Promoting a problem-posing classroom.
Children have traditionally solved our problems -- problems that we think will be of interest and relevance to them. We need to shift some of this responsibility to students and let them pose problems that they consider to he worthwhile pursuing. Child-generated problem posing is recognized as an important component of children's mathematical activity, if not at the very heart of their learning (Moses. Bjork, and Goldenberg 1993; NCTM 1989, 1991: Silver 1994). Our current vision of curriculum reform is of a balanced variety of rich problem situations, ones that reflect cultural diversity and that engage children in investigating, formulating, representing, and reasoning.Such reform necessitates a shift in our role from "telling" children what to do to "questioning and listening" (NCTM 1995, 2). This approach will require first, a commitment to creating an environment in which problem posing is a natural process of mathematical learning. Second, it will require us to have strategies for helping children pose meaningful and enticing problems. Before considering these issues, we need to define problem posing and consider some of its merits.
What is Problem Posing?
Problem posing involves generating new problems and questions to explore about a given situation, as well as reformulating a problem during the course of solving it (Silver 1994). Problem posing can occur before, during, or after solving a given problem. This article is concerned primarily with the "before" and "after" activities; however, this emphasis does not deny the importance of reformulating a problem as one attempts to solve it.
Problem posing and problem solving are obviously closely related. On the one hand, problem posing draws heavily on the processes of problem solving, such as identifying the key elements of a problem and how they relate to one another and to the goal of the problem. On the other hand, problem posing takes children beyond the parameters of the solution process. For example, problem posing can involve children in questioning the origin of the ideas in a problem, or in considering what other problems might arise if the elements of the problem were modified or extended. These ideas are illustrated in the activities presented in this article.
Why Encourage Problem Posing?
Many benefits are gained from fostering a problem-posing classroom. Probably one of the main strengths of such an environment is that it can empower all children to explore problem situations and to pursue lines of inquiry that are personally satisfying. This atmosphere creates a context for more productive and enjoyable mathematical learning. More specifically, problem posing can --
* promote a spirit of inquisitiveness and generate more diverse and flexible thinking;
* encourage children to take greater responsibility for their learning;
* alert both teachers and children to misunderstandings and preconceptions;
* enhance children's problem solving, as well as reinforce and enrich basic concepts:,
* remove erroneous views on the nature of mathematics; and
* dissipate common fears and anxieties about mathematics learning (Brown and Walter 1993; Silver 1994).
Establishing a Community of Inquiry
How, then, might we set about establishing a problem-posing classroom? I have found the Philosophy for Children program (Lipman 1985; Sharp 1987; Splitter 1988) to be a good source of starting ideas. The program shows how we can build a community of inquiry in our classes wherein meaningful dialogue, or "connected talking," among the children and with the teacher becomes the primary tool for developing children's thinking and learning skills. By encouraging children to share their ideas during mathematics activities, we can gain a better idea of their needs and interests and modify our learning contexts accordingly. In so doing, we can develop the dispositions and inquiry skills needed for effective problem posing.
A worthwhile starting point for establishing a community of inquiry is to have children analyze their mathematical thinking and to share ideas on how and what they think as they tackle problems. Questions such as the following can serve as springboards for lively discussion (English 1994).
* Do you ever think about your thinking? How would you describe the way you think?
* What about the way you think when doing mathematics activities. Is it different from your "everyday" thinking?
* When doing mathematics activities, do you think in pictures? In drawings? In words? In rules? In signs? How do these ways of thinking help you?
* If your friend thinks of an idea before you do, does that mean that he or she is a better thinker than you are?
* If your friend says that he or she is a good mathematical thinker, what do you think the comment might mean?
* What might you think about as you begin to tackle a new mathematics problem?
* Do you change your thinking often as you solve a problem? In what ways might you do so?
* Which types of problems do you like to solve? Why?
* Have you ever thought about changing a mathematics problem into one that you would really like to solve? What might you do?
* Have you ever thought of another question to ask after you have solved a problem?
Children's responses to such questions can lend valuable insights into their views on mathematics and mathematical thinking. Figure 1 lists some of the responses that I received when I presented these questions to fourth- and fifth-grade children.
These general questions lay the foundation for more focused discussion on specific problem situations. Opportunities for problem posing can arise in all our mathematics activities. It is important, though, that we adopt a problem-posing perspective ourselves and model the desired inquiry skills for our students (Moses, Bjork, and Goldenber, 1993). Questions that are generative in nature, that is, ones that lead to further ideas and questions. are important in guiding children's problem-posing efforts. Such questions need to target the important ideas in a problem situation and how they are linked -- the key mathematical relations that we need to consider in achieving the problem's goal. One of the basic requirements for posing new problems is that children understand what a problem comprises (Brown and Walter 1993). Once children appreciate the importance of recognizing the key ideas and their relationships, they are in a position to consider new situations that might arise from modifying or extending these ideas. Some of the generative questions that I have found effective in this context follow.
Examples of generative questions
* What are the important ideas in this problem?
* How are they linked?
* Where else have we seen ideas like these?
* What are we asked to do with the ideas? How might we do so? Why would we do so?
* Do we have enough important ideas to solve the problem?
* Could you have used these ideas in a different way to solve the problem?
* How might you change some of these ideas to make a different problem?
* What if you were not given all these ideas? What might the problem become then?
* What if we were to add some new ideas" What ideas might we add? What new questions might we ask then?
I posed these questions to fifth-grade children when we explored the following deductive-reasoning problem, adapted from Charles, Mason. and Martin (1985):
Carla, Suzie, and Emily live next door to one another. One
of them is a lawyer, one is a teacher, and one is a politician.
Use the clues to find each woman's occupation.
1. Suzie lives in the middle apartment
2. When Emily goes away, her cat is fed by the teacher.
3. The lawyer knocks on Carla's wall when her television
gets too loud.
I have found that fourth and fifth graders greatly enjoy solving deductive-reasoning problems (English 1996). When I asked the fifth-grade children to identify the important ideas in the foregoing problem and how they are linked, one of the children, Julian, explained, "Well, you've got three women and they each have an occupation, and you have to work out which woman does what thing." Julian later explained how he would solve the problem: "Okay, you have to work out which woman is which occupation. It says one lives next to another, and so if you look down to the bottom clue, you see that Suzie might be the lawyer, because she lives in the middle apartment and she knocks on someone's wall. That means Emily and Carla have to live on either side. When Emily goes away, the cat is looked after by the teacher. That means Carla is the teacher. Emily can't be the teacher because her cat is fed by the teacher when she goes away. That means Emily is the politician." Like other children, Julian suggested that drawing a diagram would be a good way to solve the problem.
One child gave an unexpected response when I asked what the problem might become if we were not given all of the ideas: "It would become a real problem, one that is not easy to work out." Obviously this problem was not sufficiently challenging for her. When asked what new ideas we might add to make a new problem, several children suggested that we could add some clues as to where the people lived and could then pose the question "Where do they each live in the row of apartments?" The children initially suggested adding two clues but then realized that only one would be needed, such as, "Carla lives on the far left-hand side."
Generative questions such as these can broaden children's views on the nature of mathematical situations and the different ways in which they might be approached. These experiences can also help remove the erroneous view that every questionhas only one right answer, this being the one given by the teacher or the textbook (Moses, Bjork, and Goldenberg 1993). The remainder of this article considers ways in which we might use generative questioning with some multicultural activities to promote a problem-posing classroom.
From Given Problems to New Problems
The following activities are based on historical ideas from other cultures and can help children appreciate the contributions and applications of these cultures to our mathematical knowledge (Shirley 1995). The activities open interesting avenues for problem posing and also promote children's number sense, logical reasoning, and problem-solving skills. The first set of activities deals with magic squares. The second set looks at ancient dice games and game playing. The historical information and illustrations have been taken from Mathematics from Many Cultures (Irons and Burnett 1994).
Problem posing with magic squares
Magic squares are a rich source of problem-posing activities because of their many patterns and relationships. I usually begin such activities by presenting the children with an open-ended research task conducted in the library. The goal of the task is to "find out all you can about magic squares." In addition to developing children's problem-posing skills, I find that this activity broadens their initial algorithmic perception of magic squares. For example, fourth and fifth graders told me at the beginning of the activity that "magic squares are about finding out where numbers go on a square" and that "you look for numbers going vertically, horizontally, or diagonally, and you add them together to get the magic number; ... to find the missing number in the square, you have to subtract from the total."
In implementing this research activity, I try to foster a spirit of inquiry by presenting the children with questions and issues to consider at appropriate points during the activity. Such questions might include these:
* Before you begin, think about what you already know about magic squares. Now think about some questions that you would like to explore to learn more about these squares.
* What were some of the questions that you considered first? How did you answer them? Did they change any of your ideas about magic squares? How?
* When you started to answer these questions, did more questions come into your mind? What were these?
* Did some of your questions ask about the origin of magic squares?
* Share your questions and findings with your group. See if you can present your group with a problem question that none of you has ex-plored yet.
* Let us look back over our activity. Do you think you have learned more from asking your own questions and solving them than from having the teacher pose questions for you? (This question can serve as a good debating activity.)
Following this activity, I would draw the children's attention to two very old squares, namely, the Lo Shu square (see fig. 2) and Durer's square (see fig. 3) (Irons and Burnett 1994). If the children had not discovered these squares in their research, they could spend time exploring the history of each. These squares have a colorful history and display many interesting features that serve as effective springboards for problem posing.
According to ancient myth, the Lo Shu square was
discovered 2200 B.C. by a Chinese emperor who saw it on
the back of the divine turtle that lived in the Lo (Yellow)
River. The magic square was thus named Lo Shu, with Shu
being Chinese for "water." Durer's magic square was
developed by Albrecht Durer from Germany. Like people
from many other cultures, Durer believed that magic
squares brought good luck, and so he included one in his
painting Melancholia.
These squares can be used in different ways to encourage problem posing. One approach is to have children first identify some interesting features of each square, compare their respective features, and then use these observations as a basis for creating new questions and problems. I have found that children initially need a number of "discussion starters" to help them, especially when they are inexperienced in problem posing. Such discussion points could include the following:
* What have you discovered about the Lo Shu square? About Durer's square?
* Which square do you find more interesting? Why?
* Which of the features in each square are needed to make it magic? Why?
* How did you find the magic number in each square? Could you have found the magic number in other ways?
* Did you find interesting patterns of numbers in Durer's square? Use your discoveries to create some problem questions to ask your friends about Durer's magic square.
* Why do you think each square has features that are not needed to make it magic?
* What if the red dots. in the comers of the Lo Shu square were not even numbers?
* What changes might we make to the Lo Shu square while still keeping its magic number?
* What questions might we ask about our new Lo Shu square?
* See what other magic squares you can create like the Lo Shu square. Can you make up some problems about your new squares for your friends to solve?
When I used these magic squares with fourth and fifth graders, I found that their initial responses reflected their view that magic squares are about "finding the missing number." For example, when I asked the children what interesting things they could see in the Lo Shu square, they replied, "It's all filled in, there are no gaps"; "You don't have to work anything out." However, the children soon discovered other features, commenting, "There are rectangles and squares and crosses and lines," "There are dots instead of numbers," "The dots go from 1 to 91," and "Each direction adds up to the same number." The children became quite excited as they discovered further features of the Lo Shu square, including the patterns in the colors, the positioning of the odd and even numbers, and Aaron's comment that "the odd number in the middle is pointing to each of the other odd numbers." Aaron also noted that the middle number is a factor of the magic number (15).
The children enjoyed finding similarities and differences between the Lo Shu and Durer's square, noting the inclusion of numbers 1 through 9 in both, the different magic numbers and number representations, and their different origins ("The turtle was found and the other one was made up by the painter"). The children displayed mixed opinions on which magic square they preferred, with some saying that they liked Durer's square because "It is more challenging to do" and others liking the Lo Shu "Because it has more things to think about; more details:" "It doesn't have as many numbers," and "'Cause it was made naturally."
Finding other ways of determining Durer's magic number, as well as finding other number patterns in the square, generated lively discussion among the children. They readily identified the magic number (34) in different arrangements of numbers, such as in the sums of the corner numbers (16 + 13 + 1 + 4), the middle four numbers (10 + 11 + 6 + 7), the two top and bottom middle numbers (3 + 2 + 15 + 14), and many more. They also found several number patterns, including that the number pairs in the middle columns have a common difference of 1 (3, 2; 10, 11; 6, 7; 15, 14), and that the numbers lying on the diagonals of the 2 x 2 squares have a difference of 2 (e.g., 3 and 5) or a difference of 6 (e.g., 16 and 10).
In their discussions about what changes they might make to the Lo Shu square, the children put forward several options, such as putting the odd numbers
RELATED ARTICLES: Children's responses to questions about their mathematical thinking
* When you do Math, you think about numbers, but your everyday thinking is just general.
* Sometimes I like to think differently from the teacher; I mean, the teacher usually give us hints on how to work out an answer, but I like to use different strategies from the teacher.
* If my friend is a faster thinker that me, that does not mean he is a better thinker, because it could be a different idea and everyone's different and everyone thinks differently.
* A good math thinker could be good at operations but not good at working out problems.
* A good math thinker is someone who knows a lot about math.
* Math problems are usually about someone getting something or losing something. The key words give you the clues, like "left over" is subtraction and "altogether" is addition.
* When I solve a new problem I convert it to reality, I mean the real world, or I try to think of other math problems we've solved.
* I don't like math problems when they are too easy and you can do them in your head and the teacher tells you to write down how you do it and you don't know how to write it down.
* Problems would be better if they had more interesting stories and longer stories and if they had more details in the story. You could also put in things that weren't true, like goblins or princesses. instead of the even numbers in the corners. After experimenting with these options, the children finally concluded that they needed even numbers in the corners to keep the magic number but could swap corresponding even numbers (i.e., 2 with 8 and 4 with 6) and corresponding odd numbers while keeping the middle number intact. The children also enjoyed making new magic squares, modeled on the Lo Shu, with some children creating, their own mythical creature.
Problem posing with game situations
Games are steeped in cultural history and are a motivating resource for developing children's problem-solving and problem-posing skills, as well as their number sense and mathematical thinking. At the same time, ancient games can help children appreciate the significant contributions of different cultures to our continued enjoyment of game playing. The first activity explores various dice, and the second activity looks at their use in ancient games.
Exploring dice
In the first activity, children can explore the histories of dice developed by different cultures. This activity would be conducted in a manner similar to that for the previous open-ended research task.
For many thousands of years, people have used different
kinds of objects for dice in game playing. Objects such as
shells, knuckle bones, beans, and painted sticks were
popular. The Aztec people of Mexico, for example, used
cacao beans painted with white dots, whereas the North
Africans developed dice shaped like pyramids. Cowrie
shells were used in Africa and India, with scoring
determined by how many shells landed with their "mouths up."
The ancient Egyptians used four sticks as dice, with each
stick having one side that was white and flat and one side
that was black and curved.
Following the research activity, I usually present the children with a number of discussion starters to encourage them to reflect on their findings and to pose new questions and problems about the use of dice. Such discussion points might include these:
* When you began your research on the use of dice, which questions did you investigate? Why did you decide to explore these questions? What were some of your findings?
* Did you consider why different cultures chose their particular dice (e.g., Why do you think the Africans and Indians used cowrie shells?)
* Would you change any of the features on these dice? What might you do? Why"
* How were the dice used in games?
* What scoring systems were used? Do you think the scoring systems were fair? How might you find out?
* Would you make any changes to the scoring systems used with the dice? What might you do?
As part of their investigations, at this point I present children with the problem of constructing their own dice by marking dots or numbers, 1 to 6, on the sides of wooden or plastic centimeter cubes, initially without reference to a regular die. As children examine and compare their dice constructions, I might ask them such questions as these:
* Where did you place your numbers or dots on your die? Which number did you put on the face opposite number 1? Number 2? Number 3? Why did you place your numbers in that way?
* Now take a look at a regular die. How is your die the same as, and different from, a regular die?
* Is your die a fair die" How could you find out?
Many interesting relationships appear on a regular die, for example, an even number is always on a face opposite an odd number, and the sum of the numbers on opposite faces is seven. Children can also raise questions about the relationships between numbers on adjoining sides, including the relationship between the number facing upward and the numbers displayed on the sides. To help the children pose some problem questions about the design of dice, I might ask them to consider the following:
* What do you know about the origin of their design? Do you need to investigate further questions?
* Is this particular design important to a regular die?
* What if a group of people wanted to remove or change some of the special features of a regular die? What questions might you ask these people? What problem questions might we ask about the use of their die in our own game playing?
Exploring ancient games
Numerous problem-posing situations can arise from considering how different dice were used in ancient games. The first activify here looks at the scoring system that the ancient Egyptians used with their sticks.
The ancient Egyptians used four dice sticks in their games.
Each stick had one side that was white and flat and one side
that was black and curved (see fig. 4). In playing their
games with the four dice sticks, the Egyptians worked out
their score as follows (Irons and Burnett 1994):
One white side up scored I point.
Two white sides up scored 2 points.
Three white sides up scored 3 points.
Four white sides up scored 4 points.
Four black sides up scored 6 points.
Children can initially make four sticks and experiment with them in playing one of their regular games. To help them generate some problems about the use of the sticks, children can be presented with discussion starters, such as those that follow:
* Can you create some problems about the ancient Egyptians' scoring system for your friends to solve?
* Do you like their scoring system?
* Did any questions come to your mind about their scoring system as you played your game?
* Did you have any difficulties in using their system? What did you do to fix these problems?
* Would you change their scoring system? What questions would you need to ask in deciding how to do so?
In a follow-up activity, the children could create their own game for use with the sticks.
The second activity considers the game called panda, played by the Mbuti people of central Africa. It involves estimating with multiples of four. As in the previous research activities, children can investigate the history of the game and then play it with a partner.
Panda is a game for two players using thirty-two small
counters. It comes from central Africa, where the Mbuti
people used it to outwit visitors who asked for food. The
Mbuti people used to challenge the visitors to a game of
panda instead. The Mbuti people were very skillful players
and rarely lost a game. The steps of the game follow (Irons
and Burnett 1994):
1. The first player scatters the counters on the ground and then scoops up a handful.
2. The second player must quickly estimate how many counters must be returned to the total number of counters on the ground to make a multiple of four. The player must state whether none.. one, two, or three counters must be returned.
3. The first player then returns the chosen number of counters. If the total number of counters on the ground is a multiple of four, the second player is the winner.
After playing the game, the children might talk about the special skills the Mbuti people had developed that enabled them to outwit their visitors. Children could also think about how they might extend or change the game to make it more interesting or challenging.
Conclusion
The activities presented here show some of the many ways in which we can transform simple mathematical activities into problem-posing situations. The inclusion of multicultural activities can help children appreciate the contributions of different cultures, including their own, to the development and continued growth of mathematical knowledge. In presenting these activities, I have attempted to show how generative questioning within a community of inquiry exemplifies the "art of problem posing" (Brown and Walter 1983; Whitin 1993). Such questioning can engender an inquisitive disposition in our students and empower them with the skills and confidence to pursue their own problems and questions.
References
Brown, Stephen I., and Marion I. Walter. The Art of Problem Posing. Hillsdale, N. J.: Lawrence Erlbaum Assoc., 1983.
--, eds. Problem Posing: Reflections and Applications. Hillsdale, N. J.: Lawrence Erlbaum Assoc., 1993.
Charles, Randall I., Robert P. Mason, and Linda Martin. Problem-Solving Experiences in Mathematics: Grade 4. Menlo Park, Calif.: Addison-Wesley Publishing Co., 1985.
English, Lyn D. "Philosophy for Children and Mathematics Education." In Children: Thinking and Philosophy, edited by Daniela Camhy, 437-43. Germany: Academia Verlag, 1994.
--. "Children's Problem Posing and Problem-Solving Preferences." In Research in Early Number Learning, edited by Joanne Mulligan and Michael Mitchelmore, 227-42. Adelaide, Australia: Australian Association of Mathematics Teachers, 1996.
Irons, Calvin J., and James Burnett. Mathematics from Many Cultures. Levels 2 and 3. Hawthorn, Australia: Mimosa Publications, 1994.
Lipman, Matthew. "Thinking Skills Fostered by Philosophy for Children." In Relating Instruction to Research, vol. 1, Thinking and Learning Skills series., edited by Judith Segal, Susan Chipman, and Robert Glaser. Hillsdale, N. J.: Lawrence Erlbaum Assoc., 1985.
Moses, Barbara M., Elizabeth Bjork, and E. Paul Goldenberg. "Beyond Problem Solving: Problem Posing." In Problem Posing: Reflections and Applications, edited by Stephen I. Brown and Marion I. Walter, 178-88. Hillsdale, N. J.: Lawrence Erlbaum Assoc., 1993.
National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.
--. Professional Standards for Teaching Mathematics. Reston, Va.: NCTM, 1991.
--. Assessment Standards for Teaching Mathematics. Reston, Va.: NCTM. 1995.
Sharp, Ann M. "What Is a Community of Inquiry?" Analytic Teaching 8 (April 1987): 13-18.
Shirley, Lawrence. "Using Ethnomathematics to Find Multicultural Mathematical Connections." In Connecting Mathematics across the Curriculum, 1995 Yearbook of the National Council of Teachers of Mathematics, edited by Peggy A. House, 34-43. Reston, Va.: NCTM, 1995.
Silver, Edward A. "On Mathematical Problem Posing." For the Learning of Mathematics 14 (February 1994): 19-28.
Splitter, Laurance. "On Teaching Children to Be Better Thinkers." Unicorn 14 (March 1988): 40-47.
Whitin, David J. "Number Sense and the Importance of Asking `Why?'" In Problem Posing: Reflections and Applications, edited by Stephen I. Brown and Marion I. Walter, 121-29. Hillsdale, N. J.: Lawrence Erlbaum Assoc., 1993.
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| Author: | English, Lyn D. |
|---|---|
| Publication: | Teaching Children Mathematics |
| Date: | Nov 1, 1997 |
| Words: | 4760 |
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