In Search of Patterns
Liberated from the drill and practice of the old arithmetic, teachers across the country are helping students discover the magic of mathematics in their everyday lives
When former teacher Susan Ohanian tells friends she spent the 1990-91 school year sitting in primary math classes across the country, they usually laugh. When they realize she isn’t joking, they express sympathy. Watching children work their way through endless columns of numbers brings back painful memories for too many people.
But in the classrooms Ohanian visited in more than 20 states, something new and different is happening. And in Garbage Pizza, Patchwork Quilts, and Math Magic, to be published by W.H. Freeman and Co. this month, Ohanian describes “profound shifts” in mathematics instruction. “I saw kindergartners create graphs of the types of sandwiches they brought for lunch,” writes Ohanian, “watched 3rd graders make tessellations, and heard young children talk about Fibonacci patterns, probability, and game theory.”
Ohanian was recording the work of teachers who are participating in the K-3 Math Project, launched and funded by the Exxon Education Foundation to develop “primary math specialists” in school districts across the country. What she observed and describes in her book is the beginning of a revolution in math education. It is a revolution long overdue.
A host of studies and test results have documented a steady decline in student performance in mathematics over the past several decades. Only about half of all 17-year-olds are capable of doing junior high school mathematics. U.S. students are consistently outperformed by their counterparts in foreign countries, including those in some Third World countries. The average Japanese student outperforms the top 5 percent of U.S. students in college preparatory math. The 1987 report of the Second International Mathematics Study notes that 25 million children study mathematics in the nation’s schools, but their time is “largely devoted to mastery of the computational skills which would have been needed by a shop-keeper in the year 1940—skills needed by virtually no one today.” And the National Research Council concludes that three out of four Americans leave school without enough math to cope with on-the-job demands for problem solving.
For as long as anyone can remember, schools have taught mathematics as a collection of tools—algorithms and formulas to be memorized through endless drill and practice. Though the intent is to provide children with tools that will be useful in work and life, the mathematics taught has little relevance beyond the classroom for most students. Studies have shown that students who appear to have mastered certain mathematical procedures are unable to apply them in unfamiliar circumstances. In short, they failed to understand the concepts and ideas behind the procedures.
In 1986, the National Council of Teachers of Mathematics published Curriculum and Evaluation Standards for School Mathematics and sowed the seeds of the current revolution. The goals for students in mathematics, said the NCTM report, should be “that they learn to value mathematics, that they become confident in their ability to do mathematics, that they become mathematical problem solvers, that they learn to communicate mathematically, and that they learn to reason mathematically.”
To accomplish those goals, the NCTM calls for an emphasis on problem solving, for creative classes that stress comprehension not simply memorization, for the use of calculators and computers to enhance basic skills, and for the use of manipulatives to promote understanding.
The council recognizes that the success of its revolution depends on teachers—their willingness to embrace the new standards and become proficient in implementing them. That’s a big order. “One of the very scary facts about being a teacher,” Ohanian writes, “is that every second of every day we teach who we are, and if nothing extraordinary intercedes to transform us, we teach the way we were taught.” But Ohanian insists that something is interceding in the lives of many teachers. After hundreds of hours of observing and meeting with the teachers in the K-3 Math Project, she is optimistic about the future of math instruction.
She quotes on elementary teacher from Orlando: “I hated math. That’s the reason I applied to the math specialist program. … I knew I had to get better in math because I was killing these kids with my narrow, computational view. I was the kill-drill queen. And why not? I’d had two decades to practice it.”
Ohanian believes that more and more teachers are beginning to “realize that children’s learning isn’t confined to just that 40-minute segment in the plan book labeled ‘math period.’” She writes: “Just as a teacher teaches ever second of the day, with her intonation and gesture, children never stop learning. They may not be learning the lesson the teacher thinks she is teaching, but they are nonetheless learning something all the time.” Teachers are rethinking their curriculum, Ohanian says, and “they are quick to point out that they have forsworn the old arithmetic; now, they insist, they teach mathematics.”
But proponents of the “old arithmetic,” steeped in the back-to-basics orientation of instruction, are well-entrenched. Mathematics is sequential, they insist, and children must learn first things first. They must memorize the rules and master the basic skills if they are to succeed in higher-order mathematics.
Ohanian insists that “skills are the surface issue in this battle.” She adds: “The real issue is what is best for children. Is it better that they memorize a finite set of operations and look good on paper, or is it better that they take the more circuitous route of discovery and understanding?”
Even some of the teachers in Ohanian’s book struggle with this dichotomy. Few teachers resolve it, she writes; most “are torn between implementing a constructivist believe system that insists they cannot pour, drill, or lecture skills into a child and responding to their own fears and the fears of their administrators, board of education, and community that if they do not continue to pour and drill and lecture math facts, their students will not fare well on standardized tests.”
In the end, though, Ohanian is betting on teachers doing the right thing. She writes that teachers “develop a good dose of what Keats called ‘negative capability’—the ability to exist among uncertainty, mystery, doubt, insult, affront, and indignity. They work in difficult conditions, they nurture the children in their care. I have visited a great variety of classrooms across this country, and I can testify about the teachers who tend them: They are resilient, adaptable, and tough. They will persist and endure.”
Consequently, says Ohanian, there is reason to feel optimistic about the children in these teachers’ classrooms. She calls them “problem solvers for the 21st century: children who can think mathematically, who can communicate that thinking to others, who can use mathematics to make sense of their lives, who are eager to explore math topics that most of us adults have never heard of. These are children who see themselves as mathematicians, who see math as an integral part of their lives, who are developing a flexible repertoire of problem-solving strategies and attitudes of independent thinking, children who welcome challenge. They know mathematics can be beautiful. These children are showing us that there is cause for celebration.”
The following excerpt is from chapter two of Ohanian’s new book. It’s titled “Do You See a Pattern?” Contemporary mathematics is often viewed as “the science of patterns.” A 1989 National Research Council report states: “Mathematics reveals hidden patterns that help us understand the world around us. … As a practical matter, mathematics is a science of patterns and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth.”
In the dramatic opening to A Mathematician's Apology, renowned English mathematician G.H. Hardy announces: “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” Harvard professor and MacArthur Foundation fellow Stephen Jay Gould agrees, describing science as “a search for repeated pattern.” For the ecologist, Gould points out, this search involves asking questions; he calls ecology an “explanatory science,” requiring the use of differential equations, complex statistics, mathematical modeling, and computer simulation—all in search of patterns. Along the same lines, standard 13 of the National Council of Teachers of Mathematics' standards, titled “Patterns and Relationships” and arguably its most popular, states:
“Patterns are everywhere. Children who are encouraged to look for patterns and to express them mathematically begin to understand how mathematics applies to the world in which they live. Identifying and working with a wide variety of patterns help children develop the ability to classify and organize information. Relating patterns in numbers, geometry, and measurement helps them understand connections among mathematical topics. Such connections foster the kind of mathematical thinking that serves as a foundation for the more abstract ideas studied in later grades.”
The NCTM advocates that children become immersed in the search for pattern from their first day of kindergarten. The NCTM publication Professional Standards for Teaching Mathematics provides a vignette based on principles established by Mary Baratta-Lorton. It describes how a kindergarten investigation of patterns might look and sound, providing marginal pedagogical commentary to explain what's going on. [Quotations from the NCTM vignette are in italics.]
- Pat Kowalczyk's kindergarten class enjoys activities involving continuing patterns that have been started using blocks, beads, themselves, and other items. Today Mrs. K, as the children call her, plans on having her class construct patterns using their names. She thinks that this activity will extend the work she has been doing to encourage them to reason and communicate about mathematics with one another. She has prepared a paper with a 5-by-5 grid of 2-centimeter squares for each student.
At their tables, the students fill out the grid, using one square for each letter of their name. When they finish writing their names the first time, they start over and continue until each of the 25 squares contains a letter.
Mrs. K: Select your favorite crayon and color in the squares that contain the first letter of your name.
Mrs. K walks around the room observing and listening to the students as they work. When Susan wants to know if she should color both the Ss in her name, Mrs. K responds with a question, “Are both the first letter in your name?” Susan thinks for a moment and then says, “No, only this one is,” and she colors only the first S in Susan. Mrs. K makes a mental note that Susan seems confident in her decisions and does not seek additional confirmation from her. As she continues to walk around, Mrs. K observes that some children seem to understand the activity and work independently, some are actively conferring with others, and some are asking for her to help them. She muses, not for the first time, about what more she could be doing to foster greater self-reliance by her students.
When the students complete their grids, Mrs. K asks the class if they can predict who has the same patterns of colored-in squares on their grids. She tries to phrase the question to encourage the students to reason and communicate their ideas. She notices that she is improving her ability to construct good questions on the spot.
The students quickly guess that the two Jennifers in the class should have the same pattern. Mrs. K asks several students to explain how they can be sure of this without even checking the girls' grids. When she hears Marcus say, “Cause they have the same name so their papers gotta be the same too,” she is really pleased. Calling on him more often really seems to be paying off.
Searching for the next good question, Mrs. K challenges the students to find similar patterns where the students do not have the same first name. After some checking around, the students find that Kent's and Kyle's grids have the same pattern.
Kent: Maybe names that begin with the same letters look the same.
Mrs. K: Is there anyone else whose name begins with the letter K? [Katrina, Kathy, and Kevin all jump up, waving their hands.]
Katrina: But my grid is different from Kent's and Kyle's.
Kathy: But mine is the same as Kevin's.
Mrs. K: Does this fit the rule that the names that begin with the same letter give the same pattern?
Mrs. K looks around, trying to decide on whom to call and tries to remember who has not spoken much today. She remembers that Nikki has not said anything today, though she did complete her grid quickly.
Mrs. K: Nikki, how can we change our rule so that it works?
Nikki: Well, I think it will work if they have the same number of letters and if their name begins with the same letter.
Laura: Mine matches Kathy's, but our first letters are different.
Mrs. K: Let's check it out. [She holds them up to the window, one on top of the other.] Hey, it looks like they do match.
At this point, Dave, Jane, and Jose put their patterns by Kyle's and Kent's and are surprised that the patterns match. They don't know how to express their finding. Mrs. K is a little surprised that this is hard to explain. Judy says that it has something to do with the length of the name. Short names seem to match with short names but not long names. Finally, Stanley says that the names with the same number of letters will match. Some of the other students question whether he is right. After examining many other examples, they conclude that he is correct.
After school, Mrs. K reflects on the lesson. She writes a few notes in her journal about Marcus, Nikki, and several other students. She also writes down the task so that she can remember it for the future and indicates that she thinks it could be used profitably again. She is impressed with the students' ability to reason. She thinks that letting the students use different colored crayons to color in the grids may have distracted from the lesson's primary objective. She makes a note to let students pick only one color next time she uses this activity. Although she thinks she is getting better at formulating good questions, she also thinks that she needs to find more ways to encourage students to communicate their ideas with one another and to build on one another's reasoning.
Nobel physicist Richard Feynman recalls being taught about patterns even earlier, quite literally at his father's knee. His father would set up tiles of different colors for the toddler to knock down:
“Pretty soon, we're setting them up in a more complicated way: two white tiles and a blue tile, and so on. When my mother saw that she said, `Leave the poor child alone. If he wants to put a blue tile, let him put a blue tile.' But my father said, `No, I want to show him what patterns are like and how interesting they are. It's a kind of elementary mathematics.'
In his introduction to On the Shoulders of Giants, Lynn Steen observes:
"What humans do with the language of mathematics is to describe patterns. Mathematics is an exploratory science that seeks to understand every kind of pattern—patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns. To grow mathematically, children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections.”
From Temecula, Calif., to Orlando, Fla., to Greenville, Ala., primary graders use such manipulative materials as Unifix cubes, colored tiles, and plastic links to identify and construct simple patterns, sorting them into two subsets by size, color, or shape. Children identify AB patterns and build them with a variety of materials. They also have plenty of opportunities to hear and sing patterns, to clap and stomp them. Quite noticeably, teachers do not offer extra help to children who are not clapping or stomping in rhythm; in fact, teachers make no effort to “correct” them. In her popular teacher resource Mathematics Their Way, Mary Baratta-Lorton stresses that children need time to develop skills without being singled out for having difficulty. Baratta-Lorton reminds teachers, “Keep in mind that children are supposed to evidence skill at the end of this work, not at the start.” She also insists that a child doesn't need to understand a pattern before being exposed to a different one. Pattern work is not sequential and in the long run, says Baratta-Lorton, “children benefit from being slightly overwhelmed. Given time, each child sorts out the elements in his or her own way.”
There's a famous story about Srinivasa Ramanujan, an unschooled clerk obsessed with mathematics whose talent was recognized by G.H. Hardy. The two men shared a fascination with numbers, a “feel” for numbers that almost turned those numbers into living things. When Hardy visited Ramanujan in the hospital, he commented that the taxi's identification number had been rather dull: 1729, or 1719. Ramanujan disagreed. “No, Hardy,” he said, “it's a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”
Robert Kanigel, author of The Man Who Knew Infinity: A Life of the Genius Ramanujan, offers this explanation:
“Finding numbers that were the sum of one pair of cubes was easy. For example, 23 + 33 = 35. But could you get to 35 by adding some other pair of cubes? You couldn't. And as you tried the integers one by one, it was the same story. One pair was easy, two pairs never—never, that is, until you reached 1729, which was equal to 123 + 13, but also 103+93.”
How did Ramanujan know? It was no sudden insight. Years before, he had observed this little arithmetic morsel, recorded it in his notebook, and, with his trademark intimacy with numbers, remembered it. Mathematicians relish this story for a somewhat different reason: For them, all numbers are fascinating—if you just see the pattern, you can make a connection.
Marion Pelking's 1st graders in Las Cruces, N.M., bring their teddy bears to school to help them with math. Pelking puts 1-foot-square tiles on the floor so that her students can construct living graphs. When they go apple picking, for example, they put real apples on their graph. One morning, the children decided to graph their teddy bears according to color. Once all the brown bears and all the white bears were lined up, Pelking asked, “What can we say about these bears?” Jerry replied, “Well, we could say that they're all lined up waiting to get into the bank.” Pelking has her eye on this boy: He connects number patterns to real life.
Joan Goodman wrote Patterns across New Mexico for her 1st and 2nd grade development class at the Raymond Gabaldon Elementary School in Las Lunas, N.M. To accommodate and enrich a class that is 67 percent Hispanic and about 3 percent Native American, Goodman teaches the unit in both Spanish and English and designs math projects that are culturally diverse.
Goodman's students talk about patterns a lot—describing and drawing the patterns they find on each other's clothing, for example. One child suggests a game, “Let's see who has the most patterns all over him.” They write pattern poems and end up producing a class pattern book. They beat patterns on drums, clap patterns, and stomp patterns. They draw Native American patterned pottery in art and incorporate patterns in their hand weavings made of colored yarn and strips of fabric. Using aluminum foil as “silver” plus beads of two colors, Goodman's class puts their knowledge of patterns to good work creating simulated Native American jewelry. Some patterns are simple: ABC, ABC, ABC; others are more complex: AABCCB AABCCB. But every child creates a pattern.
Joan Hammond's 1st grade classroom in Conecuh County, Ala., demonstrates that you don't need to rely on a publisher for materials to create patterns in mathematics. Hammond's classroom is filled with a wonderful variety of “things” for children's investigations: locks and keys, nuts and bolts, buttons, even dog tags that were ruined at the manufacturer and then donated to the classroom. It is apparent that this teacher must have “first refusal” on anything in the county before it gets carted to the dump. And children realize the real-world importance of mathematics when materials from their lives are made available inside the classroom.
Third graders in Penny Vincent's class in Albuquerque, N.M., are stunned to learn that the patterns they've recognized have helped them solve multiplication without even knowing it. Six children working together have the problem of figuring out how many M&M's they have in all when each has two, three, four, and five. David always counts by ones. Keith counts by twos, threes, fours, and fives. Sandra makes two groups of six, three groups of six, and three groups of eight. They talk about their strategies, noticing that there are lots of ways to get the same answer. The children are excited when their teacher tells them that by figuring out faster ways to add numbers together, they are actually multiplying.
Like many teachers who are trying to “humanize” mathematics and connect it to students' lives, Pam Hagler is on the lookout for books with mathematical potential. Hagler reads How Big is a Foot? to her students at the Eugene Field School in Albuquerque. It is a comical story about a king's troubles when he tries to give the queen a bed for her birthday. Beds as we know them haven't been invented yet, and so the king has the queen lie on the floor, wearing her crown (which she sometimes likes to wear to sleep), and he paces off her length—six feet. Problems arise, however, because the carpenter's feet aren't nearly so long as the king's.
When Hagler finishes the story she asks her students, “Did you see a pattern?” Everybody is anxious to volunteer but the conversation soon turns to other reactions the story provokes. When Hagler asks her class if anybody has ever outgrown a bed, many answers illustrate the sensitivity any teacher needs, a need exacerbated in an inner-city school. Children volunteer: “I don't sleep in a bed; I sleep on the floor. My brother sleeps in the bed with my cousin.” He continues, “I don't want to sleep there.” Several other students agree that they have a similar situation at their house, but then Joey warns, “Well, don't sleep on the floor when the heater's on.” Maria chimes in, “If your feet hang out of bed la cuccaracha will get you.” Patterns are momentarily forgotten as children make cockroach jokes, but Hagler regains their attention with an extension of the story, with teams of students inventing their own units of measurement.
Children enthusiastically measure everything in sight and begin an initial investigation of the pattern to be found between lengths of various measurement units and the distances measured. There's a parent volunteer in the room helping them measure—a father in a business suit down on his hands and knees admiring the ingenuity of primary graders and getting just as involved in the investigation as they are.
There may be primary classrooms without tubs of pattern blocks, but if there are, I haven't seen them. The collection of six geometric shapes in six colors—green triangles, orange squares, blue parallelograms, tan rhombuses, red trapezoids, and yellow hexagons—are designed so that they can be nested together. Children play with them, explore with them, and eventually work with symmetry, area, perimeter, fractions, and functions. Focused exploration usually starts with patterns.
At the Learning Center for Deaf Children in Framingham, Mass., teacher Ellen Bauman asks 7- and 8-year-olds to use pattern blocks—any blocks they want—to fill in the pattern outlines in an activity book. “See if you find any patterns,” Bauman advises. Barry soon discovers that there is more than one way to fill the space. “Three greens always equal a red,” he signs. Barry then invents a contest with himself to find the least number of pieces it takes to fill a design and then the most pieces that will fit into the same design.
Bauman leaves Barry and the others to discover on their own. After a while, she interrupts, “Now that you've solved quite a few problems, I want to stop and discuss.” Bauman encourages students to talk about what they discovered about different shapes—why some fit and some don't.
They talk about hexagons. Bauman asks how many hexagons it will take to fit the space. “Seven?” volunteers Ginny, her answer half a question, half a declaration. Bauman does not affirm or deny but asks other students, “Do you agree with Ginny?” When someone disagrees, Bauman challenges Ginny, “Can you convince him?”
Next, Bauman introduces trapezoids, and Kathy immediately signs, “There's 20.”
Bauman laughs. “You answered before I asked! Would you explain it to us?” Kathy signs, “It takes two pieces for each one, and I had 10 before.”
“What's this shape?” asks Bauman, and, when nobody knows, she offers to write the name for them.
“There's 30,” says Barry.
“How do you know? Can you explain your answer?”
“There's three in each hexagon. I don't know, but it's 30; I just know that. It just came to me.” Bauman pushes a little: “If you had two for everyone and that gave you 20.” Kathy interrupts Bauman's attempts to lead them, signing, “I got 30, too.”
Barry jumps up, “There! It must be right.”
When Bauman presses for an explanation he gives an answer familiar to every teacher, “I just know it.” Every teacher knows why when parents ask their children, “What did you learn in school today?” the answer is always “nothing.” Children seldom recognize their own intellectual breakthrough or, if they recognize it, they rarely want to talk about it. Children believe they “just know” things.
Kathy and Barry seem to be on the edge of an explanation, don't quite make the intuitive leap, and so fall back on, “It just came to me, and I just know it.” Bauman respects their struggle and does not “steal” their construction of understanding by spilling the beans, so to speak. She doesn't have a timetable and so she is able to give these children space and time to figure things out.
All this may not look like a profound teaching moment, but it is. In a flash of intuition, sensitivity, and just plain kid savvy, a teacher keeps her mouth shut.
All this may not look like multiplication, either, certainly not to a generation reared on memorizing the times tables. Bauman doesn't announce to her students that they are multiplying, but she's definitely getting them ready. It is clear that these children are on the brink of understanding multiplication, of creating their own understanding and being able to verbalize it. But while teetering on the brink, they back off, become impatient with putting things into words; they beg Bauman to let them work with the pattern pieces again. They want to figure out more patterns—and do more predicting—without having to explain how they do it.
Bauman takes her cue from her students and doesn't push. She agrees with Sheila Tobias, author of Overcoming Math Anxiety, who tells us there is a difference between not knowing and not knowing yet.
Bauman is pleased with her students' small leap. She smiles and signs her agreement that they should continue their work with patterns—the work they want to do.
First graders in Delia Hakim's classroom in Tucson, Ariz., go outside and collect leaves. They talk about what they know already about leaves—and Hakim is not surprised to learn they know a lot already. “I don't start with the premise that students are just waiting to be filled up with my knowledge,” she says. “I want them to discover their own knowledge.”
Students make leaf patterns—they group leaves according to their predominant characteristic: chiquito, grande, de colores, redonda, orlado, ban nan. “We can read a lot about trees in the textbook—and forget it next week,” says Hakim. “Instead, I encourage my students to bring in their own knowledge first, then to conduct firsthand investigation, to learn new things, to figure things out for themselves. We explore our environment first and then read about it in books.” Hakim is anxious that her students experience—and enjoy—mathematics as a relevant part of their daily lives.
Hakim points out that, in possession of a collection of wonderful buttons, she was tempted to ask her students to make a graph. Instead, she gave them a more open-ended task, one that would encourage them to create their own learning. Hakim asked, “How could we count buttons?” With such a question, Hakim takes a risk and invites her students to do likewise. She asks them to examine the universe of buttons and figure out a way to exert some sort of control over it.
Some children will look at it and see no patterns, no ways to group the buttons. Others will notice patterns: The buttons can be divided into two major groups—white buttons and colored buttons. Other children will notice that buttons have different numbers of holes in them. And so on.
When Patricia Weaver asks her 2nd graders in Tucson, “How many buttons are we wearing today, December 4, 1990?” their predictions range from 51 to 110. Then they begin to talk over how they might find out how many buttons are actually in the room. They come up with these suggestions:
1. Put the children in line and count their buttons.
2. Have each person put a slash mark for every button he or she is wearing and then add up the slash marks.
3. Have Ms. Brooks (the Tucson math coordinator and frequent visitor to the classroom, who, incidentally, walks in during the button investigation) figure it out. Brooks agrees that “ask an adult” is definitely a logical and legitimate problem-solving strategy, but she also insists they need to make another choice, one where they aren't getting an adult to do their thinking. As it turns out, Amy is the “Button Queen”; she is wearing 27 buttons. Sundie, Jane, and Julie aren't wearing any. The total number of buttons in this classroom on this day is 90.
Finally, the children figure out a way to record their data so visitors can see what they've done. Their teacher does not hand them a ready-made graph worksheet on which to put their data. Instead, she gives them the time and space to “invent” graphs. And right there on the graph, I see Amy recorded forever as the “Button Queen.”
Teachers like Bauman and Hakim and Weaver are representative of their colleagues across the country. They believe that mathematics is a process of constructing knowledge, not acquiring it.
Knowing that patterns are a powerful, unifying force in mathematical understanding, they provide many opportunities for their students to find patterns in the real world. In contrast with encouraging a more deductive approach where students are trained to draw conclusions from “given” ideas that everybody accepts as true, these teachers know that use must precede theory. They place an emphasis on inductive reasoning, encouraging their students to do math, to engage in active mathematical investigation: Their students make observations, notice patterns, and then form conclusions.
And so do their teachers. Pattern explorations reveal math's mysteries for teachers, too. When one teacher worked with rectangular arrays in a workshop, she exclaimed, “So that's what square numbers mean?!”
The classrooms of project teachers are as varied as the teachers who lead them, but after a year of visits, I have noticed common goals, similar questions that are the foundation of the mathematics curriculum in these classrooms:
1. Do my students see themselves as good mathematicians?
2. Do my students see mathematics as covering a wide range of topics?
3. Are my students developing a flexible repertoire of problem-solving strategies?
4. Are my students able to communicate their problem-solving strategies to others? Can they talk and write about how they solve mathematics problems?
5. Are students able to assess themselves? Are they able to develop and use criteria to evaluate their performance?
6. Do my students engage in mathematical thinking without a specific assignment? For example, if they have “free time,” do they choose math?
7. Are my students developing the attitudes of independent and self-motivated thinkers and problem solvers?
8. Do my students welcome challenges in math? Are they able to focus on math problems of increasing complexity for longer periods of time?
9. Do my students recognize the importance of math in the real world outside school?
10. Do my students use mathematics to solve problems outside math class? These teachers are not asking what specific math facts 4th graders need to know to please the politicians; they are asking what mathematical insights people need to be useful, creative, and productive participants in a complex world.
Vol. 04, Issue 02, Pages 20-25