Mathematization Through Application and Common Sense: Motivating Intellectual Activities of Schoolchildren with Digital Tools

Abstract

This study demonstrates how mathematical ideas can be developed through genuine applications to problems that are attractive to the learners of mathematics due to consistency with their life experiences. To this end, the paper provides several examples of digital instruments both commonly available and designed by the authors with the goal to prepare schoolchildren of different ages to mathematize basic models of computer science and engineering. The mathematization includes construction and optimization of the models by using big ideas of mathematics at the level of common sense alone as a grade-appropriate prerequisite to their formal description. Also, the paper examines computer systems that can be depicted as the prototypes of artificial intelligence since, in the context of education, they can be used as tools enabling both motivation and support of one’s conceptual development rather than simply a means to carry out thinking for the learners of mathematics. Finally, by referring to a few notable contributors to mathematical, educational, and psychological knowledgebase, this study argues for the merit of intuition in the digital age as a support system in the advancement of computational problem-solving techniques.

1. Introduction

The use of digital tools in the teaching of mathematics has long been a topic of research [1,2,3,4,5,6,7,8,9,10,11,12,13]. Computer technologies are increasingly approaching in their characteristics what usually belongs to the province of human cognition associated with mental abilities that are distinctly human. Therefore, the analysis of students’ interaction with computers is becoming relevant, especially in those areas where the tool can replace human cognitive activity.

The aim of this conceptual paper is two-fold: to provide a theoretical analysis of the framework of using external and internal intellectual tools in mathematics teaching and learning and to expand this classic framework in the context of the modern-day educational approaches. Historically, the idea of external and internal thinking tools was put forward by Vygotsky & Luria [14] and further developed in Vygotsky [15]. The essence of this idea is that young children learn to control their thinking by using the so-called external tools, i.e., objects surrounding them. Then, in the process of mastering language (and later mathematics), a child begins using the signs of tools instead of the tools themselves. Thus, the process of mathematical thinking is inseparable from the use of tools and is mediated by symbolic descriptions of the tools. At the same time, unmediated interaction of a child with the environment continues: the child is surrounded by different physical objects and observes many technical processes, some of which are under the child’s control.

Using a digital context, the Vygotskian ideas were advanced by Papert [16,17], a notable mathematician–educator, and later by Harel and Papert [18] by presenting education as a process of building significant artifacts [19], which in the pre-college learning environment are underpinned by culture, tradition, and social context [20]. Whereas these theories provided profound and lasting influence on the development of the constructivist and socio-cultural approaches to teaching and learning, modern-day education offers new perspectives that allow for their reassessment in digital contexts. In particular, these new educational perspectives bring about such theoretical frameworks as situated cognition in the digital era [21], connectivism [22], universal design for learning [23], and design-based learning [24].

In the digital era, the advent of computational tools and, especially, the rise of generative artificial intelligence (AI) manifest the emergence of a new instrumental stage in the context of learners’ interaction with the environment and with themselves. The modern-day technologies enable not only building virtual analogues of real-life objects, but they help students to master theoretical abstractions of mathematics by serving as active partners in thinking rather than as passive instruments. In particular, Santos-Trigo et al. noted that computer tools can represent geometric transformations, replace formal actions with algebraic formulas, and demonstrate that “the dragging and locus tools were crucial to examine the area and length variation of embedded figures without expressing such a variation in terms of an algebraic representation” [25] (pp. 838–839). In that way, the analysis of the evolution of educational milieus from hierarchical, teacher-centered didactics of instructivism [17] to problem-solving pedagogy of constructivism [26,27], emphasizing learners’ autonomy, critical thinking, and creativity, suggests using both classic and contemporary theories. Such conceptual integration allows for a better vision of how digital instruments enable support and amplification of human thinking in the process of learning mathematics. In other words, digital instruments change their roles from being external partners in the learners’ mathematical thinking to become internal units of their cognitive architecture.

2. Materials and Methods

Two types of materials were used by the authors when working on this conceptual paper. The first type is digital—external intellectual tools used to mediate students’ comprehension of theoretical abstractions of mathematics. Those tools include an electronic spreadsheet used for numerical modeling of two-dimensional algebraic expressions; the computational knowledge engine Wolfram Alpha, developed by Wolfram Research [28]; a computer algebra system, the Graphing Calculator [29], used to support the construction of graphs and the digital fabrication [30] of points and segments connecting any two points; and the dynamic geometry program GeoGebra [31], used to support geometric constructions in the context of problem solving and problem posing. Alternatively, the Geometer’s Sketchpad [32] can be used instead of GeoGebra; unlike the former tool, the latter one is available free on-line. Although not used in this study, Desmos [33]—a free on-line graphing calculator—can be used as a substitute for constructing graphs from any two-variable equations and inequalities and for the construction of points and digital fabrication of segments.

Table 1. Comparative analysis of modern-day educational tools.

Software	Educational Uses	Cost	Free On-Line	Used in the Paper
Wolfram Alpha https://wolframalpha.com (accessed on 25 August 2025)	Computational knowledge engine	PRO versions (for everyone, students, educators)	Available	Advanced symbolic computations
The Graphing Calculator (version 4.1(5), 2018, Pacific Tech, El Cerrito, CA, USA)	Computer graphing	License required	Paid	Digital fabrication and graphing
Desmos	Computer graphing	Free	Available	Not used
GeoGebra (version 4.2, 2025, GeoGebra GmbH, Linz, Austria)	Dynamic geometry constructions	Free	Available	Wise Tasks Geometry Center of gravity construction
The Geometer’s Sketchpad	Dynamic geometry constructions	License required	Paid	Not used
Excel spreadsheet (version 16.100.2, Microsoft Corporation, Redmond, WA, USA)	Multiple	Part of Microsoft Office package	Free for students	Numeric modeling and data analysis

provides a comparative analysis of the tools mentioned above.

The second type of materials include the results of mathematically oriented competitions for school children conducted over the last decade in the Department of Algorithmic Mathematics at “LETI”—a large technical university. The number of students participating in the competitions has been about 1500. Different scenarios were used during those years. In terms of demographics, the participants represented not fewer than 1500 schools from different areas in the Russian Federation. Internationally, the participants included schoolchildren from Bulgaria, Columbia, and Thailand. The age groups (the first grade—age seven) were as follows: elementary school (ages 7–10), middle school (ages 11–14), and secondary school (ages 15–17). Problems offered to different age groups varied according to specific parameters. For example, in the “Clock” problem (Section 4.2) elementary school students, using gears, constructed clocks by modeling the fraction 1/12; older students built a clock–calendar using gears. To ensure data protection, all submissions by the participants were anonymous.

Results of the competitions were assessed based on several expectations made known to the participants. As discussed in Section 4.2, the first criterion for the “Clock” problem was the accuracy precision of the clock. The accuracy precision was defined by the difference, measured in percent, between the fraction 1/12 and the ratio of the angular speed of the hour and the minute hands of the clock constructed using gears. The second criterion dealt with the craftsmanship of the clock (i.e., its size), which was defined through the area of the smallest circle that included all the gears involved. The students were allowed a generous amount of time to complete a task. During the week, each student was able to return to and improve their (automatically saved) solution.

Assessment protocols represented the recording of participants’ interaction with the interface of a problem. Using such protocols, automatic processing of the results included counting the number of attempts used by a student to improve the solution and recording the time and outcome for each attempt. Because each solution represented a certain algorithm developed in the framework of the computerized model of the problem, all solutions were dynamically displayed, allowing the participants to observe them in the gallery format. Numeric characteristics of the results were processed using statistical methods. In addition, the protocols were analyzed by experts through a qualitative lens.

Based on the qualitative and quantitative analysis of the solutions, it was shown that the developed approach made it possible to sustain investigative activities of those students who ceased their work on a problem due to the difficulty in understanding the conditions of the problem. Furthermore, it was shown that the developed scenarios of the problems provided students with a deeper comprehension of mathematical domains modeled through those problems and finding truly original solutions, thereby enhancing the creative component in the development of the learners of mathematics through competition.

Methods specific for this paper included the qualitative analysis of the results from the competition, which aimed at conceptualizing the structure and design features of the tasks completed by schoolchildren interested in the application of digital instruments to the problems of mathematics, science, and technology. Topics included in the competitions were selected to promote the ideas of algorithmic mathematics (e.g., a Turing machine) and differential gears used in the construction of clocks. Other methods included computational experiments, problem posing, and problem solving using digital tools capable of correct interpretation of the problem posed and verification of the solution proposed by a problem poser.

3. Computer Tools for Interiorization and Partnership

Can the manipulation of non-mathematical ideas by a student become the basis for the development of mathematical thinking? In other words, can an artificially created digital environment be considered as a means for managing the intellectual process in developing mathematical ideas? To answer these questions, the characteristics of a mathematical activity is studied in which a student, using common sense, operates with images related to concepts from the natural sciences. In the process of this activity, a student may not resort to formal mathematical language or use it as an auxiliary means for recording mathematical results. To this end, digital instruments can be created and/or identified, which can play the role of mediators in mastering mathematical thinking when operating with images that have a physical or technical basis.

A theoretical basis for substantiating this approach considers the concept of interiorization [34], which involves externalizing those cognitive actions for which internal mechanisms of thinking have not yet been formed. The psychological phenomenon of interiorization ensures that actions originating in the external plane transfer to the internal plane of human cognition. To understand exactly how tools can help support the productive (intellectual) component of a student’s activity, one can consider dynamic geometry software as an example of one of the most successful projects that uses computers to teach and learn mathematics. Dynamic geometry (DG) was born as a means of creating and visualizing geometric objects in the digital milieu [35,36]. Also, its major success was in becoming an instrument for verifying users’ hypotheses, conjectures, and intuition. For example, a DG tool does not tell a user how to construct a circle inscribed in a triangle, but through embedded programming, it provides sufficient functionality for the user to carry out such a construction. Furthermore, it allows one to easily recognize a possible incorrectness of a construction, for which it is enough to “wiggle” the vertices of the triangle. In doing so, one transfers an external action of using a DG tool and its features to the internal plane of geometric thinking. This example shows the direction of digital support for a mathematical activity in which the computer becomes a partner with a student for organizing mathematical activities. The partnership does not end with testing the construction. It continues as the student’s geometric thinking is externally motivated toward tracing the roots of the flawed hypothesis and seeking ways to improve the construction.

4. Construct–Test–Explore Problem Solving and Number Theory with Digital Tools

This section demonstrates practical applications of the ideas of external mediation of mathematical thinking using the context of technology-enhanced competitions for schoolchildren at different age levels. As mentioned above, the mediation of problem solving can be effectively supported by the notion of a human–tool partnership, as considered in the context of this section, which enables intuitive experimentation with mathematical ideas, and will lead to the development of computational algorithms. The value in using digital tools as means of computational experiments is an application of the concept of interiorization [34] to modern-day mathematics education.

4.1. From NP Problems to Mathematical Competitions

Consider NP problems, which constitute a class of problems in the theory of algorithm complexity. These problems are difficult to solve in the general case, but their special cases are easy to verify. Here, the words “difficult” and “easy” refer to one’s ability to complete the work in real time. In the theory of algorithm complexity, there is a clear distinction between problems that require a solution for which there are no effective computational algorithms and problems for which such algorithms exist, thus enabling a solution at the push of a button. However, some complex problems for a computer to solve are still somehow accessible by a student using intuition and common sense. Therefore, such a division of the problem between a student and a computer is natural and does not diminish the role of the student who is expected to develop an idea for finding a solution. Moreover, from the perspective of a research-like experience, one can test the emerging idea using a computer environment.

This is the foundation of the activities associated with the Construct–Test–Explore (CTE) competition for schoolchildren held annually since 2004 [37]; behind the concept of using computers by STEM-oriented students for theoretical–numerical research, proposed by Vavilov [38,39,40]; and for computational experiments in mathematics teacher education [41]. Using the context of number theory and the emerging computational capabilities of commonly available software tools, it is possible to make new mathematical discoveries with schoolchildren and college students, including teacher candidates. However, the question remains as to how much these new opportunities will contribute to the intellectual development of the students themselves. To answer this question, consider two aspects that are emphasized in the articles cited above and published during the last decade: (1) the possibility of conducting computational experiments and (2) the importance of formulating problems for experience in independent research.

4.2. The “Busy Beaver” Problem

Let us consider a CTE competition problem, which poses a problem that does not yet have a complete solution, but is accessible to schoolchildren in search for partial solutions and motivates them to initiate disciplined inquiry into algorithmic mathematics. The “Busy Beaver” problem [42] was first offered to schoolchildren from grades 1 to 11 in 2007. This problem is associated with a commonly known formalization of the concept of an algorithm—a Turing machine for the binary alphabet (yes—1/no—0), infinite tape, and five states. The goal is to create an algorithm that will stop the machine after placing as many units as possible on the tape. In the “Busy Beaver” problem (Figure 1), the Turing machine is represented by an algorithm for the beaver’s movement along the river as a tape to place logs as units (yes) using five states described in behavioral terms—inspired (the initial state), tired (the final state), and cheerful, ruffled, and pleased (three intermediate states).

What is interesting in the context of this study is not the analysis of the results developed by the schoolchildren per se, but the meaning of those results. It was not difficult to find a solution for five logs. Indeed, as shown in Figure 1, the far-right log is the first to be placed when the beaver is in the inspired state. The beaver then changes to become cheerful and moves toward the left, placing the second log, next becoming ruffled, moving toward the left and places the third log. The beaver then becomes pleased and places the fourth log, next becomes inspired and stays in place. If there is a log in front of him, he moves to the right, and when there is no log in front of him, he sets the fifth log, becomes tired, and stops. Those who figured out how to place six logs could usually place seven logs. Only 2 out of 83 participants managed to place eight logs. No explanation toward generalization was required; it was enough to present an algorithm like the one mentioned above for five logs.

A professional mathematician who analyzed the problem was able to explain how it was possible to place seven logs by formal reasoning but was unable to explain the solution with eight logs. Furthermore, when the mathematician wrote a program to enumerate all Turing machines with given restrictions, after working for several days, a solution with 13 logs was obtained. Moreover, it was guaranteed to be the best solution for 1000 steps only. This situation of algorithm vs. explanation is similar to the Collatz conjecture [43], when the algorithm (e.g., $10\to 5\to 3\times 5+1=16\to 8\to 4\to 2\to 1$) makes it possible to show the convergence to the triple (4, 2, 1) for almost all initial values, but explanation of why the algorithm works remains one of the unsolved problems of contemporary mathematics.

The “Busy Beaver” problem is of the NP-type because, while it is difficult to produce the general solution, special cases are easy to verify experimentally by using a computer program. This example shows what problems students might encounter in experimental mathematics [44,45]: they will obtain results that cannot be explained (like the eventual convergence to the triple (4, 2, 1) in the Collatz conjecture) from the point of view of the established system of concepts. Thus, a new task arises for the development of methods of teaching mathematics: to teach not only how to conduct research within the framework of the existing system of concepts, but also to introduce new concepts that allow students to mathematize through exploring and comprehending new situations. One can see that mathematization stems from applications in which teaching is not in solving a given problem, but rather thinking about a given situation [46]. Being encouraged to think about a problem as a situation, a student is introduced into a system of reasoning to study mathematical situations and their connections to other concepts. This method of teaching mathematics through CTE problem solving also shows that the intellectual activity of creating a new conceptual structure is always attributed to the student.

The structure of CTE problems has another important design feature. These problems have the property of practically unlimited expansion of the results of problem solving. These are problems that do not have a single correct answer. Instead, there are criteria for evaluating solutions (answers). Thus, the problem has an unlimited number of partial solutions, each of which can be verified according to the specified criteria. The optimal solution satisfying these criteria may be unknown to the author of the problem. Nonetheless, the best solution can be found by comparing different results. This approach is similar to using the Multiple Solution Tasks framework [47], which examines the relationship between mathematical creativity and the ability of solving a problem in more than one way.

4.3. The “Clock” Problem

Let us consider another example of a CTE problem proposed as part of the International Competition on the Application of ICT in Natural Sciences, Technology, and Mathematics (see http://kio-nauka.ru/, accessed on 25 August 2025). The problem was motivated by the following remark included by Papert [17] in the foreword to his seminal book Mindstorms: “Gears, serving as models, carried many otherwise abstract ideas into my head. I clearly remember two examples from school math. I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x + 4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend” (pp. vi–vii). This remark (discussed in detail in Section 4.4) shows how one “learns mathematics through experiencing some situation which is an application of mathematics” [48] (p. 98).

At the CTE—2007 competition, schoolchildren were asked to create a clock from gears (for younger students), and a clock–calendar (for older students). The size and the arrangement of the gears could be chosen by the students. The result could be seen on watches with the traditional dial.

It should be noted that, in the context of this paper, there are two important discoveries in this educational experiment. First, some of the solutions found experimentally implemented not exact, but approximate clocks by means of the relative speed of the hands’ rotation. The correct proportions could be calculated using the basic knowledge of middle school arithmetic. Thus, the introduction of experimental problems into the practice of problem solving is effective when an experiment is carried out according to a mental model already existing in the cognitive tool kit of schoolchildren. Second, in the CTE competition, the control of the participants’ activities is determined by the target function comprising a hierarchical series of criteria [49]. The first criterion in the assessment of the “Clock” problem was the accuracy of the clock. The second criterion was the clock size: smaller clocks are better. According to the second criterion, the best solutions had more gears that were smaller in size (Figure 2, left).

At the same time, there were participants who did not take into account the second criterion and received what they felt was a “beautiful” solution, which, nonetheless, according to the competition evaluation criteria, was not as good as others (Figure 2, right). This observation suggests that the assessment of real-life constructive activities of schoolchildren is often based on the life experience of the assessor, which might not belong to the experience of the assessed. In other words, assessment criteria are almost always subjective and, therefore, they must be as flexible as possible, not only in the context of competitions but also in the general context of education. That is, a “beautiful” (from a child’s perspective) solution could be proposed by someone who is not yet accustomed to many conditions of real life and thereby brings their own criteria for the “beauty” of the solution. Unfortunately, some mathematics teachers, with conservative beliefs developed during their own learning of the subject matter, may strongly oppose the idea of students bringing their own criteria of what is “beautiful” about problem solving [50].

4.4. Exploring Papert’s Foreword Comment Using Multiple Digital Tools

As noted above, in the foreword to Mindstorms, Papert [17] mentioned that due to his early childhood familiarity with the transmission system, the gearbox, and the differential, he understood the meaning of the two-variable equation 3x + 4y = 10. The peculiarity of a gear differential is that it is based on the idea of planetary mechanics [51]; unlike many mechanisms the learners of mathematics are familiar with, there are two parameters rather than one. How can this be explained?

We used to observe that our action is always followed by a certain re-action, be it an outcome of a natural event or the usage of a technical device. For example, when something is put into a jar filled with water, the water level rises. The faster one rotates the bicycle’s pedals, the higher is the speed. However, there are situations that cannot be described by such simple functional relations. For example, when a car’s engine is the source of energy rotating the wheels, the wheel which is closer to the center of rotation passes a smaller distance than the other wheel, and therefore the speed of the former is less than that of the latter. Unlike a bicycle, the increase in the car’s speed does not result in an equal increase in the speed of the wheels. In that way, one begins to study physical situations depending on two (or more) parameters.

To clarify, consider two Diophantine equations, z = 3x + 4y and z = –3x + 4y, two-variable equations with integer coefficients to be solved with integers. The spreadsheets shown in Figure 3 and Figure 4 display tables that illustrate the relationship defined by the former and the latter equations, respectively. A spreadsheet is an external intellectual tool capable of providing a numeric environment to support internal intellectual activities of the learner of mathematics. For example, by analyzing the spreadsheet in Figure 3, one can note that whereas the values of z are natural numbers greater than or equal to 7, not all such numbers serve as the values of z.

It is not difficult to explain why the expression 3x + 4y does not assume values smaller than 7. More difficult is the question: why do the numbers 8, 9, and 12 not occur as values of the expression 3x + 4y? Of course, this question can be answered by trial and error to exhaustively check the possible values of the expression. However, it can also be addressed using a conceptual shortcut [41,52] that appeals to the idea of divisibility. As an illustration, consider the equation 3x + 4y = 8 in which both 8 and 4x are divisible by 4, implying that 3x must be divisible by 4 as well. But if x = 4 (the smallest option greater than zero), we already have 3x = 12 > 8.

Another question motivated by spreadsheet modeling can be explored: How can one explain that the expression 3x + 4y may assume any integer value greater than or equal to 13, sometimes more than once? To answer this question, a different line of reasoning has to be used. Let $x_0$ and $y_0$ be sufficiently large integers so that ${3x}_0+4y_0=z_{0 }$. Setting $x_k=x_0+k$ and ${ y}_k=y_0-k$ where k = 0, 1, 2, …, $z_{0 }-13$, yields ${ z}_k=3\left(x_0+k\right)+4\left(y_0-k\right)=3x_0+4y_0-k=z_0-k$. That is, ${z_k=z}_0-k$ may assume all integer values in the range $[13,z_0]$.

Furthermore, an interesting problem is to find the number of solutions of the equation 3x + 4y = b for different integer values of b. This problem can be solved geometrically by graphing the equation for different values of b using the Graphing Calculator. Figure 5 shows the graphs of the straight lines 3x + 4y = 10, 3x + 4y = 19, and 3x + 4y = 31. The number of points with positive integer coordinates that belong to the three graphs represent the number of solutions to each of the three equations. The first graph passes through a single such point, the second graph passes through two such points, and the third graph passes through three such points. In the Graphing Calculator, a point with the coordinates ${(x}_0, y_0)$ can be digitally fabricated as a small disk defined by the inequality $(x-x_0{)}^2+(y-y_0{)}^2<\epsilon$ for any sufficiently small $\epsilon >0$. The graphical approach can be confirmed numerically using Wolfram Alpha and solving the Diophantine equation 3x + 4y = 31, as shown in Figure 6.

Let us return now to the spreadsheet of Figure 4 where the values of z = $-3x+4y$ are generated. This modeling ramification can be interpreted as generating the values of the expression 3x + 4y for negative x and positive y. For example, in cell E6 one can see the number $-7=-3\times 5+4\times 2$. At the same time, the main diagonal of the table comprises consecutive natural numbers. How can this observation be explained analytically, that is, by using a formula? Consider the relation $1=3\times (-1)+4\times 1$. Multiplying both sides of the last relation by a natural number n yields the relation $n=3\times (-n)+4\times n$, which shows that when $x=-n$ and $y=n$, the sum 3x + 4y is equal to n. Put another way, when x = y = n, the expression −3x + 4y is equal to n.

One can also explain mathematically the meaning of equal numbers generated by the spreadsheet shown in Figure 4, and develop an algorithm for finding such numbers. For example, the number 10 appears three times in the spreadsheet of Figure 4 (cells G2, J6, M10; noting that 10 − 6 = 6 − 2 = 4 and between H and J as well as between J and M there are two letters, thus requiring three steps from G to J and from J to M). Why is it so? Note that the relation $0=3\times (-4)+4\times 3$ implies $0=3\times \left(-4n\right)+4\times 3n$ and adding the latter to $10=3x+4y$ yields $10=3\left(x-4n\right)+4\left(y+3n\right)$. One can see that in the last equation the left-hand side does not change as n changes.

In other words, the maps $x\to x-4n, y\to y+3n$ represent the automorphism of the set of solutions of the equation $10=3x+4y$. In the spreadsheet of Figure 4, one can interpret this finding in terms of moving across the table as follows: four cells down and three cells to the right (see the note above about the rows and columns where the number 10 is located). Following is an algorithm for finding the appearance of a natural number (like the number 10) in the spreadsheet of Figure 4: (i) select a number on the main diagonal; (ii) starting from the selected number, move four cells down and three cells to the right; (iii) starting from the same selected number, move four cells up and three cells to the left.

Now let us return to the mechanical interpretation of the Diophantine equations in two variables. Figure 7 shows a differential gear used in cars for the movement transfer from the engine to the wheels. The left (L) and the right (R) gears are firmly connected to the left and the right wheels. The middle (M) gear with which the term differential is associated revolves freely about its axis, but is connected to the engine by means of rotating the handle (A) about the axis of the large gears (Figure 7 shows a simplified model of an actual differential gear).

It is said that such a construction has two levels of freedom because two of those gears can revolve independently of each other and the rotation of the third gear depends on the other two gears. For example, by fixing the left gear and making one turn of the handle, the middle (M) gear would push itself into the teeth of the left (L) gear, thus rotating the right (R) gear. Let us assume that the left and the right gears shown in Figure 7 each have 20 teeth and the middle gear has 10 teeth. In that case, one turn of the handle makes two rotations of the middle gear. This is similar to the movement along an escalator when one’s speed is the sum of two speeds, the rotation movements also add together. If the handle (A) were in the still position along with the left gear, then two rotations of the middle gear would result in one turn of the right gear. However, in our case, the middle gear itself moves in the direction opposite to the rotation of the right gear, thereby forcing the right gear to make a half turn.

Now, let us assume that both the left and right gears rotate in the same direction; the handle will then rotate in the same direction and the middle gear with stay still. However, if the left and the right gears are rotating in different directions, then the middle gear will be rotating at the double speed and the handle will remain in its place. Based on these experiments, one can try to guess the equation connecting the rotational speeds of the gears (setting x, y, and z as the speeds of rotation of the left, the right, and the middle gears, respectively). Taking the left, the right, and the middle gears with 40, 30, and 10 teeth, respectively, the equation –3x + 4y = z emerges. This explains Papert’s comment regarding the connection that exists between a two-variable Diophantine equation and the differential gear used in cars.

5. Digital Problem Solvers

Before the advent of digital problem solvers, teachers were usually posing problems for their students to solve. Tools capable of solving mathematical problems have changed this classroom tradition: the teacher formulates problems, and a digital problem solver provides solutions for the students. Nowadays, the most famous and commonly available digital problem solver is Wolfram Alpha (Mathematica), the paid version of which (e.g., PRO for Students/Educators) gives step-by-step solutions to many problems from the traditional curriculum. There are also programs like Universal Mathematical Solver [53], which can solve any school mathematical problem and present a detailed, step-by-step solution in the form sufficient for a student to turn it in to the teacher. Soon, artificial intelligence (AI) will be able to solve non-standard (Olympiad) problems by using a large database of existing samples of solutions to such problems.

If the approach to interaction between the teacher and the student through problem solving is not changed, then the presence of a digital problem solver makes students lazier [54]—they will solve problems using external intelligence (i.e., AI) rather than their internal intelligence. Therefore, a natural step towards maintaining the productive component of the educational process in mathematics will be through transition to problem posing. Students should become both problem posers and problem solvers; a digital tool (AI) should verify the proposed solutions, and the teacher should help students search for productive ideas and direct their investigations. As a way of balancing internal and external intelligences of students in the digital era, the use of the so-called TITE (technology-immune/technology-enabled) problems [12] can be recommended. Such problems are immune from the straightforward application of technology, and, at the same time, their solution does require the use of technology. For example, the development of a computational algorithm and its execution by a computer are, respectively, the TI and the TE parts of a TITE problem. As the TI part yields an algorithm for the general case, the TE part includes verification of self-evident special cases to confirm the correctness of or discover an error in TITE problem solving.

6. Wise Tasks Type Systems

The term Wise Tasks is used below to refer to another approach to mathematical problem solving in the digital era. Within this approach, the Wise Tasks Combinatorics [55] and Wise Tasks Geometry [56] systems were developed. The Wise Tasks Graph Theory system, as described in this section, is in the process of development.

All the Wise Tasks type systems are based on the idea of supporting independent formulation of new problems by the users of a system. Each system represents a computer model of a subject area and may be supported by a commonly available digital tool. For example, Wise Tasks Geometry is an add-on to GeoGebra. Students are given an opportunity to formulate problems described within a computational environment of a particular mathematics curricular topic such as combinatorics, geometry, or graph theory. Whereas students may pose and solve a problem, a Wise Tasks system in some way, which varies from system to system, has the capability to test the proposed solution. Thus, students have an opportunity to conduct independent research-like activities by setting subproblems, putting forward hypotheses regarding their solution, and testing them by using the system.

In the Wise Tasks Graph Theory system, the development of a problem can be carried out at two levels of generality in representing graph objects [57]. A formal description can be used by specifying graph properties, for example, indicating the number of vertices, edges, connectivity, planarity, or other characteristics used in the graph theory (Figure 8). At the same time, it is possible to develop specific problem types by using specialized editors that allow for visual graph construction in the form of a directed graph with ten vertices (Figure 8).

The key feature of the system is its ability to correctly interpret task descriptions using graph theory algorithms to verify properties. Students can explicitly specify required properties of a graph when formulating a problem. Since these conditions are described in a formal language closely resembling mathematical notation, the system can unambiguously interpret requirements and automatically verify their feasibility through algorithmic verifications. A student formulates and solves a graph construction problem, and the system verifies its validity by algorithmically testing the graph for compliance with the required properties or analyzing the structure of the constructed graph.

In the Wise Tasks Geometry system [56], a teacher first formulates a geometric problem, such as construct square ABCD with sides equal to the length of segment KL (Figure 9). A set of formal predicates that will be used to verify the solution is then added to the system along with available construction tools. The predicates consist of a set of simple binary (i.e., yes/no) relations: a is parallel to b, a belongs to b, a is perpendicular to b.

When verifying a solution, the system matches each step with the corresponding predicates. The advantage of this approach is that a teacher can flexibly combine intuitive textual descriptions with precise formal criteria. Thus, Wise Tasks Geometry connects human cognitive activities and machine verification of the activities, allowing students to focus on the problem’s essentials while maintaining mathematical rigor, and looking for alternative problem situations different from the teacher’s original formulation.

The AI of Wise Tasks Combinatorics is based on an enumeration algorithm. It should be noted that intelligently organized enumeration underlies many AI systems, such as chess programs. The system’s interface incorporates representations of combinatorial concepts studied in secondary school: permutations, arrangements, and combinations. Students can input any arithmetic expression using these concepts of enumerative combinatorics [56].

This interaction between internal (natural) and external (artificial) intelligences can be described as follows. Students gain the ability to pose problems within a formally constructed subject area environment using mathematical language natural to that domain. The system must properly interpret the input description to ensure its formal accuracy and completeness. The system then solves the problem using its inherent algorithms that do not replicate human thought processes but represent machine-native approaches. What matters to users is not the machine’s solution per se, but whether they can express the solution in their own conceptual terms, compatible with the machine’s understanding. Thus, solutions may be represented differently, and the system can compare different solutions to determine their equivalence. In such systems, experimentation serves not primarily as the end goal but as a means of verifying conceptual frameworks that yield consistent results. Nonetheless, discovering new mathematical results through computational experiments remains a valid pursuit in the digital world. Furthermore, verification is carried out using formal mathematical methods and not through brute-force algorithms.

7. Discussion

Minsky [58] drew attention to the “principle of investment”, the meaning of which is to use those ideas that people develop at an early age, and to “accept” new knowledge, they necessarily check it against the trusted ideas. From this point of view, the investment principle can be considered as the basic thinking mechanism of those who use intuition for comprehension. As an example of geometric intuition and common sense, Pozdniakov [59] describes an experiment in which students were first asked the question: Do you know that the medians of a triangle intersect at one point? Everyone always answered “yes” to this question. The second question was then asked: Can you explain why the medians intersect at one point? Typically, only about 15% of students can answer this question. Moreover, those who tried to remember the school proof almost never reached the goal. At the same time, others said that the center of intersection of the medians is the center of gravity, so the medians must intersect at one point.

This shows the way of understanding associated with the use of intuition. The intuition of the “center of gravity” is formalized by mental operations that can be attributed to common sense. Indeed, a triangle can be represented by a set of segments parallel to one of the sides of the triangle (Figure 10). The center of gravity of each segment is in its midpoint. The midpoints of the segments form a median of the triangle. If there is a thin support under a median (for example, a knife blade or the edge of a ruler), the triangle will be in equilibrium. It will also be in equilibrium if it is supported along each of the other two medians. From this, it is clear that the intersection of the medians is the center of gravity. By common sense, a triangle may not have more than one center of gravity. This means that the medians must intersect at one point. Finally, intuition and common sense can be digitally verified to show, using the dragging feature of GeoGebra, that the medians intersect at one point, which is the center of gravity (Figure 10).

Another example of an intellectual activity based on the interplay of common sense, simple arithmetic, and the use of technology can be considered. One of mathematical tasks used with elementary teacher candidates is to place a common fraction between two consecutive unit fractions not using any formal or computational comparison of fractions. For instance, if the fraction 3/13 is interpreted as fair sharing of three pizzas among thirteen people, then common sense tells us that more people implies less pizza; therefore, the fractions 3/15 and 3/12 (the denominators of which are multiples of 3) are smaller and greater, respectively, than 3/13. That is, 3/15 = 1/5 < 3/13 < 1/4 = 3/12.

One can either use the Graphing Calculator to digitally fabricate (Figure 11) the points with the coordinates being denominators and numerators of the fractions 3/15, 3/13, and 3/12, or just set those points in the context of GeoGebra (Figure 12). These constructions show that intuition is not enough to decide the order of the fractions using two-dimensional representations of fractions on the coordinate plane. At the same time, understanding which geometric characteristic correctly orders the fractions is important for using external intelligence for the epistemic development of a student. To provide an alternative representation of the order of the fractions, one can see that it is not the areas of triangles (Figure 12) but the angles between the x-axis and the segments connecting the origin with the points representing fractions (Figure 11). There are the pieces of knowledge that DiSessa [60] had in mind when using the “idea of knowledge in pieces” that form a whole from its small parts. Likewise, the pizza sharing comparison of fractions at the level of common sense represents its use for mathematization of arithmetic as the investment principle [58].

8. Conclusions

The emergence of digital tools capable of performing numeric modeling, symbolic computations, and graphic constructions, traditionally attributed to humans equipped with pencil and paper, calls for reassessing the importance of various aspects of the learning process. Nowadays, because the information environment for learning is changing rapidly, it is impossible to determine in advance whether a new digital tool designed for mathematics education will become widespread. For example, DG applications have become ubiquitous in mathematical teaching and learning due to the properties that have proven to be in demand in the new educational environment.

At the same time, the direction of changes in the information environment of education has already been determined. Briefly, it can be called the coexistence of natural (internal) and artificial (external) intelligences. AI penetrates education both spontaneously using tools created for other purposes, and through environments specifically created for learning. Under those conditions, the educational community has reached consensus about the importance of preserving intellectual potential of students in new learning contexts and the gravity of using digital technologies to increase the efficiency of education.

Nonetheless, wide application of such technologies is not without challenges. One challenge is the risk in replacing learners’ cognitive efforts by passive use of technology, e.g., when learning is reduced to simple observation instead of conceptual analysis of computer-generated results. Another challenge relates to the issues of digital inequality as teacher candidates’ access to advanced software tools varies greatly both institutionally and geographically. Furthermore, there are many in-service teachers who are reluctant to integrate digital instruments in the practice of mathematics teaching because traditional programs of professional development often miss methodology of combining intuitive practices with demonstrating technological opportunities.

In order to overcome these challenges, a complex approach is necessary. The main direction of such an approach is the development of pedagogical scenarios grounded in the use of technology as a critical supplement to the cognitive activities of a student. This direction can be pursued with the help of specially developed tasks that require comparative analysis of technology-immune/technology-enabled approaches to problem solving. Special attention should be assigned to providing equal access to educational technology and to the improvement of in-service and pre-service training of teachers. This calls for the investment in the creation of educational sites and off-line analogs of digital instruments.

In this report, the authors emphasized that mathematical reasoning used by students based on common sense and intuition must necessarily be included in the activities aimed at the construction of new knowledge. By analyzing different approaches to solving mathematical problems, one can appreciate the idea of teaching and learning mathematics to include computational experiments and real-life applications. For example, a comment about connecting differential gears to a two-variable Diophantine equation [17] demonstrates how a classic topic of the theory of numbers can be taught through application to engineering mechanics of ordinary cars. Furthermore, spreadsheet modeling of the left-hand side of such an equation provides students with a learning environment in which observation of numeric patterns leads to several conceptually rich interpretations of those patterns. One such interpretation made it possible to introduce the concept of automorphism within a spreadsheet and formulate an algorithm for locating equal numbers associated with this concept.

Using inequalities as tools of digital fabrication make it possible to enhance graphing tools that were not designed for the construction of points and segments. That is, once again, mathematization enters through the gate opened by applications. At the same time, comparing fractions through common sense helps students to mathematize within the context commonly considered difficult and confusing. This paper also demonstrated how the CTE competition provided school students with engaging learning environments through which the concept of the Turing machine (the “Busy Beaver” problem) and the pedagogical discussion of the assessment criteria (the “Clock” problem), which may not belong to the life experience of the assessed students, creates a didactical conflict between the two experiences.

Other topics discussed in this conceptual paper were associated with the development of the Wise Tasks systems, in which a student independently poses and solves problems on a digital model of a subject area and AI automatically checks the solutions. Further development of the Wise Tasks systems in which AI only verifies the correctness of students’ solutions appears valuable. However, for successful realization of such an approach, longitudinal studies are needed that will enable assessment of the long-term influence of such systems on the mathematical thinking of students, making it possible to develop an effective means for shielding students from cognitive offloading [61]. The peculiarity of the Wise Tasks system is that two intelligences—human (internal) and machine (external)—do not compete with each other and do not replace each other. Machine intelligence operates based on precise calculations on a subject model, while human intelligence interacts with a problem based on constructing concepts within the framework of human ideas.