Learning+Mathematical+Reasoning

= **Introduction** =

toc Data suggests that the supply of a STEM educated employees falls short of the demand in many parts of the workforce [1]. Difficulty with introductory STEM classes, the calculus sequence among them, contribute to a low retention rate in STEM fields. An attrition rate of nearly 50% in STEM fields justifies c ontinued research in mathematics education [2]. Mathematical achievement levels in the United States have been found to be lower than in other developed countries. Demographic factors influence these achievement levels. Procedural and conceptual knowledge are to central ideas to mathematics education research. Around these two concepts, educational theories regarding the development of mental frameworks have been studied. These educational theories need to be interpreted in the light of the results of classroom studies of mathematics education.

=Mathematical Achievement in the United States=

As noted in the introduction, many areas of STEM employment have demand for workers outstripping the rate at which STEM majors graduate in the US [1]. Increasing the rate of graduation among STEM majors could help improve this statistic. Statistics suggest that the United States suffers from a low level of mathematical achievement relative to other developed countries (see Figure 1). Additionally, a principle culprit for failure in STEM is a [|lack of sufficient preparation] for college level mathematics among freshmen (see Figure 2).



= **Demographic Factors** =

Among student groups taking college mathematics courses, some are more likely to underachieve than others (see Figure 3). [|Women are more likely to leave STEM majors] after taking calculus. Likewise, in pre-collegiate mathematics education, there are strong statistical correlations between underachievement in mathematics and ethnicity and socioeconomic status with children from wealthy families often outperforming children from poorer families and children from white or asian families often outperforming children from black or hispanic families [3].

The level of achievement in mathematics attained by an individual can be shown to be sensitive to a number of environmental factors. These include family dynamics, school characteristics, a country’s wealth, and the level of equality in a country [4]. Because underachievement in pre-collegiate mathematics is linked to success in STEM majors in college, these trends of underachievement along certain demographic makers persists in college. Additionally, the feeling of not belonging, known as [|imposter syndrome], experienced by some underrepresented groups in STEM fields is common. This lack of confidence in one’s ability to succeed could contribute to STEM dropout rates.

= **Procedural and Conceptual Knowledge** =

Two types of knowledge are often discussed in education research: conceptual knowledge and procedural knowledge. Put simply, conceptual knowledge is the understanding of mathematical ideas while procedural knowledge is the ability to solve problems. The relative importance of procedural and conceptual knowledge is of great interest because pre-collegiate math curricula are designed based on the findings of the education research.

The relationship between procedural knowledge and conceptual knowledge and their relative importance is a contentious debate in mathematics education [5]. The ability to abstract seems to be intimately related to the acquisition of deep conceptual knowledge. A core question of this research is whether mathematical fluency is gained through first acquiring procedural knowledge or first acquiring conceptual knowledge.

Conceptual knowledge organizes mathematical thought. The compression of mathematical knowledge into thinkable concepts achieved through conceptual knowledge is essential to developing sophisticated problem solving skills [6]. In order to avoid the issue of students viewing mathematics as a deluge of disconnected techniques, it is necessary to develop an overarching conceptual framework.

A mental framework represents an interconnected web of concepts. It is a student constructed model of the relationships between topics that allows one to access the knowledge required to solve a problem when prompted by familiar patterns. The richer the connections between different concepts, the more easily a student can access previously learned knowledge [7]. A mental framework matures over time. Initially a student categorizes and recognizes similarities between concepts. Later, a student will create theories to fit new concepts into their existing mental framework [8].

= **Mental Frameworks: Theoretical Perspective** =

Mathematical concepts deepen over time. While adults find the concept of numbers concrete and accessible, a sophisticated mental framework must be constructed [|in early childhood education] in order to understand this concept. In fact, success in developing a mental framework supporting the concept of numbers is a strong predictor of future success in mathematics [9]. At every stage of development, sophisticated hierarchies of knowledge must be developed to support mathematical reasoning and problem solving.

The construction of a mental framework for abstract thought through conceptual understanding has many facets. There is the way that one’s personal history and personality colors one’s perception of a concept, there are the connections one makes between objects and processes, and there is the way that conceptual hierarchies are formed based on levels of conceptual complexity [10].

Due to personal history and personality, students learn in different ways. For example, the previously mentioned issue of imposter syndrome can influence students’ confidence. Teachers need to understand that students have different backgrounds. A one-size-fits-all approach can create issues [11]. The way different students build their conceptual framework, and, perhaps more importantly, the existing mental framework students have coming into a class greatly influences the way they learn [12].

Another important mental construct is object-process duality. Mathematics is composed objects and processes (e.g. there is the object of an integer and the related process of addition). Objects and processes often have a fuzzy line between them. One classroom study found that students’ understanding of the meaning of the “equals” sign changes from a simple process based concept to a more complex object-process duality based concept over time [13]. Initially, students typically view the equals sign as indicating the completion of a sequence of calculations. However, after taking algebra, students often understand that two things on either side of an equals sign are identical.

The third notion is a hierarchy of complexity. Mathematics builds on itself. Each new topic a student learns makes connections to older, simpler concepts. Each level of the hierarchy depends on more basic concepts. Comfort with constructing such hierarchies is critical in developing a cohesive mental framework.

= **Mental Frameworks: Empirical Findings** =

Developing abstract thinking skills is a primary goal in mathematics education. This is not only valuable when doing math but some believe that it aids in reasoning skills applicable to other disciplines [14]. Research suggests that it is inappropriate to expose children to abstract concepts prematurely [15]. This would not respect the preexisting mental framework of the majority of incoming students and could alienate all but the most naturally gifted students.

The complex array of skills required for advanced work in mathematics accounts for the low frequency of mathematics prodigies [16]. A foundation of concreteness provides a student with the best foundation to build a hierarchy of mathematical knowledge. The construction of the mental framework to support abstract thinking skills was discussed from a theoretical perspective above. However, the construction of such a mental framework in a classroom setting takes a concrete form.

Researchers have found that deficient understanding of variables is intimately related with failure in calculus. Additionally, the researchers found that students are lacking a conceptual understanding of variables because curricula move to a purely symbolic based treatment of algebra too quickly [17],[18]. It aids students to motivate the usage of symbols in a more concrete manner before exclusively using symbolic representations in algebra. Further, the notion of functions is often poorly understood by students in precalculus. An earlier exposure to functions in a highly concrete context could benefit students greatly.

Researchers argue that three skills that indicate a student’s facility with abstract thought are the ability to construct, recognize, and build-with mathematical concepts [19]. The presence of these skills demonstrates a well constructed mental framework supporting mathematical thought. These skills are best developed using a mix of practice problems and worked out examples. In particular, open-ended questions and worked examples have been found to help students build a cognitive framework [20]. Drilling students with practice problems is a standard teaching device. However, studies have show that different tactics are more effective.

Studies have found that exposure to example solutions and guided practice problems aid in student development of conceptual understanding [21],[22]. Reading example solutions puts less mental strain on children allowing them to focus on the logic behind a method of problem solving. These studies found that children perform better as problem solvers after being exposed to incorrect examples and being asked to explain why an answer is incorrect. These classroom activities help children learn how to construct solution techniques, recognize the type of problem being asked, and build-with previous knowledge to find novel solutions to unfamiliar problems.

Problem solving is also an essential component of learning. Problem solving based learning makes mathematics more relevant to children, more application oriented, and can better facilitate the development of procedural and conceptual understanding [23]. A key component of a well designed problem set is spacing and composition [24]. Spacing practice problems means temporally separating problems of similar type. Mixing practice means presenting problems of different type alongside one another.

Traditionally, problem sets are presented at the end of a textbook chapter thus asking children to solve an excess of similar problems all at once. Studies have shown that mixing problems from different units in a single problem set improves retention. Many people learn mathematics as a set of disjoint techniques hindering their ability to make connections and develop a coherent mental framework. Well designed problem sets can help children connect techniques of problem solving with their big picture conceptual understanding. Computational fluency entails an integrated understanding of both methods of solution and an understanding of the concepts involved [25].

Historically, procedural knowledge has been characterized as a superficial kind of knowledge and conceptual knowledge as deep [5]. Research suggests that the development of both types of knowledge is concurrent. The results cited in the previous section suggest that knowledge of procedures should not be shortchanged in education. All aforementioned classroom studies have shown that conceptual knowledge is most effectively developed in the context of examples and well designed problem sets. Some of the most effective examples seem to be those where children are asked to explain and correct mistakes in a solution.

These methods of fostering deep conceptual understanding require that the children already understand procedures in some basic manner. Therefore, attempts to encourage growth in conceptual understanding are counterproductive if attempts are not made at fostering procedural knowledge.

= **Conclusion** =

Much educational research is based on idealogical values [5]. Educational research tells us what is effective and what is not. However, values should be argued from a basis of data rather than on ideological grounds [26]. Neglect of emphasis on procedural knowledge in favor of conceptual knowledge has been done on an ideological basis [5]. However, evidence from classroom studies suggests that procedural knowledge is a precursor to conceptual knowledge.

The interconnectedness of the two types of knowledge suggests the goal of mathematics education should be to focus on a mixture of the two—sometimes called //[|procedural fluency].// Teachers must be taught methodologies for teaching procedural fluency. Otherwise, teachers will resort to traditional programs. Studies show that traditional programs do not effectively facilitate the development of procedural fluency [27].

Success in STEM majors in college is dependent on the preparedness of incoming freshmen. As early as pre-algebra, care must be taken to motivate abstract symbols and concepts in a way that allows new material to build on the prior experiences of the student. The above research suggests that successful curricula will include worked examples, dialog based critique, and well designed problem sets that motivate new material in the context of previously mastered material.

= **References** =

[1] D. A. Koonce et al, "STEM Education vs. Occupation: a Supply and Demand Analysis," //IIE Annual Conference. Proceedings//, pp. 1, 2011.

[2] X. Chen, RTI International and National Center for Education Statistics (ED), "STEM Attrition: College Students' Paths into and out of STEM Fields. Statistical Analysis Report. NCES 2014-001," //National Center for Education Statistics//, 2013.

[3] C. Riegle-Crumb and E. Grodsky, "Racial-Ethnic Differences at the Intersection of Math Course-taking and Achievement," //Sociology of Education//, vol. 83, (3), pp. 248-270, 2010.

[4] M. M. Chiu, "Effects of Inequality, Family and School on Mathematics Achievement: Country and Student Differences," //Social Forces//, vol. 88, (4), pp. 1645-1676, 2010.

[5] J. R. Star, "Reconceptualizing Procedural Knowledge," //Journal for Research in Mathematics Education//, vol. 36, (5), pp. 404-411, 2005.

[6] E. Gray and D. Tall, "Abstraction as a Natural Process of Mental Compression," //Mathematics Education Research Journal//, vol. 19, (2), pp. 23-40, 2007.

[7] G. Michell and P. Dewdney, "Mental Models Theory: Applications for Library and Information Science," //Journal of Education for Library and Information Science//, vol. 39, (4), pp. 275-281, 1998.

[8] M. Mitchelmore and P. White, "Abstraction in Mathematics Learning," //Mathematics Education Research Journal//, vol. 19, (2), pp. 1-9, 2007.

[9] A. Leavy and M. Hourigan, "Using Lesson Study to Support the Teaching of Early Number Concepts: Examining the Development of Prospective Teachers’ Specialized Content Knowledge," //Early Childhood Education Journal//, vol. 46, (1), pp. 47-60, 2018.

[10] O. Hazzan and R. Zazkis, //Reducing Abstraction: The Case of Elementary Mathematics//. 2003.

[11] J. W. Stigler and J. Hiebert, "Understanding and Improving Classroom Mathematics Instruction: An Overview of the TIMSS Video Study," //Phi Delta Kappan//, vol. 79, (1), pp. 14, 1997.

[12] M. G. Jones, "Transfer, Abstraction, and Context," //Journal for Research in Mathematics Education//, vol. 40, (2), pp. 80-89, 2009.

[13] E. Warren and T. J. Cooper, "Developing Mathematics Understanding and Abstraction: The Case of Equivalence in the Elementary Years," //Mathematics Education Research Journal//, vol. 21, (2), pp. 76-95, 2009.

[14] E. Dubinksy, "Mathematical Literacy and Abstraction in the 21st Century," //School Science and Mathematics//, vol. 100, (6), pp. 289-297, 2000.

[15] M. C. Mitchelmore, //The Role of Abstraction and Generalization in the Development of Mathematical Knowledge//. 2002.

[16] R. Perna and A. R. Loughan, "The Complexities of Math Skills Development" //International Journal of Mathematics, Game Theory, and Algebra//, vol. 23, (3), pp. 165-165, 2014.

[17] P. White and M. Mitchelmore, "Aiming for Variable Understanding," //The Australian Mathematics Teacher//, vol. 72, (3), pp. 50-52, 2016.

[18] P. White and M. Mitchelmore, "Conceptual Knowledge in Introductory Calculus," //Journal for Research in Mathematics Education//, vol. 27, (1), pp. 79-95, 1996.

[19] R. Hershkowitz, B. B. Schwarz and T. Dreyfus, "Abstraction in Context: Epistemic Actions," //Journal for Research in Mathematics Education//, vol. 32, (2), pp. 195-222, 2001.

[20] P. White and M. Mitchelmore, //Learning Mathematics: A New Look at Generalization and Abstraction//. 1999.

[21] J. L. Booth et al, //Differentiating Instruction: Providing the Right Kinds of Worked Examples for Individual Students//. 2013.

[22] J. L. Booth et al, //Using Example Problems to Improve Student Learning in Algebra: Differentiating between Correct and Incorrect Examples//. 2013.

[23] J. Hiebert et al, "Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics," //Educational Researcher//, vol. 25, (4), pp. 12-21, 1996.

[24] D. Rohrer, "The Effects of Spacing and Mixing Practice Problems," //Journal for Research in Mathematics Education//, vol. 40, (1), pp. 4-17, 2009.

[25] S. J. Russell, "Developing Computational Fluency with Whole Numbers," //Teaching Children Mathematics//, vol. 7, (3), pp. 154-158, 2000.

[26] J. Hiebert, "What Can we Expect from Research?" //Mathematics Teaching in the Middle School//, vol. 5, (7), pp. 413, 2000.

[27] J. Hiebert, "Relationships between Research and the NCTM Standards," //Journal for Research in Mathematics Education//, vol. 30, (1), pp. 3-19, 1999.