How we created the CPR on Proving 

Development of the CPR on Proving was motivated by our investigation of mathematicians’ perspectives on students’ mathematical creativity in tertiary level courses (Karakok, Savic, Tang, & El Turkey, 2016). We interviewed six active research mathematicians (with pseudonyms Drs. AF), who teach undergraduate and graduate level mathematics courses, and asked them about the role of mathematical creativity in proving, in teaching mathematics, and in students’ learning. The mathematicians in our study also examined three studentcreated proofs of a theorem in number theory (Birky, Campbell, Raman, Sandefur, & Somers, 2011) using a domaingeneral creative thinking rubric (Rhodes, 2010) created by the American Association of Colleges and Universities (AAC&U). This domaingeneral rubric was created to record growth and value creativity in a broad range of interdisciplinary student work samples. We utilized the mathematicians’ ideas to modify the AAC&U rubric to make it domainspecific to mathematics. Our modification was also influenced by Leikin’s (2009) rubric on mathematical creativity in problem solving, since the proving process is considered a subset of the problemsolving process (Furinghetti & Morselli, 2009). We leveraged these ideas to develop a "reflection" (Savic, Karakok, Tang, & El Turkey, 2015) which we then refined using students’ interview data (Tang, El Turkey, Savic, & Karakok, 2015).
Our development and refinement of the CPR on Proving were grounded in two of the ten aforementioned perspectives of creativity: Developmental and Problem Solving and ExpertiseBased (Kozbelt, Beghetto, & Runco, 2010). The primary assertion of the creativity theories in the Developmental perspective is that creativity develops over time, and the main focus of investigation is a person’s developing process of creativity. This perspective also emphasizes the role of the environment surrounding a student, in which interactive elements occur to enhance a student’s creativity. The second perspective that helped shape our project is Problem Solving and ExpertiseBased, which emphasizes the role of an individual’s problemsolving process and also argues that “creative thought ultimately stems from mundane cognitive processes” (Kozbelt et al., 2010, p. 33). This particular idea highlights that during problem solving or proving, implementing seemingly “mundane” tasks (such as finding relevant examples or representing the same concept in multiple ways) help the development of creativity by laying the foundations for creativity in novel situations. For example, Kozbelt et al. (2010) noted that “archival study of individual creative episodes of eminent scientists has generated a number of computational models of the creative process” (p. 33). These computational models included key components such as problemsolving processes, heuristics (ways that experts solve problems), and tasks. Furthermore, this perspective underscores the use of openended problems to challenge students’ thinking processes, providing opportunities for students to use experts’ ways of solving problems to be creative in such novel situations.
Our development and refinement of the CPR on Proving were grounded in two of the ten aforementioned perspectives of creativity: Developmental and Problem Solving and ExpertiseBased (Kozbelt, Beghetto, & Runco, 2010). The primary assertion of the creativity theories in the Developmental perspective is that creativity develops over time, and the main focus of investigation is a person’s developing process of creativity. This perspective also emphasizes the role of the environment surrounding a student, in which interactive elements occur to enhance a student’s creativity. The second perspective that helped shape our project is Problem Solving and ExpertiseBased, which emphasizes the role of an individual’s problemsolving process and also argues that “creative thought ultimately stems from mundane cognitive processes” (Kozbelt et al., 2010, p. 33). This particular idea highlights that during problem solving or proving, implementing seemingly “mundane” tasks (such as finding relevant examples or representing the same concept in multiple ways) help the development of creativity by laying the foundations for creativity in novel situations. For example, Kozbelt et al. (2010) noted that “archival study of individual creative episodes of eminent scientists has generated a number of computational models of the creative process” (p. 33). These computational models included key components such as problemsolving processes, heuristics (ways that experts solve problems), and tasks. Furthermore, this perspective underscores the use of openended problems to challenge students’ thinking processes, providing opportunities for students to use experts’ ways of solving problems to be creative in such novel situations.
Overall, the CPR on Proving was developed from a relative, domainspecific approach, focusing on an individual student’s progress on tasks and the development of their creativity over time.
Two Categories  Making Connections and Taking Risks
Making Connections
During the proving process, a student should be encouraged to make connections from previously learned material and apply these connections to new tasks. This category originated from the AAC&U Creative Thinking Rubric (Rhodes, 2010) category Connecting, Synthesizing, Transforming, where a milestone level is achieved by a student who “connects ideas or solutions into novel ways” (p. 2).
We define the category, Making Connections, as the ability to connect the proving task with definitions, theorems, multiple representations, and examples from the current course that a student is in, and possible prior experiences from previous courses. In this category, we consider making connections: between definitions/theorems, between representations, and between examples. Each of these subcategories is described below.
Between Definitions/Theorems.To enhance connectionmaking abilities, students should make use of definitions/theorems previously discussed in the course and perhaps, from other courses. Dr. A, one of the mathematicians in our previous study (Karakok et al., 2015) stated, “Somehow your mind has to spread out a little bit to see…connections to other theorems you could use…That's creativity also.” The way in which students use previous definitions/theorems in their proving processes defines the Beginning, Developing, or Advancing levels.
At the beginning level, a student recognizes some relevant (or irrelevant) definitions/theorems from the course or textbook (or resources related to the course) with no evidence of explicit attempts to connect those definitions/theorems to the task during the current proving process. For example, a student might list definitions related to a concept (e.g., function, onto, 11) that s/he has read in the task without providing any evidence of connecting these definitions to the proof in his/her attempts. At the developing level, a student recognizes some relevant definitions/theorems from the course or textbooks (or resources related to the course) with evidence of an attempt to connect these to the task during the proving process. At the advancing level, a student implements definitions/theorems from the course and/or other resources outside the course in his/her proving. At the advancing level, a student not only recognizes relevant definitions and theorems, but also explicitly illustrates using them.
Between Representations.Creating or using multiple representations can be important for solving or understanding problems. The National Council of Teachers of Mathematics (2000)referred to a representation as one way a student might depict his/her mathematical thinking. Other researchers have emphasized making connections among and between representations of a concept, such as representing a function as a table, graph, verbally and symbolically (Carlson, Oehrtman, & Engelke, 2010). The connections students make between representations is also important for their development of mathematical creativity. In this subcategory, students may not always make broad connections across the two fields of geometry and algebra, but perhaps attempt to utilize representations within each field.
Representations include written work in the form of diagrams, graphical displays, and symbolic expressions. We also include linguistic expressions, either oral or written, as a form of a lexical representation. For example, a student can write, “the intersection of sets A and B,” or orally state “A intersect B is the set of common elements between A and B.” A Venn Diagram, the symbolic representation , the set notation (which is also a symbolic representation), are other possible representations a student can use to depict his/her mathematical thinking about the concept of intersection. Note the last two representations are in the same category, i.e., symbolic, but they are still considered to be two different representations.
At the beginning level, a studentis able to provide a representation with no evidence of attempting to connect it to another representation. At the developing level,a student should recognize connections between some representations and attempt to connect them to the proving task on hand. Students on this level may not recognize all the related representations of a mathematical object, but at least demonstrate connections to more than one representation. At the advancinglevel, a student should utilize and implement different representations in his/her proving process, hence making explicit connections between the different possible representations of a mathematical concept and applying these connections to their proof attempt.
Between Examples.This subcategory refers to students’ scratch work or “play time” where they experiment with different ideas to attempt the task. They can do this by creating examples, comparing and contrasting examples, or by providing counterexamples that are sufficient to disprove a claim. Students usually practice with examples as a method to understand the definition of a concept or to validate the verity of a mathematical statement. Doing so could help students to develop their creativity.
However, students need to further their example generation to see possible connections to a pattern, which is somewhat of a difficulty for students (Dahlberg & Housman, 1997). Merely asking students to create examples does not necessarily lead to a proof production. As Iannone et al. (2011) state, “[I]f example generation is to be a useful pedagogical strategy, more nuance is needed in its implementation” (p. 11). Thus, in this subcategory we aim to push students to make connections between examples to generalize to a key idea, or pattern.
At the beginning level, a student generates one or two specific examples with no attempt to connect them. However, at the developinglevel, a student recognizes a connection between the generated examples. At the advancinglevel, a student utilizes the key idea synthesized from generating examples. One way to see this is when students recognize patterns from examples and symbolize these patterns formally to assist in the proving process.
We define the category, Making Connections, as the ability to connect the proving task with definitions, theorems, multiple representations, and examples from the current course that a student is in, and possible prior experiences from previous courses. In this category, we consider making connections: between definitions/theorems, between representations, and between examples. Each of these subcategories is described below.
Between Definitions/Theorems.To enhance connectionmaking abilities, students should make use of definitions/theorems previously discussed in the course and perhaps, from other courses. Dr. A, one of the mathematicians in our previous study (Karakok et al., 2015) stated, “Somehow your mind has to spread out a little bit to see…connections to other theorems you could use…That's creativity also.” The way in which students use previous definitions/theorems in their proving processes defines the Beginning, Developing, or Advancing levels.
At the beginning level, a student recognizes some relevant (or irrelevant) definitions/theorems from the course or textbook (or resources related to the course) with no evidence of explicit attempts to connect those definitions/theorems to the task during the current proving process. For example, a student might list definitions related to a concept (e.g., function, onto, 11) that s/he has read in the task without providing any evidence of connecting these definitions to the proof in his/her attempts. At the developing level, a student recognizes some relevant definitions/theorems from the course or textbooks (or resources related to the course) with evidence of an attempt to connect these to the task during the proving process. At the advancing level, a student implements definitions/theorems from the course and/or other resources outside the course in his/her proving. At the advancing level, a student not only recognizes relevant definitions and theorems, but also explicitly illustrates using them.
Between Representations.Creating or using multiple representations can be important for solving or understanding problems. The National Council of Teachers of Mathematics (2000)referred to a representation as one way a student might depict his/her mathematical thinking. Other researchers have emphasized making connections among and between representations of a concept, such as representing a function as a table, graph, verbally and symbolically (Carlson, Oehrtman, & Engelke, 2010). The connections students make between representations is also important for their development of mathematical creativity. In this subcategory, students may not always make broad connections across the two fields of geometry and algebra, but perhaps attempt to utilize representations within each field.
Representations include written work in the form of diagrams, graphical displays, and symbolic expressions. We also include linguistic expressions, either oral or written, as a form of a lexical representation. For example, a student can write, “the intersection of sets A and B,” or orally state “A intersect B is the set of common elements between A and B.” A Venn Diagram, the symbolic representation , the set notation (which is also a symbolic representation), are other possible representations a student can use to depict his/her mathematical thinking about the concept of intersection. Note the last two representations are in the same category, i.e., symbolic, but they are still considered to be two different representations.
At the beginning level, a studentis able to provide a representation with no evidence of attempting to connect it to another representation. At the developing level,a student should recognize connections between some representations and attempt to connect them to the proving task on hand. Students on this level may not recognize all the related representations of a mathematical object, but at least demonstrate connections to more than one representation. At the advancinglevel, a student should utilize and implement different representations in his/her proving process, hence making explicit connections between the different possible representations of a mathematical concept and applying these connections to their proof attempt.
Between Examples.This subcategory refers to students’ scratch work or “play time” where they experiment with different ideas to attempt the task. They can do this by creating examples, comparing and contrasting examples, or by providing counterexamples that are sufficient to disprove a claim. Students usually practice with examples as a method to understand the definition of a concept or to validate the verity of a mathematical statement. Doing so could help students to develop their creativity.
However, students need to further their example generation to see possible connections to a pattern, which is somewhat of a difficulty for students (Dahlberg & Housman, 1997). Merely asking students to create examples does not necessarily lead to a proof production. As Iannone et al. (2011) state, “[I]f example generation is to be a useful pedagogical strategy, more nuance is needed in its implementation” (p. 11). Thus, in this subcategory we aim to push students to make connections between examples to generalize to a key idea, or pattern.
At the beginning level, a student generates one or two specific examples with no attempt to connect them. However, at the developinglevel, a student recognizes a connection between the generated examples. At the advancinglevel, a student utilizes the key idea synthesized from generating examples. One way to see this is when students recognize patterns from examples and symbolize these patterns formally to assist in the proving process.
Taking Risks
During the proving process, a student should be encouraged to explore concepts, create new ideas, and evaluate those attempts in order to ultimately create a valid proof. Those explorations require a student to take risks during the proving process. The category Taking Risks originated from the AAC&U Creative Thinking Rubric (Rhodes, 2010), where the highest level is achieved by a student who “[a]ctively seeks out and follows through on untested and potentially risky directions or approaches to the assignment in the final product” (p. 2).
Therefore, we define the category Taking Risks as the ability to actively attempt a proof, perhaps using multiple proof techniques, posing questions about reasoning within the attempts, and evaluating those attempts. The four subcategories, Tools and Tricks, Flexibility, Posing Questions, and Evaluation of the Proof Attempt, are described below.
Tools and Tricks. We found through interviewing mathematicians that creativity also can involve creating tools or tricks in the proving process. Dr. E stated, “You can be very creative about the way in which you approach the question, either with new tools or with a really good idea for a partial result.” Using these tools or tricks can be original to the student or the course, thus leading to relative creativity in their proving. A common example of a tool or trick that is original is involved in the proof of the theorem, “There are infinitely many prime numbers.” One must assume a finite amount of prime numbers, , and create a new number that is larger than the largest prime which one then shows is still prime. The usual question asked by students when presented with this proof is, “Where did this come from?” This new number is an example of an unexpected object (tool) created to assist with creation of the proof.
The creation of an entirely new tool or trick is creative, and we believe it is evidence of a risk taken; however, the tool or trick need not be original to be considered in line with creative thought. Adapting a previous tool or trick to new contexts is also considered unconventional. At the beginning level, a student uses a tool or trick that is algorithmic or conventional. Conventional solutions are “generally recommended by the curriculum, displayed in textbooks, and usually taught by the teachers” (Leikin, 2009, p. 133). For example, if an instructor presented the trick that you should “add zero” while completing a square, a student at the beginning level would employ the same trick in a proof that required completing the square. At the developing level, a student uses a tool or trick that is modelbased or partly unconventional. If the student used that trick in a new context or in a proof that did not require completing the square in the same course, the student would be considered developing. For example if the student had to prove for every natural number , and in the inductive case wrote , then the student would be considered at the developing level. Finally, at the advancing level, a student creates a tool or trick that is unconventional for the course or the student. If a student thought of “adding zero” without any prompting or previous knowledge in the course, this would be considered advancing.
Flexibility. In the category Making Connections, we discussed recognizing the need to use a proof technique used on previous proofs on a new proof. Flexibility is the ability to shift approaches in proving a theorem or claim. This idea was adapted from Silver’s (1997) definition of flexibility for problem solving. For example, a student might begin a proof using a direct proof, but then shift to a proof by contradiction if the student did not find the first technique helpful. Dr. D found this ability helpful during her proof attempts, “If it doesn’t work you say ‘let me try something different and use some information I gathered to [come] up [with] something that might be more useful.’”
In this subcategory continuum, at the beginning level, a student attempts one proof technique in his/her proof. At the developing level, a student acknowledges the possibility of using different proof techniques, but does not act on it. Finally, at the advancing level, a student acts on different proving approaches. A student at the advancing level would act on multiple proof techniques, perhaps because the student did not find the initial proof technique(s) helpful, or s/he wanted to attempt a more efficient proof.
Posing questions.In the proving process, there are certain times when a question can lead to a creative thought. Dr. B acknowledged that, while researching, he asks himself, “What do I need to do in order to make that step so the rest of it is downhill?” Pelczer and Rodriguez (2011), citing Jensen (1973), stated that if students want to be creative, they “should be able to pose mathematical questions that allow exploration of the original problem” (p. 384). Posing questions can occur throughout the proving process, but there are different qualities of questions that students can pose.
At the beginning level, a student will recognize that a question should be asked (perhaps with a question mark next to his/her proof), but will not formulate a full question. At the developing level, a student will pose a clarifying question about a statement of a definition or theorem, for example to clarify terms used within a theorem statement. Finally, at the advancing level, a student will pose a clarifying question about the reasoning in the proof.
Evaluation of the proof attempt.We define a successful proof as a correct proof which “establishes the truth of a theorem” (Selden & Selden, 2003, p. 5). A successful proof is neither a necessary nor sufficient condition for a creative proof attempt. That being said, understanding the key ideas that make a proof attempt successful or unsuccessful can provide insight for future proof attempts. Key ideas are defined by Raman (2003) as, “a heuristic idea which one can map to a formal proof with appropriate sense of rigor. It … gives a sense of understanding and conviction. Key ideas show why a particular claim is true” (p. 323). For example, Dr. D stated that she reevaluated a result to find a visual application and ended up “go[ing] back to the drawing board because the stuff that we thought we proved was wrong. My thinking about [a] different way to visualize it and seeing something completely unexpected got us there.”
At the beginning level, a student is one who only checks work locally, that is, for small errors or typos. At the developing level, a student recognizes a successful or unsuccessful proof attempt without identifying the key idea that makes the attempt successful or unsuccessful. A student may look at his/her proof, realize that it is incorrect, but not realize exactly why the proof is incorrect. Finally, an ability to recognize the key idea in a proof attempt, successful or unsuccessful, describes an advancinglevel in this subcategory.
Therefore, we define the category Taking Risks as the ability to actively attempt a proof, perhaps using multiple proof techniques, posing questions about reasoning within the attempts, and evaluating those attempts. The four subcategories, Tools and Tricks, Flexibility, Posing Questions, and Evaluation of the Proof Attempt, are described below.
Tools and Tricks. We found through interviewing mathematicians that creativity also can involve creating tools or tricks in the proving process. Dr. E stated, “You can be very creative about the way in which you approach the question, either with new tools or with a really good idea for a partial result.” Using these tools or tricks can be original to the student or the course, thus leading to relative creativity in their proving. A common example of a tool or trick that is original is involved in the proof of the theorem, “There are infinitely many prime numbers.” One must assume a finite amount of prime numbers, , and create a new number that is larger than the largest prime which one then shows is still prime. The usual question asked by students when presented with this proof is, “Where did this come from?” This new number is an example of an unexpected object (tool) created to assist with creation of the proof.
The creation of an entirely new tool or trick is creative, and we believe it is evidence of a risk taken; however, the tool or trick need not be original to be considered in line with creative thought. Adapting a previous tool or trick to new contexts is also considered unconventional. At the beginning level, a student uses a tool or trick that is algorithmic or conventional. Conventional solutions are “generally recommended by the curriculum, displayed in textbooks, and usually taught by the teachers” (Leikin, 2009, p. 133). For example, if an instructor presented the trick that you should “add zero” while completing a square, a student at the beginning level would employ the same trick in a proof that required completing the square. At the developing level, a student uses a tool or trick that is modelbased or partly unconventional. If the student used that trick in a new context or in a proof that did not require completing the square in the same course, the student would be considered developing. For example if the student had to prove for every natural number , and in the inductive case wrote , then the student would be considered at the developing level. Finally, at the advancing level, a student creates a tool or trick that is unconventional for the course or the student. If a student thought of “adding zero” without any prompting or previous knowledge in the course, this would be considered advancing.
Flexibility. In the category Making Connections, we discussed recognizing the need to use a proof technique used on previous proofs on a new proof. Flexibility is the ability to shift approaches in proving a theorem or claim. This idea was adapted from Silver’s (1997) definition of flexibility for problem solving. For example, a student might begin a proof using a direct proof, but then shift to a proof by contradiction if the student did not find the first technique helpful. Dr. D found this ability helpful during her proof attempts, “If it doesn’t work you say ‘let me try something different and use some information I gathered to [come] up [with] something that might be more useful.’”
In this subcategory continuum, at the beginning level, a student attempts one proof technique in his/her proof. At the developing level, a student acknowledges the possibility of using different proof techniques, but does not act on it. Finally, at the advancing level, a student acts on different proving approaches. A student at the advancing level would act on multiple proof techniques, perhaps because the student did not find the initial proof technique(s) helpful, or s/he wanted to attempt a more efficient proof.
Posing questions.In the proving process, there are certain times when a question can lead to a creative thought. Dr. B acknowledged that, while researching, he asks himself, “What do I need to do in order to make that step so the rest of it is downhill?” Pelczer and Rodriguez (2011), citing Jensen (1973), stated that if students want to be creative, they “should be able to pose mathematical questions that allow exploration of the original problem” (p. 384). Posing questions can occur throughout the proving process, but there are different qualities of questions that students can pose.
At the beginning level, a student will recognize that a question should be asked (perhaps with a question mark next to his/her proof), but will not formulate a full question. At the developing level, a student will pose a clarifying question about a statement of a definition or theorem, for example to clarify terms used within a theorem statement. Finally, at the advancing level, a student will pose a clarifying question about the reasoning in the proof.
Evaluation of the proof attempt.We define a successful proof as a correct proof which “establishes the truth of a theorem” (Selden & Selden, 2003, p. 5). A successful proof is neither a necessary nor sufficient condition for a creative proof attempt. That being said, understanding the key ideas that make a proof attempt successful or unsuccessful can provide insight for future proof attempts. Key ideas are defined by Raman (2003) as, “a heuristic idea which one can map to a formal proof with appropriate sense of rigor. It … gives a sense of understanding and conviction. Key ideas show why a particular claim is true” (p. 323). For example, Dr. D stated that she reevaluated a result to find a visual application and ended up “go[ing] back to the drawing board because the stuff that we thought we proved was wrong. My thinking about [a] different way to visualize it and seeing something completely unexpected got us there.”
At the beginning level, a student is one who only checks work locally, that is, for small errors or typos. At the developing level, a student recognizes a successful or unsuccessful proof attempt without identifying the key idea that makes the attempt successful or unsuccessful. A student may look at his/her proof, realize that it is incorrect, but not realize exactly why the proof is incorrect. Finally, an ability to recognize the key idea in a proof attempt, successful or unsuccessful, describes an advancinglevel in this subcategory.
The CPR on Problem Solving 

The CPR on Problem Solving is a metacognitive tool that utilized the previous work on the CPR on Proving coupled with research into creative problem solving. It was used in our NSFfunded grant as a resource for students solving problems in calculus. Teachers were using the CPR on Problem Solving in their own ways, and one aspect that we are currently researching is how the instructors are utilizing the CPR on Problem Solving.