Proportional Reasoning 1 Running head: NURSES ... · Ratio, proportion, percent Situated cognition Out of school mathematics . Proportional Reasoning 3 ... In their detailed critique

Proportional Reasoning 1

Running head: NURSES' PROPORTIONAL REASONING

Proportional Reasoning in Nursing Practice

Celia Hoyles, Richard Noss, and Stefano Pozzi

University of London, UK


Abstract

We investigate how expert nurses undertake the calculation of drug dosages on the

ward. This calculation is error-critical in nursing practice and maps onto the concepts of ratio

and proportion. Using episodes of actual drug administration gleaned from ethnographic

study, we provide evidence that experienced nurses use a range of correct proportional

reasoning strategies based on the invariant of drug concentration to calculate dosage on the

ward, as compared to the single taught method they describe outside of the practice. These

strategies are tied to individual drugs, specific quantities and volumes of drugs, the way drugs

are packaged, and the organization of clinical work.

Key words: Adult learning

Ethnography

Ratio, proportion, percent

Situated cognition

Out of school mathematics


Proportional Reasoning in Nursing Practice

The centrality of mathematics to school curricula has partly been justified in terms of

its importance for the world of work, and many researchers have attempted to articulate the

relationships between school-taught mathematics and the mathematics found in occupational

and everyday lives. We believe that the study of mathematics in work provides a particularly

fruitful setting for illuminating fundamental issues concerning the construction of

mathematical meanings more generally. In essence, it offers an opportunity to focus attention

on the ways in which professional discourses shape mathematical meanings and, reciprocally,

on how the use of mathematics--however defined--structures the discourse of work. One

might hope that such insight will afford leverage on didactical strategies within and beyond

work-based settings.

Until the middle of the 1980s, research in this field was largely undertaken within a

framework in which the researchers assumed that mathematics was unproblematically visible

in workplace settings and that it mainly consisted of calculation (see, for example, Fitzgerald,

1981). Later researchers, for example Wolf (1984) and Harris (1991), adopted methodologies

that were considerably more sensitive to the complexity of the practice/mathematics

relationship. Yet all using such approaches encountered a fundamental obstacle: Most

employees did not describe their activities in mathematical terms, and they often declared

that they used very little mathematics in their work. The fundamental difficulty, it seems, is

that the mathematics of work is hidden beneath the surface of cultures and practices, so that

any superficial classification of it in terms of school-mathematical knowledge will inevitably

result in its reduction to simple measurement and arithmetic (see Cockcroft, 1982, and for a

critique of this approach, see Noss, 1998).

More recently researchers, particularly from the United States, have aimed to seek

mathematical structures in work by providing detailed examination of particular work


settings. As a result, researchers have begun to take epistemological issues more seriously in

considering how employees approach mathematical tasks and how these are shaped by both

the purposes and the tools of workplace practice. The seminal work by Scribner and

Fahrmeier on the cognitive (including mathematical) strategies of dairy workers (Fahrmeier,

1984; Scribner, 1984a, 1984b, 1986) provided important insights into how people, using

salient features of their environment, regulate and think about their activities. Since then, a

range of occupations have been examined, with correspondingly detailed consideration

directed at how knowledge influences and is influenced by workplace activity. Workers

studied include carpenters (Millroy, 1992), carpet-layers (Masingila, 1994), seamstresses

(Hancock, 1996; Harris, 1987), automotive-industry workers (Smith & Douglas, 1997), and

civil engineers (Hall & Stevens, 1995; Hall, 1998).

These strands of research have put forward a conception of mathematical activity that is

intertwined with the complexities of working practices, but this conception raises a number

of questions concerning the status of the knowledge in use. What is its relationship with

formally taught school mathematics? Are formal and practical mathematics derived from

different epistemologies? What does it mean to analyze workplace settings from a

mathematical perspective, in as much as in most cases, practitioners deny that they are using

any mathematics? How do we avoid preconceiving what does and does not count as

mathematics? Some important pointers are provided by Smith, diSessa, and Roschelle

(1993). In their detailed critique of research on students' misconceptions in science and

mathematics, they offer a set of theoretical principles to give a more elaborate account of the

nature of knowledge and learning in such domains. Central to these principles is Smith et

al.’s notion of “knowledge in pieces,” whereby knowledge is reconceived as a distributed

network of components that are applicable only to a narrow set of problems, in contrast to the

conception of abstract knowledge more commonly identified with competence. For Smith et


al., the high value placed on general and compact propositions of mathematics and science is

an inadequate basis for analyzing people’s reasoning in these disciplines and fails to take

account of the diversity and plurality of such knowledge. The strength of this position is that

this diversity is conceived within an epistemological framework, in contrast to the position of

some situated-cognition theorists (for example, Lave & Wenger, 1991) who seem to reject

the notion of knowledge as a self-sustained entity. However, Smith et al. do not provide any

detailed analysis of the cultural practices in which the pieces of knowledge are used. We

believe that a synthesis of the cultural practice and knowledge-fragment analyses is within

reach: Our aim, therefore, is to analyze the material and social organization of knowledge-in-

use within a specific work situation.

Although we accept that persons must be studied in their communities, we also believe

it is possible to locate the mathematical knowledge that can describe what they do and indeed

know. This attempt to chart the plurality of mathematical knowledge has engaged us in

studying a number of professional settings and in attempting to tease out “the mathematics of

the practice”--how it is described, used, and conceived. To achieve this aim, we use

ethnography and interviews to capture the subtle meanings created in situ and the dialectical

relationship of these meanings with mathematical expression on the one hand and

professional expertise on the other.

In what follows, we home in on the conceptual field (Vergnaud, 1982) of ratio and

proportion. Our rationale for this choice was that it is a subdomain generally agreed to be in

use in the practice under study and one in which calculations are error-critical--that is,

precision and correctness are of paramount importance. By focusing on ratio and proportion,

we could inform our work with insights on students' understandings gleaned from relevant

research in mathematics education derived from school and laboratory settings. We first

outline the major threads in these arguments.


Research Related to Ratio and Proportion

Activities involving ratio and proportion are all-pervasive both in and out of school,

and this conceptual domain has been the subject of considerable research within mathematics

education since the seminal work of Inhelder and Piaget (1958). Placing proportional

reasoning at the center of formal operational thinking, they developed a theory in which they

distinguished four increasingly sophisticated stages of proportional thought. Although, on the

basis of their notion of horizontal bricolage, they anticipated task-related variations in

responses of students who were in the earlier stages, this variation was assumed to all but

disappear once an adolescent was able to relate the variables in a proportional problem in a

linear way. Since then, research has shown that student responses are highly sensitive to task

and context factors, such as the type of ratio quantity requested (Karplus, Pulos, & Stage,

1983), the particular numbers in the task (Clark & Kamii, 1996; Hart, 1984), and the context

of the problem (Clarkson, 1989; Lawton, 1993; Noelting, 1980a, 1980b; Vergnaud, 1983).

Moreover, proportional reasoning continues to be problematic for students even in their late

adolescence and beyond, with many adopting additive rather than multiplicative reasoning

strategies (see, for example, Adi & Pulos, 1980; Hart, 1984; Kaput & West, 1994; Simon &

Blume, 1994; Thompson, 1994).

Conceptual analyses of proportional reasoning stand in marked contrast to the broad

body of research on adults’ mathematical problem solving in the workplace or in everyday

situations, much of which involves problems of proportionality (e.g., Carraher, Carraher, &

Schliemann, 1985; Lave, 1988; Nunes, Schliemann, & Carraher, 1993; Schliemann &

Carraher, 1992). These studies suggest that adults are adept at solving proportional problems

in everyday or work situations but often employ informal strategies that are tailored to the

particular situation and are not easily identified with formal school-taught methods.

Moreover, in her study of best-buy strategies, Lave found that shoppers who performed in a


virtually error-free way in the supermarket became error-prone when given ostensibly the

same tasks in paper-and-pencil simulations. Lave’s work is complemented by studies that

indicate that school children shift to less error-prone strategies outside the mathematics

classroom, either in other curriculum subjects (Säljö & Wyndhamn, 1990) or in leisure

activity (Herndon, 1971; Spradberry, 1976).

One difficulty with many of these studies is that the ratios participants used were most

often elementary--for example, 2:1 and 3:1--so one cannot say how generalizable successful

strategies would be with more complex ratios. This shortcoming of the studies matters: In

order to understand the complex relationships between knowledge generation and cultural

practices, one needs to base analyses on sufficiently rich patterns of knowledge elements and

relationships.

These studies of out of school mathematics have spawned a number of attempts to

elaborate ways to compare activity in informal and formal settings. For example, Nunes and

her colleagues (1993) highlighted the need for “a complementary analysis of situational

models that embed the same logicomathematical invariants” (p. 70) and operationalized this

mode of operation in their later studies by using the conceptual framework developed by

Vergnaud (1982). His model of proportionality, generated by categorizing school-children's

strategies, is based on the notion of a multiplicative conceptual field, within which he cites

three subtypes of multiplicative structures. One subtype, the isomorphism of measures,

involves direct proportions between two measure spaces and characterizes many situations,

including equal sharing (person-objects), pricing (goods-cost), constant speed (distance-

time), density (mass-volume), and, particularly relevant here, concentration (mass in solution

volume). We will illustrate Vergnaud's classification with an example, central to our study,

concerning drug dosages; their calculation involves proportional reasoning. The mass, (m), of

a drug is dissolved in a volume, (v), of solution. The relationship between m and v is fixed


for a given concentration of drug, so for different amounts of drug f(m), there is a

corresponding volume f(v), where f is a multiplicative function. Now consider a concrete

(imaginary) case. A drug comes in packets of 120 mg diluted in 2 ml of fluid. One question

that might be asked is “How much diluted drug should be administered for a dose of 300

mg?” Within Vergnaud's model, the correct solution can be obtained in five ways: one

functional (between measure spaces), three scalar (within measure spaces), and one

consisting of the school-taught rule-of-three. These strategies are shown in Table 1.

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Insert Table 1 here

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In a study of proportional reasoning involving 100 children from sixth to ninth grades,

Vergnaud (1980) found that the three scalar strategies were used much more frequently than

the functional strategy (80% vs. 18% of the correct responses) even for problems for which a

functional approach was computationally easier (see also Freudenthal, 1978). Furthermore,

the rule-of-three approach was used in only 2% of the correct responses. His explanation for

the propensity for use of scalar strategies is based on an important insight: students were

working with quantities and relationships and not simply numbers. The use of functional

strategies involves operations between distinct quantities. It requires students to find

quotients across measures, producing a new quantity that has no direct relationship with

either of the original quantities but rather expresses a relationship between the two (e.g., ml

per mg). With the rule-of-three, this complexity is compounded further, in as much as

intermediary products or quotients sometimes have no clear meaning to students (e.g.,

multiplying mg by ml). In contrast, scalar strategies involve manipulation of scalar amounts

within each measure space and do not involve the introduction of new quantities.


Schliemann and Nunes (1990) used this framework to categorize fishermen’s responses

to proportionality problems involving the weights of freshly caught seafoods and their price

yields after processing. In this research, the fishermen were invited to solve problems not

only in (for them) familiar situations but also in unfamiliar contexts: for example, inverse

problems (which arose rarely in their work) and problems involving ratios with which they

were not accustomed. The results showed that the fishermen could use their proportional

reasoning strategies flexibly, never used the rule-of-three (perhaps unsurprisingly), and, most

relevantly, rarely used functional strategies. Even when a functional relationship was

numerically simple compared to the scalar one, the fishermen would pursue computationally

awkward scalar approaches. In one study, the researchers systematically compared the

performance of fishermen and students, and noticed in students the same propensity seen in

the fishermen to pursue scalar approaches when faced with problems with numerically

simpler functional solutions. Nunes et al. (1993) concluded that scalar approaches are drawn

from experiences in everyday situations, are more flexible and generalizable than the easily

forgotten algorithmic approaches taught in Brazilian schools, and, most relevant here, allow

people to preserve the meaning of the situation by keeping variables separate and not

calculating across measures.

Nursing, Drug Administration, and Mathematics

We now turn to the professional setting of our study, pediatric nursing, and consider

the place of drug administration in this practice. Nurses daily confront the possibility of

making mathematical errors with potentially serious consequences when they calculate and

administer drugs as part of their routine work. Even if not harmful, errors can still cause great

distress or result in disciplinary action for the nurses and others involved, and they may have

legal implications for the hospital (see Arndt, 1994).


The nursing profession and other healthcare organizations spend considerable time and

resources planning to avoid these errors by focusing attention on nurses’ mathematical and

practical competence and on the adequacy of health and safety procedures in hospitals. Drug

administration involves a range of mathematical concepts, including measurement and

estimation as well as ratio and proportion. To give a flavor of what it involves on the ward,

we present a simplified description of a typical scenario. An intravenous antibiotic,

vancomycin, is prescribed for a 12-year-old suffering from meningitis. The dose and timing

are given in the prescription chart, which also has background information about the patient

(see Figure 1).

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Insert Figure 1 here

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First the nurses must check whether the dose of 400 mg is appropriate for the patient’s

condition, present symptoms, and size, perhaps finding the information in a drug formulary.

The appropriate concentration of the drug must be chosen: Intravenous vancomycin comes in

vials of 250 mg per 2 ml and 500 mg per 2 ml. Next, the nurse must calculate, using an

appropriate proportional strategy, the volume of the drug to be given. Thus, if the 500 mg per

2 ml vial is chosen, the nurse needs to calculate the required volume for 400 mg, 1.6 ml in

this case. Finally, the nurse must measure the drug in an appropriately sized syringe, and, if

necessary, further dilute it before administering the drug to the patient. In some hospitals, to

minimize the occurrence of error, two nurses independently prepare and calculate volumes of

intravenous drugs.

Because of the high profile of drug administration in nursing, those within the

profession have undertaken research into the effectiveness of their practice, studying student

nurses and, to a lesser extent, qualified nurses. We reviewed 30 such studies, mainly from the


United States but also from the United Kingdom, Sweden, and Papua New Guinea. Many

involved the administration of paper-and-pencil tests on which subjects were usually

prohibited from using calculators. Clearly the paradigm adopted in the profession is that the

individual must achieve a “correct” competence. Yet the findings from these studies paint an

unequivocal picture of deeply flawed performance (see, for example, Bindler & Bayne, 1991;

Blais & Bath, 1992; Conti & Beare, 1988; Kapborg, 1995; Miller, 1992; Pirie, 1987).

In some nursing studies of drug calculations, researchers have gone beyond reporting

test scores to analyze the types of errors made. For example, Worrell and Hodson (1989)

identified 41 variations in written approaches to one item involving a prescription of 8.5 mg

of morphine packaged in 10 mg per ml. They suggested that this variation might be a factor

in resultant errors, although their examples indicated the use of only two strategies

(illustrated in Table 2)--the rule-of-three and the “nursing rule,” that we will describe later.

Blais and Bath (1992) also analyzed error types and found that 68% of errors were based on

“conceptual difficulties” rather than on erroneous computation; the former included setting

up a problem incorrectly, prescribing inappropriate fractions of tablets, or using units

incorrectly.

Even studies of registered qualified nurses have reported unacceptably high levels of

error: Perlstein, Callison, White, Barnes, and Edwards (1979) found that 95 practicing

pediatric nurses achieved an average of 76.6% (range: 45%-95%) on a 10-item test of

difficult drug calculations. They also found that scores were not correlated with the number

of years of service and suggested that less experienced nurses were no more prone to

maladminister because “the judgmental uncertainty of the inexperienced nurse may . . . serve

as a protection against the administration of doses wrongly computed” (p.378).

One obvious explanation of all these findings, based on the research reported earlier, is

that the results are simply a function of the test situation. Indeed, Perlstein et al. (1979)


pointed out that reported incidents of drug maladministration are less frequent than would be

expected on the basis of their study,2 . Conti and Beare (1988) also indicated that the

performance of newly qualified nurses on mathematics tests bore little relation to

subsequently reported drug errors. In fact, the only relationship they could identify was a

positive correlation between the performance on their test and the level of nursing

qualification.3 Furthermore, Hutton (1997) found that newly qualified nurses quickly became

accustomed to drug calculations on the ward, even when as students they had lacked

confidence in mathematics.

We therefore questioned whether results on written tests are valid for judging either the

accuracy of nurses’ drug calculations or the methods they would use to carry them out. Yet,

surprisingly, we could find few researchers who had adopted alternative methods to assess

competence, apart from Hutton (1997) and, in an early study, Sullivan (1982), who compared

performance on a written drug-administration test with a full-simulation test involving

equipment and the actual drugs. Significantly, in this latter study, the participants--all student

nurses--performed better on the written test than in the simulation. Sullivan explained this

result in terms of the greater possibility of error in the simulation compared to the written

test, because measurement errors and mistakes in determining key information from printed

formularies would be included in the simulation.

In summary, most nursing literature on drug calculation can be seen as a counterpart to

the error studies of early research in mathematics education with their focus on individual

performance on “decontextualized” written tests and on “misconceptions.” This literature

paints a consistent picture of high levels of error in these tests, the dominance of written as

opposed to mental strategies, yet improved performance once newly qualified nurses have

begun their careers. We have failed to find a study in which drug-calculation strategies used

by nurses on the ward, experienced or otherwise, were examined. More generally and


perhaps most important, we find a striking lack of problematization of the mathematics used

by nurses and the mathematical activity evident in their practices.

The Nursing Study

The research reported here, involving pediatric nurses in a children’s hospital, was part

of a larger project on workplace mathematics in which we also investigated the practices of

investment-bank employees and commercial pilots (see, for example, Hoyles, Noss, & Pozzi,

1999; Noss & Hoyles, 1996a). These three groups are similar in that they all have

mathematical entry requirements instantiated in specific training programs and they engage

with mathematics in “error-critical” activity--there is little or no room for error in their work.

According to these criteria, our workplaces and the practitioners we studied stand in contrast

to those analyzed in the cited studies of, for example, carpet layers and fishermen. Our

overarching aim was to examine (a) the relationship between practitioner and mathematical

knowledge and (b) the resources practitioners invoked to coordinate these two kinds of

knowledge. Our intention was to define mathematical knowledge in its broadest sense, to

include any activities that involved the mathematization of workplace activity. Thus, in the

present (nursing) study, we set out to examine nurses’ activities in many areas of their

practice. Initially we decided not to include the obvious mathematical practice of nurses’

drug calculation strategies, because given its high profile, we assumed that these strategies

had been well researched and that any investigation by us was unlikely to reveal anything

new. But in the early stages of the project, on the basis of observations of nursing practice on

the ward, we realized that there was more depth and complexity to this activity than had

previously been exposed, if we considered the nurses’ strategies together with the cultural

practices in which they were invoked instead of only calculations and errors.

The initial part of the nursing study involved an analysis of nursing-mathematics

textbooks and interviews with five senior nursing staff to develop a preliminary audit of the


mathematics expected in the profession. This mathematics comprised four broad areas: (a)

drug preparation (e.g., finding doses from prescriptions, drug concentrations, changing dose

frequencies), (b) infusion management (e.g., coordination of infusion rates, checking

concentrations), (c) fluid-balance monitoring (e.g., measuring hourly fluid intake and output;

recording, updating and interpreting fluid balance charts), and (d) vital-signs and laboratory-

data interpretation (e.g., measuring blood pressure, temperature, etc.; recording on time series

graphs with different (sometimes nonlinear) scales; interpreting laboratory-report data). (For

more details, see Hoyles et al., 1999; Noss, Pozzi, & Hoyles, 1999; Pozzi, Noss, & Hoyles,

1998).

From this analysis, we quickly deduced that accuracy in drug calculation was a primary

learning objective. All the texts we examined included a computational algorithm, mentioned

above as “the nursing rule.” Pirie (1985) described one version of the algorithm as follows:

Dose prescribed

Dose per ‘measure’!Number of measures

Thus, in the example described earlier, if 300 mg of a drug is prescribed and the dose is

packaged in 120 mg per 2 ml, the algorithm dictates that the volume required can be

calculated as follows:

300 mg

120mg! 2 ml =

In our preliminary interviews with senior nursing staff, this algorithm appeared again

and again, but in a slightly different form: “what you want, over what you’ve got, times the

volume it comes in,” or in the written form

What you want

What you’ve got! The amount it comes in

Note that this is a somewhat more general rule than Pirie's because it does not relate so

closely to the packaging of the drugs and, instead, focuses on the quantities involved. This


rule could be interpreted as a symbolic description of the proportional relationship between

the quantities and is generally true for all drug calculations, including the prescription of

tablets (for which “the amount it comes in” is 1). However the purpose of teaching nurses this

rule or, as many told us it was known, "the mantra" was not this: It was taught to bypass the

need to appropriate or understand any mathematical structure and to impose consistency on

what were seen to be dangerous variations in strategy. Thus it was believed that if nurses

followed the rule they would make error-free calculations provided they possessed either a

calculator or the appropriate computational skills with arithmetic. Note that the order of

operations in this rule is identical to the actions that a nurse would perform in identifying and

handling the three quantities when preparing a drug: Look at the drug dose prescribed on the

patient’s chart (“what you want”); next note the mass of the packaged drug to hand (“what

you’ve got”), and then the volume of solution (“what it comes in”). This match of rule with

action is an explanation for the fact that the nursing rule was never described in any other

order.

We learned that another aim of nursing education was that nurses should learn

estimation strategies to double-check their nursing-rule calculations. Nursing texts invariably

covered this requirement by describing the importance of judging the “reasonableness” of the

calculated quantity: For example, one rarely injects more than 5 ml of fluid or gives fractions

of a tablet. Clearly such checking strategies are important, but they are unlikely to cover all

cases, especially in pediatrics, in which the quantity of drug can vary enormously, by factors

of 10 or more, from patient to patient.

The second phase of the study comprised an ethnography of life on the ward that

involved shadowing a group of 12 pediatric nurses in a specialist children’s hospital and

building detailed profiles of their work over a number of visits. (Pediatric clinical nursing is

generally regarded as more mathematically intensive than other areas of nursing, partly


because of the greater variety of drug-dosage levels required for babies and children.) The

nurses--all volunteers for the study--had at least 3 years' experience on the ward and were

aged between 26 and 35; all but one were women. As part of the particular hospital’s

recruitment procedure, all the volunteers had passed a noncalculator mathematics test that

included drug-administration calculations. Thus, at least from this perspective, they had all

demonstrated a level of fluency with the written arithmetic procedures deemed appropriate by

the hospital administration.

During the ethnographic study we interviewed the nurses while they carried out

calculations, to ask them what they were doing and why. In addition, we conducted informal

interviews to tell them about the project and to ask them about two issues: what they judged

as mathematical in their work and their attitudes toward mathematics from school to ward.

Without exception, the only mathematics in their practice spontaneously described by the

nurses was drug calculation, and, like the senior nursing staff during the preliminary

interviews, they all repeated the drug calculation mantra described above. This consistency

highlights the very high profile that drug calculations have in nursing practice and contrasts

with other arenas of nursing activity that we interpreted as mathematical but that were never

mentioned by nurses (e.g., their conception of central tendency and variation in vital-sign

data).

We visited each nurse on the ward between two and seven times, each visit lasting 1 to

3 hours, giving more than 80 hours’ observation. We made detailed field notes of all the

nurses' activities and audiotaped discussions between nurses as well as between nurse and

researcher. We also made summary descriptions of all the resources used and copied records

and charts whenever possible. Field notes and transcripts of discussions were then combined

into 250 episodes and were categorized according to the different arenas of nursing activity

identified in the initial audit as (potentially) involving mathematics. We also categorized the


episodes into those involving routine activity and those that consisted of “breakdowns” in

which routine action was replaced by conflict, disagreement, or doubt. These latter episodes

provoked spontaneous explanations that rendered more visible both the nurses' reasoning and

the knowledge basis for their judgments.

Related to drug administration, we collected 30 episodes, all of which were classified

as routine. However, many of the episodes involved intravenous-drug administration in

which two nurses were involved, so we were able to record the spontaneous exchanges of

information between the two nurses without the need to intervene and disrupt the activity.

These paired administrations involved highly coordinated synchronous activity as well as

semiritualized dialogues. The drug calculation itself constituted only one small fragment of

action within this complex activity and could be easily missed if the researcher was not

vigilant. We were therefore usually able to collect data on the strategy of only one of the pair.

If possible, the researcher interviewed this nurse after the administration to find out more

about how the administration was carried out and the strategies used.

In the majority of episodes, we could identify nurses’ strategies by examining the

intermediary calculations--whether written or verbalized--through a qualitative analysis of the

transcribed discussion, field notes, written products, and post-episode interviews.

Administrations involving 26 different combinations of ratios were observed, with some

combinations being used in more than one episode. These ratios are shown in Table 3. The

numbers in the drug calculations observed ranged widely: Doses prescribed varied in quantity

between 1.5 and 2520; the packaged doses ranged from 8 to 1000; and the scalar ratio of

mass of drug prescribed to mass in package varied from simple values such as 1:1 and 2:1 to

2520:420. The units of drug quantity were either in milligrams or millimoles, and drugs in

solution came in a fixed set of volumes, either 1, 2, 5, 10, or 50 milliliters.


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Insert Table 3 here

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All the drug calculations we observed were carried out correctly, but the strategies

adopted were varied and exhibited a richer complexity than would be suggested either from

our interviews or from the existing nursing literature. According to Vergnaud’s model, we

identified scalar-operator strategies (8 episodes) and a form of functional strategy (8

episodes): Both types of strategy were always carried out mentally, with no recourse to

writing on paper or using a calculator. Two variants of the scalar strategy--scalar

decomposition and the unitary method--were not observed, nor was the rule-of-three, a

method of solution commonly taught in many countries, but rarely if ever taught in UK

schools, in which unitary or ad hoc approaches (doubling, finding a highest common factor)

are encouraged. The remaining strategies we observed could not be classified using

Vergnaud’s model: Four involved the nursing rule, and six involved a prescription identical

to the dose concentration. This latter situation we termed “one-one” (e.g., 10 mg required

from a concentration of 10 mg per 5 ml), which, given its simplicity, required no

computation. Finally, no clear strategy could be identified in the remaining four episodes.

Having established the existence in nursing practice of these different strategies, we

then tried to map out how each strategy was related to features of the practice, such as the

ratios in the problem, knowledge of the drug itself, or other material aspects of the activity. In

particular, we were fascinated by the apparent discrepancy between the dominance of a single

nursing rule in nurses’ description of their strategies and the rich and unarticulated mental

strategies they exhibited in practice.


The Nursing Rule

The four episodes that did involve the use of the nursing rule did not in fact show the

consistency of strategy assumed by nursing educators but rather exhibited a variety of uses of

both intellectual and material resources: The rule was used variously as a written algorithm,

as a mental strategy, and as a procedure to use with a calculator (calculators were in general

use in the wards observed). The following description of calculator use shows that it is quick

and trouble-free.

A nurse needed to infuse 180 mg of a drug that is packaged in 250 mg per 50 ml.

Janice: So . . . one hundred and eighty. [entered into calculator] one-eight-zero,

divided by two hundred and fifty, times . . . [reads] thirty-six.

Contrast Janice's strategy of setting up the division with the calculator with the

following example, in which one nurse--Moira--used the nursing rule as a written algorithm,

simplifying the arithmetic before doing any written calculation. The drug prescribed is 30 mg

of orally administered morphine, which comes in 200 mg per 5 ml. She had already given the

patient the same dose two days before and could recall the volume. She then double-checked

this quantity using three separately written calculations, describing them as shown in Figure

2.

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Insert Figure 2 here

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Clearly, Moira needed some facility with both simple mental arithmetic and the

algorithms of written arithmetic to reach a correct answer. Although she carried out the

procedure faultlessly, there are two interesting features of her strategy. First, simplification

was used only on the quotient, so a further possible simplification of 5 divided by 20 was not

made. Second, the operations were carried out in an order apparently " dictated" by the


nursing rule--dose-prescribed divided by dose-at-hand, then the result multiplied by volume;

other orders (e.g., dose-prescribed multiplied by volume then the result divided by dose-at-

hand) were not considered. Without devaluing Moira's strategy, we note that her use of the

nursing rule was inflexible, and this inflexibility rendered the calculation to be more complex

than necessary. The same inflexibility is clearly shared with the calculator strategy used by

Janice, but with less possibility of error in the latter case due to computational slips, a point

underlined later in the day by Moira. She had decided to use a calculator, explaining her

decision in terms of fatigue:

Moira: I was just being lazy, because I couldn’t be bothered to work it out on

paper. It gets to this time of night actually when I’m too tired to work

out things--even the simplest. I just can’t do it.

This statement raises an important question--why carry out the calculation on paper at

all? For Moira, the answer was clear. She wanted to maintain her fluency in these written-

arithmetic skill, and took professional pride in being able to do them. She had previously

worked in a nursing environment in which there was still disapproval of calculator use and a

certain amount of skepticism about the veracity of calculator results.

Moira: But I always like to check again . . . like sometimes if it’s one that I

can’t do in my head, because I like to check [whether] my calculator is

right. You know, I don’t like to take the calculator as gospel.

Given the possibility for computational error with written arithmetic, one might feel

that checking mechanisms other than written calculation would be safer, but this is not our

major concern here. Rather we underline how in the (relatively rare) cases in which the

nursing rule was used, it seemed to inhibit a sense of connection either with straightforward

arithmetic that would have simplified the task or with the meanings of the quantities derived

from nursing practice.


We did observe more flexible strategy use in very simple cases. For example, Agnes

needed to give 300 mg of a drug packaged in 500 mg per 5 ml. Immediately she muttered,

“Three hundred . . . five . . . that’s three over five, three mls.” Our interpretation is that, using

the nursing rule as a mental scaffold (suggested by the use of the word over), Agnes had

carried out the following calculation mentally:

300

500! 5

3

5! 5

The quantities involved allowed her to simplify the ratio quickly and easily, without recourse

to particular tools.

Scalar and Functional Strategies

All drug calculations, apart from the one described earlier involving the nursing rule,

were carried out mentally. Nonetheless, we were able to classify them into essentially scalar

and functional approaches, which we illustrate in two examples--the first involving a scalar

operator strategy, the second a functional one.

Sam and Joe were preparing a morphine prescription in which they needed to

administer 1.5 mg of morphine packaged in 20-mg ampules diluted to 10 ml. The dose had

already been calculated earlier by Joe, and Sam needed to double-check the dose for herself.

She had collected all the materials for the administration, including an appropriate-sized

syringe, the drug, and gloves that she had put on. Her strategy was first evidenced by her

verbal articulation of her mental approach:

Sam: Ten in five; five in two point five; one in point five . . . Zero point seven

five.

The volume was then drawn up in the syringe and was brought to the patient.


Sam was asked later about how she had carried out the calculation, and she initially

said that she had used the nursing rule:

Int: How did you measure out the one point five mg?

Sam: What we did is measure out the ten mils of water. So it was twenty . . .

what you want is one point five, divided by twenty, times ten. [Writes

out the calculation.]

1.5

20!10

Sam: What you want is one point five, divide it by what you’ve got, which is

twenty mg, times by what it's in--ten; so basically it was point seven

five?

Int: Right . . . but you did that in your head at the time, or didn’t you?

This question seemed to legitimate her actual approach, which she went on to describe.

Sam: We did, yeah. I did it in my head. [Joe] worked it out, then I went

twenty mg in ten mils, that’s ten mg in five mils, so that’s five mgs in

two poi’ . . . no wait, twenty mg in ten, ten in five, five in two point

five, one in point five and then one point five in point seven five.

This explanation showed that Sam had used a scalar strategy that Vergnaud would

describe as a unitary method involving a series of parallel numerical operations on both the

mass and the solution volume, with an additive final step. This sequence of operations is

summarized in Table 4. The transformations are awkward and require a fair degree of facility

in contrast to a functional approach, for which the calculation would have been a great deal

easier because the functional relationship is a simple 2:1 ratio.


------------------------------------

Insert Table 4 here

------------------------------------

But in the next example, we describe a strategy different from that proposed by

Vergnaud's model. Belinda needed to give 120 mg of an antibiotic, amakacine, prepared in

100 mg per 2 ml vials. Before performing the calculation, she prepared for the administration

and retrieved two vials of the antibiotic. At this point, she found the volume she had to give

with a fluency that was difficult to follow:

Belinda: Amakacine [reads doses on the two vials] one hundred; one hundred;

[reads year of expiry] ninety-eight; ninety-eight; [finds volume to be

given] two point four mils. . . .

A short interview with Belinda later revealed the nature of her strategy:

Int: I didn’t see you do any calculating there at all. You just drew it up. . . .

Belinda: I knew the doses. . . . I know that that one is two point four . . . two

point four mils. With the amakacine, whatever the dose is, if you just

double the dose, it’s what the mil is. Don’t ask me how it works, but it

does.

Int: Why, what’s the . . . ?

Belinda: One hundred and twenty mg, right [dose] and it comes in . . . and it goes

in one hundred milligrams per two mils. So if you double it, that makes

two hundred and forty . . . two point four mils.

Int: I’m sorry. I don’t understand.

Belinda: So if you just double it up. Double one twenty; one twenty and one

twenty is two hundred and forty. And the dose is two forty. So very


often that’s how it is with amakacine, so if you’re giving eighty . . .

eighty milligrams to give, and if you double it up it’s one point six.

Belinda’s description clearly indicates a transformation from the dose mass to the dose

volume, so in this sense the strategy is functional. But a simple classification of the strategy

as functional does not do it justice. Her description also seems to suggest that the operation

was associated with the drug itself rather than with the ratio between the mass and volume:

“That's how it is with amakacine,” says Belinda, apparently seeing the allowable arithmetic

operation and the drug itself as intimately connected. Similarly, her description of the

strategy suggests that she was neither manipulating numbers (or even quantities) nor

performing arithmetic operations: Rather, she described the transformation as “doubling up”

and effortlessly combined into a single process what would generally be recognized as the

doubling operation and the movement of the decimal point.

Belinda’s approach maintained a strong connection between the calculation and the

drug at every step, as evidenced in her intermittent references to the drug. This connection

appeared evident in all the episodes classified as involving a functional strategy. Yet further

examination revealed that these episodes were also characterized by particular ratios of dose

mass to dose volume, namely those involving 1:2, 2:1, powers of 10, or a combination of

these ratios (see Table 5). The transformations were particularly interesting because they, like

Belinda's amakacine calculation, were often described in terms of a single process and,

revealingly, were always described in association with a particular drug rather than a

particular ratio. For example, a halving strategy was referred to in relation to a specific

analgesic rather than as a feature of all concentrations with a mass:volume ratio of 2:1.

------------------------------------

Insert Table 5 here

------------------------------------


In the case of scalar strategies, no restrictions related to particular ratios of drug

package-size to dose were witnessed; scalar strategies tended to involve transformations with

a wider variety of quantities, mediated by both the concentration ratio of the drug at hand and

the prescribed dose. These strategies are summarized in Table 6.

------------------------------------

Insert Table 6 here

------------------------------------

Within the scalar strategies we could, however, distinguish a particular method that

appeared to be especially meaningful for nurses. We termed this a "chunking" strategy in

which the nurses seemed to maintain two parallel sets of numerical quantities in their minds:

A certain chunk of mass (which may not necessarily be one unit) is related to a specific

volume, added repeatedly or multiplied by an integer to reach the required dose, and the

equivalent operations are performed on the volume. In Table 6 we give two examples of

chunking. In the first example, 15 mg chunks were combined to give 60 mg, and the

equivalent chunks of pill were then calculated. Other scalar strategies included reducing the

package mass to a suitable unit that was a factor of the dose and multiplying (Examples 1 and

2) or combining to give the dose (Example 6), comparing the dose directly with the package

mass (Example 3), and reducing the volume to 1 ml and scaling up the dose appropriately

(Examples 7 and 8).

Discussion and Conclusions

Researchers on proportional reasoning in school and the workplace have distinguished two

widely used strategies, functional and scalar strategies. Nunes and her colleagues (1993), in

their discussion of scalar strategies, have suggested that they offer a mechanism for holding

on to situational meaning by keeping only one measure in view. By way of contrast,

functional strategies tend to be seen as manipulations of numerical quantities per se, devoid


of meaning. It appears that this difference is the crux of the counterposition in the literature of

scalar and functional approaches in that the privileging of the former has arisen from the

apparent necessity in the latter to relinquish meaning in the form of a situational referent.

Our findings from ethnographic investigation of pediatric nurses' methods of drug

calculation while they are on the ward add another dimension to these studies. Drug

calculation involves proportional reasoning but, unlike other adult practices, this reasoning

has a high profile in nursing and demands exact, error-free results. Also, again in contrast to

other practices, proportional reasoning in drug calculation is formally taught to nurses as part

of their training when they are introduced to what is termed the nursing rule. This rule is

taught to support nurses in their proportionality calculations on the ward by providing them

with a general and consistent procedural approach. The nursing literature shows,

unsurprisingly, that even experienced nurses make many errors on paper-and-pencil tests of

these calculations.

In our study on the ward, we found drug administration to be routine and error free. It

was characterized by effective and flexible use of a range of proportional-reasoning strategies

for many ratios of prescribed dose to packaged dose. Some strategies, although applied

flexibly, were consistent with the nursing rule and these strategies tended to follow the flow

of clinical work by mirroring the actions needed to carry out the requisite procedures

routinely.4 Overall, however, the nurses' strategies on the ward involved only limited use of

this formally taught rule, and more frequently they comprised the adoption of two mental

strategies--a within-measure scalar approach and an across-measure functional approach,

both of which we argue were sense preserving.

Scalar strategies were used over a wide variety of ratios of mass prescribed to packaged

dose. They also displayed diverse forms, including unitary methods such as reduction of the

volume to 1 ml. An approach that seemed most clearly to mirror the shaping of the drug


packaging was one we termed chunking, in which the portions of mass usually available in

standard packs were combined to give the appropriate dose and parallel calculations were

then conducted on the volumes of solution.

The striking features of the functional strategies used by nurses were that they too

appeared to be structured by familiar material units, such as typical dosages for specific drugs

and packaging conventions, but they were also shaped by the nurses’ clinical knowledge of

what “felt okay” for a given drug and patient situation. Functional strategies were invariably

associated with references to the drug itself and were accompanied with comments such as

“with ondansatron, all you need to do is halve the dose” or “that’s how it works with

amakacine.”

On the basis of examples such as these, we are tempted argue that the nurses were

attributing to the drug some implicit arithmetical knowledge derived from a realization about

the specificities of its occurrence in situ (e.g., the quantities it comes in or the dosage that is

invariably prescribed). But we do not make such an argument: On several occasions we tried

to cross-examine nurses who adopted this kind of strategy, and we became confident that we

were witnessing a kind of meaning that was neither simply arithmetical nor professional, but

a mixture of the two. The nurses attributed the calculational strategy to the drug itself, not to

any invariant numerical properties they could identify within its use. This finding is related to

the observation of Nunes et al. (1993) of how adults and children carry out addition with

money. For them, the salient objects to be manipulated are not those of decimal arithmetic

but the names of different coins.

Of course, it may be argued that the nurses’ strategies were not genuinely functional,

and we concede that this is a sensible objection, although we see no gain in reclassifying the

behaviors we describe with a new label. Whatever we call the strategies, our analysis

indicates that any simple demarcation between scalar and functional, the one holding on to


the situation and the other abstracting from it, does not seem to apply in any straightforward

sense in this case.

More generally our data on the mathematics used in nursing confirm the disjunction

between visible mathematics--whether it is school mathematics or the mathematics

introduced in training programs--and what happens in practice. In practice, abstract rules fade

into the background in favor of more holistic strategies that emerge alongside a differentiated

and textured sense of the practice and correspondingly finely tuned strategies. From a

mathematical orientation, one might find it puzzling that nurses eschewed the simplest and

most general expression of the relationships between the known ratios of a drug calculation

problem--an expression that can be used to find the dosage in all cases and is central to the

visible mathematics of the practice. Yet the nurses simply did not use this relationship,

because the specificities of routine practice mapped onto a culturally shared set of

calculational strategies that served as well as, if not better than, the abstract rule they had

been taught. Like all workplace situations, nurses' practice is one in which a considerable

fraction of knowledge is wrapped into the artifacts and social relationships that inhere within

it. We have noted elsewhere (Pozzi et al., 1998) that the physical resources of the practice

shape, and are shaped by, the routines of nursing in supportive and highly specific ways, and

we are not surprised that the routines of calculational procedures are similarly structured by

the packaging of the drugs, the social routines of administering the drugs, and the clinical

effects of the drugs.

These kinds of observations may lead to the conclusion that abstract knowledge is not

used by expert practitioners. In the sense that nurses tend not to use general mathematical

algorithms or to express their actions in terms of formal mathematical symbols, the

observation that abstract knowledge appears not to be routinely invoked in practice is true.


Yet we have seen that nurses did perform mental calculations in a variety of ways that were

error-free and respected proportionality requirements. They had indeed abstracted something.

One way to sharpen this discussion is to focus more strongly on conceptual knowledge.

To do so, we turn our attention away from strategies to the concept of ratio of mass to volume

and its instantiation in drug concentration. This shift will allow us to focus more on nurses'

mathematical knowledge and its status within the practice. We argue that, by their flexible

use of the “what you want” rule, by their scalar strategies, and by their functional approaches

for particular quantities and particular drugs, the nurses revealed an appreciation of the

invariance of drug concentration. We suggest, therefore, that they had abstracted a concept of

concentration in their administration of drugs, as evidenced by this implicit recognition of the

covariation of mass and volume. Across all the different drug administrations, the nurses

consistently “pulled out” a set of correct representations of the underlying invariants, and

their image of invariant concentration transcended any particular circumstance.

We advance the notion of situated abstraction as an analytic tool to explain how these

strategies were, on the one hand, finely tuned to the resources in the practice, yet, on the other

hand, retained the notion of invariance of drug concentration.5 Nurses had clearly

appropriated the invariant of drug concentration that underpinned their proportional

reasoning in this context. However, it would be difficult to argue from these data that they

had a formal appreciation of this intensive quantity. What are the limitations of their

understanding of concentration? How far, in other words, is it situated within their practice?

Would nurses' conceptualization of concentration enable them to solve problems involving

proportional reasoning in nonroutine settings? These questions, together with an elaboration

of the situated-abstraction idea, were investigated in further research with the nurses, which

we shall report in a later paper.


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Author Note

Celia Hoyles, Professor of Mathematics Education, Institute of Education, University

of London, 20 Bedford Way, London WC1H 0AL, UK; [email protected]

Richard Noss, Professor of Mathematics Education, Institute of Education, University

of London, 20 Bedford Way, London WC1H 0AL, UK; [email protected]

Stefano Pozzi, Research Officer, Institute of Education, University of London, 20

Bedford Way, London WC1H 0AL, UK.

Acknowledgement

We acknowledge the support of the Economic and Social Research Council (UK) for

funding the project “Towards a Mathematical Orientation Through Computational

Modelling,” Grant RO22250004. We also would like to thank the anonymous reviewers for

their helpful remarks and suggestions.


Footnotes

2 This conclusion needs to be mitigated by two issues. First, the items given in the test

were of difficult drug calculations; such calculations would occur infrequently on the ward.

Second, there is some possibility that incidents are underreported in hospitals with inadequate

procedures (see Barker & McConnell, 1962).

3 Given that the level of nursing qualification is itself partly based on the level of

entrance qualification, the correlation indicates no more than school experience being a

predictor of achievement on the mathematics test.

4 We acknowledge that we cannot rule out the possibility that, reciprocally, the

structure of the rule imposed a kind of order on the actions themselves, with inevitable results

that the procedures became routinized and the routinization served to safety-check the

procedures.

5 We first coined the notion of situated abstraction to try to understand students’

conceptualizations of mathematical structures when they are working with computer tools

(see Noss & Hoyles, 1996b). We also use the notion more generally to interpret how tool use

can shape mathematical understandings.


Table 1

Correct Strategies Within the Isomorphism of Measures Model

Strategy Example

Functional operator (FO) between

measure spaces

120÷10

! " ! ! ! 12÷6

! " ! ! 2

300÷10

! " ! ! ! 30÷6

! " ! ! 5

Scalar operator (SO) within measure

spaces

120mg!2.5

" # " " " 300mg

2ml!2.5

" # " " " 5ml

Unitary method (UM) within measure

spaces (variant of the scalar strategy)

120mg÷120

! " ! ! ! #300

! " ! ! ! 300mg

2ml÷120

! " ! ! ! #300

! " ! ! ! 5ml

Scalar decomposition (SD) within

measure spaces (variant of a scalar

strategy)

120mg+120 mg+1

2120mg( )=300mg

2ml + 2ml +1

22ml( ) = 5ml

Rule-of-three (RT) both between and

within measure spaces e.g.,

300

x=120

2!300

120" 2 = 5ml



Table 2

Examples of Written Strategies in Worrell and Hodson (1989)

Evaluation of

response

Rule of three examples Nursing rule examples

Correct Dr' sorder (D)Doseon hand (H)

=x

Quantity (Q)

8.5mgxml

=10mg1ml

10mg:1ml =8.5 mg:xml

D

H!Q

8.5mg

10mg!1ml

Incorrect 10mg

16m:8.5mg

xm 1.0

8.5


Table 3

Dosage Quantities Observed on the Ward Organized in Scalar Ratios

Scalar ratio Mass

prescribed:Packaged dose

Mass prescribed Concentration

1:1 10 mg 10 mg per 5 ml

1:1 20 mmol 20 mmol per pill

1:1 25 mg 25 mg per pill


1:1 5000 unit 5000 units per 10 ml


1:2 125 mg 250 mg per 10 ml

3:1 24 mmol 8 mmol per pill

3:1 30 mg 10mg per pill

3:1 360 mg 120 mg per 5 ml


4.1 60 mg 15 mg per pill


5:4 5 mg 4 mg per 2 ml

3:4 3 mg 4 mg per 2ml

3:5 150 mg 250 mg per ml

3:5 300 mg 500 mg per 5 ml

3:8 30 mg 80 mg per 2 ml

3:10 6mg 20 mg per 2 ml

3:20 30 mg 200 mg per 5 ml

Table continues


Table 3 continued

Scalar ratio Mass

prescribed:Packaged dose

Mass prescribed Concentration

6:5 120 mg 100 mg per 2 ml

3:40 1.5 mg 20 mg per 10 ml

15:2 750 mg 100 mg per 1 ml

22:50 22 mg 50 mg per 1 ml

18:25 180 mg 250 mg per 50 ml

83:100 830 mmol 1000 mmol per ml

Note. Some of these dosages and concentrations were observed in more than one episode.


Table 4

Arithmetic Operations in Sam’s Scalar Strategy

Procedure Dose (mg) Volume (ml)

(a) Given 20 10

(b) ÷ 2 10 5

(c) ÷ 2 5 2.5

(d) ÷ 5 1 0.5

(e) ÷ 2 0.5 0.25

(f) (d) followed by (e) or 3x(e) 1.5 0.75


Table 5

Ratios Used in Functional Strategies

Concentration Mathematical description of transformation

20 mg in 2 mls Dose Mass

÷!10

Dose Volume


÷!2

Dose Volume

100 mg in 1 ml Dose Mass

÷!100

Dose Volume

1000 mmols in 1 ml Dose Mass

÷!1000

Dose Volume

50 mg in 1 ml Dose Mass

x 2 ÷ 100

Dose Volume


x 2 ÷ 100

Dose Volume


Table 6

Ratios Used in Scalar Strategies

Example Dose Concentration Parallel transformations

1. 300 mg 500 mg per 5 ml ÷ 5 x 3

2. 150 mg 250 mg per 10 ml ÷ 10 x 6

3. 125 mg 250 mg per 10 ml ÷ 2

4. 60 mg 15 mg per pill Chunking

5. 24 mmol 8 mmol per pill Chunking

6. 1.5 mg 20 mg per 10 ml ÷ 2 ÷ 2 ÷ 5 ÷ 2 and adding

7. 6 mg 20 mg per 2 ml ÷ 2 x 0.6

8. 30 mg 80 mg per 2 ml ÷2 x 0.75


Figure Captions

Figure 1. Sections from an example prescription chart.

Figure 2. A nurse’s explanation and calculation.

FIGURES

FIRST NAME CONSULTANTSURNAME HOSPITAL NUMBER COST CODE

12763Anna A.MediciPatient B2G7SURF AREAHEIGHT D.O.B AGE WEIGHT (Kg)

Date132 cm 2/6/851.4 sq m 12 years

53.6

6

12

14

18

22

24

DOSE other ROUTE

DRUG

microgram mg

Additional

information

Start Date Signature

initials

DateVancomycin

400

1/11/97

1/11

NS !Amanda Medici

Infuse for 2 hours

2/11

NS !

30/10/97


Moira: So you cross off the noughts. So you have [simplifies the fraction] so it’s twenty into three goes… erm….

30

200! 5

30

200! 5

Moira: Zero point twenty into thirty goes… erm… (quietly) twenty into three won’t go. 20 3

0.

)

Moira: Twenty into thirty goes one. Yeah, that’s right. 20 3.0

0.1

)

Moira: Ten over… So twenty into ten won’t go; twenty into one hundred goes five. 20 3.00

0.15

)

Moira: So it’s point one-five times five. (quietly) Five times five, so that’s twenty-five; [inaudible] six seven seventy… point seven-five. That’s it.

0.15

x 5

0.75

Documents

Proportional Reasoning 1 Running head: NURSES ... · Ratio, proportion, percent Situated cognition Out of school mathematics . Proportional Reasoning 3 ... In their detailed critique