The development of high quality mathematics software should draw upon a coherent theory of mathematical cognition and learning trajectories (Clements & Sarama, 2007). At the same time, software development is not totally a theory-based or rational process. For one thing, existing theory may be limited or even wrong, and hence
should be applied with a grain of salty humility. For another, a good dose of creativity and intuition are also required. Software design is part humble science and part fanciful art.
Most importantly, to support learning, educational technology must be designed around the learner’s needs and abilities. In learner-centred design, scaffolding tools guide children from novices to experts, as opposed to user- centred design, which assumes users already know how to do a task (e.g. write) and just need to learn a new digital tool (e.g. word processing) (Quintana, Shin, Norris, & Soloway, 2006). User-centred design tries to make tools as easy and intuitive as possible. In contrast, learner-centred design assumes that the users have varying levels of expertise and need different scaffolds to learn new information. The goal is not to make the tool easy, but to provide the appropriate level of challenge. Digital tools must be designed to provide scaffolding to guide learners and clear exit strategies to remove supports as learners become more capable.
Mathematics software should attend to both developmental progression and content expectations. MathemAntics consists of seven mathematics content areas that children can learn beginning at preschool and through the third grade:
• Enumeration.
• Equivalence.
• Estimation.
• Addition and subtraction.
• Written calculation and place value.
• Multiplication.
• Negative number.
Each content area is comprised of several activities designed to teach or assess specific concepts and skills. The content roughly reflects the Common Core Standards (National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010), although some MathemAntics activities are more advanced than the Common Core. The MathemAntics age progression roughly reflects the developmental trajectories as
portrayed by cognitive research. These, like the Standards, however, may be too conservative. Our informal observations and intuitions, and also some authorities (Papert, 1980) suggest that when provided with stimulating mathematics activities, children can learn more than the trajectories designate as age appropriate. One reason is that traditional research on trajectories generally does not involve teaching experiments, and instead, in the spirit of Piaget (but not Vygotsky), focuses on development in the absence of education. Of course, we recognize that our conjectures must be subjected to empirical test, and our research undertakes that task.
Next we show how MathemAntics employs each of the key computer affordances described earlier in the chapter to achieve its goals and demonstrate how research plays an integral role in software development.
Goal-driven Activities and Virtual Objects
Software for young children often lacks helpful instructions or goals. Our review of existing mathematics software for young children suggests that the omission of instructions appears to be a growing trend. Some software has been designed so that children are able to immediately immerse themselves in the game, thus perhaps creating an instant flow state, but they are required to figure out the goal(s) of the activity on their own as they play. Although this approach works in some cases, it may jeopardize children’s understanding of fundamental mathematical concepts, especially if goals are not learned or if learning the mathematics content is not directly a part of the activity’s goals. For example, if collecting as many badges or stickers as possible as reward for completion of tasks is the child’s primary goal, how much attention is he paying to the mathematical content of the task?
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Figure 1 Scissors tool for creating doubles.
MathemAntics activities are designed so that each has a clearly defined mathematical goal. Some activity features encourage children to explore mathematical ideas in an unstructured microworld, while other activities promote the learning and practice of specific mathematical skills, for example, helping a farmer to figure out how many animals are in a field. Goals are clearly a part of initial instructions and reinforced during feedback on each activity. To achieve these goals, children are given a variety of key software features (objects, tools, scaffolds, characters, feedback, etc.) throughout the MathemAntics system.
Tools
Digital tools can provide powerful ways to demonstrate deep mathematical constructs and provide useful
representational models (Hoyles & Noss, 2009) or mathematical metaphors (analogous to physical manipulatives) that are not possible in traditional print or tangible materials. A child cannot rotate, shrink, or enlarge a triangle on a piece of paper, but can on a computer. Similarly, a child cannot ‘take away’ a picture of a bear on a piece of paper, but can on a computer. Paper is static; computers are dynamic.
Mathematical tools are a vital component in MathemAntics. Designed to help children solve problems and explore ideas about mathematics, the tools are introduced to children as young as 3 years of age and used through the third grade.
Tools are used in MathemAntics in several different ways. They are used to directly manipulate objects, thereby allowing children to organize, enumerate, compare, and alter the physical arrangement of objects. Explicit strategy tools are specially designed to teach children useful approaches to solving problems, as in the case of addition strategies for unknown number facts. For example, the scissors allows children to snip a block representation into already mastered doubles or tens facts. In the figure, the child snips off one block from the 6, which then turns into 5 + 5 with one left over, which the child easily determines is 11. The tool enables the child to use a known double fact (5 + 5) to solve the problem 5 + 6 (Figure 1).
Representations
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Figure 2 Grouping boxes and number line.
In MathemAntics, tools are often linked to various mathematical representations. For example, imagine that a child used a blue 10-box and a white 3-box to organize 13 animals. After the child indicates her response by clicking on the number line, a thin continuous blue line spans from 0 to 10 on the number line followed by a continuous white line spanning from 10 to 13. On the screen, the child now sees the two grouping boxes (10- and-3-boxes) each filled with pigs, as well as the continuous lines showing 13 segmented by magnitude bars of 10 and 3 (each the same colour as the corresponding boxes). The child may also be shown the written number 13 and the number word, and may hear the spoken word ‘thirteen.’ (Figure 2).
Scaffolds
Scaffolds (or hints) are provided to assist the learning process. In MathemAntics, children are given various scaffolds when requested (by clicking on a help button), when inactive for a period of time, or when solving tasks inaccurately. As an example, one activity in MathemAntics asks children to locate or place numbers on a number line with only the endpoints labelled. The first time a child clicks on the help button or answers incorrectly, the midpoint of the number line appears. With each subsequent request for help or error, the number line fills in more hatch marks – tens, and then units.
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Figure 3 The Ghost Box grouping tool.
Various tools available in MathemAntics are deliberately designed to help scaffold learning. For example, a ghost- box is a grouping tool that has not been completely filled with animals. This occurs if there are fewer animals shown in the field than the size of the box created. So, if there are nine animals shown on a field and the child creates a 10-box, the nine animals fill the 10-box from the bottom-up, leaving the top box empty. This provides an immediate visual scaffold for the child. She knows there cannot be 10 animals as the 10-box has one empty box and may now begin to think of numbers in terms of their relation to the 10s. This is very helpful for dealing with all number fact problems involving 9 and even 8 (Figure 3).
Pedagogical Agents
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Figure 4 Four pedagogical agents.
MathemAntics utilizes a pedagogical agent (a child farmer) to guide children through activities by providing initial instruction and dynamic feedback. Pedagogical agents who guide children through their cognitive processing have been shown to increase learning outcomes (Baylor & Kim, 2005; Baylor & Ryu, 2003; Lester, et al., 1997; Moreno, et al., 2001). Furthermore, characters are used to provide short instructional demonstrations of how to complete an activity, and use various tools and strategies to solve problems. These instructional demos provide interactive examples to guide children in learning mathematical concepts and skills, and encourage strategy awareness and flexibility through a process of cognitive apprenticeship (Collins & Brown, 1991). Strategies used by the characters increase in difficulty according to their age; thus, older characters use more advanced strategies than younger characters. Importantly, these characters highlight that some strategies are more efficient than others in certain mathematical situations. Children are then able to use the tools to solve problems on their own (Figure 4).
Feedback
Our review of existing mathematics software suggests that feedback is often limited to positive sound effects or
‘correct’ for correct responses and negative sound effects or ‘try again’ for incorrect responses. This form of feedback provides no explanation of how or why answers are correct or incorrect. It does not help teach the child concepts or skills she lacks. Most drill-based software activities are specifically designed this way.
MathemAntics offers dynamic feedback. Characters or the pedagogical agent in MathemAntics explain the accuracy of a response through audio and visual explanations. For example, in a number facts activity, characters explicitly demonstrate and explain a useful strategy after incorrect responses. In equivalence activities, the pedagogical agent demonstrates how to use a pair-up tool to compare the number of objects when sets are aligned in a physical 1–1 correspondence. If there is nothing left over, the sets are the same number; any leftovers are highlighted to indicate which set has more than the other.
Interaction, Fantasy, Challenge, and Flow
MathemAntics incorporates fantasy through the creation of a farm environment. The child is asked to help the farmer complete a variety of activities, thereby becoming an active participant on the farm. As previously
described, these activities include such things as helping the farmer feed the animals, counting how many animals are in the field, among many others.
Children are appropriately challenged through the use of automatic difficulty adjustments throughout the system.
Activity goals and available tools are matched to meet each child’s ability based on performance. This allows new tools to be introduced over time, increasing the length of activities and the possibilities for learning.
Flow is encouraged through the use of a variety of activities with which children can engage. Within activities, children are encouraged to complete several trials of the same activity, with objects or images changed each time (i.e. different colour barns, different animals, etc.). When ready to move on, children can choose another
MathemAntics task. Importantly, we utilize formative research with children to help establish the threshold for maintaining attention and engagement in the activities, and create better flow.
Communication and Collaboration
We are in the early stages of incorporating communication and collaboration into MathemAntics. Currently, researchers create mini-lesson videos by recording an audio track as they play with the software. The children hear a child character (researchers’ voice) walking them through how she uses the tools to solve the problem. We plan to work with classroom teachers to create their own video mini-lessons. Eventually, the tools for recording mini-lessons will be built into the software, giving teachers the ability to customize the software for their own classroom. We also envision students being able to record an audio track to accompany a particular problem, describing how they solved it. These solutions could be shared with teachers, parents, or other students.
Record Keeping and Reporting
A unique affordance of computers is the ability to capture large amounts of data. Textbooks and physical
manipulatives may at times be helpful for classroom instruction, but they cannot record information about student performance. A customized logging system allows us to capture a wide range of information about each child’s performance on every MathemAntics activity. This stealth assessment records such information as what level the child is on, whether the child increased or decreased levels during any given session, exact accuracy, near accuracy (e.g. how close the child’s answer was to the correct answer), time elapsed before response, time elapsed between use of a tool and response, and usage of specific tools that promote specific strategies (such as the line-up tool for determining more, less, and equal).
Once data are gathered, the next challenge is to create informative reports for teachers. It is not enough for the software to adjust difficulty within the activities or to describe the child’s frequent strategies: teachers must then be able to utilize information about student performance to inform classroom instruction. Reports can provide
information about specific strategies that the child does not use and that can help her progress. Reports can provide suggestions about use of specific MathemAntics or classroom activities that might help the child develop specific skills or concepts. Furthermore, the reports may serve as a kind of professional development for teachers who could benefit from information about the kinds of strategies children use or the difficulties they face.