Evaluation of a Design Principle for Fostering Students' Epistemological View on Models and Modelling using Authentic Practices as Contexts for Learning

Keywords: Authentic practices, Models and modelling, Context-based

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students [START_REF] Clement | Model based learning as a key research area for science education[END_REF]. Students should understand that models are primarily a representation of ideas about phenomena and can be rejected, replaced or modified in order to fit new (empirical) data. In addition, students should become acquainted with advanced model features, like reliability and validity, and with different modelling approaches and procedures.

Before discussing further, we define our use of the terms model and the process of modelling in this study. We use the term model as some structured representation, including symbolic elements, of the essential characteristics of an idea, object, event process or a system. [START_REF] Gilbert | Developing models in science education[END_REF][START_REF] Schwarz | Metamodeling knowledge: Developing students' understanding of scientific modeling[END_REF]. We use the term modelling for the process used in much of modern science and technology that involves (a) embodying key aspects of theory and data into a model, (b) evaluating that model using advanced features such as reliability and validity and (c) revising that model to accommodate new theoretical ideas or empirical findings [START_REF] Schwarz | Metamodeling knowledge: Developing students' understanding of scientific modeling[END_REF]. In this paper we concentrate on models and modelling in the domain of chemical engineering sciences. In this paper we also use the term Above argument has important implications for the design of modelling tasks conducted in schools. Science education involves more than learning theories and models devised by others, it also involves developing appreciation for the usefulness of the epistemology and assumptions of the discipline [START_REF] Enfield | A sketch is like a sentence": Curriculum structures that support teaching epistemic practices of science[END_REF]. It has been claimed that this might be realised if students are engaged in modelling processes similar as those used in real science or technology settings [START_REF] Edelson | Realising authentic science learning through the adaptation of scientific practice[END_REF][START_REF] Sadler | Situated learning in science education: Socio-scientific issues as contexts for practice[END_REF]. In this paper we refer to such settings as authentic modelling practices.

Authentic modelling practices are characterised by a community of workers working according to shared purposes, common motives and standardised procedures using relevant knowledge, tools and attitudes (Author-2, Author, Author, & Author-3, 2006;

Author-1, Author-2, Author, & Author-3, 2008). Authentic modelling practices can be adapted into contexts for learning. If this is done so that students become meaningfully engaged, the assumption is that students come to construct, test and evaluate models guided by similar motives and convictions as in the authentic practice. In addition, students will come to learn and use the epistemic notions consistent with the formal epistemology of the particular practice at hand. This vision of learning of is underpinned by the activity theory in education [START_REF] Engestroem | Learning by expanding: An activity-theoretical approach to developmental research[END_REF][START_REF] Leont'ev | Activity, consciousness and personality[END_REF].

In a previous study we adapted the authentic practice 'Modelling drinking water treatment' into a context for learning and applied the resulting curriculum unit in classroom (Author-1, 2010). In the present study, we evaluate and reflect on the learning outcomes related to models and modelling. We aim to elaborate the potential of the approach outlined above and reflect on strategies applied in the teachinglearning process to focus students on the epistemology of models and modelling.

Theoretical framework

In this section we first summarise some major epistemological views regarding models and modelling in science and technology. Secondly, we focus on the epistemology of models and modelling in the authentic practice 'Modelling drinking water treatment'. Thirdly, we outline the major characteristics of the context for learning, including a description of the instructional strategies applied in the teachinglearning process to focus students on the epistemology of models and modelling.

Epistemology of models and modelling in science and technology

Epistemology refers to one's basic beliefs about what knowledge is in science and technology, including their respective activities [START_REF] Sensevy | An epistemological approach to modeling: Cases studies and implications for science teaching[END_REF]. Although the science-technology relation is still under debate, we regard both domains as complementary to each other. Between both domains exists a weak and reciprocal interaction.

For the science domain, the epistemologist [START_REF] Giere | Explaining science[END_REF] suggest a 'naturalistic' explanation based on how science is actually done in laboratories, offices, and so on.

Giere treats scientific theories as a structured family of models, and distinguishes a perspective of discovery, in which new theories and models are created and elaborated, and a perspective of justification, where theories and models are tested against empirical evidence. In the latter, two interests are identified, namely 'evidence-based' and 'value-loaded'. The evidence-based interest values theories and models on the basis of evidence. In contrast, the value-loaded interest leads scientists to prefer one theory or model over another independently of considerations of are both valuable and do not exclude each other. In numerous occasions in science the evidence for preferring one theory or model over another purely on epistemic grounds is scant or poor, in which case value-loaded interests decide the issue. In this respect, value-loaded interests play a dual role. They structure theory and model choices within the perspective of justification under conditions of epistemic uncertainty, and they structure practice in the perspective of discovery. Value-loaded interests generate a kind of dynamics of scientific practice. Scientists do not just choose theories and models, they work on and with them. [START_REF] Lenk | Towards an Action-Theoretical and Technology-Oriented Philosophy of Science and Epistemology[END_REF] argues that Giere's combination of scientific models and their relation to real systems needs to be extended with a action-theoretical interpretation to account for the insoluble interconnection between knowledge, experimenting and action. Such view also covers technological approaches, since technology can be seen as an complex interaction between understanding and acting, requiring conceptual, procedural and methodological knowledge, expertise, skills and information [START_REF] Ankiewicz | Some Implications of the Philosophy of Technology for Science, Technology and Society (STS) Studies[END_REF]. The type of understanding differs from the analytical understanding endorsed in science [START_REF] Cajas | Public understanding of science: using technology to enhance school science in everyday life[END_REF]. As a consequence, more concrete and pragmatic approaches are used to clarify key aspects of the object under examination. As for modelling, roughly two approaches can be distinguished: an empirical and mechanistic one. The mechanistic modelling approach starts from theoretical ideas about the object at hand, while the empirical modelling approach starts from experiment. The choice which modelling approach to follow is guided by epistemic notions such as design, failures, constraints of reality (e.g. kind of materials, costs, engineering standards), trade-offs, unintended consequences and negotiations. The above view on the modelling activity in science and technology corresponds with the characteristics given by the epistemologist Hacking (1983Hacking ( /2005)). The major points are:

-Theories are not easy to define.

-Observations are not necessarily theory driven. There have been important

observations in the history of science and technology which have included no theoretical assumptions.

-Theory and experiments cannot be directly articulated. There is an 'enormously wide ranging intermediate activity best called model-building'.

-The activity of model-building consists of two processes; one starting from theory, which makes the theory more concrete or visible, and one from experiment, which makes this experiment more abstract.

In the present study we unravelled the epistemology of models and modelling by taking an actual authentic practice as point of departure, that is the practice 'Modelling drinking water treatment'. In the next section we give an overview of this practice, with emphasis on the epistemology regarding models and modelling (Author-1 et al., 2008).

Epistemology of models and modelling in the authentic practice 'Modelling drinking water treatment'

The treatment of drinking water, and the modelling thereof, is an ongoing matter of concern in many countries, since the quality of drinking water is an important area within the field of public health. Different kinds of contaminants, such as organic compounds and micro-organisms, need to be removed to produce safe drinking water. [START_REF] Rietveld | Models for drinking water treatment[END_REF].

Although drinking water treatment has a long history, the mathematical analysis of treatment processes is still young. Within this practice, many models are data driven, for example, that for coagulation/flocculation treatment. Other treatment processes, such as disinfection, filtration and activated carbon filtration, have been widely

studied and the models have a sound basis. Two modelling approaches can be distinguished, namely mechanistic and empirical (or 'black box'). Mechanistic models strive to understand and mathematically describe the mechanics underlying the processes occurring in a given system. Empirical models are simpler and are often obtained by fitting mathematical equations to a set of experimental data. Even though a mechanistic model is preferred, an empirical model is in many cases inevitable, especially when dealing with very complex processes or if the theoretical knowledge is simply lacking. In addition, empirical models are cheap and easily constructed compared to mechanistic models, and are equally powerful in describing process behaviour in response to external alterations, e.g., changes in process variables. Both modelling approaches are conceptualised in Figure 1.

INSERT FIGURE 1 ABOUT HERE

In an empirical model there is limited information from inside included in the model.

One tries to establish a relation between input and output, based on outside information alone. In a mechanistic model information from inside is included, and consists of an argued assemblage of variables, including fit parameters to calibrate the model on empirical data. In short, the major epistemic notions guiding the modelling approach are: 2. Empirical vs. mechanistic modelling approach, e.g., the available theoretical knowledge regarding the treatment process and its complexity, taken as the number of effecting process variables; 3. Modelling procedure;

4. Model features goodness of fit, reliability and validity as criteria for evaluation.

In the following section we describe how these four epistemic notions are concretised and incorporated in the context for learning.

2) is applied to find correlations between the starting turbidity, end turbidity and effecting process variables and to formalise these in mathematical models. In the context for learning, all the above process variables are identified, but only three process variables are experimentally investigated, namely dose coagulant, starting turbidity and temperature. The removal efficiency can be formalised by the formula . Figure 2 depicts a conceptualised scheme of the empirical modelling approach, as applied in the context for learning. Turbidity in denotes the incoming turbidity, while turbidity out denotes the residual turbidity.

INSERT FIGURE 2 ABOUT HERE

The modelling procedure (epistemic notion 3) consists of three major stages (Author- .

The regression models are evaluated on the following features (epistemic notion 4):

o Goodness of fit: indicated by the value of R 2 . Theoretically, R 2 can reach the maximum value of 1, denoting a perfect fit. However, all values > 0.8 are qualified as a good fit. A notable aspect is that the goodness of fit becomes more significant the more measurements are available;

o Reliability: depending on the number and accuracy of the gathered experimental data, to be judged by the (team of) researcher(s);

o Validity: the tested range of the process variables, e.g. X 1 < turbidity in (NTU) < X 2 , Y 1 < dose coagulant (mg/L) < Y 2 and Z 1 < temperature (°C) < Z 2 .

In conclusion, the applied empirical modelling approach for turbidity removal by coagulation/flocculation can be characterised as 'value-loaded' (complex process, use an additive or multiplicative regression model) within the perspective of justification (models are fitted on empirical data), starting from an experimental point of view (select significant process variables, make the data more abstract). Next, we turn to instructional strategies to apply in the teaching-learning process to ensure that students gain understanding on the epistemic notions as described above.

Strategies to focus students on the epistemology of models and modelling We expect that students are aware of the mentioned epistemic notions in a rudimentary sense at the beginning of their learning process. However, students lack the knowledge and experience to give completion to the epistemic notions, and qualitatively describe and discuss them. The challenge is to design a teaching-learning process such that students get clear sight on these notions in the intended way. To achieve this as a pedagogic effect of teaching, there is need for explicit instructional strategies [START_REF] Chinn | Epistemologically Authentic Inquiry in Schools: A Theoretical Framework for Evaluating Inquiry Tasks[END_REF] In this section we describe the designed curriculum unit and its application in the classroom, the participants, the data collection and analysis.

Design-based research

Our design-based research approach strongly resembles what Cobb, Confrey, DiSessa, Lehrer, and Schauble ( 2003) described as 'design experiments' conducted in the classroom. This approach implies the instructional design of a teaching-learning process, accompanied by a set of argued expectations of how the process is expected to take place and why it should operate according to these expectations [START_REF] Lijnse | Developmental research' as a way to empirically based 'didactical structure of science[END_REF].

These expectations are based on literature as well as empirical findings in previous research cycles. The teaching-learning process is designed in close cooperation with teachers. The testing of the process takes place in a small-scale case study, with a classroom and its teacher as the unit as analysis [START_REF] Cobb | Participating in classroom mathematical practices[END_REF]. The findings are used to evaluate the design principle, which might give reasons for a (partial) redesign to be tested in the next research cycle.

The curriculum unit

The curriculum unit consists of five separate learning phases, in which different instructional functions are to be achieved. In Table 1 the broad outline of the curriculum unit is described per learning phase. In Appendix A all the teachinglearning activities are described in detail.

INSERT TABLE 1 ABOUT HERE Application in the classroom

The curriculum unit was put into practice at four different schools in the Netherlands in the period from January to June 2008. In two of the schools, the curriculum unit

Data collection and analysis

Given the purpose of this study, the data required were primarily qualitative. The data collection and analysis are concentrated around teaching-learning activities (TLA)

that are regarded as critical as it comes to learning epistemology of models and modelling. The analysis consisted of four stages.

In the first stage, preceding the actual data analysis, two researchers (first and second authors of this paper) developed and agreed upon a frame of reference as a coding scheme. This frame of reference consists of a set of expected students' notions per (cluster of) teaching-learning activities. The teaching-learning activities are the operational construct of the strategy components A to E (see Figure 3), while the students' epistemic notions (intended pedagogic effects) refer to the epistemic notions described in the theoretical framework. In Appendix B the frame of reference is described in detail.

In the second stage, the data were analysed by both of the researchers independently, using this pre-formulated frame of reference. In the third stage, all the codes were combined to reveal major trends and findings.

Firstly, for each teaching-learning activity per student team, if at least 80% of the intended pedagogic effects were realised, the corresponding strategy component was deemed fully accomplished. If between 20 and 80%, the valuation was partial, and if between 0 and 20% it was deemed to have achieved a low level of accomplishment.

Secondly, for each strategy component, the judgements of all the student teams were combined. A particular strategy component itself was deemed fully accomplished if it scored high in at least 80% of all student teams. A strategy component was deemed low, if 80% of all student teams judged it partly accomplished or less. In case of a score somewhere between these two extremes, the strategy component was deemed partly accomplished. Thirdly, a rater consistency check was conducted by calculating the intraclass correlation coefficient using a two-way mixed effects model [START_REF] Shrout | Intraclass correlations: Uses in assessing rater reliability[END_REF].

In the fourth stage, both researchers discussed all the judgments to identify underlying considerations, to unravel students' perspectives and to reflect on the strategy components applied. Finally, all the results were discussed by the complete research team.

The collected data sources are audio-taped conversations of student teams at work, written answers of student teams, interviews with student teams and field notes.

Below, we briefly describe each data source and specific analysis procedure.

Audio-taped conversations

While working on teaching-learning activities (TLA) 18-19, the conversations of the student teams at work were audio-taped. Next, the conversations were transcribed perspective [START_REF] Smith | Semi-structured interviewing and qualitative analysis[END_REF] using the frame of reference.

Written answers

All written answers from the student teams on teaching-learning activities (TLA) 3, 5, 14-16, 18-19, and the factsheets, were coded by two researchers (first and second author) independently using the frame of reference.

Interviews

Each student team was interviewed during or shortly after having accomplished teaching-learning activity (TLA) 3. The interview was semi-structured with the aim being to reveal the students' perspectives on (a) the purpose of the modelling and (b)

the modelling procedure to apply (common sense modelling approach). The length of the interview was approximately 10 minutes. The interviews were audio-taped and transcribed verbatim. Next, the interviews were coded by both researchers independently from an interpretative perspective [START_REF] Smith | Semi-structured interviewing and qualitative analysis[END_REF] using the frame of reference.

Field notes

During the complete enactment of the curriculum unit on both schools field notes were made by the first author of this paper. The major purpose of the field notes was to check whether the strategy components, and corresponding teaching-learning activities, were enacted in class with sufficient quality.

Results

In this section we present the results for each strategy component. At the end, we factsheet 'modelling removal of trichloromethane by activated carbon filtration'

The strategy to bring in a worked-out analogous modelling problem aims to make explicit the meta-modelling knowledge. Ten teams considerably enriched their rudimentary modelling procedure, as formulated in teaching-learning activity 3. Team 9 describes the modelling procedure as follows (for reasons of length, only the main procedural steps, shortened and summarised, are presented):

-Explain the working of the treatment step: perform experiments;

o
construct scatter plots;

analyse and typify correlations;

perform single and multiple regression.

Strategy component C embodies some key aspects regarding modelling. First of all, student teams analyse scatter plots to characterise the type of correlations. Secondly, students fit regression models on the data, both linear and power. Thirdly, students evaluate the regression models on aspects goodness of fit, reliability and validity.

The results show that nine teams were able to draw correct conclusions regarding the correlations based on the scatter plots. The drawing of the scatter plots gave the student teams more insight into the quality of their measurements. The teams doubtful measurements. The process variables of dose coagulant and starting turbidity correlated clearly with end turbidity. However, the temperature was less obvious. Six teams concluded that the correlation of the temperature was not significant, and thus could be excluded from further analysis, as typified in Figure 4.

INSERT FIGURE 4 ABOUT HERE

The majority of the teams (10) did understand the arguments either to leave out or take into account the temperature. For example, here is Team 7 discussing with the teacher the construction of multiple regression model: All teams were able to determine the best line (linear or power) through the scatter plots, including the mathematical formula. While beyond the scope of the exercise, six teams also fitted exponential (and even logarithmic) regression models. These teams extensively used the advanced model feature goodness of fit as an evaluative calculate the dose of coagulant needed to produce clear water.

S4: … And
Six teams succeeded in calculating the dose coagulant and reflected upon the predictive value of the outcomes. However, the findings reveal that in total ten teams argued about the predictive value of their constructed model. Arguments brought to the fore vary from 'process variables not taken into account' to 'model based on inaccurate measurements'.

Team 5 qualifies the model outcome as follows:

This seems to us a not likely outcome to add 20.6 g/L coagulant [to produce water with 1 NTU]. Our model is also not valid in this situation: the pH and temperature are different. This means that our model should be adapted.

Four teams judge their model outcomes as unpredictable, but suggest comparing the outcomes with real-world data, as exemplified by the statement below: compare the black-box modelling approach with the mechanistic approach

The findings reveal that ten teams identified the empirical modelling approach and evaluated the approach, as shown by statements below:

The 'black box' approach is quick, easy and also effective as it comes to describing process behaviour. (Team 3)

The 'black box' is just a big experiment, you test process variables, apply regression and construct a formula. A 'mechanistic' approach is much more theoretical regarding which process variables and why. ( Team Four teams explicitly suggest following a more mechanistic approach, as they felt uncomfortable with the empirical modelling approach.

Team 1 suggest future research: More or less a follow up of teaching-learning activity 18, all teams again articulate the need to improve the quality of the measurements. Team 4 proposed implementing some kind of JAR test, in which all experiments are done in exactly the same manner.

As for typifying the general problem regarding input-output systems, the answers vary from 'account for variable input, not just one variable', 'apply black box or mechanistic approach' to 'variability of external variables'. On average, the teams emphasise 'variability' as a major problem. Four teams formulated a general procedure for modelling input-output systems, although regression as an essential step was mentioned only once. However, the exact learning gain on meta level remains unclear. It is questionable whether the learning gain is largely absent, or that it is simply not measured. In Strategy components D (let students apply the constructed model in a real-world setting) and E (let students compare different modelling approaches), finally, do function as expected and give rise to the intended pedagogic effects, but their functioning can still be improved.

Conclusion and discussion

The aim of this study was to elaborate knowledge regarding the use of authentic

Reflections on the functioning of the strategy components

Based on the findings on the pedagogical effects, it can be concluded that strategy component A (visualise and conceptualise the example problem; epistemic notions 1 and 3; see section 'Theoretical Framework') functions as expected, especially for identifying the process variables, and formulation of the purpose of the modelling as well as a rudimentary modelling procedure. The results show that nearly half of the teams suggest an empirical approach, and the others suggest a mechanistic approach.

To further underline the pros and cons of the modelling approach to apply, we add a new strategy component: point out and discuss in class the appropriate modelling approach. A goal of the strategy components A (visualise and conceptualise the example problem) and E (let students compare different modelling approaches) was to conceptualise the coagulation/flocculation treatment step as an example case of an input-output system. This was introduced in strategy component A, and recalled in strategy component E. It was intended that the students would grasp the broader applicability of the empirical modelling approach for complex input-output systems.

However, none of the teams took notice of the broader applicability. We are a bit cautious to draw explicit conclusions about this, because the broader applicability can be typified as (a sort of) meta-knowledge, which was hardly made explicit by students (see results on strategy component E). However, the findings suggest that conceptualisation alone does not automatically led students to see the broader applicability. Mentioning other input-output systems that would be recognisable from students' perspective, for which an empirical modelling approach is feasible, might support such understanding. regarding the number and accuracy of the measurements. This might be an indication that students understand that the quality of the measurements is extremely important in a black-box modelling approach. However, the actual calculation of the dose coagulant using the constructed model needs redesign. In addition, the notion that the model has been constructed in a laboratory setting, and thus needs extensive testing in an industrial environment preceding real usage, can be fostered.

F

The strategy component E (let students compare different modelling approaches; epistemic notion 2; see section 'Theoretical Framework') proved successful in evaluating the applied modelling approach. Students encountered that science and technology is not a straightforward process, but implies fundamental choices with Summarising all the implications, we come to a slightly altered and generalised filling-in of the design principle 'content modelling', as depicted in Figure 5.

INSERT FIGURE 5 ABOUT HERE

The results give rise to the incorporation of an extra strategy component, namely to discuss and to point out the appropriate modelling approach early in the teachinglearning process.

Reflections on fostering students' epistemological views of models and modelling

We started this paper with the statement that the epistemology of many school , 1993). One needs to add a "meta modelling layer", which enables students to develop explicit theories about the nature of models themselves. [START_REF] Sandoval | Understanding students' practical epistemologies and their influence on learning through inquiry[END_REF] argues that individuals can rely on multiple epistemologies to interact in different contexts, eliciting students' ideas, challenging those ideas and introducing alternative ideas. In our opinion, this can be realised by engaging students in different authentic modelling practices, in which different epistemological views are employed, such as constructing mechanistic and empirical models. By doing so, we expect that students will gain a richer, more varied perspective on science and technology. In addition, it enables students to gain some insight into socio-scientific values related to authentic practices. However, as might become clear from this study, there are still multiple (major) design challenges in using authentic practices as contexts for learning. The promising results can only be achieved by high quality in the design and enactment of the teaching-learning process.

Limitations

The conclusions of this study are subject to at least three limitations. Firstly, it should be noted that this particular authentic practice was selected after a thorough evaluation (Author-1 et al., 2008). It is important to select authentic modelling practices which, in principle, are feasible for upper secondary chemistry education. Secondly, the design principle 'content modelling' applies to (1) using an authentic practice as the context for learning, (2) the domain of models and modelling, and (3) students in grades 10 and 11 (aged 16-17 years), high school chemistry (upper secondary level).

The present filling-in of the principle emerged from the adaptation and enactment of only one authentic practice. The application of this design principle to other authentic modelling practices, either within chemistry or other science domains, needs to be examined. Thirdly, the teachers involved were all well informed about the pedagogy In this appendix the curriculum unit is described in terms of content and sequence of teaching-learning activities per learning phase.

Phase I: Orientate on the practice

Content related questions

Sequence of teaching-learning activities (TLA) including desired learning outcomes TLA 1: Broad orientation on treatment of water. Students make an outline of the treatment process of ground-and surface water to produce drinking water. In addition, for each treatment step students summarise the process variables that effect the removal efficiency.

• Students realise the societal importance of good quality drinking water (public health).

Students gain insight into the dynamics of the treatment process.

What causes turbidity?

What is coagulation/ flocculation?

TLA 2: Exceeding of chemical parameters. Students receive a list of (occasionally) exceeded chemical parameters for drinking water (source: government document). Students notice the outrun of the quality parameter turbidity. Turbidity is removed by the treatment step of coagulation/flocculation.

• Students realise that the removal of turbidity by coagulation/flocculation is one of the issues dealt with in the treatment of water.

The dosage of ferric chloride needs to be adjusted to the starting turbidity to produce clear water.

TLA 3: Zoom in on turbidity removal by coagulation/flocculation. The teacher conducts a laboratory experiment in classroom showing the clearance of turbid water by coagulation/flocculation. Next, the teacher conceptualises the problem as an exemplary case of an input-output system. Students outline a modelling procedure to find a relation between starting turbidity, dose coagulant and end turbidity.

• Students make explicit their notions regarding the purpose of the modelling and the modelling procedure. Students grasp the broader applicability of the modelling procedure to be learned.

How is this done in real practice?

TLA 4: Orientate on the modelling approach proposed by experts. Students study an adapted and shortened version of an authentic project plan concerning 'Modelling drinking water treatment'. They compare their own formulated modelling procedure (TLA 3) with the modelling approach proposed by experts.

• Students make a list of similarities and differences in both modelling approaches. Students gain a sense of direction, in terms of modelling activities to conduct.

Phase II: Zoom in on an example problem • Students make a summary. They realise that they need to identify all process variables that effect coagulation/flocculation.

TLA 7: Identify process variables that affect coagulation/flocculation. Students identify process variables and hypothesise about their possible influence. Students use the acquired knowledge in TLA 3 (demonstration) and TLA 6 (article on coagulation/flocculation).

• Students identify multiple process variables that influence the end turbidity, such as dose coagulant, starting turbidity, temperature and mixing effects.

Which process variables affect the removal of turbidity by coagulation/ flocculation? TLA 8: Extend the list of identified process variables. The teacher gives a short lecture about the chemistry underpinning coagulation processes. Next, students study an article from which they learn about coagulation mechanisms and extend their list of process variables.

• Students add process variables acidity (pH) and total salt concentration. Students realise that the influence of the process variables needs to be examined experimentally.

TLA 9: Bring up to date list of concepts and factsheet. Students bring up to date their factsheet (TLA 5) and their list of content related concepts.

• Students realise that it is important to update their newly acquired knowledge and findings regularly.

TLA 10: Investigate the influence of process variables empirically. The teacher divides the work such that three process variables are studied experimentally: dose coagulant (FeCl 3 ), starting turbidity and temperature. Students receive laboratory prescripts for the experiments.

• Students understand that everybody needs to conduct the experiments in a similar way in order to combine the results later on.

How large is the influence of each process variable? How accurate are the measurements?

TLA 12: Reflect on obtained results. Students evaluate the obtained experimental results.

• Students find out that some experimental results are suspect, due to poor performance or being deviant from the observed tendency, and reflect on possible explanations.

TLA 13: Bring up to date list of concepts and factsheet. Students bring up to date their factsheet (TLA 5) and their list of content related concepts.

• Students realise that it is important to update their newly acquired knowledge and findings regularly.

Which process variables significantly influence the turbidity?

TLA 14: Correlation between end turbidity and process variables. Students present their scatter diagram summarising their results. The teacher invokes discussion about the observed correlations. Students select the process variables with a significant effect on turbidity removal (leaving starting turbidity and dose coagulant). Students study a manual dealing with correlation (and regression).

• Students realise that the experimental results need to be analysed further in order to quantify this influence.

How to quantify the influence of the process variables?

TLA 15: Single regression of end turbidity on dose coagulant and starting turbidity. The teacher gives a short lecture about regression. Next, students perform single regression on end turbidity versus (1) dose coagulant and (2) starting turbidity. Students fit linear and non-linear (power) regression models and reflect on their 'goodness of fit'. Students work according to a manual dealing with correlation and regression, and use MS Excel software.

• Students realise that one model is needed to account for different influences on the same process simultaneously. How to couple the models to one formula?

TLA 16: Multiple regression. Students attend a short lecture about multiple regression. Next, students fit multiple linear and non-linear (power) regression models to the data, according to planned procedure using MS Excel. Students evaluate the resulting regression model on aspects 'goodness of fit' and 'validity'.

TLA 17: Bring up to date list of concepts and factsheet. Students bring up to date their factsheet (TLA 5) and their list of content related concepts.

• Students realise that it is important to update their newly acquired knowledge and findings regularly.

Phase IV: Evaluate and reflect on the findings.

Content related questions

Sequence of teaching-learning activities (TLA) including desired learning outcomes

In what way can the model contribute to turbidity removal by coagulation & flocculation?

TLA 18: Apply developed multiple regression model. Students apply the multiple regression model (TLA 16) to calculate the dosage of coagulant needed to produce clear water in a production side given a certain raw water quality. Students evaluate the outcomes on the aspects 'purpose', 'reliability' and 'validity'.

• Students realise that their recommendations should be underpinned by reporting the main findings.

What are the strengths and weaknesses of the applied 'blackbox' modelling approach?

TLA 19: Reflect on the applied 'black-box' modelling approach. The conceptualised inputoutput system (TLA 3) is recalled. Students compare the applied black-box modelling approach with the mechanistic approach and think of pros and cons. In addition, students think over future research on turbidity removal by coagulation/flocculation based on their own experiences and extended knowledge.

• Students realise that this type of modelling is an exemplary example of process modelling, and thus is worthwhile to make explicit.

Phase V: Express the findings Students write a factsheet summarising the applied modelling procedure, main findings, conclusions and advice for future work. This factsheet is assessed by the teacher.

• Students make their learned knowledge explicit.

F

Learning Phase

Description of the learning phase I:

Orientate on the practice

The first phase induces among students a motive for studying the problems posed in the practice at hand, and provides them with a sense of direction concerning where their study will lead them. Firstly, students orientate on the treatment of ground-and surface water for production of drinking water. Secondly, students take notice of occasional exceeding of

  Evaluation of a Design Principle for Fostering Students' Epistemological View on Models and Modelling using Authentic Practices as Contexts for Learning in Chemistry Education central goals of science education is to promote model-based reasoning in

  epistemology' to denote the situational nature of knowledge, its continuous development and active interference with the world and its subjects and objects.

  should keep in mind that the evidence-based and value-loaded interests

  and/or function of the model;

  Authentic practice 'Modelling drinking water treatment' as a context for learningThe authentic practice 'Modelling drinking water treatment' comprised the modelling of multiple treatment processes, numerous substances and quality parameters. For use as a context for learning, we decided to focus, or 'zoom in', on one particular treatment step and quality parameter, namely the removal of turbidity by coagulation/flocculation (Author-1, Author-2, Author, &Author-3, 2009). The unit of measure of turbidity is the nephelometric turbidity unit (NTU). The turbidity of surface water normally varies between 0 till 50 NTU. In general, the legally set maximum allowed turbidity for drinking water is about 4 NTU. The main reason for focussing on turbidity removal by coagulation & flocculation is the availability of small-scale laboratory experiments suitable for the school environment (Author-1, 2010). Turbidity is caused by suspended matter, such as clay particles and colloids.The coagulation/flocculation treatment is affected by a variety of process variables, such as type and dose of coagulants and flocculants, starting turbidity, acidity (pH), salt concentration, mixing effects and temperature. The purpose of the modelling is to be able to predict the end turbidity as a function of the 'raw' water quality and process conditions (epistemic notion 1). An empirical modelling approach (epistemic notion

  detail with respect to the functioning of the design principle 'content modelling', as case studies. Both of these schools can be characterised as rural schools with few students from ethnic minorities. In total 44 students participated, aged 16-17 years, and in grades 10 and 11. The curriculum unit compromised eight lessons of 50 minutes, excluding time for private study. The students worked in teams of four persons (11 teams in total). The teachers were well acquainted with the content and pedagogy of the curriculum unit, since they were involved in the process of designing the unit (author-1, 2010).

  coded by both researchers independently from an interpretative

  combine and summarise the findings to answer the research question. The teachinglearning activities were enacted in the classroom as planned. The analysis showed a the raters reflected in the intraclass correlation coefficient of 0.78.Strategy component A: Visualise and conceptualise for students the example problem on which they are going to work:demonstrate treatment step coagulation/flocculation conceptualise the modelling problem as an input-output systemThe results show that all teams do notice the process variables of dose coagulant and starting turbidity. Other process variables mentioned were stirring effects, temperature and type of coagulant. All teams recognised the variability of the input. As for the purpose of modelling, ten teams mention the dose coagulant in order to produce clear water. To achieve this, in general, two perspectives are brought to the fore. Six teams propose to concentrate on the relation between the end turbidity and the dose coagulant. On the contrary, five teams suggest focussing explicitly on the process, typified by statements such as 'elucidate what happens on a molecular level'. Team 7's written answer on teaching-learning activity 3 was: 'Which substances cause the turbidity? What exactly happens? What causes the particles to grow during the stirring phase?' Ten teams formulated a modelling procedure in response to the plenary demonstration. The majority of the teams (10) focussed on conducting experiments to investigate the variability of external factors, as exemplified below:

  data point (whether or not falling within the trend) and deleted

  fact brought to the fore a fundamental limitation of the empirical modelling approach: the absence of theoretical arguments to navigate to a particular regression model. We will come to this point later on.In total nine teams succeeded in constructing a multiple regression model. The remaining two teams failed in multiple regression. In valuating the regression models, seven teams used goodness of fit as the decisive criterion. Only four teams explicitly formulated the validity of the constructed regression models. In Table2the results are summarised, including some general remarks on the functioning of strategy component C. Strategy component D: Let students apply the constructed model in a real-world setting:

  7)We did not know exactly what happens during the process. (Team 2) If high (> 0.8) [value R 2 ], then alright according to the 'black box' method, but theoretically much remains unclear. (Team 8) the need for future research, using the advanced model feature goodness of fit, as well as the quality of the measurements, as primary arguments.

S3:

  Is there reason for a follow up? You could … S4: extensive … S3: …develop a mechanistic model for an exact description of the working and behaviour of the process. S2: cause … S3: well, … with use of a mechanistic model you can give an exact description.

  chemical modelling practices as contexts for learning. The knowledge involved was captured in the design principle of 'content modelling' with five strategy components.The principle of 'content modelling' deals with focussing learners on the epistemology of models and modelling. The research question was: to what extent does the current completion of the design principle of 'content modelling' lead students to acquire the intended insight into epistemology of models and modelling?The results show that four out of five strategy components (A, B, D, E) function sufficiently. Strategy component C, however, requires reconsideration. In this final section we reflect on the functioning of each strategy component, as well as the broader applicability of this design principle for adapting other authentic modelling units. In addition, we formulate implications for future research.

  . We propose to further strengthen the strategy component with an explicit focus on estimations made, assumptions and neglected variables. In our opinion, through discussing and comparing different modelling approaches earlier in the teaching-learning process, students' views on the epistemology of the applied modelling approach might be fostered.

  modelling tasks is antithetical to the epistemology of modelling in real science and technology practices. Traditionally, school science (unintentionally) presents students with a straightforward route to the development of new models, leaving out the difficulties and failures inherent in conducting research. We argue that engaging students in authentic practice-based modelling processes might offer a way out. The results obtained in the present study support the arguments and encourage further research on the design challenges. Science curricula, emphasising learning about the nature, purpose and construction of models, has the potential to enable students to develop accurate and productive epistemologies of science and technology[START_REF] Schwarz | Metamodeling knowledge: Developing students' understanding of scientific modeling[END_REF]. However, as reported in the literature, simply engaging students in developing models is not enough to achieve epistemological sophistication(Carey and

  up a modelling procedure for removal of turbidity by coagulation/flocculation. Students receive a factsheet summarising the approach and outcomes of an analogous modelling problem: removal of trichloromethane by activated coal filtration. This analogous problem serves as a leading example. Students enrich their formulated modelling procedure, and notice advanced model features to evaluate the resulting model.• Students formulate a modelling procedure:1 Find out more about coagulation and flocculation 2 Identify process variables (next to dose of ferric chloride and starting turbidity) 3 Experimentally investigate the influence of variables 4 Develop a mathematical formula for predicting end turbidity 5 Evaluate the model • Students copy the basic structure of the factsheet.

  treatment step coagulation/flocculation. Students study an article about treatment step of coagulation/flocculation. This article describes the way the treatment is conducted. It also illuminates the difficulties in removing the small clay particles and colloids causing turbidity.

TLA 11 :

 11 Draw scatter plots. Students plot scatter diagrams showing the experimental results of end turbidity versus (1) dose coagulant, (2) starting turbidity and (3) temperature. Students interpret the results and think back and forth between the hypothesised influence (TLA 7 & 8) and observations.

Figure 4 .Figure 5 .

 45 Figure 4. The analysis of student team 1 regarding the type of correlation based on the scatter plot of temperature against end turbidity.

  (chemical) parameters, including the turbidity. Thirdly, the removal of turbidity by coagulation/flocculation is demonstrated by an experiment in class. Students formulate a purpose of modelling and a (rudimentary) modelling procedure for turbidity removal by coagulation/flocculation [Strategy component A]. Fourthly, students compare their formulated modelling procedure with the modelling procedure proposed by experts, by studying an adapted and shortened version of an authentic project plan 'Modelling drinking water treatmentphase the students enrich their own formulated modelling procedure. Students make explicit the sequence of modelling activities to conduct and the advanced model features to judge the quality of the constructed model(s), and they gain sight of the (type of) end product to deliver. This process is facilitated by studying a factsheet describing a worked-out analogous modelling problem, namely modelling the removal of trichloromethane by activated carbon filtration [Strategy component B]. Students extract the modelling procedure and identify the advanced model features goodness of fit, reliability and validity. III:Solve the example problemIn the third phase the students extend and apply their knowledge related to modelling turbidity removal by coagulation/flocculation. This extending and applying of knowledge is an iterative (intertwined) process. The main stages, as already identified in learning phase II, are: 1. Identify process variables affecting turbidity removal by coagulation & flocculation, e.g., dose of ferric chloride, starting turbidity, temperature, acidity, … 2. Conduct experiments under controlled conditions to determine the influence of three process variables, namely dose of ferric chloride, starting turbidity and temperature.Students work according to laboratory prescripts. 3. Conduct regression on the acquired empirical data. Students draw scatter plots, select those process variables with significant correlation, and fit linear and non-linear (power) regression models to the data. First a single regression, followed by a multiple regression.Students evaluate the constructed multiple regression model on the advanced model features goodness of fit, reliability and validity [Strategy component C].In the fourth phase students summarise, evaluate and reflect on their findings. Students used the constructed multiple regression model to calculate the dose of ferric chloride needed to produce clear water in an industrial plant site[Strategy component D]. They judge the model outcomes in terms of reliability and validity. Next, students reflect on the applied modelling approach, as an exemplary case of black-box modelling. They compare the black-box approach with the mechanistic approach and formulate pros and cons [Strategy component E]. Finally, students give advice on future research concerning the modelling of turbidity removal by coagulation/flocculation. V:Express the findingsIn the fifth phase the students make explicit their findings in light of the particular modelling issue worked on. Students write a factsheet (as introduced in learning phase II), summarising the main results, including an outlook on further research from the students' perspective.

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	Scope and research question Method
	This research study is positioned within the broader perspective of developing and
	investigating context-based curriculum units in science education. The aim of this
	study is to contribute to a knowledge base regarding the use of authentic chemical
	modelling practices as contexts for learning. The present study builds on a previous
	regarding the details and sequence of the one, in which the authentic practice 'Modelling drinking water treatment' was teaching-learning activities. F o r selected as suitable context for learning (Author-1 et al., 2009). The selection was based upon criteria students' interest, ownership, familiarity and complexity. The F o r
	Design principles are defined as tools providing heuristic guidelines by means of P topic of water treatment is well known in Dutch curriculum. In general, students are P
	strategy components for realising pedagogic effects in class (Van den Akker, e familiar with basics treatment methods (e.g. filtration, adsorption, precipitation, e
	Gravemeijer, McKenny, & Nieveen, 2006). Design principles link strategy e r aeration) and the major concepts involved (e.g. alkalinity, oxidation, contaminants). e r
	components, e.g., what to do, how precisely, when in the sequence, with what tools and how enacted, pedagogic effects, e.g., students' epistemic notions regarding models and modelling, and arguments, e.g., literature on educational research, empirical findings from previous applications and/or practical considerations. In the adaptation of the authentic practice into a context for learning we distinguished three design principles, labelled context, content modelling and chain of activities (Author-R e v i e w However, students are unfamiliar with the employed mathematical models, as well as R the applied modelling approaches. Currently, the Dutch chemistry (Van Koten, De e Kruijff, Driessen, Kerkstra, & Meinema, 2003) and science (KNAW, 2003) curricula are under revision to make way for new topics, such as modelling. Within this respect, v i e the current Dutch developments and this research study can be positioned within the international trend to implement and investigate context-based approaches (Author-3 w
	1, 2010). In the current study we focus specifically on the functioning of the design principle of 'content modelling'. The principle of 'content modelling' deals with O n l and author-2, 2006). We designed a curriculum unit which enacted in the classroom, with students in grade O n l
	focussing learners on the essential generic content regarding models and modelling. y 10 and 11 (age 16-17 years), in upper secondary chemistry education. The evaluation y
	Figure 3 depicts the design principle 'content modelling'. was focussed on the functioning of the design principle of 'content modelling'. The
	INSERT FIGURE 3 ABOUT HERE central research question addressed here is:
	To what extent does the current completion of the design principle of 'content
	In this study we evaluate to what extent the strategy components lead to the intended
	modelling' lead students to acquire the intended insight into epistemology of models
	pedagogic effects, e.g., students' epistemic notions regarding models and modelling,
	and modelling?
	and reflect on the underpinning arguments.
	12 13

  To see whether this[outcome] is real, it should be tested for reliability by conducting this experiment in the real world [not laboratory setting] multiple times.The majority of the teams (10) conclude that more (accurate) measurements are needed and that more process variables should be researched to 'make the model fit for every situation'. Four teams argue explicitly for testing the model in a real industrial plant environment, showing that these students are aware of the present state of the model. In Table2the results are summarised, including some considerations on the functioning of strategy component D.

	If the amount of added coagulant proves to be around 132 mg/L, then the model
	outcomes are trustworthy. This, of course, should be tested multiple times under
	varying conditions …
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	22

Team 10 reflect on the model outcomes as follows:

Strategy component E: Let students compare different modelling approaches:

  Table 2 the results are summarised, including some considerations on the functioning of strategy component E.The results for the strategy components A to E are summarised in Table2. For each strategy component, we also present the major considerations (successful aspects, as well as opportunities and/or possible improvements). As becomes clear, strategy components A (visualise and conceptualise the example problem) and B (supply students with a worked-out analogous problem) function to a sufficient extent.However, strategy component C (involve students in the nature, characteristics and wording of the models at hand) needs redesign. A main aspect to consider is how to construct a continuous line of teaching-learning sequences from single to multiple regression, such that students arrive at the intended multiple regression model(s).

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	Summary of main findings on the strategy components
	24

  Other teachers willing to enact the curriculum unit and to apply this design principle should be confident with the domain specific pedagogy. the design of concrete teaching-learning activities in class. There is a need for such explicit knowledge, since in many curriculum innovations teachers are designated as the developers of teaching-learning processes. In the next stage of this project we will focus on the broader applicability of the design principle of 'content modelling'. The fact that model-based teaching and learning is widely regarded as central in science education makes it worthwhile to develop further a knowledge base about the use of authentic modelling practices as contexts for learning.Appendix A. Overview of the curriculum unit in terms of content and sequence of teaching-learning activities

	of the curriculum unit. Implications for future research
	Further development of design principles, or equivalent ways for capturing
	knowledge on educational designs, is important, because it bridges (abstract) theories
	F o r on learning with F o r F o r F o r F o r
	P P P P P
	e e e e e
	e r e r e r e r e r
	R R R R R
	e e e e e
	v i e v i e v i e v i e v i e
	w w w w w
	O n l O n l O n l O n l O n l
	y y y y y
	30

co.uk International Journal of Science Education

  Appendix B. Frame of referenceThe frame of reference describes the expected students' epistemic notions as a result of the strategic components. The epistemic notions are concretised completions of the intended pedagogic effects. The strategy components are operationalised in teaching-learning activities, which are used as data sources. Figure 1. Conceptualised scheme of an empirical and mechanistic modelling approach.

	Page 39 of 44					Page 38 of 44
	Empirical modelling approach	Mechanistic modelling approach
				Empirical modelling approach
	input	water in		output coagulation/flocculation input	water out	output
	Strategy component			Data sources	Students' epistemic notions
		Process variables -Turbidityin -Temperature	(Learning phase | Teaching-learning Dose coagulant	Process variables -Turbidityout -Temperature
				activity -TLA)
	o r A Visualise and conceptualise for Figure 2. Conceptualised scheme of the empirical modelling approach of turbidity removal by -Identify process variables dose coagulant, starting turbidity and students the example problem on which they are going to work: -demonstrate treatment step coagulation/flocculation -conceptualise the modelling problem as an input-output system I | TLA 3 stirring regime -Think of measurements to take to prevent occasional outruns of the norm -Formulate a rudimentary modelling procedure for input-output systems F o r r o F coagulation/flocculation.
	B Supply students with a worked-out			-Formulate a modelling procedure for the example problem
	analogous modelling problem as leading example: -factsheet 'modelling removal of trichloromethane by activated carbon filtration'	P e e r P P e e II | TLA 5 e r e r	1. Study coagulation/flocculation 2. Identify process variables and describe influence 3. Conduct experiments 4. Present data in scatter plots 5. Develop mathematical model to quantify influence 6. Evaluate and reflect on the constructed model 7. Write a factsheet -Identify advanced model features (goodness of fit, reliability,
	C Involve students in a series of teaching-learning activities emphasising the nature, characteristics and wording of the model(s) at hand: -perform experiments -construct scatter plots -analyse and typify correlations		R e v i e validity) to evaluate the developed models III | TLA 14 -16 Factsheet -Conduct experiments according to prescripts (see the point of R R collecting much data for each process variable) -Draw scatter plots and delete suspicious measurements e e -Select the process variables with a significant correlation (positive or negative) with the end turbidity -Conduct single regression (fit linear and power models on the data) -Conduct multiple regression (fit additive and multiplicative v i e v i e
	-perform single and multiple regression D Let students apply the constructed			w models on the data) -Evaluate the constructed models on the goodness of fit -Evaluate the calculated dose coagulant w w
	model in a real-world setting: -calculate the dose of coagulant needed to produce clear water E Let students compare different modelling approaches: -compare the black-box modelling approach with the mechanistic approach	IV | TLA 18 -19 Factsheet	O n l y -Evaluate the predictive value of the constructed model on the reliability (number and accuracy of the measurements) and validity (tested range, industrial vs. laboratory setting) -Identify the black-box modelling approach and reflect on pro and cons: • Quick and cheap O n l O n l • Suitable to describe process behaviour • No solid mechanistic foundation y y
					-Evaluate the constructed model and formulate future research
					-Formulate a generic modelling procedure for input-output
					systems

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  Figure3. Conceptualised scheme of the design principle of 'content modelling'. The capital letters connect strategy components with expected pedagogic effects and underlying arguments. The numbers in the 'Pedagogic effects' box refer to the epistemic notions as described in the theoretical framework.ArgumentsA Previous applications of the curriculum unit have proved that visualisation of the modelling problem evokes students' notions concerning a modelling procedure. By conceptualising the modelling problem, students grasp the broader applicability of the modelling procedure to be learned. B It is found that supplying students with an advanced organiser informs students' views on the activities to conduct and (type of) knowledge to learn/apply. Previous applications of the curriculum unit have revealed that a worked-out analogous modelling problem, originating from the authentic practice at hand, serves as a suitable organiser. It provides students with a view on epistemic notions to take into account. C It is found that emergent modelling, e.g. gradually formalising empirical data in mathematical models, is a suitable way to guide students through levels of abstraction. Previous enactments of the curriculum unit have revealed that multiple regression should be preceded by single regression to reduce complexity and foster students' understanding. D Previous applications of the curriculum unit have shown that it is necessary to guide students back to the purpose of the modelling. The advanced model features goodness of fit, reliability and validity become meaningful to students when evaluating the model outcomes in a real-world setting. E Previous applications of the curriculum unit have proved that comparing the empirical (black-box) modelling approach with the mechanistic one induces reflection on the pros and cons. Students become aware that multiple modelling approaches exist with different epistemologies. Moreover, students make explicit an empirical (black-box) modelling approach for input-output systems in general.

			Page 40 of 44
	Strategy components		Pedagogic effects
	A Visualise and conceptualise for students the		A Students formulate the purpose of the
	example problem on which they are going to		modelling and activate their prior procedural
	work:		knowledge base (1, 3)
	-demonstrate treatment step		B Students activate and enrich their prior
	coagulation/flocculation		conceptual and procedural knowledge base:
	-conceptualise the modelling problem as an		-mention advanced model features 'goodness
	input-output system		of fit', 'reliability' and 'validity' (4)
	B Supply students with a worked-out analogous		-draw up a modelling procedure (3)
	modelling problem as leading example:		C Students show understanding of (4):
	-factsheet 'modelling trichloromethane removal		-accuracy of measurements
	by activated carbon filtration' F C Involve students in a series of teaching-learning activities emphasising the nature, characteristics o and wording of the model(s) at hand: r P -perform experiments -construct scatter plots -analyse and typify correlations -perform single and multiple regression D Let students apply the constructed model in a real-world setting: e -calculate the dose of coagulant needed to e r produce clear water E Let students compare different modelling approaches:	… expected to result in … … because	-global trends and correlations -lines of best fit through scatter plots -single and multiple regression models -advanced model feature 'goodness of fit' D Students recall and evaluate: -purpose of the regression model (1) -advanced model features 'goodness of fit', 'reliability' and 'validity' (4) E Students reflect on the epistemology of models and modelling (2)
	w e i v e of … R -compare the black-box modelling approach with the mechanistic approach
			O n
			l y

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	'… this scatter plot shows that
	there is little correlation between
	temperature and the end turbidity.
	This can be concluded from the
	low correlation. The process
	variable temperature has hardly
	any/no influence on the end
	turbidity ….'
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

Previous applications of the curriculum unit have proved that discussing different modelling approaches for the problem at hand fosters reflection on the applied approach in learning phase IV. Students become aware that multiple modelling approaches exist

  ArgumentsA Previous applications of the curriculum unit have proved that visualisation of the modelling problem evokes students' notions concerning a modelling procedure. By conceptualising the modelling problem, students grasp the broader applicability of the modelling procedure to be learned. B It is found that supplying students with an advanced organiser informs their views on activities to conduct and (type of) knowledge to learn/apply. Application of previous versions of the curriculum unit has revealed that a worked out analogous modelling problem serves as a suitable organiser. It provides students with a view of the epistemic notions to take into account. C . D It is found that emergent modelling, e.g., gradually formalising empirical data in mathematical models, is a suitable way to guide students through levels of abstraction. E Previous applications of the curriculum unit have shown that it is needed to guide students back to the 'purpose of the modelling'. The advanced model features goodness of fit, reliability and validity become meaningful to students when evaluating the model outcomes in a real-world setting. F Previous applications of the curriculum unit have shown that comparing modelling approaches induces reflection on the pros and cons. Students become aware that multiple modelling approaches exist with different epistemologies. Moreover, students make explicit a black-box modelling approach for input-output systems in general. G

			Page 42 of 44
	Strategy components		Pedagogic effects
	A Visualise and conceptualise for students the		A Students activate their prior conceptual
	example problem on which they are going to		knowledge base and their procedural knowledge
	work:		base (1, 3)
	-demonstrate treatment step		B Students make explicit and build on their prior
	coagulation/flocculation		conceptual knowledge base and their procedural
	-conceptualise the modelling problem as an		knowledge base:
	input-output system		-advanced model features 'goodness of fit',
	B Supply students with a worked-out analogous		'reliability' and 'validity' (4)
	modelling problem as leading example:		-draw up a modelling procedure (3)
	-factsheet 'modelling removal of		C,D Students proceed through the sequence of
	trichloromethane by activated carbon		activities and learn/apply detailed knowledge
	filtration'		until a satisfactory solution for the example
	C Discuss with students different modelling		problem can be presented (2, 4):
	approaches and point out an appropriate		-accuracy of measurements
	one for the example problem at hand		-global trends and correlations
	D Involve students in a series of teaching-		-lines of best-fit through scatter plots
	learning activities emphasising the nature,		-single and multiple regression models
	characteristics and wording of the model(s) at		-advanced model feature 'goodness of fit'
	hand:		E Students evaluate the learned conceptual and
	-perform experiments		procedural knowledge:
	-construct scatter plots -analyse and typify correlations -perform single and multiple regression E Let students apply the constructed model in a	… expected to result	-purpose of the regression model (1) -advanced model features 'goodness of fit', 'reliability' and 'validity' (4) F Students reflect on the epistemology of models
	real-world setting:	in …	and modelling (2)
	-calculate the dose of coagulant needed to
	produce clear water
	F Let students compare different modelling
	approaches and reflect on the assumptions	…
	and estimations made, and the possible	because
	effect of neglected variables: -compare the black-box modelling approach	of …
	with the mechanistic approach

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	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

Table 1 .

 1 Broad outline of the content of the five learning phases in the curriculum unit. The placement of the strategy components of the design principle 'content modelling' are highlighted in bold.

Table 2 .

 2 Overview of the realisation (full, partial, low) on each strategy component and major considerations.

							Page 44 of 44
	Strategy component	# student teams Full Partial Low	Successful	Major considerations Opportunities and/or possible improvements
					Students showed
					understanding of:
	A: Visualise and conceptualise the example problem	9	2	-	-variability through -the purpose of modelling external factors -process variables	-Emphasise the appropriate modelling approach
					and a rudimentary
					modelling procedure
	B: Supply students with a worked-out analogous problem	7	4	-	Students formulated a modelling procedure	-Explicit focus on advanced model features goodness of fit, reliability and validity
					Students did show		-Notion of significance of
	C: Involve students in the nature, characteristics and wording of the models at hand	4	6	1	understanding of: -correlations between quantities -advanced model features goodness of fit and reliability	correlation -Construct a continuous line of teaching-learning activities from single to multiple regression -Emphasise the advanced
							model feature validity
	D: Let students apply the constructed model in a real-world setting	6	3	2	Students were aware of: measurements -considerations regarding predictive value of model -the number and accuracy of the empirical	-Notion that the model needs extensive testing in industrial environment
							-Make explicit learned 'meta
	E: Let students compare different modelling approaches	6	4	1	Students evaluated: -applied empirical modelling approach	knowledge' regarding black-box modelling. -Reflect on the applied modelling approach and make explicit broader
							applicability

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URL: http://mc.manuscriptcentral.com/tsed Email: editor_ijse@hotmail.co.ukInternational Journal of Science Education


K Hoorde

F Leunen

Ph. Amant

T Vandorpe

Dr Ann Smeets
email: ann.smeets@uzleuven.be

B Van Calster

K Van Hoorde

K Leunen

F Amant

Ph Moerman

K Deraedt

O Brouckaert

S Van Huffel

H Wildiers

M R Christiaens

P Neven

Lobular and non-lobular breast cancers differ regarding axillary lymph node metastasis: a cross-sectional study on 4292 consecutive patients

Keywords:

Lobular and non-lobular breast cancers differ regarding axillary lymph node metastasis: a cross-sectional study on 4,292 consecutive patients

Introduction

Lymph node involvement is the most important prognostic factor in breast cancer. It is a multifactorial event determined by tumor and patient characteristics. Variables associated with lymph node metastasis are tumor size, grade, lymphovascular invasion (LVI) multifocality, estrogen receptor (ER), progesterone receptor (PR) and HER-2 status [START_REF] Patani | Predictors of axillary lymph node metastasis in breast cancer: a systematic review[END_REF].

Invasive lobular carcinoma (ILC) is the second most common type of invasive breast cancer after invasive ductal carcinoma and accounts for 8% to 14% of all breast cancer cases [START_REF] Arpino | Infiltrating lobular carcinoma of the breast: tumor characteristics and clinical outcome[END_REF][START_REF] Cristofanilli | Invasive lobular carcinoma classic type: response to primary chemotherapy and survival outcomes[END_REF]. ILC is characterized by small, round cells which infiltrate both the stroma in an Indian file pattern and the adipose tissue without desmoplastic reaction, which makes early diagnosis challenging [START_REF] Arpino | Infiltrating lobular carcinoma of the breast: tumor characteristics and clinical outcome[END_REF]. Absence of E-cadherin expression is considered a characteristic of ILC [START_REF] Cristofanilli | Invasive lobular carcinoma classic type: response to primary chemotherapy and survival outcomes[END_REF]. The diagnosis of ILC carries distinct prognostic and biologic implications. Differences in clinical behaviour between ILCs and invasive ductal carcinomas have been reported [START_REF] Arpino | Infiltrating lobular carcinoma of the breast: tumor characteristics and clinical outcome[END_REF][START_REF] Cristofanilli | Invasive lobular carcinoma classic type: response to primary chemotherapy and survival outcomes[END_REF][START_REF] Pestalozzi | Distinct clinical and prognostic features of infiltrating lobular carcinoma of the breast: combined results of 15 International Breast Cancer Study Group clinical trials[END_REF][START_REF] Tubiana-Hulin | Response to neoadjuvant chemotherapy in lobular and ductal breast carcinomas: a retrospective study on 860 patients from one institution[END_REF][START_REF] Viale | Predicting the status of axillary sentinel lymph nodes in 4351 patients with invasive breast carcinoma treated in a single institution[END_REF][START_REF] Wasif | Invasive lobular vs. ductal breast cancer: a stagematched comparison of outcomes[END_REF]. ILCs have a propensity for multifocal and multicentric distribution but also for bilateralism and a particular pattern of metastatic spread. The incidence of ILC is increasing, especially among postmenopausal women using hormone replacement therapy.

The goal of our study was to investigate the impact of lobular histology on lymph node involvement.

Patients and methods

Design, setting and patients

This was a retrospective cross-sectional study based on data from patients of the Multidisciplinary Breast Centre of the University Hospitals Leuven, Belgium. This database contains data on 4486 female patients with a primary operable invasive breast cancer from the period between 2000 and 2009. Exclusion criteria were [START_REF] Patani | Predictors of axillary lymph node metastasis in breast cancer: a systematic review[END_REF] patients who had neoadjuvant systemic therapy (2) patients who never had breast cancer surgery (3) patients whose pathologic assessment was not performed in the University Hospitals Leuven (4) patients with primary metastatic disease. Of the 4486 patients satisfying these criteria, 19 were ER negative and PR positive. We excluded these patients because there is controversy regarding such tumors. There were 175 patients (3.9% of 4467) with missing data for one or more of the studied variables, most often due to lack of sufficient tumor tissue for correct pathologic assessment of molecular markers. These tumors were excluded for analysis, resulting in a dataset of 4292 tumors. Patients that underwent a sentinel lymph node (SLN) procedure (n=1506) were defined as a subgroup for which a separate analysis was performed following the analysis on all patients.

Variables studied

The following variables were examined: age at diagnosis, maximal microscopic tumor size (largest diameter), histology (ILC vs. non-ILC), worst tumor grade, multifocal disease, axillary lymph node (ALN) status, and the combined ER, PR and HER-2/neu status. Histology and multifocal disease were analyzed as dichotomous variables, the combined receptor status as six-level categorical. Determination of tumor grading and ER, PR, and HER-2/neu status was done according to established procedures, which have been described in more detail elsewhere [START_REF] Neven | Age interacts with the expression of steroid and HER-2 receptors in operable invasive breast cancer[END_REF]. A breast cancer was allocated to the lobular subtype if more than 90% of the tumor mass harboured morphological features of the lobular type; this was always confirmed by lack of E-cadherin expression. Tumors with mixed ductal and lobular phenotype were considered as non-ILC. Tumors with a lobular growth pattern with an E-cadherin positive phenotype were classified as non-ILCs. In case of multifocal disease, the largest tumor was used for size and histotype. In case of bilateral breast cancer, both sides were included as separate tumors.

Lymph nodes in an ALN dissection, were examined by H&E staining using 3 sections per node; according to published guideline, lymph nodes from lobular breast cancers classified as lymph node negative on H&E were additionally stained with epithelial markers. SLN were routinely examined by serial sectioning and IHC. Isolated tumor cells were classified as lymph node negative.

Statistical Analysis

First, ILC and non-ILC tumors are compared with respect to grade, combined receptor status, multifocality, age at diagnosis, and tumor size. Kendall's tau test was used for tumor grade, the Mann-Whitney test (i.e. Wilcoxon rank sum) for age and tumor size, and likelihood ratio chi-square tests for the categorical variables. Next, univariable relationships with ALN are investigated using odds ratios (OR) with 95% CIs. Finally, a multivariable logistic regression model is constructed to assess the independent relationships with ALN using adjusted odds ratios (OR) and 95% CIs. Covariates included in the multivariable analysis were age, tumor size, multifocal disease, tumor grade, and the combined ER, PR, Her-2 status. Linearity for the continuous predictor variables (age, tumor size and tumor grade) was investigated using generalized additive models based on smoothing splines [START_REF] Hastie | Generalized additive models[END_REF]. Goodness-of-fit was assessed using calibration plots that indicate the accuracy of the estimated probabilities of ALN involvement. All analyses were performed using Design package in R version 2.8.1 (www.r-project.org) and using SAS version 9.2 (SAS Institute, Cary, NC, USA).

Results

Univariable analysis

The incidence of ILC observed in the present study was 13%. Table 1 summarizes the patients and tumor characteristics according to the histologic subtype. There were several apparent differences in tumor characteristics between ILCs and non-ILCs. ILCs were more often grade II, had larger tumor sizes, were more often multifocal, tended to be present in older patients and are more frequently ER+/PR+/Her2-.

The results of the univariable prediction of ALN involvement are shown in Table 2. ILCs were slightly more likely to have ALN involvement compared to non-ILCs (42.0% vs. 38.3%; OR=1.17; 95% CI: 0.97-1.40). The overall frequency of lymph node metastasis was 39%. The following factors were univariably related to a higher probability of lymph node involvement: high tumor grade, larger tumor size, multifocal disease, and a ER-/PR-/Her2+ or ER+/PR+/Her2+ profile. No strong effect of age at breast cancer diagnosis on ALN involvement was observed.

The univariable analyses were repeated for the subgroup of 1506 patients that underwent a SLN biopsy, as shown in Table 3. In this subgroup analysis, ILCs were less likely ALN positive than non-ILCs (20.5% vs. 28.3%; OR=0.65, 95% CI: 0.41-1.00). The tumor diameter in the SLN subgroup was significantly smaller than in the non-SLN group with a median tumor size of 17 mm versus 25 mm respectively (p<0.001). Apart from the obvious loss in statistical power, the univariable ORs of age at diagnosis, tumor size, and multifocal disease were comparable to those obtained on all patients. The effect of grade was weaker. The effect of combined receptor status was difficult to compare given the large number of small counts for the SLN subgroup (Table 3).

Multivariable logistic regression

Multivariable analyses were performed using a logistic regression model to determine whether ILC is an independent predictive factor for lymph node involvement.

The results of the regression model and corresponding adjusted odds ratios are presented in Table 4. The multivariable model identified the lobular subtype as less likely to have ALN involvement. The odds ratio for ILCs versus non-ILCs is 0.66 (95% CI: 0.53-0.82) in the multivariable model. The predicted risks of ALN involvement for ILC and non-ILC tumors is illustrated in Figure 1. The plot shows the risk by tumor size while fixing the other covariates at their overall median values: grade 2, unifocal, ER+/PR+/Her-2-tumors for patients who were 58 years old at diagnosis. The plot clearly illustrates that ILC tumors have a smaller chance of ALN involvement.

Multivariable analysis of the subgroup that underwent a SLN procedure confirms the significant lower probability of ALN involvement in ILC tumors as well (OR=0.49; 95% CI: 0.30-0.78). The results are presented in Table 4.

Discussion

The incidence of ILCs observed in our series of 4292 breast tumors (13%) is in accordance with the incidence range of 8-14% reported in the literature [START_REF] Arpino | Infiltrating lobular carcinoma of the breast: tumor characteristics and clinical outcome[END_REF].

Largely in agreement with other series, patient and tumor characteristics at diagnosis are different for ILC than for non-ILC [2-4, 6, 7, 10]. ILC are larger at diagnosis, more often grade II, more often multifocal, are frequently ER+ PR+ Her2-and tend to present in older patients (Table 1). A lower proliferative rate and the lack of a desmoplastic reaction may make ILC impalpable or invisible, explaining why ILC tends to present at older age and as a larger tumor than non-ILCs [START_REF] Cristofanilli | Invasive lobular carcinoma classic type: response to primary chemotherapy and survival outcomes[END_REF].

We observed more ALN involvement in ILC tumors compared to non-ILC tumors (42.0% vs. 38.3%). Prior series comparing lobular and non-lobular tumors are not consistent regarding the incidence of lymph node involvement in lobular tumors. The largest published series including 263408 patients from SEER database, is in line with our results while a large series from Bayer college, including 4140 ILCs and 45169 IDCs, showed no difference in the incidence of lymph node involvement between both groups. Sastre even found a lower incidence of involved lymph nodes in lobular tumors.

In the univariable prediction of ALN involvement in our series, tumor size, multifocality, grade and the molecular markers were significant predictors for lymph node involvement. In contrast, age and lobular histology were not significantly correlated with lymph node involvement in our series.

In the published studies that address the predictors for ALN metastasis in the pre-SLN and SLN eras, tumor size, age, LVI, and histologic subtype were the variables most commonly described as independently associated with ALN involvement. In SLN studies, the variables associated with SLN metastases were, as expected, quite similar [START_REF] Viale | Predicting the status of axillary sentinel lymph nodes in 4351 patients with invasive breast carcinoma treated in a single institution[END_REF][START_REF] Bevilacqua | Doctor, what are my chances of having a positive sentinel node? A validated nomogram for risk estimation[END_REF][START_REF] Voogd | Changing attitudes towards breast-conserving treatment of early breast cancer in the south-eastern Netherlands: results of a survey among surgeons and a registry-based analysis of patterns of care[END_REF].

To evaluate whether the higher incidence of ALN involvement in lobular tumors is a consequence of the lobular histology or of other covariates that interact with ALN involvement, we built a multivariable model correcting for age, tumor size, tumor grade, multifocality, and the combined ER, PR and Her-2 status. All these covariables differed between lobular and non-lobular tumors and correlated with lymph node involvement. The multivariable logistic regression showed a significantly lower incidence of involved lymph nodes in ILCs compared to non-ILCs (OR = 0.66; 95% CI: 0.53-0.82).

Prior multivariable analyses are not consistent regarding ILC as an independent predictor for lymph node involvement. In the bulk of the literature, the lobular subtype is not an independent predictor for lymph node involvement [START_REF] Bevilacqua | Doctor, what are my chances of having a positive sentinel node? A validated nomogram for risk estimation[END_REF][START_REF] Gann | Factors associated with axillary lymph node metastasis from breast carcinoma: descriptive and predictive analyses[END_REF][START_REF] Olivotto | Prediction of axillary lymph node involvement of women with invasive breast carcinoma: a multivariate analysis[END_REF][START_REF] Velanovich | Lymph node metastasis in breast cancer: common prognostic markers lack predictive value[END_REF][START_REF] Voogd | The risk of nodal metastases in breast cancer patients with clinically negative lymph nodes: a population-based analysis[END_REF]. It is important to keep in mind that most of these studies only included patients eligible for the sentinel lymph node procedure (clinical node negative, smaller tumors, unifocal). A lower probability of ALN metastasis for ILC against ductal was shown by Viale in a large series of 4351 although not significant (OR=0.84; 95% CI: 0.65-1.07).

There can be several reasons why ILCs might be less likely ALN positive than non-ILCs.

Firstly, the uniform appearance of bland tumor cells that lack cellular atypia and often have a low mitotic rate, makes the lobular carcinoma cells more difficult to detect in metastatic lymph nodes [START_REF] Sastre-Garau | Infiltrating lobular carcinoma of the breast. Clinicopathologic analysis of 975 cases with reference to data on conservative therapy and metastatic patterns[END_REF]. The reported higher rate of false-negative lymph nodes by histologic examination in lobular carcinomas could cause a general understaging of the histologic type at the time of surgery. In our series IHC was used to assess ALN of all lobular tumors when routine H&E was negative. While contribution of IHC over H&E to improve lymph nodal staging in ILC remains controversial [START_REF] Cserni | Histopathologic examination of the sentinel lymph nodes[END_REF][START_REF] Patil | Keratin immunohistochemistry does not contribute to correct lymph node staging in patients with invasive lobular carcinoma[END_REF], one could assume the possibility of a higher false negative rate in the non-ILC group since IHC was not performed in these cases. Therefore, we might even be underestimating differences in ALN involvement between ILC and non-ILC.

In the subgroup analysis of patients that underwent a sentinel procedure however, all sentinel lymph nodes, ILC and non-ILC, were examined by enhanced pathology (serial sectioning and immunohistochemistry), reducing the chance of underestimating the differences in lymph node involvement due to different pathological assessment.

In this subgroup analysis, patients with a lobular tumor were less likely to be lymph node positive, strengthening our finding that the higher incidence of lymph node involvement in lobular tumors in univariable analysis might be due to confounding variables such as tumor size and occurrence of multifocal disease rather than the lobular subtype as such. This idea is supported by the fact that in the sentinel subanalysis, an inherently and at least partially stratified subgroup concerning tumor size (as mainly small tumors are included for the SLN procedure), SLN involvement is lower in ILC than in non-ILC (20.5% and 28.3% respectively, OR 0.66, 95% CI: 0.41-1.00).

According to the literature, tumor size seems by far the most powerful predictor of ALN involvement [START_REF] Patani | Predictors of axillary lymph node metastasis in breast cancer: a systematic review[END_REF][START_REF] Viale | Predicting the status of axillary sentinel lymph nodes in 4351 patients with invasive breast carcinoma treated in a single institution[END_REF][START_REF] Maiorano | Prognostic and predictive impact of central necrosis and fibrosis in early breast cancer: results from two International Breast Cancer Study Group randomized trials of chemoendocrine adjuvant therapy[END_REF][START_REF] Zurrida | Importance of the level of axillary involvement in relation to traditional variables in the prognosis of breast cancer[END_REF] but histology (ILC versus non-ILC) is at least as important in our multivariate ALN/SLN model as independent prognostic factor (Table 4).

Secondly, genes involved in the metastatic process of ILCs might be different from those involved in non-ILC.

Yoder stated that ILC are thought to arise from the same cell type as infiltrating ductal carcinoma with distinct genomic changes resulting in a different phenotype. Based on these genetic findings, loss of E-cadherin has become a widely used IHC parameter to diagnose ILCs. All our ILC cases were confirmed E-cadherin negative.

However, retention of E-cadherin in ILC is rare and loss of E-cadherin in non-ILCs has been described in 7.2% of non-ILCs [START_REF] Silva | Aberrant expression of E-cadherin in lobular carcinomas of the breast[END_REF][START_REF] Rakha | E-cadherin expression in invasive non-lobular carcinoma of the breast and its prognostic significance[END_REF] without an impact on ALN status and therefore unlikely to affect our results.

Thirdly, ALN involvement may be a function of tumor volume/surface area rather than tumor diameter.

Theoretic models have shown that the propensity for metastases depends on the total number of cells of a tumor combined with the probability of each individual cell to disseminate [START_REF] Michaelson | Predicting the survival of patients with breast carcinoma using tumor size[END_REF]. As tumors of similar diameter are different three-dimensional objects, the number of cells might be better estimated by three-dimensional parameters such as tumor volume than by a linear measurement. Tumor volume or cell load and even microvessel density may differ between ILCs and non-ILCs. Recent data does not support the assumption that three-dimensional measurements have an improved prognostic capability in comparison with traditional onedimensional measurements. Lobular carcinomas however, were excluded from analysis because in many cases, there is no grossly well-demarked mass, and therefore three-dimensional assessment of size is imprecise [START_REF] Verkooijen | Important increase of invasive lobular breast cancer incidence in Geneva, Switzerland[END_REF].

ALN is the best independent prognostic factor for disease-free and overall survival of breast cancer. Our findings of a lower likelihood for cell migration towards lymph nodes are in line with previous and more recent reports which suggested that prognosis is better for patients with ILC than for those with invasive ductal adenocarcinoma after matching for tumor stage [START_REF] Wasif | Invasive lobular vs. ductal breast cancer: a stagematched comparison of outcomes[END_REF][START_REF] Dixon | A comparison of the clinical metastatic patterns of invasive lobular and ductal carcinomas of the breast[END_REF][START_REF] Gudlaugsson | Prospective multicenter comparison of proliferation and other prognostic factors in lymph node negative lobular invasive breast cancer[END_REF][START_REF] Toikkanen | Invasive lobular carcinoma of the breast has better short-and long-term survival than invasive ductal carcinoma[END_REF]. Both types of carcinoma are usually managed with the same adjuvant strategy which may be over-treatment for ILC as they are less systemically aggressive.

In conclusion, the present study demonstrates that the lobular subtype is an independent predictor for lymph node involvement with lobular cancer having a lower incidence of involved lymph nodes. The higher incidence of lymph node metastasis in lobular cancers in univariable analysis is not due to the lobular subtype, but to confounding factors that interact with lymph node involvement.

Figure 1 :

 1 Figure 1: Multivariable logistic regression: Probability of ALN involvement vs. tumor size. Full and dashed plot lines are model predictions for respectively non-ILC and ILC tumors with the following fixed characteristics: grade II, unifocal, ER+, PR+, HER-2-for age 58.

Table 1 : Patient and tumor characteristics according to the histologic subtype Variable Statistic ILC non-ILC P Grade 1

 1

		N (col %)	7 (1.3)	633 (17.0) <0.001 a
	2	N (col %)	476 (85.5) 1445 (38.7)
	3	N (col %)	74 (13.3)	1657 (44.4)
	ER, PR, HER-2
	ER-PR-Her-2-	N (col %)	7 (1.3)	419 (11.2)	<0.001
	ER-PR-Her-2+	N (col %)	4 (0.7)	181 (4.9)
	ER+ PR-Her-2-	N (col %)	56 (10.1)	347 (9.3)
	ER+ PR-Her-2+	N (col %)	2 (0.4)	89 (2.4)
	ER+ PR+ Her-2-	N (col %)	475 (85.3) 2502 (67.0)
	ER+ PR+ Her-2+	N (col %)	13 (2.3)	197 (5.3)
	Multifocality	N (col %)	88 (15.8)	374 (10.0)	<0.001
	Age (years)	Median (IQR)	59 (50-68)	57 (49-67)	0.001 b
	Tumor size (mm)	Median (IQR)	30 (17-50)	20 (14-30) <0.001 b

Table 2 : Univariable prediction of ALN involvement Variable

 2

		Statistic	LN+	LN-	Odds Ratio of LN+ (95% CI)	P
	Histology
	ILC non-ILC	N (row %) N (row %)	234 (42.0) 1430 (38.3) 2305 (61.7) 323 (58.0)	ILC vs non-ILC: 1.17 (0.97-1.40)	0.093
	Grade
	1	N (row %)	161 (25.2)	479 (74.8)	Vs grade 1:
	2	N (row %)	720 (37.5)	1201 (62.5)	1.78 (1.46-2.19)	<0.001
	3	N (row %)	783 (45.2)	948 (54.8)	2.46 (2.01-3.01)
	ER, PR, HER-2
	ER-PR-Her-2-	N (row %)	148 (34.7)	278 (65.3) Vs ER-PR-Her-2-:
	ER-PR-Her-2+	N (row %)	86 (46.5)	99 (53.5)	1.63 (1.15-2.32)
	ER+ PR-Her-2-ER+ PR-Her-2+	N (row %) N (row %)	149 (37.0) 34 (37.0)	254 (63.0) 57 (63.0)	1.10 (0.83-1.46) 1.12 (0.70-1.78)	0.005
	ER+ PR+ Her-2-	N (row %)	1145 (38.5) 1832 (61.5)	1.17 (0.95-1.46)
	ER+ PR+ Her-2+	N (row %)	102 (48.6)	108 (51.4)	1.77 (1.27-2.49)
	Multifocal
	unifocal multifocal	N (row %) N (row %)	1448 (37.8) 2382 (62.2) Multi-vs unifocal: 216 (46.8) 246 (53.3) 1.44 (1.19-1.75)	<0.001
	Age	Median (IQR)	56 (48-66)	58 (49-67)	0.97 per 10 year (0.92-1.02)	0.174
	Tumor size (mm)	Median (IQR)	30 (20-45)	19 (12-26)	1.52 per cm (1.46-1.59)	<0.001

Table 3 : Univariable prediction of ALN involvement for SLN patients only.

 3

	Variable	Statistic	LN+	LN-	Odds Ratio of LN+ (95% CI)	P
	Histology
	ILC non-ILC	N (row %) N (row %)	27 (20.5) 389 (28.3)	105 (79.5) 985 (71.7)	ILC vs non-ILC: 0.65 (0.41 -1.00)	0.056
	Grade
	1	N (row %)	81 (25.4)	238 (74.6)	Vs grade 1:	0.429

Table 4 : Multivariable logistic regression predicting ALN involvement for all patients and for the SLN patients separately ALL PATIENTS SLN SUBGROUP

 4

	Variable	Coefficient	Odds Ratio (95% CI)	P	Coefficient	Odds Ratio (95% CI)	P
	ILC	-0.421	0.66 (0.53-0.82)	<0.001	-0.722	0.49 (0.30-0.78)	0.003
	Grade
	1	Reference level		Reference level
	2	0.314	1.37 (1.10-1.71)	<0.001	0.091	1.10 (0.80-1.51)	0.714
	3	0.550	1.72 (1.38-2.18)		0.151	1.16 (0.81-1.67)
	ER, PR, HER-2
	ER-PR-Her-2-	Reference level		Reference level
	ER-PR-Her-2+	0.436	1.55 (1.06-2.25)	0.005	0.516	1.68 (0.67-3.98)	0.183
	ER+ PR-Her-2-	0.350	1.42 (1.04-1.94)		0.557	1.75 (0.95-3.23)

BVC is a postdoctoral fellow of the Research Foundation -Flanders (FWO). KVH is supported by a grant from the Agency for Innovation by Science and Technology (IWT).