Since many years, research has demonstrated children’s difficulties to reason linearly (or proportionally), both from a technical and conceptual point of view. At the same time, recent studies in different countries show that children also tend to over-use the linear model, i.e. they start applying linear/proportional strategies also in non-linear situations.

This symposium brings together research findings on the development of students’ reasoning in linear and non-linear situations, and more specifically on their tendency to over-use linearity. The studies include students from various ages (from 5-year olds to adults) and pertain to a variety of mathematical domains. By confronting these different lines of research – each encountering the over-use of linear methods in a particular way –, we will not only show the university and diversity of this phenomenon, but also gain a deeper understanding of the psychological and educational processes involved.

Ebersbach et al. found that even 5-year-old preschoolers already discriminate between linear and non-linear growth processes in inductive reasoning. Nevertheless, they still underestimate non-linear growth in a too linear fashion. 

Van Dooren et al. focus on the solution of word problems by 9-12-year-olds. They convincingly show that the number structure of a missing-value problem strongly affects students’ tendency to (improperly) apply linear methods.

Modestou et al. further develop the knowledge on improper linear reasoning in geometrical problem solving concerning area and volume in 14-15-year-olds. These reasearchers found a weaker impact of the linear model on 15-year old students’ reasoning compared to that of younger students.

Finally, Hadjidemetriou et al. investigated the over-use of linearity in graphical contexts by 15-year-olds and their teachers. They provided strong evidence for the existence of a ‘Linear Prototype’ in students, and even in some teachers.

Many phenomena in everyday life may be conceived of as either linear or non-linear processes: Whereas the total price of chocolate bars is usually a linear function of the total number of chocolate bars one buys, the thickness of a folded paper increases non-linearly with the number of folds.

The present study investigated whether 5-year-olds (N=54) are able to differentiate between both types of processes. Children forecasted linear and quadratic growth in an inductive reasoning task, in which the information amount initially provided, was systematically varied.

As a result, the majority of children assumed a rule-based process even if only minimal information was given. Furthermore, they discriminated between linear and quadratic growth by estimating the first one correctly with lower magnitudes than the latter one. Nevertheless, the curve of their estimations exhibited for both linear and quadratic growth a linear shape.

Combining these finding with those of earlier research in this domain, it might be hypothesized that the concept of non-linearity develops in several phases. First, no differentiation between linear and non-linear processes will take place. Thereafter, estimations of non-linear processes will become higher than those of linear processes but, nevertheless, will continue exhibiting a linear shape. Later, these estimations will show a non-linear shape, but only a part of the true exponent of the underlying non-linear function is taken into account. Finally, non-linear processes will be estimated with the appropriate magnitudes. However, it was shown that even adolescents and adults achieve this last phase only with simple tasks, whereas in a variety of other tasks this sample grossly underestimated non-linear processes. The development of the concept of non-linearity will be discussed also with regard to people’s ability to estimate linear processes.

Introduction

A variety of phenomena in different domains of life may be conceived of as being based on either linear or non-linear processes. In biological growth, for example, one might consider the propagation of some animals, such as elephants, as linear, getting within a period of several years only one offspring at a time. On the other hand, a population of mice might propagate rather non-linearly with an increasing growth rate.

The main objective of our ongoing research is whether children possess some early knowledge about both linear and non-linear processes and how this knowledge develops. In contrast to linear processes, being characterized by a constant rate of change, we subsume processes that exhibit an in- or decreasing rate of change as non-linear processes. 

Methodology

The present experiment investigated the ability of 5-year-olds (N=54; 32 boys, 22 girls; mean age: 5.7 years; range: 4.3 to 6.10 years) to forecast linear and quadratic growth. Tasks were presented in an inductive reasoning format, in which only the growth quantities of the first steps of each growth process were shown. The general rule these quantities were connected with had to be detected by the children themselves and to be applied in order to forecast future quantities. In addition, the initially provided quantity of information (i.e., the number of growth steps presented) was varied to examine its effect on the quality of forecasts. Growth processes were embedded in a story in which a population of rabbits propagated either in a linear or in a quadratic manner. The quantities were displayed as filling heights of tubes in order to avoid the involvement of numbers or counting. Instead, children’s intuitive knowledge was aimed to be tapped.

The following hypotheses were pursued: (1) Provided with only minimal information (i.e., information about the growth quantities of the first and second step of the growth process), children would, if ever, assume a linear rule, which has been shown to serve often as default function in problem solving tasks (e.g., De Bock, Van Dooren, Janssens, & Verschaffel, 2002). (2) More information would lead to more appropriate forecasts and to the discrimination between linear and quadratic growth, if 5-year-olds were already able to.

In order to determine whether 5-year-olds already possessed both an early linearity and a non-linearity concept, the following indicators were used. First, it was examined whether children discriminated by means of their estimations between linear and quadratic growth. Second, the curve shapes of their estimations were investigated, which should exhibit either a linear or a non-linear shape. Finally, the mathematical appropriateness of estimations was examined.

Results and conclusions

Results revealed that the very majority of 5-year-olds, even when presented with only minimal information (i.e., the growth quantities of the first and second step of the growth process), assumed a systematic, rule-based process. In line with our hypotheses, their estimations of both growth types fitted in the minimal information condition a linear model best. This was true even when the non-linearity of quadratic growth became more salient by showing the first four steps of the growth process. This suggests that a linear function is not only a default function, serving to solve this kind of tasks, but that non-linear growth was also generally estimated in a linear manner. Though additional information about the growth process yielded more accurate estimations in terms of smaller deviations from the correct values, the detection of non-linearity in quadratic growth was not fostered.

Nevertheless, children discriminated between linear and quadratic growth. When presented with the first three steps of the growth process, they succeeded to forecast quadratic growth correctly with higher magnitudes than linear growth. This leads at least to two assumptions. First, children as young as 5-years old are sensitive towards deviations from linearity and seem to differentiate between both kinds of processes by assuming linear growth to result in smaller magnitudes than quadratic growth. Second, they nevertheless failed to produce non-linearly shaped forecasts of quadratic growth and also underestimated this process significantly.

Discussion

Relating the results to those of previous research examining the judgment and forecast of linear and non-linear phenomena, a common developmental trend with regard to the non-linearity concept becomes apparent. In the very beginning, young children appear to differentiate not at all between linear and non-linear processes, forecasting the two processes similarly (Ebersbach & Wilkening, in press). Later, young children start to discriminate between linear and non-linear processes by means of their estimations. They seem to conceive non-linearity as a process yielding high magnitudes, whereas they estimated it rather linearly. In a next phase, children do not only estimate non-linear processes with higher magnitudes but their estimations exhibit also a non-linear shape. However, the exponent of the underlying function is only partly taken into account (Ebersbach, Lehner, Resing, & Wilkening, 2006). Finally, non-linear processes are estimated correctly concerning both the magnitude of estimations as well as their curve shape. Recent research nevertheless showed that even adolescents and adults achieve this last phase only in relatively simple tasks, that is, in tasks in which non-linearity is a salient characteristic or tasks that deal with only moderate non-linear growth, resulting in rather concise magnitudes (Ebersbach & Wilkening, in press). In other tasks, in which non-linearity is an underlying feature of the task that needs to be discovered by participants themselves (De Bock, Verschaffel, & Janssens, 1998; Van Dooren, De Bock, Depaepe, Janssens, & Verschaffel, 2003) or in tasks involving rapidly developing non-linear processes that result within a short time in remarkable magnitudes, even adults fail to estimate non-linearity appropriately but remain, instead, in the previous phase by taking only a part of the actual exponent into account (e.g., Wagenaar & Sagaria, 1975; Wagenaar & Timmers, 1978, 1979).

Previous research showed that primary school pupils over-use proportional methods especially when solving non-proportional missing-value word problems­. The current study examines whether the numbers appearing in these word problems partly explain this phenomenon. In most previous studies, the numbers in the problems formed integer ratios (i.e., the outcome could be obtained by making an integer multiplicative jump). This may have stimulated pupils to use proportional methods, also in cases where these are inappropriate.

A test containing proportional and non-proportional word problems was given to 508 4^th to 6^th graders. Numbers in these problems were experimentally manipulated so that the ratios were sometimes integer and sometimes not. For example, a non-proportional problem with integer ratios was:

Ellen and Kim are running around a track. They run equally fast, but Ellen started later. When Ellen has run 16 laps, Kim has run 32 laps. When Ellen has run 48 laps, how many has Kim run?

while the version with non-integer ratios was

Ellen and Kim are running around a track. They run equally fast, but Ellen started later. When Ellen has run 16 laps, Kim has run 24 laps. When Ellen has run 36 laps, how many has Kim run?

Correct (additive) reasoning is comparably easy for both versions, but proportional reasoning is far less evident (though still possible, of course) for the version with non-integer ratios.

As expected, problems with integer ratios elicited much more (inappropriate) proportional methods in pupils than non-integer ratios. This effect was particularly strong in 4^th grade, dropped in 5^th grade to disappear in 6^th grade.

Theoretical, methodological, and practical implications of these findings are discussed.

Introduction

Proportionality (or linearity) is a key concept throughout mathematics education. Research shows, however, that pupils strongly tend to apply linear/proportional strategies also when this is inadequate. This phenomenon was demonstrated in different mathematical contexts.

Recently, we found that pupils’ tendency to over-rely on proportionality on arithmetic problems already occurred as early as in 3^rd grade, and then considerably increased until 6^th grade to decrease afterwards (Van Dooren et al., 2005). For example, the ‘runner’ problem

Ellen and Kim are running around a track. They run equally fast, but Ellen started later. When Ellen has run 16 laps, Kim has run 32 laps. When Ellen has run 48 laps, how many has Kim run?

was answered correctly (“48+16=64 laps”) by 56% of 3^rd graders, 29% of 6^th graders and 45% of 8^th graders, while an opposite trend was observed for the inadequate proportional responses (“48×2=96 laps”).

So far, research on the over-use of proportionality overlooked one issue: the nature of the numbers in the problems, and their possible impact on pupils’ responses. The proportional reasoning literature (e.g., Hart, 1984; Karplus et al., 1983) indicates that inexperienced proportional reasoners sometimes are affected by the numbers in missing-value problems. If the numbers form non-integer ratios (e.g., “Mixture A has 4 oranges to 10 parts of water. Mixture B tastes the same and has 6 oranges. How many parts of water does it have?”, in which the multiplicative jumps 4à6 and 4à10 are both non-integer) they more often respond ‘additively’ (“10+2=12 parts”) instead of proportionally than when the multiplicative jumps are integer.

The present study applies this finding to the over-use of proportional methods in non-proportional problems, aiming to find out whether the ‘easy’ integer ratios in Van Dooren et al.’s study (2005) are an additional ‘seducing’ element.

Method

508 4^th, 5^th and 6^th graders solved 8 missing-value word problems (identical to those used by Van Dooren et al., 2005): 2 proportional problems (where proportional strategies are correct) and 3 non-proportional types (requiring other strategies): Additive problems (e.g., the above-mentioned ‘runner’ item), constant problems (with no relation between the variables) and affine problems (i.e., a f(x)=ax+b model).

The numbers in the problems were manipulated so that when reasoning proportionally, one needs to work with integer ratios (I-version) or with non-integer ratios (N-version). For each problem pupils received at random the I- or N-version.

For example, compared to the I-version of the ‘runner’ item given above, the N-version was:

Ellen and Kim are running around a track. They run equally fast, but Ellen started later. When Ellen has run 16 laps, Kim has run 24 laps. When Ellen has run 36 laps, how many has Kim run?

Correct (additive) reasoning for the I-version and N-version is equally easy, but proportional reasoning is far less easy (though still possible, of course) for the N-version.

Pupils’ answers were categorised as correct (C), proportional (P), or other error (O). For proportional problems, only C and O were used as categories.

Hypotheses

With respect to the proportional problems, we expect a similar effect as reported by, e.g., Hart (1984) and Karplus et al. (1983): I-versions will elicit more correct answers than the N-versions, and this effect will be stronger in younger, less experienced proportional reasoners.

Our main interest is, however, in the non-proportional problems. As argued above, number characteristics may have a similar effect on the improper use of proportionality as on the proper use of proportionality. So, we expect that N-versions of non-proportional problems will elicit fewer unwarranted proportional (P) answers than I-versions, and, again, that this effect will be stronger in the younger pupils.

Results

Figure 1 shows percentages of correct answers to the proportional problems. As expected, N-versions elicited less correct answers (56.8%) than I-versions (82.1%), c²(1,N=508)=52.51, p<.0001. The difference between both versions was very strong in 4^th grade, smaller in 5^th grade, and no longer significant in 6^th grade (see also Figure 1), ‘number type’ × ‘age’ interaction: c²(2,N=508)=166.59, p<.0001.

This study explores the different dimensions of students’ abilities in geometrical problem solving concerning area and volume, with special emphasis on students’ behaviour while handling pseudo-proportional problems and on alteration of this behaviour with students’ age.

Students in 9^th and 10^th gradewere given a test involving three types of problems: usual computation problems, pseudo-proportional problems and impossible ones. The examined grades were deliberately chosen because they belong to two different educational levels in Cyprus with different approaches in the teaching of geometry. Confirmatory Factor Analysis was used for the analysis of the data in order to explore the structural organization of the various dimensions of geometrical problem solving in each age group. This statistical technique was employed as the application of other analyses (MANOVA) did not show a variation in students’ mean performance in pseudo-proportional tasks, with respect to grade level. Therefore, a more comprehensive analysis was necessary to further illuminate the phenomenon of pseudo-proportionality, based on the conjunctions of students’ handling the pseudo-proportional problems and the problems of different reasoning requirements on the same content.

Results suggest the existence of two different structural models - one for each age group - for the interpretation of students’ geometrical problem solving behaviour. The students of both grades did not approach the three types of problems in the same way but used different reasoning processes. For the younger students the pseudo-proportional problems were of a similar nature as the usual problems and therefore, composed a common factor. On the other hand, the pseudo-proportional problems formed a factor of their own in the case of the older students, making obvious a different reasoning approach compared to the usual ones. This is indicative of the weaker impact of the linear model on 15-year old students’ reasoning compared to younger students’ thinking.

Introduction

Geometry has always been a privileged domain of research among psychologists and researchers of Mathematics Education. Students’ abilities in problem solving concerning these concepts of length, area and volume have been studied extensively in the recent years (De Bock, Verschaffel, & Janssens, 1998; Modestou & Gagatsis, 2006, 2007) under the perspective of the phenomenon of pseudo-proportionality. This phenomenon refers to students’ tendency to apply the linear model in non-proportional situations of area and volume, which involve an enlargement or reduction of the figure’s size in relation to its side length. The objective of the present paper is to illuminate this phenomenon by articulating a structural model related to geometrical problem solving and more specifically to the abilities involved in solving area and volume problems. A main concern is also to compare the structure of the aforementioned model between students in 9th and 10th grade. The two grades examined were deliberately chosen based on the fact that grade 9 and grade 10 belong to two different educational levels in Cyprus with different approaches in the teaching of geometry. In particular, in grade 10 the Euclidean geometry is taught in a more systematic manner as a continuation and extension of the teaching of the particular topic in grade 9. 

Method

The sample of the study consisted of 653 Cypriot students of grade 9 and 10 (14- and 15-year olds). The students were administered a 40 minutes test that consisted of 9 geometrical word problems concerning the perimeter, the area and the volume of different figures. The test involved three types of tasks: usual computation problems, pseudo-proportional problems and impossible ones.

Results

A 2 (the two age groups) ´ 3 (appropriate-usual vs. pseudo-proportional vs. impossible problems) multivariate analysis of variance (MANOVA) was performed to specify the possible influence of the task variable, that is the type of the problems and the subjects’ variable, that is age, on problem solving. The effect of age F_(1,651)=3.121, p=.078, η² =0.005 was not significant, indicating that mean performances of the two age groups were not significantly different. The main effect of the type of tasks was very strong F_(2,650)= 1086.906,  p<0.0005, η² =0.770, revealing that the appropriate-usual problems were significantly easier than the pseudo-proportional problems, which in turn were easier than the impossible problems.

A more advanced analysis of the data called Confirmatory Factor Analysis (CFA) was employed in order to explore the structural organization of the various dimensions of geometrical problem solving, examined here, in each age group. Bentler’s (1995) EQS program was used for the analysis. The findings of this study illustrate that despite the invariance of the students’ mean performance in problem solving with respect to grade level, the structure of a model involving problem solving of pseudo-proportional tasks in combination with impossible and typical tasks on area and volume does show variance between grade 9 and 10. Students of the two different grades responded to the given set of tasks in a manner that resulted in different dimensions of geometrical problem solving.

The tendency of the 9^th grade students to apply the proportional model was strong and for these students the pseudo-proportional problems were almost of the same nature as the usual measurement problems. For this reason the three usual problems and the three pseudo-proportional ones constituted one factor. On the other hand, the present study indicates that students of 15 years of age started to differentiate their ways of interpreting and understanding this kind of problems. They appeared to approach the pseudo-proportional problems by activating different processes from the ones they used in usual problems, something that is clearly denoted by the fact that the pseudo-proportional problems alone constituted a distinct factor. This suggests that 15-year old students may have confronted the non-proportional problems in a less superficial way relatively to the younger students, indicating the weaker impact of the linear model on their reasoning. In other words, they started to question to some degree the deep-rooted linear model’s applicability in all the types of measurement problems.

The impossible problems were also handled differently across the two grades. Students of grade 9 dealt with these problems differently from the other two types of problems, as they formed a distinct factor. However, the abilities of 10^th grade students to tackle the impossible problems and to resolve the usual tasks established a common factor.  A hypothetical explanation for this finding is that older students were more familiar with the structure of the impossible problems, because of their more systematic involvement with problems of Euclidean geometry.

Discussion

The above results suggest that the type of reasoning that the particular geometrical tasks require does have an effect on students’ problem solving processes. However, despite the significant variation of these effects across the two age groups, certain commonalities appeared in the models of the groups, revealing that some aspects of their ways of thinking in problem solving remained invariant with development.

In the models of the both groups lower factor loadings of the solutions of usual computation tasks, relatively to the approaches in dealing with the pseudo-proportional and impossible problems, were observed. The different nature and reasoning requirements of the usual tasks compared to other two types of tasks may provide an explanation for this difference. It appears that even though good geometrical knowledge and competence in employing formal strategies, such as using formulas, meets the requirements of the usual computation tasks, these are not sufficient for the solution of the pseudo-proportional or the impossible problems. Solving the pseudo-proportional problems requires students’ overcoming of the illusion of linearity, while tackling the impossible problems requires sensible and realistic considerations for the interpretation of the situations involved and the breach of inadequate habits and beliefs about solving word problems, such as being obliged to provide an answer to all the problems given to them.

This paper describes pupils’ and teachers’ performance on a task designed to diagnose the Linearity Prototype (LP). The ‘Charity’ item required pupils to draw a graph showing that after a Charity event ‘The more people help, the sooner we finish tidying up’. The whole test, consisting of 29 items, was administered to 425 pupils and their 12 teachers. Results showed 80% of 14-15-year olds exhibiting the LP in the Charity item with no significant differences among year 9 and 10 pupils. Pupils’ responses were confirmed and enriched through group interviews in order to analyse the thinking process behind their inappropriate linear reasoning. 18 pupils were interviewed. The results indicated some mismatch between pupils’ reasoning and their graphs with the linearity answer being ‘conceptually’ but not ‘realistically’ correct for some of them. Pupils’ responses also confirmed that, under test conditions, they answer questions superficially without engaging in deep mental processes, and that they fall into the ‘linearity trap’ because they inappropriately apply the methods they used to draw linear graphs to unsuitable situations.

The teachers were asked to rate the difficulty of these items on a five-point scale, answer the questions and predict possible difficulties of their pupils. Teachers’ ratings were analysed using the Inverse Partial Credit Model. Teacher’s difficulty estimates were compared to pupils’ actual difficulty estimates and discrepancies were detected. Although teachers accurately predicted pupils’ difficulty in most of the items of the test, their prediction for this particular item was significantly underestimated. Semi-structured interviews with the teachers indicated that some carry the LP themselves. They were generally not aware that children tend to exhibit this prototype, which explains their inaccurate rating of the items difficulty for the students.

Introduction

The everyday (i.e., non-academic) development of a concept is based on examples encountered in practice and mediated by the associated discourse. When the properties that construct the concept are common to the prototypical, but not all members of a class that defines the concept, then these properties will be more representative than others and may become considered necessary (Lakoff, 1990). Therefore, a prototype invokes a collection of properties that accurately describes only ‘the best examples’ of that concept. Smith and Medin, (1981) state that, often “people use non-necessary properties in categorization because that is mainly what prototypes are made of”. Learners prototypically assume such ‘unnecessary properties’ when drawing graphs, e.g., graphs should be linear or smooth, since these graphs are most frequently encountered.  Research showed that students use linear functions as their examples (Hershkowitz, 1989; Schwarz & Hershkowitz, 1999). In this study, we investigate the connection between pupils’ and their teachers’ use of the Linearity Prototype (LP).

Method

Our diagnostic instrument (see Hadjidemetriou & Williams, 2002) was distributed to 425 pupils and contained diagnostic items from graphicacy literature. Items related to various errors, including the LP (Leinhardt et al., 1990). Two items diagnosed the LP.  An adapted test was given to their teachers (N=12). They were asked to answer all items and also:

1.       predict how difficult their children would find the items (on a five-point scale)

2.       suggest likely errors and misconceptions in the children

3.       suggest methods/ideas to help pupils overcome these difficulties

Pupils’ results (N=425) were analysed using the Rasch model in order to get a pupil difficulty estimate for each item. Teacher’s predictions of item difficulties were subjected to an Inverse Partial Credit Analysis (Hadjidemetriou & Williams, 2004), providing an item-perception difficulty measure for each item. Consequently pupils’ and teachers’ item estimates can be compared and Teacher-Pupil Discrepancies (TPD) can be detected (i.e. items pupils found easy and teachers found difficult and vice-versa).

Results and discussion

This paper focuses the ‘Charity’ item, designed to elicit pupils’ tendency towards linearity. This item – inspired by Swan (1985) - required pupils to draw a graph showing that after a Charity event ‘The more people help, the sooner we finish tidying up’ (with the x axis representing ‘the number of people’ and the y axis ‘the time to finish tidying up’). Results indicated that the tendency to draw a linear graph was present in both year 9 and 10 groups with no significant differences between both. A high percentage of pupils (80%) fell into the ‘linearity trap’ (De Bock et al., 1998) with only 15 (3.5%) pupils of high mean ability (1.43 logits) giving a prototype-free answer (not linear and not passing through the origin). Partial credit was given to pupils sketching linear graphs with negative gradients. Additionally, coding distinguished among those drawing linear graphs with negative gradients cutting the axes (32.7%, of mean ability 0.95) or not (5.6% of mean ability 1.22 logits, i.e. not much less able than those answering correctly). The largest group of the pupils (41.6%) of significantly lower ability (average 0.00 logits) drew a linear graph through the origin similar to the line ‘y=x’.

To get more insight into the thinking process of pupils using the linear model, 18 pupils were interviewed. This happened in groups of 2 or 3 in order to examine the dynamic of their discourse so in a group there were pupils with a variety of answers. Most pupils who answered ‘y=x’ changed their answer to linear with negative gradient after discussing with the interviewer and their peers. The interviews of the weak pupils who drew the ‘y=x’ graph revealed that they did not read the question properly with some of them admitting applying the ‘more A-more B’ strategy. For the more able pupils who drew a linear graph with a negative gradient, data indicated two main response categories: those whose reasoning was consistent with their linear graph, and those that was not. 3 pupils drew a linear graph but explained that ‘every person saved two minutes, so one [person] across, two [minutes] down’, which is conceptually, but not realistically, correct. 10 pupils drew linear graphs and explained that ‘if they [people] are doing the same amount of work, it would go down the same way’ or ‘it [the graph] is straight because … all people tidy at the same speed’. 3 pupils admitted that under test conditions they do not think of the problem in great detail and one pupil mentioned he drew a linear graph because he applied a strategy for drawing linear graphs (plotting only two points) instead.  

To summarise, the data confirmed previous literature that often pupils approach such items superficially (De Bock et al., 1998, 2002) and answer demanding test questions spontaneously. Pupils attributed their failure to themselves (not reading the question properly), to the ‘test conditions’ or to their selection of inappropriate strategies. The interviews confirmed the existence of the linearity tendency among 14-15-year olds, and provided a lot of information about pupils’ reasoning. Some of the pupils were able to correctly interpret their linear graph (with negative gradient) and adapt their reasoning accordingly. We next ask: a) are teachers’ able to predict the difficulty of such items and b) is the linearity prototype anticipated as a likely pupil response by their teachers?

The quantitative analysis also detected TPD, i.e., differences among teachers’ predictions of item difficulty and pupils’ actual accuracy. Results showed that teachers particularly underestimated the difficulty of this item. Teachers’ ratings and comments were examined more closely. Some of the teachers’ graphs revealed that LP was dominant in three of the teachers’ own answers as well, explaining their perception of the item as relatively easy. Only one of the teachers anticipated the LP. We suggest this is a major weakness in pedagogical knowledge, and that such test items are valuable in teacher education courses and subsequent classroom discussions.