It is important to develop these understandings through the use of manipulative and through the active involvement of students in classroom, field and outdoor experiences. Some general guidelines and specific examples relevant to the development of the foundational objectives for kindergarten mathematics follow.
At the kindergarten level, developing each students' understanding of mathematical concepts, processes and abilities emphasizes the following:
Providing students with relevant choices which allow them to explore areas in which they are interested also leads to students developing a sense of purpose for learning. This presupposes that teachers have already thought through the big ideas to be explored and have asked themselves the important question: "Why is this worth learning?"
It is important to provide a variety of manipulative when exploring any concept. For example, students may initially explore the concept of capacity at the water table with different sizes of plastic beakers, containers and cups. Further play and exploration can be encouraged through the use of other "pour able" substances such as grains, lentils, sand, coloured water, marbles, pebbles, buttons, paper clips, macaroni or rubber bands.
It is important that children have time for free exploration, experimentation and observation of materials and activities, particularly when concepts, materials or activities are new or unknown to the students.
One of the purposes of having students touch, handle, manipulate or experiment with materials is that it allows them to discuss their observations from an experiential point of view. Much of the development in these early years focuses on strengthening students' perceptual abilities through concrete experiences. Observations, manipulations and experiences with objects, visual images, sounds, music and movement contribute to the development of students' perceptual abilities.
The use of language also facilitates perceptual development. Young children learn through communicating with others both verbally and non-verbally in a positive, supportive learning climate. As children share their observations, and their ideas around these observations, they come to understand what it is they are thinking and learning about. Through expressing themselves, children frequently find out what they think and know, and so inform themselves. Classroom situations which invite students to use language to question or to explore, nurture a natural curiosity about the world. Such situations encourage the continued use of language for the purpose of inquiry. Sometimes adults are needed to provide a model for language use or to link ideas together.
Asking children to reflect upon their observations leads to the development of perceptual, procedural, conceptual and personal understanding. Children can be helped to broaden and deepen their understanding beyond these observations through teacher support.
Teachers can support the continuous growth and learning of young children through:
Some examples of developmentally appropriate activities which work toward the development of mathematical understandings related to the five strands in this Elementary Curriculum Guide follow.
In addition to developing intellectual, mathematical understanding, this strand contributes to students' social development. An important question for teachers to ask themselves is:
"What is the child learning from this intervention/process/experience?"
This strand emphasizes the collection, organization, and interpretation of data. For kindergarten students, a variety of objects, events and activities can be provided for observing, discussing, sorting, labelling and display. Such experiences should focus on:
Important questions for teachers to ask include:
"Can you tell me about the materials you chose?"
"What kinds of groups did you make?"
"How did you get that idea?"
"What other kinds of groups could you make?"
This strand emphasizes the understanding of numbers, number patterns, counting, and estimation. Such understanding is best developed through purposeful, concrete experiences and through using manipulative. The following example demonstrates the integration of this strand with the previous strand.
Two kindergarten children were observed playing at a block centre with eight plastic models of dinosaurs. The children first arranged the dinosaurs in a line according to size and counted them one, two....eight. Then they decided to arrange them in order of meanness. After deciding that the ones with the biggest teeth were the meanest, they again lined up the dinosaurs and again counted them. They both expressed considerable surprise when they found that there were still eight dinosaurs.
In another part of the classroom, the kindergarten teacher observes four children in the imaginative play area and joins their play in the "restaurant":
"May I join you?"
"Is there another chair for me to use?"
"Do you have enough?"
"Tell me about the "special" of the day..."
There are many opportunities for children to learn about numbers through play. They need many concrete experiences with different sets of objects in different situations before their understanding of number is firmly established.
Many of the objectives which kindergarten teachers can work toward developing through appropriate materials, activities and experiences overlap with objectives related to other strands. An example follows.
Providing students with cardboard shapes of different sizes and colours, attribute blocks, paper, scissors and crayons and allowing time for exploration through observing, handling, tracing, sorting, comparing and reclassifying will contribute to students' understanding in the strand. Questions which could guide students' discoveries include:
"How are these shapes alike? Different?"
"If you close your eyes, which shape are you able to find? What is
it about this shape that helps you to find it?"
"What other things in the room/gym/playground have the same shape?"
This strand emphasizes the exploration of concepts such as length, area, capacity, mass, time, and temperature. The facilitation of this exploration is accomplished through:
For example, students' sense of time can be developed by emphasizing tasks and completion of activities:
"Tell me about your picture/construction/experiment. Where did you get your ideas? If you were to change anything, what would it be? How will you know when your project is finished?"
or
"You've been working at this painting/castle/water wheel for a while. Why?"
When exploring the concept of capacity with pour able substances, questions which could guide students' observations and experimentations include:
"How can you tell which container holds more water? Less water? The same amount?"
Questions to facilitate the exploration of the concept of area with manipulative could include:
"How many pennies/hands/buttons do you think it will take to cover the circle? How could you check?"
Objectives which kindergarten teachers can work toward developing through appropriate materials, activities and experiences are outlined in the Scope and Sequence which precedes this section. This scope and sequence of specific learning objectives at the Kindergarten level is not intended to be complete, final or prescriptive. Rather, it provides a source of items from which teachers can generate additional learning objectives that work toward the development of mathematical understandings in the kindergarten program.
It is not necessary to design a "math" center in the kindergarten classroom. Mathematical understandings can be developed through materials and activities at the block center, water table, sand table, imaginative play area, discovery table, library corner, cooking center, carpentry area or through other centres, experiences and projects. Centres, activities or projects devoted to a particular area of study are not the critical component. It is the learnings that children acquire through these varied experiences that are important.
