The role of playing games in developing algebraic reasoning, spatial sense, and problem-solving.

The role of playing games in developing algebraic reasoning, spatial sense, and problem-solving. Introduction and Literature ReviewPurpose and RelevanceUsing games in the classroom to facilitate learning has been commonpractice among teachers for many years. Blum Blum? , L��on 1872-1950.French socialist politician who served as premier (1936-1937, 1938, and 1946-1947). He was imprisoned (1940-1945) by the Vichy government during World War II. and Yocum (1996) supportedinstructional game-playing in the classroom because it provided anexciting and motivating strategy for students to practice skills alreadylearned. They suggested that there are several benefits from usinginstructional games in the classroom. Games are naturally motivating andfun, games facilitate individualization individualization,n the process of tailoring remedies or treatments to cure a set of symptoms in an indiv-idual instead of basing treatment on the common features of the disease. of assessment and instruction,and games make the abstract concrete.As a teaching tool games helped students become better problemsolvers because playing games gave them a chance to work out problemsand develop strategies for solving problems in a non-threateningenvironment. (Klein Klein, Melanie 1882-1960.Austrian-born British psychoanalyst who first introduced play therapy and was the first to use psychoanalysis to treat young children. & Freitag, 1991; Olson Olson may refer to: Olson (constructor), a former racing car constructor Olson Software Olson database, also known as zoneinfo database Sigurd Olson Environmental Institute Olson (surname), people with the given name Olson & Platt n. 1. (Mining) See Lodge,n. os> , 1992 ascited in Blum & Yocum, 1996). Playing games provided opportunitiesfor students to invent and test various strategies and procedures forsolving problems. Kamii, Lewis and Livingston Livingston,family of American statesmen, diplomats, and jurists.Robert R. Livingston (1654–1728)Robert R. Livingston, 1654–1728, b. (1993) stated, "Whenchildren invent their own problem-solving problem-solvingn → resoluci��n f de problemas;problem-solving skills → t��cnicas de resoluci��n de problemasproblem-solvingn → strategies, they do not haveto give up their own thinking, their understanding of [the concept] isstrengthened and they develop better number sense" (p. 201). Thisalso afforded students time to test their theories and strategies alongwith providing practice in multi-step problem-solving. "Playinggames offers repeated use of ... strategies and invaluable practice ofskills already learned. Practice becomes more effective because studentsbecome active participants in their own learning" (Ernest Er´nestn. 1. See Earnest. , 1986;Rakes & Kutzman, 1982; Wesson Wesson may refer to, among other things: Wesson, Mississippi, a town in Copiah County Wesson, Arkansas, a township in Union County, Arkansas Wesson cooking oil, a brand now owned by ConAgra Foods, Inc. et al., 1988 as cited in Klein &Freitag, p. 303).Playing games in the classroom provided a forum for students tohave discourse with peers (Beigel, 1997; Wakefield Wakefield, estate, United StatesWakefield,family estate of George Washington, on the Potomac River, E Va.; part of theGeorge Washington Birthplace National Monument (see National Parks and Monuments, table). , 1997). Theydiscussed options, strategies and solutions and gained insight andunderstanding from each other as well. Wakefield further proposed thatthe social interaction during the playing of games not only helpedstudent understanding, but played a large role in the game etiquette etiquette,name for the codes of rules governing social or diplomatic intercourse. These codes vary from the more or less flexible laws of social usage (differing according to local customs or taboos) to the rigid conventions of court and military circles, and they offollowing rules and fair play (1997).Most studies in the field of using instructional games in theclassroom focus on students with special needs as their subjects. Blumand Yocom mentioned three such studies that yielded positive results(1996). Beattie Beattie is a surname, and may refer to: A. L. Beattie, pioneering Chief Mechanical Engineer of the New Zealand Railways Department. Ann Beattie (1947-), American writer Craig Beattie, an Scottish footballer. and Algozzine (1982) found that students with milddisabilities who practiced math facts and played instructional mathgames were on-task about 20% more than their peers and also receivedhigher grades. Delquadri, Greenwood Greenwood.1 City (1990 pop. 26,265), Johnson co., central Ind.; settled 1822, inc. as a city 1960. A residential suburb of Indianapolis, Greenwood is in a retail shopping area. Manufactures include motor vehicle parts and metal products. , Stretton and Hall (1983) found thatby using an instructional spelling game, learning-disabled students wereable to decrease their spelling errors equal to the level of theirnon-disabled peers. Mackay and Watson (1989) were able to showimprovement of communication skills with severely learning-disabledstudents by using instructional games.There were no studies found in which non-disabled students wereused as subjects. This study looked at such groups.QuestionsThe purpose of this study was to test the hypothesis that playingmath-related games played Games played (most often abbreviated as G or GP) is a statistic used in team sports to indicate the total number of games in which a player has participated (in any capacity); the statistic is generally applied irrespective of whatever portion of the game is contested. a role in developing students' ability tosolve problems involving algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind.[CACM 2(5):16 (May 1959)].2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements. reasoning and spatial sense.Researchers chose the following from the NCTM NCTM National Council of Teachers of MathematicsNCTM Nationally Certified Teacher of MusicNCTM North Carolina Transportation MuseumNCTM National Capital Trolley MuseumNCTM Nationally Certified in Therapeutic Massage Curriculum andEvaluation Standards for School Mathematics (1989), "spatial senseis an intuitive feel for one's surroundings and the objects inthem" to define spatial sense (p. 49). Having spatial sense meansunderstanding the relationships of objects, the sizes and shapes offigures and objects, and the direction, orientation and perspectives ofobjects (Liedtke, 1995). Based on these definitions, the following areexamples of the use of spatial sense. Students with spatial sense areable to manipulate manipulateTo cause a security to sell at an artificial price. Although investment bankers are permitted to manipulate temporarily the stock they underwrite, most other forms of manipulation are illegal. patterns and shapes or objects both physically andmentally in order to show an understanding of the properties of thatpattern, shape, or object. Some examples of the games used and theirnature follow. Playing the game Rush Hour[R] which involves manipulatingcars and trucks to create a path for removing a particular car, requiredthe use of spatial sense. The playing area consists of a limited numberof spaces from which no vehicles may be removed. A similar game playedby students was Stormy storm��y?adj. storm��i��er, storm��i��est1. Subject to, characterized by, or affected by storms; tempestuous.2. Seas[R]. Students also played Connect Four[R], agame where students used a vertical game board to attempt to line upfour checkers checkers,game for two players, known in England as draughts. It is played on a square board, divided into 64 alternately colored—usually red and black or white and black—square spaces, identical with a chessboard. in a row, either vertically, horizontally, or diagonallybefore their opponent succeeded in the same task. This game requiredstudents to manipulate the checkers physically in making moves, andmentally in planning strategies to create the desired pattern in orderto win the game. Students also needed to do multi-step problem-solvingto strategize strat��e��gize?v. strat��e��gized, strat��e��giz��ing, strat��e��giz��esv.tr.To plan a strategy for (a business or financial venture, for example).v.intr. about how to win each of the above games.Based on readings related to algebraic reasoning and theirexpertise, the researchers defined algebraic reasoning as the ability todevelop relationships, some abstract, between numbers and patterns andto describe, represent, and model these relationships. An example ofalgebraic reasoning is in playing the game Muggins[R]. Participants rollthree dice, for example 4, 3, and 5. Using each number once, playersmust generate a number between 1 and 36. Players use any combination ofthe four operations in order to come up with a number. For example, 3 X4 + 5 can be used to generate the number 17.MethodsTwo fifth grade classes from the same school were involved in thisstudy. Both classes consisted of upper middle-class students who scoredabove the 30th percentile percentile,n the number in a frequency distribution below which a certain percentage of fees will fall. E.g., the ninetieth percentile is the number that divides the distribution of fees into the lower 90% and the upper 10%, or that fee level in math on the Stanford Achievement Test inApril 2000. The control group was a class of 26 students who were taughtby a teacher with 39 years of teaching experience. The experimentalgroup was a class consisting of 24 students whose teacher had 14 yearsof teaching experience. Both teachers used the same traditional text.The curriculum taught by both teachers throughout the school year wasthe same: addition, subtraction subtraction,fundamental operation of arithmetic; the inverse of addition. If a and b are real numbers (see number), then the number a−b is that number (called the difference) which when added to b (the subtractor) equals , multiplication multiplication,fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. and division of wholenumbers, decimals and fractions, as well as a unit in geometry geometry[Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts. . Theexperimental group teacher also used Mad Minute daily. The Mad Minute isa timed drill in which students attempt to complete thirty to sixtybasic facts in one minute. Success in completion and accuracy advancethe student to the next level of problems. A problem situation was alsoposed for five minutes at the beginning of class every other day in thisclass.The experimental class was divided into six groups of thestudents' choosing. The groups rotated rotatedturned around; pivoted.rotated tibiasee rotated tibia. through each of six stationstwice a week, spending about 30 minutes at each station. Most stationsconsisted of specific commercial games that require students to utilizealgebraic reasoning and/or spatial sense, based on the definitions andexamples previously stated. One station was devoted to shareware Software on the "honor system." The concept is that users try a product, and if they like it, they voluntarily pay a set registration fee or make a donation to the program's creator. There are tens of thousands of shareware programs; some fantastic, some awful. gamesfrom the Internet InternetPublicly accessible computer network connecting many smaller networks from around the world. It grew out of a U.S. Defense Department program called ARPANET (Advanced Research Projects Agency Network), established in 1969 with connections between computers at the . These games also required students to utilizealgebraic reasoning, spatial sense, or problem-solving. A complete listof the games used is included at the end of the article. The teachergave instructions about how to play each game. These games and Internetsites were also available for students to use during their free timethroughout the course of this study. This provided students withapproximately 100 minutes of optional game-playing time per week, inaddition to the required 60 minutes, for a possible total of 160 minutesper week. After a few weeks of play, some stations were combined becauseindividually the games didn't hold the students' interest forthe allotted al��lot?tr.v. al��lot��ted, al��lot��ting, al��lots1. To parcel out; distribute or apportion: allotting land to homesteaders; allot blame.2. time of play. Students helped decide which games werepaired.Ms. Lach used a pretest/posttest model with items generated fromassessments used by the school district and from Test Ready Plus[TM], aQuick-Study Program published by Curriculum Associates (1994). Sampletest 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 algebraicreasoning to develop relationships between numbers and patterns and todescribe, represent, and model those relationships. Other items chosenwere those requiring students to use spatial sense to show theirunderstanding of the relationships of objects, the size and shapes ofobjects, and the direction, orientation and perspectives of objects.Independently, the researchers chose and sorted all of the items intocategories of algebraic reasoning and spatial sense based on thedefinitions. The distribution of the test items can be found in Table 1.Reliability of this measurement tool was addressed by administeringthe test to a group of fifth grade students who were not involved in thestudy. Students took the test twice, three weeks apart, and thestudents' scores were unchanged.The researchers administered the pretest pre��test?n.1. a. A preliminary test administered to determine a student's baseline knowledge or preparedness for an educational experience or course of study.b. A test taken for practice.2. to both classes the firstweek of school and the same test was given as a posttest post��test?n.A test given after a lesson or a period of instruction to determine what the students have learned. at the end ofthis study 12 weeks later. A rubric RUBRIC, civil law. The title or inscription of any law or statute, because the copyists formerly drew and painted the title of laws and statutes rubro colore, in red letters. Ayl. Pand. B. 1, t. 8; Diet. do Juris. h.t. was used for scoring the pretestsand posttests. The rubric scoring level was 0 to 4 and differentiatedbetween those problems requiring an explanation and those that requirean answer only, as shown in Table 2.The regular math instruction that took place in Ms. Lach'sroom was more traditional in terms of skill development. The teachermodeled the skill, provided opportunities for guided practice, and thenhad the students do independent practice.FindingsA one-tailed t-test t-test,n an inferential statistic used to test for differences between two means (groups) only. This statistic is used for small samples (e.g.,N t-ratio, stu-dent's t. compared the overall average pretest scores ofthe control group and the experimental group to determine whether thegroups were different. With a probability level of 0.2756, there was nosignificant difference between the two groups. A one-tailed t-testtested the null hypothesis null hypothesis,n theoretical assumption that a given therapy will have results not statistically different from another treatment.null hypothesis,n that there was no difference between thecontrol group's pretest and posttest results (p<0.2496), and nodifference between the experimental group's pretest and posttestresults (p<0.01) as shown in Table 3. From the pretest to theposttest, the control group showed no significant difference. However,the difference with the experimental group was highly significant. Thisinformation supports the hypothesis that playing math games improved thestudents' abilities in solving problems involving spatial sense andalgebraic reasoning.An item analysis of the comparison of the mean pretest score to themean posttest score of the experimental group was performed, as shown inTable 4. This indicated that responses to items 1, 5, 6, 7, 9, 10, 11,and 12 were significantly different from the pretest to posttest, andthat items 2, 3, 4, and 8 were not. In studying the item analysis of thecontrol group, it was found that two items showed a significantdifference between the pretest and the posttest. They were item 12, witha 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 aspart of the design are stated here. Student enthusiasm for thegame-playing was very high at the outset and continued to be high for avariety of the games included in the study, as observed by the teacherand as shown in their desire to continue playing. The teacher also saw ahigh level of engagement during the games as compared to other times inclass. Some students had difficulty with some of the games because theyproved to be challenging. This was exhibited by students who becamefrustrated frus��trate?tr.v. frus��trat��ed, frus��trat��ing, frus��trates1. a. To prevent from accomplishing a purpose or fulfilling a desire; thwart: by frequent losses. Strategies were developed to enable thestudents to remain involved with the games for longer periods of time,and students were able to work through multi-step problems. Othersstruggled with social issues such as not feeling successful in front oftheir peers. This manifested itself when a parent contacted the teacherabout how the parent might do more with her child at home in order forhim to become more successful against his peers. Another sociallyconnected finding was that students learned to play fairly, to help eachother, and to communicate their understanding. Students were focused onwho 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 studentsspent more time explaining their reasoning and thinking rather thanbeing confrontational. It was also found that game-playing extendedbeyond the classroom and into the home. Several families contacted theteacher about the names of games and about where they could purchasegames in order to play them at home.DiscussionWhile the benefits of using games in the classroom seemed obvious,the researchers had no statistical evidence of their benefit prior tothe study. The posttest numbers between the two groups indicate that thegroups had very different levels of success in problem-solving withalgebraic reasoning and spatial sense after the 12 week period. Therewere a few items that students did not score differently on in eithergroup. 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. Items2 and 3 were area and perimeter The boundary of a system or network, which defines the inside and outside. It is typically determined by firewalls and addresses. See DMZ. problems. Item 8 was a number sentenceproblem. 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 7and 14. Rather than doubling to get 28, the majority of the studentsjust added seven to 14 to get 21. The majority of the students had thesame wrong answer after the implementation of the use of games becausethey only looked at one row to generate the next number in the pattern.While the study did not quantitatively measure the followingaspects of student learning, the teacher noticed several changes amongstudents throughout the study. Student enthusiasm was very high withregard 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 saidthat he only liked games that made him think. Another student said shewanted to play the "fun" games and still another said that heonly plays games on the computer now that he's older. Anotherstudent said she wasn't very good at math games, but that she wouldhelp out anyway. After a few weeks, some students exhibited frustration.They needed support in developing strategies that others had figuredout. In one particular instance, the teacher worked with the student andparent in order to help the student be more successful.Some additional benefits of using games beyond those alreadyreported in this study follow. Students learned new games, some of whichwere purchased by families for use at home. Students developed greaterconfidence in mathematics as a result of being successful at playinggames. Students also learned how to strategize and how to solvemulti-step problems, as well as how to communicate about theirstrategies. An example of this was in playing Muggins[R]. In order toget more points, students needed to place marbles in a row, onsuccessive turns. Thus, placing a first marble in a good position wouldincrease the likelihood of being able to place successive marbles nextto it. This required them to look ahead to future turns and to domulti-step problem-solving in their play. The teacher saw evidence ofstudents developing the ability to place marbles in better strategicpositions as they gained more experience with the game. Students alsolearned to play fairly with each other and learned how to help eachother as well as ask peers for help.Although the data supports the hypothesis, there are points toconsider in reading these results. For example, were there other factorsthat contributed to the success of this study? Perhaps the use of theProblem of the Day, which provided problem-solving practice, was acontributing factor to the positive outcome of this study. Woulddifferent games yield different results? Would using computer softwareprograms specific to algebraic reasoning and spatial sense affect theresults differently? If the study took place over a longer period oftime, would the results be different? What would we learn about studentunderstanding if other assessments were done?The results of the study support what was thought by many teachersfor years. Playing games provides an avenue for students to developalgebraic reasoning, spatial sense, and problem-solving. Additionalfindings were that students were more motivated mo��ti��vate?tr.v. mo��ti��vat��ed, mo��ti��vat��ing, mo��ti��vatesTo provide with an incentive; move to action; impel.mo and more involved whenlearning took place through game-playing. Students were challenged tothink about strategizing and multi-step problem-solving. They were alsomotivated to discuss their thinking with peers in order to improve thegame-playing for all involved. These results along with those above areincentives for further exploration of the learning that takes place whenstudents play games in mathematics.Games used in study:Connect Four [Game]. (1990). Milton Bradley This article or section is written like a personal reflection or and may require .Please [ improve this article] by rewriting this article or section in an . Company. Guess Who?[Game]. (1996). Milton Bradley Company. Izzi [Game]. (1992). Binary Meaning two. The principle behind digital computers. All input to the computer is converted into binary numbers made up of the two digits 0 and 1 (bits). For example, when you press the "A" key on your keyboard, the keyboard circuit generates and transfers the number 01000001 to the ArtsCorporation. Mastermind [Game]. (1998). New York New York, state, United StatesNew York,Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : Pressmen ToyCorporation. Muggins [Game]. (1990). Old Fashioned n. 1. A cocktail consisting of whiskey, bitters, and sugar, garnished with with fruit slices and often a cherry.Noun 1. old fashioned - a cocktail made of whiskey and bitters and sugar with fruit slices Crafts. Rush Hour[Game]. (1996). Binary Arts Corporation. Stormy Seas [Game]. (1998).Binary Arts Corporation. Tangramables. (1987). Highland Park Highland Park.1 City (1990 pop. 30,575), Lake co., NE Ill., a suburb of Chicago on Lake Michigan; inc. 1869. It is a retail business and medical center for the North Shore area. : LearningResources. 24 [Game]. (1998). Suntex International Incorporation.Table 1. Test Item DistributionItem # 1 2 3 4 5 6 7 8 9 10 11 12Algebraic X X X X X XReasoningSpatial X X X X X XSenseTable 2. Scoring Rubric PROBLEMS PROBLEMS REQUIRING ANSCORE REQUIRING AN ANSWER WITH NOLEVEL EXPLANATION EXPLANATION4 Provides correct answer; Correct answer given explanation is correct and thorough, showing complete understanding3 Answer may be correct or N/A incorrect, but explanation shows some understanding2 Incorrect answer; N/A explanation shows little or no understanding1 Answer may be correct or Incorrect answer given, incorrect, but no explanation was given0 Problem not attempted Problem not attemptedTable 3. Pretest/Posttest Results X Pretest X Posttest P-levelControl group 24.36 25.20 0.2496Experimental 23.13 32.65 0.000029** GroupP-level 0.2756 0.00093**** P < .01.Table 4. Item Analysis of Experimental GroupItem# 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.REFERENCESBeattie, J., & Algozzine, B. (1982). Improving basic academicskills for educable educable/ed��u��ca��ble/ (ej��u-kah-b'l) capable of being educated; formerly used to refer to persons with mild mental retardation (I.Q. approximately 50�C70). mentally retarded Noun 1. mentally retarded - people collectively who are mentally retarded; "he started a school for the retarded"developmentally challenged, retarded adolescents. Education andTraining of the Mentally Retarded, 17(3), 255-58.Beigel, A.R. (1997). The role of talking in learning. Education,117(3), 445-51.Blum, H.T. & Yocum, D.J. (1996). A fun alternative: usinginstructional games to foster student learning. Teaching ExceptionalChildren, 29(2), 60-63.Delquadri, J.C.; Greenwood, C.R.; Stretton, K.; & Hall, R.V.(1983). The peer tutoring A peer tutor is anyone who is of a similar status as the person being tutored. In an undergraduate institution this would usually be other undergraduates, as distinct from the graduate students who may be teaching the writing classes. spelling game: a classroom procedure forincreasing opportunity to respond and spelling performance. Educationand Treatment of Children 6(3), 224-39.Kamii, C.; Lewis, B.A.; Livingston, S.J. (1993). Primaryarithmetic: children inventing their own procedures. Arithmetic Teacher,41(4), 200-03.Klein, J.D. & Freitag, E. (1991) Effects of using aninstructional game on motivation and performance. Journal of EducationalResearch, 84(5), 303-08.Liedtke, W.W. (1995) Developing spatial abilities in the earlygrades. Teaching Children Mathematics, 2(1), 12-18.Mackay, M. & Watson, J. (1989) Games for promotingcommunication. British Journal of Special Education, 16(2), 58-61.National Council for Teachers of Mathematics. (1989). Curriculumand 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.Tisa LachWebster Webster,town (1990 pop. 16,196), Worcester co., S Mass., near the Conn. line; settled c.1713, set off from Dudley and Oxford and inc. 1832. The chief manufactures are footwear, fabrics, and textiles. District SchoolsLynae SakshaugSUNY SUNY - State University of New York College at Brockport