics education, guiding students toward deeper understanding and fostering a supportive learning environment. This article delves into the importance of specific and actionable feedback in mathematics education and explores strategies for both giving and receiving feedback effectively.

Understanding Feedback:

In mathematics education, feedback transcends mere praise or criticism—it is a nuanced tool for academic growth. Effective feedback should be clear, and concise, and provide guidance for improvement. It should highlight students’ strengths, address any misunderstandings, and offer actionable steps for progress.

Key Components of Effective Feedback:

Specificity: Feedback should pinpoint areas for improvement and clarify the path to success. Students need to know precisely what they need to do to enhance their understanding.

Actionability: Feedback should be actionable, outlining steps for students to move forward. This empowers students to take ownership of their learning journey.

Importance of Feedback:

Feedback serves multiple critical purposes in mathematics education:

Promoting Learning: It catalyzes academic growth by guiding students towards deeper understanding and mastery.

Building Motivation: Constructive feedback inspires students to strive for excellence and fosters a growth mindset.

Fostering Relationships: Feedback provides an opportunity for educators to connect with students on a deeper level, building trust and rapport.

The Human Element: Empathy and Trust:

Effective feedback is rooted in empathy and trust. Creating a safe and supportive learning environment is essential for feedback to be received positively. Teachers should approach feedback with empathy, avoiding emotional reactions and prioritizing the emotional well-being of their students.

Integrating Feedback into Planning:

When planning lessons, educators should:

Set Clear Goals: Define learning objectives and success criteria to guide student progress.

Anticipate Misconceptions: Be prepared to address common misunderstandings and provide targeted support.

Establish Trust: Build a culture of trust and openness in the classroom to facilitate effective feedback exchanges.

Feedback Goes Both Ways:

Teachers should be open to receiving feedback from students. Seeking feedback encourages student engagement and provides valuable insights for improving teaching practices. Additionally, teachers can infer feedback by observing students’ understanding and addressing any gaps in comprehension proactively.

Conclusion:

Feedback is a cornerstone of effective mathematics education, fostering academic growth and cultivating a supportive learning environment. By prioritizing specificity, actionability, empathy, and trust, educators can create a feedback-rich classroom where every student has the opportunity to excel in mathematics.