Using games to understand children's understanding.

When I was an elementary school student, my teachers often allowed us to play games on Fridays as a reward for good behavior. Now, as a teacher myself, I use games for evaluation and instruction.

Playing games with children is a way of holding conferences with them. Just as I usually learn more about a person from a one-on-one conversation than from meeting him or her in a group, so do I usually learn more about children's thinking from individual conferences than through their class participation or even from examining their work. I can ask children during interviews to describe their thinking if it is not obvious to me, or to verify their thinking if I believe I already understand.

While written work can provide important documentation and a way of representing thinking, it does not provide the same insights into children's thinking as a conversation does. When we are playing a game and a child gives an answer, I often ask, "How did you know that?" When the child explains his or her thinking, I learn things about the child's mathematical understanding that I would not know if I only listened to class discussions or simply checked for correct answers when grading papers.

Playing the Games

When reviewing addition and subtraction with my 3rd-graders at the beginning of the year, I begin by teaching the game "O'No 99." This addition game uses cards numbered two through ten, a few "minus ten" cards, and several other cards such as "Hold" and "Double Play" to make the game more interesting. Keeping four cards in their hands at a time, students play one card each turn, adding that card value to the total that has accumulated. The object of the game is not to be the player who pushes the total over 99.

While playing with Stacey, I noticed that she counted up "ten more" each time she put down a "10" card. For her, 36 was 26 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. From this I realized that Stacey did not yet fully understand place value. In the same situation, Sara immediately said, "36," with conviction and, likewise, when she played a "-10" card she immediately said, "16."

Both Stacey and Sara made A's on tests, but their mathematical understandings obviously differed. I would not have learned about the extent of their differences by simply listening to them give the correct answer in a class discussion, or by checking their papers.

When the total reached 89, Jeff had a "3," a "5," a "10" and a "-10" in his hand. After considering all the possibilities, he decided to put down the "10" card. A less advanced student might have put down the "3" card as the safest and easiest number to add. Other children get nervous and use a "minus" card when the total is a large number. Jeff's strategy was to push the total as high as possible without going over 99, thus making it harder for his opponents to win. He knew the "10" card would not push his total over 99, and saved his "-10" card for a time when he might really need it.

When the total was 36, Elizabeth considered putting down an "8" card, saying, "Four more will be 40, and four more will make it 44." Michael agreed, but used different reasoning. "If I add ten, that makes it 46, and eight is two less than ten, so the answer is 44." When I ask children, "How did you know that?" I invariably learn more about their mathematical thinking.

Asking the Right Questions

Before I studied how learning actually takes place, I used to spend numerous hours checking students' mathematics work to diagnose their strengths and weaknesses. I thought that process let me know who could add, subtract, multiply and divide, and who was thinking and making sense of mathematics. I did not know, however, whether the children got the answer by counting on their fingers. Do they just know that 46 + 10 = 56, or do they still count out 10 more every time? Is the 6 in 67 a 60 or just 6 for them? Are they thinking about whether their answers make sense, or have they followed a procedure they may not necessarily understand? I wanted to know how my students really thought about mathematics.

The questions I asked my students changed the more I learned about constructivist theory. Knowing that as learners we proceed from what we already know to figuring out what we do not know, I became more interested in finding out about my students' current understandings. My challenge came in figuring out how and what to ask so that I would know what my students really understood.

During mathematics lessons, I used to ask mostly "test questions" to which I already knew the answers (e.g., "What is the answer to 2 x 7?"). I wanted to "test" to see if my students also knew the answers. When I began to keep in mind that learning takes place as children relate new information to what they already know, more of my questions became "honest questions," or ones to which I honestly did not already know the answers (Cazden, 1988). Examples of honest questions include, "How did you know that? Did anyone approach this problem in a different way?" or "What do you think about the fact that Mary Ann got a different answer?" The students' answers give me insights into their understanding about mathematics, which I then use to guide my teaching.

Understanding Children's Understanding

Constructivist theory has also changed what I look for in students' thinking. It is difficult to understand someone else's thinking, especially when that thinking is different from your own. Common patterns are present, however, in the way that children solve problems. Once you recognize a thinking pattern, it is easier to understand a child's thinking because you have heard it before.

In learning multiplication, for example, children advance from thinking in terms of repeated addition to thinking about combining groups of numbers, and finally to knowing multiplication combinations. In solving the problem 6 x 8, children's thinking develops from viewing the problem as adding 6 eight times, to adding four 6s and doubling it, to finally knowing 6 x 8 = 48. Knowing about patterns in the development of thinking helps me to look for them in my students and, consequently, anticipate and understand my students' understanding. Playing games with my students allows me to observe their mathematical thinking in action and to diagnose how they really think while they are making sense of mathematics.

I frequently shift back and forth between assessment and instruction when I am playing games with children. If my purpose is primarily assessment, I ask the children follow-up questions to make sure I really understand what they know. When I use games for teaching purposes, I ask questions as we play about alternative strategies. By offering counter-suggestions and playing the same game over a period of time, I learn about children's certainty in their thinking, and I can observe the development of their thinking over time.

In September, when Amanda played "O'No 99" she added each number as a series of ones (e.g., 25 + 7 was 25 + 1 + 1 + 1 + 1 + 1 + 1 + 1). Knowing about other children's more sophisticated adding strategies, I told Amanda that "another kid" broke down the numbers to find the easiest way to get a multiple of 10, because tens were easy numbers to think about. Then, she would add on what was left. This "other kid" would think about 25 + 7 by breaking the 7 up into 5 and 2, 25 + 5 = 30, and 2 more makes the answer 32. After offering this suggestion, I asked Amanda what she thought about the "other kid's" thinking. (I do not offer the alternative strategy as one of mine to prevent students from thinking they are supposed to mimic it.) While Amanda found it interesting that she and the "other child" had arrived at the same answer, she continued to use her 1 by 1 strategy.

As the game continued, I kept offering other suggestions, including showing special interest in the outcome of adding 10s. When the total was 57, for example, Amanda put down a "10" card and counted up 10 to 67. I "noticed" out loud that when she added 10, the total on the table went from 57 to 67. (Amanda knew how to count by tens, but she did not relate this information to addition of tens.) Counter-suggestions did not change Amanda's thinking at first. Over time, however, as Amanda played games, investigated many large numbers and participated in class discussions about adding, she began to think more about tens and place value.

In November, when Amanda and I played O'No 99 again, she put a "9" card on the table when the total was 56. She said, "56 + 10 = 66, so since I'm only adding 9, it's 65." Amanda was no longer using a 1 by 1 counting strategy, and it was clear that her sense of place value had developed to the point that she could now also add numbers close to 10. We progress from what we know to figuring out what we do not know. In November, I learned that Amanda knew how to add 10s to figure out what she did not know.

Another game I use for assessment and instruction is called "Multiplication War" (Kamii, 1994). It is played with a regular deck of cards and two or more players. Players split the deck, turn their stack of cards face down, turn up two cards from the top of their stack and multiply the two numbers together. Then, they compare the products. The player with the greatest product takes all the upturned cards on the table. The object of the game is to hold the most cards at the end of the game [ILLUSTRATION FOR PHOTO OMITTED].

When I play this game with children, I ask them to "think out loud" as they figure out their answers. John turned over a 9 and a 6, and Eric turned over a 4 and an 8. John said, "I know 9 x 5 = 45 because I counted by 5s, and then I added one more 9 to 45, so 9 x 6 = 54." Eric agreed with the answer, but thought about 9 x 6 as 10 x 6 = 60, and took away 6 from 60 to make 54. Another child might have explained that 9 x 3 = 27 so 27 doubled would answer 9 x 6. Other children, on the other hand, may simply "just know" that 9 x 6 = 54. Not only does playing this game allow me to assess and instruct children, it also provides an opportunity for students to share their strategies. This social interaction requires children to clarify their thinking as they explain it to others and challenges other children's understanding, pushing them to consider new strategies.

Eric said about his cards, "I just know 4 x 8 = 32 because I've done it so many times. I can prove it because 2 eights is 16, and 16 + 16 = 32." John agreed, but knew the answer was 32 because 4 x 4 is 16 and then, like Eric, he doubled 16 to get 32. A less advanced student might have counted one at a time by 4s, eight times (1 + 1 + 1 + 1; + 1 + 1 + 1 + 1; + 1 + 1 + 1 + 1 ...) or counted by 4s eight times ("4, 8, 12, 16, 20, 24, 28, 32").

After listening to these strategies, I knew that both boys were thinking multiplicatively (Clark & Kamii, 1996). They no longer thought of multiplication just as repeated addition, and they used what they knew to figure out what they needed to know. When a child has thought about certain combinations so many times that he "just knows" the answer, he has both memorized the combination and understood the underlying network of relationships.

This network of relationships (i.e., doubling 4 x 4 to figure out 4 x 8, or figuring out 9 x 6 by thinking about it as (6 x 10) - 6) can be relied upon when memory fails (Labino-wicz, 1985). If, for example, you learn multiplication combinations only by memorizing the answers, and you forget that 6 x 7 = 42, you cannot find the answer. If you know the logic of multiplication and how it works, however, and you remember 6 x 6, you can add one more 6 to figure out 6 x 7. Memorization is most useful when an underlying network of relationships is in place (Heege, 1985).

Conclusion

My former elementary teachers knew that we liked games. The teachers probably considered it an added benefit that some games allowed us to "practice our facts." My students also like games. Besides their intrinsically motivating character, games are also useful to me. They create situations that require children to consider and make mathematical relationships. They also provide opportunities for me to assess students' understanding in a way that gives meaningful information.

Learning takes place when thinking is modified (Steffe & Killion, 1989). One's thinking is challenged and new ideas are formed by exchanging points of view, both among students and between students and teacher. This new thinking is possible because something has been learned and old ideas have been altered. Teachers must create situations that require students to make sense of what they are doing, and to create an environment that encourages and supports the learning process. Good games are one tool to help meet this challenge.

References

Cazden, C. (1988). Classroom discourse. Portsmouth, NH: Heinemann.

Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 27, 41-51.

Heege, H. (1985). The acquisition of basic multiplication skills. Educational Studies in Mathematics, 16, 375-388.

Kamii, C. (1994). Young children continue to reinvent arithmetic: 3rd grade. New York: Teachers College Press.

Labinowicz, E. (1985). Learning from children: New beginnings for teaching numerical thinking. Menlo Park, CA: Addison-Wesley.

Steffe, L., & Killion, K. (1989). Children's multiplication. Arithmetic Teacher, 37, 34-36.

Ann Dominick is a Teacher, South Shades Crest School, Hoover, Alabama. Faye B. Clark is Assistant Professor, Samford University, Birmingham, Alabama.