Benjamin Friedman and Herbert Ginsburg explain how technology can bridge the gap between everyday mathematics and formal teaching
The current state of early mathematics education is troubling. Many educators believe that mathematics education is “developmentally inappropriate” for young children, and many teachers devote little effort to teaching it. As a result, many children are poorly prepared for learning mathematics in elementary school. The situation is particularly dire for children from low-income backgrounds who enter pre-school and kindergarten with lower average levels of academic achievement than their middle- and upper-income peers. Unfortunately, the gap persists and even widens over time.
|What we know|
|● Well-designed educational software can help bridge children’s everyday mathematics and formal, school-based learning.
● Computers can be used to offer powerful learning opportunities to children in the early grades.
● Data gathered in a variety of ways (including from computers) can inform design changes that make activities more effective and provide insight into children’s learning.
Fortunately, research shows that children have the cognitive competence to learn mathematics, if only we were to teach it effectively (see “Further reading”). Even infants show evidence of developing an “everyday mathematics” of surprising complexity. Early childhood mathematics education (ECME) should help children to bridge the gap between their own everyday mathematics, and the formal subject taught in schools. To do this, ECME can and should employ many methods, including focused curricula, integrative projects, games, stories, and educational software, the topic of this paper.
The promise and the perils of computer-based software
Recently, there has been an explosion in the number of mathematics software “apps” available for mobile devices and personal computers. Unfortunately, too many popular and glitzy apps are limited to drill-in facts and ignore conceptual knowledge. This restricted approach is unfortunate because soundly designed software can provide children with powerful learning opportunities that conventional teaching cannot offer. Of course, the reverse is also true: teachers can do things that computers cannot. Therefore, we should aim for an artful blending of good teachers, software, and textbook material.
The MathemAntics project aims to use computer software to promote deep learning of mathematics in ECME. We are especially interested in reaching children from low socioeconomic backgrounds who ordinarily have limited access to high-quality teaching, including educational software. MathemAntics is a comprehensive mathematics software suite for children aged 3 to 8. Activities are designed to help children build on their initial everyday mathematics through enjoyable and engaging interactions with whimsical environments. These contain virtual objects, like chicks or goats; various mathematical tools, like one that can line up objects in a row; and a variety of mathematical representations, like a number line. The capability of computers to collect and process large amounts of data enables MathemAntics to adapt to a child’s abilities, offer useful scaffolds, and provide individualized feedback to children and performance data to teachers and parents. Each MathemAntics activity grows out of cognitive research and undergoes extensive usability testing before we conduct learning and evaluation studies.
We examine the effectiveness of MathemAntics in several ways: we give children simple non-computer problems before and after use of the software; watch the children to record key aspects of their behavior as they work with MathemAntics; and we interview them as well. We also draw upon the computer’s internal record of children’s performance, including accuracy and choice of strategies, to examine their learning as it unfolds during their work with MathemAntics.
A case study: ZurGelmAntics
ZurGelmAntics is one activity in the MathemAntics suite. It is aimed at prekindergarten to first grade (children aged 4–7), and focuses on a variety of foundational number skills, including counting, cardinality, subitizing, addition, and subtraction. The inspiration came from a classroom activity originally suggested by a teacher and developed by Osnat Zur and Rochel Gelman (see “Further reading”). The activity consists of a “change” situation where objects are either added to, or taken away from, an initial set of objects, the cardinal value of which has already been determined.
In our adaptation, barn doors open to reveal a set of cute virtual animals. Children are asked to identify the number of animals in the barn (see Figure 1). After that, visible groups of animals can sneak into or out of the barn, and children are asked to determine the number of animals in the barn after the change. Children indicate their answer on a number line, or by typing numbers into a blank answer box, and then receive feedback. Next, the barn doors open so that children can check their answers. At the end of the sequence, the child sees a numerical statement that summarizes what has been learned. For example, as the numeral 5 appears on top of the barn, 5 animals in the first set are highlighted; then the numeral 2 appears as the 2 animals in the second set are highlighted; then the whole expression 5 + 2 = 7 appears as all of the animals are highlighted (see Figure 2). In this way, the series of events involving the animals is clearly linked to the symbolism of formal mathematics.
The ZurGelmAntics activity is designed to give students opportunities to use relatively advanced addition strategies. Young children often begin by using the cumbersome countall strategy to determine the set resulting from a change: they may say, “One, two, three, four, five… six, seven.” Although usually producing accurate answers, count-all is not as efficient as counting-on (“The first number is five, so five, six, seven”) or remembering that five and two is seven. At this age, children know few number facts with which to solve the problems. Also, because the barn doors are closed when children are asked to determine the sum, count-all is hard to do. For one thing, the animals are not visible, so that the child must represent them with fingers or mental images, which are difficult to count. Further, the method is tedious with large numbers and may result in counting errors. Hence, design of ZurGelmAntics indirectly encourages children to attempt the more advanced strategy of counting on.
The experience of one child, whom we will call “Jayden,” a five-year-old African-American boy attending an inner-city school, illustrates how ZurGelmAntics can help children acquire advanced strategies over time. Before using the software, Jayden performed below grade level on measures of number sense, and did not display the ability to use the count-on strategy or solve word problems involving changes to sets of objects. During his first session Jayden consistently used the count-all strategy by counting on his fingers. When asked how he obtained his answers Jayden explained “On my fingers!” as if this was the only possible strategy.
By the end of the third session, however, Jayden used some recalled facts and the count-on strategy. Importantly, he was able to verbalize this growth, explaining that in the problem 12 + 2, instead of counting all of the animals, he was able to “count in my head, because after 12 is 13 then 14.” After the last session with the MathemAntics activity, Jayden displayed the ability to count-on when doing new sums (not on the computer) and even solved a few word problems correctly.
During the next year we will take ZurGelmAntics and various other activities that make up the MathemAntics suite into classrooms in New York City public schools. The goal will be to learn how the software can supplement and integrate with several successful curricula, ranging from Singapore Math to Everyday Math. Teachers and researchers will work collaboratively to determine how to use MathemAntics to further the learning goals of the curricula. We will create lesson plans using large-group, smallgroup, and individual activities.
About the authors
Benjamin Friedman is a research assistant in the Department of Human Development at Teachers College, Columbia University. His research projects focus on elementary school students’ learning of mathematics, and science through technology-based interventions. Herbert Ginsburg is the Jacob H. Schiff Foundation Professor of Psychology and Education at Teachers College, Columbia University. His team focuses on research related to early childhood mathematics education.
Ginsburg HP (2009), Early Mathematics Education and How to Do It. In OA Barbarin & BH Wasik (Eds.), Handbook of Child Development and Early Education. New York: The Guilford Press.
Zur O, and Gelman R (2004), Young Children Can Add and Subtract by Predicting and Checking. Early Childhood Research Quarterly, 19(1), 121–137.