5 Principles of the Modern Mathematics Classroom

Creating a Culture of Innovative Thinking

Gerald W. Aungst

• Publisher: Corwin
• Year: 2015
• ISBN:
  9781506322995
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

Students pursue problems they’re curious about, not problems they’re told to solve. Creating a math classroom filled with confident problem solvers starts with challenges discovered in the real world, not a sequence of prescribed problems. In this groundbreaking book, Gerald Aungst offers five powerful principles for instilling a culture of learning in your classroom: Conjecture, Collaboration, Communication, Chaos, and Celebration. Aungst shows how to:

• Embrace collaboration and purposeful chaos to engage students in productive struggle
• Put each chapter’s principles into practice using a variety of strategies, activities, and technology tools
• I ntroduce lasting changes in your classroom through a gradual shift in processes and behaviors

1 Math for the 21st Century and Beyond

Being a mathematician is no more definable as “knowing” a set of mathematical facts than being a poet is definable as knowing a set of linguistic facts.

—Seymore Papert, MIT Mathematician, Computer Scientist, and Educator

The moment we believe success is determined by an ingrained level of ability, we will be brittle in the face of adversity.

—Josh Waitzkin, American Chess Player, Martial Arts Competitor, and Author

Mathematics as Literacy

“But, I’m just not that good at math.” How many times have you heard this? It may be said when figuring the tip on a restaurant check, or trying to make sense of some statistic reported on the news. The unstated subtext is, “It’s OK if I get this wrong, because I haven’t got the innate ability.”

This is hardly a new phenomenon. John Allen Paulos, in his classic book, Innumeracy: Mathematical Illiteracy and Its Consequences, calls it a “perverse pride in mathematical ignorance” (Paulos, 2001, p. 4). Unlike similar failings, such as struggles with reading or social skills, being bad at math is something people boast about.

We assume that we are born with whatever mathematical talent we’re ever going to have. Math is therefore something we either get or don’t get, completely dependent on that talent. I have a word for that: hogwash. Kimball and Smith (2013) point out that “for high-school math, inborn talent is much less important than hard work, preparation, and self-confidence” (para. 2). Mathematics classrooms are often built to reinforce the former perception instead of the latter truth. “One of the most damaging mathematics myths propagated in classrooms and homes is that math is a gift, that some people are naturally good at math and some are not,” says Jo Boaler of Stanford University. “This idea is strangely cherished in the Western world, but virtually absent in Eastern countries, such as China and Japan that top the world in mathematics achievement” (Boaler, 2014, para. 2).

Sousa (2008) points out another important truth, which makes the job of math educators more difficult. “Spoken language and number sense are survival skills; abstract mathematics is not. In elementary schools we present complicated notions and procedures to a brain that was first designed for survival in the African savanna. Human culture and society have changed a lot in the last 5,000 years, but the human brain has not” (p. 1).

Math has acquired a reputation as a specialized and esoteric subject. The thinking goes, if I don’t grasp it I’m clearly not one of the elite few who have what it takes. But, people who learn to read late in life are celebrated as heroes. Why shouldn’t the same be true for math?

In order for the brain to learn these important skills and concepts, teachers need to create an environment where mathematical reasoning is essential. People learn a language best when immersed in it. Math is no different. We need to immerse students in environments that provide organic opportunities to use and apply math concepts. This book lays out the framework for creating such an environment. To understand that framework, it’s important to recognize that mathematics has far more to offer our students than the ability to accurately compute a server’s tip.

We assume that we are born with whatever mathematical talent we’re ever going to have. Math is therefore something we either get or don’t get, completely dependent on that talent. I have a word for that: hogwash.

Mathematics as an Innovation Incubator

Ask most people today to name an important technology innovator and you are likely to hear names like Steve Jobs, Bill Gates, Salman Khan, and Mark Zuckerberg. Yet, their work would have looked very different without a woman who rarely gets mentioned as a technology innovator. You may not even recognize the name Grace Hopper, but the problems she solved paved the way for those future innovators.

Dr. Grace Hopper was a mathematics professor and a U.S. Naval officer. She was also among those who developed the UNIVAC computer in the early 1950s. Her great innovation in the field of programming was the idea of the compiler, a program that takes human friendly computer code and translates it into the machine code that computers use.

Prior to Hopper’s invention, computer programs were written directly in machine code, a time-consuming and labor-intensive process. Each program was also particular to the machine for which it was written. This meant if you wanted to run a program on a different computer, you had to start again from scratch. Grace Hopper’s compiler opened the door for high-level languages. Programmers could now develop code that was independent of the hardware and could be reused on multiple machines.

Although this seems a fairly mundane concept in the mid-21st century, it was a radical idea seventy-five years ago. In fact, even after it was working, Hopper’s colleagues dismissed her compiler as impossible. “I had a running compiler and nobody would touch it,” she said. “They told me computers could only do arithmetic” (Scheiber, 1987, para. 8). Fortunately for those of us who consider a smartphone to rank near air, water, and food on the “need” scale, they were wrong.

Popular media tell us innovation is essential to the future of our society and economy. But, like math aptitude, we view innovation as something intangible and fleeting, a characteristic of the rare genius. Innovation and creativity can be learned, however, and many schools are striving to become places that nurture these characteristics in all children.

Although innovative ideas and practices happen in every field of human endeavor, and therefore in every subject studied in school, I believe the mathematics classroom is the best place to create conditions that enable innovation and creativity to thrive. We do this through the practice of solving problems.

This seems counterintuitive. After all, in math, the answer is always the answer, right? Two plus two always equals four, and no amount of creativity or innovation will change that. So, how can a math class become an environment that encourages those creative and innovative mind-sets?

Consider the following statements—all of which are true, mathematically valid, and will confound most students, if not their teachers:

1. 9 + 5 = 2
2. 8 + 8 = 10
3. A triangle can have three right angles
4. 16.75 + 15% = 20
5. 375 ¸ 70 = 6

One key to understanding these is to realize that the context for the statement matters. For example, in the first, we are talking about time. If you begin at nine o’clock and move ahead five hours, you end at two o’clock. The second occurs often in the context of computer science, and the third involves non-Euclidean geometry.

The last two may seem a bit of a cheat, since the computations are not strictly precise. However, they are both valid statements in real-world situations. It is very likely you used the fourth example within the last week or two: it is a typical example of figuring a server’s tip. Rarely do we compute a tip to the penny. More often, we just round to a convenient amount because more precision isn’t necessary. The last statement is the mathematical expression of this problem: “375 fourth graders are going on a field trip. Each school bus holds 70 students. How many buses do you need?” If you consider the expression alone, the “right” answer would have a remainder or a fraction. A fraction of a bus makes no sense in this context, however, and therefore is an incorrect answer to the question.

Mathematics, when taught as a framework for problem solving and reasoning rather than as a collection of rules and algorithms to memorize, opens us to flexible ways of thinking about the world. Instead of seeing “solvability” as a fixed property, students need to see every problem as potentially solvable, and see the possibilities that could lead them toward a viable solution.

This was Dr. Hopper’s true legacy. Later in her life, she said “The most important thing I’ve accomplished, other than building the compiler, is training young people. They come to me, you know, and say, ‘Do you think we can do this?’ I say, ‘Try it.’ And I back ’em up. They need that. I keep track of them as they get older and I stir ’em up at intervals so they don’t forget to take chances” (Gilbert, 1981, p. 4). Most interesting, perhaps, is that humans are born with this “take chances and try it” mind-set. Why, then, would we need to reinstall it in our students? Two reasons are that we train this adventurousness out of kids at a very young age, and that school culture is designed to reinforce this training.

A study conducted at University of California at Berkeley illustrates what happens. Psychologists Christopher Lucas, Alison Gopnik, and Thomas Griffiths (2010) gave preschoolers and adults similar problems to solve. They had a box which would light up and play music when certain types of objects, which the researchers called “blickets,” were placed on top. The task for the participants was to determine which of the objects had the “blicketness” property and would therefore turn on the machine. Sometimes, a single object would turn the box on, and sometimes the objects needed to be placed in combination.

The researchers found that the preschoolers were much more likely than the adults to solve the problem correctly. Even when the solution involved subtle, abstract relationships between the objects, young children were consistently better. One reason, they said, is that children are much more fluid in their thinking and aren’t hampered by prior experiences and expectations. They are more willing to try many solution paths, even those that seem unlikely. Whereas adults hone in on one likely option and try it repeatedly, even when it doesn’t seem to be working.

One explanation for this adult behavior is the way we teach math. We discourage free-range thinking, and instead insist on students learning the “one true way” that a particular kind of problem should be solved. And this restrictive teaching happens very early: the same researchers found that toddlers were much more flexible problem solvers than Kindergarteners. “Our current educational system better prepares children to answer questions that are well defined and presented to them in the classroom than it does to formulate the nature of problems in the first place. Often the skills involved in solving well-defined problems are not the same as those involved in recognizing a nonobvious problem or creating a problem” (Pretz, Naples, & Sternberg, 2003, p. 9).

My explanation for the adults’ behavior in this experiment was not considered in the Berkeley study, and to my knowledge is an unanswered research question. There are likely other factors besides traditional methods of math instruction that contribute to the loss of fluid thinking. Regardless, we should be designing school environments to reverse that trend rather than accelerate it.

With the arrival of the Common Core State Standards, schools across the United States are reexamining their mathematics curricula. Although they carry political baggage, the standards give us a tremendous opportunity to think deeply about our practices. If we do nothing else, centering our math instruction around problem solving and the mental habits that go with it will go a long way toward improving learning outcomes for our students.

Mathematics, when taught as a framework for problem solving and reasoning rather than as a collection of rules and algorithms to memorize, opens us to flexible ways of thinking about the world.

Children are much more fluid in their thinking and aren’t hampered by prior experiences and expectations. They are more willing to try many solution paths, even those that seem unlikely.

Goals of This Book

This book aims toward two goals:

1. To provide a framework for K–12 mathematics education designed to create a modern, immersive environment for nurturing the problem-solving skills of all children so they can realize their potential to be the innovators of the next half century.
2. To empower all teachers, regardless of their environment and personal level of comfort with and knowledge of mathematics content, to become stronger teachers of math.

The next chapter further expands on these two ideas, creating a deeper foundation for understanding the 5 Principles framework. I discuss the habits o...