Education Research International

Volume 2019, Article ID 3745406, 13 pages

https://doi.org/10.1155/2019/3745406

## Teaching Mathematics through Concept Motivation and Action Learning

^{1}State University of New York at Potsdam, Potsdam, NY, USA^{2}University of South Florida, Tampa, FL, USA

Correspondence should be addressed to Sergei Abramovich; ude.madstop@svomarba

Received 17 August 2018; Revised 27 February 2019; Accepted 13 March 2019; Published 14 April 2019

Academic Editor: Eddie Denessen

Copyright © 2019 Sergei Abramovich et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This is a practice-led, conceptual paper describing selected means for action learning and concept motivation at all levels of mathematics education. It details the approach used by the authors to devise insights for practitioners of mathematics teaching. The paper shows that this approach in mathematics education based on action learning in conjunction with the natural motivation stemming from common sense is effective. Also, stimulating questions, computer analysis (internet search included), and classical famous problems are important motivating tools in mathematics, which are particularly beneficial in the framework of action learning. The authors argue that the entire K-20 mathematics curriculum under a single umbrella is practicable when techniques of concept motivation and action learning are in place throughout that broad spectrum. This argument is supported by various examples that could be helpful in practice of school teachers and university instructors. The authors found pragmatic cause for action learning within mathematics education at virtually any point in student academic lives.

#### 1. Introduction

Nowadays, students require both cognitive and practical experiences throughout the continua of their mathematics education to be productive 21st century citizens. The genesis of this statement can be traced back to the writings of John Dewey, who emphasized the importance of educational activities that include “the development of artistic capacity of any kind, of special scientific ability, of effective citizenship, as well as professional and business occupations” ([1], p. 307). More recently, Billett [2], based on his studies of integrating learning experiences of tertiary students in the disciplines related to nursing and like services in support of human needs, suggested that “it might be possible to fully integrate practice-based experiences within the totality of higher education experiences that are generative of developing robust and critical occupational knowledge” (p. 840). The main argument of the present paper is that in the context of mathematics education, action learning (the concept introduced in Section 3) is the very process to impart these experiences in conjunction with concept motivation (a term introduced in Section 2) when teaching mathematics across the entire K-20 curriculum. To this end, this practice-led, conceptual paper, detailing the approach used by the authors to devise insights for practitioners of mathematics teaching, offers a survey of selected means for action learning across the formal mathematics education continuum. To a certain extent, this paper promotes the idea of learning through practice [3] in the context of mathematics education. Arguments supporting the value of action learning for all individuals involved (at the college level, adding to the duo of student and mathematics instructor a third community or university nonmathematics professional) are presented (Sections 2–4). Also considered is integration of computer-assisted signature pedagogy (CASP) and nondigital technology as well as effective questioning with action learning (Sections 5 and 6).

Students may joyfully experience formal mathematics education for twenty years or more, and they can be motivated everywhere across the expansive mathematics curricula. Action learning in mathematics education combined with rote theory brings mathematical topics to the real world. Naturally, primary-level instances are of foundational importance, and this is reinforced with secondary-level action learning (Sections 4.1.1 and 4.1.2). The open problems of mathematics can often be introduced to students in primary, secondary, and tertiary education (Section 7). Traditionally, classic results and open problems serve to motivate not only the students but also the educators themselves. Since effective mathematics teachers are needed, action learning should be used promotionally at all levels of mathematics education, knowing that future instructors are amongst the current student population. Certainly, the possibility of being involved in discovery is highly motivational to all, including students and mathematics teachers, at least.

#### 2. Curiosity and Motivation

Though the necessity of mathematical learning at the primary, secondary, and tertiary schools is common knowledge, the question on how to teach mathematics is controversial. As described in more detail in [4], with references to [5–10], the controversy is due to a nonhomogeneity of teacher preparation programs, the formalism versus meaning disagreement among mathematics faculty, and various perspectives on the use of technology. We believe that an appropriate way to teach mathematics at all levels is to do it through applications rather than to use traditional lectures, emphasizing the formalism of mathematical machinery. Real-life applications keep concerned people motivated while learning mathematics. This natural motivation can be considered as an age-dependable process spanning from natural childhood curiosity in the primary school to true intellectual curiosity at the tertiary level. Regardless of the age of learners, one can see curiosity as motivation “to acquire or transform information under circumstances that offer no immediate adaptive value for such activity” ([11], p. 76). That is, curiosity and motivation are closely related psychological traits.

Most of the studies on the development of curiosity deal with the primary education. However, these studies can inform our understanding of how curiosity turns into a motivation to become high-quality professional. For example, Vidler [12] distinguished between epistemic and perceptual curiosity, which are manifested, respectively, by “enquiry about knowledge and is shown, for example, when a child puzzles over some science problem he has come across … [and] increased attention given to objects in the child’s immediate environment as, for example, when a child stares longer at an asymmetrical rather than a symmetrical figure on a screen” (p. 18). Likewise, adult learners at the tertiary level can become motivated by their mathematics instructor’s call for questions concerning information that was shared or by their experiences with the world around them as they try to interpret “the fabric of the world … [using] some reason of maximum and minimum” (Euler, cited in [13], p. 121).

Related to the tertiary level, Vidler [14] defined achievement motivation as “a pattern of … actions … connected with striving to achieve some internalized standard of excellence” (p. 67). There are also adult learners who “are interested in excellence for its own sake rather than for the rewards it brings” ([14], p. 69). Biggs [15] admits that intrinsic motivation in the study of mathematics is associated with “the intellectual pleasure of problem solving independently of any rewards that might be involved … [suggesting that] the aims of deep learning and of achievement motivation ultimately diverge” (p. 62). A classic example in support of this suggestion is a solution of the (century old) Poincare conjecture by geometer Grigory Perelman who, after almost a decade of “deep learning,” declined several international awards for his work including the Fields Medal (the mathematician’s “Nobel Prize”) and ($1 million) Clay Millennium Prize (https://www.claymath.org/).

As curiosity is the genesis of motivation to learn, Mandelbrot [16], in a plenary lecture on experimental geometry and fractals at the 7th International Congress on Mathematical Education, advised the audience of mostly precollege mathematical educators of how to pivot on curiosity when teaching mathematics: “Motivate the students by that which is fascinating, and hope that the resulting enthusiasm will create sufficient momentum to move them through that which is no fun but is necessary” (p. 86). It is this kind of motivation that the authors describe as concept motivation. More specifically, in this paper, the term concept motivation means a teaching strategy through which, using curiosity of students as a pivot, the introduction of a new concept is justified by using it as a tool in applications to solving real problems. For example, the operation of addition can be motivated by the need to record the augmentation of a large quantity of objects by another such quantity, the concept of irrational number can be motivated by the need to measure perimeters of polygonal enclosures on the lattice plane (called the geoboard at the primary level), or the concept of integral can be motivated by the need to find areas of curvilinear plane figures.

Another mathematically relevant instrument of motivation is concreteness. According to David Hilbert, mathematics begins with posing problems in the context of concrete activities “suggested by the world of external phenomena” ([17], p. 440). We believe that “concreteness” is an appropriate synonym for motivation as it relates to mathematics education. The term *concrete* itself indicates that a variety of ingredients are brought together and synthesized. The goal of learning mathematics is to concretize notions, both theoretical and applied. It is helpful to have a precise understanding of something. Humans inherently wish to have “full” knowledge of certain things. By knowing details, and concretizing ideas, we reduce anxiety associated with describing and using those ideas. Concreteness motivates all parties involved in mathematics education. Even at the administrative level, there is understanding that “the FKL [Foundations of Knowledge and Learning] Core Curriculum will provide you with the opportunity to explore a variety of vital areas of study, making you more aware and engaged in understanding the challenges that our global *realities* require” ([18], *italics* added), where the “realities” is given our emphasis. This is motivation for everybody, since we would all like to make use of mathematical theory or, at least, see it applied. Consequently, motivation is proportionally higher for adult learners over children who may not see “usefulness” in mathematics. At the University of South Florida, instructors of certain courses (the calculus sequence, for example) are asked to include the FKL statement in their syllabi.

Until recently, the terms “industrial” and “technical” had rather pejorative connotations in mathematics education. Traditional formal lecturing is still dominant in most classrooms. However, there is often some “industry” or “technique” in examining mathematical theory, so these two notions are not complimentary. It is hard to identify a part of the massive volume of K-20 mathematics curricula which precludes either theory or eventual real-world application. Furthermore, theory is implicitly included in STEM education due to its science component.

In the context of mathematics teacher education, a focus on applications gives future teachers one very important ability of exemplifying mathematical ideas in ways which are usable. This ability can then be imparted to their own students. One can recognize at the precollege level that mathematics knowledge stems from the need to resolve real-life situations of different degrees of complexity. The curriculum principle put forth by the National Council of Teachers of Mathematics [19] includes the notion that all students at this level should be offered experiences “to see that mathematics has powerful uses in modeling and predicting real-world phenomena” (pp. 15-16). This emphasis on applications goes beyond the precollege level. Indeed, mathematics has been greatly developing and penetrating all the spheres of life, making collegiate mathematics education a necessary yet controversial element of the modern culture.

#### 3. Action Learning

Many people are pragmatic by doing what works. When something does not work, one is compelled to ask questions as to how to make it work. Beginning from the 1940s, Reginald Revans started developing the action learning concept, a problem-solving method characterized by taking an action and reflecting on the results, as an educational pedagogy for business development and problem-solving [20, 21]. Since that time, action learning has come to describe a variety of forms it can take and contexts it can be observed. In the context of achieving high quality of university teaching, “the target of action learning is the teaching of the individual teacher” ([22], p. 7). In the general context of improving professional performance, Dilworth [23] argues that action learning starts with an inquiry into a real problem so that regardless whether the problem is “tactical or strategic… [the process of] learning is strategic” (p. 36). Action learning in mathematics education can be defined as learning through student individual work on a real problem followed by reflection on this work. In most cases, this work is supported by a “more knowledgeable other.”

In mathematics education, action learning, the genesis of which is in the early childhood experience, has natural levels of maturity. Before we become concerned with the day-to-day responsibilities attached to adulthood, we can freely consider action learning in a game form. Our fondness for gaming and for learning winning strategies are carried into later life, both as means of entertainment and as a tool for instructing the next generation of children. The motivation for action learning in mathematics education gradually changes from winning games to success in real-world ventures. The key to success is the ability to solve problems. Research finds that curiosity can be characterized in terms of excitement about peculiar observations and unexpected phenomena [24]. Additionally, “What children will be curious about depends in large part on the nature of the world about them and their previous experience” ([12], p. 33). Students at all educational levels seek concreteness, are naturally curious about the real world, and enjoy benefits of action learning, especially when they use it repeatedly in mathematics education. In particular, in the postsecondary mathematics curriculum for nonmathematics majors, the problems should have applicability to reality. Interestingly, we seem to return to “gaming” when we deal with pure theory, since we might seek an abstract solution for the sake of solution itself.

Max Wertheimer, one of the founders of Gestalt psychology, argued that for many children, “it makes a big difference whether or not there is some real sense in putting the problem at all” ([25], p. 273). He gave an example of a 9-year-old girl who was not successful in her studies at school. In particular, she was unable to solve simple problems requiring the use of basic arithmetic. However, when given a problem which grew out of a concrete situation with which she was familiar and the solution of which “was required by the situation, she encountered no unusual difficulty, frequently showing excellent sense” ([25], pp. 273-274). Put another way, the best strategy to develop students’ interest in a subject matter is to focus teaching on topics that are within their basin of attraction. As William James, a classic of American psychology, who was the first to apply it to the education of teachers, put it, “Any object not interesting in itself may become interesting through becoming associated with an object in which an interest already exists” ([26], p. 62). Interest can be also used to develop motivation in education as it “refers to pattern of choice among alternatives—patterns that demonstrate some stability over time and that do not appear to result from external pressures” ([27], p. 132).

Reflection is as important as action. Being able to reflect on action carried out constitutes the so-called internal control when individuals think of themselves as being responsible for their own behavior, something that is different from external control when seeing others or circumstances being the primary motivation for an individual behavior [28]. Three basic questions commonly begin the action learning process in addressing a real problem. We ask: First, what should be happening? Second, what is stopping us from doing it? Third, what can we do?

Action learning (often referred to in academia as action research [29, 30]) has been traditionally used for teaching business management and the social sciences [31, 32], conducting scientific research [33], and teacher development [22, 34–36]. In mathematics education [4, 37], action learning, as a teaching method, has been adopted as pedagogy oriented on self-solving real problems followed up by reflection. Learning is the primary goal, even though the problem-solving is real and important. Learning is facilitated by breaking out well-established mind-sets, thereby presenting a somewhat unfamiliar setting for the problem. We now have the technology-assisted, action learning pedagogy for teaching mathematics through real-world problems, guided by STEM instructors and community professionals, employing a project component [4]. Digital technology is seen at least within the requisite typology of the manuscripts. It may go much further, of course, and include an essential utility (e.g., a numerical integrator, a spreadsheet, or specialized software). Finally, *action* learning (with origins in business education [20, 21]) provides an effective and clear approach to mathematics education. This approach was developed out of different (and, as mentioned at the beginning of Section 2, sometimes controversial) *active* learning techniques which are ubiquitous among mathematics educators across a variety of constructivist-oriented, student-centered teaching contexts [38–41].

#### 4. Action Learning in the Practice of Mathematics Education

Our USF-SUNY team [4] has established that action learning is a positive pedagogical feature throughout all grade levels (K-20). One may argue that since many people are lifelong learners, some of us may employ action learning (perhaps as mathematics instructors) beyond K-20. Our motivation to action learning mathematics can give young students a taste of the interesting things known of mathematics. The underlying concepts can be quite sophisticated and students may return to the ideas and take them further as they gather experience. Examples of action learning are presented in the subsections below by instruction level. These examples are given with an emphasis on the goal of concreteness, which in turn motivates the learners. Employing a project component makes the “one + two” Mathematics Umbrella model available at the tertiary level (Section 4.2.2).

##### 4.1. Motivation and Action Learning at the Primary and Secondary Levels

At the primary school level, mathematical concepts can be motivated through the appropriately designed hands-on activities supported by manipulative materials. Such activities have to integrate rich mathematical ideas with familiar physical tools. As was mentioned above, an important aspect of action learning is its orientation towards gaming. A pedagogical characteristic of a game in the context of tool-supported mathematics learning is one’s “thinking outside the box,” something that in the presence of a teacher as a “more knowledgeable other” opens a window to students future learning. Nonetheless, the absence of support can be observed, as Vidler [12] put it, “when a child stares longer at an asymmetrical rather than a symmetrical figure” (p. 18) recognizing intuitively, through perceptual curiosity, that stability of a figure depends on its position. That is, perceptual curiosity combined with creative thinking often transcends activities designed for one level and merges into the study of more advanced ideas at a higher cognitive level. The following two sections demonstrate how the use of two-sided counters and square tiles, physical tools commonly used nowadays in the elementary mathematics classroom, can support, respectively, the introduction of Fibonacci numbers, allowing one through the use of computing to open a window to the concept of the Golden Ratio, and to connect the construction of rectangles (out of the tiles) to the discussion of special numeric relationships between their perimeters and areas. In both cases, the transition from the primary level to the secondary one can be facilitated by the use of digital technology. That is, mathematical ideas, born in the context of action learning with physical tools, can be extended to a higher level through computational experiments supported by digital tools.

###### 4.1.1. From Two-Sided Counters to the Golden Ratio through Action Learning

Consider the following action learning scenario:

Determine the number of different arrangements of one, two, three, four, and so on two-sided (red/yellow) counters in which no two red counters appear consecutively.

Experimentally, one can conclude that a single counter can be arranged in two ways, two counters in three ways, three counters in five ways, and four counters in eight ways (Figure 1). In particular, Figure 1 shows that all the arrangements with four counters can be counted through recursive addition 3 + 5 = 8 as they can be put in two groups so that in the first group (with cardinality three), the far-right counter is red, and in the second group (with cardinality five), the far-right counter is yellow. By putting this idea into action under the guidance of a teacher, a young student can discover that the next iteration (five counters–13 ways, as 13 = 5 + 8) agrees with the description of Figure 1. Augmenting, for consistency, the sequence 2, 3, 5, 8, 13 by two ones (assuming that an empty set of counters has only one arrangement) allows one to describe the completion of the above action learning scenario (that is, reflecting on the results of acting on concrete materials according to a certain rule) through the sequence 1, 1, 2, 3, 4, 5, 8, 13, …, (in which the first two numbers are equal to one and every number beginning from the third is the sum of the previous two numbers)—one of the most celebrated number sequences in the entire mathematics named after Fibonacci (1270–1350), the most prominent Italian mathematician of his time. As part of reflection on the scenario, young students can be told that as esoteric as Fibonacci numbers might seem, they are likely to encounter them again.