Ways to Help Children Communicate and Share Mathematical Ideas

Group Sharing and Discussion Format

As children come together as a group during the course of the day, they should be encouraged to express the discoveries and observations they have made. The figure below lists some sample prompts that teachers can use to assist students with collecting their thoughts and expressing their ideas.

Questions that encourage children to organize and express comparisons and measurements.

What did you see? What happened first? Tell us what you think about... How do you know? What do you think will happen next? How tall is your plant? How did your shadow change? How much does your rock weigh?

Journaling

As children become more skilled at writing their thoughts, they can keep daily journals of their observations. They can keep track of the changes that they are observing and the phenomena that are occurring in their world. For example, if they are exploring their shadows, they may journal about how their shadow changes length at different times during the day. When children make such deliberate attempts at making their observations and their thinking visible, like scientists, they are able to make predictions and develop conceptual schemes about scientific phenomena.

Making a Book

As children have opportunities to observe, explore, and experiment with many phenomena of change and growth, they will be able to build theories about their world. As these theories become evident, a book of “Amazing Discoveries” can be created. For example, when children explore and examine what happens to shadows at different times of the day, they can begin to make claims about the position of the sun in the sky as it relates to the length of a shadow. Class statements can be made such as “When the Sun is high in the sky above, our shadows are shorter than they are late in the evening when the Sun is low.” This concept is certainly an “Amazing Discovery.”

These experiences are valuable for young children because they are the basis for much more complex and abstract concepts. The concrete experiences such as comparing the length of their shadows to the position of the Sun will, over time, expand to an understanding of the very abstract concepts related to the movements of the Earth and sun.

With each Concept Exploration, children’s theories and discoveries should be celebrated and recorded. At the end of the year, you will have constructed a wonderful book that captures the rich problem-solving experiences of the children across many topics of interest. They will have, in fact, written their own textbook, chapter by chapter, of each Concept Exploration.

Modeling Appropriate Mathematical and Scientific Language

As children observe and measure change and growth, it is important for teachers and collaborating adults to model appropriate mathematical and scientific language. As children describe phenomena of change and growth in their own language, we can supply different levels of description reflecting a more specific vocabulary. For example, “Our bean grew six centimeters in ten days,” “Our shadows were the shortest at noon,” “The wind was blowing thirty miles an hour,” “The water in the rain gauge was more than three inches high,” and “The oak tree was more than one hundred feet high.” Examples of measurement problem-solving strategies for young children are shown in the figure below.

Measurement problem-solving strategies for young children.

• Use all modes of instruction—concrete, iconic, and symbolic—and supply language.
• Pose questions and get children to explore and wonder.
• Dialogue with children about what they are measuring, how they are measuring, and what tools they could use to measure.
• Discover and invent together and independently at school and at home.
• Share and see other viewpoints and expressions.
• Make sense out of a lesson.
• Model standard mathematics.
• Have fun!

Relating Rational Counting of Units to the Process of Measurement

In order for young children to solve problems in mathematics, they need to be able to count with meaning. Young children solve mathematical problems with rational counting. As children count how many building blocks long a plant is, it is important for them to count rationally and in a stable order, knowing the appropriate sequence of numbers. Using one-to-one correspondence (that is, touching one object at a time when counting objects in a set and keeping track are important aspects of rational counting. When a child can consistently keep track of which object he touches and which sequences of numbers is used, he will be counting with meaning. When children count with meaning, they know that the last number named in a counting sequence tells how many in the set, that is, it tells the cardinal property of the set.

Rational counting is not simply naming a sequence of numbers. Children who do not count rationally use numbers as they would name people’s names in a group. That is, the counting sequence of “one, two, three, four” is done with the same purpose and understanding as naming “Carlos, Ben, Judy, and Nicole.” When children know that “one, two, three, four” tells how many are in the set, they are rationally counting.

Providing Discrete Items to Count When Measuring

Children need real objects to use when measuring things. If they are measuring how tall their teddy bear is, they need to lay objects end to end so that they can count how many of them fit the distance from the top of the teddy bear to the bottom, or from head to toe. Objects such as blocks, large paper clips, plastic ants or bugs, craft sticks, pom-poms, small toy cars, or tracings ofchildren’s feet can all be used to measure linear distances.

Links or blocks that snap together in a straight line or flexibly linked blocks can be used to measure straight or curved linear distances. The important thing to remember is that linear distances are made of discrete, countable units, whether they are inches, centimeters, blocks, clips, or cutout tracings of feet. Units of measure are the emphasis in this process.

Likewise, for mass or weight, objects such as blocks, metric weights, marbles, or any other fairly dense material can be used to fill a pan balance. Like linear distance, mass is measured with discrete units of measure. An item weighs so many grams, blocks, or marbles. So, counting units of weight is an important part of the process of learning the underlying concept of how much something weighs.

For capacity, any pourable substances such as water, sand, rice, or small beans can be used to fill and refill containers. Ample exploration with toys and tools that are volumetrically related is essential. Toys or tools are volumetrically related if they are designed to be iterations of each other. For example, measuring cups and measuring spoons are good examples of how containers are volumetrically related. That is, if you fill the quarter cup container four times, it will fill the cup measure. If stacking cup toys are volumetrically related, they will be designed to be iterations of each other. In the photograph, you can see the stacking cups with numbers on them. In this set, if you fill the number 3 cup with rice and pour it in the number 6 cup, the rice will fill it halfway. Another filling of the number 3 cup will fill the number 6 cup. When buying stacking toys, this is an important feature to determine. Playing with water, sand, or rice and such containers helps young children to conceptualize about size and proportion. Making these discoveries in a natural, inviting, and interesting setting builds a strong conceptual base for units of measure and eventually for standardization of measurement.

A good way to transition children from measuring with manipulatives to measuring with rulers is to have them construct their own iconic, pictorial rulers. This idea was developed by a group of first graders who were not quite ready for rulers. On strips of stiff poster board cut out in a shape similar to a ruler (that is, a rectangle approximately 2 inches by 12 inches), the children used a large ink-pad stamper to continually stamp an image of a rainbow, in a line across the poster board. With the images in a line next to each other, the children knew that to measure something, they could simply count how many rainbow images long it was. The children were not merely looking at an end point on a centimeter or inch ruler; they were counting discrete units of measure, in this case, rainbows.

As another example, the ink-stamped frogs in a row on the iconic ruler shown in Figure 4.8 can be used for measurement.

With this particular frog ruler, the child can count how many frogs long an item is. This closely mimics the process they used when they measured with manipulative nonstandard units. The only difference is that the units are icons stamped on a strip of paper, the iconic ruler. This process can also be done by using large stickers on the poster board strips. Other resources for constructing the iconic rulers are Ellison cutouts, potato prints, large erasers used as stampers, sponge prints, and so forth. The important concept here is that whatever stamp or sticker is used has to be identical across the poster board. For measurement, iteration of identical units is the concept that underlies the process of measurement.

As children continue to measure by counting units—first manipulative, then iconic—they begin to have a logical, clear understanding about what the lines on a ruler represent: discrete units. The first graders who invented the iconic rulers helped themselves keep track of the measuring process with clothespins to mark the beginning and ending points of the item they were measuring. If you use an iconic ruler, children measure items at different starting points to show the static nature of an item’s length. For example, measuring a toy car will yield the same length whether you place it at the beginning of the ruler or somewhere else on the ruler. This is an important concept for young children to discover and learn.

Developing Referents for Measurement

When children are ready to use formal tools to measure, such as rulers, they should begin to develop referents for the units they will be using. For example, knowing that a centimeter is about the distance of the width across a child’s little finger at the first crease helps that child to “know” how big or how long a centimeter is. Putting the child’s finger across a centimeter ruler to check to see how close her finger’s width is to a centimeter helps the child to conceptualize the distance more precisely.

Next, you could ask the children to try to draw a line segment 1 centimeter long. They will look at their little fingers and try to replicate the distance on paper. Then take the centimeter ruler and check to see how close the drawn estimate of that length is. Next, ask the children to draw a line segment 5 centimeters long. Follow the same procedure of checking and reflecting on the results. Use many different lengths. Find other similar referents for 10 centimeters and so forth.

After some experience in constructing line segments of different lengths, you can engage the children in estimation activities. Place a few items in front of the children and ask them to estimate how many centimeters long each item is. They can use their fingers to help with this process. Placing their fingers across the length of an item can help them to see the concreteness of the linear distance.

After children have clearly developed a referent for a centimeter, another approach to estimation can be taken. You can ask the children questions such as “Ifound a bug on the sidewalk today. Do you think it was one centimeter long, 50 centimeters long, or one hundred centimeters long?” Such questions get at conceptualization of what a centimeter is. Of course, fun stories can be told when they choose a “silly” answer. After all, silliness occurs when there is a break in what they understand at a logical level. A 100-centimeter-long bug, oh my! What fun stories and art could be constructed about such a creature?

A similar approach could be taken with a milliliter. If children could hold a cubic centimeter (a base-10 unit) in their hands and imagine it filled with water, they would have a referent for how much a milliliter is, because they are the same thing (Figure 4.9). Another referent for milliliter is how much water a scooped-out jelly bean will hold. Using a very small container, say, 10-milliliter capacity, children could estimate how many times they would have to fill the centimeter cube or the jelly bean, and pour it into the container to fill it up. After they estimate how many milliliters will fill it, they should actually fill it up and measure it with a milliliter measure. This small container could be labeled and used as a referent itself.

After more of this type of experimentation, you could ask questions that get at conceptualization of the amount of a milliliter. For example, “Today I washed my hair. Do you think that I used eight milliliters of shampoo, 60 milliliters of shampoo, or five hundred milliliters of shampoo?” Again the logical answer is 8 milliliters, but we could conjure up interesting scenarios about what would happen if we used 500 milliliters of shampoo in our bathtubs. Imaginations can take charge when answering these types of measurement questions.

It is a good practice to apply a child’s understanding of referent to estimation and problem solving. For example, if we have a referent developed for millimeter, centimeter, and meter, we can ask children fun questions such as these:

• I found a bug on my driveway this morning. It was about 8 _______ long! (mm,cm, m)
• If I wanted to measure how tall my pencil is, I would use _______. (cm, m)
• I want to know how much water my aquarium holds. I would measure the water with a _______ measure. (l, ml)

For young children, it is recommended that you use only common metric measurements—measurements that are often referred to in their conversations, activities, or their listening or speaking vocabulary. Solid experience with commonly used metric measurements will build a solid foundation for using other, less common increments of metric measurements in later experiences.

It is important to make your own referents based on your own situations and environment. When possible and appropriate, measure out these amounts and construct a “Referent Chart” with the children.