Preservice teacher beliefs about proofs
Mingus, Tabitha T Y
This article describes a study of backgrounds, beliefs, and attitudes of teachers about proofs. Thirty preservice elementary teachers enrolled in a mathematics content course and 21 secondary mathematics teachers in an abstract algebra course were surveyed. The study explored four issues: preservice teachers’ experiences/exposure to proof, their beliefs about what constitutes a proof and the role of proof in mathematics, and their beliefs about when proof should be introduced in grades K-12. Results of the survey are described as a means for discussing the backgrounds and beliefs future teachers hold with regard to teaching proofs in their own classrooms. Finally, a short collection of sample explorations and questions, which could he used to encourage the thinking and writing of proofs in grades K-12, is provided. One ofthese questions was posed to 215 secondary students; examples of their reasoning and a discussion of the various techniques employed by the students are included.
Mathematics is frequently referred to as both the queen and servant of the sciences. Proof could be considered both the queen and servant of mathematics. Mathematical proof has long been the means by which mathematicians convey their findings and reasoning to one another. Hermann Hankel best illustrated the need for a means to communicate mathematical ideas when he said, “In most sciences one generation tears down what another has built, and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure” (as cited in Dunham, 1990, p. vii). Proof not only provides the foundation upon which mathematical ideas are built, but also the way for each generation to learn about and extend what has already been accomplished.
Two important aspects of mathematical proof are reasoning and communicating that reasoning to others. According to Hersh (1993), “In the classroom, the purpose of proof is to explain. Enlightened use of proofs in the mathematics classroom aims to stimulate the students’ understanding.” It is helpful to temporarily adopt the broad definition that proof is a complete explanation, a convincing argument. Amore complete definition of mathematical proof will appear later in this article. The National Council of Teachers of Mathematics Standards (NCTM, 1989) emphasized the need to teach children the basics of proof early in their education. In fact, two standards present for all grade levels are “Mathematics as Communication” and “Mathematics as Reasoning.” For elementary students, the Standards suggested children be taught that justifying and explaining their reasoning as they arrive at a solution is as important as the solution. Furthermore, the Standards recommended that children he provided opportunities to practice communicating their thinking to assist them in forming links between their intuition and formal mathematical reasoning. The blending of formalism and understanding (along with an extensive treatment of the many aspects of mathematical proof) can be found in Hanna (1983).
Understanding the process of proof is not a phenomenon that takes place during a single isolated time frame in a student’s life, and for most students this understanding does not come without considerable nurturing. While the Standards called for the twocolumn proofs in geometry to receive decreased attention, they also suggested that secondary students should be able to “make and provide arguments for conjectures” (p. 125). An early and broad introduction to proofs that parallels students’ cognitive development from concrete to formal reasoning may be the best way to foster their understanding of the process of proof and increase their willingness to read and write mathematical proofs at the collegiate level.
A lack of understanding about the process of proof is not the only barrier preventing undergraduate students from successfully reading, writing, and analyzing proofs. Students’ attitudes present a significant blockade in their progress. Students taking beginning undergraduate courses fear proofs and are convinced that they cannot do proofs. Frequently, students throw up their hands in disgust without even attempting a proof stating, “I hate proof.” The “attitude barrier” prevents students from taking the risk of justifying or explaining their reasoning to others in a class. Even after students have overcome their fear and dislike and engage in the process, they are minimally prepared to articulate relatively straightforward proofs in mathematics. As part of a pretest given by one of the authors in fall 1997 in a sophomore-level discrete mathematics course at the University of Northern Colorado, students were asked to prove that the sum of the squares of two odd numbers is even. These students – the majority of whom were preservice secondary teachers stuggled with the preliminary step of representing two different odd numbers as 2h + 1 and 2k + 1.
Study Design and Survey Results
Students’ antipathy toward proof and inability to articulate their reasoning are serious deficiencies in their mathematical training. When these deficiencies are present in preservice teachers and left uncorrected, they are amplified and passed down to future generations of students (Martin & Harel, 1989). Hoyles (1997) sought to determine how the curriculum, teachers, and schools shaped secondary students’ conceptions of proof. Her study concluded that schools need to focus on the influences of curriculum organization and sequencing. Resisting the temptation to impose a linear sequencing action on situations that engage students with proof, the curriculum should “scaffold a coherent and connected conception of proof.” The development of “ways of thinking” should act as a guide in the process of developing such a curriculum.
In the study described in this article, a survey was developed to determine some of the beliefs of preservice teachers about proofs. Information was sought on four issues:
1. What was their previous experience with proofs?
2. When and in what manner should proof be taught in grades K-12?
3. What did they believe constitutes a proof?
4. What is the role of proofs in mathematics and in teaching mathematics?
This survey was given to 51 students – 30 majoring in elementary education and 21 majoring in mathematics with an emphasis in secondary education midsemester in two different mathematics content courses. One course was required for all preservice elementary teachers (ET) and the other, abstract algebra, was required for all preservice secondary teachers (Sl) majoring in mathematics.
The survey results provide a means for discussing the backgrounds and beliefs future teachers have with regard to teaching proof in their own classrooms. A list of sample explorations/questions that could be used to encourage the thinking and writing of proofs in grades K- 12 appears in Appendix A, as an indicator of the level and language and not as a cure all for the problem addressed. Also examined are the proofs developed when one of these questions was posed to 215 middle and high school students.
Survey Question #l: What Is Your Experience With Proofs in Mathematics?
One of the primary assumptions was that undergraduate students have negative attitudes about proofs and an inability to articulate their mathematical reasoning because their backgrounds at the secondary level in mathematical proof were limited to the traditional two-column proofs taught in 10th-grade geometry. A number of the students commented on their attitudes and feelings about proofs and their ability to write proofs, in addition to detailing their backgrounds. One of the ET stated, “I had a geometry class in high school where we did proofs – I hated proofs. I never knew the purpose for doing them. Maybe our teacher never told us why. We just had to do them.”
Sixty-nine percent (35 out of 51) of the preservice teachers involved in the survey had either no exposure to proofs or only the traditional 10th-grade geometry course at the secondary level. Surprisingly, however, 24% of the ST had no experience with proofs prior to entering college. Table I summarizes their precollegiate experiences with proof.
While the ET took few collegiate-level courses requiring them to work with mathematical proofs, the ST’s programs were filled with classes that not only required them to understand but also to construct their own mathematical proofs. As expected, the ST (85.7%) majoring in mathematics had taken more courses at the collegiate level that required reading and writing proofs than the ET (13.3 %). The ST students who had limited prior experiences with mathematical proof found this reintroduction to proof difficult.
Student 1: 1 have had minimal contact and experience with proofs. I have done a couple in linear algebra and modem geometry. I still feel very uncomfortable doing proofs and am very unsure of what I need to do to do the proofs correctly.
Student 2: 1 never had to do a single proof in high school. The first proofs I did were in modem geometry [college geometry course]. The number of proofs we did was almost overwhelming. I struggled with the proof process and as a result, we did all of our proofs in groups of two or more students.
Student 3: I had some experience with proofs in discrete mathematics. I remember there was a long process for proving things in this class, but I did not understand what I was doing. I had lots of experience with proofs in modern geometry just last semester. Just as in Discrete, we had a long process to follow, but I understood what I was doing and soon learned to prove things on my own.
This latter comment demonstrates how repeated experiences in courses requiring proofs can help students gain confidence in constructing their own proofs. In discrete mathematics, this student was required to learn a variety of different proof techniques (induction, contradiction, contrapositive, etc.). Once she had gained experience with these techniques, applying them to the proofs in modem geometry became a bit easier. She was then able to concentrate on the material to be learned rather than on the process of proof.
Survey Question #2: What Exposure to Proofs Is Appropriate in K-12?
These students’ experiences with proof in their own schooling affected what they thought was appropriate exposure for K-12 students. Table 2 summarizes the grade level at which the preservice teachers believed exposure to proofs should begin in elementary and secondary classrooms.
Combined, one third (17) of the preservice teachers suggested that the idea of a proof should be presented as early as possible (K-6) and that the proofs should become more formal as students progress into high school. An overwhelming majority of 69% of the preservice teachers wanted proofs introduced prior to taking 10th-grade geometry. All of the ST respondents (including those from the unclear response category) believed that introducing proof at the I I thor 12th-grade level was too late. The preservice teachers taking higher level collegiate mathematics courses wanted proof concepts to be introduced earlier in elementary school, whereas those who took only geometry in high school wanted the introduction of proof to be postponed until later. One of the ET stated,
From personal experience, I can honestly say I have no recollection of using proofs in grades K8. Which could be one reason why I had such a problem with them in high school [and now in college]. Perhaps they should eventually be introduced in these grades but not as a serious course. Nearly four times as many of the ST compared to the ET (63% to 17%) wanted exposure to proofs to begin in grades K-6. They may have recognized that a lack of exposure to formal reasoning in their middle and high school backgrounds affected their ability to learn how to read and construct proofs. One ST stated, “I think that informal proofs can be investigated in elementary school and that formal proofs should be presented in high school. It was hard for me to both learn difficult material and formal proof at the same time.”
Survey Question #3: What Constitutes a Proof?
A more precise working definition of mathematical proof may be helpful at this point in examining the preservice teachers’ beliefs about what constitutes a proof. Content knowledge includes facts, conceptual understanding, and problem solving abilities. Content knowledge cannot, however, be divorced from pedagogy and neither can proof. Proof provides a means for explaining why mathematics works and for conveying knowledge from one person (or generation) to the next. Hersh (1993) suggested, “Mathematical proof can convince, and it can explain. In mathematical research, its primary role is convincing. At the high-school or undergraduate level, its primary role is explaining.” Thus, for this study, a mathematical proof is defined as a collection of true statements linked together in a logical manner that serves as a convincing argument for the truth of a mathematical statement.
The following summarizes the survey responses of both the ET and ST preservice teachers as to what constitutes a proof:
Making sense of data, showing relationships between concepts.
A way of explaining why something works.
Something made up of other simpler proofs.
An argument that is convincing to its audience.
Like a map from point A to point B with
directions for each smaller step, there may be more than one way to get there; the end result is B, however.
The definition of mathematical proof offered by the ET seemed to be a reflection of their prior scholastic experiences with proof. Most of them concentrated on the use of proof to demonstrate geometric relationships or as a means for confirming a relationship between data gathered inductively. The relatively sophisticated definitions provided by the ST and their more extensive and varied experience with proofs lends more credence to such a conclusion. The ST focused on the explanatory power of proof in their definitions; however, they also described the use of logical and convincing arguments as necessary components of proof. One of the ST students stated, “A proof consists of understanding the question being asked or the statement being made and showing in a step-by-step process how and why the statement being proved is true. Each step in the proof relates directly to one another.”
Survey Question #4: What Is the Role of Proof in Mathematics?
The views of these preservice teachers with regard to what constitutes a proof can be seen in what they believed is the purpose of proof in mathematics. The majority of the responses focused on how proofs explain why concepts work the way they do in mathematics and how constructing proofs helps students understand the mathematics they are doing.
Student 1 (ET): It helps the student think abstractly and be able to explain reasons why things work out the way they do. Proofs show that mathematical ideas and the way things work is not just coincidental. They show that there is reasoning and sense to mathematics.
Student 2 (ET): A proof teaches students rules of mathematics and makes the student think about the data given. I also think proof makes the student question the information given to them. Why is this so or why is this this way? It is easier to remember and understand how to do math if you know why it works and explore why it works for yourself instead of having someone just tell you how to do it.
The ST also considered the role of proof for maintaining and advancing the structure of mathematics. This notion was lacking in the ET beliefs.
Student 1 (ST): I believe a proof helps firm up the foundations of my mathematical knowledge. It makes a math problem answer the “why question,” and helps the knowledge stick longer in my mind. When I understand something, it’s easier to remember.
Student 2 (ST): Proofs help advance a mathematical idea for an individual or the mathematics community. Proof is the basis of mathematics; because so much of mathematics is dependent upon a prior framework or basis, then that basis must have a solid acceptable foundation. Proofs create a basis for expansion of mathematics. Without understanding we can go no further,
Implications for Classrooms
Can Secondary Students Produce Quality Proofs?
In an attempt to judge student ability to produce a convincing argument, the following was requested of 215 middle and high school students (grades 6-12): “Show that there are just as many even numbers as there are odd numbers.” The request was presented in written form during school hours at a local secondary school; 170 students provided a written justification of their reasoning, and the remaining students either refuted the statement or provided no answer. Their backgrounds were varied; however, all of the students had completed pre-algebra. The majority of the students had taken algebra, and the older students were currently taking geometry. Their responses ranged from elegant arguments involving one-to-one correspondence, the role of digits in the units position, and a proof by contradiction to the more routine but still valid specific counts of up to the first -50 positive integers.
What was most interesting about the proofs was that students in sixth through eighth grades (not the 11th- through 12th-grade students who had more exposure to “formal” proof) constructed the most creative arguments. Figure 1 contains examples of a proof using the role of digits in the unit’s position and a proof using a one-to-one pairing between even and odd integers.
About 20% of the students did not respond to the question at all. Based on the 170 responses, the following conclusions may be reached: (a) Students can creatively analyze and explain mathematical phenomena at grade levels as early as sixth or seventh grade; (b) this creativity is somehow lost or stifled as students advance to the 10th through 12th grade. Of particular interest is that the majority of early grade level responses did not fall into the three basic proof schemes of external conviction, ritual, and authoritarian (as described by Harel & Sowder, 1998).
From the Secondary Teacher’s Perspective
A careful study of the survey comments, personal experiences of the authors, and many of the authors’ conversations and classroom encounters with pre- and in-service teachers clarify the secondary teachers’ perspectives. In particular, a three-summer National Science Foundation Teacher Enhancement grant connected the authors with 55 practicing secondary teachers most holding strong ideas about this issue. These views surfaced during formal group discussions and through the ongoing treatment of proofs in their content and pedagogy courses. They maintained that teachers should provide opportunities to students to engage in writing and verbalizing their mathematical arguments as early as possible. The teachers’ beliefs were consistent with those expressed in the NCTM Standards, which support the teaching of informal proof as early as grades KA This support does not mean classroom teachers should be instructing elementary school students in the finer points of constructing proofs by contrapositive, for example. On the contrary, the focus at the elementary level should be on students talking in class about their ideas, developing language and notation, explaining (justifying, defending) their reasoning, and finally, writing about the processes they used in solving problems.
What becomes important to the classroom teacher is having appropriate and interesting questions for students to explore, as illustrated above. Teachers need questions containing significant mathematical content and yet whose solutions are attainable by elementary students, like most of those found in Appendix A. If teachers use questions emphasizing a common theme or incorporate a particular technique (such as Gauss’ reverse and add technique for summing arithmetic sequences) year after year, students will then have the opportunity to reflect on their previous explanations and see if they can improve on their reasoning (Galbraith, 1995).
It is important to offer questions that allow for a variety of approaches, both from the point of view of the nature of the proof and the grade level of the student. Szombathelyi and Szarvas (1998) discussed how to spiral thematic questions into the curriculum year after year. They presented a series of ideas used successfully in Hungary and the United States that might help develop reasoning ability. They advocated early intervention and suggested that the earlier the process begins the sooner confidence and ability to prove develops, frustrations diminish, and students engage more in justifying responses, thus moving toward formal proof.
Conclusions and Recommendations
The two groups of preservice teachers proposed that the idea and structure of proofs should be encountered as early as possible in the curriculum. A broad view of the definition of proof should be encouraged and accepted; more precision and breadth should be expected as students advance through the K- 12 arena. It is incumbent upon teachers to first demonstrate appropriate proofs and then to constantly reinforce any student attempt at a proof at any level and to be ready to “nudge” in the right direction to ensure correct reasoning and proper conclusions.
The study of the 170 middle and high school students who provided responses to the questions indicates that most students are capable of delivering a proof at some level. If teachers and students are encouraged to ask “why” or to “explain,” a classroom dialogue will develop into the actual underpinnings of a proof. What remains Then is practice on writing out this argument. Finally, it is important to maintain a tolerant attitude in the formative stages of proof development. Different language, approaches, presentations (to include verbal, geometric, written), and levels should be fostered. In addition, mathematics educators need to continue to conduct research in the field of mathematics education with the goal of applying these findings to the classroom. A good model of such is found in Maher and Martino (1996). This longitudinal case study documents the mathematical thinking of one child advancing from first through fifth grade on several combinatorial tasks. The study provides significant insight into the process by which the student learned to make proofs, and reports how a student was successfully empowered in building the idea of mathematical proof over time.
Editor’s Note: Tabitha T. Y. Mingus, Department of Mathematics and Statistics, Western Michigan University; Richard A Grass], Chair, Department of Mathematical Sciences, University of Northern Colorado.
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Tabitha T. Y. Mingus
Western Michigan University
Richard M. Grassl
University of Northern Colorado
Correspondence concerning this article should be addressed to Tabitha T. Y. Mingus, Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008.
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