Dosage calculation skills have not traditionally been taught to medical students. One explanation for this is that they require proportional reasoning skills; mathematics taught in school is thought to transfer to situations encountered outside school (Stasz, 2001). Consequently, it is assumed that these proportional reasoning skills are acquired at school, although it is not expected that the specific context of dosage calculation will have been encountered there. However, research suggests that mathematical knowledge transfer does not happen readily, partly because of differences between formal knowledge, which may involve mathematical algorithms, and contextual knowledge, which often involves contextual cues as part of the processing (Stasz, 2001).

The objectification of skills and knowledge in this way has been criticised because it fails to consider the unique ability of the human ‘to continually shape and be shaped by their social contexts’ (Roth & Lee, 2007, p. 4). Because the relevance of context is ignored, cultural diversity is not taken into account. It also fails to consider the unique ability of the human to interpret and represent experiences and it ignores or presumes its subjects’ interpretation of situations (Cohen & Manion, 1980). Consequently, the role of the relationship between teacher and student is regarded as limited to an opportunity for skills to be passed from the teacher to the student. The control over what is learnt, and how this is done, is with the teacher, the student not seen as needing a voice or requiring an opportunity to be heard. The teacher assumes a student is available to acquire skills if they have been delivered, that the student has then acquired these skills, or will if they make sufficient effort, and has sufficient ability to reinforce learning. The responsibility for the acquisition of the skills, once the teacher has delivered them, is believed to lie with the student.

In contrast, according to the Vygotskyan sociocultural view of learning, skills are learnt naturally within the context of social activity and within a culture using artefacts (such as language or, for dosage calculation, vials, syringes and treatment guidelines) developed within that culture. Accordingly, the skill cannot be understood without reference to the context, namely the development of skill with the socio-historically developed tools involved that mediate thinking, for example the language and equipment relevant to the activity (Rogoff, 1990, p. 35). These contextual factors give the skill its meaning and importance. This assists with motivation to acquire the skill. They provide pragmatic cues which improve performance and assist with retrieval of the skills in future use.

Within the context and culture of social activity there are relationships with other participants. Vygotsky’s theory emphasises development as a process of learning to use the intellectual tools provided through social history (Roth & Lee, 2007). Social interaction is expected to foster learning through the guidance provided within relationships with people who are skilled in the use of those intellectual tools (Rogoff, 1990). Accordingly, in a one-on-one relationship a novice will learn from an expert who guides as they participate in a meaningful activity. Key to the learning process within this paradigm is the relationship between novice and guide who engage together in dialogues, which may be tacit or non-verbal. Through these, a guide is able to hear the novice’s voice and to be heard and to vary the amount of control needed to ensure the novice moves towards expertise. Within this paradigm, a student is recognised to be more or less available for learning and factors such as anxiety, motivation and distractibility are taken into account. The dialogues allow a guide to be responsive, in other words to provide a balance between support and challenge to keep the student optimally receptive to learning. Dialogues allow the guide to give just the right amount of challenge so that the activity is slightly beyond the learner’s competence, within the learner’s zone of proximal development. This Vygotskyan concept refers to the range of tasks that can be completed: the lower level of this zone corresponding to the level of skill that the learner can reach independently whilst the higher level refers to the potential skill level that can be completed with the assistance of the expert (Rogoff, 1990, p. 14). Learning is thought to occur through the relationship, which builds a bridge from the mind of the teacher to that of the learner so that the learner can borrow the perspective of the teacher to develop his own skills (Rogoff, 1990, p. 19).

Consequences of the prevailing positivist paradigm to students is that (1) the context particular to dosage calculation has not been provided, hampering retrieval, (2) although proportional reasoning skills may have been acquired at school, these skills may have been taught in the early school years only, and half-forgotten as the students focused on more abstract mathematics, and (3) skills may not have been adequately taught or acquired leaving students incompetent in this area of mathematics, but without a guiding relationship within which to communicate the need for support, possibly leading to anxiety. Aversion towards mathematics is a common phenomenon amongst students in higher education (Taylor & Galligan, 2006) and such anxiety might lead to increased defensiveness, which might prevent them from seeking help at medical school and later from checking answers with colleagues in a clinic situation.

According to the PBL approach, our students, in small groups and guided by a facilitator, develop common learning goals from a paper case related to a medical theme and report back. Students are expected to reach these goals individually, with support from useful literature, structured learning sessions, tutorials, practicals and clinic or hospital visits.

To some extent, the problem-based learning approach has remedied students’ lack of exposure to context because the paper cases provide students with some of the contextual factors of real-life health problems. However, the paper cases focus on diagnosis and less attention is paid to the details of treatment, including the technicalities of prescribing, such as calculating the correct dose. The failure of curricula to emphasise prescribing and ‘teaching the skill to treat the patient’ is a cause of concern internationally (Hogerzeil et al., 2001). As a therapeutics department, we aimed to improve prescribing competence by offering a programme of therapeutics learning opportunities relevant to the medical themes being discussed, largely in the form of large group sessions and assignments because these fit most readily into complicated timetables.

In light of the inadequate dosage skills amongst doctors and students (Simpson et al., 2009; Wheeler et al., 2004) we (the authors) felt that dosage skills training should be introduced to the medical curriculum. We believed we were best placed to provide the context relevant to dosage calculations and, like Huijser, Kimmins and Galligan (2008), that students ‘would learn better if the mathematical skills required for dosing were taught as an integral part of our programme, rather than separated and remediated in a content vacuum‘ (pp. A-24). Accordingly we set about including dosage calculation training in some of our therapeutic learning opportunities so that students could have opportunities to access prior mathematical skills as well as to determine and remediate missing mathematical skills.

For example, within the endocrine theme, where students are introduced to the body’s hormones, one of the students’ PBL cases focused on a patient who developed hypercalcaemia (abnormally high levels of calcium in the blood) as a result of overproduction of the hormone involved in controlling the level of calcium in the blood (parathyroid). Our learning session initially focused on which medicines would be used to manage this condition and how these medicines worked. In this case, an appropriate option is the administration, by intravenous infusion, of the medicine pamidronate.

In order to bring in the dosage calculation skills students needed to manage this case, students were provided with dosage information and asked to consider how this drug should be reconstituted from a powder to a solution, what volume of this solution should be added to a litre of normal saline and what drip rate should be set in order for this infusion to be given over 6 hours.

Problem:

A child is to be sedated with 1.5 mg of midazolam. A vial of midazolam has 15 mg in 3 mL. How many millilitres of midazolam are required?

Solution:

What you want = the amount of drug prescribed = 1.5 mg.

What you’ve got = the amount of drug dissolved in the formulation = 15 mg.

The volume it comes in = 3 mL

So when the formula is applied:

when presented in a particular context with a prescription with a specified dose, an ampoule with a particular strength and volume and a choice of syringes with which to draw it up, a student can manipulate these to produce the correct prescribed dose. (p. 4)

A formula may be forgotten when it is needed in practice or the vague terms involved may lead to the inclusion of incorrect values; for example, ‘what you’ve got’ may be understood as the concentration rather than the amount of drug and this wrong information would lead to calculation of the wrong dose.

Various educational interventions have been implemented and found to improve dosage skills amongst nursing students. In comparison with traditional lectures, where formulae (such as the nursing rule) were taught, significantly higher sustained learning was reported with methods that built on students’ existing mathematical problem-solving skills and focused on avoiding conceptual errors (Gillies, 2004; Koohestani & Baghcheghi, 2010). With this in mind, we avoided the mechanistic use of formulae.

Vergnaud’s (1982) model of proportionality includes a multiplicative structure involving direct proportion between two measure spaces. Determination of an appropriate volume of a drug in a solution could be characterised by this structure, the measure spaces being the mass of the drug and the volume of the solution. According to this model, the correct volume of a drug could be obtained either by rule-based, functional or scalar approaches. The first two approaches involve manipulating figures across measured spaces, whilst the last (that is the scalar approach) involves working only within measure spaces by adopting various strategies. Nurses, although generally taught rule-based strategies, most commonly adopt scalar strategies in a clinical setting (Hoyles et al., 2001). Scalar approaches to solving proportionality problems are more flexible and generalisable to the workplace, preserving the meaning of the quantities and their relationship by keeping variables separate (Nunes, Schliemann & Carraher, 1993).

In view of these findings, we adopted a scalar approach to solving dosage problems in our large group sessions to help keep students in touch with the meaning of the quantities. We called it the ‘two column stepwise method’ in which ‘whatever was done’ to one column (one measure space: mass) ‘was done’ to the other (the second measure space: volume). In other words the same arithmetical calculation was applied to each measure space.

Corresponding with the unitary method (one of the scalar strategies described by Vergnaud (1982), the initial step is to reduce the quantity in the first space to unity (one). This is illustrated in Figure 1 using the problem and solution below.

FIGURE 1: Illustration of the ‘two column stepwise method.’

A child is to be sedated with 1.5 mg of midazolam. A vial of midazolam has 15 mg in 3 mL. How many millilitres of midazolam are required?

Solution:

In order to get from 15 mg to 1.5 mg the first ‘step’ is to get to 1 mg by dividing the number by itself (in this case 15). The same divisor is then applied to the second column (volume). Next, the resultant numbers in both columns are multiplied by 1.5.

After this, students are required to be pragmatic and to check, in a different way, that the answer ‘makes sense’. This means appraising whether the result obtained is feasible. For example, in the above example a different scalar approach can be used to see if the answer ‘looks’ sensible: 1.5 mg is a tenth of 15 mg, so a tenth of 3 mL is needed and a tenth of 3 mL equals 0.3 mL.

The ‘two column stepwise method’, followed by the pragmatic check, was used in all large group sessions, report backs and model answers.

Accordingly, we aimed to improve learning from the sociocultural perspective, within the constraints of the teaching interactions we could readily provide for students, namely the large group session and assignment for self-assessment. One of the goals relevant to large group sessions would be to provide meaningful tasks that closely resemble those that must be undertaken in the workplace as a focus for joint attention. Although resource and safety restraints precluded the use of real tasks, we developed paper dosage calculation problems of patients receiving drug treatment for conditions relevant to the current theme, using formulations and dosage regimens that would be encountered in practice in order to preserve sufficient meaning for learning to occur and be transferable to a clinic setting, as advocated by Coben (2010, p. 10).

We also took guidance from the seven hallmarks of good practice in undergraduate education, as described by Chickering and Gamson (1987). These were developed from 50 years of student and teacher experience and research and are widely regarded as a gold standard for measuring the quality of undergraduate education. These guidelines include encouraging student-faculty contact (improving student-teacher communication would be an important part of this contact), developing cooperation and reciprocity amongst students, participating in active learning, providing prompt feedback, communicating high expectations, respecting diversity and emphasising the necessity of dedicating time to the acquisition of a skill (‘time on task’). These guidelines fit well with the sociocultural paradigm because the seven principles would be expected to occur within the social dialogues between a learner and a teacher or a more competent learner working together on a meaningful activity.

Most of the students started medical school immediately after leaving school and the median age of our group was 21 years with a range of 18–34, as a few students already had other degrees. There is considerable diversity in terms of both culture and educational background, with students drawn from homes with eleven different languages other than English. The majority of students come from government schools, some of which were historically disadvantaged in terms of resources and access to skilled teaching. Other students come from advantaged privately funded schools, both in South Africa and neighbouring African countries.

Every assessment comprised four questions, one of each type. These were randomly selected from a bank of appropriate questions. Students were deemed competent at the 100% level if they got all four calculations correct at any one time. At least three of the four questions correct meant they were competent at least at the 75% level.

After a baseline test at the beginning of third year, students were assessed repeatedly and the cumulative number of students attaining competence by their final assessment in their third and fourth year was measured. In order to investigate the effect of ‘time on task’ advocated by Chickering and Gamson (1987), competence acquired in the third year was compared with that achieved in the fourth year.

In the third year, students were given an hour-long introductory lecture and an assignment, followed a week later by a report-back session and a model answer for self-assessment. In fourth year, in an effort to improve ‘peer contact’ as advocated by Chickering and Gamson (1987, p. 3), they were offered a further lecture followed by a tutorial in which they worked together through examples. It was hoped that this peer-assisted teaching strategy would help to overcome language barriers, as students would be able to seek assistance from more able same-language peers. A sample of 200 of these students were given formal assessments both before and after the extra tuition. The change in calculation competence was compared between those who attended and those who did not.

In an attempt to test the value of increasing ‘contact between students and faculty’ (Chickering & Gamson, 1987, p. 3), 23 students were offered individual tuition prior to the start of the fourth year. After giving written consent, each was asked to work through dosage calculations that they had previously answered incorrectly. If they faltered, the researcher would provide just sufficient information or explanation to enable the student to continue and would make a note of the difficulty experienced by the student. The interview transcripts, written calculations and case notes were analysed to determine common key problems experienced by the students. The influence of one-on-one teaching on student competence was assessed retrospectively by comparing their results for assessments immediately before and after this tuition.

There were two subgroups in this group of 23 students. The first (N = 13) was sampled from those who in previous tests had omitted questions or made errors where the student’s reasoning could not be followed. The sample was stratified across the range of student ability. The second subgroup (N = 10) was a convenience sample from the poorest performing students. There were 38 such students all of whom were invited but only ten chose to attend. Details of the training offered to the different groups of students are provided in Table 1.

QSR NVivo 8, a software package for qualitative data analysis (QSR International, 2008), was used to analyse the material related to the students given individual teaching sessions to determine the key problems they experienced.

The group comprising very poor students had the additional difficulty that, despite our efforts, they still did not understand that the concentration of a solution is actually giving information about the quantity or mass of a substance relative to the volume, in other words two variables. This group was also overly reliant on calculators. They lacked an ability to simplify numbers or see patterns and had no insight into an unrealistic number generated by the calculator.

There was significant improvement in those students who attended the group tuition. Besides affording students dedicated time to focus on calculations away from competing priorities, the small groups provided more intimacy as there were fewer students present than in lectures. This gave students the opportunity to learn from more competent peers who may have guided them through difficult steps and clarified areas of misunderstanding. This finding is in line with the advantage of peer-assisted learning proposed by Chickering and Gamson (1987). In a group of nursing students, guidance from mentors and informal opportunities to engage with slightly more competent peers were also found to be helpful (Penman & White, 2006). Help and support from peers also afforded an opportunity for theorisation of learning in a more accessible language as, because there are 11 official languages and limited personnel within these language groups with appropriate teaching skills, teaching and assessments are entirely conducted in English. It is interesting to note that, by the end of our study, not speaking English at home had been lost as a risk factor for poor calculation competence.

One-on-one teaching markedly improved competence. All randomly selected students achieved 100% and even in the group selected from the poorest students, most improved. Our success with individual tuition is consistent with Vygotsky’s theory of learning: when the informal, disorganised concepts of the learner meet with the formal, methodical reasoning of the teacher this allows for co-regulation between the teacher and the student (Fogel, 1993). This state of inter-subjectivity between the student and the teacher enables the teacher to guide the student forward toward a better understanding of a situation (Kolikant & Broza, 2011).

Students who participated in the one-on-one interventions also had good attendance at the group tuition, so their results are a reflection of both group and individual tuition. It may be that the competence they developed during the individual tuition motivated them to attend the group tuition.

During the one-on-one interviews, it was apparent that students’ most common problems related to selecting the appropriate units to express a concentration given as a ratio or percentage. This was in line with findings from the written assessment, where such questions were more problematic for students achieving at the 75% level of competence but failing to attain full competence. In the randomly selected group of interview students, once this was clarified they were able to develop competence and master the calculations. The next most common problem, namely that with unit conversion, was also improved with individual teaching.

The group comprised of the very weakest students had additional conceptual problems that benefitted in particular from the one-on-one tuition. This was consistent with Jackson and De Carlo (2011), who found that nursing students with conceptual deficiencies required more rigorous remediation. Similarly, Kolikant and Broza (2011) contend that low-achieving students need appropriate interventions to improve their conceptual understanding.

The very weak group struggled with the concept of the strength of a solution and the idea that a ratio or percentage can describe the relationship between the amount of drug in a solution and the volume of that solution. This suggests that they had difficulty handling intensive quantities, which are measured by a relation between two variables. Concentration (an intensive quantity) is measured in terms of units of mass per units of volume. Reasoning about intensive quantities has been shown to be difficult for school children who lack opportunities in school to develop this understanding (Nunes, Desli & Bell, 2003).

Although one-on-one teaching may seem like a luxury in a world where cost-cutting exercises make it a rarity, it does have benefits beyond the obvious; Huijser et al. (2008) noted that ‘meanings negotiated during one-on-one consultations are not a one-way street, but rather part of a mutual learning experience during which valuable insight is gained’. They noted that individual teaching provides the opportunity for teacher responsiveness not only to the learner receiving the individual tuition but, by allowing misconceptions to be identified, to the group as a whole. Likewise, once our individual teaching had revealed that understanding related to intensive quantities could not be taken for granted, we changed our overall practice. We now include visual representations of particles in solution in our large group learning. We also make the analogy that when a large cup of sweet tea is poured into smaller cups it maintains the same sweetness (concentration), although the amount of sugar and volume of liquid in each cup is less.

Once students are reminded of their understanding of intensive quantities in this way, the idea that drug concentrations describe the relationship between two quantities (or Vergnaud’s two measure spaces) is introduced. The teacher now describes a hypothetical 1% solution, asking students to imagine 1 g of powdered active ingredient dissolved in 100 mL of water and then poured into each of one hundred 1 mL ampoules. Attention is drawn to the fact that the concentration has not changed; it remains 1%.

The very weak students were also overly reliant on calculators. This may have contributed to their lack of insight into an unrealistic result generated by a calculator. McMullan et al. (2010) found that young nurses who had relied on calculators since school had a false sense of security, resulting in more conceptual errors. Gillies (2004) has also expressed concern that unquestioning learning of a formula as well as over-reliance on a calculator could prevent students from monitoring their calculations thoughtfully causing them to blindly accept results, abandoning common sense and resulting in the acceptance of clearly unreasonable answers (Gillies, 2004).

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