Instruction, repetition, discovery: restoring the historical educational role of practice
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Abstract
This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion that mathematics should be taught like swimming. The general claim being made is that “direct instruction” and “discovery” are not opposite but complementary, linked by repetitive yet explorative practice. This claim is elaborated through an empirical case of martial arts instruction. That repetitive practice can nonetheless be a fountainhead of discovery is explained using Bernstein's notion of repetitionwithoutrepetition. Finally, we attend to parallels in canonical mathematics practice. Implications are discussed, with a focus on reconceptualizing direct instruction, repetition, and discovery as complementary and synergistic.
Keywords
Mathematics education Martial arts pedagogy Learning theory Direct instruction Discovery Embodied cognition Awareness Drill Explorative practice ExerciseIn mathematics education, “direct instruction” and “discovery learning” have been depicted as opposing sides at war (Schoenfeld 2004, Math Wars). On one side, advocates claim students learn through direct instruction and repetition. On the other side, advocates claim students learn by discovering key underlying principles. While a simplification, this depiction can be useful: if not for what it says about learning, for what it reveals about how we think about education.
Direct instruction carries with it the notion of teacherasgiver. Teachers impart knowledge: they pass on or transmit knowledge to students. This remains a commonplace, influential, and persistent metaphor for instruction. In this article, for the sake of simplicity, direct instruction means explicitly telling. For example, a teacher might tell a student that the sum of any two odd integers is an even integer, attempting to directly transmit this element of knowledge, as it were. This particular type of telling we call teachingbyexplaining, or just explaining.
Discovery learning, on the other hand, carries with it the notion of studentasexplorer. There, knowledge is discovered by the student. Knowledge is not found as one might a set of lost keys, of course.^{1} The explorer is an active organizer of experiences, constructing stable understandings by repeatedly constructing them anew (Piaget 1977). For example, students might work on summing integers and, while engaged in this activity, find that every summand of odd integers is even. Through this discovery, students begin constructing knowledge of this arithmetic principle.
A central dispute of math wars (arguably “the” dispute) may be captured by the following question. If the educational metaphor of teacherasgiver is appropriate, why would anyone ask students to discover knowledge on their own? Searches are by definition unreliable. Why not simply teach in a way “that fully explains the concepts and procedures that students are required to learn” (Kirschner et al. 2006, p. 75)? These questions are at the heart of the “instructivistconstructivist debate” (see, e.g., Kapur 2016), which spills out of mathematics education into education more broadly. To clarify, both sides agree that learning^{2} is a constructive activity: the disagreement is about the pedagogical use of discovery. If students construct knowledge regardless, why ask them “to discover the fundamental and wellknown principles” (Kirschner et al. 2006, p. 76) of the disciplines?
In an attempt to answer this question, numerous leads could be pursued. For example, there is converging empirical evidence that asking students to first explore and discover can result in more powerful learning than first explaining whatever it is that students are expected to learn. These results come from controlled experiments (e.g., Kapur 2014; Schwartz et al. 2011), and quasiexperimental, classroomsituated studies (e.g., Kapur and Bielaczyc 2012; Schwartz and Martin 2004). A recent metaanalysis (225 studies) of STEM teaching methods supports these findings: the authors ask, “In the STEM classroom, should we ask or should we tell?” (Freeman et al. 2014, p. 8410), and conclude that “a constructivist ‘ask, don’t tell’ approach may lead to strong increases in student performance” (p. 8413).
In addition to empirical results, we might turn to theoretical frameworks supportive of a focus on discovery and knowledgebuilding: constructivism (e.g., Kamii and DeClark 1985), radicalconstructivism (von Glasersfeld 1983), constructionism (Papert 1980), or “learning by doing” (Dewey 1944). We could investigate the history and philosophy of the disciplines, noting that the arc of progress tends to travel from use to understanding to formalism (Kleiner 1988; Nathan 2012; Wagner 2017), the reverse of what is advocated by instructivists. We may even turn to the writings and accounts of professional mathematicians. For instance, William Thurston, a Fields Medal awardee, was critical of the idea that mathematics should be treated as a set of facts to be explained (1994). Mathematics is an ongoing and interconnected subject, Thurston points out, and if we were to attempt compiling a list of all possible mathematical explanations of any “particular piece of mathematics,” we would quickly realize the futility of our task: “the list continues; there is no reason for it ever to stop” (p. 4). Thurston’s perspective demonstrates that some mathematicians, at least, disagree with the assumption that mathematical ideas can be “fully explained,” as advocated by instructivists (Kirschner et al. 2006), and even some constructivists (Spiro and DeSchryver 2009).
both sides in the debate are able to cite research that compares constructivist instruction with direct instructions: and the outcome is that constructivist instruction is either more, or less, effective—depending which studies are considered. (p. 24)
The aim of this paper is not to support one side or the other, but look for an alternative. In a nutshell, I argue for direct instruction (i.e., telling students what to do) as a means of fostering discovery via guided, repetitive practice. In doing so, I offer that the metaphors of teacherasgiver and studentasexplorer are not contradictory, but complementary. I build my argument on an empirical case study of an existing yet underresearched pedagogical approach that appears to incorporate these metaphors. In this approach, direct instruction is primarily used not to transmit knowledge, but to instruct students on, and guide them through, a repetitive practice. In turn, practicing provides learners with opportunities for discovery—something explicitly expected of the student. Practice is needed for discovery to take place, as it allows for patterns to become salient to the practitioner. Finally, these discoveries are contextualized and signified in the semiotics of the discipline. This approach, which I call explorative practice, acknowledges but seeks to refine and integrate both teacherasgiver and studentasexplorer metaphors. In a sense, something is imparted (or, “given”) to the student. What is imparted is how to practice; then, via this practice, and guided by the teacher, the student is expected to make a discovery.
This pedagogical approach is readily available outside of mathematics (and STEM) education, where we now turn.
Towards an outside perspective: sidestepping outside of mathematics
Let us consider the example of learning how to swim. Imagine a scenario where a swimming teacher provides her novice students with an elaborate worksheet detailing the biomechanics of freestyle, an extensive howtoswim manual. Then, after a clear and detailed lecture, she announces to the class that they are now swimmers. After all, she explained how to swim, to the fullest extent possible! Presumably, the very idea seems preposterous (also see Lockhart 2009). We could similarly note that teaching swimming by having students “discover a way of not sinking” appears unreasonable, as well. This case, at least, suggests that learning without discovery is depthless; learning without guidance, miraculous. Some aspects of swimming resist straightforward teachingbyexplaining as much as they resist easy discovery. A solution involves practice^{3}: swim instructors provide students with a supervised swimming practice. Practice is implemented through direct instruction, demonstrations, and other forms of guidance.
Education researchers have previously made a connection between mathematics learning and physical education.
In my interpretation, while Freudenthal’s metaphor of teachingmathematicslikeswimming emphasizes discovery, it (perhaps surprisingly) does not deny elements of instructivism, such as direct instruction and other forms of guidance, with the possible exception of teachingbyexplaining. One way to apply lessons learned from teaching an activity like swimming to teaching mathematics, then, is to engage students in (repetitive) mathematical exercises, much like swimming necessarily involves a period of flailing one’s arms and legs in water, preferably under the guidance of a qualified teacher. What I hope to make clear is that teaching mathematics as swimming—as an activity—does not mean forsaking instruction, guidance, or repetition.Children should be granted the same opportunities as the grownup mathematician claims for himself. Telling a kid a secret he can find out himself is not only bad teaching, it is a crime. (p. 148, emphasis added)
Freudenthal was not the only prominent scholar (and advocate of constructivist teaching) to have called for the pursuit of parallels between mathematics and physical education. One of the most remarkable instances of this would have to be von Glasersfeld’s (1983) plenary address at the 5th Annual Meeting of the North American Group of Psychology of Mathematics Education, titled Learning as Constructive Activity. In his address, von Glasersfeld urged mathematics education researchers to look to the teaching of physical skills for insight and inspiration that may be applied to mathematics education.
Like Freudenthal, von Glasersfeld was surely aware that physical disciplines are not “the same” as mathematics. For one, the former lack the sophisticated symbolic formalisms employed by the later. Nonetheless, he argued, “the primary goal of mathematics instruction has to be” bringing mathematical knowledge to “students’ conscious understanding” (p. 51). This, he claimed, was “in principle” similar to education in the physical disciplines—a claim this article supports.
Martial arts pedagogy
Responding to Freudenthal and von Glasersfeld, I attend to the teaching of physical disciplines in order to rethink the teaching of academic disciplines, such as mathematics, and specifically to reconceptualize the role of practice in those disciplines. The goal is to sidestep from mathematics to motor skill, learn over there about practices and processes of teaching and learning, and then consider these observations in the context of mathematics and education more broadly (see Abrahamson and Trninic 2015, for more on this research program; see also Foster 2013). In this article, we venture into a domain rarely explored in the learning sciences: that of neijia, or Chinese internal martial arts, and report on a pedagogical approach characterized by direct instruction, a high degree of guidance and repetition, yet a clear expectation (requirement) of discovery.
I do not mean to suggest that all physical activities are interchangeable as pedagogical subjects. Rather, my interpretation of Freudenthal and von Glasersfeld’s call is that it is an invitation to think more fundamentally about what it means to teach not a rolodex of facts, but to teach an activity. Thus, to think more fundamentally about teaching an activity, we investigate a clearcut example of “an activity that could be learned, not in lectures and from books, but by acting it out” (Freudenthal 1971, p. 139). Martial arts are a suitable case: it is evident that merely hearing or reading about combat does not make a better fighter.
Beyond this suitability, studies of physical disciplines provide a methodologically advantageous research context. Studies of motor skill learning have the advantage of being an easier research problem than mathematics learning, for within these domains of practice, expert and novice actions are overtly physical, affording researchers greater transparency and insight—when compared to domains like mathematics, where substantial stretches of activity take the form of Rodin’s Le Penseur.^{4}
Furthermore, the pedagogies present in internal martial arts have arguably stood the test of time. These disciplines flourished during a time when combat was still a reality (Draeger and Smith 1981), and martial artists would obtain lucrative yet dangerous positions (e.g., bodyguard, or caravan guard). Poor teaching would lead to poor students and, because martial skill mattered, these circumstances presumably conspired to eliminate not only ineffective styles, but also ineffective teachers and pedagogies.
Finally, martial arts are rarely investigated in the learning sciences, which are traditionally more concerned with conceptual than motor skill learning. Therefore, one contribution of this paper is to introduce the field to a particular martial arts pedagogy.
Following our case study of a martial arts pedagogy, we turn to Bernstein (1996) for an explanation of its method. Bernstein’s ideas are then developed through Vygotsky’s discussion of kinesthetic sensations (Vygotsky 1997), and later through the contemporary perspective of embodied cognition. We consider parallels in mathematics, thinking through a contemporary classroom activity where students discover certain features of rational numbers revealed by the practice of the division algorithm.
Martial arts pedagogy: a case study
Chen Fake, a renowned Chen style taijiquan grandmaster, is reported to have said, “There are three steps to learn taijiquan: first to learn the moves, then to practice often, and finally understand the details.” Consider the account of education implied by this statement: one’s learning begins not ends with “learning the moves.” But taijiquan—an internal Chinese martial art, better known in the West under its abbreviated WadeGiles transliteration Tai Chi (I will use taiji)—is best known for its forms, or choreographed sequences of movements. Chen Fake’s statement thus suggests that taiji training involves more than memorizing movements.
 Author

How do you see your role as a teacher in your discipline?
 Edward

…I do not speak for the whole art, only myself. My goal as a teacher is to help students discover the appropriate technique through personal repetition. Early on, a beginner must be told what to do over and over, while being shown the correct way. But once a student has done the technique numerous times… I feel the student must learn to feel something is incorrect.
 Author

Would you be able to say more about repetition in practice?
 Edward

I would be happy to… Repetition is crucial because we are training our body to understand the appropriate movements.
I offer this pedagogical approach as explorative practice: direct instruction and guided repetition oriented towards students’ learning through the discovery and subsequent development of relevant internal sensations. To contextualize and elaborate on this view, we’ll consider an empirical example of instruction broadly representative of student–teacher interactions in taiji.
Practicing taijiquan pushhands
Like any complex and culturally rich discipline, taiji resists easy summaries. It is said to be the most popular martial art in the world, with estimates of over 250 million practitioners worldwide. As a martial art, taiji utilizes awareness of one’s balance for a martial purpose; however, for historical reasons tied to the Chinese Communist Revolution, and because developing awareness of one’s balance is useful more broadly, the vast majority practice taiji exclusively as a health exercise.
We focus on a relatively basic taiji principle that might be shared with beginners. We could call it redirectnotresist. Attending to redirectnotresist helps undo a person’s natural instinct to resist force with force, teaching the practitioner to accept, lead, and redirect, rather than resist, incoming force. A telltale indication of resisting force is leaning into the incoming force, in anticipation of being pushed off balance. This intuitive response compromises the defender’s balance: if the aggressor stops applying force, the defender will stumble forward.
Learning not to instinctively resist incoming force with force is a substantial challenge for most if not all beginners. This development goes handinhand with an oftrepeated admonition for beginning taiji practitioners to “relax more!” Relax is an imperfect translation of the Chinese word song (松), which my teachers fluent in Chinese described as an appropriate state of bodily alertness, neither limp nor rigid. When in this relaxedyetalert state, a more skilled taiji practitioner would align her body structure with the incoming force, redirecting it to the ground (rather than leaning in or pushing against the opponent); if the incoming force proves too strong, she would move her body minimally such that the incoming force no longer threatens to compromise her balance. It is difficult (arguably impossible) to verbally explain this action, relatively easy to demonstrate it to a student, and challenging for the student to get a sense of how to recreate the action at all.
The following excerpt^{6} will showcase a Eureka! moment of discovery one student experiences after the teacher guides him to discover the principle of redirectnotresist. Prior to this moment, the master consistently (and patiently) directs the student’s attention towards a particular dimension of the interaction, waiting for a moment of insight. One method of directing a student’s attention, frequently used in such training sessions, is for the master to conspicuously offer the student an opportunity to redirectnotresist and offbalance the master; if the student instead resists and pushes against the master (a common, habitual response), the master will appropriately redirectnotresist and offbalance the student instead. This sequence repeats many times, as is typical of such training sessions. Occasionally, the student will successfully perform redirectnotresist. This was the case here, even before the excerpt we will focus on. In other words, the highlighted interaction sequence will not be the first time the student succeeds in demonstrating redirectnotresist.
For instance, as early as two minutes before the upcoming excerpt, the master offbalances the student once; then, as he is about to do so again, he freezes the interaction to redirect the student’s attention:
Master: When you [inaudible] … feel arms here… yeah!
The word “here” is emphasized, occurring at the same time the master conspicuously moves to offbalance the student. At that very moment, the master purposefully overextends, so as to give the student an opportunity to redirectnotresist and offbalance him. The student successfully does so, which the master acknowledges (“yeah!”). The master then briefly looks up at the student’s face, as if searching for something. Practice continues. For now, the point to keep in mind is that what we are about to see, below, is not the first time, neither in this particular session, nor his overall training, that the student successfully performed redirectnotresist.
 Master

When you feel force increase, change in direction. [Figure 2b; the master squeezes his hand in front of the student’s face, then opens his hand and wiggles sidetoside while uttering the last three words, with emphasis]
 Student

Change in direction.
 Student

Ah, yeah. Wow. [laughs] OK. Thank you.
 Master

OK, See. In mind, never use force, never use force. (Figure 2d)
During this interaction, the teacher introduces a problem situation into an ongoing push hands routine and states the general principle. This intervention, I argue, provides the student with an opportunity not merely to perform but to experience the principle of redirectnotresist. Phenomenologically, this experience takes the form of discovery. Once the student manages to redirectnotresist—and appears aware of this action—the teacher again verbally describes the principle: “In mind, never use force, never use force.” The student’s physical sensation is signified within a disciplinary context; the student now has a sense of—or, more precisely, is beginning to make sense of—what it means to redirectnotresist.
While the above account concerns the student’s experience, the teacher’s role is indispensable. During the entire training session, the master remained patient, providing additional guidance by specifically and repeatedly highlighting the very microevents during which the student should redirect the incoming force—and then allowing him to do so.
Remember that the above excerpt was not the first time the student successfully performed redirectnotresist. It is, however, the first time the student appears to recognize this principle in his performance. Recall also that 2 min prior to the excerpt, once he was offbalanced (“feel arms here… yeah!”), the master looked up at the student’s face. This brief action was possibly done to check whether the student appears to have not only performed but felt the action. Note how much effort the master put into guiding the student towards this single moment of insight: repetition alone, without this guidance, has only a small chance of success.
The primary justification behind this pedagogy is that practice is used as a vehicle for exploration and discovery of what is important, and what is not; of what works, and what does not. Recall an earlier quote: “…the student must learn to feel something is incorrect.” This feel is revealed through a repetitive, explorative practice: in the course of exploration, teacher’s guidance brings forth and highlights features of the activity that may otherwise remain invisible. This is what Chen Fake implied by “learning the moves, practicing, and finally understanding,” and what Edward meant by “repetition is crucial because we are training our body to understand the appropriate movements.” Trninic (2015) describes another taiji master’s stance on such explorative practice:
In my experience, it is not uncommon for internal martial arts teachers to restrict or even withhold explanations, believing that explanations are of little use to students who (my words) have nothing to ground them in. Thus explorative practice, as I present it here, involves direct instruction, a generally high degree of guidance and repetition, but minimal explanations, and an emphasis on student’s discovery.Fong [the master] emphasized practice to what seemed an extreme degree, sometimes outright refusing to verbally elaborate on underlying principles… He expected all beginners to spend at least one hundred hours practicing standing postures, “That’s the minimum to change your thinking.” Early on in my training, I asked him to elaborate on how it is possible for a taiji student to strike if we are told to always follow the maxim of “no collision,” one of his preferred expressions. After all, I told him, taiji is supposed to be “soft,” so how could anyone actually strike while being relaxed and without colliding? His response was, “You want to figure [out] everything by understanding it intellectually. That won’t help here. Do the work for at least one hundred hours and you’ll start to understand.” A couple of years later, when I asked him the same thing, he [explained]… There was no contradiction in Fong’s teachings, but it was not something that could be put into words, either. (p. 27)
Bernstein’s repetitionwithoutrepetition and the development of kinesthetic sensations
To make sense of explorative practice, I draw on the work of Soviet neurophysiologist Nikolai Bernstein, who argued that skill development follows the process of “repetition without repetition.” Bernstein was against the then (and now) prevalent belief that repetitive practice is merely a means of “beating down the same path” over and over. If repetition is merely about “beating down the same path,” he wondered, how does anyone transition from novice to expert? A novice, by definition, cannot perform as well as an expert. Thus a novice will, by definition, practice the wrong thing. The following observation by Kirsh comes to mind: “it is odd, to say the least, that practicing (literally) the wrong thing can lead to better performance of the right thing” (2013, p. 4; cf. Schmidt and Bjork 1992). Yet that we perceive this as “odd” might say more about our models of cognition and learning than about this commonplace, everyday phenomenon.
He later elaborates on the limits of teaching:During the [first] phase, a novice decides how the movements involved in a motor skill look from the outside; during the [second] phase, the novice learns how these movements and their sensory corrections feel inside. (pp. 184–185)
Insofar as there are secrets in the disciplines, they are in practice. Through practice, the discovery of these secrets takes the form of “special sensations” rising to the level of consciousness, as the novice learns how movements “feel inside.”Any movement can be taught by demonstration… The fact that the ‘secrets’ of swimming or cycling are not in some special body movements but in special sensations and corrections explains why these secrets are impossible to teach by demonstration. (p. 187)
Every movement—for example, holding up one’s right hand—is associated with a particular kinesthetic sensation, and Vygotsky claims that these are the most essential aspects of our thinking.those sensations and experiences that are associated with the natural movement… [and] that seem to make the individual aware of his own actions. … [These] sensations are the most essential aspects of our thinking. In order to perform some movement… the sensations associated with these movements must first be present in our consciousness. (Vygotsky 1997, p. 161)
According to Freeman (2000), an organism’s understanding of an action (its meaning, essentially) develops through a neural combination of sensory messages with past experiences. Movement and sensations are thus intertwined, and intentional actions require that the “organism has some idea, whether correct or mistaken, of what it is looking for.” (p. 109, emphasis added). How does the organism know what to look for?
As Vygotsky states, above, “to perform some movement… the sensations associated with these movements must first be present in our consciousness.” That is, the development of voluntary action is contingent on the awareness of associated kinesthetic sensations (see Kimble and Perlmuter 1970). Turning intentionality upside down, one could argue that voluntary actions first occur, in a sense, involuntarily, serendipitously, and through a (sometimes guided) search process (also see Trninic and Abrahamson 2012, 2013). Subsequent learning depends on consciously recognizing the appropriate internal sensation(s) associated with a given movement, in order for it to be repeated as a voluntary act. Thus practice, or repetitionwithoutrepetition, is the process of becoming aware of relevant kinesthetic sensations and developing conscious control by recreating these sensations.
Revisiting taiji pushhands practice
With this in mind, let us briefly return to our earlier case of the taiji student practicing pushhands. Recall that this is done for the purpose of developing a sense of redirectnotresist. Merely performing redirectnotresist without an awareness of this sensation, as the student did prior to the highlighted excerpt, takes place in the first stage of Bernstein’s motor skill development. The performance works from the outside or, at least, it “works” when the teacher signals when to redirect. For discovery to take place, the student needs to experientially differentiate an instance of redirectnotresist from his previous actions, identifying the appropriate kinesthetic sensation. This is a critical shift towards the second stage of motor skill development (see also Haken et al. 1985, on phase transitions). The student’s joyful series of utterance—“Ah, yeah. Wow. OK. Thank you.”—indicates he was not merely thanking the master for his patience, or for allowing himself to be offbalanced. If that were the case, the student would have thanked the teacher with that same wonder every time this happened, which he did not. Again, recall that the student successfully offbalanced the teacher just minutes before. That time, however, the student’s facial expression remained fairly neutral. The teacher’s gaze briefly lingered on the student’s face, perhaps searching for something. Did the student feel it? It seems that he did not, and neither stopped to signify the event, merely continuing the practice. Why? Something was missing—the student’s awareness of the relevant kinesthetic sensation.
In my interpretation, the student was thankful not only that he “changed in direction” but that he experienced what it feels to “change in direction.” This principle becomes grounded in experience, grounded as a kinesthetic sensation. Of course, this is merely the beginning of skill development. Once brought to conscious awareness, this sensation becomes something the student could consciously reenact and adapt. He now knows what to look for (or, rather, what to feel for, and enact).
Parallels of explorative practice in mathematics
So far, we have looked at a case of taiji push hands instruction, where I posited discovery emerges through practice—a case of action before concept (or, actionbecomesconcept). There, the teacher’s role involves helping students bring to awareness their subjective construction of the action, signifying it within the semiotics of the discipline. In this section, we return to mathematics education and entertain the idea that explorative practice holds implications for education more broadly. We think through an illustrative^{9} example of a teacher employing direct instruction in the service of discovery. I do not argue that this is “a case of” explorative practice in mathematics education; rather, it can help us think through some parallels with the pushhands case.
The example comes from a lesson implemented in a high school classroom at a charter school in the San Francisco Bay Area. The specific practice in question is the division algorithm. At the start of the lesson, the teacher, Cristina, reminded students how to perform the division algorithm, then instructed them to perform a series of computations.
My interpretation of this particular lesson was that Cristina was, in effect, providing students with opportunities to make mathematical discoveries. Specifically, she wanted students to observe that not every number can be expressed as a fraction. This observation may seem obvious to educated nonmathematicians, but it is actually an important (and nontrivial) moment in the history of mathematics, redefining the very definition of a number (Knorr 1975).
Cristina’s explicit pedagogical approach was to “minimize her work” in the classroom, so as to maximize the time students actually worked on something. Yet Cristina’s teaching in general exhibited a high degree of guidance, a characteristic also reflected in her goodnatured references to being “the captain of this ship” (i.e., the classroom). During the lesson, the practice ranged from simple division problems, such as \(1025/2 = 512.5\), to more involved, such as \(2/13 = 0.153846153846153846 \ldots = 0.\overline{153846153846}\). These problems were provided in sets, each set resulting in answers containing a terminating decimal (e.g., 1025/5), or a repeating decimal (e.g., 2/13, where the decimal becomes periodic). Separating fractions into those with terminating or repeating decimals was meant to make these features more apparent to the students. In other words, this was something for students to discover. Despite this, Cristina did not consider it a crime to “tell secrets”—like the author, she believed that students have to work to figure them out anyway.
While students worked and conversed with each other, as was the classroom norm, Cristina walked around the room, answering questions and checking in on individual students. On occasion, she would ask students whether they found anything interesting while performing the computations. Towards the end of the second day of this twoday lesson, most of them had. Initially, the “interesting things” students noticed were the more obvious patterns, made salient by the structure of the worksheet: specifically, that each answer had either a terminating or repeating decimal expansion. This led students to wonder—and if they did not, for Cristina to explicitly ask—Do all fractions follow one of these patterns?
Jamal, the student whose solution procedure we are looking at, used an abbreviated and modified version of the division algorithm. We follow along with this procedure. To start, 1 divided by 7 equals 0 with 1 remainder, and we move into decimals. 10 divided by 7 equals 1 with 3 remainder. Next, 30 divided by 7 equals 4 with 2 remainder. And so on. In this particular case, we can see that the procedure comes to loop in on itself once we reach 50 divided by 7 equaling 7 with 1 remainder—leading, once again, to 10 divided by 7 (note the arrow). From then on, this pattern will keep repeating, infinitely, and so the decimal expansion can be expressed as \(0.\overline{142857}\) (expressed unconventionally by Jamal, with an underline).
Jamal’s initial solutions did not have the loopback arrow: he started adding it after an intervention. Specifically, Jamal’s attention was directed to the cyclic nature of the algorithm in the case of periodic decimal expansion. This was pointed out by Cristina as a “shortcut”—in the sense of, look, from here onward, you will run into the same numbers all over again and thus no longer have to go through each step to determine the solution. In the example above, Jamal not only utilized the “shortcut” on his own initiative, but appeared excited as he explained it to the teacher that there will always be a shortcut.
Let’s consider why this observation of there “always” being a shortcut might be mathematically interesting.
Every time we divide by 7 (see Fig. 3, above), we always end up with a remainder smaller than 7. This remainder determines the next step in the division algorithm: in this case, remainder 1 (which becomes 10), then 3, 2, 6, 4, 5, and once again 1 (which becomes 10), in that order. Because each remainder is always smaller than 7, there are only so many remainders we can go through before one of them repeats. Once a remainder repeats, we find ourselves in a cycle. But there’s nothing unique about 1 divided by 7 that makes this true, and the same claim can be made for any fraction. Consequently, every division algorithm with either terminate, or some remainder will repeat, causing a cycle. Furthermore, while all fractions either terminate or become periodic because of the inevitable loopback pattern, this does not exhaust all the possibilities of a decimal expansion. Namely, numbers that neither terminate nor become periodic are the irrational numbers.
If you found this last paragraph difficult to follow, I invite you to try performing the division algorithm procedure a few times on your own (e.g., 1 divided by 13) while paying attention to the remainders after each step. Once you complete a few algorithms, you may find yourself surprised with how quickly the loopback, cyclic pattern reveals itself. In a very real sense, this captures the gap between explaining and doing—and, perhaps, between doing once and a practice. A practice can make patterns salient, even when these patterns, put into words or symbols, appear esoteric or ephemeral—senseless.
Unfortunately, we do not have data that more precisely identifies the moment when Jamal first suspected that there will always be a shortcut. It is impossible to determine whether Jamal’s own reasoning followed ours, as presented above. It would be unreasonable to suppose that his justification for there “always being a shortcut” was particularly sophisticated. However, what does seem to be the case, based on Jamal’s excitement when sharing this observation, is that he recognized in his own actions something new (or anew). Perhaps he experienced this moment as a surprise, much like the martial arts student in the pushhands case.
Like the student working on push hands practice, Jamal’s performance and understanding did not occur simultaneously. I offer that both students performed an action, and, through repeated performances (action before concept), experienced something new (Trninic and Abrahamson 2012, 2013). Like the martial artist developing a sense of what it means to “change in direction,” Cristina’s students had the opportunity to develop a sense of a principle. The claim is that this matters: first, that developing a sense of an action is an integral not incidental part of mathematics and, second, that a sense of something is difficult (perhaps impossible) to convey through an ungrounded explanation.
Kinesthetic sensations in mathematics
One goal of the paper was to show how direct instruction and discovery learning can be linked through practice. Toward this, I analyzed examples from both “physical” (martial arts) and “conceptual” (math) domains. I offered a possible mechanism through which explorative practice led to a student’s discovery of a taiji principle, couching it in terms of repetitionwithoutrepetition leading to a discovery—that is, awareness—of a kinesthetic sensation. In this section, I offer a tentative explanation for why and how the mathematics example might be understood using a similar set of constructs.
This may come as a surprise to the reader. In a straightforward reading of the paper, the taiji example can be thought of as a useful analogy, a whatif inspiration for the design of mathematics education tasks. Even repetitive exercises, sometimes derided as drills, can nonetheless be a fountainhead of insight and discovery—this paper can certainly be read with that interpretation.
However, I offer that there is more at hand than a productive analogy. Specifically, I entertain the possibility that the construction of even abstract mathematical ideas involves kinesthetic sensations. This conjecture depends upon a particular reading of the embodied cognition perspective, one positing shared neural systems underlying both motor skill and conceptual development, as we now discuss.
Embodied cognition and neural reuse
Even the most abstract thoughts of relations that are difficult to convey in the language of movement, like various mathematical formulas, philosophical maxims, or abstract logical laws, even they are related ultimately to particular residues of former movements now reproduced anew. (Vygotsky 1997, p. 162)
The many perspectives under the umbrella of embodied cognition agree that an organism’s cognition is grounded in its sensorimotor capacities (Chemero and Turvey 2011; Kiefer and Barsalou 2013; O’Regan and Noë 2001). Of interest to us is the neuroanatomical argument that all mental actions, including the processing of abstract concepts, utilize the same systems that support perception and action. This hypothesis, which Anderson (2010) calls neural reuse, explains why thinking about a hammer activates the very same neurons used when hammering (Martin 2007). For us, it provides a justification for why even mathematical concepts might contain elements of kinesthetic sensations—kinesthetic sensations may enter through “the reuse of neural circuitry” also used in the support of perception and action.
Why would cognitive processes reuse the same systems that support perception and action? Natural selection works with whatever is available; in this case, our capacity to manipulate abstractions appears to be cobbled together from preexisting biological features of the nervous system. The biological mechanisms that enable our thinking have evolved over hundreds of millions of years. Since the nervous systems of earlier multicellular animals provided the advantage of coordinating movement through perceptually guided action (Bucher and Anderson 2015; also see Llinas 2002; Varela et al. 1991), our ability to think (say, of abstract mathematics) is built through the reuse of these existing capacities. Barsalou (2016) summarizes the current stateoftheart on neural reuse:
Reuse might explain, for example, why even purely symbolic manipulations—that is, “mental manipulations” of symbols—appear to be processed as literally (i.e., physically) manipulating symbols (Goldstone et al. 2009).[Neural] reuse may not be complete and may vary considerably across tasks and contexts. Nevertheless… higher cognitive processes are grounded in more basic processes by virtue of reusing their neural resources. (p. 1130)
Tao’s admission—“you just use whatever tools you have available”—strikes me as a commonsense way of phrasing the neural reuse hypothesis.recalls the day his aunt found him rolling around her living room floor in Melbourne with his eyes closed. … He was trying to visualise a “mathematical transform”. “I was pretending I was the thing being transformed; it did work actually, I got some intuition from doing that.” His aunt is likely still puzzled. “Sometimes to understand something you just use whatever tools you have available.”
Revisiting mathematical practice as developing kinesthetic sensations
Neural reuse is thus used to argue for the plausibility of even abstract mathematical concepts having associated kinesthetic sensations. (Perhaps Tao’s “intuition” arising from rolling on the floor is an example of such sensations, though one need not suppose that an overt physical action is necessary.) This supports a reading of the long division episode as more closely aligned with push hands training. Both the martial arts student and math student are first copying movements (from the outside: the initial stage of motor development) and then, through repetitionwithoutrepetition, becoming aware of something previously unnoticed in their own actions (from within: the later stage of motor development).
In a nutshell, the conjecture is that when students “make sense” of mathematics—for example, grasping a connection between a loopback pattern in the division algorithm and the construction of repetitive decimals—they do so through literal sensemaking. Mathematical constructions, like motor skill development, may involve kinesthetic sensations of the sort Vygotsky describes. I offer that mathematicians do this all the time, whether rolling on the floor like Tao, or sitting still: thanks to neural reuse, one need not act overtly.
[T]here is relatively little evidence (but not none) that verbal instruction helps… pure verbal instruction is effective only to the extent that it helps students understand real or imagined examples (p. 463).
This perspective goes some way towards explaining an observation^{10} shared among mathematicians, as captured by von Neumann’s quote: “Young man, in mathematics you don’t understand things. You just get used to them.” This was, reportedly, von Neumann’s reply to an inquiry asking for an explanation. An interpretation of von Neumann’s reply is that one understands mathematics by doing mathematics. No amount of explanation will help otherwise.
Conclusions
This paper aimed to broaden the discourse around direct instruction, repetitive practice, and discovery. Repetitive practice is sometimes viewed as supportive of “getting the basics” or, alternatively, ridiculed as “drill and kill” (Schoenfeld 2004; Fletcher 2009). Yet, this paper strives to reconceptualize repetitive practice as an exploratory activity where direct instruction and discovery are seen as intimately integrated. This integration is observed in explorative practice, a pedagogical approach wherein the teacher instructs and guides a student through a practice for the explicit purpose of having the student develop certain internal sensations. So doing, the teacher aims to bring to the student’s awareness their subjective construction of the action, signifying it within the semiotics of the discipline.
Embedded in explorative practice is the idea that learning depends on becoming aware of certain bodily sensations. This aligns with von Glasersfeld’s argument that the primary goal of mathematics instruction has to be bringing mathematical knowledge to students’ conscious understanding. The development of conscious awareness of mathematical knowledge can be gleaned through the comparatively simpler development of awareness of kinesthetic sensations, which emerges through guided, repetitive practice (also see Trninic and Abrahamson 2012, 2013). This practice is necessary in order to make the principles of the discipline salient, that is, noticeable. I conjecture that principles of the discipline are instantiated as patterns in and of practice: through practice, we follow and become aware of the pattern (in that order). As we incorporate the pattern into our repertoire, we simultaneously begin to develop our understanding of the principle.
Finally, instruction and guidance are needed to provide both the activity that is the context of discovery and the guidance that focuses students’ attention in a way that supports discovery. From the perspective presented in this paper, the problem with unguided discovery is not that it expects students to discover underlying principles of the disciplines, but that it expects students to also discover how to go about doing this in the first place.
Embodied design for mathematics
Before returning to the instructivistconstructivist debate, I wish to briefly highlight a possible design direction for explorative practice in the context of mathematics education. Over the last two decades, embodied cognition has made an impact in the mathematics education community, and particularly relevant for us are examples of researchers making a link between mathematical knowing and kinesthetic sensations (whether or not they use that term).
The pioneering work of Nemirovsky has a recurring theme of internal sensations playing a role in mathematics knowing: “kinesthetic experience can transfer or generalize to the building and interpretation of formal, highly symbolic mathematical expressions” (Nemirovsky and Rasmussen 2005, p. 15). Canadian scholars Radford (2003) and Roth (Roth and Thom 2009) similarly argue that mathematical conceptions are better understood as networks of experiences that emerge from (re)organizations of embodied sensations.
One designbased approach with evident parallels to explorative practice is to build learning environments that elicit motor schemes and challenge the learner to signify them within a discipline’s semiotic system (Abrahamson 2009). In one design (see Howison et al. 2011), students begin with an ostensibly simple game of moving their arms in front of a monitor to “make the screen green.” Through interaction, they come to recognize that only certain proportional, bimanual movements (e.g., one arm moves twice as fast as the other) result in a green screen. As students make further observations, the task shifts from a movementbased game to a mathematics problem. Abrahamson and collaborators argue that such shifts of perspective in their design enable students to signify embodied experiences in mathematical formalisms (Abrahamson et al. 2016). This parallels explorative practice, where the teacher guides the student toward awareness of their subjective construction of the action, even while signifying it within the semiotics of the discipline. Future work in this direction, coupled with increasingly sophisticated behavioral and neural signatures, may provide the empirical evidence needed for a definitive evaluation of the ideas raised in this paper.
Revisiting the instructivistconstructivist debate
We now return to the “sides at war” metaphor we started with as a means of situating explorative practice within the broader debate. The previously mentioned article by Kirschner et al., Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problembased, experiential, and inquirybased teaching (2006), serves as an apt starting place, having become something of a landmark (or a lightning rod) since its publication. According to Kirschner et al., noninstructivist teaching approaches are a failure, because they involve minimal guidance and thus ignore the cognitive load theory. In their analysis, asking students to make discoveries burdens their working memory, which is viewed as detrimental to learning. Instead, students should be provided with “information that fully explains” whatever it is we think they should know.
In the followup debate (e.g., Tobias and Duffy 2009), instructivists continued to criticize those advocating minimally guided instruction, while constructivists criticized instructivists for confusing guidance with instruction (see, e.g., Gresalfi and Lester 2009). Putting aside the accuracy of these critiques and the productivity of such dialogues (see Taber 2010), their existence hints at how difficult it is to situate this paper. Explorative practice is a pedagogical approach with a high degree of guidance but a minimal degree of explaining. Depending on one’s side in the debate, such an approach could be interpreted as either constructivist, because explaining is minimized in favor of personal discovery, or instructivist, because guidance is maximized at the expense of unguided exploration.
Moreover, the reason why explaining is generally minimized in explorative practice—as we saw in the martial arts section, where teachers like Fong actually withhold explanations^{11}—is precisely because it is believed that explanations could unduly increase the student’s cognitive burden. Or, as Fong might phrase it, “I could tell you, but you wouldn’t understand.” To Fong, an explanation of a principle is not something the expert gives, but something the student constructs.^{12}
We can, however, identify analogs of explorative practice within Kirschner et al.’s oppositional landscape by paying closer attention to what is included and what is excluded in their article. Of note is their simultaneous inclusion and omission of Ann Brown’s work. Specifically, Brown and Campione (1994) is included as evidence of established scholars in the field arguing against pure discovery.^{13} What is omitted, however, is what Brown and Campione argued for, namely guided discovery, which they defined as a “middle ground… where the teacher acts as a facilitator, guiding the learning adventures of his or her charges” (p. 230).
If forced to pick a position, we could attempt to situate explorative practice somewhere in this “middle ground” advocated by Brown and Campione. Fittingly, this metaphor is used by both sides of the debate—see, for example, Kirschner and Lund’s Finding a middle ground: Wars never settle anything (2017), where the authors advocate for a ceasefire between “proponents of traditional mathematics” and “proselytizers of… reform mathematics” (p. 36).
However, is it necessary to choose a position in this imaginary conflict? It has been my experience that our current discourse on instruction and discovery treats these two approaches as oil and water. Just as oil and water are immiscible liquids, an assumption underlying this metaphor is that these approaches do not mix. Consequently, we find ourselves forced into a dilemma: How much instructional time should we devote to one at the expense of the other?
If instead we entertain the possibility that instruction and discovery are not oil and water, that instruction and discovery coexist and can work together, we may find a solution^{14} to this impasse in the field. Perhaps our way out of the instructivistconstructivist impasse thus involves not a “middle ground” compromise but an alternative conceptualization of instruction and discovery. This article proposes that one alternative can be found by examining the teaching methods found in the physical disciplines, where instruction and discovery—and their attendant metaphors of teacherasgiver and studentasexplorer—are seen as interconnected, complementary halves of the same educational activity.
In contrast to other papers in this special issue, the ideas presented here are not novel, but ancient, yet their longevity might also speak to their usefulness. While we cannot directly impart the secrets of our disciplines, we can instruct students on, and guide them through, practices that lead to meaningful discovery and construction of knowledge. In this sense, “direct instruction” and “discovery” are not antagonistic but synergistic, linked by practice.
Footnotes
 1.
Our present focus on “discovery” is a matter of convenience for the sake of a pedagogical argument. By discovery I do not also mean “minimally guided,” but merely that the learner is expected to make a discovery, in the standard dictionary definition of become aware of, or recognize the potential of.
 2.
For example, while arguing against “discovery teaching,” Kirschner et al. (2006) unequivocally state that “the constructivist description of learning is accurate” (p. 78).
 3.
In the standard dictionary sense of repeated exercise in or performance of an activity so as to develop or maintain proficiency in it.
 4.
Also known as The Thinker, a famous bronze sculpture depicting a nude man who appears deep in thought despite visible muscle tension.
 5.
Evidence used in this paper comes from data collected in New York City and the San Francisco Bay Area. They include semistructured interviews, video data, and training notes. Other practitioners saw the author as an inquisitive member of the community rather than a scholar.
 6.
Video recorded by Novell Bell of NYC.
 7.
For one comparison, see Watson and Mason (2006), on the pedagogical benefit of structuring varied exercises in mathematics education.
 8.
According to Bernstein’s students, the two not only worked at the same institute, but played chess and were friendly. There is also some evidence that Bernstein assisted Vygotsky in at least one experiment with photography.
 9.
Because no video or audio records were taken, this illustrative case was put together from observational notes, recollections, and discussions with the teacher.
 10.
The philosopher and historian of mathematics, Roi Wagner, called this “a commonplace” observation among mathematicians present and past (personal communication, 2017).
 11.
Asking students to engage in practice while apparently withholding information from them requires trust or, as one anonymous reviewer put it, requires that students collude with the teacher—not exploring this dimension is a potentially significant limitation of present work.
 12.
Explanations are instead reserved for conveying what Hewitt (1999) might call the “arbitrary” aspects of the discipline.
 13.
“Discovery learning is often contrasted with didactic instruction, and given that choice, I vote for discovery” (Brown 1992, p. 168). While Brown noted problems with pure discovery, her explicit stance in this debate was for discovery.
 14.
E.g., ethanol and water. Alternatively, we might think of explorative practice as an emulsifying agent bringing together water and oil.
Notes
Acknowledgements
For their highly constructive comments on earlier drafts, I wish to thank Hillary Swanson and José Gutiérrez.
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