|Observing Children’s Mathematical Problem Solving with 21st C Technology|
|Written by Prof. Thomas O'Brien on Friday, 10 March 2006|
Inference — the deriving of information with logical certainty — is at the heart of mathematical thinking. Indeed, it can be argued that inference is at the heart of thinking in general (Piaget, 1987; Bruner, 1968). The new technology, Personal Data Assistants (PDAs), provides anywhere, any time access to the Internet and provides a portable tool for developing and enhancing learning, including the development of inference making capabilities in environments that, for reasons of cost or space, are not suitable for “traditional” PCs.
The following discussion of activities designed to provoke inference making is based on software developed for the Palm PDA. Research on children’s strategies for solving the inference problems posed by the software suggests that the software is effective for a wide variety of children, even for children diagnosed with Asperger’s syndrome who have had no academic success.
Higher-order thinking skills, especially in math, need to be emphasized not only in classrooms but at home and in the wider community. The National Association of State Boards of Education in the United States, for example, says that learning must no longer be limited to formal classroom settings and must challenge students young and old (NASBE, 2001).
The increasingly complex, dynamic, and powerful systems of information required by a knowledge-based economy call for a work force that can interpret complex systems involving important mathematical processes — constructing, explaining, justifying, predicting, conjecturing and representing, quantifying, coordinating, and organising data — that are under-emphasized in numerous mathematics curricula (English & Watters, 2004).
Educators and policy makers are recognising that children, as well as the adults in the community who interact with them, must have access to learning devices that will equip them for the challenges of the 21st century (Passey et al. 1999; Abfalter, Mirski & Hitz n.d.). New technologies such as the Palm PDA are cost effective, portable, high capacity devices that provide for anywhere, any time learning in school, home and the broader community. They are capable of providing the kinds of mathematical activities that textbooks are unable to stimulate and their portability ensures that learning can be a full time engagement.
One of the basics in thinking is inference — the deriving of new information based on old information. The elementary school years are the beginning of the time for the construction of inference abilities. It is our sense that inference is an oft-neglected topic in elementary school mathematics. In order to provide challenging inference activities a series of software applications (Treasure Hunt, Find It and Mystery Three) for the Palm PDA was developed. In ongoing research that is still underway, the activities were introduced to children at ages 8-12 in several schools in the Midwestern United States. The sections that follow (1) describe the activities and (2) present data on children’s responses..
2. TREASURE HUNT: EMERALDS WITH SIXTH GRADERS
At odd intervals, once a week or so over a period of several months, the inference activities were first introduced to two classes of sixth graders (age 11-12) on a chalkboard because the software was being developed and later, when development was completed, with a PDA and a Margi projection device.
The students were told, “An Emerald (E) is hidden at random in a 4 by 4 grid and your job is to find the Emerald. Ask about the boxes. For each choice you make, you will be given the distance from the box you choose to the box where the Emerald is hiding. You will not be given diagonal distances, only left-right and up-down. For example, box A-1 is 2 from the Emerald and C-3 is 3 from the Emerald.” (See Figure 1.)
Here is a sample of the three or four games that were played. As children asked their questions, for example, “How far is the Emerald from A-1?” the answers were marked in a grid on the chalkboard. (See Figure 2.)
When children were asked, “Can you tell where the Emerald is?” (A-3) they were able to solve problems after a half hour’s experience. Having mastered this activity, children were anxious to go on to two Emeralds.
After two or three fifty-minute sessions, the Palm version of the Treasure Hunt activities was used for one
more session with the distance game. The children did not use the device; the teacher used it to generate
data for the hand-drawn grids on the board and to check children’s responses when they said, for example, “The Emeralds are in boxes A-4 and C-2.” After a week without playing any Treasure Hunt game, an investigation was begun to look at children’s work in a much more detailed way.
Pairs of children were interviewed as they played the distance game using the Palm. Each session began with a warm-up period. Children had not handled the Palm device before and they were given as much time as needed to become efficient in its use. This took about two minutes.
Children were asked to play the Two Emeralds game until they came to a point where they felt they could “play as well as possible.” This took from five to fifteen minutes.
When they announced that they were ready, children worked together in pairs to find two Emeralds. Records of individual keystrokes were made and their work was timed. Nine pairs of children were interviewed. Each pair played three Emerald games.
As in the classroom work, children played thoughtfully, cooperatively and efficiently.
Over all 27 games, the average time to successful completion was 2 minutes and 18 seconds. The average number of clues was 5.03.
3.2 Paper-and Pencil Test
A week after the interviews took place, the three grids shown below were given on paper to individual children in the second class. Data here would provide information in the generalisability of the interview findings. (See Figure 4.)
1. One Emerald
2. Two Emeralds
3. Three Emeralds
Children’s work was not individually timed, but the entire test session took less that 15 minutes.
All nineteen children located the Emerald correctly in Question 1. Eighteen located both Emeralds correctly in Question 2, and one child located one of the Emeralds. For Question 3, 16 of 19 children located both Emeralds with the remaining three children locating one Emerald correctly. That is, 100%, 97% and 92% of the Emeralds were correctly located in Questions 1, 2, and 3 respectively.
3.3 Assuming Leadership in a Problem-solving Situation:
An Asperger’s Syndrome Child’s Responses Another activity presented to a class of children that included a child with Asperger’s Syndrome was an inference game we called “The Bunny Game.”
Teacher: “Imagine that a Bunny is hiding in one of the boxes of a 4 by 4 grid. You can ask about various
boxes. If the box you ask about is touching the Bunny sidewise (i.e., on any of its four sides) you’ll get the
answer ‘Hot.’ If the box is touching the Bunny at a corner, the answer is ‘Warm.’ Otherwise, the answer is ‘Cold.’ ” (See Figure 5.)
Teacher: “Can you show me another box that’s Hot?” The children said A-2.
Teacher: “And what about C-2?” The children said “Cold.” “And D-4?” The children said “Cold.”
Children: “What about the box the Bunny is in? Hot or Cold?”
Teacher: “The Bunny’s box does not touch itself sidewise or cornerwise so the answer is ‘Cold.’”
As before, we played the game with one Bunny hidden. This was easy and children asked for two-Bunny
games. The two-Bunny activities followed the same rules except that if a box was hot to one Bunny and
warm to another, the feedback was “Hot” (because hot over rules warm). Similarly, hot over rules cold and warm over rules cold.
As children played the game and moved toward efficiency, we challenged them to explain why this or that
question was a good question. We commonly asked children to take a minute or two to decide upon the
best possible question. And occasionally we would halt questions and ask children to put an X in all the
boxes that had been eliminated by the available data.
Not far along in the session Ivan, a child diagnosed with Asperger’s Syndrome and who, according to the
teacher had had virtually no academic or social success in his six years of schooling, showed complete
understanding. At a certain point, he decided he would take charge of the two-Bunny game, reached for the chalk and introduced a variant. “Let’s play the same game,” he said. “I will give you a full grid but I will make one mistake.” Can you find the mistake Ivan introduced? (See Figure 6.)
Next Ivan said, “Let's play with three Bunnies, “ and he led the class in several games where the data were (a) given out box by box and then (b) given out all at once.
Figure 7 shows a grid of the second type. Can you find the three Bunnies?
The members of the class solved both of Ivan’s problems. Can you?
Full reports of the sessions involving the activities and the use of the Palm are reported in O’Brien and
Barnett, 2003; 2004. In general, in all classes all children were engaged by the activities. The high-
achievement children were enthusiastic but equally so — often even more so — were many low-achieving
children, many of whom who were often unsuccessful and inattentive in traditional classroom activities.
Children almost always worked as mixed-ability friendship groups. When someone asked for a clarification
of rules or suggested another method of solution or challenged someone else’s solution (or support for a
solution), the interaction was attentive and respectful.
Often it was the “low-achieving” children who eagerly volunteered a solution or an explanation. Almost
always the solution was correct.
Striking at this age, children were almost always willing to relinquish their deeply-held explanations in
response to another child’s convincing correct or more efficient explanation. Many children, including
some who were almost pathologically shy or reluctant, including Ivan, the child with Asperger’s Syndrome
who had had virtually no academic success or social interaction in five years of schooling, acted as the
class’s teacher in solving and inventing problems, in finding more efficient solutions and in helping other
children. Laura, a shy pupil whose voice never rose beyond a whisper, commonly took over the class in a
quiet and confident way with elegant insights and parsimonious explanations.
Some children reported bringing activities home to their parents to solve, thereby bridging the home/school gap. Most children owned their solutions and knew that they owned their solutions despite (or because?) virtually no teaching took place by the adults.
Children tended to be economical. For example, they tended to ask for clarification at just the right time
and were adept at asking the best next question and in distinguishing between the best question, a weak
question and a wasted question.
They invented tactics that went to the heart of the issue. For example, they came to an early discovery that in the Bunny game Warm was very helpful and that if Warm occurred in one of the four corners of the grid one Bunny was determined with no need for further data. Such a success is of interest in light of research showing that the hallmark of good mathematical thinking is the ability to get to the heart of an issue and not be sidetracked by details.
It was striking that children were capable of such advanced thinking involving logical necessity, one of the
basics of mathematical thinking, all the more so because some of the children still had difficulty with the
traditional arithmetic curriculum — for example, remembering their multiplication tables and carrying out
rote procedures for changing a mixed number (e.g., 1 5/8) into an improper fraction.
One of the classroom teachers summarized the situation by saying, “Why did they succeed? Because they
were thinking, not spouting memorized facts. Because there was no pressure … no boring worksheets
asking kids to do the same thing over and over. Because there were no wrong answers; that is, a child could say anything sensible and the others would react to it and comment on it and the child would grow stronger in his or her conclusion or alter it in response to what other kids said. Because they were thinking and the judgment of their thinking was whether it made sense to them and to others rather than a teacher’s judgment and red marks on a test paper.” (O’Brien and Barnett, 2003)
The results give support to the notion that learning takes place as provoked adaptation. No teaching took
place except for imparting the rules of the games. The heart of the issue was challenge. Faced with a
problem to solve, children constructed their own complex fabric of ideas. That a wide range of children
were successful—not just the high achievers—is noteworthy. The fact that an Asperger’s child (Ivan)
participated not merely successfully but inventively shows the power of problem solving for children who
differ from “neuro-typicals.” The fact that the activities were brought home by children and engaged them
and their parents shows that “Schools without Walls” can be achieved through the combination of engaging problems and portable devices that are economical and easy to use.
Abfalter D, Mirski PJ & Hitz M n.d. Mobile learning – knowledge enhancement and vocational training on
the move. www.ofenhandwork.com/ohlc/pdf_files/G-3_abfalter.pdf.
Bruner J 1968 Toward a Theory of Instruction. Norton, New York.
English Lyn D. & Watters James J 2004 Mathematical modelling with young children. Proceedings of
Psychology of Mathematics Education 28.
National Association of State Boards of Education 2001 Any Time, Any Place, Any Path, Any Place:
Taking the Lead on E-learning Policy.
O’Brien, TC 2003 Treasure Hunt, Find It and Mystery Three: software for the Palm OS.
O’Brien TC & Barnett J 2003 Fasten your seat belts. Phi Delta Kappan November. Reprinted in
Mathematics Teaching December 2003.
O’Brien TC & and Barnett J 2004 Hold on to your hat. Mathematics Teaching May 2004.
Passey D et al. 1999 Anytime anywhere learning pilot programme: A Microsoft UK supported programme
in 28 pilot schools.
Piaget J 1987 Possibility and Necessity: The Role of Necessity in Cognitive Development. University of
Minnesota Press, Minnesota.
About Professor Tom O'Brien
Thomas C. O'Brien, one of the first to be named a North Atlantic Treaty Organization Senior Research Fellow in Science in 1978, is professor emeritus at Southern Illinois University, Edwardsville, Illinois. His work in education is three-fold: teacher education, curriculum development, and research on children's thinking. As a curriculum developer, he has authored more than fifty books for children. He has also authored and edited some eighty papers on children's thinking and education published through the Teachers' Center Project at SIUE, wherehe was the director for more than 20 years.
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|Last Updated ( Sunday, 19 March 2006 )|
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