*Thoughts from Kathy Richardson*

There was a time when teachers were thought to be successful if they could get everyone in their class to do the same task correctly. The focus was on accomplishing the tasks rather than on developing an understanding of concepts. However, when we focus on teaching for understanding and try to find out what children really know and understand, we see that our children don’t all learn or understand the same things at the same time. The range of needs becomes clearer and clearer. Meeting this range of needs becomes one of our greatest challenges as it requires that we look closely at each of our children and provide the appropriate experiences for them.

I would like to share two of the twelve Guiding Principles for Teaching for Understanding that I wrote for the California Model Curriculum Guide that help us look at meeting our student’s needs.

- We will not expect all students to get the same thing out of the same experience. What students learn from any particular activity depends in large part on their past experiences and cognitive maturity. We should try to provide activities that have the potential for being understood at many different levels.
- We must know that the understandings we seek to help the students gain are developed, elaborated, deepened, and made more complete over time. We must provide a variety of opportunities to explore and confront any mathematical idea many times.

If we recognize that understanding of concepts evolves and develops over time and that children do not all learn the same thing from an activity, then the answer to meeting the range of needs lies in the learning environment we create and in the type of tasks we present our children. The tasks we present need to be appropriate for children who are working with that concept at various levels of understanding and competence – and have value in being repeated. The learning environment must support the development of concepts over time rather than exist as a set of unrelated activities or individual lessons. Once we establish a learning environment that provides opportunities for children to work with tasks in a variety of ways, then our job is to watch our children at work and look for clues that tell us what they are getting from the activity.

If we take time to watch our children, we will see how they approach a task and from that get an idea of their level of development. Allowing for children to get different things from the same activity does not mean we don’t care what they are learning. It is not enough to say, “They get what they get.” We still need to be very intentional in our teaching and interactions with the children. We need to know what we expect our children to learn from the experience, and what indicates whether they are or are not making progress.

If we look at even a simple task like determining how many objects are in a jar, we can see how this idea applies to children at many different levels of understanding.

**The task: How many in the jar?**

**Can you determine how many without counting every object in the jar? Estimate first and then find out how many.**

This task will be experienced differently depending on the age and concept level of each child. When presenting this task, the teacher should change the number of objects in the jar, depending on the developmental level of the children. However, no matter how various children approach this task, we want all of them to be developing a sense of number, learning to make reasonable estimates, beginning to look for relationships between numbers and beginning to use those relationships to figure out what is not known. We want them finding ways of organizing and keeping track as they work. Let’s look at how different children can get different things from the same experience and still be making progress towards developing number sense:

Three primary aged children, Kara, Ivan and Lou are each given a jar containing 34 pecans. They are to estimate first and then determine how many pecans in the jar. They are asked to see if they can figure out the amount without counting each one and to see if they can organize them in any way that will help them keep track.

Kara looks at the jar and says, “I think there’s 13.” She then dumps out the pecans and begins to count them one by one. While she is counting, her teacher asks, “How many do you have so far?” Kara looks up at the teacher with a puzzled look. “I forgot,” she says and begins to start over. “Do you think you could find a way to keep track of them as you count?,” her teacher asks. “I could put the ones I counted over here and then I would know I counted them,” she replies.

Ivan estimates the jar holds 20 pecans. He dumps them out and begins to group them into piles of ten. When he has counted 2 piles of ten, his teacher asks, “Do you have another idea of how many pecans the jar holds now?” “I have 20 so far, so maybe it will be 50,” he says. “Why do you think 50?” his teacher asks. “Because its bigger than 20,” he says.

Lou also estimates the jar holds 20 pecans. He dumps out a few of them and examines the jar. Then he dumps out a few more and begins to organize them into groups of ten. “It’s going to be more than 20,” he says. His teacher asks, “How come you didn’t dump all the pecans out? Aren’t you planning to count the ones still in the jar?” Lou responds, “You said to see if we could figure it out without counting them all. I tried to just dump out half of them. That way I can see how many half is and then add it two times to see how many.”

These three children are working on the same task, but are focused on different things. Kara still needs to count each one. Her attention is on the process of counting and accomplishing that and she isn’t yet able to think about how many she has as she goes along. She still needs practice counting and is getting that from this experience. She has trouble remembering what she counted. This problem is preparing her to see the need for a system for keeping track sometime in the future.

Ivan is applying what he knows about grouping into tens as a way of keeping track of what he is counting as he goes. He has not developed enough number sense as yet to use that information to make a reasonable guess. This experience will help develop that sense of quantity.

For Lou, this task is much more than just a counting task. He is intrigued with thinking of how to solve the problem without counting. He has enough sense of number to believe he can figure it out by knowing half and doubling it.

Two intermediate aged children, Rita and Eli have also been given a jar containing about 450 pinto beans and the task of determining how many the jar holds. Eli suggests they dump out half for each of them to count. “Then it wouldn’t take so long,” he says. Rita says she thinks that would still take too long. “The teacher said we didn’t have to actually count every bean,” she says. “I think we should measure them.” Eli isn’t sure about that. “How can you measure beans?” he asks. Rita replies, “I don’t mean with a ruler. I mean like using a measuring cup.” She gets some plastic cups and scoops out some of the beans. “Here, you count the beans in this cup and I am going to see how many cups.” Eli begins to count, making piles of ten beans as he goes. Rita begins to pour beans into the cups. Mr. Torres comes by and talks to each of the children. He asks Eli what he is doing. Eli is comfortable with the job of figuring out how many in the cup and tells Mr. Torres that he is guessing there are going to be more than 120 beans in the cup because he has made 6 piles of ten so far and he thinks he has about half to go. Mr. Torres then asks Rita what she is doing. “I thought I could fill all these cups and figure out how many beans but the cups seem too big and they take too long to count.” “Do you have any other ideas?” asks Mr. Torres. “I want to find a smaller scoop or something else to measure with. Then I could see how many scoops in the plastic cup. That would be easier.” Rita says. “Do you think all the scoops will hold the same number of beans?” Mr. Torres asks. “I’m not sure,” says Rita. “But I am going to find out.”

These two students are working together to solve a problem but are approaching it in their own way. They will need to discuss what they found out with each other before they can agree on the number of beans. Both will need to be satisfied with the answer before they finish with the task.

When we consider supporting children’s development over time, we must recognize that one experience is not enough. With continuing opportunities to work with number in a variety of settings, all the children will grow in their developing understanding.

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