Whole math through investigations.

Ken Goodman (1986) says it is easy to learn language when: "It's real and natural. It's whole. It's sensible. It's interesting. It's relevant. It belongs to the learner.... The learner chooses to use it. It's accessible to the learner. The learner has power to use it". This observation is not just applicable to language and literacy. In this article, the authors describe some of the ways children in a K-1-2 combination class learn math by choosing to do it in real, sensible and interesting ways. For example, when a child decides to estimate how much her classmates weigh, then checks and records their actual weights, she has chosen a real measurement activity that interests her and that belongs to her.

There are strong parallels between the whole in whole language and the whole in a child-centered approach to learning mathematics. For several years, educators in New Zealand and England have been exploring more "natural learning processes" for teaching mathematics. Central to the natural learning processes, as described by Stoessiger and Edmunds (1992), are open-ended problem-solving challenges similar to the investigations described in this article.

Many adults approach mathematics with a feeling of panic and fear (Buxton, 1991). Children, on the other hand, begin school confident in their abilities and interested in learning about numbers. By later elementary school, however, math is not a favored subject (Boling, 1991). The way math is taught in the early years of school affects not only math achievement and skill development, but also a child's disposition to learn (National Association for the Education of Young Children, 1988). Continued exploration of mathematics requires a feeling of competence and a favorable disposition toward problem-solving.

Many theorists insist that the only meaningful and genuine learning is that which is constructed by the learner from within (Baroody, 1987; Elkind, 1989). Kamii (1985) states that in order to understand and enjoy mathematics, children must literally reinvent it through their own daily explorations and with number games. Traditional math instruction with drills, flash cards and work sheets may, in fact, lead to math anxiety. The following account describes a class where, instead of disliking math, children develop a disposition to enjoy problem-solving and mathematical activities.

THE TEACHER AND THE CLASSROOM

Louise Burrell has been teaching for more than 20 years in a small rural school in the mountains of western North Carolina. She has developed a unique classroom for kindergarten, 1st- and 2nd-grade children. Instead of desks in rows, tables and centers are full of learning materials. Children learn to read, write and use mathematics through rich experiences that engage them in problem-solving and real life applications.

Investigations

Investigations are the official math activities in Burrell's class. Every day each child is expected to design and carry out some form of investigation that varies week by week. The areas for investigation are based on North Carolina Standard Course of Study (North Carolina Department of Public Instruction, 1989) as well as National Council of Teachers of Mathematics standards (1989), including geometry, numeration, estimation, measurement, patterns, classification, sequencing and money. The class focuses on a different area of concentration each week. Many strands of mathematics, however, are evident throughout the day. As the children grasp the concepts, the topic is expanded. Occasionally, children are expected to work on areas where they feel they need improvement. In order to choose the areas that need work, children must practice self-evaluation.

During large group meetings, Burrell introduces a specific topic and gives sample investigations that can be done in that area. Investigations combine math, science, problem-solving, reading and language arts. As one child in the class explained, "Investigations would be like what you call a kinda mix of math and science together. Like graphing and shapes and experiments and stuff." Investigations also always include recording of results and thus are a part of the children's writing experiences.

Just as math is a set of processes that apply to most areas of life, investigations are not confined to one area of the classroom and are not limited to computation. One student, Ronnie, noted that "Most of our centers have to do with math, like creative play and wood-working."

Planning Investigations

Burrell finds it important to keep directions general rather than specific. When assignments are spelled out in detail, children follow instructions carefully and all produce similar products. When the instructions are general and creativity and variety are emphasized, children produce more detailed, creative work. Ownership of the projects means the children work harder to please themselves as well as the teacher.

As a group, children decide on the minimum amount of work to be done, although many children do more than the minimum. Group planning of the work thus involves real life math as the children estimate the time they have and the amount of work they can do in that time. They discuss possibilities about how to explore the topic. Children are challenged to find different ways to conduct their investigations. Children often come up with unusual methods that the teacher had not anticipated. Creativity in planning is encouraged. At least twice a week, after the work period, children share their investigations with the rest of the class. Group sharing of work provides ideas for other investigations. Throughout the year, the children measure, add, subtract, estimate, conduct surveys, graph, design and conduct experiments.

This method of planning investigations has powerful implications beyond the mathematics that children learn. Burrell's aim is to empower children to build critical thinking skills by making choices. Through this empowerment, children develop a positive attitude toward work--an "I can do it!" attitude. Children learn to voice their opinions and try new ideas. The emphasis in this approach is not on right or wrong answers, but rather on thinking (National Council of Teachers of Mathematics, 1989).

Risk-taking

This approach is exciting for some students and scary for others. For some children, investigations are their favorite work. Sam likes investigations because "You get to do activities in it!" For others, investigations are the hardest assignment. Sarah says "It's harder than anything!" Thinking by yourself is not always easy. Some children thrive on choice, but for others the challenge of making up what they are supposed to do is difficult. The teacher as facilitator must teach the unsure child how to take risks.

AREAS FOR INVESTIGATION

There are many areas for investigation that will challenge children. In the following section some investigation activities from Burrell's classroom are described. Remember that these children are 5, 6 and 7 years old. Their math concepts continue to develop as they investigate problems.

Measurement

Many investigation assignments involve measurement. During group brainstorming sessions children are reminded that volume can be measured using sand, water or blocks. The following example illustrates how measurement happens in these investigations:

Zachary, a 1st-grader, and Jane Perlmutter, one of the authors, spent approximately 45 minutes at the water table filling and emptying various size containers. The containers were marked with standard measures, but Zachary was not concerned with the accuracy of measurements. Spillage occurred frequently and did not bother him. The pair checked how many pints they could pour into a quart and how many quarts would fit into a gallon. They also compared quarts and liters. After drying off, Zachary announced that he needed to do his investigation. Perlmutter thought that was what they had been doing. But Zachary let her know that investigations must be written down. He moved to the sand table, poured and then wrote the following: "I WS IN SAD I FOD OUT TT + 2 AND 4 OF A COP FOLS UP T HESTR" (I was in sand and I found out that 2 and a fourth of a cup fills up the sifter). Investigations are both process and product.

Vicki, another 1st-grader, planned to do her work at the water table. Burrell was helping her plan and asked her what she wanted to do. Vicki decided to find out how many cups of water were in the water table. She wrote in her plan book, "I will see how many cups of water are in the tub."

When Vicki went to the water table, she found Sandy and Tim already there. The limit at the water table is two children. Burrell helped Vicki, Sandy and Tim decide that Vicki could have special permission to stay. Then Vicki explained her plan to Sandy. "Is that all right?" Vicki asked. Sandy agreed. The children used a siphon apparatus to squeeze the water out of the tub. Each squeeze constituted a unit of one. Vicki reported back to the teacher that they had counted 207 cups.

Water is good for measuring and also for learning about how items sink or float. Several times Jenny estimated how many things would float or sink and then compared her estimates to what actually happened. After her work was done, she recorded her results. She told the observer that this was the hardest part of her work.

Anywhere or anything in the classroom is fair game for linear measurement. The computer, tables and books could be measured with a variety of nonstandard measures. Ashley explained, "What you do is like you measure a book or you can measure a book shelf, you measure a person." When asked what she uses to measure these items, she said, "Anything, a paper clip or eraser, a pencil, a ruler or something like that." Vicki said she mostly uses a pencil to do nonstandard measurement. Lisa measured the Macintosh with a paper clip. "It was 7."

Classmates were interesting to measure. Mollie measured people with pencils and found that Laura at 8 pencils was tallest and Ashley at 6 pencils was shortest. When a bathroom scale was available for estimating and checking weight, most children were willing to serve as subjects. Many of the adults in the classroom were not so helpful! One child told the assistant that he was trying to find out how wide she was and how thick.

One particular challenge was to develop ways to measure time. One child did jumping jacks while another child tied his shoe. One hundred and forty jumping jacks were done in the time it took for a beginner to tie his shoes. Some children constructed pendulums to measure how many swings it took for another child to read a short book. Pendulums can be made using pencils, scissors, crayons and a block of wood as weights. Pendulums of various kinds were hanging all over the room. Choices of how to measure time varied, as did the event being measured. Any activity--reading a book, singing a song, drawing a picture--can be measured.

Development of Operations

Investigations vary. Sometimes the children used dice to make addition and subtraction problems. After rolling two or more dice, the children added the resultant numbers. Depending on the number of dice used, problems can be solved with one, two or more digits. Fingers and other manipulatives can be used as necessary. Here again, the teacher can pose problems that the children do not already know how to solve. Doing two-digit subtraction requires figuring out what to do when the bottom number is larger than the top number (i.e., 16 - 32). Noting "It can't be done" is a first step toward understanding the need for borrowing.

Flash cards in the math centers are also used for practice in adding and subtraction. Computer programs are available to help children with drill and practice. One possible investigation assignment is to find a way to practice number facts. Children choose how to practice learning their combinations.

Experiments

A kindergarten student described one of his experiments: "I did it on magnets. I saw what magnets stick to and I put a magnet right here and see if it can move like that. And see if the magnets can do that and if they don't, I just write it down. That's a experiment. I learned about a magnet doesn't stick onto a car. It was a wooden car." Experiments overlap with science and general curiosity. Pendulums and measuring time are forms of experiments. Counting how many times the gerbils come up and down the ladder is measurement, science and a math experiment.

Geometry, Shapes and Patterns

For beginning geometry lessons, the teacher collected real objects for the class to describe in terms of shape. Children were challenged to find other objects in the classroom with matching shapes. They used charts to record what they found. Other charts involve comparisons of different shapes: "How do rectangles differ from triangles and how are they alike?" Questions such as "Why do we need these kinds of shapes? How are triangles and circles important as parts of basic machines?" stimulate thought and encourage application of math to other areas of the curriculum. Children can create projects to share with the class, including simple machines, drawings that show the use of shapes in art, and sculptures of a town, city or person.

Surveys

Another frequent and popular investigation topic is surveys. Implementation of process varies considerably by age level. Older students can create a graph beginning with identifying a question. The question is usually recorded at the top of a page and then the surveyor polls classmates about their preferences. Responses can be counted using tally marks. After everyone available has been polled, the surveyor totals the responses and shares the results.

For example, Lisa conducted a survey and then made a graph to show her classmates' preferences for big or little fossils. She drew lengths of fossils she had at home and asked people which size they liked. Sandy did a survey asking what learning center people liked best. On another day, Jenny wanted to find out which was the most popular letter.

These surveys have the potential to cause emotional concerns. Some of the "Which do you like best?" surveys may result in hurt feelings. Maryann decided to do a graph of boys' names. She wrote, "Do you like Dave, Jack, Buddy Name." She tried to explain that she was asking which name rather than which boy. Another child used teachers' names and the older children expressed concern. On another day, Jenny asked students if they liked girls or boys best. Frequently during this activity, she kept the observer posted as to who was "winning."

Many different materials can be used to develop a graph. String, leaves, beans or bottle caps can be used to record the answers. Every bean might stand for two people. Language skills are practiced as children name the graph, set up the problem and write a summary of their findings. After the question is asked, children try to predict the results. "Who do you think will be the tallest person?" After the data are collected, the student checks his/her prediction against the end result.

Other Mathematical Activities in the Class

As children decide issues by voting, put numbers on the calendar and determine when to present projects, they are using mathematical concepts. The calendar math in Burrell's room follows Mathematics Their Way (Baratta-Lorton, 1976) and employs a daily tally, straws bundled to show how many days they have been in school, counting by twos, using the words "today, tomorrow and yesterday" and writing the date as 4/11. Many mathematical concepts are reviewed each morning during calendar time. During group time, quick game-like drills on number combinations can be used to assess progress and provide incentive for computation work in investigations. Math baseball is a group game that helps children gain skill in computation.

Math and Play

Children learn about numbers and quantity while they play. Most of the children are unaware they are doing math when they play in the store, but a few of the children can actually tell the kinds of things they learn through play. "In creative play we learn what's a quarter, what's a nickel, what's a dime," Beth said. "We learn how much sand there is and we just measure stuff. We measure and we count. We sometimes find out how many animals there are in blocks."

At each center, a designated number of children can play at any one time. This provides concrete practice in simple addition, especially for kindergarten and 1st-grade children. When Pam saw three children in creative play, she had to ask for special permission to allow four so that she could join the group. She was adding for a real purpose.

When children are in creative play, they often buy and sell. "How much for all this?" asked one child. "That's $10.99," responded another. Laura came to the cash register and said, with authority, "Payday! My boss told me I could have ten dollars." She then pulled out several bills and put them in her bag. "I want to buy some Tylenol, pretend the Tylenol was 10 cents," she said to Ronnie. Ronnie looked hard at the calculator, and punched buttons. He seemed to be trying to find the right number. "One dollar," he said. She argued that the price should be 10 cents. He told her to keep the change.

Accuracy here did not seem as important as purchasing items and exchanging money. Laura knew you get paid for work and that things have prices that must be agreed upon. Ronnie knew change was often involved in money transactions. The children were acting out some of the functions money serves in our society.

CONCLUSION

Children who design their own investigations for the day have ownership of what they are learning. When children decide what to measure, survey and explore, they are pursuing activities that are interesting and relevant to them. A math activity that children help design is math that belongs to them. Math that is not confined to a math workbook or squeezed into 40 minutes of teacher-directed math class is whole math. The children in Burrell's class are excited about mathematics and do not suffer from "math panic." These children not only learn the math covered in the North Carolina curriculum guide, but also feel comfortable using mathematics as a tool to help them explore the world. Teaching whole math through purposeful activities, rather than isolated skill and drill practices, helps make math meaningful. Whole math is math that belongs to children.

References

Baratta-Lorton, M. (1976). Mathematics their way. Menlo Park, CA: Addison Wesley.

Baroody, A. J. (1987). Children's mathematical thinking. New York: Teachers College Press.

Boling, A. N. (March 1991). They don't like math? Well let's do something! Arithmetic Teacher, 38, 17-19.

Buxton, L. (1991). Math panic. Portsmouth, NH: Heinemann.

Elkind, D. (1989). Developmentally appropriate practice: Philosophical and practical implications. Phi Delta Kappan, 70(2), 113-117.

Goodman, K. (1986). What's whole in whole language? Portsmouth, NH: Heinemann.

Kamii, C. K. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York: Teachers College Press.

National Association for the Education of Young Children. (1988). Position statement on developmentally appropriate practice in the primary grades serving 5- through 8-year-olds. Young Children, 43(2), 64-84.

National Council of Teachers of Mathematics, Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

North Carolina Department of Public Instruction. (1989). Mathematics K-8 standard course of study. Raleigh, NC: Author.

Stoessiger, R., & Edmunds, J. (1992). Natural learning and mathematics. Portsmouth, NH: Heinemann.

Jane C. Perlmutter is Assistant Professor, Department of Elementary Education and Reading, and Lisa Bloom is Assistant Professor, Special Education, Western Carolina University, Cullowhee, North Carolina. Louise Burrell is a primary grades Teacher, Camp Laboratory Elementary School, Cullowhee, North Carolina.