Children are splendid little mathematicians ...

“Before the onset of formal  schooling, young children do not only memorize … and they do not only employ mechanical skills. They do not operate only on a ‘concrete’ level. Instead, we can say fairly that young children are splendid little mathematicians. They deal spontaneously and sometimes joyfully with mathematical ideas. This is what real mathematicians do.” (Ginsburg, 2008, p. 55)

When children first come to school, they bring inquisitiveness, energy, a wide range of social, intellectual and emotional experiences and an abundance of mathematical knowledge gleaned from their everyday experiences. This is not surprising since studies have shown that mathematical ability is evident in humans as early as infancy when they are able to discern between sets of objects that vary in number (Lipton & Spelke,

2003). Furthermore, mathematical abilities continue to develop into beginning mathematical understandings along a roughly consistent developmental path, with or without adult intervention. Ginsburg and colleagues discovered that when children are given free play opportunities there are no significant differences in the complexity of the exhibited mathematics, regardless of children’s cultural or socioeconomic backgrounds (2003, p. 235). Although each child acquires mathematics knowledge through experience, and each child comes to school with a range of prior experience, all children have the potential to productively engage in mathematical activities. Honouring children’s starting points enables educators to build on students’ mathematical knowledge with an inquiry-based approach, developing purposeful and meaningful mathematical experiences in the classroom. It is also important to realize that the ways in which young children think in mathematical situations can be quite unique. Educators “must be particularly careful not to assume that children see situations, problems, or solutions as adults do. Instead, good teachers interpret what the child is doing and thinking and attempt to see the situation from the child’s point of view” (Clements & Sarama, 2009, p. 4).

Since play is integral to a child’s world, it becomes the gateway to engaging in mathematical inquiry. Sarama and Clements suggest that mathematical experiences can be narrowed down into two forms, play that involves mathematics and playing with mathematics itself (2009, p. 327). Further, it is the adult present during the play who is able to recognize how the children are representing their mathematics knowledge and then build on their understanding through prompting and questioning. Sarama and Clements stress that “the importance of well-planned, free-choice play, appropriate to the ages of the children, should not be underestimated. Such play … if mathematized contributes to mathematics learning” (2009, p. 329).

Mathematization is a critical learning process which involves “redescribing, reorganizing, abstracting, generalizing, reflecting upon, and giving language to that which is first understood on an intuitive and informal level.” (Clements & Sarama, 2009, p. 244)

Educators also provide experiences in playing with mathematics itself by using a repertoire of strategies, including open and parallel tasks that provide differentiation to meet the needs of all students and ensure full participation. Moreover, students do not have to see mathematics as compartmentalized, but instead as it mirrors their life experiences through other subject areas like science and the arts. As such, “high quality instruction in mathematics and high quality free play need not compete for time in the classroom. Engaging in both makes each richer and children benefit in every way” (Sarama & Clements, 2009, p. 331). This equity of opportunity is essential so all students can fully develop their mathematical abilities.

A carefully planned mathematics environment enables the use of manipulatives whether commercial products or found objects, sometimes brought in by the students themselves. Ideally, manipulatives serve as learning tools to help students build their understanding and explain their thinking to others. Research has shown, however, that “manipulatives themselves do not magically carry mathematical understanding. Rather, they provide concrete ways for students to give meaning to new knowledge” (Ontario Ministry of Education, 2003, p. 19). Students need the opportunity to reflect upon their actions with manipulatives, and through discussion, articulate the meaning they generate, so that the link between their representations and the key mathematical ideas is apparent (Clements & Sarama, 2009, p. 274).

Computer manipulatives can sometimes be more powerful than concrete manipulatives. For example, some applications offer flexibility “to explore geometric figures in ways not available with physical shape sets” (Clements & Sarama, 2009, p. 285). However, “effectively integrating technology into the curriculum demands effort, time, commitment and sometimes even a change in one’s beliefs” (Clements, 2002, p. 174). In their three-year study of the use of computers in primary classrooms,Yelland and Kilderry found “that most of the tasks the children experienced in early years mathematics classes were unidimensional in their make-up. That is, they focus[ed] on the acquisition of specific skills and then they [were] practised in disembedded contexts” (2010, p. 91). Such tasks were comparable in complexity to traditional pencil and paper mathematics tasks, were teacher directed and anticipated correct answers with narrow solution strategies. In other words, while technology was being used in many classrooms, its potential to promote student thinking was being greatly underutilized. Yelland and Kilderry recommend that technology in primary classes should offer multidimensional mathematical tasks ensuring both student input into the direction of their learning and supporting more varied learning outcomes (2010, p. 101). Such an approach to technology enables students to use powerful yet familiar media to express and extend their learning.