Chapter 4 The Nature of Mathematics and Its Impact on K-12 Education PDF

Chapter 4 The Nature of Mathematics and Its Impact on K-12 Education Rick A. Hudson, Mark A. Creager, Angela Burgess, and Alex Gerber 4.1 Introduction What is mathematics? This question elicits controversy among both mathematicians and the general public alike. Although mathematical content is a...

Chapter 4 The Nature of Mathematics and Its Impact on K-12 Education Rick A. Hudson, Mark A. Creager, Angela Burgess, and Alex Gerber 4.1 Introduction What is mathematics? This question elicits controversy among both mathematicians and the general public alike. Although mathematical content is accepted as an important component of the K-12 curriculum, what students should learn about the nature of mathematics is not well defined, and the experiences K-12 students have in learning about the nature of mathematics varies greatly. These differences in students’ experiences have an impact on how students develop conceptions about what mathematics is and how it is valued. Boaler (2015) asserts that students’ understanding of mathematics is different than their under- standing of other fields. She states that children describe science and English in ways that are similar to how professors in these fields describe the subject. However, children view mathematics as focused on numbers or “a set of rules.” In contrast, mathematicians often describe mathematics as “a study of patterns.” Although much of the K-12 school curriculum focuses on the study of numbers and their properties, ironically a great many mathematicians are not concerned with numbers at all. Such great differences between the mathematics experienced by students and profes- sional mathematicians may help us to account for the lack of interest that many students have in mathematical study. Mathematicians often classify mathematics into two separate, but complemen- tary fields – pure mathematics and applied mathematics. Pure mathematics is con- cerned with mathematics for its own sake, whereas applied mathematics seeks to R. A. Hudson (*) · M. A. Creager University of Southern Indiana, Evansville, IN, USA e-mail: [email protected] A. Burgess · A. Gerber Indiana University, Bloomington, IN, USA © Springer Nature Switzerland AG 2020 45 V. L. Akerson, G. A. Buck (eds.), Critical Questions in STEM Education, Contemporary Trends and Issues in Science Education 51, https://doi.org/10.1007/978-3-030-57646-2_4 46 R. A. Hudson et al. use mathematics at the service of other disciples, such as biology, physics, or finance. These two separate branches of the field of mathematics contribute further to misunderstandings about what the nature of mathematics is and its relationship to STEM. A commonality between the two branches is that both pure and applied mathematicians seek to use reasoning to solve problems. However, the branch of applied mathematics is most often associated with the definition of STEM because applied mathematics displays an interdependence on the contexts in which it is situ- ated, including science, technology, and engineering. In this chapter, we will describe some of the philosophical underpinnings that motivate the nature of mathematics. We will follow with a discussion of how stu- dents experience the nature of mathematics in today’s K-12 classroom and provide suggestions about what K-12 students should learn about the nature of mathematics and the role that STEM may have in progressing this discussion. 4.2 Philosophical Underpinnings: Contrasting Absolutist and Fallibilist Perspectives Although other theories about the nature of mathematics exist, two contrasting per- spectives regarding the nature of mathematics are the fallibilist and absolutist view- points. The absolutists believe that mathematical knowledge is certain and unchallengeable (Davison & Mitchell, 2008). Furthermore, an absolutist views mathematics as a construct that has always been present and humans just discovered it. In other words, they view math as a divine gift that never has error or contradic- tion (White-Fredette, 2010). Alternatively, individuals who hold a fallibilist philos- ophy view mathematics as a human construct, and therefore, as susceptible to falsifiability as any human endeavor may be. Fallibilists believe that mathematics is built upon the needs of the society and limited by cultural boundaries that dictate its certainty and applicability (Hersh, 1997). It should come as no surprise that these two distinct philosophies manifest themselves in mathematics teaching in differ- ent ways. Educators with an absolutist view often see mathematics as a set of rules and procedures that are valuable for solving the problems. These teachers often work to break large ideas down into their most elementary ideas and then construct connec- tions between these elementary ideas that build the larger idea. However, it is often argued that teaching in this way causes students to miss out on opportunities for critical thinking and exploration of the mathematical concepts, because students are simply applying ready-made formulas or procedures often without thought or rea- son (White-Fredette, 2010). From an ontological standpoint, fallibilists are more likely to view reality as socially constructed, and therefore hold epistemologies supportive of social theories of knowledge and teaching (Ernest, 2018). Teaching mathematics as a human con- struct encourages students to invoke critical thinking, socially construct mathemati- cal knowledge and, explore and investigate concepts using mathematical inquiry 4 The Nature of Mathematics and Its Impact on K-12 Education 47 (White-Fredette, 2010). Critics of this philosophy often argue that it is wasteful of the precious little time teachers have to force students to “re-invent/discover” the mathematics that took society centuries or millennium to discover. Moreover, teach- ers will argue that the mathematics that mathematicians do vary from what students will do, and although mathematicians’ conjectures are certainly tentative, the theo- rems K-12 students typically encounter are correct and beyond falsification. This concern is common enough to warrant an example of what is meant by the tentative nature of mathematics before diving too deeply into the implications for teaching this philosophical debate has. Consider a conversation among a group of third graders who are exploring the concept of multiplication where they have developed an understanding of A × B as the total number of objects when there are A groups with B objects in each group. Child 1: Multiplication will always make a bigger number. Child 2: That’s interesting. Let’s check 2 times 3. I get 6, which is bigger. Child 3: I tried 1 times 3 and got 3, which isn’t bigger. Child 1: It will work with all numbers except 1. This child’s conjecture displays thinking that a falibilist would value, as mathe- maticians regularly look for patterns and make conjectures. However, this student’s conjecture is certainly false as many numbers, including 1, refute the conjecture (e.g., zero, negative numbers, and real numbers between 0 and 1). Adults regularly make claims about all numbers when they really mean counting numbers, but more importantly, third graders are likely to not have learned about operations with nega- tive numbers or fractions yet. So, this child’s conjecture, if modified to exclude 1, is true in the reality of numbers to that child. The child’s conjecture fails because the claim is for all numbers, but the conjecture can be made to be correct, by properly restricting the domain of numbers the conjecture covers (e.g., real numbers greater than 1). In describing the nature of mathematics as tentative, we do not mean to infer that 3 times 2 will someday equal something other than 6. Instead, like in the case of the child’s conjecture, what makes mathematics potentially false is that all theorems in mathematics are based on either assumed to be true statements or state- ments proven to be true. It is through these assumed and foundational statements, like our fictional child’s understanding of the concept of number, that the potential for falsity arises. In fact, the mathematical historian, Lakatos (1976) suggested that by wrestling with terms like multiplication, we gain a better understanding of what students mean and in turn create more advanced mathematics. Having clarified these two contrasting arguments of what is mathematics, we turn to describing the impact of these philosophies on mathematics education, which is an important, yet neglected aspect of math education research. There is little evidence in the literature about what philosophical standpoints mathematics educators hold and how they accommodate curriculum expectations, teaching prac- tices and their own math epistemologies. The philosophy of mathematics is rarely examined by researchers and it is important to discern the understanding of the basic principles and concepts that teachers hold before sustainable reform can be initiated. 48 R. A. Hudson et al. Despite prominent philosophers of mathematics education (e.g., Ernest, 2018) advocating that a dichotomy exists between fallibilist and absolutist philosophies of mathematics, others suggest that there is no such sharp distinction and that teachers’ philosophies of math run along a continuum with fallibilism and absolutism lying at the extremities (Davison & Mitchell, 2008). An important question to ask is: are teachers even thinking about the nature of mathematics when determining how to teach a given topic? Are they consciously making decisions about the global prac- tices and viewpoints that students will develop about mathematics, or are they look- ing locally at meeting the content objectives within a single lesson? It is important to draw a distinction between a teacher’s philosophy of teaching and learning and a teacher’s philosophy of mathematics. Simply because a teacher believes that knowledge can be appropriated via social interaction, it does not nec- essarily follow that they hold a fallibilist view that mathematics is a socially-­ constructed, tentative discipline. Equally, a teacher who employs traditional procedurally-focused instructional strategies may do so, not because they hold an absolutist philosophy, but because they believe this is the most effective strategy to help their students learn specific mathematical content. 4.3 What Should K-12 Students Know About the Nature of Mathematics? What K-12 students should come to understand about the nature of mathematics is not clearly defined. Although research has offered little advice on how the absolutist and fallibilist philosophies impact teachers’ decisions, several curriculum efforts have sought to describe what mathematical practices and processes students should come to enact when doing mathematics. For example, NCTM’s Principles and Standards for School Mathematics described five process standards to “highlight ways of acquiring and using [mathematical] content knowledge” (2000, p. 29). These processes included problem solving, reasoning and proof, communication, connections, and representations. These processes and similar mathematical practices have been incorporated into curriculum documents in a number of countries. For example, the Australian Curriculum incorporates the key ideas of understanding, fluency, problem-solving, and reasoning (Australian Curriculum, Assessment, and Reporting Authority, n.d.). The Singapore Mathematics Framework identifies mathematical problem solving as its central focus, but also emphasizes other mathematical processes, including rea- soning, communication, and modeling (Ministry of Education, Singapore, 2012). In the United States, the Common Core State Standards for mathematics [CCSSM] (NGA/CCSSO, 2010) include the Standards for Mathematical Practice (SMPs), which “describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.” For example, students should be able to “make sense of problems and preserve in solving them,” which they describe as having a 4 The Nature of Mathematics and Its Impact on K-12 Education 49 variety of skills like analyzing “givens, constraints, relationships, and goals.” The SMPs require students to be the active agents in doing mathematics. Although a teacher can tell students the Pythagorean Theorem, they cannot tell students how to build a logical progression of statements to prove such a theorem. A teacher may demonstrate a proof of the Pythagorean Theorem, but doing so does not guarantee her students will learn how to construct proofs. In fact, the literature is filled with evidence of students at all levels of mathematics achievement who struggle to con- struct even basic proofs (Healy & Hoyles, 2000; Stylianides, Stylianides, & Weber, 2017). In the sections that follow, we will describe three fundamental ideas about math- ematics that we believe K-12 students should come to recognize about mathematics. These descriptions transcend the mathematical processes and practices described above which focus on students’ ability to engage in the work of mathematics. These three ideas describe aspects of mathematics that will support students’ understand- ing of the nature of the field. 4.4 Mathematics Is a Way of Knowing When approaching a problem from an absolutist perspective, it might seem as if one simply needs to select the appropriate mathematical tool for the job. For example, if one were working a problem involving a right triangle, perhaps trigonometry or the Pythagorean Theorem will be helpful. Absolutists might assume that students must first master procedures, like applying the Pythagorean Theorem to find the length of a hypotenuse, and after doing so they can apply their skills in problem solving situ- ations. By doing this, teachers hope to instill how to reason in mathematics. This absolutist perspective has guided the development of many textbooks where stu- dents repeat a procedure multiple times, and then use this procedure to solve a small number of word problems. However, problem-solving situations are rarely this straightforward. In the right triangle example, there are so many options for tools from trigonometry that selecting an appropriate tool is far from a trivial task (Duval, 2006; Weber, 2001). Unfortunately, reasoning is not like the procedural knowledge one finds in traditional textbooks, because there is no pre-determined procedure for how to reason in mathematics. This suggests that reasoning cannot be taught directly. Not shockingly, performance on problems involving reasoning are among the poorest on large-scale national and international assessments (Roach, Creager & Eker, 2016). From a fallibilist perspective, students need to be encultured to what it means to reason in mathematics. Namely, students need to be given problem-solving tasks, but solving those tasks is not the end goal. Instead the goal is making the processes used explicit when solving the problem, so students might learn what constitutes reasoning in mathematics. Today, many curriculum materials are questioning the popular belief that students must first learn a set of procedures before applying them to solve problems. In one research study, Nathan and Petrosino (2003) asked 48 50 R. A. Hudson et al. pre-­service high school teachers to rank six related algebra tasks by level of diffi- culty. One of the ways the tasks varied is that some were purely algebraic (i.e., solve this equation) while others were contextually-based problems (i.e., a story problem that required the student to solve an equation). Some tasks were similar in that they involved the same numbers and operations and solving them produced the same answer. What they found is that most teachers felt that the contextually-based prob- lems would be the hardest for their students because the teachers felt their students would have to first write an equation for the contextually-based problem. This equa- tion was the same equation as the purely algebraic problem, however they felt the contextually-based problem required two steps whereas the purely algebraic prob- lem only required one. This seems logical, however research into how students solve these problems have consistently showed that the students were more likely to solve the contextually-based problem correctly (Nathan & Petrosino, 2003). This is because they did not set up an equation like their teachers imagined. They instead reasoned about the context to help them solve the problem. Studies like these and others provide support for the National Council of Teachers of Mathematics (NCTM) who proposed that procedural fluency should be built from conceptual understanding (NCTM, 2014). Similarly, education researchers have also examined ways that skills and procedures can be co-developed in students with processes like reasoning (Kobiela & Lehrer, 2015). Like defining what mathematics is, defining reasoning in mathematics is a simi- larly difficult task. Many education researchers have made efforts to do so (e.g., Jeannotte & Kieran, 2017; Russell, 1999). A summary of these definitions described reasoning as a set of interrelated practices that include generalizing/conjecturing, investigating why, and proving/refuting (Lannin, Ellis, & Elliott, 2011). Although it may appear that these processes happen linearly as they are written, it is often not the case. Instead, it is important for students to be able to participate in the zig-zag nature of mathematical reasoning. This is more aligned with Lakatos’s (1976) description of how mathematics developed historically. Although proof might be thought of as the culminating activity of mathematics, and in many cases this is true, Lakatos noted that refutations play a far greater role in mathematical discoveries than proofs. Although teachers might be concerned about a child being wrong, it can be argued that being wrong positions the child to learn a more important lesson that reasoning is the foundation of mathematical ideas. 4.5 Mathematics Is Tentative, Because Mathematical Claims Are Based on Assumptions One of the challenges of helping students to appreciate the tentative nature of math- ematics is that much of the mathematical content that is the focus of K-12 mathe- matics has been settled among mathematicians for centuries, if not millennia. In contrast, the public regularly has opportunities to understand the tentative nature of 4 The Nature of Mathematics and Its Impact on K-12 Education 51 science when a new experimental drug shows promise in curing a chronic disease or a new plant hybrid is used to boost crop production. However, the groundbreaking work that is happening in mathematics today is rarely featured in the media, unless someone calculates a few more digits of pi. Mathematical research, especially in pure mathematics, is often inaccessible to an undergraduate student studying math- ematics, let alone a member of the general public. As a result, there are fewer oppor- tunities for the public to understand the tentative nature of mathematics, which may be one reason the absolutist views of mathematics are perpetuated. In mathematics, axioms are described as statements that are assumed to be true. Such assumptions underlie the tentative nature of mathematics. In pure mathemat- ics, axioms are the fundamental ideas with which we build mathematical proofs. Throughout the history of mathematics, there have been a number of discoveries that have occurred as a result of questioning assumptions made. For example, for more than 2000 years, Euclid’s five postulates formed the basis of geometric think- ing. Euclid’s fifth postulate is equivalent to the statement “Given a line and point not on it, at most one line parallel to the given line can be drawn through the point.” Although this statement was commonly agreed upon, during the nineteenth century, mathematicians began to assume it was false and investigated the results. Consequently, new Non-Euclidean geometries, such as hyperbolic and elliptic geometries were introduced. Such moments in the history of mathematics provide arguments for the fallibilist perspective. The commonly cited van Hiele (1985) levels describe students’ typical progres- sion of thinking in geometry, and the highest level describes students’ ability to compare axiomatic systems. Few students encounter Non-Euclidean geometries before postsecondary mathematics, however K-12 students should come to recog- nize that one’s mathematical claims must be rooted in the assumptions that they make. For example, in creating mathematical models of complex situations, stu- dents should reflect on the choices and assumptions that they make throughout the modeling cycle. How do teachers help students understand the tentative nature of mathematics? Drawing connections to the history of mathematics and their personal explorations of numbers may help to support this understanding. Throughout the K-12 curricu- lum, students’ access to the number system gradually expands. Initially students focus on counting numbers, but as they progress, they develop their view of num- bers to include rational, irrational, and eventually complex numbers. This sequence corresponds to how number systems developed historically. Ancient Egyptians used 5 unit fractions to describe rational numbers. (For example, they viewed as the sum 1 1 8 of and.) The introduction of irrational numbers was so controversial among 2 8 the ancient Greeks that some historians believe the Pythagoreans murdered one of their own who discovered their existence. The work of mathematicians often involves making an attempt to solve a prob- lem, finding a refutation to that line of reasoning, and then seeking a new route to solve the problem. These erroneous attempts are in stark contrast to the polished 52 R. A. Hudson et al. examples that students see in textbooks or worked out by teachers. Devlin (2010) suggested using historical letters between the notable mathematicians Pascal and Fermat can provide students with insights about how uncertainty exists in problem solving, even among great mathematical minds. Another promising method for teaching the tentativeness of mathematics is the use of rough-draft talk during the act of problem solving. Jansen (2009, Jansen, Cooper, Vascellaro, & Wandless, 2016/2017) discusses how using “rough-draft talk” can provide opportunities for students to express their false starts and uncer- tainty in their thinking. Jansen’s intent of introducing rough-draft talk is to create practices in which all students’ thinking is valued and to increase students’ partici- pation in mathematical discussions in the classroom. However, when a student engages in rough-draft talk, they also have the opportunity to understand the tenta- tive nature of mathematics. Jansen et al. (2016/2017) identify that one of the under- lying principles for rough-draft talk should be the promotion that learning mathematics involves revising one’s understanding over time. 4.6 Mathematics Is Creative An absolutist might argue that mathematics is inherently not creative. If one sees mathematics through the eyes of pre-determined certainty, then creativity in math- ematics would be unnecessary. The way that mathematics has traditionally been taught reflects this viewpoint where a teacher describes a procedure or algorithm, and students learn to mimic that procedure. In such a classroom, there is little need for creativity, or what Pair (2017) refers to as “the exploration of ideas.” However, engaging in true problem-solving experiences can help students appre- ciate the creativity involved in mathematical thinking. For example, a group of teachers was posed the following mathematical task (adapted from Wilburne, 2014) during a professional development workshop: “A fast food restaurant sells chicken tenders in packs of 4 and 7. What is the largest number of tenders that you cannot buy? How do you know this is the largest number that you cannot buy?” This was a task that was unfamiliar to the teachers, and initially the teachers were uncertain whether there would be a largest number that cannot be bought in packs of either 4 or 7. The teachers worked in small groups to solve their problem and document their thinking. One group of teachers (whose work is shown in Fig. 4.1) documented lists of numbers that could be bought and numbers that were impossible to buy using packs of 4 and 7. They realized that four consecutive numbers of 18, 19, 20, and 21 chicken tenders were possible to buy. Given that any of these were possible the next four consecutive numbers – 22, 23, 24, and 25 – could be bought by buying an addi- tional pack of 4 chicken tenders. Since this line of reasoning could be extended, they determined the largest number that could not be bought was 17. A second group (work shown in Fig. 4.2) took a more methodical approach to arrive at the answer 4 The Nature of Mathematics and Its Impact on K-12 Education 53 Fig. 4.1 Group 1’s Work on the Task Fig. 4.2 Group 2’s Work on the Task of 17. They listed all the numbers from 1–100 and began by crossing through the multiples of four and multiples of seven to represent that these numbers of chicken tenders could be bought. They continued by crossing through multiples of 11, 15, and 18. After a conversation with the facilitator, they reasoned that since they had crossed off 15 previously, they could also cross of 19 by buying another pack of four. Other numbers were eliminated using similar reasoning. A third group of teachers (work shown in Fig. 4.3) reasoned about the problem by using the expres- sion 4x + 7y and created a table of values for x, y, and the total number of tenders. 54 R. A. Hudson et al. Fig. 4.3 Group 3’s Work on the Task The type of thinking required to solve this problem stands in stark contrast to the thinking that is typical of traditional school mathematics. The work completed to craft these solutions suggest that the solving required ingenuity. They required cre- ativity and innovation within the minds of the teachers engaged in solving them, and permitted the solvers to take ownership of their creative solution strategies. The differences in the three groups’ methods demonstrate how creativity can be used in mathematical problem solving, but creativity can also play a role in compu- tational tasks. In recent years, the introduction of Number Talks (Parrish, 2014) in mathematics classrooms has provided a way to express students’ personal creativity in computation. Although many variations to Number Talks exist, a common rou- tine includes (1) a teacher poses a computational problem; (2) students mentally solve the problem in more than one way; (3) the teacher solicits possible solutions; and (4) students defend their solutions by describing different strategies. These solutions often correspond to a student-created mental strategy, rather than an algo- rithm. For example, when solving 84–68, one student might say she took 84 minus 60 to get 24, and then subtracted another 8 to get 16. A second student might say he took 84 minus 70 to get 14, and compensated by adding 2 to get 16. A third student might use an adding-up strategy by starting at 68, then adding 2, then adding 10, then adding 4 to arrive at 84. She would recognize that 2 + 10 + 4, or 16, is the difference. 4 The Nature of Mathematics and Its Impact on K-12 Education 55 4.7 Future Directions for the Nature of Mathematics Within the Context of STEM Education In 2017, the National Council of Teachers of Mathematics published the Compendium for Research in Mathematics Education (Cai, 2017), an edited volume with 1000+ pages and 38 chapters summarizing the major research completed in mathematics education for the past 10+ years. Although the index is quite detailed, there was no reference to “nature of mathematics,” suggesting that the research regarding this topic has not become mainstream among researchers in mathematics education. There is still much to learn about what students and teachers think about the nature of mathematics and how these views impact their work. STEM provides further opportunities for students to use mathematics in produc- tive ways, but does STEM activity provide opportunities for students to reflect on the nature of mathematics? The answer is inconclusive and is likely context specific. Although STEM has the potential to create rich problems to apply mathematical thinking, there are some mathematics educators with reservations about STEM. They believe that science, technology and engineering may overshadow the mathematical thinking, or that mathematics will be integrated into STEM using mathematics that students have learned previously (e.g., computation, measurement skills). Furthermore, the different disciplines that contribute to STEM may have competing interests. A concern may exist that students will not come to accept mathematics as a way of knowing. However, such assertions can be refuted by emphasizing the defi- nition of STEM as an interdependence of its constituent fields. As such, mathemat- ics should complement and support the other disciplines, just as the other STEM disciplines should complement and support the learning of mathematics. Such hypotheses about the role that mathematics may play in STEM activity provides an important area for further research endeavors. Whether our teaching falls on the side of the absolutist, fallibilists or somewhere in between, STEM’s hope for improving mathematics education lies in deliberate opportunities within a STEM context for students to explore and reason about big mathematical ideas. In doing so, they will come to see that their arguments have to be based on assumptions. By examining their assumptions and listening to alternatives, they might come to see mathematics as a creative outlet. References Australian Curriculum, Assessment and Reporting Authority. (n.d.). Key ideas. Australian Curriculum. https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/ key-ideas/ Boaler, J. (2015). What’s math got to do with it?: How teachers and parents can transform math- ematics learning and inspire success. New York: Penguin. Cai, J. (Ed.). (2017). Compendium for research in mathematics education. Reston, VA: National Council of Teachers of Mathematics. 56 R. A. Hudson et al. Davison, D. M., & Mitchell, J. E. (2008). How is mathematics education philosophy reflected in the math wars? The Mathematics Enthusiast, 5(1), 143–154. Devlin, K. (2010). The Pascal-Fermat correspondence: How mathematics is really done. Mathematics Teacher, 103, 578–582. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathemat- ics. 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Principles to actions: Ensuring mathemati- cal success for all. Reston, VA: Author. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Pair, J. D. (2017). The nature of mathematics: A heuristic inquiry. Unpublished doctoral disserta- tion. Middle Tennessee State University, Murfreesboro. Parrish, S. (2014). Number talks: Helping children build mental math and computation strategies, Grades K-5. Sausalito, CA: Math Solutions. Roach, M., Creager, M., & Eker, A. (2016). Reasoning and sense making in mathematics. In P. Kloosterman, D. Mohr, & C. Walcott (Eds.), What mathematics do students know and how is that knowledge changing? Evidence from the National Assessment of Educational Progress (pp. 261–293). Charlotte, NC: Information Age Publishing. Russell, S. J. (1999). Mathematical reasoning in the elementary grades. In L. V. Stiff (Ed.), Developing mathematical reasoning in grades K-12, 1999 Yearbook of the National Council of Teachers of Mathematics (NCTM) (pp. 1–12). Reston, VA: NCTM. Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in math- ematics education (pp. 237–266). Reston, VA: NCTM. Van Hiele, P. M. (1985). The child’s thought and geometry. In D. Fuys, D. Geddes, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp. 243–252). Brooklyn, NY: Brooklyn College, School of Education. (Original work published 1959). 4 The Nature of Mathematics and Its Impact on K-12 Education 57 Weber, K. (2001). Student difficulties in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119. White-Fredette, K. (2010). Why not philosophy? Problematizing the philosophy of mathematics in a time of curriculum reform. The Mathematics Educator, 19(2), 21–31. Wilburne, J. (2014, September 15). What is the largest number you cannot make? [Blog Post] Retrieved from https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/ What-is-the-Largest-Number-You-Cannot-Make_/ Rick A. Hudson is an Associate Professor and Chair of Mathematical Sciences at the University of Southern Indiana, and a former secondary teacher. His research interests include the teaching and learning of data analysis and statistics in K-12 schools, mathematics teacher education, and mathematical teaching practices. He is currently co-PI on the NSF-funded grant titled Enhancing Statistics Teacher Education with E-Modules. Mark A. Creager is an Assistant Professor of Mathematics at the University of Southern Indiana, and a former secondary teacher. His research focuses on preservice and inservice teachers’ math- ematical knowledge and in particular about how teachers use their mathematical knowledge in classroom settings. Additionally, he investigates students’ understandings of proof in geometry and ways of developing an understanding of proof in geometry that is meaningful for students. Angela Burgess is a Doctoral Student of Science Education at Indiana University and an informal environmental educator. Her research focuses on environmental literacy, environmental education practices and participant outcomes. Alex Gerber is a Doctoral Candidate in Science Education at Indiana University. He has experi- ence teaching in a variety of informal contexts as well as college courses in the School of Education at Indiana University. His research focuses on informal science education and, more recently, the use of representations in science teaching.