For more than a year, we had been team-teaching 1st graders at South Tucson's Ochoa Elementary School. After this day's combined morning session, we were preparing to return the children to their respective classrooms as usual when Yvonne, one of the exuberant members of the class, stopped us:
“Wait, we want to show you a game.”
Her “we” referred to Ysidro, a quiet student who seemed to be enjoying his compatriot role. On this morning, Yvonne took charge, while Ysidro lent moral support and helped get the other children's attention. Yvonne politely asked everyone to sit down on the carpet in front of the chalkboard, and all 38 of us did as she requested (we teachers at opposite ends of the group).
Yvonne and Ysidro had built an incredibly long string of attached colored cubes, which now snaked along a wall of the classroom. It became apparent that both students had acquired certain knowledge or skills that they wanted to share in a leadership role with their classmates and teachers. When necessary, Yvonne raised her voice to be heard, and she exercised another management technique by raising her hand for quiet. That proved to be contagious: Ysidro and the rest of the children also raised their hands to signify they wanted it quiet.
Autonomy and Authenticity
The instance of “play learning” that we were about to see was typical of the real experiences we allow our 6-year-olds to have as part of an alternative education program we've participated in for the past five years. In 1990, Paul E. Heckman of the University of Arizona conceived the program—known as the Educational and Community Change Project—to promote “indigenous invention.” That is, to let those most involved in the school— teachers, parents, students, principals, and community residents— create and implement curriculum, teaching practices, and classroom structures.
The program was launched at our school, Ochoa Elementary, which is in a predominantly Spanish-speaking, low-income area where students have traditionally scored low on norm-referenced tests. The goal of the changes we're making is not only to educate the children, but also to keep them interested so that they won't become dropouts.
We believe that as children identify and solve problems or address actual challenges, their efficacy and self-confidence grow. They are doing the work of real scientists, mathematicians, or artists. Often, we let the students guide our combined class's direction, and we attempt to help them realize the significance of their problem solving.
In this example, a student initiated a contextual lesson in mathematics. It incorporated number sense, counting, estimation, prediction, and data analysis—all critical strands that the National Council of Teachers of Mathematics has promoted in its new Curriculum and Evaluation Standards for School Mathematics.
Let the Game Begin
As Yvonne waited for everyone's attention, Delia and I looked at each other, wondering what would come next. We work so closely that sometimes we can communicate without speaking. Conditioning in traditional teaching practices can make some strange thoughts run through a teacher's mind when a student exerts his or her authority. Is this the 1st grade version of a coup d'état? Can there be discipline if we teachers are not in charge? What if the children tell their parents that Yvonne taught the class? What if a teacher with a traditional approach walks in? What if nobody learns anything? What if nobody likes the game—how do we help Yvonne save face and not suffer hurt or embarrassment?
Our fears, fortunately, proved groundless. Although several years ago we taught thinking processes and skills according to a curriculum guide, our recent efforts to involve the children in the development of their own thinking processes had paid off.
Once everyone quieted down, Yvonne explained that the game was to count the number of cubes that made up the “cube train.” The two children had intended to play this game in the morning but ran out of time. In describing the game, Yvonne alternated between Spanish and English, while other students offered translations of one language or the other, as well as their own suggestions.
Yvonne then asked the class for predictions: “How many cubes do you think there are?” As each child guessed a number, Yvonne, with the help of Ysidro and the other children sounding out the letters, wrote the names and predictions on large chart paper taped to the chalkboard. This was time-consuming, so the children decided to help by writing their own names and predictions. That took only about 15 minutes.
Counting the Lessons
Also in the interest of time, we suggested counting the cubes by 10's, using a number chart that hung on the wall. Some children made other suggestions, but there was consensus that we should count by 10's. Delia and I were becoming restless, but the children were fine; they were still on task and completely interested. Yvonne asked us to review the predictions, so together we read the numbers and names and then identified the lowest and highest predictions.
To further expedite the game, we suggested that the children get in groups of five or six. Yvonne began dismantling the train, and Ysidro handed bunches of cubes to each group. We communicated in English and Spanish and helped one another count, referring to the number chart when necessary. One group began to see patterns in groupings of 10, and one student suggested that they arrange these groupings by color.
Each group was to report its final count, which Yvonne would record on the chart. The first group, setting aside a few extra blocks, reported that it had counted 100. Oh, no! Yvonne didn't know how to write a three-digit number. We all helped her. The next group also reported 100 after setting aside some extras.
Now Yvonne knew how to write 100, but she didn't know how to align the two numbers. We verbally guided her and also talked about the plus sign. The children were looking and listening. Their social interaction was exemplary, with expert and novice relationships developing everywhere.
The children in the third group announced that they, too, had counted 100 cubes, and this time several students reminded Yvonne how to place the numbers and add the 200 cubes to the 100. A final group of 100 cubes was added, but then there were the extra cubes—a total of 14. What to do? Delia and I facilitated the situation by demonstrating how a two-digit number could be added to the three-digit number for a total of 414.
Yvonne's final step in the game was to see how close the students' estimates had been. Most were extremely low or excessively high, but Andrea had guessed 400 and was declared the winner.
What We Learned
We suggested that the children write a “sentence story” about our findings. They put forth many ideas, but the final decision was, “We were `plussing' the numbers”—or, as they dictated it, “WE WR Rlusing The NumBrs.”
We felt it was important for the children to recognize the significance of the hour, so we all reflected on what we had learned. And we had learned a lot: how to write names and form letters, how to write three-digit numbers and add them, how to behave (self-regulation), how to communicate ideas in both Spanish and English, how to work as a mathematician would work, and how to work as both teacher and learner.
As teachers, we feel that we are accountable and should be able to justify the learning experiences in our classrooms. We recognized that the entire lesson was energized by the magnetism of the game, which stimulated the children's cognitive strengths. It had been fun; it was educational; it was active. The students felt empowered and stayed on task. Further, we, as teachers, were able to reveal ourselves as learners as well.
Guided by the new mathematics curriculum standards, we then identified some of the mathematical concepts that the children had processed:
- Problem solving. The train game presented a variety of possibilities for exploring mathematics and counting. The key was to allow each problem to surface and challenge the children as they created it. The exercise was not teacher-imposed, nor did it require children to sit with meaningless work sheets and memorize meaningless formulas. Mathematics and manipulatives were useful tools.
- Estimating/predicting. The children had fun guessing numbers. And their guesses provided an opportunity to discuss and reflect on their theories of probability as they perfected their sense of numbers.
- Number sense. The children explored numbers in many ways and at many levels. Some developed skill in one-to-one correspondence; some counted by rote and turned to the number chart for more accuracy. Students helped one another count by 2's and 3's and 5's and 10's. And they conceptualized part-to-whole relationships.
- Data collection and analysis. As children continued to interpret the data they collected, they could see the game in different ways. They represented their train symbolically using manipulatives and number symbols. They added up single-, double-, and triple-digit numbers, thereby solidifying their sense of place value and their ability to recognize and read two- and three-digit numbers. They learned about two-digit addition and calculation.
- Patterns. Students found patterns in the abstract and tangible symbols that they used to represent the train. They sorted by size, color, and numbers, and made comparisons using their data. They discovered patterns in sequences of numbers: 20, 21, 22, 23 ... 30, 31, 32, 33, and so on to 100.
A Child's Eye View
Before participating in the Educational and Community Change Project, we wouldn't have seriously considered invitations such as Yvonne's. After all, as teachers, we were the decision makers; we decided what games were to be played and when. Any change in schedule would have thrown us out of control. And besides, everyone knows that games are fun, and therefore a waste of valuable learning time.
But our perspective has changed. Now, children's ideas are valued as much as adults'. We are each a teacher and a learner, and sometimes Delia and I become novices and the students become experts. Students take the initiative in organizing, presenting, and sharing. We assess our lessons—and life—alone, with a partner, or all together.
Children learn using their own language, unique learning styles, and thought processes; and at their own levels of development. We now see learning through a much more powerful lens, viewing the whole child in relation to his or her prior knowledge and dominant language and culture. The more we look, the more we see.
We're always planning and trying new things that are departures from traditional teaching. The children are flexible and spontaneous, and we adults have become flexible and spontaneous as well. We are no longer hung up on one thematic unit at a time. In general, we are free to determine what's best for our classes, and we're loving every minute of it.
So, too, are the children and parents. We told Yvonne to be sure to tell her mom what she had done in class. “You were fantastic! Tell her what a great job you did.” Yvonne gave us a smile and a shy look as we walked away, but we could hear mother and daughter moving into an animated conversation. When we saw her mother the next day, we explained how we saw the game and assessed its value. We agreed that the hour spent on the train game was right on track.
Ana Maria Andrade and Delia Hakim, who teach at Ochoa Elementary School, can be reached through the Educational and Community Change Project, The University of Arizona, College of Education Annex, 1415 N. Fremont Ave., Tucson, AZ 85719.