Chapter 1 — Disconnect

When dividing fractions, don't ask why — just flip the sucker and multiply.

Math teachers, of course, know this isn't optimal. Students who follow the procedure produce correct answers, and for a teacher under pressure to cover a curriculum, correct answers are the deliverables. This chapter is about two students who show why focusing on outcomes shortchanges students.

Angie

Angie could produce correct answers with the procedure, but it always bothered her. When I met her, she had just left traditional school for one of the democratic schools where I was doing my Ph.D. research. When I asked her why she switched, she didn't hesitate: "I hate it when teachers don't listen to your ideas and just make you do it their way."¹ She'd been socializing with her friends, who were nearby. The moment we started talking and drawing her ideas on the chalkboard, not only was she engaged, but her friends drifted over to follow our conversation. These are all students who weren't required to pay attention, yet were pulled into a lesson they hadn't asked for, and completely free to leave whenever they chose to.

Angie reasoned that in dividing fractions, if you divide across the top and then the bottom numbers you get to the right answer: ½ ÷ ½ = 1. That's right, but locally bounded: dividing across top and bottom (1÷1 over 2÷2 = 1) works whenever the numerators match and the denominators match, because dividing equal numbers gives 1. It breaks at a case like ½ ÷ ¾. Angie hadn't yet tested her rule against enough cases to find its limits, which would have been the teacher's follow-up had she taken Angie's idea and worked with her on it.

But look at what her reasoning was doing. She was drawing on what she knew about division, and thinking mathematically even when her procedure only worked in limited cases. Her teacher, instead, saw Angie draw out the wrong procedure, re-explained the algorithm, and moved on. What Angie learned was: your way of solving this doesn't count. Her working model was treated as an error to fix rather than the beginning of an inquiry.

Angie exhibited what I'll call engaged agency: a student's effort to understand from their own position as a knower, through their own thinking, however partial or wrong. The origination of the idea is key here. Was the student doing the procedure as assigned, or was the student the originator of their own thinking? Their beginning thinking rarely looks like mastery; it often looks messy, partial, or wrong. But note, Angie did not arrive at the lesson unengaged. She came motivated, and then it was squashed. Angie was demotivated by the teacher's very typical classroom instruction. How often do we try to motivate a disengaged student? We try to fix it; supply more of it — relevance, incentives, activities, rapport — rather than ask, "What sucked the air out of this student's motivation?"

It's worth seeing exactly what the jingle drops mathematically. The procedure preserves the number's value and erases the unit that gives it meaning. Take ½ cup ÷ ¾ cup. The algorithm yields ⅔ — correct, but ⅔ of what? What the division actually asks is: how many ¾ cups fit in half a cup? The answer is ⅔ of a ¾-cup. Restore the unit and the relationship is intelligible; drop it, and the student is left holding a number that they can't visualize in a real-life process. Mathematically Angie's version is partly accurate because units cancel units, but Angie's resistance was an attempt to tie the symbols back to something she could see and make sense of.

Nancy

Nancy's story shows the other side of the same problem — not a student whose partial thinking was visible, but one whose thinking was illegible to her teacher.

Nancy was a fourth-grader in a mixed third- and fourth-grade classroom at a progressive school that prized hands-on learning.² Her teacher, John, posed a yard-sale problem to the class: three students earned a dollar and had to split it evenly. He handed each child 100 pennies, allowed them to trade pennies for other coins from his bin, and then circulated to see if they successfully solved it. Students worked in several ways: some made three piles, some made strands and counted off, and some exchanged pennies for higher-value coins. Nancy traded pennies for quarters and then arranged her coins into three symmetrical groups of thirty-three cents, with a single penny at the center.

When John reached her, he was unsettled; he was looking for something that matched 100 ÷ 3 = 33 r 1. He asked Nancy, "Where is the remainder?" She first looked startled, then searched her mat, and began counting on her fingers. Her flower model didn't have a remainder — the center penny anchored a balanced arrangement. Because Nancy was known as being artistic, John assumed she'd drifted into art and forgot she was doing math. He moved on.

Nancy's arrangement inherently carried a mathematical orientation — a geometric one — whether or not she could yet name it. The coins were amounts, but they were also things she could arrange, so the relationships between them changed: the total value held steady while the configuration made "evenly" mean radial and balanced rather than a count with a leftover. What was missing was a teacher to surface that difference.

Surfacing it would have brightened Nancy rather than startled her — and would have shown her, and the class, that her way of seeing through art can also be understood as mathematical. Instead, she learned to file art under "not math," and reproduce the lesson.

Nancy was not the only student with a paradigmatic difference with John's intention. Several of her classmates brought out a different aspect of the math problem: social fairness. Taking the three-friends framing seriously, they asked: if it's split equally, who gets the extra penny? Do you rotate it? Give it to the youngest? These weren't off-task questions.

I went home and experimented on my then five-year-old youngest, who had not experienced school division yet. She had the same dilemma. We put out three teddy bears, I gave her the coins, and after dividing them, she held the last penny; who gets it? The biggest? Mommy bear? A remainder that's obvious on paper becomes a real question once you ask to whom it belongs. One student said it plainly: I don't know who gets the remainder, but I know what the teacher wants. That sentence should stop us. A child who has set aside his questions has already begun to disconnect from himself as a knower in order to give the teacher the thinking he wants (von Duyke & Matusov, 2016).

Nancy's flower and the students' social questions could have been part of the lesson: Does Nancy's arrangement show something mathematical? Is the center penny a remainder? What does "evenly" mean here? The answer 33 r 1 becomes sensible and memorable precisely when a class has worked through the questions students are already asking about it. John's lesson worked, but only in the narrow sense — students were able to construct an answer, but their questions were left hanging. To be fair to John, he did hesitate over Nancy's mat, precisely because he saw that her arrangement did seem to work. But the constructivist training he likely carried had no move for what she had done; it takes a dialogic lens to hear that she'd changed the question (to be discussed in Chapter 5).²ᵇ

With classroom discussion, Nancy and her classmates would've been able to move between their perceptual orientations (aesthetic, algorithmic, social) and choose which is appropriate for which problem, and note the limitations in the algorithm alone. Nancy's aesthetic orientation would have been built on as a legitimate mathematical move. She could have connected it to the social and algorithmic models in the rest of the class. The class would be exposed to differences in understanding, surfacing questions that hadn't yet been formulated.

What Nancy was doing, in the taxonomy I present in Chapter 2, was a bid towards epistemic agency: not engagement inside the teacher's frame, but a change of the frame. Where Angie's engaged agency worked inside the teacher's question, Nancy's move proposed a different one — that "evenly" could be a pattern rather than a count. Epistemic agency is easy to miss because it often looks like off-task behavior, and because an epistemic bid only develops in class when it is recognized by the teacher. The thinking pattern exists without teacher recognition, but an unread bid leaves the student with no confirmation and no added value to her ideas. The rest of this book builds a language for seeing these differences in student thinking, so that what happened to Angie and Nancy becomes less likely — not because teachers try harder to engage students, but because they can recognize what has already engaged them. Did Nancy learn that an artistic orientation towards math has no value? Did Angie learn that thinking through alternative procedures has no value? When student thinking is discounted often enough, it becomes a kind of epistemic gaslighting.⁵ No wonder Angie was pissed off enough to finally leave.

Nancy's coin arrangement was a proposal from outside the task frame — the kind of frame-shift that only a dialogic lens recognizes, and that even the theories I most admire, radical constructivism and Magdalene Lampert's discourse work, have no move for.⁶

What both students show

These are not idiosyncratic misses, nor bad teachers; they are predictable outcomes of classrooms organized around producing legible answers rather than interpreting student thinking. Schooling trains students to adapt to the teacher's framing rather than meet learning on their own terms — to fit meaning rather than make it. It teaches them that their questions, models, and frameworks don't count. It's teaching with one foot on the gas and the other on the brake: we stop students exactly when they're engaged, then ask them to engage in a way they aren't curious about.

What standard practice misses

Standard practice misses three levels of student agency simultaneously:

1. The answer level: it evaluates correctness without examining the reasoning.

2. The agency level: it doesn't recognize bids for engagement, epistemic reframing, or authorial direction.

3. The chronotope level: the misreading is systematized by policy decisions around legible outputs.

When policy demands legibility, teachers can't afford time for agency. When they can't afford agency, they can't see diverse reasoning. When they can't see reasoning, they teach to the answer. The teacher who could read Nancy's flower as mathematics would be doing something schools don't support: developing mathematical thinking from aesthetic perception, not just "allowing" it as a warm-up before getting to the "real" math.

To be a knower is to be received as someone whose sense-making counts — whose partial, half-formed account is treated as the beginning of understanding rather than the absence of it. This opens the door for meaning-making: a different way of understanding, seeing, or personally relating to a lesson. Students originate meanings → they make those meanings available through questions, models, objections, arrangements, and behaviors → classroom discourse either recognizes and develops those contributions or redirects and suppresses them → repeated patterns of failing to take up their ideas shape the engagement students experience. This dialogic approach also requires a shift in what school knowledge is taken to be: not a finished object transferred to students intact, but something a person originates in their own understanding as they engage with it with others.³ Students' ideas can then be tested, revised, and articulated more powerfully — making them answerable to each other.

To be fair, the research literature already uses the term epistemic agency, for students' active participation in inquiry; I use it to denote a different shift — the point at which the student's thinking begins to exceed the lesson frame.⁴

What in the structure of schooling makes this kind of disconnection nearly inevitable rather than exceptional, especially in high-poverty schools?

Notes — Chapter 1

1. Angie appears in my dissertation research on democratic schools (von Duyke, 2013). Angie is a pseudonym, and identifying details have been altered to protect her privacy. von Duyke, K. S. (2013). Students' autonomy, agency, and emergent learning interests in two open democratic schools [Doctoral dissertation, University of Delaware]. ProQuest Dissertations & Theses (Order No. 3595010).

2. Nancy and John are pseudonyms from my research in a progressive school classroom; identifying details have been altered. von Duyke, K., & Matusov, E. (2016). Flowery math: A case for heterodiscoursia in mathematics problems solving in recognition of students' authorial agency. Pedagogies: An International Journal, 11(1), 1–21. https://doi.org/10.1080/1554480X.2015.1090904. In that article, I described Nancy's work as authorial, but as my thinking on agency has evolved, I no longer think her bid was as intentional as authorial agency; I see it as originating with her but not fully realized in the classroom. The research exists for a framework that would have supported Angie's model (cf. von Glasersfeld, 1995), but would have missed Nancy's reframe. This incident was drawn from my field notes, but didn't appear in the article.

2b. John was likely trained in an earlier register of constructivist pedagogy — the idea that students should be left to construct their own understanding, with the teacher's job mainly to step back and let that happen. Classic constructivist pedagogy holds that students can construct their own learning because doing so makes the problem internally sensible; in John's class, the construction in fact raised more uncertainty. A later refinement, radical constructivism, sharpens the teacher's role instead of removing it: the standard isn't whether a student's model matches the teacher's answer, but whether it's viable — whether it holds up under further testing. Nancy's frame shift is a step removed from even that. Cf. von Glasersfeld, E. (1995). Radical Constructivism: A Way of Knowing and Learning. Falmer Press.

3. I want to point out that even a textbook is in contact with others, as the author(s) who conveyed those ideas were also in contact with other interpretations back to the origination of the idea. Bakhtin, M. M. (1981). The Dialogic Imagination: Four Essays (M. Holquist, Ed.; C. Emerson & M. Holquist, Trans.). University of Texas Press.

4. My use of engaged agency is much closer to how the educational literature often defines epistemic agency. In the science-ed strand (Miller et al., 2018; Stroupe, 2014; González-Howard & McNeill, 2020), students act within the teacher's frame, and their moves are treated as worth following. In the knowledge-building strand, students take over what Scardamalia calls "problems of goals, motivation, evaluation, and long-range planning" normally left to teachers (Scardamalia, 2002; Damşa et al., 2010), whereas Nancy's bid was based on the criteria for how knowledge itself is constructed. The distinction matters because the literature often conflates student activity and teacher response under the assumption that the teacher is always shaping the student toward the lesson outcomes, rather than using student understandings to deepen or even reconsider lesson outcomes. Miller, E., Manz, E., Russ, R., Stroupe, D., & Berland, L. (2018). Addressing the epistemic elephant in the room: Epistemic agency and the Next Generation Science Standards. Journal of Research in Science Teaching, 55(7), 1053–1075. Stroupe, D. (2014). Examining classroom science practice communities: How teachers and students negotiate epistemic agency and learn science-as-practice. Science Education, 98(3), 487–516. González-Howard, M., & McNeill, K. L. (2020). Acting with epistemic agency: Characterizing student critique during argumentation discussions. Science Education, 104(6), 953–982. Scardamalia, M. (2002). Collective cognitive responsibility for the advancement of knowledge. In B. Smith (Ed.), Liberal Education in a Knowledge Society (pp. 67–98). Open Court. Damşa, C. I., Kirschner, P. A., Andriessen, J. E. B., Erkens, G., & Sins, P. H. M. (2010). Shared epistemic agency: An empirical study of an emergent construct. Journal of the Learning Sciences, 19(2), 143–186.

5. "Gaslighting" captures the experiential quality for the student: not just "my idea was ignored" but "my capacity to trust my own perception is eroded." That's different from, and more damaging than, simple correction. A teacher who corrects a math error while validating the student's reasoning preserves epistemic confidence. A teacher who treats the error as evidence the student wasn't doing math at all erodes it.

6. Lampert's research in mathematics aimed to open the discourse: students are encouraged to conjecture and disagree, but the direction of fit runs one way, toward disciplinary norms. The teacher is answerable to mathematics. Wouldn't it have been better if the teacher were answerable to the student's bid? Lampert's repertoire takes up wrong-but-coherent reasoning within the lesson frame; it has no move for a bid that reframes it. See also the Introduction, note 2.