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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.

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” ([

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

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 [

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 [

Related to the tertiary level, Vidler [

As curiosity is the genesis of motivation to learn, Mandelbrot [

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” ([

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 [

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 [

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 [

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” ([

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 [

Action learning (often referred to in academia as action research [

Our USF-SUNY team [

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 [

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

Pictorial representation of Fibonacci recursion [

Indeed, at the secondary level, Fibonacci numbers

In connection with the use of two-sided counters in the context of Fibonacci numbers, it should be noted that many teacher candidates believe that concrete materials can only be utilized at the elementary level and beyond that level, they are of no use. With this in mind, the authors would like to argue that, just as with Fibonacci numbers, concrete materials can be used to introduce rather sophisticated concepts in order to add the factor of concreteness to the study of abstract ideas. In particular, two-sided counters can serve as an embodiment of binary arithmetic in an introductory computer science course. More specifically, if one writes down the first 16 natural numbers in the binary form, then, with the support of two-sided counters, one can see the following. There are two one-digit numbers with no 1’s appearing in a row (no red counters back to back), three two-digit numbers with no 1’s appearing in a row, five three-digit numbers with no 1’s appearing in a row, and eight four-digit numbers with no 1’s appearing in a row. The numbers 2, 3, 5, and 8 are consecutive Fibonacci numbers which, thereby, can be used as bits of students’ previous knowledge in developing new ideas through action learning. For more secondary (and tertiary)-level explorations with Fibonacci numbers, see [

Evidently, motivation becomes connected to an anticipated future success as a consequence of adolescence. Students now seek greater concretization of concepts. When students at the secondary level have strong motivation for action learning, they can, and do, produce undergraduate-quality projects, as described for undergraduates in Section

Humans are creative when they are motivated, and one may be more creative following general, formative concretizations of ideas. It is important to recognize student creativity early. Educators see creativity as “one of the essential 21st century skills … vital to individual and organizational success” ([

An elementary teacher candidate, working individually with a second-grade student (under the supervision of the classroom teacher), asked him to construct all possible rectangles out of ten square tiles (a real problem for grade two), expecting the student to construct two rectangles, 1 by 10 and 2 by 5, each of which representing a multiplication fact for the number 10, something that would be studied later (in grade three). The teacher candidate was surprised to see three rectangles as shown in Figure

Eight tiles—two rectangles with no windows.

Setting

A rectangle (square) with a square window [

To conclude this section, note that the troika, an elementary student, a classroom teacher, and a teacher candidate, can be compared in the context of action learning with that of an undergraduate student, a mathematics faculty, and a subject area advisor as described below in Section

Mathematics language is abstract with greater abstraction at higher levels. Traditionally, university mathematics for nonmathematics majors is taught by distancing it from reality with no connection to students’ professional interests. In this setting, quite a number of soon-to-be professionals do not see the importance of mathematics in their prospective fields [

The entire collegiate mathematics curriculum for nonmathematics majors can benefit from action learning. It is found that, particularly at the collegiate level, there should be a “middle-of-the-road” stance on the relative weights given to theory and application. The Mathematics Umbrella Group (MUG) at the University of South Florida (USF), initiated by Arcadii Grinshpan in 1999 [

The hallmark of MUG is its stratagem of interconnecting

Applied mathematics projects connect students with academic and industrial STEM professionals [

Another strong feature is the community ties which are possible or the interdisciplinary connection that at least takes place beyond the institution’s mathematics faculty. Action learning brings “reality” to the abstractions of mathematics. Even when mathematics instructors try to supply problems with applications, the usefulness is not known firsthand until the students put it to use. This is a motivational approach for all parties in the trio. Students may later elect to conduct research in connection with their project experiences. Also, they are likely to retain the concepts involved longer than they might have in the “pure lecture” approach.

Action learning is a strong motivating factor for all participants involved in the Mathematics Umbrella Group. This factor seems to be a common thread throughout the K-20 action learning spectrum. The participants’ interest in action learning may be proportional to individual experience. Mathematics instructors may potentially get the biggest benefit, but students are expected to know enough of the theory to be motivated as well. For the undergraduate mathematics courses such as calculus II and III, it is deemed sufficient for students to prevail on several smaller tests and homework assignments and then to devote their energies toward action learning, rather than requiring them to succeed on the final examination. In particular, this action learning pedagogy helps students who are “marginally successful” by allowing their final grades to include an action learning component which is justifiably given significant weight in the overall grading for the course.

More often, there are “standard achievers” who may be very productive with their action learning projects. There is the potential for students’ work to be published, or perhaps even honored [

Notably, students are naturally motivated by success in their mathematics courses. The influence of action learning has been analyzed at the University of South Florida in courses of engineering calculus involving thousands of students enrolled in these courses and follow-up courses from Spring 2003 to Spring 2015 [

Mean pass rates and 95% confidence intervals for Engineering Calculus II by race/ethnicity [

Motivation for mathematics instructors derives from exposure to new experiences with action learning. There are now many hundreds of action learning projects on record, representing a wide range of topics. Additionally, there is always some fine action learning going on, which is never documented. Of those projects which are available in the Undergraduate Journal of Mathematical Modeling: One + Two (UJMM) [

In addition to the many published undergraduate projects, there are “action learning scenarios,” which might be viewed as amalgams of different action learning experiences. Several idealistic problems have this mixed experience derivation. The problems might be considered typical of what

Questions posed generally become more sophisticated as students mature. Instructors at all levels of mathematics education use knowledge and experience to answer questions. Concrete and confident responses are desired, with the occasional possibility (generally at higher levels) that questions may require additional reflection prior to their exposition. In the context of problem-posing and problem-solving, it is important that one distinguishes between two types of questions that can be formulated to become a problem: questions seeking information and questions requesting explanation of the information obtained [

What does it mean that teachers need to possess “deep understanding” of mathematics? Why do they need to have such understanding? There are several reasons for prospective teachers to be thoroughly mathematically prepared in order to have positive effects on the progress of young learners of mathematics. First, in the modern mathematics classroom, students of all ages are expected and even encouraged to ask questions. In the United States, the national standards already for grades pre-K-2 suggest, “Students’ natural inclination to ask questions must be nurtured… [even] when the answers are not immediately obvious” ([

Just across the border with the United States, the Ontario Ministry of Education in Canada, through their mathematics curriculum for early grades, sets expectations for teachers to be able to “ask students open-ended questions… encourage students to ask themselves similar kinds of questions… [and] model ways in which various kinds of questions can be answered” ([

At the undergraduate level, second-order questions are often discussed. Mathematics instructors are aware that such questions can be valuable for stimulating further inquiries. It may be true that mathematics encountered at the primary and early secondary levels should be unimpeachably understood by mathematics instructors and that students can be “sure” of what is taught. When we begin dealing with, say set theory or two/three-dimensional geometry, there can be enigmatic results which truly stimulate learners to consider studying higher mathematics. The curiosities of mathematics are the things which learners are likely to find attractive. Certainly, it is good for the mathematics instructor to have deep understanding of the topic; however, there may be details to an answer which defy immediate conjuring. In a few rare cases, an answer is not even available. It is expected that students’ maturity will allow them to accept that at the higher mathematics levels they are not to lose faith and respect for the instructor, if an explanation is deferred. At earlier stages in mathematics education, learners believe that mathematics is perfect. However, mathematics is just as imperfect as anything else devised by human beings. Students should know this.

Curiosity and motivation can also be supported by the use of digital tools as instruments of action learning. As it was shown through examples from precollege mathematics education, computers can facilitate a transition from one cognitive level to another (higher) one. This is consistent with the modern-day use of computers in mathematics research when new results stem from computational experiments. For example, the joy of transition from visual to symbolic when two-sided counters were suggested as means of recursively developing Fibonacci numbers, which could then be modelled within a spreadsheet where, perhaps by serendipity, a definitive pattern in the behavior of the ratios of two consecutive terms could be discovered. This discovery motivates the formal explanation of why the ratios behave in a certain way. Likewise, the transition from numeric description of rectangles in terms of perimeter and area leads to their formal representation. While a rectangle with a hole was discovered by thinking “out of the box,” the availability of a digital tool facilitates the transition from visual to symbolic with the subsequent use of the latter representation in a mathematical modeling situation.

The power of computational modeling can serve as a motivation for developing and then exploring more complicated recurrence relations than that of Fibonacci numbers. As discussed in [

All this leads to the notion of computer-assisted signature pedagogy (CASP) when encouraging reflection and supporting analysis of the action taken by a student in the context of action learning provides CASP with the deep (rather than surface) structure of

The student of mathematics (at any level of education) is likely to encounter exposure to the “futility” of mathematical perfection. In mathematics, there are easily expressed questions (conjectures) which defy answers (proof). It seems to be analogous to the Heisenberg uncertainty principle where there are “limits to precision” in finding both position and momentum, for example. The important notion is that there are not always “standard” solutions to mathematical problems. Knowing this, students can possibly develop further mathematics to resolve some problems. There is a “nonstandard” action learning at work in these cases. The initial pondering is largely theoretical, but eventually an application will be summoned. Notice that the problem need not even be solved, much is bound to be learned in the attempt. This process is motivational. Also, the reflection brings concreteness to the concepts within the problem and relates to the overall “nature” of problems and problem-solving.

Real-life applications of mathematics provide a great deal of stimulation for various kinds of research in the subject matter field, involving professional mathematicians and students of different majors alike. This is not to say that applied mathematics is the only meaningful source of the development of mathematical thought. Indeed, there are many problems within mathematics itself that used to motivate and keep motivating those who seek to gain full appreciation of mathematics as a fundamental science. Some of these problems (sometimes referred to as conjectures) can be recommended to be a part of mathematics curriculum for nonmathematics majors as well as for teacher candidates. The authors’ experience indicates that theorems and conjectures with origins in both pure and applied mathematics have the potential to trigger imagination and thought process of those whose mind is open to challenge.

For example, the statements and historical details of such exciting problems as Fermat’s Last Theorem proved by Andrew Wiles [

Fermat’s Last Theorem states that

The Bieberbach conjecture states that for each

There is also famous Goldbach’s conjecture [

Another famous yet easy to understand problem is the Palindrome conjecture [

It appears that using technology for meaningful experimentation with numbers under the umbrella of CASP has the potential to inspire and motivate students already at the precollege level towards new discoveries in elementary number theory. By expanding our understanding of mathematics in any way, we potentially expand our ability to “flourish.” This is the inherent value and motivation for action learning. All of mathematics is conjectured to provide applications. We only need be motivated to devise those applications.

This paper, using the authors’ experience in mathematics teaching and supervising applications of the subject matter in the practice of public schools and industry, introduced the framework of the joint use of action learning and concept motivation in the context of K-20 mathematics education. Different examples of the action learning—an individual work on a real problem followed by reflection under the supervision of a “more knowledgeable other”—have been provided. Such supervision may include a “duo of others”—a classroom teacher and a teacher candidate in a K-12 school, and mathematics faculty and subject area advisor at a university. The paper has demonstrated that action learning of mathematics goes hand in hand with concept motivation—a teaching methodology where the introduction of mathematical concepts is motivated by (grade appropriate) real-life applications which may include student action on objects leading to formal description of this action through the symbolism of mathematics. This approach is based on notable recommendations by mathematicians [

The main concluding message of the paper is that by repeatedly utilizing concept motivation and action learning at all levels of mathematics education, overall student success has great potential to improve. This message is supported by examples of creative thinking of young learners in the classroom grounded in comprehensive collaboration of school teachers and university faculty (in the spirit of the Holmes Group [

At the onset of formal mathematics education, schoolchildren should begin experiencing action learning and concept motivation pedagogy enhanced, as appropriate, by asking and answering questions and learning to use technology. As was shown in the paper, not only K-12 mathematics curricula of many countries support student learning through asking questions but also their future teachers appreciate that kind of mathematical learning. Likewise, computer-assisted signature pedagogy [

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.