The role of playing games in developing algebraic reasoning, spatial sense, and problem-solving
Introduction and Literature Review
Purpose and Relevance
Using games in the classroom to facilitate learning has been common practice among teachers for many years. Blum and Yocum (1996) supported instructional game-playing in the classroom because it provided an exciting and motivating strategy for students to practice skills already learned. They suggested that there are several benefits from using instructional games in the classroom. Games are naturally motivating and fun, games facilitate individualization of assessment and instruction, and games make the abstract concrete.
As a teaching tool games helped students become better problem solvers because playing games gave them a chance to work out problems and develop strategies for solving problems in a non-threatening environment. (Klein & Freitag, 1991; Olson & Platt, 1992 as cited in Blum & Yocum, 1996). Playing games provided opportunities for students to invent and test various strategies and procedures for solving problems. Kamii, Lewis and Livingston (1993) stated, “When children invent their own problem-solving strategies, they do not have to give up their own thinking, their understanding of [the concept] is strengthened and they develop better number sense” (p. 201). This also afforded students time to test their theories and strategies along with providing practice in multi-step problem-solving. “Playing games offers repeated use of … strategies and invaluable practice of skills already learned. Practice becomes more effective because students become active participants in their own learning” (Ernest, 1986; Rakes & Kutzman, 1982; Wesson et al., 1988 as cited in Klein & Freitag, p. 303).
Playing games in the classroom provided a forum for students to have discourse with peers (Beigel, 1997; Wakefield, 1997). They discussed options, strategies and solutions and gained insight and understanding from each other as well. Wakefield further proposed that the social interaction during the playing of games not only helped student understanding, but played a large role in the game etiquette of following rules and fair play (1997).
Most studies in the field of using instructional games in the classroom focus on students with special needs as their subjects. Blum and Yocom mentioned three such studies that yielded positive results (1996). Beattie and Algozzine (1982) found that students with mild disabilities who practiced math facts and played instructional math games were on-task about 20% more than their peers and also received higher grades. Delquadri, Greenwood, Stretton and Hall (1983) found that by using an instructional spelling game, learning-disabled students were able to decrease their spelling errors equal to the level of their non-disabled peers. Mackay and Watson (1989) were able to show improvement of communication skills with severely learning-disabled students by using instructional games.
There were no studies found in which non-disabled students were used as subjects. This study looked at such groups.
The purpose of this study was to test the hypothesis that playing math-related games played a role in developing students’ ability to solve problems involving algebraic reasoning and spatial sense.
Researchers chose the following from the NCTM Curriculum and Evaluation Standards for School Mathematics (1989), “spatial sense is an intuitive feel for one’s surroundings and the objects in them” to define spatial sense (p. 49). Having spatial sense means understanding the relationships of objects, the sizes and shapes of figures and objects, and the direction, orientation and perspectives of objects (Liedtke, 1995). Based on these definitions, the following are examples of the use of spatial sense. Students with spatial sense are able to manipulate patterns and shapes or objects both physically and mentally in order to show an understanding of the properties of that pattern, shape, or object. Some examples of the games used and their nature follow. Playing the game Rush Hour[R] which involves manipulating cars and trucks to create a path for removing a particular car, required the use of spatial sense. The playing area consists of a limited number of spaces from which no vehicles may be removed. A similar game played by students was Stormy Seas[R]. Students also played Connect Four[R], a game where students used a vertical game board to attempt to line up four checkers in a row, either vertically, horizontally, or diagonally before their opponent succeeded in the same task. This game required students to manipulate the checkers physically in making moves, and mentally in planning strategies to create the desired pattern in order to win the game. Students also needed to do multi-step problem-solving to strategize about how to win each of the above games.
Based on readings related to algebraic reasoning and their expertise, the researchers defined algebraic reasoning as the ability to develop relationships, some abstract, between numbers and patterns and to describe, represent, and model these relationships. An example of algebraic reasoning is in playing the game Muggins[R]. Participants roll three dice, for example 4, 3, and 5. Using each number once, players must generate a number between 1 and 36. Players use any combination of the four operations in order to come up with a number. For example, 3 X 4 + 5 can be used to generate the number 17.
Two fifth grade classes from the same school were involved in this study. Both classes consisted of upper middle-class students who scored above the 30th percentile in math on the Stanford Achievement Test in April 2000. The control group was a class of 26 students who were taught by a teacher with 39 years of teaching experience. The experimental group was a class consisting of 24 students whose teacher had 14 years of teaching experience. Both teachers used the same traditional text. The curriculum taught by both teachers throughout the school year was the same: addition, subtraction, multiplication and division of whole numbers, decimals and fractions, as well as a unit in geometry. The experimental group teacher also used Mad Minute daily. The Mad Minute is a timed drill in which students attempt to complete thirty to sixty basic facts in one minute. Success in completion and accuracy advance the student to the next level of problems. A problem situation was also posed for five minutes at the beginning of class every other day in this class.
The experimental class was divided into six groups of the students’ choosing. The groups rotated through each of six stations twice a week, spending about 30 minutes at each station. Most stations consisted of specific commercial games that require students to utilize algebraic reasoning and/or spatial sense, based on the definitions and examples previously stated. One station was devoted to shareware games from the Internet. These games also required students to utilize algebraic reasoning, spatial sense, or problem-solving. A complete list of the games used is included at the end of the article. The teacher gave instructions about how to play each game. These games and Internet sites were also available for students to use during their free time throughout the course of this study. This provided students with approximately 100 minutes of optional game-playing time per week, in addition to the required 60 minutes, for a possible total of 160 minutes per week. After a few weeks of play, some stations were combined because individually the games didn’t hold the students’ interest for the allotted time of play. Students helped decide which games were paired.
Ms. Lach used a pretest/posttest model with items generated from assessments used by the school district and from Test Ready Plus[TM], a Quick-Study Program published by Curriculum Associates (1994). Sample test items are shown in Figure 1.
[FIGURE 1 OMITTED]
Issues of validity were addressed in the following ways. Researchers selected items that required students to use algebraic reasoning to develop relationships between numbers and patterns and to describe, represent, and model those relationships. Other items chosen were those requiring students to use spatial sense to show their understanding of the relationships of objects, the size and shapes of objects, and the direction, orientation and perspectives of objects. Independently, the researchers chose and sorted all of the items into categories of algebraic reasoning and spatial sense based on the definitions. The distribution of the test items can be found in Table 1.
Reliability of this measurement tool was addressed by administering the test to a group of fifth grade students who were not involved in the study. Students took the test twice, three weeks apart, and the students’ scores were unchanged.
The researchers administered the pretest to both classes the first week of school and the same test was given as a posttest at the end of this study 12 weeks later. A rubric was used for scoring the pretests and posttests. The rubric scoring level was 0 to 4 and differentiated between those problems requiring an explanation and those that require an answer only, as shown in Table 2.
The regular math instruction that took place in Ms. Lach’s room was more traditional in terms of skill development. The teacher modeled the skill, provided opportunities for guided practice, and then had the students do independent practice.
A one-tailed t-test compared the overall average pretest scores of the control group and the experimental group to determine whether the groups were different. With a probability level of 0.2756, there was no significant difference between the two groups. A one-tailed t-test tested the null hypothesis that there was no difference between the control group’s pretest and posttest results (p<0.2496), and no difference between the experimental group's pretest and posttest results (p<0.01) as shown in Table 3. From the pretest to the posttest, the control group showed no significant difference. However, the difference with the experimental group was highly significant. This information supports the hypothesis that playing math games improved the students' abilities in solving problems involving spatial sense and algebraic reasoning.
An item analysis of the comparison of the mean pretest score to the mean posttest score of the experimental group was performed, as shown in Table 4. This indicated that responses to items 1, 5, 6, 7, 9, 10, 11, and 12 were significantly different from the pretest to posttest, and that items 2, 3, 4, and 8 were not. In studying the item analysis of the control group, it was found that two items showed a significant difference between the pretest and the posttest. They were item 12, with a p-level of 0.01913, and item 7, with a p-level of 0.02287.
Other findings related to this study, but that were not included as part of the design are stated here. Student enthusiasm for the game-playing was very high at the outset and continued to be high for a variety of the games included in the study, as observed by the teacher and as shown in their desire to continue playing. The teacher also saw a high level of engagement during the games as compared to other times in class. Some students had difficulty with some of the games because they proved to be challenging. This was exhibited by students who became frustrated by frequent losses. Strategies were developed to enable the students to remain involved with the games for longer periods of time, and students were able to work through multi-step problems. Others struggled with social issues such as not feeling successful in front of their peers. This manifested itself when a parent contacted the teacher about how the parent might do more with her child at home in order for him to become more successful against his peers. Another socially connected finding was that students learned to play fairly, to help each other, and to communicate their understanding. Students were focused on who went first and who might be cheating at the beginning of the study. As the study continued, going first became less of an issue and students spent more time explaining their reasoning and thinking rather than being confrontational. It was also found that game-playing extended beyond the classroom and into the home. Several families contacted the teacher about the names of games and about where they could purchase games in order to play them at home.
While the benefits of using games in the classroom seemed obvious, the researchers had no statistical evidence of their benefit prior to the study. The posttest numbers between the two groups indicate that the groups had very different levels of success in problem-solving with algebraic reasoning and spatial sense after the 12 week period. There were a few items that students did not score differently on in either group. Those items were 2, 3, 4, and 8. Students scored well on items 2, 3, and 8 before and after the implementation of the use of games. Items 2 and 3 were area and perimeter problems. Item 8 was a number sentence problem. Clearly, the students understood the problems at the outset. Item 4 was a grid that had a pattern established in the first two rows. The pattern was a doubling one. The third row contained the numbers 7 and 14. Rather than doubling to get 28, the majority of the students just added seven to 14 to get 21. The majority of the students had the same wrong answer after the implementation of the use of games because they only looked at one row to generate the next number in the pattern.
While the study did not quantitatively measure the following aspects of student learning, the teacher noticed several changes among students throughout the study. Student enthusiasm was very high with regard to playing games. The students knew they were being studied, which might have been a factor in some of the enthusiasm.
Among student responses several were insightful. One student said that he only liked games that made him think. Another student said she wanted to play the “fun” games and still another said that he only plays games on the computer now that he’s older. Another student said she wasn’t very good at math games, but that she would help out anyway. After a few weeks, some students exhibited frustration. They needed support in developing strategies that others had figured out. In one particular instance, the teacher worked with the student and parent in order to help the student be more successful.
Some additional benefits of using games beyond those already reported in this study follow. Students learned new games, some of which were purchased by families for use at home. Students developed greater confidence in mathematics as a result of being successful at playing games. Students also learned how to strategize and how to solve multi-step problems, as well as how to communicate about their strategies. An example of this was in playing Muggins[R]. In order to get more points, students needed to place marbles in a row, on successive turns. Thus, placing a first marble in a good position would increase the likelihood of being able to place successive marbles next to it. This required them to look ahead to future turns and to do multi-step problem-solving in their play. The teacher saw evidence of students developing the ability to place marbles in better strategic positions as they gained more experience with the game. Students also learned to play fairly with each other and learned how to help each other as well as ask peers for help.
Although the data supports the hypothesis, there are points to consider in reading these results. For example, were there other factors that contributed to the success of this study? Perhaps the use of the Problem of the Day, which provided problem-solving practice, was a contributing factor to the positive outcome of this study. Would different games yield different results? Would using computer software programs specific to algebraic reasoning and spatial sense affect the results differently? If the study took place over a longer period of time, would the results be different? What would we learn about student understanding if other assessments were done?
The results of the study support what was thought by many teachers for years. Playing games provides an avenue for students to develop algebraic reasoning, spatial sense, and problem-solving. Additional findings were that students were more motivated and more involved when learning took place through game-playing. Students were challenged to think about strategizing and multi-step problem-solving. They were also motivated to discuss their thinking with peers in order to improve the game-playing for all involved. These results along with those above are incentives for further exploration of the learning that takes place when students play games in mathematics.
Games used in study:
Connect Four [Game]. (1990). Milton Bradley Company. Guess Who? [Game]. (1996). Milton Bradley Company. Izzi [Game]. (1992). Binary Arts Corporation. Mastermind [Game]. (1998). New York: Pressmen Toy Corporation. Muggins [Game]. (1990). Old Fashioned Crafts. Rush Hour [Game]. (1996). Binary Arts Corporation. Stormy Seas [Game]. (1998). Binary Arts Corporation. Tangramables. (1987). Highland Park: Learning Resources. 24 [Game]. (1998). Suntex International Incorporation.
Table 1. Test Item Distribution
Item # 1 2 3 4 5 6 7 8 9 10 11 12
Algebraic X X X X X X
Spatial X X X X X X
Table 2. Scoring Rubric
PROBLEMS REQUIRING AN
SCORE REQUIRING AN ANSWER WITH NO
LEVEL EXPLANATION EXPLANATION
4 Provides correct answer; Correct answer given
explanation is correct and
3 Answer may be correct or N/A
incorrect, but explanation
shows some understanding
2 Incorrect answer; N/A
explanation shows little or
1 Answer may be correct or Incorrect answer given,
incorrect, but no
explanation was given
0 Problem not attempted Problem not attempted
Table 3. Pretest/Posttest Results
X Pretest X Posttest P-level
Control group 24.36 25.20 0.2496
Experimental 23.13 32.65 0.000029**
P-level 0.2756 0.00093**
** P < .01.
Table 4. Item Analysis of Experimental Group
Item# Type X Pretest X Posttest p-value
1 AR 2.0434 2.8761 0.0116*
2 SS 3.1739 3.3043 0.3651
3 SS 3.8695 3.7391 0.2876
4 AR 2.6956 2.4347 0.2876
5 AR 1.7826 2.8695 0.0139*
6 AR 0.4782 1.7826 0.0004**
7 SS 0.8260 1.9565 0.0003**
8 AR 2.3913 2.7391 0.1678
9 SS 1.9130 2.9565 0.0012**
10 SS 1.7826 3.3913 0.0000**
11 SS 1.3913 2.1304 0.0019**
12 AR 0.7826 2.5652 0.0001**
Note. AR = algebraic reasoning; SS = spatial sense
*p < .05. **p < .01.
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National Council for Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. 49.
Test Ready Plus. (1994). MA: Curriculum Associates, Inc. Reprinted/Adapted by permission.
Wakefield, A.P. (1997). Supporting math thinking. Phi Delta Kqppan, 79(3), 233-36.
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