Journal of Mathematical Behavior20 (2002) 263–266

Editorial

A garland of teaching experiments

Children learn important mathematics in the course of interaction with each other andwith adults.

On the one hand, this sentence seems so obvious that it hardly needs an explanation. On theother hand, if we try to clarify in detail what the two key nouns,mathematicsandinteraction,might mean in specific, given situations, the sentence becomesvery interesting. In particular,we think that it becomesespeciallyinteresting, when we consider the five papers broughttogether in this issue.

Consider, for example, the wordinteraction. One way to discover whether a child might beready to learn a given piece of mathematics is to try to teach it to her, and then see what happens— another statement whose simplicity can be deceptive! The investigator comes into theinteraction with a mindset:1 certain cognitive schemes, already in place in the investigator, havebeen activated, both for the mathematics and the work and thinking of the child. On the basisof these schemes, the investigator seeks to assimilate the actions of the child. Frequently, atleast for the growth of understanding, these schemes prove inadequate, and so the investigatoris challenged toaccommodate, reshaping the preliminary understanding, and then testing theaccommodation in the course of further interaction with the child.

Now look at this interaction, for a moment, from the child’s position. The child, too,has schemes in place, both for the mathematics at hand and for her interactions with theinvestigator. The investigator might try to postulate these, but the child’s actions, the reasonsfor these actions, and the conclusions that she draws from them, nonetheless, remain her own.

With these ideas in mind, we come tochildren’s mathematics. The investigator might beginby modifying her personal (in other words, hypothesized) account of the child’s mathematicalthinking, but often, soon enough, the investigator, inspired through interaction with the child,begins tothink within the mathematics she has builtin order to account for the child’s actions.In this way, a kind oflocal mathematicsnow emerges, a mathematics which is in somewaysnew, as a result of the learner–teacher interaction. More precisely, both the child and

1 Sometimes the observer’s mindset might be called acontext. See Langer (1989).

0732-3123/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved.PII: S0732-3123(02)00073-1

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the investigator might be interpreted as jointlybuilding2 chunks of mathematics as they goalong, a mathematics whose content, as well as its developmental history, requires carefulexplication.

All the papers in this issue share related themes, because they work with learners’ math-ematics as emerging in the course ofgoal-directed human interactions. Each treats both themathematics and the interactions in a special way. Two major research studies frame thisissue: one on fractions, one on probability. Both raise major questions, challenge widespreadpractices, and report interesting data.

In his study of the learning of fractions, Steffe follows pairs of children, in third and fourthgrade, as they interact with each other and with individual investigators. They work with aparticular software environment3 which facilitates the construction, modification and com-parison of linear models; these present quantities as sub-divided lengths. The software allowsboth children and investigators to produce visual images quickly, to facilitate investigationand discussion. Steffe’s main working hypothesis, exactly as he states it, isthat fractionalschemes emerge as accommodations of numerical counting schemes. This hypothesis guidesthe interactions with the children, as well as Steffe’s own interpretations of the children’s math-ematics in the making.4 A remarkable amalgam emerges, in which investigators, postulatingschemes, and children, building images (based upon, but not restricted to their interactionswith the software) and reasoning about them, jointly build new understandings of the mathe-matics they explore. Steffe’s detailed analysis puts special emphasis onspecific actions taken,as key elements within the schemes which the investigators see as guiding how the childrenbuild, reflect upon, and then rebuild their understanding. In particular, the ways in which thechildren count,and how they draw conclusions from their counting, anchor the main findingsof the study, and also motivate new research questions.

As Steffe carefully explains, his work shares common features with some recent investi-gations about children’s fraction learning, while challenging others. His guiding hypothesis,indeed, directly questions the widespread, perhaps even dominant, position that whole-numberunderstanding interferes with learning fractions. Instead, Steffe maintains that children’s workwith fractions, under suitable conditions, can trigger areorganizationof whole-number un-derstanding, to underpin a more powerful, more general, and more flexible arithmetic. Hisargument is detailed, evidenced with care, and, we feel, well worth the effort to assimilate.

The extensive study which concludes this issue, by Brousseau, Brousseau and Warfield,reports for the first time a series of whole-class experiments conducted at Bordeaux in theearly 1970s. In these lessons, fifth-grade students actively explore, for extended periods oftime, situations designed to help them reason about randomness. The children, in a spiritedcollaboration with their teacher, conduct experiments which they in part design. Then, again

2 Or perhapsdiscovering, inventing, rediscovering, or reinventing, depending on one’s viewpoint.3 TIMA: sticks (Biddlecomb, 1994).4 See also Speiser and Walter (2000), where the distinction between mathematics in the making and readymade

mathematics is pivotal for a discussion, related in some ways to Steffe’s, about how learners (there preserviceelementary teachers) construct number systems.

Editorial / Journal of Mathematical Behavior 20 (2002) 263–266 265

with strong participation by the teacher, they reflect, focus and sometimes reformulate ideasthat they are building. The research design, like Steffe’s but perhaps more formally, evolvedthrough the ongoing interaction. Here, the focus was not primarily cognitive, as in Steffe’swork, but rather toachieve a way of workingwith the thinking of these students which could,more or less directly, inform later teaching. The mathematical emphasis was on fundamentalprobabilistic and statistical ideas, especially sampling, sampling distributions, and predictionbased on the analysis of samples. The ongoing research design, and hence the study’s impli-cations for instruction, build from quite specific questions that the students and their teachersaw as central. These questions and the bold experiments they led to, make a very lively story.Like Steffe’s work on fractions, the Brousseaus and Warfield challenge widespread views,here about children’s statistical and probabilistic thinking. In particular, ideas5 like samplespace, event, andformal calculations with eventsdo not begin the story here, but rather mightemerge as useful constructs later, possiblymuch later, once other, still more fundamentalunderstandings have been reached. This paper, too, we think, has much to offer for its readers.

Framed by these two studies, the three remaining papers also help to build our understandingof the mathematics built by learners and their teachers, and suggest new ways to think aboutthat mathematics as a subject in the making.

Brizuela and Lara-Roth report on a part of teaching experiment designed to emphasizesome algebraic aspects of arithmetic. They focus on how second-graders work with addi-tive relations, in particular how they represent such relations by means of tables they inventthemselves. Based on very interesting student data, from the way the children build and usetheir tables, the authors suggest interesting conclusions about which issues these learners havefound relevant. Based on such conclusions, more responsive teaching might emerge.

The next paper, by Christou and Philippou, reports on the work of fourth- and fifth-graderson Cyprus, as they solve proportion problems. The researchers focus their attention onstudents’ solution strategies, as well as how certain intuitions developed by the students influ-ence their strategies. The students’ intuitions might support, or perhaps also constrain, theirstrategies.

In a teaching experiment, much in Steffe’s style, Hines works with one-eighth-grader. Thestudent and the interviewer work with dynamic physical models, such as a spool elevatingsystem, to investigate linear relations between pairs of variables. Such physical devices arebelieved to facilitate a shift from thinking about multiplication as an operation on two givennumbers, to thinking about multiplicative relationships between two quantities that vary. Hinesinvestigates in detail, proposing further tasks based on her student’s work and thinking. Notsurprisingly, different ways of representing quantitative variation turned out to be helpful.This paper’s interest may lie rather in the ways that Hines has chosen to conceptualize (andevidence) her student’s growth of understanding.

Every paper in this issue treats the interactive building of important mathematics in newways. Both the mathematics that emerges, and the ways that people work with it, are rich,

5 Some might call them “big ideas.”

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complex, and, we believe, fascinating — fascinating in Piaget’s (1964) specific sense: thatthere is something here that we are eager to assimilate. Further, we invite readers to entertainthe hopeful possibility that each experiment reported here gives just aglimpseof what ispossible, that the mathematical potential of learners and their teachers is still, to a greatextent,unfathomed.

Beethoven’s Birthday

16 December, 2001

References

Biddlecomb, B. D. (1994). Theory-based development of computer microworlds.Journal of Research in ChildhoodEducation, 8(2), 87–98.

Langer, E. (1989).Mindfulness. Reading, MA: Addison-Wesley.Piaget, J. (1964). Development and learning. In: R. E. Ripple, & V. N. Rockcastle (Eds.),Piaget rediscovered:

a report of a conference on cognitive studies and curriculum development(pp. 7–19). Ithaca, NY: CornellUniversity Press.

Speiser, R., & Walter, C. (2000).Five women build a number system. Stamford, CT: Ablex.

The Editors