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PRIMUS: Problems,Resources, and Issues inMathematics UndergraduateStudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20
UNIVERSITY STUDENTS'PROBLEM POSING ABILITIESAND ATTITUDES TOWARDSMATHEMATICSTodd A. Grundmeier aa Department of Mathematics and Statistics ,University of New Hampshire , Durham, NH,03824-3591, USAPublished online: 13 Aug 2007.
To cite this article: Todd A. Grundmeier (2002) UNIVERSITY STUDENTS' PROBLEMPOSING ABILITIES AND ATTITUDES TOWARDS MATHEMATICS, PRIMUS: Problems,Resources, and Issues in Mathematics Undergraduate Studies, 12:2, 122-134, DOI:10.1080/10511970208984022
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UNIVERSITY STUDENTS'PROBLEM POSING ABILITIESAND ATTITUDES TOWARDS
MATHEMATICS
Todd A. Grundmeier
ADDR ESS: Depar tm ent of Mathematics and Statistics, University of NewHampshire, Durham NH 03824-3591 USA.
ABSTRACT : This study exp lored the problem posing abilit ies and at ti tudes towards math emat ics of students in a university pre-calculusclass and a university mathematical proof class. A measure of attitudetowards mathematics and corresponding two sample t-test revealed asignificant (p = .001) difference in the at t itude levels of st udents in thetwo classes . A measure of problem posing ab ility explored studentsproblem posing ab ilit ies in both num eric and non- num eric contexts.Two sample t-tests showed no significant difference in the problem posing ability (numeric or non-numer ic) of st udents in the two classes. Amatched pair s t-test showed a significant difference in numeric posingversus non-numeric posing ability in both classes. Lastly there wasno corre lation between students attitudes towards mathematics andproblem posing abilit ies.
KEYWORDS: Problem posing, attitudes, und ergraduat e mathematics.
IN TRODU CTION
P roblem posing is becoming recognized in the United St at es as a necessar y component of mathemat ics teaching and learning [5,8,9,10]. Allowing st udents to pose their own mathematics problems can influence, amongot her things, attitudes towards mathematics, ownership of mathematics andmathematics achievement [3,10]. As st at ed by Silver , "contemporary constructivi st theor ies of teaching and learning requ ire that we acknowledge
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the importance of st udent -genera ted problem posing as a component of inst ructional activity [10, p. 19]." Researchers and educators have begun toincorporate problem posing into mathematics teach ing and learn ing [3,4,14].English and Grove st ud ied the implementation of a problem posing programin third, fifth and sevent h grade classrooms and showed that it is feasibleto incorporate pro blem posing into instruction at these grades [4] . Winograd and Higgins have ut ilized problem posing as a too l for interdisc iplina ryEnglish and mathemat ics classes [14].
Resear ch has also examined problem posing abilities ra nging from elementar y school st udents to middle school teachers [6,11,12]. Winograd hasshown that elementary school st udents are both willing and able to posemathemat ical problems that challenge themselves and classm at es [12,13].Leung and Silver showed that prospective elementary school teachers wereabl e to pose mathematical problems but in many cases their problems lackedmathematical complexit y [6]. As problem posin g is beginning to be incorporated into mathematics classrooms it is important to cont inue to documentst udents capabilit ies as problem posers .
Resear ch has also examined th e relationship between student s mathematics achievement and mathematics anxiety and between mathematicsachievement and at t itudes towards mathematics [2,7]. These results combined wit h pas t resear ch on problem posing have lead to the quest ion, "Isthere a relationship between students' problem posing ability and at t itudestowards mathematics?"
The resear ch presented here document s and ana lyzes the problem posingabilit ies and attitudes towards mathematics of st udents in two universitymathematics classrooms. The following aims motivat ed this research,
1. To recor d the problem posing abilit ies and attitudes towards ma thematics for both Pre-calculus and Mathematical Proof st udents andt ry to identify quantitative differences in problem posing abilit ies andat t itudes .
2. Make conclusions relat ed to how problem posing ability and attitudetowards mathematics differ between students in introductory and advanced mathematics classes.
The remainder of thi s pap er will focus on the resear ch methods undertaken in this project , results, implications of these results , and suggestionsfor future research.
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RESEARCH METHODS
This was an observationa l st udy wit h the int ention of examining problempos ing abilit ies and at t it udes towards math ematics of uni versity st udents .The resear ch was undertaken at a medium sized state uni versity in thenor th east and was done with approval from the institutions rev iew boardfor the use of human subjects. Data was collecte d in both a pre-calculusand ma thematical proof class and the students responded to measures oftheir at t it ude towards math ematics and prob lem posing ab ility.
Directi ons: Respond to th e st atements on a sca le from 1 to 5,1 being strongly disagree, 3 being neutral and 5 being st ronglyagree . Do not spend time thinking about each question in detail,please answer with your initial instinct [1].
1. I enjoy going beyond th e ass igned work and t rying to solve new problems in mathematics.
2. Mathematics is enjoyable and stimulating to me.
3. Mathematics makes me feel uneasy and confused .
4. I am interested and willing to use mathematics outside of school andon the job.
5. I have never liked mathematics, and it is my most dr eaded subject.
6. I have always enjoyed studying mathematics in school.
7. I would like to develop my mathematical skills and study this subjectmor e.
8. Mathematics makes me feel uncomfortable and nervous.
9. I am int erest ed and willing to acquire further knowledge of mathematics.
10. Mathematics is dull and boring because it leaves no room for personalop inion.
11. Mathematics is very interest ing and I have usually enjoyed courses inthis subject.
Figure 1. Sample statements (from the complete set of 21 statements) from the measureof attit ude towards mathematics.
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Materials
St udents completed a measure of attitude toward s mathemati cs based on theAiken At ti tude Scales [1]. The measure included both positive and negativestatements related to learning mathematics and the nature of mathematics.Aiken showed that this scale measures st udents enjoyment of mathematicsand the value they place on mathemat ics [1]. An example of the measurefollows in Figure 1.
Directions: Consider the possible combina tions of pieces of information given below and pose as many ma themati cal problemsas you can think of. Do not solve your problems and t ry to poseproblems that you think your peers may not be able to create .
Item #1: You have decided to purchase a computer for college.The new top of th e line laptop costs $2500. You have two optionsfor purchasin g th e computer, you can use your credit card , whichhas an annual interest rate of 13.99% or you can finance it throughth e University computer store for 48 months at $70 a month. Youhave saved $500, bu t you need to be able to pay for your booksnext semester.
Item #2: The University has decided to build a parkin g garagefor the use of students and staff. T he University has a maximumamount of land that they can use and also have a minimum number of faculty /staff spots and a minimum number of st udent spotsthat are needed at certain hours of the day. The university hasdone research th at shows that a fixed numb er of facul ty / staff anda fixed number of st udents arr ive at 8am and 12 noon. Also theuniversity is restricted by a fixed bud get for paving and genera lconstruct ion.
Figure 2. Measure of problem posing ability. Adapted from [6].
Students also compl eted a measure of problem posing ability based onpast work by Leung and Silver [6]. Leung and Silver jointly developedthe TAPP (Test of Arithmetic Problem Posing) to examine the problemposing ability of prospective elementary school teachers [6] . As part oftheir research Leung and Silver reported that "the TAPP functioned in areasonable way as a measure of arithmetic problem posing [6, p. 18]" andthey suggested that the TAPP could be utilized for other problem posingresearch and with other audiences. Based on these conclusions and the factthat all participants were familiar with arithmetic the TAPP was adapted
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for use in this study by making the sets of information more relevant to theparticipants.
Leung and Silver found that performance on the TAPP was affected bywhether or not numeric information was included in the given [6] . Therefore,in this study, students were asked to read two sets of given information andpose as many mathematical problems as possible from each. In Item 1in Figure 2 numeric information that could be used in posing problemswas given. Alt ernatively in It em 2 in Figure 2 no numeric information wasgiven to assist in problem posing. The measure did not ask students to solvetheir problems, but to focus solely on posin g problems . It was explained tostudents before problem posing began that they were to try to pose solvablemathematics problems. Also students were instructed that th ey could addinformation to ass ist in posing problems and they were asked to not e anynew information added along with the problem it was added to .
D a t a Collection
Dat a collect ion occur red on two occasions and the setting was a universityclassroom . Data was collected in a Pre-calculus class of eightee n studentsand a Mathematical Proof class of nineteen students. All students involvedconsented to being part of the research and willingly filled out both themeasure of attit ude and measure of problem posing ability. Student s inboth classes were given approxima te ly twenty-five minutes to fill out bothmeasures and were asked to pose as many mathematical problems as theycould in that amount of time.
The maj ority of st udents in the pre-calculus class were freshman in theCollege of Engineering and Physical Sciences, whereas the majority of students in the mathematical proof class were sophomores who had alreadydeclared mathematics and or computer science maj ors within the Collegeof Engineering and Physical Sciences. Thus the maj or difference in thesetwo groups was experience with mathematics, a year of calculus, and somepre-calculu s experience.
D a t a Analysis
The measure of at t it ude towards mathemati cs was given in the form of aLikert Scale with twent y-one statements for students to respond to on asca le from one to five (one being "st rongly disagree", 3 being "neut ral"and five being "strongly agree") . Before coding the data from this measurest atements were classified as either positive (i.e. 1,2,4,6,7,8,9 and 11 in Figure 1) or negative (i. e. questions 3,5,8 and 10 in Figure 1). Each student
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was given a score from one to five for each statement, one was awarded forstrongly disagreeing with a positive statement or strongly agreeing with anegative statement and five was awarded for strongly agreeing with a positive statement or strongly disagreeing with a negative statement. Studentsscores on this measure could range from 21 (one on all 21 statements) to105 (five on all 21 statements) with 64 implying a neutral attitude towardsmathematics.
The measure of problem posing ability was coded by awarding students apoint per st ep of mathematical problem posed. It took careful examina t ionto det ermine if each problem was solvable and, if so, how many ste ps (bythe authors judgment ) were necessary to solve it. Examples of thi s codingcan be seen in Figure 3. In Figure 3 the problems appear exact ly as posedby the students.
Problem ScoreExample 1: Why would it not be beneficial to ente r 0a four - year payment plan on a personal computer?Example 2: How many months would it take to pay 1off the computer if you used your credit card andpaid $70 a monthExample 3: If the university has determined th ey 2have a 5:1 student to faculty ratio for the need ofa parking spot, and they have a budget of 150,000,if they also determine that each parking spot willcost approximate ly 133.33 per spot, how many fac-ulty spots will become available from the building ofthe largest facility with the given bud get.
Figure 3. Examples of problem posing coding.
Example 1 in Figure 3 is not a solvable mathematical problem becausethere is missing informat ion and received a score of O. Example 2 is a onest ep problem; solve for the numb er of months in the proper formula forcont inually compounded interest . In this coding arithmet ic st eps were notconsidered separate steps. On the other hand Example 3 would be solved byfirst determining th e number of spots possible within th e given bud get andthen equating this with the student faculty ratio to determine th e numberof faculty spots th at will be built , a two step problem.
Aft er coding students problems the points were tot aled and each studentreceived a score for numeric problem posing (from Item 1 in Figure 2) , ascore for non-numeric problem posing (from Item 2 in Figur e 2) and a total
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problem posing score (numeric plus non-numer ic). It was these scores thatwere used in data ana lysis.
Afte r data collect ion and data coding were comp leted a statistical software package was used to ana lyze the data. Two sample t-test s were used tocompare means between classes on the at t itude measure and the numeric,non-numeric, and total score from the problem posing measure.
Results
T he data collected was ana lyzed between classes to look for differences inproblem posing abilities and attit udes towards math emat ics. The discussion of data ana lysis between classes will begin with the means , standarddeviations and 95% confidence int ervals for the means (see Fig ure 4) foreach of the four factors involved in this research; attitude , numeric prob lempos ing, non-numeric problem posing, and total prob lem posing .
Attit ude by ClassLevel Number Mean Std Dev St d Er r Mean Lower 95% Upper 95%P re-calc 18 75.1111 13.2704 3.1279 68. 761 81.461Proof 19 88 .7368 9.6023 2.2029 84 .265 93.209
Numeric Posi ng by ClassLevel Number Mea n Std Dev Std Er r Mean Lower 95% Upper 95%Pre-calc 18 3.38889 1.64992 0.38889 2.5994 4.1784Proo f 19 3.52632 2.14394 0.49 185 2.52 78 4.5248
Non-Nume ric Posi ng by ClassLevel Number Mean Std Dev Std Err Mean Lower 95% Upper 95%Pre-calc 18 2.22222 1.76754 0.41661 1.3765 3.0680Proof 19 2.21053 1.96013 0.44968 1.2976 3.1234
Posing Total by ClassLeve l Number Mea n Std Dev Std Err Mean Lower 95% Upper 95%Pre-calc 18 5.6 1111 2.45282 0.578 14 4.4374 6.7848Proof 19 5.73684 3.6944 7 0.8475 7 4.0162 7.4575
Figure 4. Means and standard deviations for data collected.
i-tests
Two samp le t-t ests were performed to compare the means, bet ween classes,of the four factors stated previously, at the 95% confidence level. For eachtest the null hyp oth esis tested was that the mean of the pre-calculus class
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equals the mean of the mathematical proof class. The results of the fourtwo sample t-tests (attitude, numeric posing, non-numeric posing, and totalposing) only produced a rejection of the null hypothesis on the measureof attitude (p = .001). Therefore it can be concluded with greater than99% confidence that th e two classes studied had different attitudes towardsmathematics, with the mathematical proof class having a better attitudeas a whole than the pre-calculus class. But with regards to the measureof problem posing ability it cannot be concluded that the classes abilitiesdiffer .
Pre-calc
-0.20290.0612
Numeric Posing Non-Numeric Posing Total AttitudePosing
0.0409Numeric PosingNon-Numeric Posing 0.0409Posin g TotalAttitude -0.2029 0.0612
partialled with resp ect to all other vari abl es
Proof
-0.45 300.5196
Numeric Posin g Non-N umeric Posing Total AttitudePosing
0.7076Numeric PosingNon-N umeric Posing 0.7076Posing TotalAttitude -0.4 530 0.5196
partialled with respec t to all ot her variables
Figure 5. Partial Correlations between classes
Correlations
Figur e 5 shows the partial corre lat ions for th e two classes and the fourfactors being discussed. Partial correlat ions are reported here becau se totalposing, num eric posing and non- numeric posing are dir ectly relat ed andpar ti al corre lat ions account for this relationship .
As can be seen by examining Figure 5 there ar e no significant corre lationsin the dat a collected in the pre-calculus class , th e largest being th e negativecorrelat ion (-0.203) between attitude and numeric posing ability. On th eother hand th ere are st ronger correlat ions in the data from the proof class,the strongest corr elation being between numeric posing ability and nonnumeric posing ability (0.708). From this it can be concluded that in both
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classes students were able to pose more problems from the item includingnumeric information.
Matche d Pairs
In all the reported results parametric methods of data analysis have beenutilized , under the assumption that the dat a follows a normal distribution .Normal quantile plots using th e statisti cal software helped justify the normality assumption.
A mat ched pairs z-t est was performed on the difference between numericproblem posing ability and non-numeric problem posing abili ty for eachclass. This test was conducted with the null hyp othesis that the meandifference between the non-numeric and numeric problem posing score waszero in each class. This test in the pre-calculus class resulted in a p-value(p = .053) and in th e proof class a p-value (p = .005). Thus the nullhypothesis would be rejected in the pre- calculus class with greate r than94% confidence and in the pro of class with grea te r than 95% confidence.
DISCU SSION
Attitude Towards Mathematics
The mean on the measure of attitude towards mathematics was 88.74 forthe mathematical proof class and 75.11 for the pre-calculus class . It canbe concluded with greater th an 99% confidence th at in thi s situation themathematical proof students have a better att itude towards mathematicsthan the pre-calculus students, with both being higher than neutral. Thisresult may be explained by the fact that the mathematical proof studentshave had more exposure to mathematics, have alr eady decided on majors inmathematics or a related field, and may appreciate the value of mathematicsmore.
Problem Posing Ability
A hypothesis of this pro ject was that as students gain experience with mathematics their attitude towards mathematics increases and in turn they become better problem posers . Thus, a statistically significant difference inproblem posing ability between the two classes was expected. Instead thet-tests for numeric problem posing ability (p = .83), non-numeric problemposing abi lity (p = .98) and total problem posing ab ility (p = .90) result in
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clear nonrejections of the null hypothesis that the mean of the two populations are different. In fact as can be seen in Figure 4 the means for thepopulations with respect to all three factors are almost identical.
Students are not often asked to pose their own mathematical problems,in not only univers ity settings, but throughout their educational experience.Therefore students with any level of undergraduate mathematics experiencehave similar, limited, problem posing experience. Neither pre-calculus normathematical proof student s have a frame of reference for where to beginor how to pose mathematics problems when they are confronted with aproblem posing task. As shown in Figure 4 both populations were only abl eto pose, on average, problems tot aling 6 steps in approxima tely twenty-fiveminutes.
The combination of these results may imply the necessity to begin incorporating more problem posing early into mathematics classrooms to allowstudents to think about mathematics from this perspective and to engagethem more in mathematics. Future mathematicians , engineers and computer scientist s will sure ly be confronted with problem posing tasks in theirprofessional activit ies.
Another interest ing result from this ana lysis is that both classes showeda statistically significant difference in their ab ility to pose problems froma numeric versus non-numeric set of given inform ation. This may implythat students are able to perform problem posing tas ks better in less openended situations. It has been a goal of mathematics educatio n reform to increase student s performance in solving open-end ed mathematics problems[8,9] . Increasing students abilities to pose problems in open-end ed situations may influence their problem solving ability. This again may implythe need to incorp orat e more problem posing into mathematics classroomsand curriculums as future engineers, mathematicians, and computer scient ists will have to pose and solve problems in open-ended situations in theirprofessional activit ies.
Correlations
It was a hypothesis of this research that there may be a correlation betweenthe problem posing ability and attitude towards mathematics of universitystudents. However no significant correlation was found in this research andit seems that predicting problem posing ability based on attitude level isdifficult. This may also be a result of all of the students lack of experienceposing mathematical problems. Problem posing is such a new idea to moststudents that their performance may be based on their first instincts andtheir attitude towards mathematics may be less likely to influence their
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per formance. T his may not be the case with resp ect to problem solvingab ility as students may reflect on th eir past exper iences and t hus th eirattit ude may be more likely to play a role as they already see themselvesas either good or bad problem solvers [7] . It may be that as st udents gainexperience with probl em posing their attitude towards mathematics willshed light on their problem posing ability or vice versa .
There were a few interesting results with regards to correlations in themath ematical proof class. A positive correlat ion (0.7076) between numericand non-numeric problem posing ability (see Figure 5) implies tha t st udentsin the proof class were either able to pose problems in both contexts orhad an overall difficul ty with problem posing. If they could pose numericproblems they could pose non-numeric problems and vice versa. T here wasalso a posit ive correlat ion between non-numeric posing score and attit ude(0.5196) . This may beg in to justify the hypoth esis that as st ude nts havemore experience with mathematics they begin to become bet ter problemposers in more open-ended situ ations and that engag ing st udents in problemposing act ivit ies may influence their attit ude towards mathematics.
POSSIBLE FUTURE DIRECTIONS
A few questions for resear ch rela ted to problem posing abilit ies and attit udestowards mathematics arise from the resul ts of this study. Will the incorporation of problem posing into university mathematics classrooms influencest udents att it udes towards mathemati cs, prob lem posing ability, and problem solving ability? Would similar results have been found if th e measureof problem posing was changed to a measure of problem solving? If so whatare the impli cations? Would the attit udes of pre-ca lculus students resembl ethose of st udents with more mathematics experience if they were exposed toprob lem posing in the pre-calculus curriculum? Is the cor relation betweennon-numeric problem posing ab ility and attit ude towards mathematics thesam e in other adva nced mathematics classes?
These are just a few questions to begin th e explorat ion of problem posingin uni versity mathematics teaching and learning. Slowly problem posing isbecoming recognized as a necessary comp onent of mathematics teachingand learning, hopefully the resear ch reported here and these questions canbeg in a dialog about its imp ortance.
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REFERENCES
1. Aiken , L. R. 1974. Two sca les of a t t it ude towards mathem atics. J ournal for Research. in Ma th em atics Ed ucation . 5: 67 71.
2. Brassell, A., S. Petry, and D. M. Brooks . 19S0. Abili ty grouping,mathem atics achievement , and pupil attit udes towards mathemat ics. Journa l f or Researcli in M ath em ati cs Education. 11: 22-2S.
3. Brown , S. 1. and M, 1. Wal ter. 1993. Problem posing: Refl ect ions andapplications. Hillsdale NJ : Lawr ence Erlbaum Associates.
4. English, L. D. and K. Grove. 1995. Problem posin g ill middle-schoolclassrooms. Pap er presented at t he Annual Co nference of thc AmericanEducational Research Association , San Dicgo, CA.
5. Kilpatrick, J . 19S7. Problem formul ation : where do goo d problemscome from ? In A. H. Schoenfeld (Ed .) Cogni tive sci ence an d math em aticseducation. pp . 123 147, Hillsdalc N.J: Erlbaurn .
6. Leung, S. S., and E. A. Silver. 1997. The role of task format , ma thcmatics knowledge, and creative thinking Oil thc ari t hmetic problem posin gof prosp ecti ve elementary schoo l teachers , M atliem atics Educat ion Research.Jo urn al. 9(2): 5 24.
7. 1\la, X. 1999. A meta-analysis of thc relationship between anxietytoward mathematics and achievement in mathematics. Journal for Rcsearcliin Mathematics Educat ion . 30(5): 521-540 .
S. National Council of Teachers of Mathematics. 19S9. Curriculum andEvaluation Suuulards fo r School Mathematics. Rest on VA: Author
9. National Council of Teachers of Mathematics. 2000 . Principl es andStandards for S chool Mathematics. Reston VA: Author
10. Silver , E . A. 1994. On mathematical problem posing. For the Learning of Mathematics. 14(1) : 19 2S.
11. Silver , E. A. , and J. Mamona, 19S9. Problem posing by middleschool mathematics teachers. In C. A. Maher, G. A. Goldin, and R. B. Davis(Eds), Proc eedings of the eleven th annual meeting of the North AmericanChapter of the International Group for the Psychology of Mathematics Education .
12. Winograd, K 1992. What fifth graders learn when they write theirown math problems. Educational Leadership. 49(7): 64 67.
13. Winograd, K 1997. Ways of sharing student authored story problems . Teaching Children Mathematics . 4(1) : 40 47.
14. Winograd, K and KM. Higgins. 1994/1995. Writing, reading,and talking mathematics: One interdisciplinary possibility. Th e ReadingTeacher. 4S(4): 310 317.
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BIOGRAPHICAL SKETCH
Todd A. Grundmeier is a fourth year graduate student at the Universityof New Hampshire, pursuing his doctorate in mathematics education. Hisresearch interests include prob lem posing, attitudes towards mathematics,and beliefs about the teaching and learning of mathematics. He is currentlyworking on his dissertation research incorporating problem posing into amathematics content class for pre-service teachers.
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