PARKER PENGUIN’S MATH ADVENTURE

Chika Ando -

Jiraorn Assarat -

Andrea Kulkarni -

Lee Martin -

Amanda Mathias -

Parker Penguin’s Math Adventures (hereafter the toy) is a product designed to be used by young children (ages 4 and up) both to entertain and to provide a productive context to improve counting and addition and subtraction skills. The expected use is in the home, both with and without adult participation.

Physical specification: The toy is shaped more or less like a laptop computer, but without the sharp corners. When closed, the toy is extremely portable. The outside of our case protects the inner workings and also provides a handy pouch for carrying along an accompanying storybook or two. When opened, it provides a wealth of opportunity for the child to play with numbers.

The top panel consists of an exciting series of slides that lead from a platform above to a number line / counting area below. In the course of using the toy, little plastic penguins hop from the top platform down to the slide, sliding noisily down until they take their place on the penguin number line. Also included is an automatic escalator that takes the penguins from the number line back up to the top, where they will be ready to slide down again.

The bottom panel consists of a number of different areas. Along the top there are numerals from 1 to 20. Each number can light up, in order to show the child which numbers are "active", and each one is also a button that can be used for controlling the toy or providing answers to questions. Below these numbers are three screens. On the left is the animated character Parker, the cheerful penguin character that asks questions and provides feedback to the children about their math activities. To the right of Parker are two panels for displaying numbers and numerical expressions. The panel in the middle shows the terms of addition and subtraction problems. The panel on the right shows the results of these equations. This space is also used for displaying individual numbers during counting activities.

Below these panels are a number of disks on a wire, similar to an abacus. These provide a tangible way for children to manipulate quantities, while preventing the problem of loose pieces which can get lost and can present a choking hazard. We believe this part of the toy would be particularly useful if an adult wanted to show counting or addition and subtraction principles using concrete objects. It is less likely that a young child would spontaneously use this part of the toy to puzzle out numerical questions, although they might enjoy sliding the beads back and forth. Below the abacus is a five position slider. This slider chooses the mode of operation. We will discuss the various modes and their functions below. On the underside of the toy there is a volume control that can be set by an adult using a coin or other flat object. This is to allow parents to set the volume to a safe and comfortable level.

The toy also comes with two storybooks, and more stories may be purchased or downloaded and printed from a web site. The stories provide fun ways to put mathematics in context, and they are meant to be used by the parent and child together while using the toy. The stories feature Parker Penguin and his friends and relate their adventures and a variety of mathematical puzzles that they encounter. A typical question that would be presented in the text would be the following: “Parker is throwing a party for his friend’s birthday, and he wants to be sure he has enough fish for everyone. Yesterday, he invited 3 friends. Today, he invited 4 more. How many friends did he invite altogether?” The books contain suggestions in the margins as to how to use the toy to help explain or figure out the mathematical dilemmas presented in the text.

The toy comes back from the toy store, still in its box, while little Jimmy is sitting watching TV. His mom says "I've got something for you!" and he jumps up at the thought of a new toy. With his mom's help, he opens the box and pulls out the toy. She pops in the batteries and opens it up (laptop-style). It springs to life with a friendly little tune while the numbers on the front panel flash and blink for a moment.

A bit about the people in this scenario: Jimmy is about four years old. His family is interested in helping him learn, so they bought him this learning toy, but they also want learning to be fun. They want to use the toy with him sometimes, but they also want to it to be able to entertain him while on car trips or when he has to play alone. Jimmy can count to 10, and knows which number symbols go with which words, but his concept of ordering is still developing, and he doesn't have a lot of experience with addition and subtraction problems.

Jimmy starts to push the buttons and slide the sliders to explore the toy. He's not paying a lot of attention to what's going on, but his eyes light up as he sees one of the penguins drop out of the penguin-holder and slide down the ramps. He gets quite excited and presses the same buttons again several times to get it to happen again. His mom is happy to see the toy is engaging for him right away, and she reaches into the box and pulls out the storybook and the parents’ guide that come with the toy. Jimmy is still quite absorbed in his play. She has switched the toy to toy-directed learning mode, and the toy begins to ask him addition and subtraction problems. He is intrigued by the voice, but he doesn't entirely understand what the questions mean, so he looks to his mom. She has been glancing through the parents’ guide and has gotten some idea of the capabilities of the toy and the design rationale of the various features.

She switches the toy to child-directed counting mode, and asks, “do you want to read a story together about the penguins?"

Jimmy nods in agreement, so his mom sits down next to him and they put the book out in front of them, next to the toy. The story begins by describing the main characters and setting up the simple storyline they are involved in. Before long, a situation arises where the penguins need to be able to do addition in order to solve a problem: "Percy invited three friends to the clambake, and Pamela invited four. Help them plan for their party. How many friends have they invited altogether?" Notes in the margin of the book suggest using the toy to help solve this problem. His mom asks, "How many penguins do we start with?" Jimmy answers, "three," and with the nod from his mother, presses the three button, sending a cascade of three penguins whizzing down the slides while making happy noises.

“And how many do we add?” asks his mom. Jimmy doesn't remember, so his mom helps by whispering, “four.”

He reaches over and presses four, sending four more penguins down the slide. "So how many do we have altogether," asks his mom. Jimmy counts the penguins, answering "7". "That's right!", exclaims his mom, “ three plus four equals 7." She suggests he presses the “Count” button to check his answer.

When he presses the "Count" button, the toy says "1, 2, 3, 4, 5, 6, 7, there are 7 Penguins," in a cheerful voice. As it counts the numbers above each penguin lights up, so at the end the numbers 1 through 7 are all lit.

“Let's get back to the story,” his mom says, and they go back to reading.

***

Later, in the car, Jimmy is playing with the toy by himself. At this point it is set to "toy-directed counting mode." In this mode, the toy asks questions and provides feedback to the child in a way meant to mimic (to the extent possible) how a parent scaffolds the child's activity. These include "guess my number games," questions about what is one more than or one less than a given number, clap along games, and “greater than less than” activities. The activities are motivated by the animated character Parker, who asks questions and provides hints and feedback. The feedback system is built around an intelligent algorithm that allows the questions to get easier or harder depending on how well Jimmy has been doing. This way, Jimmy usually sees problems that he can get correct, so that he does not get frustrated, but there are enough challenging questions interspersed so that he does not get bored. This feature is designed to keep the toy attending to the child’s zone of proximal development.

  1. Parker says, "let's play a counting game."
  2. Twelve penguins, slide down, one by one, and line up along the bottom.
  3. Parker says, "Can you count the penguins? How many are there?"
  4. Jimmy starts counting the penguins, one by one, pointing at each one with his finger as he says the number.
  5. He accidentally skips one of the penguins when he is counting, so he presses the number 11 on the toy.
  6. Parker responds, "Close, but not quite right. Try counting again, or tap on my picture if you want help."
  7. Although he does not always do so, Jimmy decides to take Parker up on his offer, and he taps on Parker's image.
  8. Parker says, "count along with me as I count the penguins, one by one." He begins to count, "one, two," and as he does so, the numbers above the penguins light up, one at a time, in time with Parker's counting.
  9. Jimmy follows along, pointing at each penguin as its number is called.
  10. When Parker finishes counting, he says, "There are 12 penguins altogether. Let's try another counting problem." He then starts to ask a new problem. This time, Jimmy gets it right the first time.

Various modes explained:

There are four play modes, plus off, resulting in a five-position slider for mode. The four play modes result from a two by two distinction. In one dimension, there is distinction between counting mode, and learning (add-subtract) mode. In the other dimension, the distinction is between toy-directed play, and child-directed play. These distinctions can be summarized as follows:

Counting / Add-subtract
Child-directed / Child-count / Child-learn
Toy-directed / Toy-count / Toy-learn

On the slider in the toy, these modes are depicted graphically, so that the child does not need to read to be able to use the slider.

In the toy-directed modes, if the child does not know how to complete a problem, he or she can turn to the animated helper, Parker. The child presses the help button on Parker's screen, and Parker provides hints appropriate to the task. For example, if the child is having difficulty counting a number and asks Parker for help, Parker would ask the child to count along with him, lighting up each penguin as it is

counted. This is meant to be similar to the way a parent would help the child to count to set of objects.

Toy-directed count: In this mode, play is centered on counting activities. The toy prompts the child, asks questions, provides support when a problem is difficult, and attempts, to the degree possible, to scaffolds the child's learning. The activities in counting mode include:

  • guess my number games
  • what number comes next or before
  • how many are there questions
  • greater than less than questions

In this and in all modes, the difficulty of the questions would be modulated by the intelligent difficulty-monitoring system described below.

Child-directed count: In this mode, the toy behaves in a way consistent with counting activities, but it does not actively prompt the child. This mode is most appropriate when counting questions are being created externally. It may be that the child is playing independently and inventing his or her own questions to investigate, or it may be that an adult is working with the child and suggesting activities, possibly within the context of one of the storybooks. This mode would also be appropriate if the child just wanted to play with the more physical aspects of the toy. We believe that even during such play, the mathematical content and representations shown by the toy would still help the child to gain an understanding of numeracy. When one of the numbers is pressed in this mode, the corresponding number of penguins slides down and lines up, ready to be counted. A number line corresponding to the slots that the penguins occupy lights up simultaneously. If the “count” button on the toy is pressed, Parker counts out the penguins for the child. When a button is pressed again, any penguins that are currently in the number line are sent back to the top, and a number of penguins corresponding to the number just pressed slide down the slide and line up in the number line.

Toy-directed learn: In this mode, the toy provides questions, activities, feedback, etc., to the child for activities centered on addition and subtraction. Other than the changing content, this mode is fairly similar to the toy-count mode. Activities in this mode include addition and subtraction problems in multiple wordings. Some of these would be explicit, asking for example "what is 3 plus 4," while others would be less direct,

asking for example, "If I had 3 apples, and my friend gave me 4 more, how many would I have altogether?"

Child-directed learn: In this mode, the toy behaves in a manner consistent with addition and subtraction problems, but does not actively prompt the child. As with the child-directed counting mode, this mode can be used for independent play when the child does not want to be prompted, times when the child is playing along with adults, and, in particular, times when the child is working with adults and reading an associated story. In this mode, the addition and subtraction buttons on the toy become active. If the child presses the three button, three penguins come tumbling down. If they then press the plus button, and then the two button, two more penguins will come down, for a total of five. The toy will inform the child if he or she has tried to do something that the toy is not capable of doing, such as adding 20 and 19 (there are not enough penguins included to perform this sum), or subtracting 12 from 5.

Intelligent difficulty-level modulating software: an important feature of the toy is the software that chooses the difficulty of the problems presented to the child (in toy-directed mode). The software monitors the child's success rate in answering problems, including which problems the child requires hints for. It uses this information to assess the child's location both with regard to specific skills and in terms of a more general developmental framework. The goal is to keep the success rate high enough so that the child does not become unduly frustrated, while presenting enough challenging problems so that the child has an opportunity to learn and does not become bored. In other words, the goal is to present activities that are in the child's zone of proximal development.

Because the software is based on developmental theory, the toy should provide developmentally appropriate questions for the child. It is worth noting that a research-version of this toy could be created that would keep track of the child's performance and development over time. The information stored by the toy could then be used by researchers of child development to assess the validity of their models.

Research Findings on Young Children’s Number and Arithmetic Concepts

In 1965, Jean Piaget published his influential work on children’s mathematical competence. Piaget found that young children between the ages of three and five develop their mathematical concepts from everyday experiences while interacting with their physical and social worlds. Children, he observed, possess an innate curiosity about naturally occurring patterns and a strong desire to resolve problems, both essential qualities of mathematical thinking. As children explore their surroundings they detect counting patterns, devise counting rules, and sometimes apply these rules too broadly (Baroody, 2000). Piaget named these processes of childhood mathematical apprehension, interaction, and application, “constructivism.”

The most important findings of Piaget and his colleagues at the International Center of Genetic Epistemology were the following (cf. Kamii & DeClark,1985):