“If you are an educator and want to efficiently teach collaborative mathematics to your class, this is the book for you! It is a fun, challenging, and playful way to introduce problem-based learning by providing all the tools and problems necessary to get started.”
Michaela Hlasek,
Math Teacher and Combinatorics Instructor, Awesome Math Summer Program 2017
“Awesome Math, appears to be about math, but really has lessons for education in general and even for re-skilling in the corporate world—an effective approach to educate and prepare the next generation for a YouTube + machine learning world. It gives new meaning to the phrase ‘the journey is the reward.’ The biggest danger? This book could convince you that math can be fun.”
Raj Varadarajan,
Senior Partner and Managing Director, Boston Consulting Group
“This book is a brilliant road map that delights in its own theorem of authenticity and relevance. Full of the philosophical groundwork, expert insights, and plenty of practice problems, Awesome Math: Teaching Mathematics with Problem-Based Learning is a must-read for any Math or STEM educator concerned with the relevance and joy of a beautiful and expansive discipline.”
Ben Koch,
Co-founder and CEO, Numinds Enrichment
“Awesome Math makes a strong case for ditching rote memorization and turning to collaborative problem-solving and mastery-based learning instead. This book is a must-read for parents and educators in all subject areas who wish to develop their students’ creative and critical thinking skills.”
Jaime Smith,
Founder and CEO of OnlineG3.com
“The book is an excellent source for educators interested in problem-based learning through student-centric approach. Students and teachers will find some ‘secrets’ of how math circles, math competitions, experiences of other math educators, and even math games along with wonderful and challenging problems can be used for an entire lesson or just as a mini-unit.”
Dimitar Grantcharov,
Professor at University of Texas at Arlington
“Through playful problem solving, mastery learning, the three C's, and more, Awesome Math challenges the idea of a traditional, teacher-centric classroom. Kathy, Alina, and Titu are visionaries in the field of math education, and their book has sparked new inspiration for strategies that I am eager to utilize in my own math classroom.”
Hannah Keener
“Awesome Math emphasizes the importance of collaborative problem solving in a classroom setting, featuring interesting and carefully chosen concepts and problems that can be used in a regular classroom and enrichment academic mathematics programs such as math circles or summer camps.”
Zvezdelina Stankova,
Teaching Professor of Mathematics at University of California at Berkeley
“This inclusive book speaks in voices of the many. It has the irresistible flow of a well-curated social feed. There are shiny treasures to repost, ‘today-I-learned’ surprises to ponder, wise checklists to save, heartfelt polemics to debate—and so many kind math friends to meet!”
Dr. Maria Droujkova,
Founding Director of Natural Math
“I believe the most important goal of education is acquiring the ability to learn on your own. This book is mainly aimed at this goal and will help teachers and students improve their logical thinking, making them more independent learners and scholars.”
Dr. Krassimir Penev,
Bergen County Academies
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FIRST EDITION
To our awesome community of colleagues, family, and friends who inspire us daily and made this publication possible.
Special thanks to Navid Safaei and Alessandro Ventullo for their time in reviewing the mathematical content for this book.
– Titu and Alina
My heartfelt appreciation goes to my closest community, my family, for their support, advice, and contributions to this effort. To my husband, David, whose ideas and insights have added value not only to this book, but to our family for over 25 years. To my oldest son, Jacob, for his incredible gift of explaining complex concepts elegantly and easily, which helped improve sections of the book. To my youngest son, Adam, for his content corrections and positive support that kept me on track and enjoying the process. And lastly, to my mother-in-law, Sandy, and my sister, Kelly, for being early readers and emotional support. Thank you.
– Kathy
We'd like to thank Amy Fandrei, our executive editor, for her kind guidance and for providing us with the opportunity to share our love of problem-based learning. Many thanks also to Pete Gaughan, the content enablement manager for this project, who helped us every step of the way to create a quality publication.
– Titu, Kathy, and Alina
Dr. Titu Andreescu has been coaching, teaching, and training students and teachers for most of his exemplary career. Starting as a high school mathematics teacher in Romania and later in the United States, Titu became coach and leader of the United States International Mathematics Olympiad team, director of the Mathematical Association of America's AMC tests (American Mathematics Competitions), and an associate professor at University of Texas at Dallas in the Science and Mathematics Education department training mathematics teachers. His passion for problem solving and mathematics teaching has extended to the following noteworthy accomplishments.
For Kathy Cordeiro, innovation, problem solving, and team collaboration have been the leading constants throughout her varied career. A degree in communications, coupled with an MBA, has given Kathy a unique skill set to create and market customized education initiatives, in business and/or academia, which allows her customers and students to reach their goals and realize success. Kathy began her own enrichment school, Eudaimonia Academy (2006–2012), where she coached math teams, taught a philosophy/creative writing course, and co-led speech and debate teams.
Kathy is the marketing and communications director for the AwesomeMath organization. In this role, she has had various speaking engagements as well as managed multiple communication channels online, where she discusses mathematics education with parents, teachers, students, and businesses.
Beyond being connected with multiple math groups, Kathy is also a part of a network that includes parents, teachers, and students, such as
Alina Andreescu was born and raised in Romania, at a time when mathematics education was exceptionally strong. She participated successfully in Romanian mathematics competitions. She completed her finance degree in the United States and later obtained an M.A. in management with emphasis on leadership. Alina was never afraid of change and challenges, embarking on lifetime journeys from moving to the United States to becoming a successful cofounder and leader of the AwesomeMath and XYZ Press organizations.
As operations director of the AwesomeMath programs for the past 12 years, Alina has been integral in every facet of creating the opportunities/resources that fulfill the mission of providing enriching experiences in mathematics for intellectually curious learners. She fosters a community of staff, students, and instructors that values critical thinking, creativity, passionate problem solving, and lifetime mathematical learning. Since the AwesomeMath community is international, she must meld a diverse background of individuals into a thriving learning environment.
In writing this book, we hope to lead you to what you already know: that problem-based learning is an effective method for raising tomorrow's thinkers by collaborating over interesting and relevant problems. Through the AwesomeMath Summer Program,1 the inspiration for this book, we've had the privilege to work with thousands of the brightest minds from around the globe for over 10 years. We've seen first-hand the leaps in skills, growth of curiosity, and joy of problem solving that arises when individuals are immersed in a kind, collaborative, and challenging environment where students create positive life-long memories and form valuable friendships.
So, how do you raise out-of-the-box thinkers in a check-the-box world? Teaching is an opportunity to inspire and guide, but that means diverging from the conformity required in today's education system and allowing students to take intellectual risks and, yes, fail. The outdated criterion of identifying top students through grades is flawed; it's evaluating someone's worth based on an outcome and not the process, which sets up situations where students avoid intellectual risks so they can maximize grades. Students aren't learning how to think, work together, or find challenging opportunities.
Furthermore, they aren't being prepared to face the current challenges in today's workforce, which values innovation, leadership, collaboration, resilience, and critical thinking. We need students who can do more than solve mere exercises for a check mark; they need to be able to tackle difficult problems and also be able to notice problems worthy of solving by seeking patterns, reframing information, and asking the right questions. Students are all different and have different strengths to offer in every setting. We need to value them for who they are with a student-centric approach as opposed to evaluating them with standardized conformity and false metrics.
When Randy Pausch gave his Last Lecture at Carnegie Mellon University,2 he explained that the moment someone lowers their expectations for what you can accomplish, they've stopped caring about you. Students need to have challenges that we, as educators, know they can overcome and master. When we allow students to work toward mastery instead of grades, then the journey becomes about the process and not the outcome. This approach, however, requires facilitators, helpers, and guides along the way so that each student can recognize their value and be their best version.
Problem-based learning approaches education with a deep respect for the value, abilities, and strengths of each student by raising expectations beyond the standard and providing guidance in a supportive environment.
The main goals of this book are as follows:
Here are the top five characteristics of a problem-based learning curriculum as detailed in this book:
In this section:
Today's kids are busier than ever! Juggling their schedules inside and outside of school requires major planning, and as a result, enticing them to focus in a mathematics class can be difficult. That is not to say that they are incapable of deep thought, but rather, asks how mathematics can compete with all the other distractions that life throws their way. What makes activities such as sports or video games so much more appealing? How can we construct a mathematics environment so that students are engaged with the subject and work together to achieve a superior understanding for mathematics?
The common thread is playful problem solving. Play is an integral part of life. Even as adults, we love to play and compete and solve problems with friends. You can challenge yourself to move up levels and share your experiences with peers – plus, there is no fear of losing, whereas in mathematics, there is fear. Fear of appearing stupid, fear that if you are slow to understand that you just aren't good at math, fear that doing poorly in math means you won't get into college. We need to erase that fear and help kids take thought risks with problem solving.
While teachers want each student to excel, in reality, great teachers work on improving the abilities of their entire class every day, spotting areas that are weak, celebrating strengths, and being a cohesive unit. When all of those areas come together, then success will happen. Children are not outcomes and need to be guided by a great educator to think critically and creatively.
Currently, math education in middle and high schools is a series of exercises with easily obtained answers, e.g., find the perimeter of a square, training students to do what a computer can do better. Problem solving goes much deeper and taps into what makes us human, namely, multiple creative approaches with a string of steps to solving meaningful and interesting problems. It takes the shift away from outcome-based learning (grades/test scores, rank, grade point average [GPA]), which is a fixed-mindset approach, to learning for mastery, where students challenge themselves to improve every day (growth mindset).
What exactly is problem solving? Even mathematicians and researchers haven't come up with a definitive answer, but in this book, we believe problem solving has the following characteristics:
Problem solving is the strategy, and math competitions are the vehicle to train your math class to be stellar thinkers. Since the current school curriculum delivers a narrow path of mathematics knowledge, climbing aboard the math competition train will expose students to a greater array of topics, including discrete mathematics, an area that incorporates both number theory and combinatorics (counting and probability). Discrete math, along with finite mathematics and linear algebra, are necessary to work in the modern world of computing. Mathematical modeling and a strong understanding of statistics is also critical. The level of deep thinking required to solve hard problems in the areas of discrete mathematics, algebra, geometry, and the areas in between (e.g., geometric inequalities), transfers to future careers in STEM (science, technology, engineering, mathematics) fields, and beyond. Mathematics competitions provide exposure to all these topics while working with peers to solve challenging problems.
Just as every football player cannot be the quarterback, not every student is going to excel in the same way with mathematics competitions, but this brings us back to the focus being placed on the process and not the outcomes. The reason to engage in math competitions is to have something to work toward where each student can get a little better every day and be motivated in a collaborative and supportive environment. Some students may enjoy working through lots of different types of problems while others may prefer to look at the methods employed and want to write their own problems based on their discoveries. Every type of student can play an important role in your mathematics class, and as the teacher, you want to look at every student as a collection of strengths as opposed to a collection of weaknesses that need to be fixed.
Regardless of the role a student chooses, all students grow their skills faster when collaborating toward a common goal than they would on their own, because when you engage in the trade of ideas, everyone improves.
The learning environment for the game is critical to bringing out the best in the players and the rewards are range, rigor, and resilience.
As educators, we understand the importance of depth and breadth in learning. For beginning piano players, listening to a concert pianist perform can ignite curiosity and inspire them to practice more. In mathematics, there seems to be a reticence to hear the symphony for fear that it will be too much, too soon, and by limiting the range, we limit curiosity and growth.
Even the terms used for learning piano and learning mathematics are different: Students play piano and work on math problems. There needs to be a fundamental shift in approach and exposure to a range of problems that are harder and more interesting so that students can see where math can take them. So much of math education today is about waiting:
And so on….
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
G.H. Hardy
Students need to work together to weave a pattern of ideas with what they know and can add more as their knowledge base grows. To fully appreciate the full tapestry of mathematics is not by adding one color of thread at a time, but weaving a picture with various thread colors within a group of learners who are just as excited by the beauty as you are.
Collaborative problem-based learning holds the key to unlocking abilities and mathematical growth. The collaboration nurtures interpersonal skills, leadership traits, and conflict resolution, while providing interesting problems gives students a common goal to work toward where they can connect and share ideas. Students can see the fun and beauty in mathematics, start to play, and see where math can take them long term.
A fun problem that works well in a group is the following:
Students are allowed to be transported by a musical symphony before understanding individual notes, so why not provide the symphony of mathematics and introduce students to its wonders and challenges? Parents and educators will read books beyond a student's personal reading level so that they can hear the richness of language and be exposed to more intricate sentence structure. There are so many wonderful places an enriching math education can take you.
When educating young math students, you can let them know that they are the captains of their ship, but as their navigator, you can guide them to really interesting destinations and expose them to a wider range of mathematics.
There is a world of mathematics to discover, and the playful pursuit will ignite their interest and provide them with the introspection to know their strengths and passions. Context while explaining mathematical concepts is also important, so it's not just range within the topic of mathematics but range outside the topic as well. Knowing the history and story of what they are learning makes a huge difference. Mathematical discoveries were made to solve real-life problems, and if students learn the story, they are more connected with the material.
For example, students will often ask, “When will I ever really need to use algebra?” It's easy to give them the F.U.D. answer (a marketing term that means Fear, Uncertainty, and Doubt to nudge consumers into decisions): “If you don't learn algebra, you won't do well on the SAT, and then you won't get into a good college.” That's a lousy answer, and unfortunately is the message heard by many students, either directly or indirectly.
Below is an algebra problem that students can also use logic to figure out. These problems help engage the class and get them thinking!
Providing a range in topics and connecting mathematics with other disciplines, such as history, will ignite students' curiosity to dig deeper and find the beauty and relevance of the mathematics they learn.
When exploring a range of topics, this is not something to be done alone. Exploration should, of course, be a shared experience. Students need to present what they've learned to the class so that they can all grow together and gain the confidence for bigger challenges.
With the high-pressure stakes of standardized testing, rigor is starting to take on a negative connotation. It's been used to pump and dump, namely, shove as much information as possible into a student so that they, in turn, can dump that information on to a bubble sheet of multiple-choice answers.
In this book, we refer to rigor as a way to tap critical thinking so that students can have meaningful experiences and try novel approaches to solve problems. That means presenting the student with problems instead of exercises. Math education must be more than a series of easily obtained answers (exercises) (e.g., find the perimeter of a square, training students to do what a computer can do better). Problem solving goes much deeper and taps into what makes us human, namely multiple creative approaches with a string of steps to solving challenging and interesting problems.
Teaching through challenging problems requires rigor on both the part of the educator and the student, but also requires maintaining a balance between providing these problems and learning foundational concepts. If students can be exposed to the tough problems in mathematics early on, this rigorous training will make other challenges easier to face. Just as it's easier to jog a mile if you've been training for harder races, when you learn the critical thinking that comes along with problem solving, then every other topic that requires math (and most do, in one way or another, e.g., economics) will be easier to understand.
Rigor is both the result of critical thinking and creativity as well as the discipline to learn the foundational concepts necessary to be successful. This rigor is easier to manage when you have a collaborative team (the teacher and fellow students) offering support and encouragement along the way. This leads to the next reward: resilience.
Making the simple complicated is commonplace; making the complicated simple, awesomely simple, that's creativity.
Charles Mingus
Problem solving requires creativity to reach a solution, as there isn't always just one clear-cut approach. Creativity requires bravery and resilience. People feel that creativity just happens, when in actuality it is the product of trial and error – requiring the resilience to persevere. Pushing through failures to the other side is a worthy goal because the benefits are so high, such as developing the resilience to take on harder and harder challenges.
Students, through rigor and range, grow their courage for trying new things as well as hone their creativity and resilience through constraints. How are creativity and constraints related? Math, in and of itself, can be a beautiful and creative pursuit – elegant proofs, creative problem solving, and being connected to a global community where ideas can flourish. That creativity, of course, operates within the constraints of mathematical reasoning and rigorous proofs.
There are myriad problems where you have to jump in headfirst and creatively play before a solution will present itself to you. They take time, patience, and resilience to continue and chip away at them, and that process can be the most exciting part.
Following is a problem that requires a lot of thought and needs a well-conceived plan of attack, building resilience while also providing the thrill of solving!
Because of these constraints, people will reach the conclusion that mathematics is dry and formulaic, but anyone that has ever delved deeper beyond the typical classroom approach to math understands how these constraints free the mind and lead to amazing discoveries. Certainly, math isn't the only area where constraints can be creative. One only has to listen to minimalistic music, read poetry, or see modern art for examples of limiting choices to create simple and elegant forms. Many interesting and delectable meals have been invented by people who are constrained by budget, choices, or geography.