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How to use the AOPS books?

I just acquired the art of problem solving prealgebra book.

I would like to ask, how does one use the AOPS books, are they meant to be supplementary to a full textbook? Or are they used for introducing new concepts that you would have leant in a textbook etc. I would like advice on how to get the most out of the Art of problem solving books, as a self learner that is.

Soon I am also hoping to use the AOPS Introduction to algebra book alongside a full textbook on algebra 1 once I am finished with prealgabra.

Also, would anyone be able to recommend a complementary full algebra 1 textbook to buy for my AOPS introduction to algebra book?

• self-learning
• book-recommendation

[Disclosure/context: I teach online classes for AoPS which use their books as the textbooks. However, the opinions expressed here are my own and I am not speaking for AoPS.]

The AoPS subject books (Prealgebra, Introduction to Algebra, etc) are meant to be stand-alone textbooks which you can learn from without using another textbook. That said, they are intended for dedicated students who will work hard and learn relatively quickly. They do not have many routine practice problems, and instead have mainly more challenging problems which force students to think hard and explore the concepts more deeply. A weaker student may prefer a more traditional textbook, to use either instead of or in conjunction with an AoPS book. AoPS books also tend to be less useful as definitive references than traditional textbooks: you will find fewer well-organized statements of important facts and identities and theorems in them (though I think this is more relevant to the more advanced books than it is to Prealgebra).

To get the most out of an AoPS book (or pretty much any other math book), do the problems . Seriously, do them. I can't stress this enough. Spend a good chunk of time trying all the problems at the beginning of each section, before you move on to read their solutions. Do the exercises at the end of each section, and the review and challenge problems at the end of each chapter. There are a lot of problems, and some of them are quite difficult, so you should not expect to be able to solve every single one of them. But you should take the time working on the problems and reviewing the material until you can solve most of them.

I would also recommend you get the Solutions Manual which has full, detailed solutions to all the problems, if you don't have it yet. That allows you to verify your solutions and also learn a lot more by reading solutions that might approach the problems in different ways than you did. (Just resist the temptation to peek at the solutions before solving or at least working hard on the problems on your own, since you'll learn a lot less if you don't work through them on your own.)

If you learn better from oral instruction than just reading a book, there are also videos on the AoPS site which are associated with each section of the book which you can watch to see Richard Rusczyk explaining the ideas.

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The Math Evangelist Who Preaches Problem-Solving

September 13, 2022

Richard Rusczyk, 50, at the Art of Problem Solving campus in San Diego.

Philip Cheung for Quanta Magazine

Introduction

When Richard Rusczyk became interested in math competitions as a middle schooler in the early 1980s, the contest problems looked nothing like the ones in his math classes. He couldn’t find any book to guide him — there were only the problems themselves.

In some of the more advanced competitions he participated in as he moved on to high school, he couldn’t solve a single problem. Gradually, though, he figured out how to “kind of connect the dots, and back out what was actually going on,” he said. He learned a lot of math, but also something he considers even more important: the art of problem-solving.

Later, as an undergraduate at Princeton University, he saw classmates struggling in math classes despite having gotten perfect scores in high school. Their earlier classroom experiences had taught them to memorize a grab bag of tricks, he said. “When you get to college, that doesn’t work anymore.”

So Rusczyk and a competition-loving classmate, Sandor Lehoczky, set out to write the book their 13-year-old selves would have devoured. The resulting two-volume series, The Art of Problem Solving , opens by addressing readers: “Unless you have been much more fortunate than we were, this book is unlike anything you have used before.” From the start, the books sold 2,000 copies per year — “enough to cover rent,” Rusczyk said. Word of mouth grew, and over the 30 years since, well over 100,000 math enthusiasts have bought copies.

Today, Rusczyk’s company, Art of Problem Solving (AoPS), offers not just a large array of textbooks but also online and in-person math classes for “ambitious problem solvers” that serve nearly 25,000 students each year. These courses include both contest prep classes and subject-matter courses, but they have a common goal of fostering a problem-solving mentality. The company is currently expanding its elementary school materials, called Beast Academy, into a full curriculum, with the goal of bringing the problem-solving mindset to more than just self-selected math lovers.

This mindset “should be baked into the curriculum,” Rusczyk said. “It shouldn’t be the thing you do on every third Friday.”

Quanta spoke with Rusczyk about how to turn math learners into problem solvers. (In the interest of full disclosure, our interviewer’s child has taken AoPS classes, and her sister taught AoPS summer camps online in the first year of the pandemic.) The interview has been condensed and edited for clarity.

Your Beast Academy textbooks are comics, and you introduce concepts through story. The characters are talking about their math homework on the school bus, or they’re in woodworking class, or they’re on a field trip. What made you choose that approach?

You can’t lecture a third grader. You need to have a back-and-forth. The comic book structure we use has little kid monsters in conversation with each other, parents, teachers, the different characters in the universe.

Beast Academy’s illustrated math guides.

So you can model exploration, you can model overcoming challenges, you can model being OK with being wrong. You can create the environment for the child emotionally and intellectually. Every year we have parents sending in pictures of their kids dressing up as various characters for Halloween. They are putting themselves in these spaces.

We spent months trying to figure out: What is our delivery mechanism? We had 150 pages of worksheets, and we’re like, “No, this doesn’t work.” And then in one five-minute stretch, someone said comic books, and someone else said monsters. And we got a fantastic artist and started building out the books.

The lessons you’re trying to teach seem to go far beyond any specific math content, or even specific problem-solving techniques.

One of the main things we’re trying to get across is just the mindset of openness and willingness to engage with things we don’t understand at first. This is something kids are naturally inclined to do. But then something happens during elementary school, particularly in math classes, and we train that out of them.

We’re trying to encourage kids not to lose this curiosity or get into a mindset where the goal is to do everything perfectly. Because we have machines for that now. When we set kids up to compete with computers, we’re setting them up for failure, because anything a computer can do, it’s going to do better.

Within Beast Academy, the kids have different strengths. There’s one that’s wacky and does outlandish things that are sometimes not right, but sometimes really insightful. There are characters that are very precise and organized. And there’s a character who emerges over time as just plain brilliant. These are all different aspects of approaching different types of problems.

Video : Richard Rusczyk, founder of Art of Problem Solving, discusses how to bring out the joy, creativity and beauty in math.

Photo by Philip Cheung for Quanta Magazine; Video by Emily Buder/Quanta Magazine and Noah Hutton and Jesse Aragon for Quanta Magazine

Your materials for older students don’t incorporate a storytelling framework. But one striking thing about them is how each new chapter or class session begins not by introducing concepts, but with a collection of problems. What made you choose that format?

This was how I learned math. It was a pretty powerful way to learn.

When I started experiencing high school math Olympiads, it was two years of getting zero right on every single test. That was really frustrating. But it was safe, because it was a math contest, and who really cares? It wasn’t the first-year math class in college, staring at four problems and thinking, “I am not going to be able to do this, I am not going to be a scientist, I’m not going to be an engineer.”

That’s the experience our educational system gives to a lot of students. They think they’re not good enough, because the first time they’ve had this experience is when they get to college. They’re good enough, they just haven’t been prepared.

So we show the problems first. If a student discovers math for themselves, it becomes their math, instead of just something that was told to them. They’re not always going to get there, and that’s fine. Or sometimes they’re going to do it very differently than we did. That’s great too.

Your classes tend to attract kids who are already excited about math, and that in turn attracts teachers with strong math backgrounds. It’s one thing to make a system that works well for such enthusiastic and experienced participants, and another to make something that will work in classrooms everywhere. What challenges do you anticipate in scaling up your Beast Academy materials to a full curriculum?

We are approaching it first as a learning experience for us. We have a strong perspective on a certain type of student, and a strong conviction about some of the approaches we think should be taught to students. As to how to best deliver those resources to teachers and students in different environments, that’s something we’re more than humble about.

I’ll step back further and say I believe a lot of the troubles in education right now are technology companies going to schools and saying, “This is how you should do things.” It has to be a partnership between the content providers and the most important delivery mechanism these kids will ever have, which is the teacher in the room and the other kids.

Two or three years ago, we started working with schools using Beast Academy as a supplement, and that’s been pretty successful. But to reach more students and have a deeper impact on them, you really want to be the entire experience.

Rusczyk with his staff at the San Diego office.

When you say that Beast Academy has been successful as a classroom supplement, how do you measure that?

We just had a study completed in a school district in Minnesota. It was a little over 1,000 students in three groups: a “gifted” group, that passed some test; “Rising Scholars” students, who I think are defined as kids from diverse communities that didn’t pass this test but were close; and other students. They looked at the students’ performance on the Minnesota [standardized] test, and how that varied with the number of lessons they did on Beast Academy online. And they found a very strong relationship — the students who did more than, like, 150 or 200 lessons grew by a much larger margin than the kids who did 15 lessons, or no lessons. One really interesting thing is, the effect size was largest in the Rising Scholars group.

Who chose how many lessons kids did — the teachers, or the kids themselves?

It was during the pandemic, so my guess is a little of both. The outliers are almost certainly kids choosing it themselves. Whether this is revealing that the material teaches the kids or the material unlocks the kids, I’m not sure it matters, right? You have to give them material that’s going to make them want to do it. Getting the student to a place where they are interested in struggling with whatever you’re showing them, that for a lot of kids is the whole game.

There’s a lot of debate in educational circles about whether kids at both the high and the low end of performance are best served by being put on separate tracks or the same track. It sounds like you feel pretty strongly about giving extra challenge to kids who are ready for it.

We want to give students the materials that are most suited to help them realize their potential. If you give students material that is not speaking to them, you’re not giving them the opportunity to realize that potential.

When you remove advanced programs, you remove them for all students. So there’s going to be some kid who’s brilliant, but she will never know. And that’s a missed opportunity for her and for us, because these are the highest-leverage people in terms of making medical and technological advances.

Rusczyk at Torrey Pines State Natural Reserve in San Diego.

Creating those experiences also helps the students find their people. Part of what we do with Art of Problem Solving is our online community. For some students, it’s the only place where they feel safe expressing a love of math and science, because it is not part of the culture of their schools.

When I went to math competitions for the first time, the thing that resonated with me was, not only were there other kids who liked the same geeky stuff I did, there were adults who were excited about me being good at math, and they weren’t my parents, they weren’t my teachers. They weren’t required by profession or relation to be happy that I could do math. I had never seen that before.

Math competitions can be great for kids who are naturally competitive, but that’s not all kids. What can we offer the other kids?

It’s one of the great failings of the math community that the primary way you can explore deep interest in math is through competitions. When I was a student, contests were the only game in town.

This has gotten less true in the last 10 to 15 years, which is great. Now there are summer camps that are not contest-focused, and math circles that came out of the Eastern European tradition where professors work with the top students in their city.

I started one of these math circles at UCSD here in San Diego before I started Art of Problem Solving. And we had Efim Zelmanov, a Fields medalist, come give a talk. This was joyous, beautiful math — he was just so magnetic and happy to be there. So I thanked him for coming, and his answer was, “Well, I’m here to do this because this is what people did for me growing up.” And I’m sitting here thinking, I have exactly the opposite answer. We’re building these things because we didn’t have this sort of stuff.

It seems like Beast Academy, the imaginary school in the comic books, is the kind of place you would have dreamed of attending as a kid. You’ve said that some kids dress up as their favorite Beast Academy monster for Halloween, but what about you? Is there a monster you especially identify with?

Bits and pieces of various characters. But I might have identified most with Fiona [the math team coach]. In her day, she was pretty strong. But her interest is in sharing beautiful, interesting things with students, and helping them become stronger than she was.

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The Power of Teamwork in Entrepreneurship

The power of teamwork in entrepreneurship is a subject of increasing interest and importance. Contrary to the conventional notion of entrepreneurship as an individual pursuit, contemporary understanding emphasizes the role of collective effort. Successful entrepreneurs recognize the merits of assembling a robust and diverse team, as collaboration within a team fosters innovative ideas and effective problem-solving. Team dynamics significantly influence entrepreneurial endeavors, with effective communication, trust, and mutual respect being vital for cohesive teams. Nonetheless, teamwork in entrepreneurship presents challenges, including conflicts and disagreements that must be resolved for sustained productivity. Despite these challenges, the benefits of teamwork in entrepreneurship are manifold, including enhanced creativity, problem-solving abilities, and motivation in the face of adversity. Building and managing a successful entrepreneurial team necessitates careful selection, diversity in skills and perspectives, effective leadership, regular evaluation and feedback, and continuous learning and development opportunities. This article explores the power of teamwork in entrepreneurship, its role in entrepreneurial success, and the strategies to foster and maintain effective teamwork.

Table of Contents

Key Takeaways

• Entrepreneurship is not just about individual achievement, but also about the collective effort of a team.
• Collaboration within a team can lead to innovative ideas and problem-solving.
• Effective communication and trust among team members foster a positive working environment.
• Diversity in skills, backgrounds, and perspectives can contribute to a well-rounded and successful entrepreneurial team.

The Role of Teamwork in Entrepreneurial Success

The role of teamwork in entrepreneurial success is crucial as it allows for the integration of diverse skills, perspectives, and experiences within a well-functioning team, fostering collaboration, innovative ideas, and problem-solving abilities. Leveraging team dynamics for entrepreneurial innovation is a key aspect of this process. Effective communication plays a vital role in entrepreneurial teams, as it enables the exchange of ideas, information, and feedback among team members. Clear and open communication channels facilitate the sharing of knowledge and expertise, leading to improved decision-making and problem-solving. Moreover, effective communication helps in building trust and mutual understanding among team members, creating a supportive and cohesive work environment. By leveraging team dynamics and promoting effective communication, entrepreneurial teams can enhance their performance, adaptability, and overall success in achieving their goals.

Leveraging Team Dynamics for Entrepreneurial Growth

Leveraging team dynamics can significantly contribute to the growth and success of entrepreneurial ventures. Maximizing collaboration and harnessing diversity within a team can lead to innovative ideas and problem-solving. Effective communication and collaboration are essential for a cohesive team that can bring together different skills, perspectives, and experiences. By establishing clearly defined roles and responsibilities, team members can work together towards common goals. Trust and mutual respect among team members foster a positive working environment, enabling the team to overcome challenges and conflicts. A well-balanced team, with a diverse range of skills and expertise, enhances its overall performance. The benefits of teamwork in entrepreneurship include increased creativity and problem-solving abilities. A supportive team can provide motivation and encouragement during challenging times, contributing to the long-term success of the entrepreneurial venture.

The Importance of Clear Roles and Responsibilities in Entrepreneurial Teams

Establishing clear roles and responsibilities within entrepreneurial teams is crucial for effective collaboration and goal achievement. Clear roles enable team members to understand their individual responsibilities and contribute to the team’s overall objectives. Effective communication is essential in ensuring that everyone is on the same page and understands their role in the team’s success. By clearly defining roles, team members can avoid confusion, duplication of efforts, and conflicts that may arise due to overlapping responsibilities. Furthermore, clear roles facilitate effective decision-making processes, as team members know who is responsible for what aspects of the venture. Effective communication within the team ensures that information flows smoothly, enabling timely decision-making and problem-solving. Overall, establishing clear roles and promoting effective communication are vital for successful collaboration and the achievement of entrepreneurial goals.

Communication and Collaboration: Key Pillars of Entrepreneurial Teamwork

Effective communication and collaboration are essential components of successful entrepreneurial teamwork. In order for a team to work cohesively and efficiently, members must employ effective communication strategies and collaboration techniques. Communication strategies involve clear and concise information sharing, active listening, and mutual understanding. This can be achieved through regular team meetings, open and honest communication channels, and the use of appropriate communication tools. Collaboration techniques involve working together towards a common goal, leveraging each member’s strengths, and fostering a sense of shared responsibility. This can be achieved through effective delegation, regular feedback and brainstorming sessions, and encouraging a culture of collaboration and trust. By implementing these communication strategies and collaboration techniques, entrepreneurial teams can enhance their overall performance and achieve successful outcomes.

Fostering Trust and Mutual Respect in Entrepreneurship Teams

Fostering a culture of trust and mutual respect is crucial for creating a harmonious and productive work environment within entrepreneurship teams. Effective communication, building trust, and collaboration are key elements in achieving this culture.

• Open and transparent communication allows team members to express their ideas, concerns, and suggestions freely.
• Building trust involves creating an environment where individuals feel safe to take risks and share their opinions.
• Collaboration encourages teamwork and the pooling of diverse skills and perspectives to solve problems and innovate.
• By fostering effective communication, team members can better understand each other’s strengths and weaknesses, leading to better task allocation and performance.
• Building trust and collaboration within entrepreneurship teams promotes a sense of unity, motivation, and shared responsibility towards achieving common goals.

Overall, creating a culture of trust and mutual respect is essential for enhancing team dynamics and driving the success of entrepreneurial ventures.

Enhancing Performance Through Skillful Team Composition

Team composition plays a crucial role in maximizing productivity and optimizing collaboration in entrepreneurship. The selection of team members should be based on their skills, expertise, and diverse backgrounds. A well-rounded team with a balance of complementary skills can enhance the overall performance of the team. Effective leadership is essential in guiding and managing the team towards its goals. Regular evaluation and feedback help identify areas for improvement within the team and contribute to its long-term success. By carefully selecting team members and promoting diversity, entrepreneurship teams can tap into the collective knowledge and expertise of its members, leading to increased creativity, problem-solving abilities, and innovation. The skillful composition of teams in entrepreneurship is vital for achieving success and realizing the full potential of collaborative efforts.

Overcoming Challenges: Resolving Conflicts in Entrepreneurial Teams

Resolving conflicts within entrepreneurial teams requires open communication and a willingness to address divergent viewpoints. Conflict resolution strategies play a crucial role in promoting a positive work environment. Here are some effective strategies for resolving conflicts in entrepreneurial teams:

• Active listening: Team members should actively listen to each other’s perspectives without interrupting or judging.
• Collaboration: Encouraging team members to work together and find mutually agreeable solutions.
• Mediation: Involving a neutral third party to facilitate discussions and help find common ground.
• Constructive feedback: Providing feedback in a constructive manner to address issues and promote growth.
• Conflict resolution training: Offering training sessions to team members to enhance their conflict resolution skills.

Adaptability and Feedback: Essential Traits for Entrepreneurial Team Members

In the context of entrepreneurial teams, adaptability and feedback are essential traits for team members. Adaptability refers to the ability to adjust to new situations, challenges, and ideas. It enables team members to respond effectively to changing market conditions and evolving business environments. In the entrepreneurial context, adaptability plays a crucial role in the success of ventures as it allows teams to be flexible and responsive to emerging opportunities. Feedback, on the other hand, provides valuable insights and information that can help team members refine their strategies and improve their performance. It facilitates learning and growth within the team, allowing members to identify areas for improvement and make necessary adjustments. Ultimately, both adaptability and feedback contribute to the overall success of entrepreneurial teams by fostering continuous improvement and innovation.

Unleashing Creativity and Problem-Solving Through Teamwork

Collaboration among individuals with diverse skills and perspectives can lead to the generation of innovative ideas and effective problem-solving. In the context of entrepreneurship, teamwork plays a crucial role in unleashing creativity and enhancing problem-solving abilities. To maximize team synergy and foster a culture of collaboration, the following strategies can be implemented:

• Encourage open and inclusive communication within the team.
• Create a supportive and trusting environment where team members feel comfortable sharing their ideas.
• Emphasize the value of brainstorming and encouraging different perspectives.
• Implement collaborative problem-solving techniques, such as design thinking or agile methodologies.
• Provide opportunities for cross-training and skill development to enhance the team’s overall capabilities.

The Power of Support: Motivation and Encouragement in Entrepreneurship

Motivation and encouragement are essential factors that contribute to the success of individuals in entrepreneurial endeavors. In the context of entrepreneurship, individuals often face numerous obstacles and challenges. A supportive environment that fosters motivation and encouragement can greatly assist individuals in overcoming these obstacles. Such an environment provides emotional support, guidance, and resources necessary for entrepreneurs to persevere and achieve their goals. Motivation is crucial in helping individuals maintain focus and drive, while encouragement instills confidence and a positive mindset. It creates a sense of belief in one’s abilities, leading to increased resilience and determination in the face of adversity. Moreover, a supportive environment can provide valuable feedback and mentorship, helping entrepreneurs navigate through challenges and learn from their experiences. Overall, motivation and encouragement within a supportive environment play a pivotal role in empowering individuals to overcome obstacles and succeed in their entrepreneurial endeavors.

Building Successful Entrepreneurial Teams: The Art of Selection

The previous subtopic emphasized the significance of support, motivation, and encouragement in entrepreneurship. In line with this, the current subtopic focuses on the critical aspect of building successful entrepreneurial teams through a careful selection process and team composition. This process involves several key factors that contribute to the overall effectiveness and performance of the team.

The selection process entails evaluating potential team members based on their skills, expertise, and compatibility with the team’s goals and values. It is essential to consider diversity in terms of skills, backgrounds, and perspectives to create a well-rounded team. Effective leadership plays a vital role in managing and guiding the team towards its objectives. Regular evaluation and feedback provide insights into areas that require improvement within the team. Additionally, continuous learning and development opportunities contribute to the long-term success of the team.

Embracing Diversity: Fueling Innovation in Entrepreneurial Teams

Diversity within entrepreneurial teams contributes to increased innovation and creativity. Fostering inclusion and harnessing diversity are essential for fueling innovation in entrepreneurial teams. Research has shown that diverse teams bring together a variety of perspectives, experiences, and skills, which can lead to the generation of novel ideas and approaches to problem-solving. By embracing diversity, entrepreneurial teams can tap into a broader range of knowledge and expertise, enabling them to adapt to changing market dynamics and develop innovative solutions. Moreover, diverse teams are more likely to challenge conventional thinking and avoid groupthink, leading to better decision-making processes. To fully harness the benefits of diversity, it is important for team leaders to create a supportive and inclusive environment where all team members feel valued and empowered to contribute their unique perspectives and insights.

Effective Leadership: Guiding Entrepreneurial Teams Towards Success

Effective leadership plays a crucial role in guiding and facilitating the success of entrepreneurial teams. Developing effective leadership skills is essential for entrepreneurs who aim to lead their teams effectively towards success. Here are five key aspects of effective leadership in guiding entrepreneurial teams:

Clear vision and direction: Effective leaders provide a clear vision and direction for the team, ensuring that everyone understands the goals and objectives.

Effective communication: Leaders should possess strong communication skills to effectively convey information, expectations, and feedback to team members.

Empowering and motivating: Leaders should empower team members by delegating responsibilities and providing autonomy while also motivating them to perform at their best.

Conflict resolution: Effective leaders are skilled in resolving conflicts within the team, addressing disagreements, and fostering a positive and collaborative work environment.

Continuous development: Leaders should continuously develop their own leadership skills and encourage the growth and development of their team members through training and mentorship opportunities.

Evaluating and Improving: Continuous Growth in Entrepreneurial Teams

Continuous growth in entrepreneurial teams involves evaluating and improving various aspects of team performance to enhance overall success. Evaluating the effectiveness of a team is essential to identify strengths, weaknesses, and areas for improvement. This evaluation process typically involves analyzing team dynamics, communication patterns, and individual contributions. By assessing these factors, team leaders can gain insights into the team’s performance and identify areas that need attention. Once weaknesses are identified, strategies can be implemented to improve team effectiveness and promote continuous growth. These strategies may include team-building activities, training programs to enhance skills, and fostering a culture of open communication and collaboration. Additionally, regular feedback and performance evaluations can help track progress and ensure continuous improvement in team performance. By continuously evaluating and improving various aspects of team performance, entrepreneurial teams can enhance their overall effectiveness and increase their chances of success.

Learning and Development: Nurturing Long-Term Success in Entrepreneurial Teams

In the context of entrepreneurial teams, continuous learning and professional development play a crucial role in nurturing long-term success. This subtopic explores the importance of ongoing learning and development opportunities for entrepreneurial teams.

Key points to consider include:

• Continuous learning enables team members to stay updated with industry trends and developments.
• Professional development programs provide opportunities for acquiring new skills and knowledge.
• Ongoing learning fosters innovation and adaptability within the team.
• Professional development initiatives enhance the team’s problem-solving abilities and decision-making skills.
• Nurturing a culture of continuous learning and professional development contributes to the long-term success of entrepreneurial teams.

Frequently Asked Questions

How can conflicts within an entrepreneurial team be effectively resolved.

Conflict within an entrepreneurial team can be effectively resolved through conflict resolution techniques, such as active listening, mediation, and compromise. Effective communication, including clear and open dialogue, is crucial in addressing conflicts and fostering a positive team environment.

What Traits Are Essential for Entrepreneurial Team Members to Be Adaptable and Open to Feedback?

Traits essential for entrepreneurial team members to be adaptable and open to feedback include a willingness to embrace change, flexibility in response to new information, humility to accept constructive criticism, and a growth mindset that views feedback as an opportunity for improvement.

How Does Teamwork in Entrepreneurship Contribute to Increased Creativity and Problem-Solving Abilities?

Increased collaboration in entrepreneurship fosters innovative thinking, leading to enhanced creativity and problem-solving abilities. Research shows that teams that effectively collaborate and leverage diverse perspectives are more likely to generate novel ideas and find innovative solutions to challenges.

What Strategies Can Be Used to Build a Successful Entrepreneurial Team?

Strategies for building a successful entrepreneurial team include building trust among team members, fostering effective communication, and ensuring clear roles and responsibilities. These factors contribute to a cohesive and productive working environment, enhancing the team’s overall performance.

How Does Diversity in Skills, Backgrounds, and Perspectives Contribute to the Success of an Entrepreneurial Team?

Diversity in skills, backgrounds, and perspectives contributes to the success of an entrepreneurial team by enabling synergy and collaboration. It allows for a wider range of ideas, problem-solving approaches, and innovation, resulting in enhanced team performance and outcomes.

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I am an Art of Problem Solving Drop-out.

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By Jenn in Mo , September 21, 2010 in High School and Self-Education Board

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We're a mathy family here, so I'm feeling kind of like a failure here, but I closed the book today and said, "Enough." My math-loving son was diligently struggling through hours of math a day and getting nowhere. It took nearly two months to get through two chapters and I know he couldn't pass any review tests. He understands algebra, but this is....different. My husband and I are struggling to understand half of the problems ourselves...and this is our *thing*, ya know?

I haven't heard a negative word about AoPS yet, so it's kind of embarrassing to admit, but I just do not "get" some of these questions. Anyone that gets it feel like helping to clear the mud on this one?

2.38 What number must be in the blank in the expression 3(x+7)-_(2x+9) if the expression is the same for all values of x?

The solution manual says that "if the expression is the same for all values of x, then the x's in 3(x+7) must cancel out with those in _(2x+9).

My question is...WHY must they cancel out? It's a random, unfinished number sentence. What is in the original problem that tells me they must cancel? What am I missing here? Knowing they must cancel, we can work the problem, but I'm not grasping why I should know they cancel each other out. :svengo:

My other question is, what do I do with this book now? Do I take him through NEM and then come back to this or avoid it forever? I've never given up on a program before. I chose it because the other algebra books were too easy. We've ran into hard things in other books before, but stopped and worked through and moved on. Every day I see him patiently ramming his head into the AoPS wall and it's not something I can help him just "work through" this time. Even though I know that, I still hear Frankie Avalon in my head singing AoPS Drop-Out. :blush:

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2.38 What number must be in the blank in the expression 3(x+7)-_(2x+9) if the expression is the same for all values of x?   The solution manual says that "if the expression is the same for all values of x, then the x's in 3(x+7) must cancel out with those in _(2x+9).   My question is...WHY must they cancel out? It's a random, unfinished number sentence. What is in the original problem that tells me they must cancel? What am I missing here? Knowing they must cancel, we can work the problem, but I'm not grasping why I should know they cancel each other out. :svengo:   My other question is, what do I do with this book now? Do I take him through NEM and then come back to this or avoid it forever? I've never given up on a program before. I chose it because the other algebra books were too easy. We've ran into hard things in other books before, but stopped and worked through and moved on. Every day I see him patiently ramming his head into the AoPS wall and it's not something I can help him just "work through" this time. Even though I know that, I still hear Frankie Avalon in my head singing AoPS Drop-Out. :blush:

(Okay LOL on the Frankie Avalon - I love that song!)

But if the value of the expression is the same for all values of x, that means there won't be any x's left when you simplify it. Otherwise the value of the expression would change for each different possible x.

Have you considered the classes? I know it's a lot of money for a maybe (although I think you have three weeks to bail and get a refund) but I find that there are things I should know but can't always blurt out right when DS needs them... and I like the idea of having someone else who is prepared to do it and other kids who can add to the discussion. Even one math jam left me going "wow - this guy rocks"... :) DS is starting two AoPS classes this year, so I'm not really speaking from experience yet, but I know I'd be stuck going "uhhhhhh...." even on things I really should be able to get. It is hard stuff, and not the "I can teach this in my sleep" of other algebra and geometry programs. I've tutored those for years, but AoPS really is quite a bit more.

8filltheheart

Well, I was in a serious minority (a group of 1, me ;) ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it. Everyone else was saying that they were accessible to all students.

I have a lot of good math students (I had 1 take alg in 5th, 1 in 6th grade, and 3 others in 7th), but I only have 1 child that I would use AoPS with. He is not just good at math; he is passionate about math. He lives and breathes math. He has taken several AoPS classes and they are tough. He spent hours this summer on their alg 3 challenge sets. Thing is........he wants the challenge. That sort assignment literally makes him smile and jump right in.

My other kids do not desire that type of math depth. They used/are using programs like Foerster. I do not think they are being deprived at all. My ds that loves AoPS thinks mathematically and sees the world that way. The others are just good at math. :)

FWIW......I don't think you have anything to be ashamed of. The books are designed for the top 2-3% of all math students (not all students). They just aren't for everyone. (I couldn't teach them if I tried. He either takes their online classes or has a math coach for them. They are wayyyyyyyyyy beyond my abilities and my dh's as well (and he is an engineer! ;) )

ETA: Thought I should share that if I had known about AoPS umpteen yrs ago, I would have wanted my oldest to try them. I do think they stretch mathematical comprehension beyond the norm......basing that on the fact that my 14 yos now talks some strange language that I don't have a clue about what he is saying. All that said, I **know** that the rest of my older kids would say no way. (as a matter of fact my 11th and 6th graders did!! )

Guest Cheryl in SoCal

Well, I was in a serious minority (a group of 1, me ;) ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it. Everyone else was saying that they were accessible to all students.   I have a lot of good math students (I had 1 take alg in 5th, 1 in 6th grade, and 3 others in 7th), but I only have 1 child that I would use AoPS with. He is not just good at math; he is passionate about math. He lives and breathes math. He has taken several AoPS classes and they are tough. He spent hours this summer on their alg 3 challenge sets. Thing is........he wants the challenge. That sort assignment literally makes him smile and jump right in.   My other kids do not desire that type of math depth. They used/are using programs like Foerster. I do not think they are being deprived at all. My ds that loves AoPS thinks mathematically and sees the world that way. The others are just good at math. :)   FWIW......I don't think you have anything to be ashamed of. The books are designed for the top 2-3% of all math students (not all students). They just aren't for everyone. (I couldn't teach them if I tried. He either takes their online classes or has a math coach for them. They are wayyyyyyyyyy beyond my abilities and my dh's as well (and he is an engineer! ;) )

:iagree:I haven't used it yet but have been looking through the Intro to Counting and Probability and a bit of the Intro to Algebra and from what I have seen I wholeheartedly agree. No math curricula is the answer for every student but IMHO AoPS is going to be a good fit for the minority of students. One of my ds's would DIE if I had him use AoPS and another I think will like it because he thinks that way, though we'll only know after we try it.

I had a look at the algebra and geometry and found the layout visually confusing, which is kind of weird since in some ways it is not as busy or distracting as textbooks in general. But it is packed solid, and hard to read as a text, if that makes any sense. And algebra is my own strength. My dd went for Discovering Algebra instead.

The geometry book struck me similarly as user-unfriendly, but that may be again just my visual reaction to such a dense text.

However, that similar packing did not bother dd with the data and probability books, which she greatly enjoys. So maybe it depends on the kid (and grown-up) as well as on the area of math involved.

(Okay LOL on the Frankie Avalon - I love that song!)   But if the value of the expression is the same for all values of x, that means there won't be any x's left when you simplify it. Otherwise the value of the expression would change for each different possible x.

Ha! Lightbulb! Thank you! Regardless of what x is, the expression will equal the same thing. You're speaking my language. The book is not. :tongue_smilie:

I am relieved to hear these responses. He loves math and enjoys a challenge, but I couldn't at all say that he is passionate about math. I had forgotten that Foerster was on my list of maybes before I heard that AoPS was the crème de la crème of algebra texts. So....NEM or Foerster....thoughts?

I know absolutely nothing about NEM, so I can't offer any insight there. I have used Foerster with all of my older kids, including my ds that now uses AoPS. He used Foerster for both alg 1 and alg 2. His first AoPS courses were Intro to Counting and Prob and their alg 3 courses online and he owns several of their other "off normal track" math books that he works through on his own.

My oldest used Foerster and it more than prepared him for university level cal and an engineering major.

I'm not sure I can explain the differences very easily. Foerster is a solid math program and they absolutely learn the math skills they need to perform high level math problems, etc. No question.

AoPS helps them process math in a completely different fashion. Ds goes around "discovering" concepts and "proving" things like why x equals some bizarre thing or how an indefinite number of fractions add.......I don't know......I just nod my head!!! :lol: He thinks he wants to double major in astrophysics and math. For him it is a match. I have only ever met 2 other people that think that way.......one is now majoring in math and the other has a phD in math. For the rest of mere mortals, I think we are fine w/o it!

In The Great White North

Dc used/are using Foerster. They arde very solid math books. I have the AoPS Problem Solving, Counting & Probability and Number Theory books. Dc don't like them. They are not straight forward. Dc like to have an example, an explanation and get the problems done. Foerster does this for them. AoPS does not.

AoPS is for people who like to PLAY with their math. If your idea of a great time is to go round and round with a math problem til you figure it out, AoPS is for you. Yes, the top 2-3% of math students in the country are more likely to enjoy that, but even they don't always like that.

I've seen but not used Foerster, so I can't compare... but NEM is good challenging work. Most of it is more straightforward than AoPS (examples first, exercises second) but there are Challenge sections and Mathematical Investigations that could be frustrating in the same ways AoPS can be. They do (in those sections) throw you in and expect you to dig yourself out. And to some extent they leave some gaps in the regular explanations too -- nothing huge, but enough that sometimes the student (or the teacher) has to connect the dots.

C_l_e_0..Q_c

I'm in the opposite situation. We're a NEM drop-out. I'm so much happier as a mom with AoPS *and* the online class. Someone else is teaching math to my math kid!

(and I"m good at math too, I did math all the way to a Master's level, although it was not my major - I'm an engineer, we need math )

My son is truly having a blast with the online class. He did Algebra 1 over the summer. The kids at the summer pool who learned he was doing a math class assumed it was remedial, until they saw a problem! LOL. Sweet nerd revenge. :)

The math classes are totally worth the cost. Plus they force the kids to move forward, and not spend 2 months on 2 chapters. In 15 weeks, they did 14 chapters (possibly 13, I'm not sure) so the interest is kept high.

at the beach

I agree that AoPS is for people who enjoy playing around with math. I think I am one of those who said something along the lines that more people can be successful with it than the top few percent. But I think I also said that sometimes the solutions/explanations gloss over things or skip a step, assuming readers will see that step in the problem when they might not. I felt it could be used by more than just the top few percent because I think that kids often underestimate their abilities. I also agree with the pp that even kids in the top few percent who can do it may not want to do it because it does take a lot of time and puzzling.

I was terrified of math from 8th grade on. I never took math in college but took logic to avoid it. When I signed my daughter up for AoPS this summer, I decided to give the book a try. I did the first three chapters. I wanted to help my daughter if she needed it, but she was the one helping me :001_huh: and the class moved so fast that I couldn't keep up.

The problem that you mention, to me, looks like it is asking you to make the 2x the same as the 3x so the only thing that works is 3/2. How do you know it will cancel? I think you know that because it's a subtraction problem. Maybe I am oversimplifying this.

Maura in NY

NEM is an integrated program - Algebra/Geometry/Trig and rolled together. It's challenging, in a good way, but it doesn't fit the typical Algebra/Geometry/Algebra II/Trig-PreCalc/Calc track.

FWIW, my ds used NEM 1 in 7th, thinking he would stick with it for 4 years. Instead, we switched. He didn't use Foerster's Algebra I, but he did use Foerster for Algebra II and PreCalc. Great real world application problems.

We used AoPS for several months and also found some of the problems strangely worded. But, our biggest problem with AoPS was not enough repetition.

We are using Foerester's Algebra now. It has more than enough problems for us, but it's much easier to eliminate problems than try to add more in.

Miss Marple

Not a drop-out yet, but we are doing AoPS very slowly and I've added NEM2 back as the primary math. Ds is still working through the geometry book (45 minutes per day), but we've put the algebra book away because he already did LOF Beginning Algebra and the NEM should cover the bases thoroughly.

I'm finding that hormones cause my son to be less inquisitive and less dedicated right now. He does great with math, so I chose the other program I love - Singapore NEM, and it is going very well.

If we never finish the AoPS books that will be OK because NEM is the primary curriculum (which we will finish). I know that any exposure ds has had with AoPS has helped fine tune his mathematical abilities.

Well, I was in a serious minority (a group of 1, me ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it

:iagree: I cannot imagine how frustrated a student and a parent would be trying to work through this program and thinking that it is accessible for all students!

The problems in NEM can be plenty challenging at times. My son did NEM 1 before switching to AoPS.

Regarding AoPS Algebra 1, they introduce Algebra 2 concepts, too. That doesn't make it easier! Ds thought the middle of the text was difficult and often trudged through the problems, but then he flew through the last few chapters. He thought they were easier, even many of the challengers. I don't know if they really were, or if he just started to get the hang of the text and problems.

When it comes to math, it's so important to pick what works and then bite off and chew well at a pace that's suitable for the student. I wouldn't hesitate to step back and reconsider if something wasn't working.

  AoPS is for people who like to PLAY with their math. If your idea of a great time is to go round and round with a math problem til you figure it out, AoPS is for you. Yes, the top 2-3% of math students in the country are more likely to enjoy that, but even they don't always like that.

I agree entirely with the bolded.

If anybody is deciding whether to go with the AoPS classes, there's a Math Jam on the subject this afternoon/evening (7:30 Eastern, I believe). It will include sample questions from the class, so you can try them out.

Math Jams are free informational sessions, usually hosted by Richard Rusczyk, who is always responsive to questions. If he can't answer your question within the confines of the session, he'll likely invite you to e-mail him and he'll help you decide whether a class is right for your family.

We're mathy. . . and we're AoPS drop outs as well.

Dd & I tried Number Theory for a couple/few months. It actually went OK so long as *I* worked through *every section* and *every problem* on my own before dd tried it (or simultaneously sitting together).

The exploration and problem solving was interesting. I rather enjoyed it. But, I just *don't have time* to do it right now.

We dropped it after a few chapters -- which were successful but slow b/c I just couldn't make myself find the time to do it often enough to keep up w/ dd's needed pace.

So sad. . .

Now y'all are scaring me. My dd loved the AoPS Number Theory book so much this summer, I bought Alg I for her (but we don't start for another year...) I had been planning on Foerster's, so I've got both ready to go (well, I still need that pesky Foerster's Solutions manual...)

We're mathy. . . and we're AoPS drop outs as well.   Dd & I tried Number Theory for a couple/few months. It actually went OK so long as *I* worked through *every section* and *every problem* on my own before dd tried it (or simultaneously sitting together).

Dd and I worked through all the examples together, but then she did the problem sets pretty much on her own (except I let her skip the challenge problems at the end of the chapters). Was your dd just needing you to sit with her for the "lesson" part, or also the problem sets? I didn't mind working through the lesson with her... I actually learned a lot myself and had fun... :tongue_smilie: For some reason I really like the layout and presentation of these books... OTOH the Lial's layout really drove me crazy when I first saw it (I'm getting used to it now that my other dd's working through BCM).

But now you guys are intimidating me a bit... :001_huh: I do think the Algebra text looks more challenging than the Number Theory, but I'm hoping after she's gotten through Singapore DM1, she'll be preparerd (or I guess we'll do Foerster's after all...)

I found the solutions manual for Foerster this afternoon! I finally called the publisher and it turns out that they renamed the book and it is now called Classic Edition instead of Foerster's. The ISBN # is 0201861003. I can't find it used anywhere, but it is available new at pearsonschool.com.

I'm still torn though. I loved Singapore so well and would like to try going on with NEM. But I don't want to commit to it being his math for all of high school. I hear that Foerster does an excellent job with proofs, which I think are important. I may end of tossing a coin before this is over.

This is exactly what I was trying to express in the other thread! It's just as much about the approach as the difficulty, imo. Yes, you need to be a good math student, but I dont' think you need to be a TOP math student. Not in the sense of the top 2% to 3%, at any rate. I can't imagine my dd would score anywhere near that high, but she loves AoPS.

And you can be a top math student, but if you hate this approach, it will be very hard to persevere and succeed.

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Art of Problem Solving: Least Common Multiple Slick Trick

Video tutorial uses examples to illustrate easy ways of finding the least common multiple. [7:18]

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