A problem-solving based mathematics course and elementary teachers’ beliefs
In both An Agenda for Action: Recommendations for School Mathematics in the 1980s (NCTM, 1980) and Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) the National Council of Teachers of Mathematics (NCTM) has identified problem solving as the most important topic in the mathematics curriculum. Thompson (1989) indicates that while “helping teachers learn ways to improve their students’ problem-solving competence and enhance their mathematical thinking” (p. 233) a number of hindrances have been encountered. She states:
main difficulties I have encountered have been related to the teachers views of what constitutes a problem in mathematics, their views about the nature of mathematics in general and of problem solving in particular, their attitudes towards problem solving, and their beliefs about what it means to do mathematics. (p. 234)
The beliefs these teachers hold about the nature of mathematics and what it means to do mathematics actually interfere with their ability to help students become successful problem solvers. Thompson’s statement suggests that before these teachers can help their students become successful problem solvers, they need to rethink some of their beliefs and perhaps gain a new perspective on mathematics. This article examines the impact a problem-solving based course has in prompting preservice elementary school teachers (PSTs) to rethink several commonly held limiting mathematical beliefs. Limiting beliefs are those seen as limiting or hindering mathematical performance. Attention is also given to determine if this approach to teaching is equally influential in challenging hindering mathematical beliefs for students at all levels of achievement. Lastly, consideration is given to identifying those aspects of the course that were particularly beneficial in helping to challenge beliefs.
Addressing the beliefs preservice elementary school teachers hold toward mathematics is critical to improving the mathematical performance of students, because these beliefs can have a strong influence on his/her approach to teaching mathematics. Hersh (1986) states that “one’s conception of what mathematics is affects one’s conception of how it should be presented” (p. 13). Raymond, Santos, and Masingila (1991) go even further and state that “teaching actions are directly influenced by teachers’ beliefs, and in turn those teacher actions have a tremendous impact on students’ belief systems” (p. 2). The work of Confrey (1984), Thompson (1984), and others also supports this position.
As evidenced above, what goes on in mathematics classrooms is directly related to the beliefs teachers hold toward mathematics. Schofield (1981) as well as Kloosterman, Raymond, and Emenaker (1993) indicate that elementary teachers have a significant role in students’ achievement, as well as their formulation of beliefs and attitudes toward mathematics. Karp (1991) found that teachers with negative mathematical attitudes actually “encouraged a learned helplessness response” (p. 267) from students, whereas the students of teachers with positive beliefs toward mathematics “explored and discovered mathematics meanings and interrelationships rather than passively receiving information” (p. 268).
PSTs’ beliefs clearly have an influence on their students, however are poor beliefs a problem with the PST population as a whole? Thompson (1984) has found that many of the beliefs preservice elementary teachers hold are very similar to severely math-anxious people attending math-anxiety clinics. No data is presently available about the mathematical beliefs of PSTs as a whole, however, Rech, Hartzell, and Stephens’ (1993) found that PSTs have poorer attitudes toward mathematics than the general college population. Although the Rech et al. study addresses mathematical attitudes, it is important to remember that “the role of beliefs is central in the development of attitudinal and emotional responses to mathematics” (McLeod, 1992, p. 579). If beliefs are central to development of attitudes as McLeod suggests, it seems reasonable to conclude that poor attitudes towards mathematics can also serve as an indicator of poor mathematical beliefs. Because the beliefs of a teacher can play a significant role in the formation of the students’ beliefs, as indicated by the research findings, the mathematical beliefs of teachers and PSTs cannot be ignored if positive mathematical beliefs are going to be fostered in students.
At present a lot of information exists that identifies a variety of negative beliefs and their impact, however, few studies have been conducted to determine the effect of non-traditional methods of mathematics education on students’ mathematical beliefs. This study intends to provide insight into the impact a non-traditional mathematics course, T104, has on students’ mathematical beliefs. This course is offered for preservice elementary education teachers (PSTs) at Indiana University (IU) and is taught via a problem-solving approach.
In addition to studying the impact of this problem solving approach on students’ beliefs, this study also focused on determining if this method of mathematics instruction has a greater influence on high- or low-achieving mathematics students. This information will provide insight as to whether a problem-solving approach to mathematics instruction is more effective with one of the two groups or perhaps of equal benefit to both groups. The insight gained can then be used to direct future mathematics curriculum development.
T104 was developed by the members of Indiana University’s Mathematics Education program area as part of a National Science Foundation (NSF) grant (TEI8751478) “to better prepare all prospective elementary teachers at Indiana University to teach with a problem-solving focus” (Kloosterman, 1992, p. 5). The course uses a problem-solving approach to help preservice teachers gain an understanding of the mathematics they will be required to teach in elementary schools. In a typical class, which meets three times a week for two hours, the PSTs spend about one and a half hours working in small groups on problems that are designed to help develop their conceptual understanding of mathematics. The areas covered in T104 are problem solving, numeration, operations, number theory, geometry and measurement, and rational numbers. In T104, understanding involves more than memorizing mathematical facts and equations and being able to apply them. It includes being able to actually describe, discuss, and apply the underlying concepts in the various content areas covered. The Jordan River Problem given below is an example of a problem that is in the geometry section.
Your group’s task is to choose a location on the Jordan River (the creek which runs through the campus) and determine the width of the river at this point without directly measuring across the river. All of the members of your group must stay on one side of the creek and all measuring must be done on that one side. Develop a strategy and make a plan before going outdoors.
Typically five or six sections of T104 are offered each semester by the Department of Mathematics. The course is normally taught by either mathematics or mathematics education graduate students. At the time of the study, all sections of T104 were taught by graduate students from the Department of Mathematics, none of whom had any part in the development of the course or any affiliation with the study other than allowing access to the classes for the collection of data.
All of the research took place on the main campus of Indiana University located in Bloomington Indiana. the study focused primarily on three questions. They
1. Does a problem-solving approach to mathematics help to challenge the limiting beliefs many PSTs hold toward mathematics?
2. Are efforts to challenge limiting beliefs more effective with high or low mathematical ability students?
3. What experiences in T104 proved to be most effective in challenging limiting beliefs and fostering the development of positive beliefs?
In order to determine the effects of T104 on the beliefs that the PSTs hold toward mathematics and themselves as doers of mathematics the following strategy was employed. A Likert-style survey was developed and tested the semester prior to the study. Part of the survey was based on survey questions from either Kloosterman and Stage (1992) or Schoenfeld (1989), other questions were written specifically for this survey. The survey was administered and modified several times to improve reliability. After each modification of the survey was complete, the survey was administered to PSTs in a different section of T104. In the end, reliabilities (Cronbach’s alpha) ranged from .71 to .85. In its final form, the survey contained three positively and three negatively stated positions for each of the five beliefs listed below.
1. If a math problem takes more than 5 – 10 minutes, it is impossible to solve (TIME).
2. Math is mostly memorization (MEMORY).
3. All problems can be solved using a step-by-step algorithm or a single equation (STEP).
4. Only geniuses are capable of creating or understanding formulas and equations (UNDERSTAND).
5. There is only one correct way to solve any problem (SEVERAL).
The survey also included several open-ended questions related to the usefulness of mathematics, the importance of memorization in mathematics, the reasonableness of expecting people to discover some mathematics on their own, and the amount of time that should be spent on a mathematics problem. A copy of the final survey is found in Emenaker (1993).
The survey in its final form was administered in all five sections of T104 in early January and late April to provide the students maximum exposure to a problem-solving approach to learning mathematics. Participation in the study was strictly voluntary for the instructors and students. After the January and April data were collected, paired sample t-tests were conducted to measure changes in the students’ beliefs from the start to the end of the semester.
In addition to the surveys, interviews were conducted with nine PSTs who completed the course. The purpose of these interviews was to gain further insight into possible changes in the students’ mathematical beliefs and their level of confidence in their mathematical abilities. Attention was also given to identifying particular aspects of T104 that were most influential in prompting change. All interviews were audio taped and later transcribed. The data was then chunked and sorted into themes using Glaser and Strauss’ constant comparison method (Lincoln & Guba, 1985).
The t-test results are listed in Table 1. (Tables omitted) These results indicate positive shifts on each of the belief scales although the shift in the TIME scale did not achieve significance. A mean score of 18 for a scale can be interpreted as “undecided” and scores higher than 18 indicate increasingly stronger agreement with a more positive position for a particular belief. The highest score possible for any scale is 30 and the lowest is 6.
The students demonstrated a significant increase on the SEVERAL scale. This gain suggests that T104 helped the PSTs realize that the a many ways to solve most problems. A significant improvement in the STEP scale was also observed, indicating that the PSTs shifted toward the belief that many mathematics problems can be solved without having to rely on memorized step-by-step procedures.
The significant increase in the MEMORY scores suggests that T104 helped the students see that memorization does not play as large of a role in mathematics as they might have expected coming into the course. Without a group of formulas and algorithms to memorize, the students were placed in a situation where something other than memorization was the key to success. In this case the key was understanding the concepts that were covered.
With the students placed in a situation where they were being asked to understand the concepts and not just rotely memorize formulas and algorithms, a number of them found that they were actually able to understand the concepts underlying many areas of mathematics. Experiencing an understanding of the concepts challenged their belief that the average person is unable to understand mathematics which in turn could have resulted in the significant increase in the UNDERSTAND scores.
There was no significant change in the last scale, TIME. The fact that the was no significant change in the TIME scale suggests that most of the students still held to the notion that no more than five to ten minutes should be spent on any mathematics problem. Further support for this was found in the short answer responses to the survey question, “If you understand the material, how long should it take to solve a typical homework problem?” The mean increase in time from January to April was only 0.42 minutes (SD 7.54, n = 101), which does not represent a significant (p
A general trend was found to exist between level of achievement and change in beliefs. All significant shifts in beliefs occurred among the higher level achievers. Significant changes (p
Careful examination of the interview transcripts resulted in the identification of a number of themes. Some of these themes are directly related to specific questions in the interview and some are the result of additional comments provided by the interviewees. Although there were many more, five of the themes a listed below.
* There is more than one way to solve a problem and some problems have more than one correct answer.
* Understanding concepts in mathematics is more important than memorizing procedures.
* It is reasonable to expect people of average mathematical ability to discover some mathematical concepts on their own.
* T104 caused the PSTs to consider and, in some cases, change their ideas of how to teach mathematics.
* Over half of the PSTs voluntarily indicated an increased confidence in their problem-solving abilities.
In addition to developing these findings, attention will be given to how T104 had an impact in shaping or influencing the changes that were observed.
Several Ways to Solve a Problem. When asked if the idea that there could be several ways to solve a particular problem would surprise them, the PSTs provided a diversity of responses. Four PSTs indicated that this would come as no surprise to them. One even made the comment, “I’ve always come up with different ways than my teachers” (106AC7). Two of the PSTs stated that they had some idea that there were different ways to solve a particular problem, but T104 served to reinforce this belief.
During the interviews three of the people indicated that the idea of there being more than one way to solve a mathematical problem was something new to them as a result of T104. One of the people for whom this was a new idea indicated that coming into the class she was definitely of the mindset that there is only one correct way to solve any problem as evidenced in her comment, “I was definitely, you do math this way. Definitely structured, very structured” (113S5). Two PSTs stated that they were considerably more comfortable with the idea of there being more than one way to solve a problem as a result of T104. The other four PSTs stated that they were comfortable with the idea that there was more than one way to solve a problem before taking T104.
When asked what aspect or aspects of T104 led to either a change in or strengthening of their belief that there are several ways to solve a problem, three points were identified. The first was the group work. Working in a group allowed the students to see, firsthand, people taking different approaches to the same problem. A second point that was brought up by two of the students was the exercise in the Geometry/Measurement chapter where the class is asked to develop five different proofs of the Pythagorean Theorem. A third aspect, mentioned by one person, was a required reading of an excerpt from Rebecca Brown Corwin’s (1989) article Multiplication as Original Sin. The respondent indicated that this article really helped her to see the importance of being open to alternate methods of solution as a teacher.
When asked about the possibility that there may be more than one correct answer to a problem, three of the PSTs indicated that this was realistic possibility in some cases. Rachel and Amy both said that the existence of more than one correct answer to a problem was something they had experienced prior to T104. Lauri indicated that it was something that she was first exposed to in T104 and now realized was a possibility. Surprisingly, several of the students interviewed indicated that they did not recall seeing problems with more than one answer even though T104 actually incorporates many such problems. From the information obtained from the interviews, it seems that the instructor played a key role in helping students overcome the misconception that problems have only one correct answer. If the instructor pointed out to the class that there were several correct answers to some of the problems, the students were more likely to have their belief challenged The differences in PSTs’ responses could be attributed to differences in instructors.
Memorization Versus Understanding. Because T104 focuses on teaching for understanding of concepts instead of memorization of facts, it was hoped that the course would influence the value PSTs place on memorization. The interviews revealed that seven of the nine students placed less emphasis on memorization after completing T104. In addition to diminishing the importance of memorization in the eyes of these students, T104 also helped to cultivate a greater value for understanding the concepts that undergird the formulas, theorems, and algorithms that would normally be committed to memory. Lauri’s comment provides good illustration of a change in position. She stated that:
I think before
I would have said it
was really important and now I don’t believe it is so important. … there are so many other things you can put in your head besides memorizing … the are better ways to do it
than just straight sit down and memorize …
by understanding the concepts
… it’s going to stick with them a lot longer than just straight memorization (901T5).
Students attributed changes in beliefs about memorization to several aspects of their T104 experience. A primary feature seemed to be the problems in the student packet. Six PSTs all voluntarily indicated that the problems were key to challenging their ideas about the importance of memorization. The reason they gave for the problems being instrumental in challenging their belief was that many of the problems were not solvable by application of a memorized formula. Depending on the mathematics they had memorized no longer worked and they were then ready to consider something new.
A second aspect of the course that helped to challenge their belief about the importance of memorization was the group work. Lisa observed that: one of my group members sometimes will know something, … sometimes she’ll just know it, but she won’t know why. Like she’ll know it’s right but she won’t really know it, it won’t be solid in her head. Seeing that helped strengthen it
the need to understand the concepts instead of just memorize the facts
, seeing how other people do math and not do math sometimes helps to strengthen it (018T8).
The group work provided the PSTs opportunities to observe other students trying to actually do mathematics and observe the shortcomings of using a strict memorization approach These types of observations throughout the semester seemed to have a cumulative effect of challenging their beliefs on the importance of memorization for success in mathematics.
Preparing for tests helped to further challenge the importance of memorization for some students. This is illustrated by Susan’s comment that:
I found that I didn’t have to study very hard. I didn’t have to go over things and memorize them if I could just know the idea of how to do them, then I was OK. It was a lot less stress (106S5).
One of the PSTs even related a situation from her life that, by her own admission, had a major influence in helping her to see the value in understanding concepts as opposed to memorizing formulas. She and a friend were trying to find the sum of the numbers on the face of a clock. Realizing it was similar to a summation problem she had derived a formula for two months earlier in T104, she was able to re-derive the needed formula and find the sum. Having the ability to re-derive a needed formula really surprised her and resulted in her placing a greater value on understanding concepts. It also resulted in additional confidence in her mathematical abilities.
A final observation that was made in analyzing the interview data with respect to memory versus understanding was that memorization and understanding of concepts seemed to be at opposite ends of a continuum. As the students came to place greater value on the understanding of concepts, they seemed to move away from placing as great of a value on memorization of facts, algorithms, and formulas. This observation also seems to be supported by the inter-scale correlational data found in Table 2. The inter-scale correlation between MEMORY and UNDERSTAND was computed to be .66 (p
Personal Confidence and Average People. It is not unusual for people to view mathematics as a discipline that needs extensive explanation. Mathematics is seldom viewed as an area where people can discover concepts, formulas, generalizations and theorems on their own. Because T104 places the students in a position of discovering many concepts an their own, a topic of interest was to determine the impact this would have on the PSTs’ belief that the average person is not capable of making mathematical discoveries. Four of the PSTs stated that they felt they could make some mathematical discoveries due to T104. Two PSTs indicated that coming into the course they were confident they could make some discoveries in mathematics, but that they were more confident in their own abilities to discover mathematics due to T104. The other three stated that they felt they could discover mathematical concepts on their own prior to taking T104.
Overall, T104 seemed to have a positive impact on the PSTs’ belief that they can discover some mathematical concepts on their own. All nine PSTs who were interviewed voluntarily reported an increased confidence in their abilities to discover mathematics on their own. For example, Amy stated, “there are some things that I could figure out on my own if I had to” (121A1). When asked if she would have been comfortable with the idea of discovering concepts on her own prior to taking T104 she said, “probably not. I always avoided math” (121B1).
In addition to exploring changes in their belief about personal abilities to discover mathematical concepts, the PSTs were also asked about the reasonableness of expecting average people, in general, to make their own discoveries of concepts in mathematics. They all indicated that they felt this was a reasonable expectation. Seven of the PSTs indicated that this expectation for others to discover mathematics on their own was a result of or strengthened by T104.
When asked to identify what it was about T104 that resulted in an increased confidence for discovering mathematics, two factors were commonly mentioned. One was the process of having to actually make their own discoveries every class and the other was the process of working in groups. When Rachel was asked what led to an increased confidence in her ability to discover some mathematical concepts, she stated:
The fact that I had to figure everything out on my own instead of being able to go up and ask someone to explain it to me, I had to figure it out. I never really thought I could do that (108C1).
As with the findings in the previous two sections, regularly encountering situations that conflicted with their belief was key to rethinking their belief.
The group work in T104 also seemed instrumental in several ways for challenging their beliefs. First, it facilitated the students being able to discover mathematics collectively that they could not have discovered individually, thereby increasing their confidence. Second, it provided the students with the opportunity to see other students making discoveries on their own. It was no longer a situation where a student could take the position of “Hmm, I can discover some mathematics on my own,” but a situation where “I can discover some mathematics on my own and so can the other people in my group.” This seemed helpful in prompting the students to rethink their position on average people being able to make mathematical discoveries on their own.
Changes in Teaching. When asked if T104 had an influence on the way they were going to teach, every student responded affirmatively. The number of changes were identified, each will be treated briefly. A first change was that several PSTs said they were going to be more empathetic to the difficulties their students will encounter. This was the result of difficulties experienced with elementary operations in other bases.
T104 also resulted in several of the PSTs reporting greater inclination to teach concepts instead of just facts. They expressed an interest in providing students with a more conceptually based understanding of the material instead of focusing on memorization of the rules. In addition to placing less emphasis on memorization, all of the PSTs indicated that they will expect their students to discover some mathematical concepts on their own. For eight of them this idea was directly a result of T104. They were able to see firsthand that discovering mathematics on your own is both reasonable and, in many cases, a better way to learn than rote memorization.
Some of the most interesting comments about changes in teaching came came when PSTs were asked about incorporating group work into their mathematics class. With regard to group work, Lisa stated:
I think without the class
… without having the experience or the knowledge, I wouldn’t have that
to choose from. Because it is like a new color, you think about all of the colors you know. If you don’t see a color and you never have seen it you wouldn’t choose that color because its nonexisting in your mind. So it’s kind of opened up another way of teaching (018X8).
Lauri was also very favorably disposed to incorporating group work into her own class after experiencing T104. She commented that:
It really helps to get input from other people and there were times when
would explain something and it would just completely go over my head and someone else from the group would say, “no this is what
meant.” It was just like the sun coming out, I understood it. So I think working in groups can help kids a lot because I’m not going to be able to reach every student with everything. A lot of times it helps them to learn from each other. It builds up confidence too (901Z6).
Another area of interest regarding changes in teaching pertained to allowing the children to struggle with a problem for a period of time instead of immediately providing the needed information. Five of the PSTs indicated that they were comfortable with this as a result of T104. This change came from personally experiencing a better understanding of many problems in T104 when allowed to struggle. For example, when Susan was asked why she was more comfortable with allowing her students to struggle, she referred back to her own T104 experiences:
When I was allowed to struggle with
a lot of problems that I had and I struggled with and was frustrated and just laid it to the side and
came back to it
, it would make sense. That was better than having someone tell us. … Because we figured it out for ourselves and then it really made sense… Thinking it out and doing it your own way or having somebody give you clues so that you can work through it and possibly make up your own problem works better because you have it straight in your head (106AN11, 106AP11, 106AQ11).
For many students T104 is the first place they have ever seen or used manipulatives to help solve problems. Four of the students interviewed indicated that their desire to use manipulatives in their own classrooms was the direct result of T104. It seemed that actually using the manipulatives to solve problems afforded the PSTs an insight into their usefulness that would not have been possible by just reading about them. In addition to allowing the PSTs to see that the manipulatives are useful, they were able to gain experience in how to use them in specific situations.
Aspects of T104 that were Influential. When asked to identify an aspect(s) of T104 that was instrumental in prompting observed changes, the PSTs consistently identified the group work and personal experience of regularly working problems in T104. The group work allowed the PSTs to see a variety of methods of solution from others in the group. It also allowed the PSTs to occasionally see the limitations of merely having a formula memorized, but not having a real understanding of the underlying concepts. This was illustrated earlier by a quote from Lisa.
Regularly working mathematics problems in T104 that conflicted with their beliefs was also important to challenging the PSTs’ beliefs. For example, one problem required the PSTs to develop five different proofs to the Pythagorean Theorem. Having five solutions, all of them equally correct, certainly conflicted with their belief that there is only one correct method of solution. The PSTs’ regular exposure to mathematics problems that conflicted with their existing beliefs had a significant impact in challenging their beliefs. It was a major part of every PST’s response when discussing facets of the course that were most influential in promoting change. The findings that personally experiencing mathematical situations that conflict with the students’ beliefs and small-up work are important in challenging students’ mathematical beliefs further confirms similar observations made by Buerk (1985) and Schram, Wilcox, Lanier, and Lappan (1988).
Several conclusions seem warranted based on the data. The first conclusion is that a problem-solving approach to teaching mathematics to PSTs does have a positive impact on the mathematical beliefs of the group as a whole. Significant (p
Although the increase for each scale is modest, it is important to remember that the beliefs studied formed over a period of many years. Obtaining significant positive changes in four of the five beliefs after only 15 weeks of class was unexpected. The changes were expected to be much smaller, with few of them reaching statistical significance because many researchers have suggested that changing the cognitive beliefs of students is difficult and requires an extended period of time (Clement, 1982; Driver, Guesnes, & Tiberghein; 1985, Owens, 1987; Sowder, 1986).
A second goal of this study was to determine if the presentation of mathematics from a problem-solving perspective would have a greater impact on high- or low-achieving PSTs. Analysis of the data by achievement level indicates that the high-achieving PSTs were more strongly influenced than low-achieving PSTs. Significant changes (p
The data suggest that this problem-solving approach to teaching mathematics had a positive influence on the mathematical beliefs of high-achieving PSTs, but no significant influence on the beliefs of low achieving PSTs. While the changes did not achieve statistical significance, it is interesting to note that PSTs in the “C” level had negative changes in the mean scale values for SEVERAL and STEP and the “D/F” level PSTs had negative changes in the mean scale values for SEVERAL and MEMORY. Presently, literature contains very little with respect to the impact a problem-solving approach to teaching mathematics has in challenging the mathematical beliefs of high- versus low-achieving PSTs, thus no comparison or contrast with current literature is possible.
The only belief that was studied for which no significant change in the mean scale values was observed for either the study group as a whole or by achievement level was TIME. This result was quite surprising. The PSTs worked on mathematics problems that required more than 15 minutes to solve virtually every class period, consequently a positive shift in this belief was expected. It is possible that the PSTs were thinking about the individual homework problems from the textbook, problems requiring less than ten minutes to solve, when responding to the questions related to TIME.
The third search question that was addressed focused on identifying the aspects of T104 that were instrumental in challenging the PSTs’ beliefs. The PST interviews led to the identification of two particularly important facets of the course. The first was group work. The second facet of the course that was consistently identified as important to challenging their beliefs was regularly working problems that conflicted with existing beliefs. The findings that both personally experiencing mathematical situations that conflict with the students’ beliefs and small-group work are important in challenging students’ mathematical beliefs further confirm similar observations made by Buerk (1985) and Schram et al. (1988).
While conducting the study several other issues arose, but were beyond the scope of the data collected and the resources available. One issue relates to the stability of the observed changes in beliefs over a period of time. It would be interesting to see if some beliefs are more resilient to reverting back than others. If the changes seem stable then a logical next step would be to determine if these changes are reflected in the PSTs’ teaching.
Clearly there is need for further research in the area of challenging PSTs’ hindering mathematical beliefs. Raymond, Santos, & Masingila’s (1991) statement that “teaching actions a directly influenced by teachers’ beliefs, and in turn those teachers’ actions have a tremendous impact
on the students’ belief systems” (p. 2) illustrates the importance of making efforts to improve PSTs’ mathematical beliefs, especially if the desired improvements in mathematics education suggested by the National Council of Teachers of Mathematics (1989) are going to be realized.
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