1

Reasoning About Shape as a Pattern in Variability

Arthur Bakker

Freudenthal Institute

Utrecht University

The Netherlands

arthur@fi.uu.nl

SRTL 3, July 23-28, Lincoln, Nebraska

2

Reasoning about shape as a pattern in variability

Background of the research

The first time I visited an American classroom I attended a statistics lesson in grade 5. When the

teacher asked a question that sounded statistical but did not require a measure of center, a boy

thoughtlessly muttered “meanmedianmode,” as if it were one word. My impression was that

these students had been drilled to calculate mean, median, and mode, and to draw bar graphs, but

did not use their common sense in answering questions. This small incident exemplifies what a

litany of research in statistics education reports on: too often, students learn statistics as a set of

techniques that they do not apply sensibly. Even if they have learned to calculate mean, median,

mode, and to draw histograms and box plots, they mostly do not understand that they can use a

mean as a group descriptor when comparing two data sets—to give just one example that is well

documented (Konold & Higgins, 2003; McGatha, Cobb, & McClain, 2002; Mokros & Russell,

1995). The problem that students do not know well how to use statistical techniques in data

analysis is not typically American; it also applies to the Dutch context, but to a lesser extent. The

reason for this is probably that Dutch students learn most statistical concepts and graphs such as

median, mode, histogram, and box plot about three years later than in the USA.

Despite differences between the curricula in different countries, the underlying problem remains

the same: students generally lack the necessary conceptual understanding for analyzing data with

the statistical techniques they have learned. Konold and Pollatsek (2002) argue that students need

to develop a conceptual understanding of signal and noise in order to understand what an average

value or a distribution is about in relation to the variation around that value or distribution. In this

context, it is useful to distinguish two types of patterns in variability, or signals in noisy

processes. First, the signal can be a true value with error as noise around it. Such signals are

apparent in repeated measurements of one item (Petrosino, Lehrer, & Schauble, 2003). The

“center clump” is then an indication of where the true value probably is. Second, the signal might

be a distribution, such as the smooth bell curve of the normal distribution, with which we model

data. The noise in that case is the variation around that smooth curve. In either type of pattern, it

is evident that students need good conceptual understanding before they can recognize signals in

noisy processes. The focus of recent research seems to shift to the key concepts that students

3

need to develop in order to analyze data in a meaningful way. This paper focuses on the key

concepts of variability, sampling, data, and distribution. The main question of the research

presented here was, how do we promote coherent reasoning about variability, sampling, data,

and distribution in a way that is meaningful for students with little statistical background?

As a partial answer to this question, two instructional activities that support such reasoning are

presented in this paper. The first, growing a sample, is an elaboration of what was presented at

SRTL 2 (Bakker, 2002). It is similar to what Konold and Pollatsek (2002) suggest as a way to

give students the opportunity to experience stable features in variable processes. The second

activity, reasoning about shape, is a sequel to the growing samples activity in the sense that

students came to reason with the shapes they predicted during the growing samples activity for

very large samples of weight data (pyramid, semicircle, bell shape). Two skewed shapes were

added to direct the discussion toward skewness.

Key concepts of statistics at middle school level

There is no unique list of the key concepts of statistics. The list for the middle school level

presented in this section is inspired by a list of Garfield and Ben-Zvi (in press) for statistics

education in general. No particular order is suggested, because these key concepts are only

meaningful in relation to each other (cf. Wilensky, 1997).

• Variability

• Sampling

• Data

• Distribution

• Covariation1

Variability, the phenomenon that things are variable, is at the core of all statistical investigation

(Wild & Pfannkuch, 1999). If students do not expect any variability in a particular context, such

as the life span of batteries or other industrial contexts, they neither have an intuition of why one

would take a sample or look at a distribution. The terms “variability” and “variation” are

1 Covariation is not discussed further in this paper because it is not addressed in the Dutch mathematics curriculum for secondary education. See for instance Cobb, McClain, & Gravemeijer (2003).

4

sometimes used interchangeably, but here “variability” is used for the general phenomenon of

being variable, and “variation” for described or even quantified variability (Reading &

Shaughnessy, in press). In this paper, “variation” is taken as a characteristic of the distribution of

a data set, and “spread” is used as an informal student term for “variation.”

Sampling. To describe or predict a particular variable phenomenon, we need data and data are

mostly created by taking a sample. Students’ notions of sampling have been studied at the middle

school level, but not extensively. Though students seem to have useful intuitions about sampling,

it turns out to be a complex notion for students to learn (Jacobs, 1999; Konold & Higgins, 2003;

Rubin, Bruce, & Tenney, 1990; Schwartz, Goldman, Vye, Barron, & Vanderbilt, 1998; Watson,

2002; Watson & Moritz, 2000).

Data. Sampling and measurement lead to data, that is numbers with a context (Moore, 1997). As

indicated by Latour (1990) and Roth (1996), data have a history and a meaning in a context.

Understanding the key concept of data includes insight into why data are needed and how they

are created. This implies that the idea of data relies on knowledge about measurement. Too often,

however, data are detached from the process of creating them (Wilensky, 1997), so many

researchers and teachers advocate that students collect their own data (National Council of

Teachers of Mathematics, 2000; Shaughnessy, Garfield, & Greer, 1996). This is time consuming,

so “talking through the process of data creation,” as Cobb (1999) calls it, can sometimes

substitute for the real creation of data. A class discussion on why and how data are created is

necessary for students to experience a data set as meaningful. In fact, such a process of talking

through the data creation process is a way to address the sampling issue. For analyzing the data

we can manipulate data icons in a graph to find relations and characteristics that are not visible

from the table of numbers (Lehrer & Romberg, 1996).

Distribution. The problem with finding such relations and characteristics, however, is that many

students tend to perceive data just as a series of individual cases, and not as a whole that has

characteristics that are not visible in any of the individual cases. Hancock et al. (1992) note that

students need to mentally construct such an aggregate before they can perceive a data set as a

whole. Many researchers encountered the same problem and experienced the persistency of it

5

(Ben-Zvi & Arcavi, 2001; Wilensky, 1997). To move from a case-oriented view to an aggregate

view on data, students need to develop a conceptual structure with which they can conceive data

sets as aggregates. The concept of distribution is such a structure (Cobb, 1999; Gravemeijer,

1999). In the words of Petrosino et al. (2003, p. 132): “Distribution could afford an organizing

conceptual structure for thinking about variability located within a more general context of data

modeling.” Of course, distribution is a very advanced concept (Wilensky, 1997) that, in its full

complexity, is far beyond the scope of middle school students. Nevertheless, it is possible to

address the issue of how data are distributed from an informal situational level onwards by

focusing on shape (Bakker & Gravemeijer, in press; Cobb, 1999; Russell & Corwin, 1989). In

the research of Cobb, McClain, and Gravemeijer (2003), students came to reason with hills to

indicate majorities of data sets. Bakker and Gravemeijer (in press) report on how students

reasoned how a “bump” changed if older students would be measured, and what would happen

with the bump if the sample would grow. Students in this Dutch research tended to structure a

real or hypothetical data set into three categories of low, “average,” and high values. In other

words, they soon came to see a pattern in the variability of different phenomena such as weight,

height, and wingspan of birds. These “average” groups and “majorities” correspond to what

students in other studies called “clumps” or “clusters.” In fact, students use such “modal clumps”

(Konold et al., 2002) to indicate the center of a distribution and they can use the range of these

clumps to convey something about the spread of the data. This brings us to two core aspects of

distribution, center and spread.

Center. Traditionally, measures of center such as mean and median are taught before students

have developed a notion of center. Zawojewski and Shaughnessy (2000) reason that students can

find mean and median difficult because they have not had sufficient opportunities to make

connections between center and spread; that is, they have not made the link between measures of

central tendency and the distribution of the data sets (p. 440). It is only possible to choose

sensibly between mean and median if one takes the context and the distribution of the data into

account. If the distribution is skewed and there are outliers, one might choose the median as a

measure of center, but there can also be reasons to use the mean—depending on the context. It is

therefore important to give students opportunities to learn ways to describe how data are

distributed, perhaps even before teaching more formal measures of center.

6

Spread or variation. Variation did not receive much attention in statistics education research

until about 1997. The reason for neglecting variation in education has probably been the focus on

the measures of center (Reading & Shaughnessy, 2000; Shaughnessy, Watson, Moritz, &

Reading, 1999). Noss, Pozzi, and Hoyles (1999) report on nurses who view variation as crucial

information when investigating blood pressure, and Lehrer and Schauble (2001) report on

variation in a physical and biological contexts. For an overview of the research literature on

variation see, for example, Meletiou (2002).

The rationale of the learning trajectory used in the research presented in this paper was that

students would be engaged in coherent reasoning about variability, data, sampling, and shape

from the outset. The learning process aimed at was characterized as “guided reinvention”

(Freudenthal, 1991). Students were stimulated to contribute their own ideas, strategies, and

language in solving statistical problems (reinvention), but they were also provided with

increasingly sophisticated ways to describe how data were distributed and to characterize data

sets (planned guidance). The software tools used were two so-called “minitools,” which Bakker

and Gravemeijer (in press) also used in teaching experiments in grade 7. In grade 7, it turned out

to be difficult for students to reason with shape, except for the high achievers who reasoned with

bumps. The next teaching experiment was therefore carried out in a higher grade. The next

section provides information on the methodology used and the subjects involved in this eighth-

grade teaching experiment. After the analysis of the two instructional activities, implications for

research, teaching, and assessment are discussed.

Methodology and subjects

For answering the main question of how coherent reasoning about variability, sampling, data,

and distribution could be promoted, a design research study was carried out. In line with the

principles of Realistic Mathematics Education (Freudenthal, 1991; Gravemeijer, 1994) and the

NCTM Standards (2000), we looked for ways to guide students in being active learners, in this

case dealing with increasingly sophisticated computer minitools and students’ own

representations. Design research typically involves a preparation and a design phase (of

instructional materials for example), teaching experiments, and retrospective analyses (Cobb,

Confrey, diSessa, Lehrer, & Schauble, 2003; Gravemeijer, 1994).

7

Preparation and design phase. In the research presented here, the preparation phase consisted of

the reformulation of a hypothetical learning trajectory (Simon, 1995) that was developed during

design experiments in grade 7 (Bakker & Gravemeijer, in press). In the last seventh-grade

experiment, the activity of growing a sample turned out to be promising as a way to give students

the opportunities to reason about shapes of distributions (in particular “bumps”). This activity

was again used in grade 8 with the expectation that a more advanced type of reasoning about

shape could be fostered than in grade 7. In the seventh-grade study, it turned out that one of the

solutions to assist students overcome a purely case-oriented view of data, and develop an

aggregate view, was to let them predict hypothetical situations without any data. Examples

questions were, “What would the graph look like if a larger sample would be taken? What would

a weight graph of eighth graders look like compared to one of seventh graders?” To stimulate an

aggregate view we so to speak asked about forests instead of the trees and often stayed away

from data. Letting students invent and compare their own graphs had also appeared to be useful

in engaging them in meaningful mathematical activity. Because we conjectured that this was

easier for students to reason about data in well-known contexts, we again chose weight as the

context for the two activities described here.

Teaching experiment and data collection. The collected data include audio and video recordings

of class activities, student work, field notes, and interviews after every second lesson. An

essential part of the data corpus was the set of mini-interviews held during the lessons. Mini-

interviews varied from about twenty seconds to four minutes, and were meant to find out what

concepts and graphs meant for students. This influenced their learning, because the mini-

interviews often stimulated reflection. However, the validity of the research was not in danger,

since the aim was to find out how students learned to reason with shape or distribution, not

whether teaching the sequence in other eighth-grade classes would lead to the same results in the

same number of lessons. The teaching experiment was carried out in an eighth-grade class with

30 students in a public school in the center of the Dutch city of Utrecht in the fall of 2001. It

lasted for ten lessons of 50 minutes each, half of which were carried out in a computer lab. The

students in this study were being prepared for pre-university (VWO) or higher vocational

education (HAVO)—types of education that the top 35-40% of Dutch students attend. The

remaining 60-65% of Dutch students are prepared for other types of vocational education

(VMBO). In the practice of Dutch mathematics education, the school textbooks play a central role.

8

Students are expected to be able to work through the tasks by themselves, with the teacher

available to help them if necessary. As a consequence, tasks are broken down into very small

steps and real problem solving is rare. Students’ answers tend to be superficial because they have

to deal with about eight contexts per lesson (van den Boer, 2003). The students in the class that is

reported on here were not used to whole-class discussions, but rather to be “taken by the hand”

as the teacher called it. The eighth-grade students had had no prior instruction in statistics; they

were acquainted with bar and line graphs, but not with dot plots, histograms, or box plots. In the

research reported here, students already knew the mean from calculating their report scores

(grades), but mode and median were not introduced until the second half of the instructional

sequence after variability, data, sampling, and shape had been topics of discussion.

Retrospective analysis. For the retrospective analysis, the transcripts were read, the videotapes

watched, and conjectures formulated on students’ learning on the basis of the read and watched

episodes. The generated conjectures were tested against the other episodes and the rest of the

collected data (student work, field observations, tests) in the next round of analysis

(triangulation). Then the whole generating and testing process was repeated. This method

resembles Glaser and Strauss’s constant comparative method (Cobb & Whitenack, 1996; Glaser

& Strauss, 1967). About one quarter of the episodes (including those discussed in this paper) and

the conjectures belonging to these episodes were judged by three assistants who attended the

teaching experiment. Only the conjectures that all of us agreed upon were kept, and this was

more than 95%. The interrater reliability was therefore high. An example of a conjecture that

was confirmed was that students tended to group data sets (real or imagined) into three groups of

low, “average,” and high values.

Growing a sample

The overall goal of the growing samples activity as formulated in the hypothetical learning

trajectory was to let students reason about shape in relation to sampling, center, and spread. The

activity of growing a sample consisted of cycles of making sketches of a hypothetical situation

and comparing those sketches with graph displaying real data sets. In the fourth lesson of the

experiment, three such cycles took place as described below.

9

Figure 0. Minitools 1 and 2: value-bar graphs and dot plots with the same data set of battery life

spans.

First cycle of growing a sample

The text of the activity sheet started as follows:

Last week you all predicted graphs for a balloon driver. During this lesson you will get to see

real weight data of students from another school. We are going to investigate the influence of the

sample size to the shape of the graph.

a. Predict a graph of ten data values, for example with the dots of minitool 2.

The sample size of ten was chosen because the students found that size reasonable after the first

lesson in the context of testing the life span of batteries. Figure 1 shows different diagrams

students made to show their predictions: there were three value bar graphs (such as in minitool 1,

Figure 0), eight with only the endpoints (such as with the option to “hide bars”) and remaining

nineteen plots were dot plots (minitool 2).

Rikkert

Christel

10

Figure 1a. Student predictions (Rikkert, Christel, and Susan) for ten data points (weight in kg).

Figure 1b.Three real samples in minitool 2.

To stimulate the reflection on the graphs, the teacher showed three samples of ten data points on

the blackboard (the overhead projector had just broken down) and students had to compare their

own graphs (Figure 1a) with the graphs of the real samples (Figure 1b).

b. You get to see three different samples of size 10. Are they different than your own prediction?

Describe the differences.

The reason for showing three small samples was to show the variation among these samples.

There are no clear indications, though, that students conceived this variation as a sign that the

sample size was too small for drawing conclusions, but they generally agreed that larger samples

were more reliable. There was just a short class discussion on the graphs with real data before

students worked for themselves again. Please note that a grammatical translation into English of

ungrammatical spoken Dutch does not always sound very authentic.

Teacher: We’re going to look at these three different ones [samples in Figure 1b].

Can anyone say something yet? Give it a try.

S1: In the middle one, there are more together.

Teacher: Here there are many more together, clumped or something like that.

Susan

11

[pointing to the middle graph of Figure 1b] Who can mention other

differences?

S2: Well, uh, the lowest, I think it’s all the furthest apart.

Teacher: Those are all the furthest apart. Here they are in one clump. Are there

any other things you notice, Ricardo?

S3: Yes, the middle one has just one at 70. [This is a case-oriented view.]

Teacher: There’s only one at 70 and the rest are at 60 or lower? Yes?

Teacher: Can you say something about the mean perhaps?

S4: The mean is usually somewhere around 50.

For the remainder of this section, the figures and written explanations of three students are

demonstrated, because their work gives a representative impression of the variation of the whole

class: their diagrams represent all types of diagrams made in this class and the learning abilities

of these students varied considerably. Rikkert and Christel’s report scores were in the bottom

third of the class and Susan had the best report score of the class on the total of all subjects.

Rikkert: Mine looks very much like what is on the blackboard.

Christel: The middle-most [diagram on the blackboard] best resembles mine because

the weights are close together and that is also the case in my graph. It lies

between 35 and 75 [kg].

Susan: The other [real data] are more weights together and mine are further apart.

Rikkert’s answer is not very specific, like most of the written answers in the first cycle of

growing samples. Christel used the predicate “close together” and added numbers to indicate the

range, probably as an indication of spread. Susan used such terms as “together” and “further

apart,” which address spread. The students in the class used common predicates such as

“together,” “spread out” and “further apart” to describe features of the data set or the graph. Van

Oers (2000) calls the process of attributing predicates to features “predication.” “Predication is

the process of attaching extra quality to an object of common attention (such as a situation, topic

or theme) and, by doing so, making it distinct from others” (p. 150). For the analysis it is

important to note that the students used predicates (together, apart) and no nouns (spread,

average) in this first cycle of growing samples. Spread can only become an object-like concept,

something that can be talked about and reasoned with, if it is a noun. In the semiotics of Peirce

12

(1976), such transitions from the predicate “the dots are spread out” to “the spread is large” are

important steps in the formation of concepts.

Second cycle of growing a sample

With the feedback of the samples of ten data points in dot plots, students had to predict the

weight graph of a whole class of 27 students and of three classes with 67 students (27 and 67

were the sample sizes of the real data sets of eighth graders of another school.

c. We will now have a look how the graph changes with larger samples. Predict a sample of 27

students (one class) and of 67 students (three classes).

d. You now get to see real samples of those sizes. Describe the differences. You can use words

such as majority, outliers, spread, average.

During this second cycle, all of the students made dot plots, probably because the teacher had

shown dot plots on the blackboard, and because dot plots are less laborious to draw than value

bars (only one student started with a value-bar graph in the sample of 27 switched to a dot plot

for the sample of 67). The hint on statistical terms was added to make sure that students’ answers

would not be too superficial (as often happened before) and to stimulate them to use such notions

in their reasoning. It was also important for the research to know what these terms meant for

them (see transcripts of video fragment 4.1 at the end of the paper). When the teacher showed the

two graphs with real data, there was once again a short class discussion in which the teacher

capitalized on the question of why most student prediction now looked pretty much like what

was on the blackboard, whereas with the earlier predictions there was much more variation

(transcripts of video fragment 4.2). No student had a reasonable explanation, which indicates that

this was an advanced question. The written answers from the same three students were:

Rikkert: My spread is different.

Christel: Mine resembles the sample, but I have more people around a certain weight

and I do not really have outliers, because I have 10 about the 70 and 80 and

the real sample has only 6 around the 70 and 80.

Susan: With the 27 there are outliers and there is spread; with the 67 there are more

together and more around the average.

13

Figure 2a. Predicted graphs for one and for three classes by Rikkert, Christel, and Susan.

Figure 2b. Real data sets of size 27 and 67of students in another school.

Rikkert addressed the issue of spread. Christel was more explicit about a particular area in her

graph, namely the high values. She also correctly used the term “sample,” which was newly

introduced in the second lesson. Susan used the term “outliers” in this stage, by which students

meant “extreme values”. She also seemed to locate the average somewhere and to understand

that many students are about average. These examples illustrate that students used statistical

notions for describing properties of the data and diagrams. From a statistical point of view, these

Susan

Rikkert Christel

14

terms were not very precise. With “mean” students generally meant “about average” or “the

middle typical group”; with “spread” they meant “how far the data lie apart.” And with “sample”

they seemed to mean just a bunch of people, not necessarily the data as being representative for a

population (cf. Schwartz et al., 1998).

In contrast to the first cycle, students used nouns instead of just predicates for comparing the

diagrams. Rikkert (like others) used the noun “spread,” whereas students earlier used only

predicates such as “spread out.” Of course, this does not always imply that if students use these

nouns that they are thinking of the right concept; nor is it meant as a linguistic trick. Statistically,

however, it makes a difference whether we say, “the dots are spread out” or “the spread is large.”

In the latter case, spread is something that can have particular aggregate characteristics that can

be measured (for instance by the range, the interquartile range, or the standard deviation). Other

notions, outliers, sample, and average, are now used as nouns, that is as conceptual objects that

can be talked about and reasoned with.

Third cycle of growing a sample

In contrast to what was intended in the hypothetical learning trajectory, no student made a

continuous shape or talked about one yet. This was to change in this last cycle of growing the

sample, when the task was to make a graph showing data of all students in the city, not

necessarily with dots. The intention of asking this was to stimulate students to use continuous

shapes and dynamically relate samples to populations without making this distinction between

sample and population explicit yet. The conjecture was that this transition from a discrete

plurality of data values to a continuous entity of a distribution is important to foster a notion of

distribution as an object with which students could model data and describe aggregate properties

of data sets. During teaching experiments in the seventh-grade experiments (Bakker &

Gravemeijer, in press), in two American sixth-grade classes, and a visit to an American group of

ninth graders, reasoning with continuous shapes turned out difficult to accomplish, even if it was

asked for. It often seemed impossible to nudge students toward drawing the general, continuous

shape of data sets represented in dot plots. At best, students drew spiky lines just above the dots.

This underlines that students have to construct something new (a notion of signal, shape, or

distribution) with which they can look differently at the data or the variable phenomenon. The

task proceeded as follows:

15

e. Make a weight graph of a sample of all eighth graders in Utrecht. You need not draw dots. It

is the shape of the graph that is important.

f. Describe the shape of your graph and explain why you have drawn that shape.

Figure 3. Predicted graphs for all students in the city by Rikkert, Christel, and Susan.

The written explanations of the same three students were:

Rikkert: Because the average [values are] roughly between 50 and 60 kg.

Christel: I think it is a pyramid shape. I have drawn my graph like that because I found

it easy to make and easy to read.

Susan: Because most are around the average and there are outliers at 30 and 80 [kg].

Rikkert’s answer resembles that of students in seventh grade who indicated a range of the

average values or the majority. His answer focuses on the average group, or “modal clump” as

Konold and colleagues call such groups in the center (Konold et al., 2002). During an interview

after the fourth lesson, Rikkert literally called his graph a “bell shape,” though he had probably

not encountered that term in a school situation before (three other students also described their

graphs as bell shapes). This is probably a case of reinvention. Christel’s graph was probably

inspired by line graphs that the students made during mathematics lessons. She introduced the

vertical axis with frequency, though such graphs had not been used before in the statistics course.

Susan

Christel

Rikkert

16

Susan’s graph shows both dots and a continuous shape. It could well be that she started with the

dots and then drew the continuous shape.

In this third cycle of growing samples we see a first remark about shape as was asked for (an

example of guidance). The shapes that students proposed were pyramid (three students),

semicircle (one), and bell shape (four); 23 students drew a bump shape. Based on prior

experience that students were not inclined to make continuous sketches of distributions, it was a

pleasant surprise that students in this eighth-grade class drew continuous sketches during this

growing samples activity. This could support the sense of doing such an activity, though there

may be another reason why students in this class made continuous sketches. They had a better

mathematical background than the students in other experiments, for instance knowledge about

line graphs. If students draw continuous shapes, we do not exactly know what these shapes mean

for them. Therefore, in the next section, students’ reasoning with such shapes is analyzed.

Reasoning about shapes

In the fourth lesson, almost all student graphs looked roughly symmetrical, which is not

surprising when the history of distribution is taken into account (Steinbring, 1980). In real life,

however, the phenomenon of weight shows distributions that are skewed to the right because of a

“left wall effect” (two students had in fact drawn a left wall in the fourth lesson). By a left wall I

mean that the lower limit (say about 30 kg) is relatively close to the average (53 kg) and the

upper limit (sumo wrestlers are sometimes 350 kg) is relatively far away from the average. The

lower limit of 35 kg serves as a left wall, because adults can hardly live if they are lighter than 30

kg. This left wall in combination with no clear right wall causes the distribution to be skewed to

the right. Skewness is another important characteristic of a distribution and once there are

different shapes to talk about, for example symmetrical or skewed, students can characterize

shapes with different predicates. According to the hypothetical learning trajectory, skewness

therefore had to become a topic of discussion as well.

In collaboration with the teacher, the following activity was designed. To focus the students’

attention on shape and skewness, the three student shapes were drawn on the blackboard together

with two skewed shapes, which resulted in a pyramid, a semicircle, a bell shape, a unimodal

17

distribution that was skewed to the right, and one that was skewed to the left. Students had to

explain which shapes could not match the context of weight. The teacher expected that it would

be easier for students to engage in the discussion if they could argue which shapes were not

correct, instead of defending the shape they had chosen.

Figure 4. Five shapes as on the blackboard (1) semicircle, (2) pyramid, (3) normal distribution,

(4) distribution skewed to the right, (5) distribution skewed to the left.

The teacher chose students from the groups who thought that a particular shape on the

blackboard could not be right. For all shapes except the normal shape, many students raised their

hands. Apparently, most students expected a “normal” shape.

1. First, Ricardo explained why the semicircle (1) could not be right (video fragment 6.1).

Ricardo: Well, I thought that it was a strange shape (...) For example, I thought that the

average was about here [a little to the right of the middle] and I thought this

one [top of the hill] was a little too high. It has to be lower. And I thought that

here, that it was about 80, 90 [kg], and I don’t think that so many people

weigh that much or something [points at the height of the graph at the part of

the graph with higher values].

Teacher: (...) Does everybody agree with what Ricardo says?

Tobias: Yes, but I also had something else. That there are no outliers. That it is

straight and not that [he makes a gesture with two hands that looks like the

tails of a normal distribution]. I would expect that it would be more sloping if

it goes to the outside more [makes the same gesture in the air].

These students used statistical notions such as “outliers” and height to explain shape issues

(especially frequency). They clearly used their knowledge of the context to reason about shape.

18

2. Because all of the students seemed to agree that the semicircle was not the right shape, the

teacher wiped it off the blackboard and turned to the pyramid shape (2). This discussion involved

outliers and the mean in relation to shape (video fragment 6.2).

Maarten: Well, I thought this one not, because yes, I think that a graph cannot be so

square, or so rectangular.

Teacher: The graph is not so rectangular? [inviting him to say more]

Maarten: No, there are no outliers or stuff.

Sander: It does have outliers; right at the end of both it does have outliers.

Student2: That is just the bottom [of the graph].

Sander: At the end of the slanting line, there is an outlier, isn’t it? (...)

Annemarie: But the middle is the mean and everything else is outlier. [Other students

say they do not agree, e.g. Iris:]

Iris: Who says that the middle is the mean?

Annemarie: Yes, yes, roughly then.

Teacher: Tobias, you want to react.

Tobias: Look, if you have an outlier, then it has to go straight a bit [makes a

horizontal movement with his hands]; otherwise it would not be an outlier

(...) but that is not what I wanted to say. I wanted to react, that it [this

graph] could not be the right one, because the peak is too sharp and then

the mean would be too many of exactly the same.

Michiel: He just means that of one weight exactly all these kids have the same

weight, so if the tip is at, I don't know how many kilos, maybe 60 kilos,

that all these kids are exactly 60 kilos.

This transcript shows that students started to react to each other. Before this lesson they mainly

reacted to questions from the teacher—a type of interaction that is very common unfortunately

(van den Boer, 2003). In other words, the activity stimulated students to participate and their

passive attitude started to change. Because the students agreed that the pyramid was not the right

shape, the teacher wiped this shape off the blackboard also.

2 If we could not find out from the video who said this, we just put “student” as the speaker.

19

3. Next, Mariëlle was to explain why the bell shape (3) could not be the right shape (video

fragment 6.3). Before the discussion, almost all of the students thought this was the right shape

(one girl did not know).

Mariëlle: I had it that this was not the one, because there are also kids who are

overweight. Therefore, I thought that it should go a bit like this [draws the

right part a little more to the right, thus indicating a distribution skewed to the

right, like figure 4.4]. (...)

Rogier: That means that there are more kids much heavier, but there are also kids

much less, so the other side should also go like that [this would imply a

symmetrical graph].

Tobias: Guys, this is the right graph!

Because there was no agreement, the teacher did not wipe the graph off the board.

4. Next, Michiel had to explain why he thought that the fourth, skewed graph could not be right

(video fragment 6.4).

I thought that this was not it because... if the average is perhaps, if this it the highest point,

then this [part on the left] would be a little longer; then it would have a curve like there

[left half of the third graph]. I think that this cannot be right at all, and I also find it

strange that there are so many high outliers. Then you would maybe come to 120 kilos or

so. [Note that there were no numbers in the graphs.]

5. Last, Elaine about the fifth graph, which was skewed to the left (video fragment 6.5):

Well, I think this one is also wrong because there are more heavy people than light

people. And I think that eighth graders are more around 50 kilos. That’s it.

Tobias then objected, “it says 50 nowhere,” and a lively discussion between the two evolved.

Thus, as intended, skewness became a topic of discussion, even in relation to center and

“outliers.” Some students argued that the mean need not be the value in the middle. Still students

20

seemed to make no clear distinctions between midrange, mean, and mode. Because the mode is

not a measure that is often used in statistics, it was not the intention to address the mode unless

students were already reasoning with it. But since students at this point argued about the mean

versus the value that occurred the most, it was decided to introduce a name for the mode, which

these students had not learned before.

Researcher: The value that occurs the most often has a name also; it is called the mode

[pointing at the value where the distribution has its peak]. (...) Who can

explain in this graph [skewed to the right] whether the mean is higher or lower

than the mode? (...)

Rogier: There are just more heavy people than light people, and therefore the mean is

higher.

In this way, there were opportunities to introduce statistical terms and relate them to each other,

because students already talked about the corresponding concepts or informal precursors to them.

Traditionally, the mode is just introduced as the value that occurs the most, but here it was

introduced as a characteristic of a distribution, be it informally. The median was introduced in

the ninth lesson as the value that yields two equal groups. In fact, this introduction in relation to

continuous distributions was inspired by the history of the mode and median (Bakker, 2003;

Walker, 1931), because these notions were first defined in relation to distributions, not in relation

to data sets. Moreover, in my view the mode only makes sense in relation to continuous

distributions or for categorical variables.

The purpose of this activity of reasoning about shapes was that students would come to reason

about skewed distributions, and they did. They were even more engaged than expected. The

satisfactory thing about this activity was that they came to reason with notions in a way they had

not demonstrated before, and that they better engaged in the discussion than ever before, even

the students with low scores for mathematics. I conjecture that the lack of data, the game-like

character, and students’ knowledge about the context were important factors, but also the fact

that they had to argue against certain shapes. Such reasoning is safer than choosing the shape

they think is right and defending that one. I also conjecture that the lack of formal definitions

makes it easier for low-achieving students (such as Tobias) to engage in the discussion. This

issue can be illustrated with a metaphor that Frege, who is seen as one of the first modern

21

logicians and philosophers of language, wrote to Hilbert, who is seen as a formalist. The topic

was using and making symbols in mathematical discourse.

I would like to compare this with lignification [transformation into wood]. Where the tree

lives and grows, it must be soft and sappy. If, however, the sappiness does not lignify, the

tree cannot grow higher. If, on the contrary, all the green of the tree transforms into wood,

the growing stops. (Frege, 1895/1976, p. 59; translation from German3)

On the one hand, if statistical concepts are defined before students even have an intuitive idea of

what these concepts are for (such as mean, median, mode), then the tree transforms into wood

and students’ conceptual development is hindered. On the other hand, if teachers and textbooks

do not guide students well in a process of reinvention, the tree stays weak and cannot grow

higher. It is evident that the notions of average, outliers, distribution, and sample of students in

the present research needed to be developed into more precise notions, but at least they

developed a language that was meaningful to them, an image that could be sharpened later on or

a sappy part of the tree that can be lignified soon.

Implications for research, teaching, and assessment

In this discussion, two questions are raised. First, why do almost all school textbooks follow the

same routes? Second, what are the challenges of the approach taken in this paper for research,

teaching, and assessment?

Why do almost all school textbooks first introduce mean, median, and mode as a trinity, and

provide students with graphical tools such as histogram and box plot long before students have

the conceptual understanding to use such tools sensibly? G. Cobb (1993, nr. 53c) compared the

situation with a night picture of a city: “if one could superimpose maps of the routes taken by all

elementary books, the resulting picture would look much like a time-lapse night photograph of

car taillights all moving along the same busy highway.” Apart from the phenomenon of copying

what others do, one important reason could be that mean, median, mode, and graphs seems so

22

easy to teach and, even more importantly, to assess. The approach taken in this paper is much

harder for the teacher because notions stay informal for a while. The learning that results from

such an approach might also be harder to assess than whether students can calculate average

values or draw a histogram. In the approach taken here, the teacher has to accept that students’

notions stay informal for a while, but in my view this is better than building castles in the air. In

countries with a many tests, this might be difficult to accomplish (cf. Makar & Confrey, in

press). Research is needed to find out how and to what extent the approach propagated in this

paper is feasible. Moreover, research is needed into the question of how students can develop

their own informal notions, such as center clumps, spread, and shapes, into conventional

measures of center, variation, and other distribution aspects.

The hypothetical learning trajectory that is envisioned after the design study reported here is the

following. The main goal is still that students enhance their case-oriented views with aggregate

views on data sets and develop a coherent understanding of key concepts such as variability,

sampling, data, and distribution. In well-known contexts students can develop a view on data sets

as consisting of three groups of low, average, and high values. The middle group can be

informally called a clump, cluster, or majority and it can serve both as a measure of center

(mean, median, mode are generally in the modal clump) and as a measure of spread (some

clumps have a larger range than others). By such activities as growing samples reported in this

paper, students can be stimulated to reason about stable features within this process of sampling

(stabilizing clump or shape). If opportunities appear to introduce conventional definitions

(median, mode for example), teachers can take advantage of them. It is important to start with

case-value plots (Konold & Higgins, 2003) that are meaningful for students (bar graph, dot plot)

and wait with aggregate plots such as histogram and box plot until students have enough

conceptual basis. Once students have developed an understanding of center (e.g. by reasoning

with clumps), they can appreciate more formal measures of center such as mean and median.

When comparing distribution, the need can be acknowledged to use a convention of comparing

the middle 50% instead of various locally motivated proportions. Not until then, box plots can

function as meaningful tools in students’ reasoning about distributions. Further research is

needed to test this hypothetical learning trajectory and design instructional means supporting

such learning.

3 Ich möchte dieses [Symbolisieren] mit dem Verholzungsvorgange vergleichen. Wo der Baum lebt und wächst, muss er weich und saftig sein. Wenn aber das Saftige nicht mit der Zeit verholzte, könnte keine bedeutende Höhe

23

Acknowledgments

I thank Corine van den Boer for teaching the eighth-grade class, and Carolien de Zwart, Sofie

Goemans, and Yan-Wei Zhou for assisting in multiple ways. I also thank Nathalie Kuijpers for

translating the transcripts and correcting the English, and Jantien Smit, Anneleen Post, and Katie

Makar for their editing help. The research was funded by the Netherlands Organization for

Scientific Research under number 575-36-003B. The opinions expressed in this chapter do not

necessarily reflect the views of the Organization.

References

Bakker, A. (2002). From data via 'bump' to distribution. Statistics Education Research

Journal, (http://fehps.une.edu.au/serj), 1(1), 35.

Bakker, A. (2003). The early history of statistics and implications for education.

www.amstat.org/publications/jse/v11n1/bakker.html. Journal of Statistics Education,

11(1).

Bakker, A., & Gravemeijer, K. P. E. (in press). Learning to reason about distribution. In D.

Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy,

reasoning, and thinking. Dordrecht, the Netherlands: Kluwer Academic Publishers.

Cobb, G. (1993). Considering Statistics Education: A National Science Foundation

Conference. Journal of Statistics Education [Online], 1(1),

www.amstat.org/publications/jse/v1n1/cobb.html

Cobb, P. (1999). Individual and collective mathematical development: The case of statistical

data analysis. Mathematical Thinking and Learning, 1(1), 5-43.

Cobb, P., Confrey, J., diSessa, A. A., Lehrer, R., & Schauble, L. (2003). Design experiments

in educational research. Educational Researcher, 32(1), 9-13.

Cobb, P., McClain, K., & Gravemeijer, K. P. E. (2003). Learning about statistical covariation.

Cognition and Instruction.

erreicht werden. Wenn dagegen alles Grüne verholzt ist, hört das Wachstum auf.

24

Cobb, P., & Whitenack, J. (1996). A method for conducting longitudinal analyses of

classroom videorecordings and transcripts. Educational Studies in Mathematics, 30(3),

213-228.

Frege, F. L. G. (1976). Wissenschaftlichter Briefwechsel [Scientific correspondence] (Gabriel,

G., Ed.) (First ed. Vol. 2). Hamburg, Germany: Meiner.

Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht, the

Netherlands: Kluwer Academic Publishers.

Garfield, J., & Ben-Zvi, D. (in press). Research on statistical literacy, reasoning, and thinking:

Issues, challenges, and implications. In D. Ben-Zvi & J. Garfield (Eds.), The challenge

of developing statistical literacy, reasoning, and thinking. Dordrecht, the Netherlands:

Kluwer Academic Publishers.

Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory; Strategies for

qualitative research. Chicago: Aldine Publishing Company.

Gravemeijer, K. P. E. (1994). Educational development and developmental research. Journal

for Research in Mathematics Education, 25, 443-471.

Gravemeijer, K. P. E. (1999). An instructional sequence of analysing univariate Data sets.

Paper presented at the AERA 1999. Unpublished manuscript, Montréal, Canada.

Hancock, C., Kaput, J. J., & Goldsmith, L. T. (1992). Authentic enquiry with data: critical

barriers to classroom implementation. Educational Psychologist, 27(3), 337-364.

Jacobs, V. R. (1999). How do students think about statistical sampling before instruction?

Mathematics Teaching in the Middle School, 5, 240-263.

Konold, C., & Higgins, T. (2003). Reasoning about data. In J. Kilpatrick & W. G. Martin &

D. Schifter (Eds.), A research companion to Principles and Standards for School

Mathematics (pp. 193-215). Reston, VA: National Council of Teachers of

Mathematics.

Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A. D., Wing, R., & Mayr, S. (2002).

Students' use of modal clumps to summarize data. In B. Phillips (Ed.), Proceedings of

the International Conference on Teaching Statistics [CD-ROM], Cape Town.

Voorburg, the Netherlands: International Statistics Institute.

Latour, B. (1990). Drawing things together. In M. Lynch & S. Woolgar (Eds.), Representation

in scientific practice (pp. 19-68). Cambridge, MA: MIT Press.

Lehrer, R., & Romberg, T. (1996). Exploring children's data modeling. Cognition and

Instruction, 14(1), 69-108.

25

Lehrer, R., & Schauble, L. (2001). Accounting for contingency in design experiments. Paper

for AERA, April 2001, Seattle, WA.Unpublished manuscript, Madison, WI.

Makar, K., & Confrey, J. (in press). Secondary Teachers’ Reasoning about Comparing Two

Groups. In D. Ben-Zvi & J. Garfield (Eds.), The Challenges of Developing Statistical

Literacy, Reasoning, and Thinking. Dordrecht, the Netherlands: Kluwer Academic

Publisher.

McGatha, M., Cobb, P., & McClain, K. (2002). An analysis of students' initial statistical

understanding: Developing a conjectured learning trajectory. Journal of Mathematical

Behavior, 21, 339-355.

Meletiou, M. (2002). Conceptions of variation: A literature review. Statistics Education

Research Journal (http://fehps.une.edu.au/serj), 1(1), 46-52.

Mokros, J., & Russell, S. J. (1995). Children's concepts of average and representativeness.

Journal for Research in Mathematics Education, 26, 20-39.

Moore, D. S. (1997). New pedagogy and new content: The case for statistics. International

Statistical Review, 65, 123-165.

Noss, R., Pozzi, S., & Hoyles, C. (1999). Touching epistomologies: Meanings of average and

variation in nursing practice. Educational Studies in Mathematics, 40, 25-51, 1999.

Peirce, C. S. (1976). The new elements of mathematics (Eisele, C., Ed.) (Vol. I-IV). The

Hague-Paris/Atlantic Highlands, N.J.: Mouton/Humanities Press.

Petrosino, A. J., Lehrer, R., & Schauble, L. (2003). Structuring error and experimental

variation as distribution in the fourth grade. Mathematical Thinking and Learning,

5(2&3), 131-156.

Reading, C., & Shaughnessy, J. M. (2000). Student perceptions of variation in a sampling

situation. In T. Nakahara & H. Koyama (Eds.), Proceedings of the 24th Conference of

the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 89-

96). Hiroshima, Japan: Department of Mathematics Education, Hiroshima University.

Reading, C., & Shaughnessy, J. M. (in press). Reasoning about variation. In D. Ben-Zvi & J.

Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and

thinking. Dordrecht, the Netherlands: Kluwer Academic Publishers.

Roth, W.-M. (1996). Where is the context in contextual word problems? Mathematical

practices and products in grade 8 students' answers to story problems. Cognition and

Instruction, 14, 487-527.

26

Rubin, A., Bruce, B., & Tenney, Y. (1990). Learning about sampling: Trouble at the core of

statistics. Paper presented at the Third International Conference on Teaching Statistics,

Dunedin, New Zealand.

Russell, S. J., & Corwin, R. B. (1989). Statistics: The shape of the data. Used numbers: Real

data in the classroom. Grades 4-6. Washington, DC: National Science Foundation.

Schwartz, D. L., Goldman, S. R., Vye, N. J., Barron, B. J., & Vanderbilt, C. a. T. G. a. (1998).

Aligning everyday and mathematical reasoning: The case of sampling assumptions. In

S. P. Lajoie (Ed.), Reflections on statistics: Learning, teaching, and assessment in

grades K-12 (First ed., pp. 233-273). Mahwah, NJ: Lawrence Erlbaum Associates.

Shaughnessy, J. M., Garfield, J., & Greer, B. (1996). Data handling. In A. J. Bishop & K.

Clements & C. Keitel & J. Kilpatrick & C. Laborde (Eds.), International handbook of

mathematics education (pp. 205-237). Dordrecht, the Netherlands: Kluwer.

Shaughnessy, J. M., Watson, J. M., Moritz, J., & Reading, C. (1999). School mathematics

students' acknowledgment of statistical variation. NCTM Research Presession

Symposium: There's more to life than centers. Paper presented at the 77th Annual

NCTM Conference, San Francisco, California.Unpublished manuscript.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist

perspective. Journal for Research in Mathematics Education, 26, 114-145.

Steinbring, H. (1980). Zur Entwicklung des Wahrscheinlichkeitsbegriffs - Das

Anwendungsproblem in der Wahrscheinlichkeitstheorie aus didaktischer sicht [On the

development of the probability concept --The applicability problem in probability

theory from a didactical perspective. Bielefeld: Institut für Didaktik der Mathematik

der Universität Bielefeld.

van den Boer, C. (2003). Als je begrijpt wat ik bedoel. Een zoektocht naar verklaringen voor

achterblijvende prestaties van allochtone leerlingen in het wiskundeonderwijs [If you

know what I mean. A search for an explanation of lagging results of mathematics

education among ethnic minority students] CD Beta Press: Utrecht, the Netherlands.

van Oers, B. (2000). The appropriation of mathematics symbols: A psychosemiotic approach

to mathematics learning. In P. Cobb & E. Yackel & K. McClain (Eds.), Symbolizing

and communicating in mathematics classrooms; Perspectivies on discourse, tools, and

instructional design (pp. 133-176). Mahwah, NJ: Lawrence Erlbaum Associates.

Walker, H. M. (1931). Studies in the history of statistical methods with special reference to

certain educational problems. Baltimore: Williams & Wilkins Company.

27

Watson, J. M. (2002). Creating cognitive conflict in a controlled research setting: Sampling.

In B. Phillips (Ed.), Developing a Statistically Literate Society; Proceedings of the

Sixth International Conference of Teaching Statistics Cape Town [CD-ROM].

Voorburg, the Netherlands: International Statistics Institute.

Watson, J. M., & Moritz, J. (2000). Developing concepts of sampling. Journal for Research in

Mathematics Education, 31, 44-70.

Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International

Statistics Review, 67, 223-265.

Wilensky, U. (1997). What is normal anyway? Therapy for epistemological anxiety.

Educational Studies in Mathematics, 33, 171-202.

Zawojewski, J. S., & Shaughnessy, J. M. (2000). Mean and Median: Are They Really so

Easy? Mathematics Teaching in the Middle School, 5(7), 436-440.

Translations of transcripts

Lesson 4 on growing samples

Video fragment 4.1: students drawing graphs plus two interviews [43.30 to 49.32]

Interv: Why do you have that shape, Joshua? [dotplot of 67 students]

Joshua: Because it’s easier that way to see what the average of 67 children is

[This answer is typical; it occurs quite often.]

Interv: Can you see more than just the mean? You’re talking about the mean

now, but I can see a lot more.

Joshua: Yes, there are outliers in there too, that they don’t all weigh the same.

Interv: And, eh, I can see, the top is more to the left than to the right, right?

Joshua: Yes, because around average children [makes a more-or-less gesture] are

something like around 50 kilos, in eighth grade. So that would be the

highest, that’s where most of the dots are.

Interv: Yes, but why isn’t there more to the left or the right or in the middle?

Joshua: I think most will be between 50 and 60 , if you measure it [looks around]

I think most are around 50 and 60.

Video fragment 4.2: discussion after second cycle [49.46 to 53.13]

[Lynn draws while teacher starts discussion]

28

Teacher: Tobias, what was yours like? Do they look like that?

Tobias: It’s almost the same, see. [It was remarkable how good their predictions

were.]

Teacher: That’s really quite remarkable, isn’t it, that, well, not with Rikkert, but for

most say: ‘Here there was some variation and here they begin to match.’

Does anyone have an explanation for it? Rikkert, how would you explain

it?

Rikkert: Well, if there are a lot more, then uh it’ll be spread out more, you could

say.

Teacher: You think it will be more spread out when there are more.

Rikkert: Yes, they will all be from 50 to somewhere up at 80, will be a lot of

students, because there are so many

Teacher: We had been talking about this one, many students said that this one

looked a lot like the one they drew. How could it be that we all found so

many different ones, and here, that looks a bit like it for everyone. What

would you think of that?

Niels: Well because, see, because. Yes, many are a bit in the middle. Also

around 60 and 50 there are many, with like, for the weight. So if you do

more, you will get more and more.

Teacher: Elaine, do you have anything else?

Elaine: Well, uh, between those 3 you have there, there can be small people, light

people or something. And when you have a large group, you have more

small people and more people who are a bit heavier, so then it would be

.... [is talking softer and softer] [good argument which I expected to

appear: wider range?]

Teacher: So it would be more noticeable or something like that, is that what you

mean?

Elaine: Yes, it doesn’t get noticed now, because there are more between 40 and

60.

Teacher: Good, we have heard all kinds of arguments now about comparing these

things. You will now continue with the problem. We have 10 minutes

left. You go to d again, describe the difference. The same you had to do

29

for question b and then you will do e, make a weight graph of the sample

of all eighth grade classes in Utrecht. What might that look like?

You don’t have to draw dots now of course. It’s about the SHAPE

(emphasis) of the graph on? Do you understand what they mean? What

would the shape be like here?

Student: Pyramid, something like that.

Lesson 6 Reasoning about shapes

The teacher’s explanation of the activity

Teacher: Now we will look at the homework. There were 5 graphs and then it said:

Which of these do you think matches the weight of many eighth graders

the best? So I’m going to turn that question around. If one is the best

match according to you, that means four don’t have the best match.

Yes? That’s logical, isn’t it?

Who says this isn’t the best match, it certainly isn’t this one. [many

students raise their hands] [the semicircle] Ricardo.

Who says it certainly isn’t this one? [the pyramid] Maarten.

Who says it certainly isn’t this [the normal (symmetrical) distribution]

Marielle.

Who thinks it certainly isn’t this one? [with the hump on the left] Michiel.

And who thinks it certainly isn’t this one [with the hump on the right]

Elaine.

We’ll talk them through one by one, Ricardo will come over here with

this graph. And what is the plan? Ricardo will explain to you why this

certainly isn’t the right one. After he’s finished his explanation, and you

can do this in any way you like, you have to convince the class that this

isn’t the right one. Because I didn’t see everyone raise their hands. When

you are done, I will ask the class who agrees with you. When someone

says ‘no, I don’t agree with him,’ you will sit down and the other person

will come to the front. Yes? We’ll do all five graphs like that. So the

others already know ‘I’ll have to think of good arguments.’

Give it a try; this is difficult, because we’ve never done something like

this before.

30

Video fragment 6.1: semicircle [25.18 to 26.41]

Ricardo: Well, I thought that it was a strange shape and I thought the third one was

the best (...)

Teacher:

No, I said to turn it around, you can’t say which one you thought was the

best, but why is it not this one? What’s wrong with it?

Ricardo: For example, I thought that the average was about here [a little to the right

of the middle] and I thought this one [top of the hill] was a little too high.

It has to be lower. And I thought that here, that it was about 80, 90 [kg],

and I don’t think that so many people weigh that much or something

[points at the height of the graph at the part of the graph with higher

values].

Teacher: Don’t look at me, you don’t have to convince me [makes a gesture to the

class] (...) Does everybody agree with what Ricardo says?

Tobias: Yes, but I also had something else. That there are no outliers. That it is

straight and not that [he makes a gesture with two hands that looks like

the tails of a normal distribution]. I would expect that it would be more

sloping if it goes to the outside more [makes the same gesture in the air].

Teacher: Ha, Tobias says more outliers. We’ll take these two arguments together.

Ricardo’s story and Tobias’s story. Are here any more people who still

say ‘but I can still think of something that could be wrong. Anyone who

says: ‘But it could still be the right one’ [someone says no] Okay, we’ll

get to that [?] Excellent

Video fragments 6.2: pyramid shape [27.17 to 28.00; 28.20 to 29.12; 29.39 to 31.08]

Maarten: Well, I thought this one not, because yes, I think that a graph cannot be so

square, or so rectangular.

Teacher: The graph is not so rectangular? [inviting him to say more]

Maarten: No, there are no outliers or things like that.

Sander: Yes, there are

Maarten: Like here [?]

Sander: Yes, there are.

Teacher: It is, a distribution is never that rectangular, you say, and I don’t see any

31

outliers. Raise your hands please, because I think there are people who

want to react. Sander, when Maarten said there are no outliers, you said

there were.

Maarten: No, there are no outliers or stuff.

Sander: It does have outliers; right at the end of both it does have outliers.

Teacher: What’s an outlier.

Sander: One that/ [interrupted]

Student: That is just the bottom [of the graph].

Sander: At the end of the slanting line, there is an outlier, isn’t it? (...)

Maarten: But that’s all, yes, do you have an outlier at the bottom in the middle.

Sander: [sigh]

Maarten: Well, you come to the front and explain it then.

Teacher: Let’s have a look, Annemarie.

Annemarie: It does have outliers, because that point is the mean and everything that’s

there. The outsides are the outliers.

Student b: Yes, Maarten...

Tobias: No, nooooo, because when you suddenly...

Teacher: So you’re saying outliers are in the middle, they don’t have to be on the

outside at all [class in confusion].

Annemarie: This is the mean, and if you go more to the [?]/

Teacher: I think you’re working really well here, but I say who gets to speak or it’ll

be a mess. Annemarie can finish her story and then Tobias gets to react

first, and then Maarten will defend his position, or not.

Annemarie: But the middle is the mean and everything else is outlier. [Other students

say they do not agree, e.g. Iris:]

Iris: Who says that the middle is the mean?

Annemarie: Yes, yes, roughly then. [A number of students make it quite clear that

they don’t agree with Annemarie by calling out loud ‘yeah right’ and

similar comments. No clear arguments can be discerned as yet.]

Teacher: Tobias, it’s your turn to reply. Anyone else who wants to react, raise your

hand, or no one will be able to follow what is going on

Tobias: Look, if you have an outlier, then it has to go straight a bit [makes a

horizontal movement with his hands]; otherwise it would not be an

32

outlier/

Student: Two.

Tobias: (...) but that is not what I wanted to say. I wanted to react, that it [this

graph] could not be the right one, because the peak is completely sharp

and then the mean would be too many of exactly the same, or something.

Michiel: So what he means is that all the children at one weight all weigh exactly

the same, so when the peak is at who knows how many kilos [he makes

an upward gesture], maybe 60 kilos, immediately all those children weigh

exactly 60 kilos.

Teacher: Does anyone have something to add to this? Anyone who says it could

still be the right one? Well done.

Video fragments 6.3: normal shape [31.20 to 31.55]

Mariëlle: I had it that this was not the one, because there are also kids who have

overweight. Therefore, I thought that it should go a bit like this [draws the

right part a little more to the right, thus indicating a distribution skewed to

the right, like figure 4.4]. (...)

Teacher: Why don’t you draw roughly how it should be with another color?

Rogier: That means that there are more kids much heavier, but there are also kids

much lighter so the other side should also go like that [this would imply a

symmetrical graph].

Tobias: Guys, this is the right graph! [general agreement from the group to

Tobias’ comment]

Teacher: Who agrees with Marielle that this might not be the right one? [little

reaction]

Teacher: So I won’t wipe it off the blackboard yet. It’s a bit doubtful, isn’t it.

Rogier: Doubt, I just think it’s the right one. [people are calling “doubt???”]

Student: It IS the right one.

Video fragments 6.4: skewed to the right [32.40 to 33.06; 35.32 to 36.X; 37.40 to 39.20]

Teacher: Does anyone want to react to this? Or are you saying ‘no, that is

completely wrong?’ [pause] Everyone agrees.

Researcher: Michiel, maybe you want to put in the numbers, what you think it should

33

be.

Michiel: What do you mean?

Researcher: You said 120, where the average is and things like that. [the group calls

out all kinds of suggestions and numbers. Michiel’s numbers are

remarkably close to reality]

Teacher: Okay, the numbers are there now.

Teacher: Michiel, are you done? You believe you have convinced the class.

Everybody agrees that this might not be the right one? By the way, what

is going on with that kind of peak?

Rikkert: Yes, the mean, that most people weigh that much.

Teacher: Now you’re saying two different things. You’re saying mean, and you’re

saying that most people weigh that much.

Student z: That isn’t the mean.

Student q: Yes, it is. [discussion about whether or not it is the mean, impossible to

follow]

Teacher: Okay, back to raising your hands. Take a breather. Is this the mean or

not? Rogier.

Rogier: No, it’s just, it’s just what most people weigh.

Michiel: You calculate the mean by just taking all the people, adding their weight

and divide that by the number of people.

Teacher: And it doesn’t necessarily have to be this? [Points at the highest point in

the graph; Teacher is in a hurry, is looking at the clock, even appears to

find the discussion not as valuable as researcher does, taking too long,

loss of concentration?]

Boy: No, it could be a bit more or less.

Researcher: When the mean and what occurs most are different, in the fourth graph.

Would the average be higher or lower?

Student: Lower.

Class: [the others all call ‘higher’] Higher.

Researcher: So some are saying that the mean is something else than that which occurs

most. The value that occurs the most has a name also; it is called the

34

mode [pointing at the value where the distribution has its peak]. (...)

Teacher: Mode, is what occurs most [‘modus’ in Dutch]. And you can remember

that because there is an English word in it. What word does it resemble?

Class: Modern.

Teacher: Most [the class is not enthusiastic about this] So now we will use the

word mode every time. Mode is the weight that occurs most. So when you

say that most eighth graders weigh exactly 48 kilos. You’re saying that 48

kilos is the mode.

Researcher: So how do you see most are here anyway?

Maarten: That’s where it’s highest.

Researcher: So the most are where the graph is highest. My question is, I’ll repeat it, is

the mean higher or lower than the mode. Who says higher [most students

raised their hands for higher], who says lower [no one for lower. So there

were some hands missing].

Teacher: Who doesn’t know? [one student admits]

Researcher: Who can explain in this graph [skewed to the right] why the mean is

higher or lower than the mode? (...)

Anthony: Most of what comes after, it’s more than is on the left, on the low side. So

there are more people with a higher weight than with a lower weight.

Researcher: And why would the mean be higher?

Anthony: Because uh. There are more people so you do [...]

Researcher: Does anyone understand what Anthony means? [general laughter from

the class, because no one understood] Ah, there must be some. Can

anyone repeat it in their own words?

Rogier: There are just more heavy people than light people, and therefore the

mean is higher.

Researcher: In that graph you mean? [graph with clump on the left, right-skewed]

Rogier: Yes.

Researcher: Who understands what Rogier is saying?

Tobias: Me!

Researcher: Who doesn’t [no hands?].

35

Video fragments 6.5: skewed to the left [39.48 to 41.15; 41.45 to 42.26]

Elaine: Well, I think this one is also wrong, because there are more heavy people

than light people. And I think that eighth graders are more around 50

kilos. That’s it. [Elaine goes back to her place.]

Teacher: Wait, I don’t know if there are people who want to react to this. Does

everyone agree with Elaine?

Tobias: I didn’t even hear what she said, it was a bit short, but I think I know why

that one isn’t the right one

Teacher: Wait wait wait, first Elaine again. Elaine, could you repeat it?

Elaine: I think it’s not that one because there are more heavy people, that one has

more heavy people and I think most eighth graders would weigh around

50.

Tobias But it doesn’t say anywhere that that isn’t 50.

Researcher: Yes, which numbers do you think go with this? Write it down briefly.

[While Elaine is busy on the blackboard, Tobias has a discussion with

someone. The class is fairly noisy anyway.]

Tobias: But they can also be a short distance.

Student: What do you mean, a short distance?

Tobias: Well, it could be between 54 and 64. Then it would be possible.

[There is a vehement reaction to this, but it is inaudible because

everybody is talking at once. Teacher stops the discussion and gives the

last word to Tobias.]

Tobias: Now, look, if it’s a whole for example, if there in the beginning, it’s 40 or

whatever and there at the end is 60. That there isn’t that much distance

between. Then it could be right.

Elaine: Yes, but look, there it goes suddenly, here it would be 40 and suddenly it

would be 50 here.

Tobias: Yes, because a lot of people weigh 50.

Elaine: Yes, but then it suddenly goes mmmm [sound of a racing car], all the way

to 50 here and then suddenly 60.

Tobias: Yes, because many weigh a lot less than a few.

Elaine: Yes, but that isn’t possible.

Boy: That’s impossible, because 50 goes to 60 very fast and that’s impossible.

36

Tobias: Well, it could be 70 as well, I don’t know. But I’m not saying it’s right.